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+arXiv:2301.13597v1  [hep-ph]  31 Jan 2023
+The scalar exotic resonances X(3915),
+X(3960), X(4140)
+A.M.Badalian and Yu.A.Simonov
+NRC “Kurchatov Institute”
+Moscow, Russia
+February 1, 2023
+Abstract
+The scalar resonances X(3915), X(3960), X(4140) are considered
+as exotic four-quark states: cq¯c¯q, cs¯c¯s, cs¯c¯s, while the X(3863) is proved
+to be the c¯c, 2 3P0 state. The masses and the widths of these reso-
+nances are calculated in the framework of the Extended Recoupling
+Model, where a four-quark system is formed inside the bag and has
+relatively small size (<∼ 1.0 fm).
+Then the resonance X(3915) ap-
+pears due to the transitions: J/ψω into D∗+D∗− (or D∗0 ¯D∗0) and
+back, while the X(3960) is created due to the transitions D+
+s D−
+s into
+J/ψφ and back, and the X(4140) is formed in the transitions J/ψφ
+into D∗+
+s D∗−
+s
+and back. The characteristic feature of the recoupling
+mechanism is that this type of resonances can be predominantly in the
+S-wave decay channels and has JP = 0+. In two-channel case the reso-
+nance occurs to be just near the lower threshold, while due to coupling
+to third channel (like the c¯c channel) it is shifted up and lies by (20–
+30) MeV above the lower threshold. The following masses and widths
+are calculated: M(X(3915)) = 3920 MeV, Γ(X(3915)) = 20 MeV;
+M(X(3960)) = 3970 MeV, Γ(X(3960) = 45(5) MeV, M(X(4140)) =
+4120(20) MeV, Γ(X(4140)) = 100 MeV, which are in good agreement
+with experiment.
+1
+
+1
+Introduction
+In the region (3.9–4.2) GeV there are now three scalar resonances and the
+X(3915) was the first, observed by the Belle in the e+e− → J/ψωK process
+[1].
+Later this resonance was confirmed by the BaBar [2] and in several
+other experiments [3]), in particular, in two-photon collisions [4, 5].
+For
+some years this resonance was assumed to be the conventional c¯c meson
+– χco(2P), although this interpretation has called out some doubts [6, 7]
+(see discussion in the reviews [8, 9]) and does not agree with predictions
+in different relativistic potential models (RPM) [10]-[13]. The experimental
+masses of the X(3915) and χc2(2P) were found to be almost equal, while
+in the RPMs a smaller mass, M(2 3P0) ∼= 3870 ± 30 MeV, and much larger
+mass difference, δ20(2P) = M(χc2(2P) − M(χc0(2P) ∼= (70 − 100) MeV,
+were predicted.
+Notice that large mass difference δ20 is kept even if the
+coupling of the χc0(2P) to open channels is taken into account [14, 15]. Such
+theoretical expectations were supported by the Belle observation of the wide
+scalar X(3860) resonance [16], both in e+e− → J/ψD+D− and e+e− →
+J/ψD0 ¯D0 decays, which has the mass M = 3862+26
+−32
++40
+−82 MeV and large width
+Γ ∼= 200 MeV. The existence of the scalar X(3860) resonance is confirmed
+by the analysis of two-photon production, γγ → D ¯D in [17].
+Very recently the LHCb [18] has observed two more scalar resonances
+X(3960), X(4140) in the D+
+s D−
+s mass spectrum in the B+ → D+
+s D−
+s K+ de-
+cays with the parameters: M(X(3960)) = (3956±5±10) MeV, Γ(X(3960)) =
+(43 ± 13 ± 8) MeV, M(X(4140)) = (4133 ± 6 ± 6) MeV, Γ(X(4140)) =
+(67 ± 17 ± 7) MeV, both with JP C = 0++. These new scalar resonances evi-
+dently look as exotic states and the X(3960) was interpreted as the molecular
+D+
+s D−
+s state within the QCD sum rules approach [19, 20] and in a coupled-
+channel model [21]; in [22] it appears due to the triangle singularity, while
+in [23] the parameters of the X(3960), as a diquark-antidiquark state, were
+obtained in a good agreement with experiment, using the QCD sum rules
+approach. Notice that the masses of the X(3960) and X(4140) resonances
+lie by ∼ 20 MeV above the thresholds: D+
+s D−
+s and J/ψφ, respectively.
+In our paper we assume that the X(3915) and both the X(3960), X(4140)
+belong to exotic four-quark states cq¯c¯q and cs¯c¯s and to define their parame-
+ters we will use the Extended Recoupling Model (ERM), recently suggested
+in [24], which develops the Recouplimg Model, presented earlier [25]. The
+ERM allows to calculate the mass and width of a scalar four-quark states,
+however, within suggested mechanism such resonances cannot exist in the
+2
+
+systems with two identical mesons, like D+
+s D+
+s , D∗+
+s , D∗+
+s . This theoretical
+prediction is supported by the Belle experiment [26]. In the ERM the system
+of two mesons, e.g. (J/ψ + φ), can transfer into another pair of the mesons
+(D+
+s , D−
+s ) by rearranging confining strings and back in the infinite chain of
+transformations, like J/ψφ → (D+
+s ¯D−
+s ) → J/ψφ → .... Note that such se-
+quences can also be treated, for example, in the standard OBE approximation
+with the meson exchanges, which, however, does not produce the singulari-
+ties near the thresholds. In the coupled-channel models (CCM) [27, 28] the
+interaction between hadrons, like D+
+s D−
+s , J/ψφ, is usually neglected, while
+in the ERM such interaction is taken into account, introducing the four-
+quark bag. It is important that all hadrons involved have rather small sizes,
+∼= (0.40 − 0.55) fm and only ω(1S) has a bit larger r.m.s. ∼ 0.7 fm. We
+would like to underline the characteristic features of the ERM [24]: first, due
+to the string rearrangement of a four-quark system the singularity lies close
+to the lower threshold; second, this mechanism produces the resonance in
+the S-wave hadron-hadron system and therefore, the quantum numbers of
+these resonances JP C = 0++, 1++, 2++; third, a resonance does not appear,
+if hadrons are identical.
+In the literature there are still a controversy, concerning the X(3915), and
+different interpretations were proposed. This resonance was considered in
+tetraquark model within the Born–Oppenheimer approach in [29, 30, 31, 32],
+due to the triangle singularity [22] and the threshold effects [33], as the
+molecular Ds ¯Ds bound state [34] or the lightest cs¯c¯s state [35] and as the
+diquark-antidiquark state, using the QCD sum rule method [23, 36]. In con-
+trast to a molecular structure of four-quark states in the ERM these systems
+are assumed to be compact systems, similar to the diquark-antidiquark states
+studied in [37]. In such compact systems their wave functions at the origin
+are not small and therefore they can be produced in the γγ transitions.
+In our paper we will shortly discuss the higher scalars, X(4500), X(4700),
+observed by the LHCb [38], which admit different interpretations.
+The structure of the paper is as follows. In next section we shortly remind
+the basic formulas in two-channel case and give the values of the parame-
+ters, needed to define the masses and widths of the recoupled four-quark
+resonances. In section 3 more general matrix representation of the ERM is
+presented. In section 4 we calculate the transition amplitudes and give the
+masses and widths of the scalar resonances, and compare them with exper-
+imental data. In section 4 the masses of high X(4500), X(4700) resonances,
+as the c¯c states, are discussed. Our conclusions are presented in section 5.
+3
+
+2
+The two-channel approach in the Extended
+Recoupling Model
+We study the experimental process where, among other products, two hadrons
+are produced and one pair of hadrons (the pair 1) can transfer into another
+pair of hadrons (the pair 2). In [24] the probability amplitude of this tran-
+sition was denoted as V12(p1, p2), with p1, p2 – relative momenta of the
+hadrons, referring to the pair 1 and 2.
+If an infinite set of the transfor-
+mations was supposed and the total production amplitude A2 of the pair
+2 was written as a product of the slowly varying function F(E) and the
+singular factor f12(E) =
+1
+1−N , then the amplitude A2 = F(E)f12(E). This
+definition of the transition amplitude V12 = V21 differs of that in other ap-
+proaches, where one or more the OBE diagrams with meson exchanges are
+taken. In the ERM [24] the process occurs through the intermediate stage of
+the Quark Compound Bag (QCB) [39, 40], where all quarks and antiquarks
+of two hadrons are participating in the string recoupling and, possibly, the
+spin recoupling. Denoting the QCB wave functions as Φ(qi) (i = 1, 2, 3, 4)
+and the two-hadron wave functions as Ψi(h1, h2), the amplitude V12 can be
+written as,
+V12 = (Ψ1(ha1hb1)Φ(qi))(Φ(qi)Ψ2(ha2hb2) = V1(p1)V2(p2),
+(1)
+i.e. the amplitude V12 =
+1
+1−N acquires the factorized form: V12(p1, p2) =
+v1(p1)v2(p2) with the factor N, written as
+N = z(E)I1(E)I2(E).
+(2)
+Here z = z(E) can be called the transition probability, while I1(E), I2(E)
+are the following integrals (see [24]):
+Ii(E) = viGivi =
+�
+d3pi
+(2π)3
+v2
+i (pi)
+E′(pi) + E
+′′(pi) − E ,
+(3)
+where the hadron energies E′(pi), E
+′′(pi) in the i-th pair near thresholds,
+E′(p) =
+p2
+2m′ + m′, include corresponding thresholds Eth
+i
+and the reduced
+masses µi, namely,
+Eth
+i
+= m′(i) + m
+′′(i),
+µi =
+m′(i)m
+′′(i)
+m′(i) + m
+′′(i).
+(4)
+4
+
+The result of the integration in Ii(E) can be approximated by the form:
+Ii = consti
+1
+νi − i
+�
+2µi(E − Eth
+i )
+.
+(5)
+with µi, defined in (4), while νi is expressed via the parameters of the hadron
+wave functions, which were calculated explicitly in [24]. Here we would like to
+underline that the transition probability z(E) appears to be the only fitting
+parameter in the ERM.
+The whole series of the transitions from the pair 1 to 2 and back is summed
+up to the amplitude f12,
+f12(E) =
+1
+1 − zI1I2
+,
+Ii =
+1
+νi − i
+�
+2µi(E − Eth
+i )
+,
+(6)
+where νi are found from the four-quark wave functions, as in [37, 40]. The
+form of Eq. (6) takes place for the energies E > E1, E2, while for E <
+E1, E2, i.e.
+below thresholds, the amplitude f1 =
+�
+1
+ν1+√
+2µ1(|E−E1|)
+�
+.
+It
+is important that in the ERM the process proceeds with the zero relative
+angular momentum between two mesons, L = 0, otherwise the transition
+probability z12(E) is much smaller and a resonance may not appear.
+Note also that if the recoupling mechanism is instantaneous, or the tran-
+sition from one pair of the mesons to another proceeds instantaneously, then
+the transition amplitude V (12) does not factorize into V (1)V (2); such an
+assumption was used in the original Recoupling Model [25]. However, in this
+approximation, e.g. for the Tcc resonance agreement with experiment was
+not reached [25]. On the contrary, in the ERM [24] the recoupling mecha-
+nism proceeds in two stages: at first stage the hadrons h1, h2 collapse into
+common “compound bag” [39, 40], where the four quarks are kept together
+by the confining interaction between all possible quark pairs. This compound
+bag has its own wave function Φi(q1, q2, q3, q4) and the probability amplitude
+of the h1, h2 → Φ transition, which defines the factor V1(p1) in Eq. (2). In
+a similar way the transition from the Bag state to the final hadrons h3, h4
+defines the factor V2(p2) and we obtain the relation:
+v1(pi) =
+�
+d3q1...d3q4ψh1ψh2Φi(q1, ..q4),
+(7)
+and similar equation for v2(p2), replacing h1, h2 by h3, h4. From vi(pi) the
+function Ii (3) is defined and using (6), one obtains νi.
+5
+
+Now we give experimental data and corresponding the ERM parame-
+ters, referring to the four-quark systems, cq¯c¯q for X(3915) and cs¯c¯s for the
+X(3960), X(4140). We give also the threshold energies E1, E2.
+The parameters of the four-quark resonances
+1) X(3915), JP = 0+, Γ(exp .) = 20(5) MeV [1, 3], J/ψω → D∗ ¯D∗, E1 =
+3.880, E2 = 4020, µ1 =
+M(J/ψ)M(ω)
+M(J/ψ)+M(ω) = 0.624,
+µ2 =
+M(D∗)M( ¯D∗)
+M(D∗)+M( ¯D∗) =
+1.050 (all in GeV). From [24] ν1(J/ψω) = 0.21 GeV, ν2(D∗ ¯D∗) =
+0.44 GeV.
+2) X(3960), JP = 0+, Γ(exp .) = 43(21) MeV [18], [J/ψφ] → [D−
+s D+
+s ], E1 =
+3.936, E2 = 4116, µ1 =
+MJ/ψMφ
+MJ/ψ+Mφ = 0.767,
+µ2 =
+M(D+
+s )M(D−
+s )
+M(D+
+s +M(D−) =
+0.984; ν1(J/ψφ) = 0.265,
+ν2 = 0.424 (all in GeV).
+3) X(4140), JP = 0+, Γ(exp .) = 67(24) MeV[18], [J/ψφ] → [D∗−
+s D∗+
+s ], E1 =
+4.116, E2 = 4.224, µ1 = 0.767,
+µ2 = 1.056,
+ν1 = 0.265,
+ν2 = 0.410
+(all in GeV).
+Here q can be u, d quarks. To define the structure of the cross sections
+we start with the value of the recoupling probability z = 0.2 GeV2 and the
+parameters from the item 1) to obtain the distribution |f12(E)|2; the values
+of |f12(E)|2 will be given in Section 4. In the amplitude f12(E) the resulting
+singularity can be found in the form of (6) and for equal threshold masses
+it produces a pole nearby thresholds; however, real distance between the
+thresholds is large, ∼ 100 MeV and the actual singularity structure can be
+more complicated.
+3
+The matrix approach in the ERM
+In previous Section we have presented the ERM equations in the case of two
+channels, which are convenient to define the mass of a resonance. However,
+they do not allow to study some details of the process, or to consider a larger
+number of channels, which can have a influence at the properties of a four-
+quark system. Therefore here we present a more general representation of
+the amplitude using the unitarity relation, when the standard form of the
+transition amplitudes fij(E) (for L = 0) is
+fij − f ∗
+ji =
+�
+n
+2iknfinf ∗
+jn,
+(8)
+6
+
+or the unitarity relation can be realized through the M-matrix representation,
+ˆfM =
+1
+ˆ
+M − iˆk
+,
+(9)
+where ˆf, ˆ
+M, ˆk are the matrices in the channel numbers [28]. In some cases
+instead of the ˆ
+M it is more convenient to use the ˆK matrix, ˆ
+M = − ˆK−1,
+where the matrix elements (m.e.) Mik(E) are the real analytic functions of
+E with the dynamical cuts. For two-channel system ˆfM can be written as
+ˆfM =
+1
+ˆ
+M − iˆk
+=
+ˆN
+D(E),
+(10)
+with
+ˆN =
+�
+M22 − ik2
+−M21
+−M12
+M11 − ik1
+�
+.
+(11)
+Here
+D(E) = (M11 − ik1)(M22 − ik2) − M12M21.
+(12)
+One can easily establish the relation between the equations (10)- (12) and
+the amplitude f12(ERM) (6) in two-channel case, which is a partial case of
+these equations:
+f12(ERM) = N11N22
+D(E) ,
+D(E) = (ν1 − ik1)(ν2 − ik2) − z,
+(13)
+and
+z = M12M21,
+νi ≡ Mii(E).
+(14)
+One can see that for z > 0 the values νi = Mii are real analytic functions
+of E. In the ERM [24] νi were positive constants (defined via the parameters
+of the compound bag model), while in general case Eqs. (12)-(14) include
+other transition m.e.s fik. Later in our analysis we will be interested only in
+the denominator D(E) (12) and the factors in (13), (14), which fully define
+the position of a resonance.
+The value of z, in principle, can be calculated within the ERM, however,
+it can depend on many unknown parameters, and at the present stage we
+prefer to keep z as a single fitting parameter. It can be shown that z depends
+on the width of a resonance, but weakly depends on the resonance position.
+Now we consider three channels case to study more realistic case and
+choose the situation, when a resonance lies above the threshold 3. Here we do
+7
+
+not need to specify the channel 3, which for example, may be a conventional
+c¯c state with JP C = 0++. We introduce the 3 × 3 amplitude ˆfM(E) with
+three thresholds Ei (i = 1, 2, 3) and the momenta ki =
+�
+2µi(E − Ei), µi =
+m1im2i
+m1i+m2i, and Ei = m1i + m2i. Here m1i, m2i are the masses of two hadrons
+in the channel i. In this case the form of Eq. (9) is kept,
+ˆf3(E) =
+ˆN3
+D3(E), D3(E) = ((M11−ik1)(M22−ik2)−M12M21))(M23−ik3)+∆M,
+(15)
+where ∆M is
+∆M = M31M12M23+M32M21M13−M13M31(M22−ik2)−M32M23(M11−ik1).
+(16)
+For the energy E below the thresholds, 1 and 2, −ik1 = |k1|, −ik2 = |k2|, and
+the factor ∆M is a real function of E. For the threshold 3 below thresholds
+of 1 and 2 one can define the poles of the amplitude ˆf3, or the zeroes of
+D3(E), and rewrite the Eq. (15) as,
+D3 = (M11 − ik1)(M22 − ik2) − ˜z(E),
+(17)
+where the transition probability ˜z(E)
+˜z(E) = M12M21 − ∆M(M33 + ik3)
+M2
+33 + k2
+3
+(18)
+One can see that ˜z(E) acquires imaginary part, which can be of both signs.
+Therefore the influence of the third (or more) open channels, lying below
+the thresholds E1, E2 in the 2 × 2 matrix f12(E), may be important in some
+cases. The channel 3 can be taken into account, introducing complex values
+of z(E), which can depend on the energy as in Eq. (18).
+4
+The masses and widths of the scalar reso-
+nances
+We start with the X(3915) resonance and consider the following recoupling
+process: J/ψω → D∗ ¯D∗. At first we look at two-channel situation and choose
+the recoupling parameter z2 = 0.18 GeV2. For the X(3915) structure – cq¯c¯q
+the parameters µi, νi, Ei are given in the item 1) of section 2. Then inserting
+8
+
+all parameters to the Eq. (13), one obtains the distribution |f12(E)|2 (f2 ≡
+f12). Its values for different E are given in Table 1, which show that the
+maximum takes place at E = 3880 MeV, just near the lower threshold, and
+Γ2 = Γ(2 − channels) ∼= 15 MeV. In experiment for this resonance, observed
+by the Belle group in the process e+e− → e+e−J/ψω [1], the larger mass
+M(exp .) = (3918.4 ± 1.9) MeV and Γ(exp .) = (20 ± 5)
+MeV [3] were
+obtained.
+In the case of 3-channels, when e.g. the coupling to the c¯c channel is
+taken into account, the factor z3(E) acquires an imaginary part. In this case
+we calculate the amplitude f3(E), taking z3 = (0.18−i0.20) GeV2; the values
+of |f3(E)|2 are given in Tab. 1.
+Table 1: The values of the |f12(E)|2 for X(3915)
+E(GeV)
+3.85
+3.86
+3.88
+3.89
+3.90
+3.91
+3.915
+3.93
+|f2(E)|2
+3.04
+3.68
+63.08
+25.02
+8.33
+2.13
+1.65
+1.72
+|f3(E)|2
+1.82
+1.79
+1.03
+1.50
+3.30
+348.4
+360
+243
+From Table 1 one can see that in the 3-channel case the peak is shifted
+up by ∼ 35 MeV and corresponds the mass ER ∼= 3.915 GeV and the width
+Γ3 ∼= 20 MeV, which are in good agreement with the experimental mass and
+Γ(exp.) = 20(5) MeV [3].
+The scalar resonance X(3960) with JP C = 0++ was recently observed by
+the LHCb in the B+ → J/ψφK+ [18] and within the ERM it can be explained
+due to the infinite chain of the transitions: J/ψφ → D+
+s D−
+s and back. In
+two-channel approximation the X(3960) parameters (νi, µi, Ei, (i = 1, 2) are
+given in the item 2) (Section 2), which are used to define the amplitude (13).
+First, we choose z2 = 0.30 GeV2 and calculate the transition amplitudes
+|f12(E)|2; their values are given in the Table 2.
+In the two-channel approximation the numbers from Table 2 show the
+peak at E = 3940 MeV, near D+
+s D−
+s threshold, and Γ(2 − ch.) ∼= 15 MeV.
+In the 3-channel case the mass of the X(3960) resonance is shifted up to
+the position M(3 − ch.) = 3970 MeV and the width increases to the value
+Γ(th.) ∼= 45(5) MeV; these values are in agreement with the experimental
+numbers: M(X(3960)) = 3956(15) MeV, Γ(X(3960)) = (43 ± 21) MeV [18].
+In [18] the LHCb has reported about another, the X(4140) resonance,
+with JP C = 0++, in the B+ → D+
+s D−
+s K+ decay. Its mass M(X(4140) =
+9
+
+Table 2: The transition probability |f12|2 as a function of the energy E for
+the X(3960) resonance
+E(GeV)
+3.85
+3.88
+3.89
+3.92
+3.95
+3.97
+4.00
+4.05
+|f12|2(z = 0.30)
+3.93
+28.6
+7.89
+3.20
+2.28
+2.00
+1.38
+1.50
+|f3|2(z = 0.30 − i0.30)
+2.0
+1.43
+4.02
+23.7
+198
+500
+142.3
+42.2
+4133(12) MeV is close to the J/ψφ threshold. We consider this resonance as
+the cs¯c¯s system and first calculate the squared amplitudes |f12(E)|2 in two-
+channel case, taking the parameters µi, νi, Ei from the item 3) of Section 2. In
+this 2-channel case: J/ψφ and D∗+
+s D∗−
+s
+the transition probability z2 = 0.35
+is taken and the calculated values of |f12|2 are given in Table 3.
+In three-channel case the channel D+
+s D−
+s is added as the third one, then
+the values |f3|2 are calculated for z3 = 0.20 − i0.20 and given in Table 3.
+Table 3: The values of the |f12(E)|2 and |f3(E)|2 for the X(4140)
+E(GeV)
+4.00
+4.07
+4.12
+4.17
+4.22
+|f12(E)|2(z = 0.35)
+3.40
+8.67
+3.86
+1.27
+0.45
+|f3|2(z = 0.2 − i0.2)
+4.54
+12.87
+32.12
+13.7
+0.66
+From Table 3 one can see the peak at ER = (4.09 ± 0.01) GeV, Γ(th.) =
+60 MeV in two-channel approximation and the peak at ER = (4.12±0.02) GeV
+with the width Γ(th.) ∼= 100 MeV in tree-channel case, which are in good
+agreement with the experimental mass M(X(4140)) = (4133 ± 12) MeV and
+Γ(X(4140)) = (67 ± 24) MeV [18].
+Our numbers in Tables 1–3 show that in two-channel case the resonance
+always lies just near the lower threshold, however, if the coupling to the third
+channel is taken into account, then it is shifted up and its position occurs to
+be close to the experimental number. The masses and widths of the exotic
+resonances, X(3915), X(3960), X(4140), defined in the ERM, are given in
+the Table 4 together with experimental data.
+From Table 4 one can see that in the ERM the predicted masses and
+the widths of the scalar four-quark resonances are in good agreement with
+10
+
+Table 4: The ERM predictions for the masses and widths (in MeV) of exotic
+resonances with JP C = 0++
+Resonance
+M(th.)
+M(exp.)
+Γ(th.)
+Γ(exp.)
+X(3915)
+3920
+3918 (2)
+20
+20(5) [3]
+X(3960)
+3970
+3956(15)
+45(5)
+43(21) [18]
+X(4140)
+4120(20)
+4133(12)
+100
+67(24) [18]
+experiment, if besides two channels, which creates the resonance, the coupling
+of the resonance to third channel is taken into account.
+Comparing our results with those in literature, one can notice that our
+conclusions on the four-quark structure of the X(3915), X(3960, X(4140))
+also agree with the analysis in the paper [33], based on the coupled channel
+model of the c¯c and meson-meson systems. Notice that the general structure
+of the channel-coupling matrix elements in both approaches is similar.
+5
+The scalar X(4500), X(4700) resonances
+High scalar resonances X(4500), X(4700), or χc0(4500), χc0(4700), [38], were
+studied in many papers and for them two interpretations were suggested.
+First, the X(4500) and X(4700) are considered as the c¯c states – 4 3P0 and
+5 3P0 and their masses were calculated in relativistic quark models, where
+coupling to open channels was taken into account [14, 15, 41]. In [41] the
+influence of open channels is studied using the so-called screened potential
+[11], while in [13] the spectrum was calculated using the relativistic string
+Hamiltonian [42] with the flattened confining potential [43]; this flattening
+effect arises due to creation of virtual q¯q pairs. Notice that the flattened
+confining potential appears to be universal for all types of the mesons and it
+produces the hadronic shifts down ∼ (100 − 130) MeV for the 4P, 5P char-
+monium states and gives the masses of the 4 3P0, 5 3P0 states in a reasonable
+agreement with experiment [13]. On the contrary, in [44], within the
+3P0
+model, much smaller shifts due to the coupled-channel effects, <∼ 30 MeV ,
+were obtained for the 4 3P0, 5 3P0 states, while in [41] these states acquire too
+large mass shifts for the chosen screened potential.
+Model-independent analysis of the c¯c spectrum can also be done by means
+11
+
+of the Regge trajectories, if they are defined not for the meson mass M(nL)
+but for the excitation energy: E(nL) = M(nL) − 2 ¯mQ [45], where ¯mQ is the
+current heavy quark mass [13]:
+(M(n 3P0)−2 ¯mc)2 = 1.06+1.08nr, (inGeV2); n = nr +1,
+¯mc = 1.20 GeV2.
+(19)
+This Regge trajectory gives M(4 3P0) = 4.474 GeV and M(5 3P0) = 4.719 GeV,
+in good agreement with the LHCb data [38] (see Table 5).
+Table 5: The Regge trajectory predictions for the masses of the charmonium
+n 3P0 states (in MeV)
+state
+M(nP)
+exp. mass
+1 3P0
+3429
+3414.8(3))
+2 3P0
+3863
+3862+26
+−32 [16]
+3 3P0
+4194
+abs.
+4 3P0
+4473
+4474 ± 6 [38]
+5 3P0
+4719
+4694 ± 4+16
+−3 [38]
+6 3P0
+4941
+abs
+In Table 5 the masses M(2 3P0) = 3863 MeV, M(4 3P0) = 4473 MeV and
+M(5 3P0) = 4719 MeV, show very good agreement with those of χc0(3862)
+[16], X(4500) and X(4700) [38].
+At present other high excitations with
+JP = 1+, 2+ (n = 4, 5) are not yet found and their observation would be
+very important to understand the fine-structure effects of high charmonium,
+in particular, the fine-structure splitting have to decrease for a screened GE
+potential.
+Notice that the resonance X(4700) lies very close to the ψ(2S)φ threshold
+and this fact indicates a possible connection between the c¯c and the cs¯c¯s
+states. The four-quark interpretation of the X(4500), X(4700) was discussed
+in different models [19],[46]-[49], where in the mass region (4.4–4.8) GeV the
+radial or orbital excitations of a diquark-antidiquark systems can exist.
+12
+
+6
+Conclusions
+In our paper the scalar resonances X(3915), X(3960), X(4140) are assumed
+to be the four-quark states, produced due to recoupling mechanism, when
+one pair of mesons can transform into another pair of mesons infinitely many
+times. These resonances do not exist in the c¯c spectrum. As the four-quark
+states they have several specific features:
+1. The resonance appears only in the S-wave decay channel.
+2. Within the ERM it lies rather close to the lower threshold.
+3. The scalar four-quark resonance can be created in two channel case due
+to transitions between channels, but it can also be coupled to another
+channel 3, e.g. the c¯c channel.
+4. These resonances have no large sizes, being the compact systems, and
+this fact may be important for their observation. In the case of the
+X(3915) this statement is confirmed by the Belle analysis of the Q2
+distribution of the X(3915) → J/ψω decays in [50].
+The masses and widths of the X(3915), X(3960), X(4140), presented in Ta-
+ble 4, are obtained in a good agreement with experiment.
+The authors are grateful to N. P. Igumnova for collaboration.
+References
+[1] S. K. Choi et al. ( Belle Collab.), Phys. Rev. Lett. 94, 182002 (2005);
+arXiv: hep-ex/0408126.
+[2] B. Aubert et al. (BaBar Collab.),Phys. Rev. lett. 101, 082001 (2008).
+[3] P. A.Zvla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020,
+083 (2020).
+[4] S. Uehara et al. (Belle Collab.), Phys. Rev. Lett. 104, 092000 (2010);
+arXiv: 0912.4451 [hep-ex].
+[5] J. P. Lees et al. (BaBar Collab.), Phys. Rev. D 86, 072002 (2012); arXiv:
+1207.2651 [hep-ex].
+13
+
+[6] M. X. Duan et al. Phys. Rev. D 101, 054029 (2020); arXiv: 2002.03311
+[hep-ph], and references therein.
+[7] S. L. Olsen, arXiv:1904.06130 [hep-ex] and references therein; Phys. Rev.
+D 91, 057501 (2015), arXiv: 1410.6534 [hep-ex].
+[8] S. L. Olsen, T. Skwaricki, and D. Xiemnska, Rev. Mod. Phys. 90, 015003
+(2018); arXiv: 1708.04012 [hep-ex]. and references there in.
+[9] N. Brambilla et al., Phys. Rept. 873, 1 (2020); arXiv: 1907.07583 [hep-
+ex].
+[10] T. Barnes, S. Godfrey, and E. S. Swanson, Phys. Rev. D 72, 054026
+(2005); arXiv: hep-ph/0505002.
+[11] B. Q. Li, C. Meng, and K.T. Chao, Phys. Rev. D 80, 014012 (2009);
+arXiv: 0904.4068 [hep-ph].
+[12] D. Ebert, R. N. Faustov, and V. O. Galkin, Eur. Phys. J. C 71, 1825
+(2011); arXiv: 1111.0454 [hep-ph].
+[13] A. M. Badalian and B. L. G. Bakker, Phys. Rev. D 100, 054036 (2019);
+arXiv: 1902.09174 [hep-ph].
+[14] P. G.Ortega, J. Segovia, D. R. Entem, and F. Fernandez, Phys. Rev. D
+94,114018 (2016), arXiv:1608.01325 [hep-ph].
+[15] E. J. Eichten, K. Lane, and C. Quigg, Phys. Rev. D 73, 014014 (2006);
+arXiv: hep-ph/0511179.
+[16] K. Chilikin et al. (Belle Collab.), Phys. Rev. D 95, 112003 (2017); arXiv:
+1704.01872 [hep-ex].
+[17] E. Wang, H. S. Li, W. H. Liang, and E. Oset, Phys. Rev. D 103, 054008;
+arXiv: 2010.15431 [hep-ph].
+[18] LHCb Cllaboration, arXiv: 2210.15153 [hep-ex].
+[19] Q. Xin, Z. G. Wang, and X. S. Yang, arXiv: 2207.09910 [hep-ph].
+[20] H. Mutuk, arXiv:2211.14836 [hep-ph].
+[21] T. Ji et al. arXiv: 2212.00631 [hep-ph].
+14
+
+[22] J. M. Xie, M. Z. Liu, and L. S. Geng, arXiv:2207.12178 [hep-ph].
+[23] S. S. Agaev, K. Azizi, and H. Sundu, arXiv:2211.14129 [hep-ph].
+[24] Yu. A. Simonov, EPJ C 82, 1024 (2022), arXiv: 2209.03697 [hep-ph];
+A. M. Badalian and Yu. A. Simonov, arXiv: 2205.02576 [hep-ph].
+[25] Yu. A.Simonov, JHEP 04, 51 (2021); arXiv: 2011.12326 [hep-ph].
+[26] X. Y. Gao et al. (Belle Collab.) Phys. Rev. 105, 032002 (2022), arXiv:
+2112.02497 [hep-ex].
+[27] I. V. Danikin and Yu. A. Simonov, Phys. Rev. lett. 105, 102002 (2010),
+arXiv:1006.0211 [hep-ph]; Phys. Rev. D 81, 074027 (2010), arXiv:
+0907.1088 [hep-ph]; V. I. Danilikn, V. D. Orlovsky, and Yu. A.Simonov,
+Phys. Rev. D 85, 034012 (2012), arXiv: 1106.1552 [hep-ph].
+[28] A. M. Badalian, L. P. Kok, M. I. Polikarpov and Yu. A. Simonov, Phys.
+Rept. 82, 32 (1982).
+[29] N. Brambilla, G. Krein, J. T. Castella and A. Vairo, Phys. Rev. D 97,
+016016 (2018), arXiv:1707.09647 [hep-ph].
+[30] L. Maiani, A. Pilloni, A. D. Polosa, and V. Riquet, arXiv: 2208.02730
+[hep-ph].
+[31] D. Ebert, R. N. Faustov and V. O. Galkin, EPJC 58, 399 (2008) ; arXiv:
+0808.3912; [hep-ph].
+[32] E. Braaten, C. Langmarck, and D. H. Smith, Phys. Rev. 90, 014040
+(2014); arXiv: 1402.0438 [hep-ph].
+[33] P. G. Ortega, J. Segovia, D. R. Entem, and F. Fernandez, Phys. Lett.
+B 778, 1 (2018), arXiv: 1706.02639 [hep-ph].
+[34] X. Li and M. B. Voloshin, Phys. rev. D 91, 114014 (2015), arXiv:
+1503.04431 [hep-ph].
+[35] R. F. Lebed and A. D. Polosa, Phys. Rev. D 93, 094024 (2016), arXiv:
+1602.08421 [hep-ph]
+[36] W. Chen, et al. Phys. Rev. D 96, 114017 (2017), arXiv: 1706.09731
+[hep-ph].
+15
+
+[37] A. M. Badalian, B. L. Ioffe, and A. V. Smilga, Nucl. Phys. B 281, 85
+(1987).
+[38] R. Aaij et al. (LHCb Collab.) Phys. Rev. Lett. 118, 022003 (2017),
+arXiv:1606.07895 [hep-ex]; Phys. Rev. D 95, 012002(2017), arXiv:
+1606.07898 [hep-ex];
+Phys. Rev. Lett. 127,082001 (2021), arXiv:
+2103.01803 [hep-ex].
+[39] R. L. Jaffe, Phys. Rev. D 15, 267; ibid. 281(1977).
+[40] Yu. A. Simonov, Nucl. Phys. A 416, 103 (1984); Sov. J. Nucl. Phys. 36,
+99 (1982). Yad. Fiz. 36, 722 (1982).
+[41] M. X. Duan and X. Liu, Phys. Rev. D 104, 074010 (2021), arXiv:
+2107.14438 [hep-ph]
+[42] A. V. Dubin, A. B. Kaidalov, and Yu. A. Simonov, Phys. lett. B 343,
+310 (1995); Phys. Atom. Nucl. 56, 213 (1993); arXiv: hep-ph/9311344.
+[43] A. M. Badalian, B. L. G. Bakker, and Yu. A. Simonov, Phys. Rev. D
+66, 034026 (2002), arXiv: hep-ph/0204088.
+[44] S. Ferretti and E. Santopinto, Front. in Phys. 9, 76 (2021); arXiv:
+2104.00918 [hep-ph].
+[45] S. S. Afonin, Mod. Phys. Lett. A 22, 1369 (2007); S. S. Afonin and
+I. V. Pusenkov, Phys. Rev. D 90, 094020 (2014); arXiv: 1308.6540
+[hep-ph].
+[46] D. Ebert, R. N. Faustov, and V. O. Galkin, Eur. Phys. J. C 58, 399
+(2008); arXiv: 0808.3912 [hep-ph].
+[47] J. Wu, Phys. Rev. D 94, 094031 (2016), arXiv: 1608.07900 [hep-ph].
+[48] Q. F. L´’u and Y. B. Dong, Phys. Rev. D 94, 074007 (2016); arXiv:
+1607.05570 [hep-ph].
+[49] Y. Xie, D. He, X. Luo, and H. Sun, arXiv:2204.03924 [hep-ph].
+[50] Y. Teramoto et al. (Belle collab.), arXiv:2301.09421 [hep-ex].
+16
+

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+Deep learning for full-field ultrasonic characterization
+Yang Xu1, Fatemeh Pourahmadian1,2∗, Jian Song1, Conglin Wang3
+1 Department of Civil, Environmental & Architectural Engineering, University of Colorado Boulder, USA
+2 Department of Applied Mathematics, University of Colorado Boulder, USA
+3 Department of Physics, University of Colorado Boulder, USA
+Abstract
+This study takes advantage of recent advances in machine learning to establish a physics-based data analytic
+platform for distributed reconstruction of mechanical properties in layered components from full waveform
+data. In this vein, two logics, namely the direct inversion and physics-informed neural networks (PINNs), are
+explored. The direct inversion entails three steps: (i) spectral denoising and differentiation of the full-field
+data, (ii) building appropriate neural maps to approximate the profile of unknown physical and regularization
+parameters on their respective domains, and (iii) simultaneous training of the neural networks by minimizing
+the Tikhonov-regularized PDE loss using data from (i). PINNs furnish efficient surrogate models of complex
+systems with predictive capabilities via multitask learning where the field variables are modeled by neural
+maps endowed with (scaler or distributed) auxiliary parameters such as physical unknowns and loss function
+weights. PINNs are then trained by minimizing a measure of data misfit subject to the underlying physical
+laws as constraints.
+In this study, to facilitate learning from ultrasonic data, the PINNs loss adopts (a)
+wavenumber-dependent Sobolev norms to compute the data misfit, and (b) non-adaptive weights in a specific
+scaling framework to naturally balance the loss objectives by leveraging the form of PDEs germane to elastic-
+wave propagation. Both paradigms are examined via synthetic and laboratory test data. In the latter case, the
+reconstructions are performed at multiple frequencies and the results are verified by a set of complementary
+experiments highlighting the importance of verification and validation in data-driven modeling.
+Keywords:
+deep learning, ultrasonic testing, data-driven mechanics, full-wavefield inversion
+1. Introduction
+Recent advances in laser-based ultrasonic testing has led to the emergence of dense spatiotemporal datasets
+which along with suitable data analytic solutions may lead to better understanding of the mechanics of complex
+materials and components. This includes learning of distributed mechanical properties from test data which is
+of interest in a wide spectrum of applications from medical diagnosis to additive manufacturing [1, 2, 3, 4, 5,
+6, 7]. This work makes use of recent progress in deep learning [8, 9] germane to direct and inverse problems in
+partial differential equations [10, 11, 12, 13] to develop a systematic full-field inversion framework to recover the
+profile of pertinent physical quantities in layered components from laser ultrasonic measurements. The focus is
+on two paradigms, namely: the direct inversion and physics-informed neural networks (PINNs) [14, 15, 16, 17].
+The direct inversion approach is in fact the authors’ rendition of elastography method [18, 19, 20] through the
+prism of deep learning. To this end, tools of signal processing are deployed to (a) denoise the experimental
+data, and (b) carefully compute the required field derivatives as per the governing equations. In parallel,
+the unknown distribution of PDE parameters in space-frequency are identified by neural networks which are
+then trained by minimizing the single-objective elastography loss. The learning process is stabilized via the
+Tikhonov regularization [21, 22] where the regularization parameter is defined in a distributed sense as a
+separate neural network which is simultaneously trained with the sought-for physical quantities. This unique
+∗Corresponding author: tel. 303-492-2027, email [email protected]
+Preprint submitted to Elsevier
+January 9, 2023
+arXiv:2301.02378v1  [math.NA]  6 Jan 2023
+
+exercise of learning the regularization field without a-priori estimates, thanks to neural networks, proved to
+be convenient, effective, and remarkably insightful in inversion of multi-fidelity experimental data.
+PINNs have recently come under the spotlight for offering efficient, yet predictive, models of complex
+PDE systems [10] that has so far been backed by rigorous theoretical justification within the context of linear
+elliptic and parabolic PDEs [23]. Given the multitask nature of training for these networks and the existing
+challenges with modeling stiff and highly oscillatory PDEs [12, 24], much of the most recent efforts has been
+focused on (a) adaptive gauging of the loss function [12, 25, 26, 27, 28, 29, 13], and (b) addressing the gradient
+pathologies [24, 13] e.g., via learning rate annealing [30] and customizing the network architecture [11, 31, 32].
+In this study, our initially austere implementations of PINNs using both synthetic and experimental waveforms
+led almost invariably to failure which further investigation attributed to the following impediments: (a) high-
+norm gradient fields due to large wavenumbers, (b) high-order governing PDEs in the case of laboratory
+experiments, and (c) imbalanced objectives in the loss function.
+These problems were further magnified
+by our attempts for distributed reconstruction of discontinuous PDE parameters – in the case of laboratory
+experiments, from contaminated and non-smooth measurements. The following measures proved to be effective
+in addressing some of these challenges: (i) training PINNs in a specific scaling framework where the dominant
+wavenumber is the reference length scale, (ii) using the wavenumber-dependent Sobolev norms in quantifying
+the data misfit, (iii) taking advantage of the inertia term in the governing PDEs to naturally balance the
+objectives in the loss function, and (iv) denoising of the experimental data prior to training.
+This paper is organized as follows.
+Section 2 formulates the direct scattering problem related to the
+synthetic and laboratory experiments, and provides an overview of the data inversion logic. Section 3 presents
+the computational implementation of direct inversion and PINNs to reconstruct the distribution of L´ame
+parameters in homogeneous and heterogeneous models from in-plane displacement fields. Section 4 provides
+a detailed account of laboratory experiments, scaling, signal processing, and inversion of antiplane particle
+velocity fields to recover the distribution of a physical parameter affiliated with flexural waves in thin plates.
+The reconstruction results are then verified by a set of complementary experiments.
+2. Concept
+This section provides (i) a generic formalism for the direct scattering problem pertinent to the ensuing
+(synthetic and experimental) full-field characterizations, and (ii) data inversion logic.
+2.1. Forward scattering problem
+Consider ultrasonic tests where the specimen Π ⊂ Rd, d = 2, 3, is subject to (boundary or internal)
+excitation over the incident surface Sinc ⊂ Π and the induced (particle displacement or velocity) field u: Π ×
+[0 T] → RNΛ (NΛ ⩽ d) is captured over the observation surface Sobs ⊂ Π in a timeframe of length T. Here, Π
+is an open set whose closure is denoted by Π, and the sensing configuration is such that Sinc ∩ Sobs = ∅. In
+this setting, the spectrum of observed waveforms ˆu: Sobs × Ω → CNΛ is governed by
+Λ[ˆu; ϑ](ξ, ω) = 0,
+ˆu := F[u](ξ, ω),
+ξ ∈ Sobs, ω ∈ Ω,
+(1)
+where Λ of size NΛ×1 designates a differential operator in frequency-space; F represents the temporal Fourier
+transform; ϑ of dimension Nϑ×1 is the vector of relevant geometric and elastic parameters e.g., Lam´e constants
+and mass density; ξ ∈ Rd is the position vector; and ω > 0 is the frequency of wave motion within the specified
+bandwidth Ω.
+2.2. Dimensional platform
+All quantities in (1) are rendered dimensionless by identifying ρ◦, σ◦, and ℓ◦ as the respective reference
+scales [33] for mass density, elastic modulus, and length whose explicit values will be later specified.
+2.3. Data inversion
+Given the full waveform data ˆu on Sobs × Ω, the goal is to identify the distribution of material properties
+over Sobs.
+For this purpose, two reconstruction paradigms based on neural networks are adopted in this
+study, namely: (i) direct inversion, and (ii) physics-based neural networks.
+Inspired by the elastography
+2
+
+method [18, 19], quantities of interest in (i) are identified by neural maps over Sobs × Ω that minimize a
+regularized measure of Λ in (1). The neural networks in (ii), however, are by design predictive maps of the
+waveform data (i.e., ˆu) obtained by minimizing the data mismatch subject to (1) as a soft or hard constraint.
+In this setting, the unknown properties of Λ may be recovered as distributed parameters of the (data) network
+during training via multitask optimization.
+In what follows, a detailed description of the deployed cost
+functions in (i) and (ii) is provided after a brief review of the affiliated networks.
+2.3.1. Waveform and parameter networks
+Laser-based ultrasonic experiments furnish a dense dataset on Sobs × Ω. Based on this, multilayer per-
+ceptrons (MLPs) owing to their dense range [34] may be appropriate for approximating complex wavefields
+and distributed PDE parameters. Moreover, this architecture has proven successful in numerous applications
+within the PINN framework [15].
+In this study, MLPs serve as both data and property maps where the
+input consists of discretized space and frequency coordinates (ξi, ωj), i = 1, 2, . . . , Nξ, j = 1, 2, . . . , Nω, as
+well as distinct experimental parameters, e.g., the source location, distilled as one vector τk on domain T
+with k = 1, 2, . . . , Nτ, while the output represents waveform data Dijk = [Rˆu, Iˆu](ξi, ωj; τk) ∈ RNΛ × RNΛ,
+and/or the sought-for mechanical properties Pijn = [Rϑn, Iϑn](ξi, ωj) ∈ R × R, n = 1, 2, . . . , Nϑ. Note that
+following [35], the real R and imaginary I parts of (1) and every complex-valued variable are separated such
+that both direct and inverse problems are reformulated in terms of real-valued quantities. In this setting, each
+fully-connected MLP layer with Nl neurons is associated with the forward map Υl : RNl−1 → RNl,
+Υl(xl−1) = tanh(W lxl−1 + bl),
+xl−1 ∈ RNl−1,
+(2)
+where W l ∈ RNl×Nl−1 and bl ∈ RNl respectively denote the lth layer’s weight and bias. Consecutive compo-
+sition of Υl for l = 1, 2, . . . , Nm builds the network map wherein Nm designates the number of layers.
+2.3.2. Direct inversion
+Logically driven by the elastography method, the direct inversion approach depicted in Fig. 1 takes advan-
+tage of the leading-order physical principles underpinning the test data to recover the distribution of relevant
+physical quantities in space-frequency i.e., over the measurement domain.
+The ML-based direct inversion
+entails three steps: (a) spectral denoising and differentiation of (n-differentiable) waveforms ˆu over Sobs × Ω
+according to the (n-th order) governing PDEs in (1), (b) building appropriate MLP maps to estimate the
+profile of unknown physical parameters of the forward problem and regularization parameters of the inverse
+solution, and (c) learning the MLPs through regularized fitting of data to the germane PDEs.
+Note that synthetic datasets – generated via e.g., computer modeling or the method of manufactured
+solutions, may directly lend themselves to the fitting process in (c) as they are typically smooth by virtue
+Figure 1: Direct inversion: (a) FFT-based spatial differentiation of the full-field data as per operator Λ, (b) MLP-based approx-
+imation of the unknown PDE and regularization parameters (ϑ, α) on their respective domains, and (c) training the MLPs via
+minimizing the elastography loss Lε according to (3).
+3
+
+MLP
+ultrasonic test data
+u(S,w; T)
+N(s,w)
+spectral differentiation
+3
+Vu(s,w; T)
+Mα(s, w)
+VVu($, w; T)
+:
+(a)
+(b)
+M
+(9*,α*) := (Ng, )
+) = arg min L(u, *;α*)
+(c)
+9*,α*of numerical integration or analytical form of the postulated solution. Laboratory test data, however, are
+generally contaminated by noise and uncertainties, and thus, spectral differentiation is critical to achieve the
+smoothness requirements in (c). The four-tier signal processing of experimental data follows closely that of [36,
+Section 3.1] which for completeness is summarized here: (1) a band-pass filter consistent with the frequency
+spectrum of excitation is applied to the measured time signals at every receiver point, (2) the obtained
+temporally smooth signals are then differentiated or integrated to obtain the pertinent field variables, (3)
+spatial smoothing is implemented at every snapshot in time via application of median and moving average
+filters followed by computing the Fourier representation of the processed waveforms in space, (4) the resulting
+smooth fields may be differentiated (analytically in the Fourier space) as many times as needed based on the
+underlying physical laws in preparation for the full-field reconstruction in step (c). It should be mentioned
+that the experimental data may feature intrinsic discontinuities e.g., due to material heterogeneities or contact
+interfaces. In this case, the spatial smoothing in (3) must be implemented in a piecewise manner after the
+geometric reconstruction of discontinuity surfaces in Sobs which is quite straightforward thanks to the full-field
+measurements, see e.g., [36, section 3.2].
+Next, the unknown PDE parameters ϑ are approximated by a fully connected MLP network ϑ⋆ := Nϑ(ξ, ω)
+as per Section 2.3.1. The network is trained by minimizing the loss function
+Lε(ˆu, ϑ⋆; α) = ∥Λ(ˆu; ϑ⋆)∥2
+L2(Sobs×Ω×T )NΛ + ∥α1ϑ ⊙ ϑ⋆∥2
+L2(Sobs×Ω)Nϑ ,
+(3)
+where 1ϑ indicates an all-ones vector of dimension Nϑ × 1, and ⊙ designates the (element-wise) Hadamard
+product. Here, the PDE residual based on (1) is penalized by the norm of unknown parameters. Observe
+that the latter is a function of the weights and biases of the neural network which may help stabilize the MLP
+estimates during optimization. Such Tikhonov-type functionals are quite common in waveform tomography
+applications [37, 38, 39] owing to their well-established regularizing properties [21, 22]. Within this framework,
+R ∋ α > 0 is the regularization parameter which may be determined by three means, namely: (i) the Morozov
+discrepancy principle [40, 41], (ii) its formulation as a (constant or distributed) parameter of the ϑ⋆ network
+which could then be learned during training, and (iii) its independent reconstruction as a separate MLP
+network α⋆ := Nα(ξ, ω) illustrated in Fig. 1 (b) that is simultaneously trained along with ϑ⋆ by minimizing (3).
+In this study, direct inversion is applied to synthetic and laboratory test data with both α = 0 and α > 0,
+based on (ii) and (iii). It was consistently observed that the regularization parameter α plays a key role in
+controlling the MLP estimates. This is particularly the case in situations where the field ˆu is strongly polarized
+or near-zero in certain neighborhoods which brings about instability i.e., very large estimates for ϑ⋆ in these
+areas. In light of this, all direct inversion results in this paper correspond to the case of α > 0 identified by
+the MLP network α⋆.
+2.3.3. Physics-informed neural networks
+By deploying the knowledge of underlying physics, PINNs [14, 15] furnish efficient neural models of complex
+PDE systems with predictive capabilities.
+In this vein, a multitask learning process is devised according
+to Fig. 2 where (a) the field variable ˆu – i.e., measured data on Sobs × Ω × T , is modeled by the MLP
+map ˆu⋆ : = Nˆu(ξ, ω; τ) endowed with the auxiliary parameter γ(ξ, ω; τ) related to the loss function (4),
+(b) the physical unknowns ϑ could be defined either as parameters of ˆu⋆ as in Fig. 2 (i), or as a separate
+MLP ϑ⋆ : = Nϑ(ξ, ω) as shown in Fig. 2 (ii), and (c) learning the MLPs and affiliated parameters through
+minimizing a measure of data misfit subject to the governing PDEs as soft/hard constraints wherein the spatial
+derivatives of ˆu⋆ are computed via automatic differentiation [42]. It should be mentioned that in this study
+all MLP networks are defined on (a subset of) Sobs × Ω × T where Sobs ∩ ∂Π = ∅. Hence, the initial and
+boundary conditions – which could be specified as additional constraints in the loss function [15], are ignored.
+In this setting, the PINNs loss takes the form
+Lϖ(ˆu⋆, ϑ⋆|γ) = ∥ˆu − ˆu⋆∥2
+N(Sobs×Ω×T )NΛ + ∥γ1Λ ⊙ Λ(ˆu⋆; ϑ⋆)∥2
+L2(Sobs×Ω×T )NΛ, N = L2, �Hι, ι ⩽ n, (4)
+where 1Λ is a NΛ× 1 vector of ones; n is the order of Λ, and �Hι denotes the adaptive Hι norm defined by
+4
+
+Figure 2: Two logics for the physics-informed neural networks (PINNs) with distributed parameters: (i) the test data ˆu(ξ, ω; τ)
+are modeled by a MLP map, while the unknown physical parameters ϑ – on Sobs × Ω, and the loss function weight γ – on
+Sobs × Ω × T , are defined as network parameters, and (ii) ˆu(ξ, ω; τ) and ϑ(ξ, ω) are identified by separate MLPs, while γ is a
+parameter of Nˆu. The MLP(s) in (i) and (ii) are then trained by minimizing Lϖ of (4) in the space of data and PDE parameters.
+∥ · ∥ �
+Hι :=
+�
+�
+1⩽|e|⩽ ι
+γe ∥∇e(·)∥2
+L2 + ∥·∥2
+L2,
+∇e =
+∂|e|
+∂ξe1
+1 ∂ξe2
+2 ··· ∂ξed
+d
+,
+|e| :=
+d
+�
+i=1
+ei.
+(5)
+Here, e:= {e1, e2, . . . ed} is a vector of integers ei ⩾ 0. Provided that ∀e, γe = 1, then �Hι is by definition
+equal to Hι [43]. Note however that at high wavenumbers, Hι is dominated by the highest derivatives ∇eˆu⋆,
+|e| = ι, which may complicate (or even lead to the failure of) the training process due to uncontrolled error
+amplification by automatic differentiation particularly in earlier epochs. This issue may be addressed through
+proper weighting of derivatives in (5). In light of the frequency-dependent Sobolev norms in [44, 37], one
+potential strategy is to adopt the wavenumber-dependent weights as the following
+γe =
+�
+1
+κe1
+1 κe2
+2 ··· κed
+d
+�2
+,
+1 ⩽ |e| ⩽ ι,
+wherein κi is a measure of wavenumber along ξi for i = 1, . . . , d.
+In this setting, the weighted norms of
+derivatives in (5) remain approximately within the same order as the L2 norm of data misfit. Another way to
+automatically achieve the latter is to set the reference scale ℓ◦ such that κi ∼1. Note that the �Hι norms directly
+inform the PINNs about the “expected” field derivatives – while preventing their uncontrolled magnification.
+This may help stabilize the learning process as such derivatives are intrinsically involved in the PINNs loss via
+Λ(ˆu⋆; ϑ⋆). It should be mentioned that when N = �Hι in (4), the “true” estimates for derivatives ∇eˆu may
+be obtained via spectral differentiation as per Section 2.3.2.
+The Lagrange multiplier [45, 46] γ(ξ, ω; τ) in (4) is critical for balancing the loss components during
+training. Its optimal value, however, highly depends on (a) the nature of Λ [12], and (b) the distribution
+of unknown parameters ϑ.
+It should be mentioned that setting γ = 1 led to failure in almost all of the
+synthetic and experimental implementations of PINNs in this study. Gauging of loss function weights has
+been the subject of extensive recent studies [12, 25, 47, 26, 27, 28]. One systematic approach is the adaptive
+SA-PINNs [12] where the multiplier γ(ξ, ω; τ) is a distributed parameter of ˆu⋆ whose value is updated in
+each epoch according to a minimax weighting paradigm. Within this framework, the data (and parameter)
+networks are trained by minimizing Lϖ with respect to ˆu⋆ and ϑ⋆, while maximizing the loss with respect to
+γ as shown in Fig. 2.
+Depending on the primary objective for PINNs, one may choose nonadaptive or adaptive weighting. More
+speci��cally, if the purpose is high-fidelity forward modeling via neural networks where ϑ is known a-priori and
+PINNs are intended to serve as predictive surrogate models of Λ, then ideas rooted in constrained optimization
+e.g., minimax weighting is theoretically sound. However, if the inverse solution i.e., identification of ϑ(ξ, ω)
+from “real-world” or laboratory test data is the main goal particularly in a situation where any assumption on
+the smoothness of ϑ and/or applicability of Λ may be (at least locally) violated e.g., due to unknown material
+5
+
+MLP
+network parameters
+(i)
+(ii)
+9*($, w)
+9*:=
+(S,w; T)
+N(s, w)
+?
+↑
+automatic
+E
+3
+differentiation
+α*:=
+V*(S,w; T)
+α*:=
+T
+3
+Na(S, w; T)
+VVu*($, w; T)
+Na(S, w; T)
+T
+:
+MLP
+↑
+(S,w; T)
+u*
+= arg min max Lw(u*, *I)
+*,9*heterogeneities or interfacial discontinuities, then trying to enforce Λ everywhere on Sobs × Ω × T (via point-
+wise adaptive weighting) may lead to instability and failure of data inversion. In such cases, nonadaptive
+weighting may be more appropriate. In light of this, in what follows, γ is a non-adaptive weight specified by
+taking advantage of the PDE structure to naturally balance the loss objectives.
+3. Synthetic implementation
+Full-field characterization via the direct inversion and physics-informed neural networks are examined
+through a set of numerical experiments. The waveform data in this section are generated via a FreeFem++ [48]
+code developed as part of [49].
+3.1. Problem statement
+Plane-strain wave motion in two linear, elastic, piecewise homogeneous, and isotropic samples is modeled
+according to Fig. 3 (a). On denoting the frequency of excitation by ω, let ℓr = 2π
+ω
+�
+µr/ρr, ρr = 1, and µr = 1
+be the reference scales for length, mass density, and stress, respectively. In this framework, both specimens
+are of size 16×16 and uniform density ρ = 1. The first sample Π1 ⊂ R2 is characterized by the constant Lam´e
+parameters µ◦ = 1 and λ◦ = 0.47, while the second sample Π2 ⊂ R2 is comprised of four perfectly bonded
+homogenous components Π2j of µj = j and λj = 2j/3, j = {1, 2, 3, 4} such that Π2 = �4
+j=1 Π2j. Accordingly,
+the shear and compressional wave speeds read c◦
+s = 1, c◦
+p = 1.57 in Π1, and cj
+s = √j, cj
+p = 1.63√j in Π2j.
+Every numerical experiment entails an in-plane harmonic excitation at ω = 3.91 via a point source on Sinc
+(the perimeter of a 14 × 14 square centered at the origin). The resulting displacement field uα = (uα
+1 , uα
+2 ),
+α = 1, 2, is then computed in Πα over Sobs (a concentric square of dimension 8 ×8) such that
+µα∆uα(ξ) + (λα + µα)∇∇ · uα(ξ) + ρω2uα(ξ) = δ(ξ − x)d,
+ξ ∈ Πα, x ∈ Sinc,
+�
+λα∇ · uα(ξ)I2 + 2µα∇symuα(ξ)
+�
+· n(ξ) = 0,
+ξ ∈ ∂Πα,
+(6)
+where x and d respectively indicate the source location and polarization vector; n is the unit outward normal
+to the specimen’s exterior, and
+�
+µα = µ◦, λα = λ◦,
+α = 1
+µα = µj, λα = λj,
+α = 2 ∧ ξ ∈ Π2j∈{1,2,3,4}
+.
+Figure 3: synthetic experiments simulating plane-strain wave motion in homogeneous (top-left) and heterogeneous (bottom-left)
+specimens: (a) testing configuration where the model is harmonically excited at frequency ω by a point source on Sinc, and the
+induced displacement field u is computed over Sobs along ξ1 and ξ2 as shown in (b) and (c), respectively.
+6
+
+TT1
+μo,\。
+u1
+μ3,^3
+μ4，^4
+TT2
+W2
+(a)
+(b)When α = 2, the first of (6) should be understood as a shorthand for the set of four governing equations
+over Π2j, j = {1, 2, 3, 4}, supplemented by the continuity conditions for displacement and traction across
+∂Π2j\∂Π2 as applicable.
+In this setting, the generic form (1) may be identified as the following
+Λ = Λα := µα∆ + (λα + µα)∇∇ · + ρω2I2,
+α = 1, 2,
+ˆu = uα(ξ, ω; τ),
+ϑ = [µα, λα](ξ, ω),
+ξ ∈ Sobs, ω ∈ Ω, τ ∈ T ,
+(7)
+wherein I2 is the second-order identity tensor; τ = (x, d) ∈ Sinc × B1 = T with B1 denoting the unit circle
+of polarization directions. Note that ρ is treated here as a known parameter.
+In the numerical experiments, Sinc (resp. Sobs) is discretized by a uniform grid of 32 (resp. 50×50) points,
+while Ω and B1 are respectively sampled at ω = 3.91 and d = (1, 0).
+All inversions in this study are implemented within the PyTorch framework [50].
+3.2. Direct inversion
+The three-tier logic of Section 2.3.2 is employed to reconstruct the distribution of µα and λα, α = 1, 2,
+over Sobs, entailing: (a) spectral differentiation of the displacement field uα in order to compute ∆uα and
+∇∇ · uα as per (6), (b) construction of three positive-definite MLP networks µ⋆, λ⋆, and α⋆; each of which
+is comprised of one hidden layer of 64 neurons, and (c) training the MLPs by minimizing Lε as in (3)
+and (7) by way of the ADAM algorithm [51]. To avoid near-boundary errors affiliated with the one-sided FFT
+differentiation in ∆uα and ∇∇·uα, a concentric 40×40 subset of collocation points sampling Sobs is deployed
+for training purposes. It should also be mentioned that in the heterogeneous case, i.e., α = 2, the discontinuity
+of derivatives across ∂Π2j∈{1,2,3,4} calls for piecewise spectral differentiation. According to Section 2.3.1, the
+input to P⋆ = NP(ξ, ω), P = µ, λ, and α⋆ = Nα(ξ, ω) is of size NξNτ × Nω = 1600Ns × 1 where Ns ⩽ 32
+is the number of simulations i.e., source locations used to generate distinct waveforms for training. In this
+setting, since the physical quantities of interest are independent of τ, the real-valued output of MLPs is of
+dimension 1600 × 1 furnishing a local estimate of the L´ame and regularization parameters at the specified
+sampling points on Sobs. Each epoch makes use of the full dataset and the learning rate is 0.005.
+In this work, the reconstruction error is measured in terms of the normal misfit
+Ξ(q⋆) = ∥q⋆ − q ∥L2
+∥q ∥L∞
+,
+(8)
+where q⋆ is an MLP estimate for a quantity with the “true” value q.
+Let Sinc be sampled at one point i.e., Ns = 1 so that a single forward simulation in Πα, α = 1, 2, generates
+the training dataset. The resulting reconstructions are shown in Figs. 4 and 5. It is evident from both figures
+that the single-source reconstruction fails at the loci of near-zero displacement which may explain the relatively
+high values of the recovered regularization parameter α⋆. Table 1 details the true values as well as mean and
+standard deviation of the reconstructed L´ame distributions ϑ⋆ = (µ⋆, λ⋆) in Π1 (resp. Π2j for j = 1, 2, 3, 4)
+according to Fig. 4 (resp. Fig. 5).
+This problem may be addressed by enriching the training dataset e.g., via increasing Ns. Figs. 6 and 7
+illustrate the reconstruction results when Sinc is sampled at Ns = 5 source points. The mean and standard
+deviation of the reconstructed distributions are provided in Table 2. It is worth noting that in this case the
+identified regularization parameter α⋆ assumes much smaller values – compared to that of Figs. 4 and 5. This
+is closer to the scale of computational errors in the forward simulations.
+To examine the impact of noise on the reconstruction, the multisource dataset used to generate Figs. 6
+and 7 are perturbed with 5% white noise. The subsequent direct inversions from noisy data are displayed in
+Figs. 8 and 9, and the associated statistics are presented in Table 3. Note that spectral differentiation as the
+first step in direct inversion plays a critical role in denoising the waveforms, and subsequently regularizing the
+reconstruction process. This may substantiate the low magnitude of MLP-recovered α⋆ in the case of noisy
+data in Figs. 8 and 9. The presence of noise, nonetheless, affects the magnitude and thus composition of terms
+in the Fourier representation of the processed displacement fields in space which is used for differentiation.
+This may in turn lead to the emergence of fluctuations in the reconstructed fields.
+7
+
+Figure 4: Direct inversion of the L´ame parameters in Π1 using noiseless data from a single source: (a) MLP-predicted distributions
+µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µ◦ = 1 and λ◦ = 0.47, (c) MLP-recovered distribution of
+the regularization parameter α⋆, and (d) loss function Lε vs. the number of epochs Ne in the log = log10 scale.
+Figure 5: Direct inversion of the L´ame parameters in Π2 using noiseless data from a single source: (a) MLP-predicted distributions
+µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µj = j and λj = 2j/3, j = {1, 2, 3, 4}, (c) MLP-recovered
+regularization parameter α⋆, and (d) loss function Lε vs. the number of epochs Ne.
+Table 1: Mean ⟨·⟩D and standard deviation σ(·|D) of the reconstructed L´ame distributions in D = Π1, Π2j=1,2,3,4. Here,
+the direct inversion is applied to noiseless data from a single source as shown in Figs. 4 and 5.
+D
+Π1
+Π21
+Π22
+Π23
+Π24
+µ
+µ◦ = 1
+µ1 = 1
+µ2 = 2
+µ3 = 3
+µ4 = 4
+⟨µ⋆⟩D
+0.998
+0.991
+1.983
+2.825
+3.835
+σ(µ⋆|D)
+0.024
+0.083
+0.182
+0.441
+0.325
+λ
+λ◦ = 0.47
+λ1 = 0.67
+λ2 = 1.33
+λ3 = 2
+λ4 = 2.66
+⟨λ⋆⟩D
+0.376
+0.615
+0.850
+1.746
+1.412
+σ(λ⋆|D)
+0.128
+0.161
+0.399
+0.486
+0.864
+8
+
+(a)
+(b)
+1.2
+0.2
+1.1
+0.15
+0.1
+×10-2
+(c)
+(d)
+1.4
+log(Le)
+0.9
+0.05
+1
+1
+0
+0.8
+0
+0.7
+0.2
+0.6
+-1
+（?
+0.6
+0.15
+-2 
+0.2
+0.5
+0.1
+0
+0.5
+×104
+1
+Ne
+0.4
+0.05
+0.3
+0(a)
+(b)
+三(μ*)
+0.8
+3
+(c)
+×10-2
+(d)
+0.4
+2
+log(Le)
+1
+2
+0
+0.8
+0
+0.5
+1 ×104
+0.4
+NeFigure 6: Direct inversion of the L´ame parameters in Π1 using noiseless data from five distinct simulations: (a) MLP-predicted
+distributions µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µ◦ = 1 and λ◦ = 0.47, (c) MLP-recovered
+regularization parameter α⋆, and (d) loss function Lε vs. the number of epochs Ne.
+Figure 7: Direct inversion of the L´ame parameters in Π2 using five noiseless datasets: (a) MLP-predicted distributions µ⋆ and
+λ⋆, (b) reconstruction error (8) with respect to the true values µj = j and λj = 2j/3, j = {1, 2, 3, 4}, (c) MLP-recovered
+regularization parameter α⋆, and (d) loss function Lε vs. the number of epochs Ne.
+Table 2: Mean and standard deviation of the reconstructed L´ame distributions from five distinct noiseless datasets
+according to Figs. 6 and 7.
+D
+Π1
+Π21
+Π22
+Π23
+Π24
+µ
+1
+1
+2
+3
+4
+⟨µ⋆⟩D
+1.000
+0.999
+2.003
+2.999
+3.999
+σ(µ⋆|D)
+0.001
+0.012
+0.011
+0.012
+0.016
+λ
+0.47
+0.67
+1.33
+2
+2.66
+⟨λ⋆⟩D
+0.464
+0.660
+1.302
+1.997
+2.635
+σ(λ⋆|D)
+0.012
+0.039
+0.071
+0.048
+0.068
+9
+
+(a)
+(b)
+2
+u*
+1.02
+(×)m
+1
+(c)
+×10-3
+(d)
+1
+log(Le)
+5
+1
+0.98
+×10-2
+0
+3
+-1
+7.5
+^*
+0.5
+三(\*)
+-2 
+5
+0.45
+0
+0.5
+1 ×104
+Ne
+2.5
+0.4
+/×10-2(a)
+(b)
+4
+1.75
+3
+(c)
+×10-3
+(d)
+0.75
+1.4
+2
+log(Le)
+×10-2
+0
+0.8
+0.2 -2
+2
+0.5
+×104
+0
+0.4
+Ne
+×10-1Figure 8: Direct inversion of the L´ame parameters in Π1 using five datasets perturbed with 5% white noise: (a) MLP-predicted
+distributions µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µ◦ = 1 and λ◦ = 0.47, (c) MLP-recovered
+regularization parameter α⋆, and (d) loss function Lε vs. the number of epochs Ne.
+Figure 9: Direct inversion of the L´ame parameters in Π2 using five datasets perturbed with 5% white noise: (a) MLP-predicted
+distributions µ⋆ and λ⋆, (b) reconstruction error (8) with respect to the true values µj = j and λj = 2j/3, j = {1, 2, 3, 4}, (c)
+MLP-recovered regularization parameter α⋆, and (d) loss function Lε vs. the number of epochs Ne.
+Table 3: Mean and standard deviation of the reconstructed L´ame distributions from noisy data according to Figs. 8
+and 9.
+D
+Π1
+Π21
+Π22
+Π23
+Π24
+µ
+1
+1
+2
+3
+4
+⟨µ⋆⟩D
+1.001
+1.002
+2.005
+2.996
+3.996
+σ(µ⋆|D)
+0.005
+0.016
+0.035
+0.054
+0.088
+λ
+0.47
+0.67
+1.33
+2
+2.66
+⟨λ⋆⟩D
+0.462
+0.650
+1.263
+2.006
+2.654
+σ(λ⋆|D)
+0.042
+0.051
+0.225
+0.182
+0.300
+10
+
+(a)
+(b)
+1.05
+(r)=
+L￥
+2
+×10-3
+(d)
+(c)
+1
+4
+log(Le)
+3
+0.95
+×10-2
+0
+2
+0.5
+三()*)
+0.35
+1
+-2
+0.25
+0.45
+0
+0.5
+1 ×104
+0.15
+Ne
+0.05
+0.4(a)
+(b)
+4
+E(μ*)
+0.8
+3
+(c)
+×10-3
+(d)
+0.4
+1.4
+2
+log(Le)
+1
+×10-1
+0
+0.5
+0.6
+-1
+0.2
+2
+0.3
+0
+0.5
+Ne
+0.13.3. Physics-informed neural networks
+The learning process of Section 2.3.3 is performed as follows: (a) the MLP network uα⋆ = Nuα(ξ, ω, x|γ, ϑ⋆)
+endowed with the positive-definite parameters γ and ϑ⋆ = (µ⋆, λ⋆) is constructed such that the input x labels
+the source location and the auxiliary weight γ is a nonadaptive scaler, (b) µ⋆ and λ⋆ may be specified as scaler
+or distributed parameters of the network according to Fig. 2 (i), and (c) uα⋆ is trained by minimizing Lϖ
+in (4) via the ADAM optimizer using the synthetic waveforms of Section 3.1. Reconstructions are performed
+on the same set of collocation points sampling Sobs×Ω×T as in Section 3.2. Accordingly, the input to uα⋆ is
+of size Nξ×Nω×Nτ = 1600×1×Ns, while its output is of dimension (1600×1×Ns)2 modeling the displacement
+field along ξ1 and ξ2 in the sampling region. Similar to Section 3.2, each epoch makes use of the full dataset for
+training and the learning rate is 0.005. The PyTorch implementation of PINNs in this section is accomplished
+by building upon the available codes on the Github repository [52].
+The MLP network u1⋆ = u1⋆(ξ, ω, x|γ, ϑ⋆) with three hidden layers of respectively 20, 40, and 20 neurons
+is employed to map the displacement field u1 (in Π1) associated with a single point source of frequency
+ω = 3.91 at x = x1 ∈ Sinc.
+The L´ame constants are defined as the unknown scaler parameters of the
+network i.e., ϑ⋆ = {µ⋆, λ⋆}, and the Lagrange multiplier γ is specified per the following argument. Within
+the dimensional framework of this section and with reference to (7), observe that on setting γ =
+1
+ρω2 (i.e.,
+γ = 0.065), both (the PDE residue and data misfit) components of the loss function Lϖ in 4 emerge as some
+form of balance in terms of the displacement field. This may naturally facilitate maintaining of the same scale
+for the loss terms during training, and thus, simplifying the learning process by dispensing with the need to
+tune an additional parameter γ. Keep in mind that the input to u1⋆ is of size 1600×1×1, while its output is
+of dimension (1600×1×1)2. In this setting, the training objective is two-fold: (a) construction of a surrogate
+map for u1, and (b) identification of µ⋆ and λ⋆.
+Fig. 10 showcases (i) the accuracy of PINN estimates based on noiseless data in terms of the vertical
+component of displacement field u1
+2 in Π1, and (ii) the performance of automatic differentiation [42] in capturing
+the field derivatives in terms of components that appear in the governing PDE 7 i.e., u1
+2,ij = ∂2u1
+2/(∂ξi∂ξj),
+i, j = 1, 2.
+The comparative analysis in (ii) is against the spectral derivates of FEM fields according to
+Section 2.3.2. It is worth noting that similar to Fourier-based differentiation, the most pronounced errors
+in automatic differentiation occur in the near-boundary region i.e., the support of one-sided derivatives. It
+is observed that the magnitude of such discrepancies may be reduced remarkably by increasing the number
+of epochs. Nonetheless, the loci of notable errors remain at the vicinity of specimen’s external boundary or
+internal discontinuities such as cracks or material interfaces. Fig. 10 is complemented with the reconstruction
+results of Fig. 11 indicating (µ⋆, λ⋆) = (1.000, 0.486) for the homogenous specimen Π1 with the true L´ame
+constants (µ◦, λ◦) = (1, 0.47). The impact of noise on training is examined by perturbing the noiseless data
+related to Fig. 10 with 5% white noise, which led to (µ⋆, λ⋆) = (0.999, 0.510) as shown in Fig. 12.
+Next, the PINN u2⋆ = u2⋆(ξ, ω, x|ϑ⋆) with three hidden layers of respectively 120, 120, and 80 neurons
+is created to reconstruct (i) displacement field u2 in the heterogeneous specimen Π2, and (ii) distribution of
+the L´ame parameters over the observation surface. In this vein, synthetic waveform data associated with five
+point sources {xi} ∈ Sinc, i = 1, 2, . . . , 5 at ω = 3.91 is used for training. Here, ϑ⋆ is the network’s unknown
+distributed parameter, of dimension (40×40)2, and the nonadaptive scaler weight γ = 0.065 in light of the
+sample’s uniform density ρ = 1. In this setting, the input to u2⋆ is of size 1600×1×5, while its output is
+of dimension (1600×1×5)2. Fig. 13 provides a comparative analysis between the FEM and PINN maps of
+horizontal displacement u1
+2 in Π2 and its spatial derivatives computed by spectral and automatic differentiation
+respectively.
+Table 4: Mean and standard deviation of the PINN-reconstructed L´ame distributions from five distinct noiseless datasets
+according to Fig. 14.
+D
+Π21
+Π22
+Π23
+Π24
+⟨µ⋆⟩D
+0.975
+1.973
+2.941
+. 3.918
+σ(µ⋆|D)
+0.054
+0.123
+0.135
+0.226
+⟨λ⋆⟩D
+0.686
+1.250
+2.045
+2.065
+σ(λ⋆|D)
+0.247
+0.400
+0.520
+0.857
+11
+
+Figure 10: PINN vs. FEM maps of vertical displacement and its derivatives in Π1: (a) MLP estimates, from noiseless data, for
+{u1
+2
+⋆, u1⋆
+2,11, u1⋆
+2,22, u1⋆
+2,12} wherein the derivatives u1⋆
+2,ij, i, j = 1, 2, are obtained by automatic differentiation, (b) FEM displacement
+solution and its spectral derivatives for {u1
+2, u1
+2,11, u1
+2,22, u1
+2,12}, and (c) normal misfit 8 between (a) and (b).
+Figure 11: PINN reconstruction of L´ame constants in the homogeneous plate Π1 from noiseless data: (a) µ⋆ vs. number of epochs
+Ne, (b) λ⋆ vs. Ne, and (c) total loss Lϖ and its components (the PDE residue and data misfit) vs. Ne in log scale.
+Figure 12: PINN reconstruction of L´ame constants in Π1 from noisy data: (a) µ⋆ vs. number of epochs Ne, (b) λ⋆ vs. Ne, and
+(c) total loss Lϖ and its components (the PDE residue and data misfit) vs. Ne in log scale.
+12
+
+2
+0.2
+? 
+U2,22
+1
+1
+0.5
+0.1
+(a)
+0
+0
+0
+0
+-0.1
+-0.5
+-1
+-0.2
+2
+0.2
+I
+u2,11
+u2,22
+2,12
+1
+1
+0.5
+(b)
+0
+0
+0
+0
+-0.5
+-1
+-1
+-0.2
+三(u2
+7
+E(u2,11)
+三(u2,22)
+0.3
+三(u2,12)
+0.2
+?L
+1.2
+5
+0.2
+(c)
+0.8
+0.1
+3
+0.1
+0.4
+1
+×10-2
+×10-1(a)
+(b)
+(c)
+0.8
+\*
+- PDE loss
+1.2
+0
+.- data loss
+ total loss
+0.8
+0.4
+-2 
+0.4
+-4
+Ne
+Ne
+0
+0
+×105
+×105
+×105
+0
+0.4
+0.8
+1.2
+1.6
+2.
+0
+0.4
+0.8
+1.2
+1.6
+2
+0
+0.4
+0.8
+1.2
+1.6
+2(a)
+(b)
+(c)
+\*
+PDE loss
+L*
+1.2
+0.6
+ data loss
+0
+ total loss
+0.8
+0.4
+-2 
+0.4
+0.2
+Ne
+Ne
+UN
+0
+0
+×105
+×105
+×105
+0
+0.4
+0.8
+1.2
+1.6
+2.
+0
+0.4
+0.8
+1.2
+1.6
+2
+0
+0.4
+0.8
+1.2
+1.6
+2Figure 13: PINN vs. FEM maps of horizontal displacement and its derivatives in Π2: (a) PINN estimates, from noiseless data, for
+{u2
+1
+⋆, u2⋆
+1,11, u2⋆
+1,22, u2⋆
+1,12} wherein the derivatives u2⋆
+1,ij, i, j = 1, 2, are obtained by automatic differentiation, (b) FEM displacement
+solution and its spectral derivatives for {u2
+1, u2
+1,11, u2
+1,22, u2
+1,12}, and (c) normal misfit 8 between (a) and (b).
+Figure 14: PINN reconstruction of L´ame parameters in Π2 using five noiseless datasets: (a) PINN-predicted distributions µ⋆ and
+λ⋆, (b) reconstruction error (8) with respect to the true values µj = j and λj = 2j/3, j = {1, 2, 3, 4}, (c) total loss Lϖ and its
+components (the PDE residue and data misfit) vs. Ne in log scale.
+The PINN-reconstructed distribution of PDE parameters is illustrated in Fig. 14 whose statistics is
+detailed in Table 4.
+It is worth mentioning that the learning process is repeated for a suit of weights
+γ = {0.01, 0.025, 0.1, 0.25, 0.5, 1.5, 2, 5, 10, 15}. In all cases, the results are either quite similar or worse than
+that of Figs. 13 and 14.
+13
+
+2 *
+0.4
+2*
+2*
+2*
+ui
+ui,11
+ui,22
+ui,12
+3
+3
+2
+0
+1
+1
+(a)
+0
+-0.4 
+-1
+-1
+2
+-3
+-3
+-0.8 
+0.4
+ui,11
+2
+ui,12
+3
+3
+2
+1
+1
+(b)
+0
+-0.4
+-1
+-1
+-2 
+-3
+-3
+-0.8
+5
+三(ui
+三(ui,11)
+2
+三(ui,22)
+2*
+E(ui,12)
+2*
+2
+2
+3
+(c)
+1
+L
+1
+×10-3
+×10-2
+×10-2
+×10-2(a)
+(b)
+4
+三(μ*)
+0.4
+3
+(c)
+0.2
+2
+PDE loss
+0
+data loss
+total loss
+0
+-2
+三(\*)
+-4
+0.4
+2
+-6
+×106
+0
+0.4
+0.8
+1.2
+1.6
+2
+0.2
+Ne
+14. Laboratory implementation
+This section examines the performance of direct inversion and PINNs for full-field ultrasonic character-
+ization in a laboratory setting. In what follows, experimental data are processed prior to inversion as per
+Section 2.3.2 which summarizes the detailed procedure in [36]. To verify the inversion results, quantities of
+interest are also reconstructed through dispersion analysis, separately, from a set of auxiliary experiments.
+4.1. Test set-up
+Experiments are performed on two (homogeneous and heterogeneous) specimens: Π
+exp
+1
+which is a 27 cm
+×27 cm×1.5 mm sheet of T6 6061 aluminum, and Π
+exp
+2
+composed of (a) 5 cm×27 cm×1.5 mm sheet of Grade
+2 titanium, (b) 2.5 cm×27 cm×1.5 mm sheet of 4130 steel, and (c) 5 cm×27 cm×1.5 mm sheet of 260-H02
+brass, connected via metal epoxy. For future reference, the density ρµ, Young’s modulus Eµ, and Poisson’s
+ratio νµ for µ = {Al, Ti, St, Br} are listed in Table 5 as per the manufacturer.
+Ultrasonic experiments on both samples are performed in a similar setting in terms of the sensing config-
+uration and illuminating wavelet. In both cases, the specimen is excited by an antiplane shear wave from a
+designated source location Sinc, shown in Fig. 15, by a 0.5 MHz p-wave piezoceramic transducer (V101RB by
+Olympus Inc.). The incident signal is a five-cycle burst of the form
+H(fct) H(5−fct) sin
+�
+0.2πfct
+�
+sin
+�
+2πfct
+�
+,
+(9)
+where H denotes the Heaviside step function, and the center frequency fcis set at 165 kHz (resp. {80, 300} kHz)
+in Π
+exp
+1
+(resp. Π
+exp
+2 ). The induced wave motion is measured in terms of the particle velocity vβ, β = 1, 2, on the
+scan grids Gβ sampling Sobs where Sobs ∩Sinc = Sobs ∩∂Π
+exp
+β = ∅. A laser Doppler vibrometer (LDV) which is
+mounted on a 2D robotic translation frame (for scanning) is deployed for measurements. The VibroFlex Xtra
+VFX-I-120 LDV system by Polytec Inc. is capable of capturing particle velocity within the frequency range
+∼ DC − 24 MHz along the laser beam which in this study is normal to the specimen’s surface.
+The scanning grid G1 ⊂ Π
+exp
+1
+is identified by a 2 cm×2 cm square sampled by 100×100 uniformly spaced
+measurement points. This amounts to a spatial resolution of 0.2 mm in both spatial directions. In parallel,
+G2 ⊂ Π
+exp
+2
+is a 2.5 cm×7.5 cm rectangle positioned according to Fig. 15 (b) and sampled by a uniform grid of
+180×60 scan points associated with the spatial resolution of 0.42 mm. At every scan point, the data acquisition
+is conducted for a time period of 400 µs at the sampling rate of 250 MHz. To minimize the impact of optical
+and mechanical noise in the system, the measurements are averaged over an ensemble of 80 realizations at
+each scan point. Bear in mind that both the direct inversion and PINNs deploy the spectra of normalized
+velocity fields vobs for data inversion. Such distributions of out-of-plane particle velocity at 165 kHz (resp. 80
+kHz) in Π
+exp
+1
+(resp. Π
+exp
+2 ) is displayed in Fig. 15.
+It should be mentioned that in the above experiments, the magnitude of measured signals in terms of
+displacement is of O(nm) so that it may be appropriate to assume a linear regime of propagation. The nature
+of antiplane wave motion is dispersive nonetheless. Therefore, to determine the relevant length scales in each
+component, the associated dispersion curves are obtained as in Fig. 19 via a set of complementary experiments
+described in Section 4.4.1. Accordingly, for excitations of center frequency {fc1, fc2, fc3} = {165, 80, 300} kHz,
+the affiliated phase velocity cµ and wavelength λµ for µ = {Al, Ti, St, Br} is identified in Table 6.
+Figure 15: Test set-ups for ultrasonic full-field characterization: (a) an Al plate Π
+exp
+1
+is subject to antiplane shear waves at 165
+kHz by a piezoelectric transducer; the out-of-plane particle velocity field is then captured by a laser Doppler vibrometer scanning
+on a robot over the observation surface, and (b) a Ti-St-Br plate Π
+exp
+2
+undergoes a similar test at 80 kHz and 300 kHz.
+14
+
+exp
+2
+exp
+1..239
+Ti
+St
+Br
+(a)
+(b)4.2. Dimensional framework
+On recalling Section 2.2, let ℓr : = λAl = 0.01 m, µr : = EAl = 68.9 GPA, and ρr : = ρAl = 2700 kg/m3 be
+the reference scales for length, stress, and mass density, respectively. In this setting, the following maps take
+the physical quantities to their dimensionless values
+(ρµ, Eµ, νµ) → (ρµ, Eµ, νµ) :=
+� 1
+ρr
+ρµ, 1
+µr
+Eµ, νµ
+�
+,
+µ = {Al, Ti, St, Br},
+(fcι, λµ, cµ) → (fcι, λµ, cµ) :=
+�
+ℓr
+� ρr
+µr
+fcι, 1
+ℓr
+λµ,
+� ρr
+µr
+cµ
+�
+,
+ι = 1, 2, 3,
+(h, f, vβ) → (h, f, vβ) :=
+� 1
+ℓr
+h, ℓr
+� ρr
+µr
+f,
+� ρr
+µr
+vβ�
+,
+β = 1, 2,
+(10)
+where h = 1.5 mm and f respectively indicate the specimen’s thickness and cyclic frequency of wave motion.
+Table 5 (resp. Table 6) details the normal values for the first (resp. second) of (10). The normal thickness and
+center frequencies are as follows,
+{fc1, fc2, fc3} = {0.33, 0.16, 0.59},
+h = 0.15.
+(11)
+Table 5: Properties of the aluminum, titanium, steel and brass sheets as per the manufacturer. Here, χµ := Eµ/ρµ.
+physical
+µ
+Al
+Ti
+St
+Br
+Eµ [GPA]
+68.9
+105
+199.95
+110
+quantity
+ρµ [kg/m3]
+2700
+4510
+7850
+8530
+νµ
+0.33
+0.34
+0.29
+0.31
+normal
+Eµ
+1
+1.52
+2.90
+1.60
+value
+ρµ
+1
+1.67
+2.91
+3.16
+χµ
+1
+0.91
+1
+0.51
+Table 6: Phase velocity cµ and wavelength λµ in µ = {Al, Ti, St, Br} at {fc1, fc2, fc3} = {165, 80, 300} kHz as per Fig. 19,
+and their normalized counterparts according to (10).
+physical quantity
+µ
+Al
+Ti
+St
+Br
+λµ(fc1) [cm]
+1
+−
+−
+−
+cµ(fc1) [m/s]
+1610.4
+−
+−
+−
+λµ(fc2) [cm]
+−
+1.4
+1.4
+1.17
+cµ(fc2) [m/s]
+−
+1140
+1126
+936
+λµ(fc3) [cm]
+−
+0.65
+0.64
+0.5
+cµ(fc3) [m/s]
+−
+1960.8
+1929
+1501.6
+normal value
+µ
+Al
+Ti
+St
+Br
+λµ(fc1)
+1
+−
+−
+−
+cµ(fc1)
+0.32
+−
+−
+−
+λµ(fc2)
+−
+1.4
+1.4
+1.17
+cµ(fc2)
+−
+0.23
+0.22
+0.19
+λµ(fc3)
+−
+0.65
+0.64
+0.5
+cµ(fc3)
+−
+0.39
+0.38
+0.3
+4.3. Governing equation
+In light of (11) and Table 6, observe that in all tests the wavelength-to-thickness ratio λµ
+h ∈ [3.33 9.33],
+µ = {Al, Ti, St, Br}. Therefore, one may invoke the equation governing flexural waves in thin plates [53] to
+approximate the physics of measured data. In this framework, (1) may be recast as
+Λ = Λβ :=
+χβh3
+12(1 − ν2
+β)∇4 − h(2πf)2,
+χβ := Eβ
+ρβ
+, β = 1, 2,
+ˆu = vβ(ξ, f; τ),
+ϑ = χβ(ξ, f),
+ξ ∈ Sobs, τ ∈ Sinc, f ∈ [0.8 1.2]fcι, ι = 1, 2, 3,
+(12)
+where ρβ, Eβ, νβ respectively denote the normal density, Young’s modulus, and Poisson’s ratio in Π
+exp
+β , β =
+1, 2, and τ indicates the source location. Note that νβ ∼ 0.32 according to Table 5 and Λ, related to 1 − ν2
+β,
+15
+
+shows little sensitivity to small variations in the Poisson’s ratio. Thus, in what follows, νβ is treated as a
+known parameter. Provided vβ(ξ, f; τ), the objective is to reconstruct χβ(ξ, f).
+4.4. Direct inversion
+Following the reconstruction procedure of Section 3.2, the distribution of χβ in Gβ, β = 1, 2, is obtained
+at specific frequencies. In this vein, the positive-definite MLP networks χ⋆
+β = Nχβ(ξ, ω) and α⋆ = Nα(ξ, ω)
+comprised of three hidden layers of respectively 20, 40, and 20 neurons are constructed according to Fig. 1.
+In all MLP trainings of this section, each epoch makes use of the full dataset and the learning rate is 0.005.
+When β = 1, the inversion is conducted at f1 = 0.336. Sinc is sampled at one point i.e., the piezoelectric
+transducer remains fixed during the test on Al plate, and thus, Nτ = 1, while a concentric 60×60 subset
+of collocation points sampling Sobs is deployed for training. In this setting, the input to χ⋆
+1 and α⋆ is of
+size NξNτ × Nω = 3600 × 1, and their real-valued outputs are of the same size. The results are shown in
+Fig. 16. When β = 2, the direct inversion is conducted at f2 = 0.17 and f3 = 0.61. For the low-frequency
+reconstruction, Sinc is sampled at one point, while a 40×120 subset of scan points in G2 is used for training
+so that the input/output size for χ⋆
+2 and α⋆ is 4600×1. The recovered fields and associated normal error are
+provided in Fig. 17. Table 7 enlists the true values as well as mean and standard deviation of the reconstructed
+distributions χ⋆
+β in Π
+exp
+β , β = 1, 2, according to Figs. 16 and 17. For the high-frequency reconstruction, when
+β = 2, Sinc is sampled at three points i.e., experiments are performed for three distinct positions of the
+piezoelectric transducer, while the same subset of scan points is used for training. In this case, the input to
+χ⋆
+2 and α⋆ is 13800×1, while their output is of dimension 4600×1. The high-frequency reconstruction results
+are illustrated in Fig. 18, and the affiliated means and standard deviations are provided in Table 8. It should
+be mentioned that the computed normal errors in Figs. 16, 17, and 18 are with respect to the verified values
+of Section 4.4.1. Note that the recovered α⋆s from laboratory test data are much smoother than the ones
+reconstructed from synthetic data in Section 3.2. This could be attributed to the scaler nature of (12) with a
+single unknown parameter – as opposed to the vector equations governing the in-plane wave motion with two
+unknown parameters. More specifically, here, α⋆ controls the weights and biases of a single network χ⋆
+β, while
+in Section 3.2, α⋆ simultaneously controls the parameters of two separate networks µ⋆ and λ⋆. A comparative
+analysis of Figs. 17 and 18 reveals that (a) enriching the waveform data by increasing the number of sources
+remarkably decrease the reconstruction error, (b) the regularization parameter α in (3) is truly distributed
+in nature as the magnitude of the recovered α⋆ in brass is ten times greater than that of titanium and steel
+which is due to the difference in the level of noise in measurements related to distinct material surfaces, and
+(c) the recovered field χ⋆
+2 – which according to (12) is a material property E2/ρ2, demonstrates a significant
+dependence to the reconstruction frequency. The latter calls for proper verification of the results which is the
+subject of Section 4.4.1.
+4.4.1. Verification
+To shine some light on the nature discrepancies between the low- and high- frequency reconstructions in
+Figure 16: Direct inversion of the PDE parameter χ1 in Π
+exp
+1
+using test data from a single source at frequency f1 = 0.336: (a) MLP-
+predicted distribution χ1(ξ, f1) in ξ ∈ G1, (b) reconstruction error (8) with respect to the true value χ1 = χAl = 1, (c) MLP-
+recovered distribution of the regularization parameter α⋆, and (d) loss function Lε vs. the number of epochs Ne in log scale.
+16
+
+(a)
+(b)
+(c)
+×10-3
+(d)
+三(x1)
+X1
+α*
+6
+1.06
+0.06
+log(Le)
+4
+1.04
+4
+0.04
+-5
+1.02
+0.02
+2
+-6
+Ne
+ELLLEFE
+0
+2
+×103
+4Figure 17: Direct inversion of the PDE parameter χ2 in Π
+exp
+2
+using test data from a single source at frequency f2 = 0.17: (a) MLP-
+predicted distribution χ2(ξ, f2) in ξ ∈ G2, (b) reconstruction error (8) with respect to the true value χ2 ∈ {χTi, χSt, χBr} =
+{0.91, 1, 0.51} as per Table 5, (c) MLP-recovered distribution of the regularization parameter α⋆, and (d) loss function Lε vs. the
+number of epochs Ne in log scale.
+Table 7: Mean and standard deviation of the reconstructed distributions in Figs. 16 and 17 via the direct inversion of
+single-source test data.
+β
+1
+2Ti
+2St
+2Br
+χβ
+1
+0.91
+1
+0.51
+⟨χ⋆
+β⟩Πexp
+β
+1.041
+0.872
+0.978
+0.443
+σ(χ⋆
+β|Πexp
+β )
+0.017
+0.044
+0.060
+0.052
+Figure 18: Direct inversion of the PDE parameter χ2 in Π
+exp
+2
+using test data from three source locations at frequency f3 =
+0.61: (a) MLP-predicted distribution χ2(ξ, f3) in ξ ∈ G2, (b) reconstruction error (8) with respect to the related estimates
+{0.57, 0.59, 0.24} as per Fig. 20, (c) MLP-recovered distribution of the regularization parameter α⋆, and (d) loss function Lε
+vs. the number of epochs Ne in log scale.
+Table 8: Mean and standard deviation of the reconstructed distributions in Fig. 18 via the direct inversion applied to
+high-frequency test data from three distinct sources.
+β
+2Ti
+2St
+2Br
+χ′
+β
+0.57
+0.59
+0.24
+⟨χ⋆
+β⟩Πexp
+β
+0.585 0.606 0.227
+σ(χ⋆
+β|Πexp
+β )
+0.015 0.029 0.016
+Figs. 17 and 18, a set of secondary tests are performed to obtain the dispersion curve for each component of
+the test setup. For this purpose, antiplane shear waves of form (9) are induced at fcj = 50j kHz, j = 1, 2, . . . , 7,
+17
+
+(a)
+x2
+0.9
+(d)
+0.7
+(c)
+log(Le)
+2
+-3
+0.5
+*Φ
+(b)
+1
+-3.5
+三(x2)
+-4
+0.2
+×10-3
+0
+8
+×103
+0.1
+Ne(a)
+0.6
+x2
+(d)
+0.4
+(c)
+log(Le)
+0.2
+*0
+1.2
+(b)
+0.6
+-2
+三(x2)
+0.08
+×10-3
+0
+4
+8
+×103
+0.04
+NeFigure 19: Experimental vs. theoretical dispersion curves f(λ−1
+µ ) for µ = {Al, Ti, St, Br}. Analytical curves (solid lines) are
+computed from (13) using the pertinent properties in Table 5.
+Figure 20: Discrepancy in the balance law (12) at f3 = 0.61: (a) elastic force field T1
+µ, µ = {Ti, St, Br}, according to (14) with
+adjusted coefficients {χTi, χSt, χBr} = {0.57, 0.59, 0.24}, (b) the inertia field T2
+µ, and (c) normal discrepancy Dµ.
+in 60 cm × 60 cm cuts of aluminum, titanium, steel, and brass sheets used in the primary tests of Fig. 15.
+In each experiment, the piezoelectric transducer is placed in the middle of specimen (far from the external
+boundary), and the out-of-plane wave motion is captured in the immediate vicinity of the transducer along
+a straight line of length 8 cm sampled at 400 scan points. The Fourier-transformed signals in time-space
+furnish the dispersion relations of Fig. 19. In parallel, the theoretical dispersion curves affiliated with (12) are
+computed according to
+f = 2π(λµ)−2
+�
+χµh2
+12(1 − ν2µ),
+χµ = Eµ
+ρµ
+,
+µ = {Al, Ti, St, Br},
+(13)
+using the values of Table 5 for χµ and νµ and h = 1.5mm. A comparison between the experimental and
+theoretical dispersion curves f(λ−1
+µ ) in Fig. 19 verifies the theory and the values of Table 5 for χµ in the low-
+frequency regime of wave motion. This is also in agreement with the direct inversion results of Figs. 16 and 17.
+Moreover, Fig. 19 suggests that at approximately fµ = {170, 200, 120, 110} kHz for µ = {Al, Ti, St, Br} the
+governing PDE (12) with physical coefficients fails to predict the experimental results which may provide an
+insight regarding the high-frequency reconstruction results in Fig. 18. Further investigation of the balance
+law (12), as illustrated in Fig. 20, shows that the test data at 312 kHz satisfy – with less than 10 − 20%
+discrepancy depending on the material – a PDE of form (12) with modified coefficients. More specifically,
+Fig. 20 demonstrates the achievable balance between the elastic force distribution T1
+µ and inertia field T2
+µ
+in (12) by directly adjusting the PDE parameter χ′
+2 to minimize the discrepancy Dµ according to
+T1
+µ :=
+χ′
+2h3
+12(1 − ν2
+2 )∇4v2,
+T2
+µ := h(2πf)2v2,
+Dµ := |T1
+µ − T2
+µ|
+max |T2µ| .
+(14)
+18
+
+×106
+1
+T
+Br
+Al
+0.8
+0.6
+f [s-1]
+0.4 
+0.2
+0
+0
+0.2
+0.4
+0.6
+0.8
+1 0
+0.2
+0.4
+0.60.810
+0.2
+0.6
+0.8
+10
+0.2
+0.4
+0.60.8
+1
+×103
+入=1 [m-1](a)
+(c)
+2
+①μ
+0
+(b)
+Ti
+St
+Br
+2
+×10-1
+Ti
+St
+BrWith reference to Table 8, the recovered coefficients χ′
+2 at f = f3 = 0.61 verify the direct inversion results of
+Fig. 18. This implies that the direct inversion (or PINNs) may lead to non-physical reconstructions in order to
+attain the best fit for the data to the “perceived”” underlying physics. Thus, it is imperative to establish the
+range of validity of the prescribed physical principles in data-driven modeling. Here, the physics of the system
+at f3 is in transition, yet close enough to the leading-order approximation (12) that the discrepancy is less
+than 20%. It is unclear, however, if this equation with non-physical coefficients may be used as a predictive
+tool. It would be interesting to further investigate the results through the prism of higher-order continuum
+theories and a set of independent experiments for validation which could be the subject of a future study.
+4.5. Physics-informed neural networks
+Following Section 3.3, PINNs are built and trained using experimental test data of Section 4.4. The MLP
+network v1⋆ = v1⋆(ξ, f, x|γ, χ⋆
+1) with six hidden layers of respectively 40, 40, 120, 80, 40, and 40 neurons is
+constructed to map the out-of-plane velocity field v1 (in Π
+exp
+1 ) related to a single transducer location x1 and
+frequency f1 = 0.336. The PDE parameter χ1 is defined as the unknown scaler parameter of the network, and
+following the argument of Section 3.3, the Lagrange multiplier γ is specified as a nonadaptive scaler weight of
+magnitude
+1
+h(2πf1)2 = 1.5. The input/output dimension for v1⋆ is Nξ×Nω×Nτ = 3600×1×1, and each epoch
+makes use of the full dataset for training and the learning rate is 0.005. Keep in mind that the objective here
+is to (a) construct a surrogate map for v1, and (b) identify χ⋆
+1.
+Fig. 21 demonstrates (a) the accuracy of PINN-estimated field v1⋆ compared to the test data v1, (b)
+performance of automatic differentiation in capturing the fourth-order field derivatives e.g., v1⋆
+,1111 that appear
+in the governing PDE (12), and (c) the evolution of parameter χ⋆
+1. The comparison in (b) is with respect to the
+spectral derivates of test data according to Section 2.3.2. It is no surprise that the automatic differentiation
+incurs greater errors in estimating the higher order derivatives involved in the antiplane wave motion compared
+to the second-order derivatives of Section 3.3.
+In addition, the PINN v2⋆ = v2⋆(ξ, f, x|γ, χ⋆
+2) with seven hidden layers of respectively 40, 40, 120, 120, 80,
+40, and 40 neurons is created to reconstruct (i) particle velocity field v2 in the layered specimen Π
+exp
+2 , and (ii)
+distribution of the PDE parameter χ2 in the sampling area. The latter is defined as an unknown parameter
+of the network with dimension 40×120, and the scaler weight γ is set to
+1
+h(2πf2)2 = 5.84 for the low-frequency
+reconstruction. In this setting, the input/output dimension for v2⋆ reads 4800×1×1. Fig. 22 provides a
+comparative analysis between the experimental and PINN-predicted maps of velocity and PDE parameter.
+The associated statistics are provided in Table 9. It is evident from the waveform in Fig. 22 (a) that the most
+pronounced errors in Fig. 22 (d) occur at the loci of vanishing particle velocity. Similar to Section 3.2, this
+could be potentially addressed by enriching the test data.
+5. Conclusions
+The ML-based direct inversion and physics-informed neural networks are investigated for full-field ultra-
+sonic characterization of layered components. Direct inversion makes use of signal processing tools to directly
+compute the field derivatives from dense datasets furnished by laser-based ultrasonic experiments. This allows
+for a simplified and controlled learning process that specifically recovers the sought-for physical fields through
+minimizing a single-objective loss function. PINNs are by design more versatile and particularly advantageous
+with limited test data where waveform completion is desired (or required) for mechanical characterization.
+PINNs multi-objective learning from ultrasonic data may be more complex but can be accomplished via
+carefully gauged loss functions.
+In direct inversion, Tikhonov regularization is critical for stable reconstruction of distributed PDE param-
+eters from limited or multi-fidelity experimental data. In this vein, deep learning offers a unique opportunity
+to simultaneously recover the regularization parameter as an auxiliary field which proved to be particularly
+insightful in inversion of experimental data.
+In training PINNs, two strategies were remarkably helpful: (1) identifying the reference length scale by the
+dominant wavelength in an effort to control the norm of spatial derivatives – which turned out to be crucial in
+the case of flexural waves in thin plates with the higher order PDE, and (2) estimating the Lagrange multiplier
+by taking advantage of the inertia term in the governing PDEs.
+19
+
+Figure
+21:
+PINN
+vs.
+experimental
+maps
+of
+particle
+velocity
+and
+its
+derivatives
+in
+Π
+exp
+1
+:
+(a)
+PINN
+estimates
+for
+{v1⋆, v1⋆
+,1111, v1⋆
+,2222, v1⋆
+,1122} wherein the derivatives are obtained by automatic differentiation, (b) normalized LDV-captured par-
+ticle velocity field v1 and its corresponding spectral derivatives, (c) normal misfit 8 between (a) and (b), (d) PINN-reconstructed
+PDE parameter χ⋆
+1 vs. the number of epochs Ne, and (e) total loss Lϖ and its components (the PDE residue and data misfit)
+vs. Ne in log scale.
+Laboratory implementations at multiple frequencies exposed that verification and validation are indis-
+pensable for predictive data-driven modeling. More specifically, both direct inversion and PINNs recover the
+unknown “physical” quantities that best fit the data to specific equations (with often unspecified range of va-
+lidity). This may lead to mathematically decent but physically incompatible reconstructions especially when
+the perceived physical laws are near their limits such that the discrepancy in capturing the actual physics
+is significant. In which case, the inversion algorithms try to compensate for this discrepancy by adjusting
+the PDE parameters which leads to non-physical reconstructions. Thus, it is paramount to conduct comple-
+mentary experiments to (a) establish the applicability of prescribed PDEs, and (b) validate the predictive
+capabilities of the reconstructed models.
+Authors’ contributions
+Y.X. investigation, methodology, data curation, software, visualization, writing – original draft; F.P. con-
+ceptualization, methodology, funding acquisition, supervision, writing – original draft; J.S. experimental data
+curation; C.W. experimental data curation.
+20
+
+1×
+.1*
+1 *
+1*
+0.4
+2
+1
+0.4
+(a)
+0
+0
+0
+0
+-2
+-0.4
+-0.4
+v,1111
+V,2222
+v,1122
+0.4
+2
+1
+0.4
+(b)
+0
+0
+0
+0
+-2
+-0.4
+-0.4
+-1
+8
+三(v,1111)
+3
+?L
+1 *
+6
+6
+6
+4
+4
+(c)
+4
+2
+2
+2
+×10-3
+×10-1
+×10-1
+×10-1
+1
+x1
+PDE loss
+2
+ data loss
+0.8
+total loss
+0.6
+(d)
+(e)
+-4
+0.4
+MA
+0.2
+-6 
+Ne
+Ne
+0
+×105
+0
+0.2
+0.4
+0.6
+0.8
+0
+0.2
+0.4
+0.6
+0.8
+1Figure 22: Low-frequency PINN reconstruction in Π
+exp
+2
+using test data from a single source at f2 = 0.17: (a) PINN-predicted distri-
+bution of particle velocity v2⋆, (b) normalized LDV-captured particle velocity v2, (c) normal misfit between (a) and (b), (d) PINN-
+predicted distribution of the PDE parameter χ⋆
+2, and (e) total loss Lϖ and its components (the PDE residue and data misfit)
+vs. the number of epochs Ne in log scale.
+Table 9: Mean and standard deviation of the PINN-reconstructed distributions in Fig. 22 from a single-source, low-
+frequency test data.
+β
+2Ti
+2St
+2Br
+χβ
+0.91
+1
+0.51
+⟨χ⋆
+β⟩Πexp
+β
+0.790
+0.890
+0.414
+σ(χ⋆
+β|Πexp
+β )
+0.214
+0.356
+0.134
+Acknowledgments
+This study was funded by the National Science Foundation (Grant No. 1944812) and the University of
+Colorado Boulder through FP’s startup. This work utilized resources from the University of Colorado Boulder
+Research Computing Group, which is supported by the National Science Foundation (awards ACI-1532235
+and ACI-1532236), the University of Colorado Boulder, and Colorado State University. Special thanks are
+due to Kevish Napal for facilitating the use of FreeFem++ code developed as part of [49] for elastodynamic
+simulations.
+References
+[1] X. Liang, M. Orescanin, K. S. Toohey, M. F. Insana, S. A. Boppart, Acoustomotive optical coherence
+elastography for measuring material mechanical properties, Optics letters 34 (19) (2009) 2894–2896.
+[2] G. Bal, C. Bellis, S. Imperiale, F. Monard, Reconstruction of constitutive parameters in isotropic linear
+elasticity from noisy full-field measurements, Inverse problems 30 (12) (2014) 125004.
+[3] B. S. Garra, Elastography: history, principles, and technique comparison, Abdominal imaging 40 (4)
+(2015) 680–697.
+[4] C. Bellis, H. Moulinec, A full-field image conversion method for the inverse conductivity problem with
+internal measurements, Proceedings of the Royal Society A: Mathematical, Physical and Engineering
+Sciences 472 (2187) (2016) 20150488.
+[5] H. Wei, T. Mukherjee, W. Zhang, J. Zuback, G. Knapp, A. De, T. DebRoy, Mechanistic models for
+additive manufacturing of metallic components, Progress in Materials Science 116 (2021) 100703.
+21
+
+0.8
+2
+0.8
+0.4
+(a)
+(c)
+0
+0.4
+0.4
+0.8
+PDE loss
+0.4
+(b)
+2
+data loss
+0
+total loss
+-0.4
+4
+(d)
+6
+-6
+4
+(c)
+8
+2
+×105
+0
+1
+2
+3
+×10-3
+4[6] C.-T. Chen, G. X. Gu, Learning hidden elasticity with deep neural networks, Proceedings of the National
+Academy of Sciences 118 (31) (2021) e2102721118.
+[7] H. You, Q. Zhang, C. J. Ross, C.-H. Lee, M.-C. Hsu, Y. Yu, A physics-guided neural operator learn-
+ing approach to model biological tissues from digital image correlation measurements, arXiv preprint
+arXiv:2204.00205.
+[8] C. M. Bishop, N. M. Nasrabadi, Pattern recognition and machine learning, Vol. 4, Springer, 2006.
+[9] Y. LeCun, Y. Bengio, G. Hinton, Deep learning, nature 521 (7553) (2015) 436–444.
+[10] S. Cuomo, V. S. Di Cola, F. Giampaolo, G. Rozza, M. Raissi, F. Piccialli, Scientific machine
+learning through physics-informed neural networks:
+Where we are and what’s next, arXiv preprint
+arXiv:2201.05624.
+[11] S. Wang, H. Wang, P. Perdikaris, Improved architectures and training algorithms for deep operator
+networks, Journal of Scientific Computing 92 (2) (2022) 1–42.
+[12] L. McClenny, U. Braga-Neto, Self-adaptive physics-informed neural networks using a soft attention mech-
+anism, arXiv preprint arXiv:2009.04544.
+[13] Z. Chen, V. Badrinarayanan, C.-Y. Lee, A. Rabinovich, Gradnorm: Gradient normalization for adaptive
+loss balancing in deep multitask networks, in: International conference on machine learning, PMLR, 2018,
+pp. 794–803.
+[14] M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics informed deep learning (part i): Data-driven solutions
+of nonlinear partial differential equations, arXiv preprint arXiv:1711.10561.
+[15] M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework
+for solving forward and inverse problems involving nonlinear partial differential equations, Journal of
+Computational physics 378 (2019) 686–707.
+[16] E. Haghighat, M. Raissi, A. Moure, H. Gomez, R. Juanes, A physics-informed deep learning framework
+for inversion and surrogate modeling in solid mechanics, Computer Methods in Applied Mechanics and
+Engineering 379 (2021) 113741.
+[17] A. Henkes, H. Wessels, R. Mahnken, Physics informed neural networks for continuum micromechanics,
+Computer Methods in Applied Mechanics and Engineering 393 (2022) 114790.
+[18] R. Muthupillai, D. Lomas, P. Rossman, J. F. Greenleaf, A. Manduca, R. L. Ehman, Magnetic resonance
+elastography by direct visualization of propagating acoustic strain waves, science 269 (5232) (1995) 1854–
+1857.
+[19] P. E. Barbone, N. H. Gokhale, Elastic modulus imaging: on the uniqueness and nonuniqueness of the
+elastography inverse problem in two dimensions, Inverse problems 20 (1) (2004) 283.
+[20] O. A. Babaniyi, A. A. Oberai, P. E. Barbone, Direct error in constitutive equation formulation for plane
+stress inverse elasticity problem, Computer methods in applied mechanics and engineering 314 (2017)
+3–18.
+[21] A. N. Tikhonov, A. Goncharsky, V. Stepanov, A. G. Yagola, Numerical methods for the solution of
+ill-posed problems, Vol. 328, Springer Science & Business Media, 1995.
+[22] A. Kirsch, et al., An introduction to the mathematical theory of inverse problems, Vol. 120, Springer,
+2011.
+[23] On the convergence of physics informed neural networks for linear second-order elliptic and parabolic
+type pdes, Communications in Computational Physics 28 (5) (2020) 2042–2074.
+22
+
+[24] S. Wang, X. Yu, P. Perdikaris, When and why pinns fail to train: A neural tangent kernel perspective,
+Journal of Computational Physics 449 (2022) 110768.
+[25] Z. Xiang, W. Peng, X. Liu, W. Yao, Self-adaptive loss balanced physics-informed neural networks, Neu-
+rocomputing 496 (2022) 11–34.
+[26] R. Bischof, M. Kraus, Multi-objective loss balancing for physics-informed deep learning, arXiv preprint
+arXiv:2110.09813.
+[27] H. Son, S. W. Cho, H. J. Hwang, Al-pinns: Augmented lagrangian relaxation method for physics-informed
+neural networks, arXiv preprint arXiv:2205.01059.
+[28] S. Zeng, Z. Zhang, Q. Zou, Adaptive deep neural networks methods for high-dimensional partial differ-
+ential equations, Journal of Computational Physics 463 (2022) 111232.
+[29] J. Yu, L. Lu, X. Meng, G. E. Karniadakis, Gradient-enhanced physics-informed neural networks for
+forward and inverse pde problems, Computer Methods in Applied Mechanics and Engineering 393 (2022)
+114823.
+[30] S. Wang, Y. Teng, P. Perdikaris, Understanding and mitigating gradient flow pathologies in physics-
+informed neural networks, SIAM Journal on Scientific Computing 43 (5) (2021) A3055–A3081.
+[31] A. D. Jagtap, K. Kawaguchi, G. Em Karniadakis, Locally adaptive activation functions with slope recovery
+for deep and physics-informed neural networks, Proceedings of the Royal Society A 476 (2239) (2020)
+20200334.
+[32] Y. Kim, Y. Choi, D. Widemann, T. Zohdi, A fast and accurate physics-informed neural network reduced
+order model with shallow masked autoencoder, Journal of Computational Physics 451 (2022) 110841.
+[33] G. I. Barenblatt, Scaling (Cambridge texts in applied mathematics), Cambridge University Press, Cam-
+bridge, UK, 2003.
+[34] K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural networks 4 (2) (1991)
+251–257.
+[35] Y. Chen, L. Dal Negro, Physics-informed neural networks for imaging and parameter retrieval of photonic
+nanostructures from near-field data, APL Photonics 7 (1) (2022) 010802.
+[36] F. Pourahmadian, B. B. Guzina, On the elastic anatomy of heterogeneous fractures in rock, International
+Journal of Rock Mechanics and Mining Sciences 106 (2018) 259 – 268.
+[37] X. Liu, J. Song, F. Pourahmadian, H. Haddar, Time-vs. frequency-domain inverse elastic scattering:
+Theory and experiment, arXiv preprint arXiv:2209.07006.
+[38] F. Pourahmadian, B. B. Guzina, H. Haddar, Generalized linear sampling method for elastic-wave sensing
+of heterogeneous fractures, Inverse Problems 33 (5) (2017) 055007.
+[39] F. Cakoni, D. Colton, H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, SIAM, 2016.
+[40] V. A. Morozov, Methods for solving incorrectly posed problems, Springer Science & Business Media,
+2012.
+[41] R. Kress, Linear integral equation, Springer, Berlin, 1999.
+[42] A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga,
+A. Lerer, Automatic differentiation in pytorch.
+[43] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer Science &
+Business Media, 2010.
+23
+
+[44] T. Ha-Duong, On retarded potential boundary integral equations and their discretization, in: Topics in
+computational wave propagation, Springer, 2003, pp. 301–336.
+[45] R. T. Rockafellar, Lagrange multipliers and optimality, SIAM review 35 (2) (1993) 183–238.
+[46] H. Everett III, Generalized lagrange multiplier method for solving problems of optimum allocation of
+resources, Operations research 11 (3) (1963) 399–417.
+[47] D. Liu, Y. Wang, A dual-dimer method for training physics-constrained neural networks with minimax
+architecture, Neural Networks 136 (2021) 112–125.
+[48] F. Hecht, New development in freefem++, Journal of Numerical Mathematics 20 (3-4) (2012) 251–265.
+URL https://freefem.org/
+[49] F. Pourahmadian, K. Napal, Poroelastic near-field inverse scattering, Journal of Computational Physics
+455 (2022) 111005.
+[50] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein,
+L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner,
+L. Fang, J. Bai, S. Chintala, Pytorch: An imperative style, high-performance deep learning library,
+Advances in neural information processing systems 32.
+[51] D. P. Kingma, J. Ba, Adam: A method for stochastic optimization, arXiv preprint arXiv:1412.6980.
+[52] Pytorch implementation of physics-informed neural networks, https://github.com/jayroxis/PINNs
+(2022).
+[53] K. F. Graff, Wave motion in elastic solids, Courier Corporation, 2012.
+24
+

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+Astronomy & Astrophysics manuscript no. dust_filtering
+©ESO 2023
+January 16, 2023
+Leaky Dust Traps: How Fragmentation impacts Dust Filtering by
+Planets
+Sebastian Markus Stammler1, Tim Lichtenberg2, Joanna Dr˛a˙zkowska3, and Tilman Birnstiel1, 4
+1 University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679, Munich, Germany
+2 Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands
+3 Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany
+4 Exzellenzcluster ORIGINS, Boltzmannstr. 2, D-85748 Garching, Germany
+January 16, 2023
+ABSTRACT
+The nucleosynthetic isotope dichotomy between carbonaceous and non-carbonaceous meteorites has been interpreted as evidence for
+spatial separation and coexistence of two distinct planet-forming reservoirs for several million years in the solar protoplanetary disk.
+Rapid formation of Jupiter’s core within one million years after CAIs has been suggested as a potential mechanism for spatial and
+temporal separation. In this scenario, Jupiter’s core would open a gap in the disk and trap inwards-drifting dust grains in the pressure
+bump at the outer edge of the gap, separating the inner and outer disk materials from each other. We performed simulations of dust
+particles in a protoplanetary disk with a gap opened by an early formed Jupiter core, including dust growth and fragmentation as well
+as dust transport using the dust evolution software DustPy. Our numerical experiments indicate that particles trapped in the outer
+edge of the gap rapidly fragment and are transported through the gap, contaminating the inner disk with outer disk materials on a
+timescale that is inconsistent with the meteoritic record. This suggests that other processes must have initiated or at least contributed
+to the isotopic separation between the inner and outer Solar System.
+Key words. Meteorites, meteors, meteoroids — Methods: numerical — Protoplanetary disks — Planets and satellites: formation –
+Planets and satellites: composition
+1. Introduction
+Recent high-precision isotopic measurements reveal a di-
+chotomy between carbonaceous and non-carbonaceous mete-
+orites indicating that both have been formed in separate reser-
+voirs within the early Solar System (Trinquier et al. 2007, 2009;
+Leya et al. 2009; Warren 2011; Mezger et al. 2020; Kleine et al.
+2020). Kruijer et al. (2017) and Desch et al. (2018) argued that
+these reservoirs must have been well separated for at least two
+million years without interchanging solid material, proposing the
+rapid formation of Jupiter’s core opening a gap in the protoplan-
+etary disk as possible mechanism to prevent the mixing of both
+reservoirs. The physical origin of the isotopic separation is a po-
+tential critical clue to the timescales of planet formation in both
+the inner and outer Solar System (Nimmo et al. 2018), and thus
+ultimately the origin of the chemical abundances in the terres-
+trial planets and similar exoplanets (Krijt et al. 2022; Lichten-
+berg et al. 2022).
+This concept of a gap-opening Jupiter preventing dust reser-
+voir mixing, however, intimately depends on the evolution of the
+dust flux during the evolution of the protoplanetary disk. Dust
+particles in protoplanetary disks are subject to gas drag and drift
+(Whipple 1973; Weidenschilling 1977; Takeuchi & Lin 2002).
+The radial dust velocity is given by:
+vd = vg
+1
+St2 + 1
++ 2vP
+St
+St2 + 1
+.
+(1)
+The Stokes number St is an aerodynamic measure and propor-
+tional to the particle size. Small particles with small Stokes num-
+bers are dragged along with the gas with velocity vg as can be
+seen by Equation 1. The gas is, in contrast to the dust, pressure
+supported and orbits the star with sub-Keplerian velocities in a
+typical smooth disk with inward pointing pressure gradient. The
+dust particles, on the other hand, are not pressure supported, ex-
+change angular momentum with the gas and drift in direction of
+pressure gradients. Intermediate particle sizes are most affected
+by this effect. Small particles are well coupled to the gas, while
+large particles are completely decoupled. From Equation 1 it can
+be seen that particles with Stokes number of unity will experi-
+ence maximum drift in direction of the pressure gradient with
+velocity vP, which is given by:
+vP = 1
+2
+c2
+s
+vK
+∂ log P
+∂ log r ,
+(2)
+with the sound speed cs, pressure P, and the Keplerian velocity
+vK. Particles typically grow to maximum sizes with Stokes num-
+bers between 10−2 to 10−1 (see Birnstiel et al. 2012), depending
+on the disk parameters, and are therefore affected by radial drift.
+Growing planets can perturb the pressure structure in the disk
+by opening a gap in the gas (Paardekooper & Mellema 2006;
+Rice et al. 2006). At the outer edge of the gap the pressure gradi-
+ent reverses and is pointing outward. If the pressure pertubation
+is large enough, large dust pebbles that are affected by drift can
+be prevented from crossing the gap. The planetary mass at which
+the pressure pertubation is large enough to stop particle dift is
+called pebble isolation mass (see Lambrechts et al. 2014; Bitsch
+et al. 2018) and is given by Drazkowska et al. (2022) as:
+Miso ≃ 25 M⊕
+�HP/r
+0.05
+�3 M⋆
+M⊙
+.
+(3)
+Article number, page 1 of 8
+arXiv:2301.05505v1  [astro-ph.EP]  13 Jan 2023
+
+A&A proofs: manuscript no. dust_filtering
+From NASA’s Juno mission Jupiter’s core is estimated to
+have a mass of up to 25 M⊕ (Wahl et al. 2017) and would have
+therefore been able to open a gap and stop the flux of dust peb-
+bles in the disk. A rapid formation of Jupiter’s core could there-
+fore explain two isolated dust reservoirs with the dust in the outer
+disk forming the carbonaceous and the dust in the inner disk the
+non-carbonaceous bodies in the Solar System.
+Dr˛a˙zkowska et al. (2019), however, showed in two-
+dimensional hydrodynamic simulations of gas and dust includ-
+ing collisional dust evolution, that the pressure bump at the outer
+edge of planetary gaps does not only show an accumulation of
+large dust pebbles, but also of small dust particles. But in con-
+trast to large pebbles, these small particles are not trapped by
+the pressure bump, they are produced in situ by collisions of
+large particles leading to fragmentation. These small fragments
+can escape the pressure bump due to diffusion and gas drag. The
+equations of motion of the dust particles are given by:
+∂
+∂tΣd + 1
+r
+∂
+∂r
+�
+rΣdvd − rDΣg
+∂
+∂r
+�Σd
+Σg
+��
+= 0,
+(4)
+with the dust diffusivity given by Youdin & Lithwick (2007) as
+D = δrc2
+s
+ΩK
+1
+St2 + 1
+.
+(5)
+with δr being a free parameter that defines the strength of radial
+dust diffusion. Small particles are therefore most affected by dif-
+fusion. If the diffusivity is high enough, these small particles can
+diffuse out of the pressure maximum and are dragged with the
+gas through the gap. If this is the case the inner disk would be
+contaminated with dust from the outer disk negating the idea of
+two distinct dust reservoirs separated by an early formed Jupiter
+core.
+In this letter we test this hypothesis. In section 2 we present
+a toy model which initially has dust placed only outside of the
+planet to show as a proof of concept, that solid material can pen-
+etrate planetary gaps if the dust is subject to fragmentation and
+diffusion. In section 3 we investigate the influence of the plan-
+etary mass and the dust diffusivity on the dust permeability of
+planetary gaps. In section 4 we present models with a realis-
+tic evolution of the planetary mass, as it has been suggested for
+Jupiter, for models with both fragmentation and bouncing. Fi-
+nally, in section 5 we discuss our results, before we conclude in
+section 6.
+2. Toy Model
+To investigate the influence of a planet on the dust flux in the
+inner disk, we model dust coagulation and transport in a proto-
+planetary disk with a planet opening a gap at 5 AU using the dust
+evolution software DustPy1 (Stammler & Birnstiel 2022). In a
+first simplified toy model, we initialize the disk only with dust
+outside of a Saturn mass planet. Therefore, any dust flux mea-
+sured inside the planet must have crossed the gap. We use this
+simplified model to investigate different scenarios: dust growth
+limited by fragmentation, dust growth limited by bouncing, and
+unlimited dust growth. Furthermore, we compare the toy model
+to a model without a gap.
+We initialize the gas surface density with the self similar so-
+lution of Lynden-Bell & Pringle (1974):
+Σg (r) = Mdisk
+2πr2c
+� r
+rc
+�−1
+exp
+�
+− r
+rc
+�
+(6)
+1 DustPy v1.0.1 has been used for the simulations presented in this
+work.
+with a cutoff radius of rc = 30 AU and an initial disk mass of
+Mdisk = 0.05 M⊙. We impose a gap onto this gas surface density
+profile originating from a Saturn mass planet located at 5 AU, for
+which we use the gap profile fits provided by Kanagawa et al.
+(2017). To maintain this gap profile F (r) throughout the sim-
+ulation we impose the inverse of this profile onto the turbulent
+viscosity parameter α, since the product of gas surface density
+and viscosity is constant in quasi steady-state:
+α (r) =
+α0
+F (r).
+(7)
+In the default setup, we use α0 = δr = 10−3. Please note, that this
+change in α (r) does not affect the turbulent diffusion of the dust
+particles, since δr is a constant in our models.
+We initialize the dust surface density with a constant gas-to-
+dust ratio of 100 and the dust size distribution according Mathis
+et al. (1977) as n (a) = a−3.5 with a maximum initial particle size
+of 1 µm. In the toy model we initially have dust only outside of
+15 AU.
+DustPy simulates dust growth by solving the Smoluchowski
+equation of a dust mass distribution. Dust transport is simulated
+by solving Equation 4 for every dust size individually.
+The gas surface density is evolved by solving the viscous
+advection-diffusion equation
+∂
+��tΣg + 1
+r
+∂
+∂r
+�
+rΣgvg
+�
+= 0
+(8)
+with the gas velocity given by Lynden-Bell & Pringle (1974) as
+vg = −
+3
+Σg
+√r
+∂
+∂r
+�
+Σgν √r
+�
+(9)
+and the kinematic viscosity given by
+ν = αcsHP
+(10)
+with the sound speed cs =
+�
+kBT/µ, the pressure scale height
+HP = cs/ΩK, and the viscosity parameter α given by Equation 7.
+We run five different flavors of the toy model: one with a
+fragmentation velocity of vfrag = 10 m/s (fiducial), one with no
+fragmentation at all, one with a fragmentation velocity of 1 m/s,
+one with bouncing as described by Windmark et al. (2012), and
+one without a gap, i.e. F (r) = 1. In the default collision model
+used by DustPy particles fragment once their relative collision
+velocities exceed the fragmentation velocity. Fragmenting colli-
+sions of equal size particles lead to catastrophic fragmentation
+of both collision partners. If the target particle is significantly
+larger, only the projectile particle fragments entirely while erod-
+ing mass off the target particle (Schräpler et al. 2018; Hasegawa
+et al. 2021). The transition between pure sticking and fragmenta-
+tion is smooth, since DustPy is assuming a velocity distribution
+of possible collision velocities. For details on the collision model
+we refer to Stammler & Birnstiel (2022).
+Panel A of Figure 1 shows the initial dust distribution with
+dust located outside of 15 AU with particles sizes up to 1 µm.
+The white lines are contour lines of Stokes numbers St =
+�
+10−3, 10−2, 10−1, 100�
+with the bold white line corresponding to
+St = 1. Panel B shows the fiducial simulation with a Saturn mass
+planet at 5 AU and the fragmentation velocity vfrag = 10 m/s
+after 1 Myr. Particles trapped in the pressure bump outside the
+planetary gap can reach sizes with Stokes numbers of up to
+St = 10−1 corresponding to particle sizes of a few centimeters.
+It can be seen that even small particles are accumulated in the
+pressure bump, even though their Stokes numbers are too small
+Article number, page 2 of 8
+
+Sebastian Markus Stammler et al.: Leaky Dust Traps: How Fragmentation impacts Dust Filtering by Planets
+101
+102
+Distance from star [AU]
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+102
+Particle size [cm]
+A: initial
+101
+102
+Distance from star [AU]
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+102
+Particle size [cm]
+B: fiducial
+101
+102
+Distance from star [AU]
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+102
+Particle size [cm]
+C: without planet
+101
+102
+Distance from star [AU]
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+102
+Particle size [cm]
+D: no fragmentation
+101
+102
+Distance from star [AU]
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+102
+Particle size [cm]
+E: vfrag = 1 m/s
+101
+102
+Distance from star [AU]
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+102
+Particle size [cm]
+F: bouncing
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+dust [g/cm²]
+Fig. 1. Panel A: Initial dust distribution. The white lines correspond to Stokes numbers of St =
+�
+10−3, 10−2, 10−1, 100�
+with the bold white line
+corresponding to St = 1. All other panels show snapshots of models at 1 Myr. Panel B: The fiducial toy model with a Saturn mass planet at
+5 AU and a fragmentation velocity of 10 m/s. Panel C: Model without a planet. The vertical dashed lines are the location at which the dust flux
+is measured in Figure 2. Panel D: Model without fragmentation. Panel E: Model with a reduced fragmentation velocity of 1 m/s. Bottom right:
+Model with bouncing instead of fragmentation.
+to be affected by drift. These small dust particles are produced
+by collisional fragmentation of larger particles trapped in the
+bump. They diffuse out of the bump and are dragged with the
+gas contaminating the inner disk with outer disk material. It can
+be seen that particles with Stokes numbers of about St = 10−2,
+corresponding to particle sizes of a few millimeter, can diffuse
+through the gap into the inner disk. Particles in the inner disk
+can again grow to centimeter sizes and can contribute to phe-
+nomena like the streaming instability or pebble accretion. Panel
+C shows a simulation with identical initial conditions but with-
+out a planet opening a gap. The vertical yellow and green dashed
+lines in panels B and C are the locations at which the dust fluxes
+shown in Figure 2 are measured.
+The dust fluxes at the outer disk are identical in both sim-
+ulations with the solid and dashed green lines overlapping in
+Figure 2. The fluxes in the inner disk, however, differ in both
+simulations. The onset of dust flux in the inner disk in the sim-
+ulation with a planet is delayed by about 20 000 yr compared to
+the simulations without a planet. Without a planet, the large dust
+particles can freely drift into the inner disk. With a planet, how-
+ever, they are first trapped in the pressure bump at the outer edge
+of the gap, fragment down to smaller sizes, and diffuse out of the
+pressure bump before the gas can drag them into the inner disk
+where they grow to larger particles again. Due to this delayed
+processing the maximum dust flux is reduced by about one or-
+der of magnitude. The duration, however, is prolonged such that
+Article number, page 3 of 8
+
+A&A proofs: manuscript no. dust_filtering
+103
+104
+105
+106
+107
+Time [yrs]
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+10
+2
+Dust flux [M
+/yr]
+r =   2 AU
+r =  15 AU
+with planet
+w/o planet
+103
+104
+105
+106
+107
+Time [yrs]
+10
+2
+10
+1
+100
+Fraction of total dust mass accreted
+Fig. 2. Top: Comparison of the dust flux in the inner disk (at 2 AU)
+and outer disk (at 15 AU) in the toy model with a Saturn mass planet
+at 5 AU (panel B in Figure 1) and a model without a planet (panel C
+in Figure 1). Both green 15 AU lines overlap. Bottom: Total dust mass
+accreted through the inner disk over time.
+the total mass of dust flowing through the inner disk is identical
+after 10 Myr as can be seen in the bottom panel of Figure 2. The
+Saturn mass planet did not separate the inner from outer disk
+material, but only delayed the material transport.
+Panel D of Figure 1 shows a simulation without fragmen-
+tation. In this scenario, particles sizes are limited only by the
+radial drift, consistent with the model presented by Kobayashi &
+Tanaka (2021). In the center of the pressure bump, the pressure
+gradient is zero and the growth is in principle unlimited until the
+particles accumulate at the upper end of the simulation grid. This
+scenario most closely represents the separation of inner and outer
+dust reservoirs with only very few particles being able to diffuse
+through the gap, because they were not able to grow to large par-
+ticles quickly enough. It is, however, rather unlikely that the dust
+particles do not fragment or get eroded at some point given the
+relative velocities they typically experience (see Blum & Münch
+1993; Wada et al. 2009; Schräpler et al. 2018).
+Panel E of Figure 1 shows a model with a fragmentation ve-
+locity of 1 m/s as indicated by recent experiments (see e.g. Blum
+2018; Gundlach et al. 2018; Musiolik & Wurm 2019). In this
+case, the particles cannot reach particles sizes large enough to be
+efficiently trapped in the pressure bump.
+The objective is therefore to halt particle growth without pro-
+ducing small particles. This can be achieved if the growth is lim-
+ited by bouncing, when particles simply bounce of each other
+without growing or fragmenting. Panel F of Figure 1 shows a
+simulation with the bouncing barrier implemented as described
+by Windmark et al. (2012). In this model, bouncing starts when
+103
+104
+105
+106
+107
+Time [yrs]
+10
+7
+10
+6
+10
+5
+10
+4
+10
+3
+Pebble flux [M
+/yr]
+103
+104
+105
+106
+107
+Time [yrs]
+10
+4
+10
+3
+10
+2
+10
+1
+100
+Fraction of dust mass accreted
+no planet
+30 M
+50 M
+Msat, 
+r = 10
+2
+Msat, 
+r = 10
+3
+Msat, 
+r = 10
+4
+Msat, 
+r = 10
+5
+200 M
+Mjup
+Fig. 3. Top: Dust flux through the planetary gap in models with different
+planet masses. The blue line is for a model without a planet. The dashed,
+dotted, and dash-dotted red lines show additional simulations with a
+Saturn mass planet for different radial dust diffusivity parameters δr.
+Bottom: Total fraction of outer dust mass accreted through the planetary
+gap.
+the relative velocity reaches a few centimeters per second. In this
+case, however, the particles only reach sizes of a few 100 µm
+corresponding to Stokes numbers lower than 10−3, which is too
+small to be efficiently trapped in the pressure bump created by
+the planet. The particles can diffuse through the gap and contam-
+inate the inner disk.
+3. Full Disk Models
+The toy model in section 2 served as a proof of concept that
+planets do not prevent dust flux if particles are subject to frag-
+mentation. In this section we discuss full disk models with dif-
+ferent planet masses in which dust is initialized in the entire disk
+to investigate the dust permeability of the gap. The top panel of
+Figure 3 shows the dust flux through the planetary gap for differ-
+ent planet masses from 30 Earth masses to one Jupiter mass. In
+the case of a Saturn mass planet we additionally performed sim-
+ulations with different dust diffusivity parameters δr (see Equa-
+tion 5). The bottom panel of Figure 3 shows the total fraction of
+outer disk dust material that has been accreted through the gap
+over time. In all models the planets have their respective masses
+already from the beginning of the simulations.
+The smallest planetary mass considered here is 30 M⊕, which
+is already higher than the upper estimate of Jupiter’s core mass.
+The largest mass considered is 1 Mjup. The smallest planetary
+mass is not capable of efficiently suppressing the dust flux
+through the gap. After about 300 000 yr almost the entire dust
+Article number, page 4 of 8
+
+Sebastian Markus Stammler et al.: Leaky Dust Traps: How Fragmentation impacts Dust Filtering by Planets
+mass (horizontal line in bottom panel) of the outer disk has been
+accreted through the gap. Increasing the planetary mass simply
+delays the accretion time, but is not able to prevent accretion.
+The maximum delay of accretion seems to be achieved already
+with a 200 M⊕ planet. Increasing the planet mass further to a
+Jupiter mass planet does not significantly change the accretion
+history. At the end of the simulation at 10 Myr about 80 % of the
+dust mass has been accreted through the gap.
+The dust diffusivity δr has a more significant influence on the
+accretion. Increasing the diffusivity by a factor of 10 to δr = 10−2
+in the Saturn mass simulation has the same effect as reducing the
+planet mass by a factor of about 2, mimicking the accretion his-
+tory of a 40 M⊕ planet with diffusivity of δr = 10−3. Note that we
+only changed δr, while keeping α0 = 10−3 and therefore keeping
+the shape of planetary gap. The relative collision velocities of
+the dust particles are not affected by this change in δr. Decreas-
+ing δr by a factor of 10 is more efficient in retaining the dust
+than having a Jupiter mass planet with the standard diffusivity.
+In this case only about 10 % of the dust mass has been accreted at
+the end of the simulation after 10 Myr. Lowering the diffusivity
+even further to δr = 10−5 reduces the dust permeability further
+to a about 5 % of the outer disk mass after 10 Myr. It is however
+noted that the fraction of outer disk material present in the inner
+disk is usually significantly larger, since the inner disk material
+is accreted onto the star on short timescales and only re-supplied
+with outer disk material.
+4. Time-dependent planet mass
+In the previous models we assumed that the planets are fully
+formed from the beginning of the simulation and the planet mass
+does not evolve over time. Kruijer et al. (2017) argue that the
+two dust reservoirs have been separated from about 1 Myr to
+3−4 Myr after CAI formation. They therefore claim that Jupiter’s
+core must have been massive enough to open a gap at 1 Myr
+and must have reached a mass of about 50 M⊕ after 4 Myr to be
+able to scatter planetesimals from the outer disk to the inner disk
+where they are observed today in the asteroid belt. We there-
+fore performed simulations with a time-dependent planet mass
+as shown in the top left panel of Figure 4. The solid blue line
+shows an evolutionary track where the planet reaches 30 M⊕ af-
+ter 1 Myr, 50 M⊕ after 4 Myr and a final mass of Mjup at the end
+of the simulation after 10 Myr.
+The bottom left panel of Figure 4 shows the fraction of mass
+accreted through the planetary gap normalized to the dust mass
+in the outer disk at 1 Myr when the planet was massive enough
+to open a gap. We performed simulations with different values of
+the dust diffusivity δr between 10−5 and 10−3. In the standard run
+with δr = 10−3 about 80 % of the dust mass has been accreted
+through the gap after 4 Myr (vertical solid line) when the assem-
+bly of the meteorite parent bodies has been completed. Even in
+the low diffusivity run with δr = 10−5 about 60 % of the mass has
+been accreted though the gap between 1 Myr and 4 Myr, strongly
+contaminating the inner disk with dust from the outer reservoir
+on a system-wide scale. Lowering the dust diffusivity to very low
+values does not help keeping both reservoirs separated, since the
+planet mass is too low in this scenario.
+The bottom right panel of Figure 4 shows a model with
+bouncing instead of fragmentation. The solid blue line shows a
+model with radial dust diffusivity δr = 10−3. As already shown in
+section 2, this is not sufficient to stop dust accretion through the
+gap. Only after 7 Myr when the planet already reached a mass
+of about 200 M⊕ the gap is deep enough and accretion is halted.
+Allowing the planet to reach these masses at earlier times would,
+however, not change the dust redistribution, since these massive
+planets are able to scatter planetesimals from the outer disk into
+the inner disk, which is inconsistent with observations from the
+meteoritic record at these early times (Deienno et al. 2022).
+The green solid line shows a model with δr = δt = δz = 10−5.
+The parameters δt and δz are similar to δr and parametrize the
+strength of turbulent motion and vertical settling of the particles
+(see Stammler & Birnstiel 2022; Pinilla et al. 2021, for details).
+In that way the relative velocities between the particles are re-
+duced, allowing them to grow to larger sizes before being lim-
+ited by bouncing. They can therefore be trapped by gaps created
+by smaller mass planets. However, even in that case accretion is
+only halted after abut 3 Myr, when the planet reached a mass of
+about 40 M⊕.
+The dashed green line shows a model where the planet
+reaches a mass of 40 M⊕ already after 1 Myr (dashed line in top
+left panel of Figure 4). In this case accretion of dust through the
+gap is efficiently stopped at 1 Myr. The top right panel of Fig-
+ure 4 shows a snapshot of this simulation after 4 Myr. The inner
+disk is heavily depleted in dust, all of which has been accreted
+onto the star. The dust mass in the inner disk at this stage was
+all supplied from the outer disk. Meteoritic bodies formed in the
+inner disk would therefore be entirely made out of outer disk
+material.
+5. Discussion
+Isotopic measurements of meteoritic material indicate that me-
+teorites must have formed in two dust reservoirs, that coexisted
+spatially separated for several million years. The early formation
+of Jupiter’s core has been proposed as natural explanation for
+the observed separation. A planet exceeding the pebble isolation
+mass opens a gap in the gas disk creating a pressure bump at the
+outer edge of the gap, which can trap large dust particles. Two-
+dimensional hydrodynamical simulations by Dr˛a˙zkowska et al.
+(2019) including dust coagulation and fragmentation showed an
+overabundance of small dust particles at the location of the pres-
+sure bump, which should be too small to be efficiently trapped.
+These particles were created in fragmenting collision of large
+dust pebbles that have been trapped in the pressure bump. These
+small dust fragments can diffuse out of the bump and can be
+dragged by the gas through the gap.
+Our simulations in this work suggest that collisional frag-
+mentation of dust pebbles in pressure bumps and subsequent dif-
+fusion of small fragments can act as a leak for dust traps. As
+can be seen by Figure 3, gaps opened by planets can only de-
+lay but not fully prevent dust accretion if particles are subject to
+fragmentation. To act as an efficient dust barrier, particles need
+to grow to large pebbles that can be trapped without producing
+small particles as shown in the panel D of Figure 1.
+We investigated different planet masses and showed in Fig-
+ure 3 that no planet mass was able to completely isolate the inner
+disk from outer dust material on timescales that are relevant for
+the assumed reservoir separation. Even an initial gap formed by
+a fully-grown Jupiter mass planet would leak 20 % of the outer
+disk material into the inner disk within 1 Myr. Smaller proto-
+Jupiter masses typically lead to complete homogenization within
+∼ 105 to at maximum a few 106 yr. This presents a problem
+for the suggestion that the age differences in carbonaceus and
+non-carbonaceous meteorites may be used as a tracer to track
+the growth timescale of proto-Jupiter within the disk (Kruijer
+et al. 2017; Alibert et al. 2018): the initial spatial distribution of
+nucleosynthetic isotopes at the end of disk infall is degenerate
+Article number, page 5 of 8
+
+A&A proofs: manuscript no. dust_filtering
+0
+2
+4
+6
+8
+10
+Time [Myr]
+0
+50
+100
+150
+200
+250
+300
+Planet mass [M
+]
+default model
+rapid early growth
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Time [Myr]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Fraction of dust mass accreted
+Fragmentation
+r = 10
+3
+r = 10
+4
+r = 10
+5
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Time [Myr]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Fraction of dust mass accreted
+Bouncing
+i = 10
+3
+i = 10
+5
+i = 10
+5
+101
+102
+Distance from star [AU]
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+102
+Particle size [cm]
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+dust [g/cm²]
+Fig. 4. Top left: Evolution of the planetary mass in the time-dependent model. The solid line shows the default model where the planet reaches
+20 M⊕ at 1 Myrs. The dashed line shows the evolution in a model with rapid early growth in which the planet reaches 40 M⊕ at 1 Myr. Bottom
+left: Fraction of outer disk dust mass accreted through the gap after 1 Myr in the default planetary mass evolution model for different values of
+dust diffusivity δr with fragmentation limited growth. Bottom right: The solid lines show the fraction of outer disk material accreted through the
+gap after 1 Myr for bouncing limited growth for different values of the δi parameters in the default planetary growth model. The dashed green line
+shows a model of bouncing limited growth with δi = 10−5 and rapid early growth of the planet (dashed line in top left panel). The vertical lines
+mark 4 Myr until which both reservoirs need to be separated. Top right: Snapshot of the dust distribution at 4 Myr for the model with bouncing
+limited growth and δi = 10−5 (dashed green line in bottom right panel). The inner disk is depleted in dust and only supplied with small amounts of
+outer disk material.
+with different Jupiter growth tracks in the Jupiter barrier hypoth-
+esis. Only significantly lowering the dust diffusivity to a value
+of δr = 10−5 could decrease the dust permeability such that the
+inner disk is only contaminated with a few percent of outer disk
+material. However, isolating the inner disk from dust flux would
+quickly drain the inner disk from solids that got accreted onto
+the star, which was also previously noted by Liu et al. (2022). At
+later stages the dust in the inner disk then consists to large parts
+of outer disk material that has been slowly diffused through the
+gap, which is inconsistent with the meteoritic record.
+The situation gets worse when using a more realistic evo-
+lution of the planetary mass, assuming Jupiter’s core reached a
+mass of 20 M⊕ after 1 Myr and 50 M⊕ after 4 Myr. These masses
+are not large enough to isolate the inner disk even in models
+with very low diffusivity. Even in the most optimistic cases at
+least 60 % of the outer disk dust has been accreted through the
+planetary gap after 4 Myr as can be seen by Figure 4. However,
+increasing the core mass even more and earlier would enable
+Jupiter to scatter outer disk planetesimals into the inner disk pol-
+luting the inner dust reservoir, which has not been accounted
+for in this simple model. Only in models with bouncing lim-
+ited growth without small particles, early planetary growth and
+reduced relative particle collision velocities, the inner disk can
+be efficiently isolated from the inner disk as seen by Figure 4.
+In these cases, however, the inner disk is quickly depleted from
+dust and only re-supplied from small amount of outer disk ma-
+terial. Meteoritic bodies formed in the inner disk after this point
+would therefore consist almost entirely of outer disk material.
+Dr˛a˙zkowska et al. (2019) noted that the shape of planetary
+gaps in two-dimensional simulations is not axisymmetric, which
+is ignored in the simple one-dimensional model in this publi-
+cation. They further noted, however, that the asymmetry at the
+planet location would increase the dust flux through the gap.
+Weber et al. (2018) compared one- and two-dimensional simula-
+tions of dust transport through planetary gaps and indeed found
+that gaps in two-dimensional simulations are more permeable to
+dust particles. Our one-dimensional simulations, therefore, need
+to be considered more conservative. If it is not possible to sepa-
+rate two reservoirs in one-dimensional models, it is less likely to
+do so in higher dimensions.
+We furthermore assumed a dust fragmentation velocity of
+10 m/s, which may be rather high even for icy particles as in-
+dicated by recent laboratory experiments which are suggesting
+values of 1 m/s (see Blum 2018; Gundlach et al. 2018; Musiolik
+& Wurm 2019). Lowering the fragmentation velocity, however,
+generally decreases the particle sizes making them even less
+likely to be trapped in pressure bumps (see panel E in Figure 1).
+Other experiments indicate a significantly higher fragmentation
+Article number, page 6 of 8
+
+Sebastian Markus Stammler et al.: Leaky Dust Traps: How Fragmentation impacts Dust Filtering by Planets
+velocity (e.g. Kimura et al. 2020) than the 10 m/s used in this
+work. The exact value of the fragmentation velocity, however,
+does not significantly influence the problem of inner disk con-
+tamination. Either the fragmentation velocity is exceeded, which
+will lead to pollution of the inner disk with outer disk material
+(see panel B of Figure 1). Or the fragmentation velocity is larger
+than the maximum collision velocity of dust particles in the disk,
+in which case the particles will efficiently grow to larger parti-
+cles, that are being trapped in the outer edge of the disk, which
+will quickly deplete the inner disk (see panel D in Figure 1).
+Similarily, the porosity evolution may have an effect on the
+collisional physics of dust particles (e.g. Suyama et al. 2008;
+Krijt et al. 2015; Kobayashi & Tanaka 2021). However, as for
+the fragmentation velocity the details of the collision model do
+not have a strong effect on the outcome of the simulation. Either
+the particles fragment and the inner disk is polluted with outer
+disk material, or the particles grow unhindered to large particles
+that are trapped in the pressure bump, which is quickly depleting
+the inner disk.
+We furthermore did not consider the formation of planetes-
+imals in the pressure bump in this work. Previous publications
+have shown that the conditions in pressure maxima at the outer
+edges of gaps can facilitate planetesimal formation (Stammler
+et al. 2019; Miller et al. 2021) or even the formation of planets
+(Lau et al. 2022; Jiang & Ormel 2023). One could conceive that
+small dust fragments could not penetrate the inner disk because
+they are quickly converted into planetesimals before they could
+transverse the gap. This would, however, require a nearly per-
+fect planetesimal formation efficiency to efficiently isolate both
+dust reservoirs, which has not been observed in previous simu-
+lations. Additionally, planetesimals formed at gap edges quickly
+have been shown in simulations to quickly ablate (Eriksson et al.
+2021). Enstatite and ordinary chondrites would thus have to be
+explained by planetesimal formation where the dust is replen-
+ished by, for instance, late-stage planetesimal collisions in the
+NC reservoir (Dullemond et al. 2014; Lichtenberg et al. 2018;
+Bernabò et al. 2022).
+This suggests that it is unlikely that the formation of Jupiter
+could have solely separated both dust reservoirs in the Solar Sys-
+tem if the dust particles were subject to fragmentation. This does
+not only apply to gaps created by planets, but also to other sub-
+structures of non-planetary origin where particles are trapped
+in pressure maxima as described in Brasser & Mojzsis (2020).
+Other suggested mechanisms to explain the observations include
+a temporal change in the isotopic content of inward-streaming
+dust grains (Schiller et al. 2018), and the formation of multi-
+ple distinct planetesimal populations in the inner and outer disk
+(Lichtenberg et al. 2021; Morbidelli et al. 2022; Izidoro et al.
+2021; Liu et al. 2022). How these physical mechanisms are con-
+nected to the structures and gaps seen in ALMA disks (Miotello
+et al. 2022) and the underlying mechanisms of protoplanet for-
+mation (Drazkowska et al. 2022) and differentiation (Lichten-
+berg et al. 2022) remain to be explored.
+6. Conclusions
+Protoplanet-induced gaps in circumstellar disks are not able to
+efficiently separate dust in the inner disk from dust in the outer
+disk on million-year timescales if the particles are subject to
+fragmentation. Particles limited by bouncing without producing
+small fragments are usually too small to be trapped by pressure
+bumps. Only significantly reducing the relative collision veloci-
+ties allows particles to be efficiently trapped in pressure bumps
+within 1 Myr, if the planet grew to 40 M⊕. In this case, however,
+the inner disk is quickly depleted from dust making it difficult to
+form meteoritic bodies in situ. Our simulations suggest that other
+physical mechanism must have initiated or at least substantially
+contributed to the large-scale separation of nucleosynthetic iso-
+topes observed in the planetary materials of the inner and outer
+Solar System.
+Acknowledgements. This project has received funding from the European Re-
+search Council (ERC) under the European Union’s Horizon 2020 research and
+innovation programme under grant agreement No 714769. This project has re-
+ceived funding by the Deutsche Forschungsgemeinschaft (DFG, German Re-
+search Foundation) through grants FOR 2634/1 and 361140270. This research
+was supported by the Munich Institute for Astro-, Particle and BioPhysics
+(MIAPbP) which is funded by the Deutsche Forschungsgemeinschaft (DFG,
+German Research Foundation) under Germany’s Excellence Strategy – EXC-
+2094 – 390783311. JD was funded by the European Union under the Euro-
+pean Union’s Horizon Europe Research & Innovation Programme 101040037
+(PLANETOIDS). Views and opinions expressed are however those of the au-
+thors only and do not necessarily reflect those of the European Union or the Eu-
+ropean Research Council. Neither the European Union nor the granting authority
+can be held responsible for them. TL was supported by grants from the Simons
+Foundation (SCOL Award No. 611576) and the Branco Weiss Foundation.
+References
+Alibert, Y., Venturini, J., Helled, R., et al. 2018, Nature Astronomy, 2, 873
+Bernabò, L. M., Turrini, D., Testi, L., Marzari, F., & Polychroni, D. 2022, ApJ,
+927, L22
+Birnstiel, T., Klahr, H., & Ercolano, B. 2012, A&A, 539, A148
+Bitsch, B., Morbidelli, A., Johansen, A., et al. 2018, A&A, 612, A30
+Blum, J. 2018, Space Sci. Rev., 214, 52
+Blum, J. & Münch, M. 1993, Icarus, 106, 151
+Brasser, R. & Mojzsis, S. J. 2020, Nature Astronomy, 4, 492
+Deienno, R., Izidoro, A., Morbidelli, A., Nesvorný, D., & Bottke, W. F. 2022,
+ApJ, 936, L24
+Desch, S. J., Kalyaan, A., & O’D. Alexander, C. M. 2018, ApJS, 238, 11
+Drazkowska, J., Bitsch, B., Lambrechts, M., et al. 2022, arXiv e-prints,
+arXiv:2203.09759
+Dr˛a˙zkowska, J., Li, S., Birnstiel, T., Stammler, S. M., & Li, H. 2019, ApJ, 885,
+91
+Dullemond, C. P., Stammler, S. M., & Johansen, A. 2014, ApJ, 794, 91
+Eriksson, L. E. J., Ronnet, T., & Johansen, A. 2021, A&A, 648, A112
+Gundlach, B., Schmidt, K. P., Kreuzig, C., et al. 2018, MNRAS, 479, 1273
+Hasegawa, Y., Suzuki, T. K., Tanaka, H., Kobayashi, H., & Wada, K. 2021, ApJ,
+915, 22
+Izidoro, A., Dasgupta, R., Raymond, S. N., et al. 2021, Nature Astronomy, 6,
+357
+Jiang, H. & Ormel, C. W. 2023, MNRAS, 518, 3877
+Kanagawa, K. D., Tanaka, H., Muto, T., & Tanigawa, T. 2017, PASJ, 69, 97
+Kimura, H., Wada, K., Yoshida, F., et al. 2020, MNRAS, 496, 1667
+Kleine, T., Budde, G., Burkhardt, C., et al. 2020, Space Sci. Rev., 216, 55
+Kobayashi, H. & Tanaka, H. 2021, ApJ, 922, 16
+Krijt, S., Kama, M., McClure, M., et al. 2022, arXiv e-prints, arXiv:2203.10056
+Krijt, S., Ormel, C. W., Dominik, C., & Tielens, A. G. G. M. 2015, A&A, 574,
+A83
+Kruijer, T. S., Burkhardt, C., Budde, G., & Kleine, T. 2017, Proceedings of the
+National Academy of Science, 114, 6712
+Lambrechts, M., Johansen, A., & Morbidelli, A. 2014, A&A, 572, A35
+Lau, T. C. H., Dr˛a˙zkowska, J., Stammler, S. M., Birnstiel, T., & Dullemond, C. P.
+2022, A&A, 668, A170
+Leya, I., Schönbächler, M., Krähenbühl, U., & Halliday, A. N. 2009, ApJ, 702,
+1118
+Lichtenberg, T., Dr˛a˙zkowska, J., Schönbächler, M., Golabek, G. J., & Hands,
+T. O. 2021, Science, 371, 365
+Lichtenberg, T., Golabek, G. J., Dullemond, C. P., et al. 2018, Icarus, 302, 27
+Lichtenberg, T., Schaefer, L. K., Nakajima, M., & Fischer, R. A. 2022, arXiv
+e-prints, arXiv:2203.10023
+Liu, B., Johansen, A., Lambrechts, M., Bizzarro, M., & Haugbølle, T. 2022,
+Science Advances, 8, eabm3045
+Lynden-Bell, D. & Pringle, J. E. 1974, MNRAS, 168, 603
+Mathis, J. S., Rumpl, W., & Nordsieck, K. H. 1977, ApJ, 217, 425
+Mezger, K., Schönbächler, M., & Bouvier, A. 2020, Space Sci. Rev., 216, 27
+Miller, E., Marino, S., Stammler, S. M., et al. 2021, MNRAS, 508, 5638
+Miotello, A., Kamp, I., Birnstiel, T., Cleeves, L. I., & Kataoka, A. 2022, arXiv
+e-prints, arXiv:2203.09818
+Morbidelli, A., Baillié, K., Batygin, K., et al. 2022, Nature Astronomy, 6, 72
+Musiolik, G. & Wurm, G. 2019, ApJ, 873, 58
+Article number, page 7 of 8
+
+A&A proofs: manuscript no. dust_filtering
+Nimmo, F., Kretke, K., Ida, S., Matsumura, S., & Kleine, T. 2018,
+Space Sci. Rev., 214, 101
+Paardekooper, S. J. & Mellema, G. 2006, A&A, 453, 1129
+Pinilla, P., Lenz, C. T., & Stammler, S. M. 2021, A&A, 645, A70
+Rice, W. K. M., Armitage, P. J., Wood, K., & Lodato, G. 2006, MNRAS, 373,
+1619
+Schiller, M., Bizzarro, M., & Fernandes, V. A. 2018, Nature, 555, 507
+Schräpler, R., Blum, J., Krijt, S., & Raabe, J.-H. 2018, ApJ, 853, 74
+Stammler, S. M. & Birnstiel, T. 2022, ApJ, 935, 35
+Stammler, S. M., Dr˛a˙zkowska, J., Birnstiel, T., et al. 2019, ApJ, 884, L5
+Suyama, T., Wada, K., & Tanaka, H. 2008, ApJ, 684, 1310
+Takeuchi, T. & Lin, D. N. C. 2002, ApJ, 581, 1344
+Trinquier, A., Birck, J.-L., & Allègre, C. J. 2007, ApJ, 655, 1179
+Trinquier, A., Elliott, T., Ulfbeck, D., et al. 2009, Science, 324, 374
+Wada, K., Tanaka, H., Suyama, T., Kimura, H., & Yamamoto, T. 2009, ApJ, 702,
+1490
+Wahl, S. M., Hubbard, W. B., Militzer, B., et al. 2017, Geophys. Res. Lett., 44,
+4649
+Warren, P. H. 2011, Earth and Planetary Science Letters, 311, 93
+Weber, P., Benítez-Llambay, P., Gressel, O., Krapp, L., & Pessah, M. E. 2018,
+ApJ, 854, 153
+Weidenschilling, S. J. 1977, MNRAS, 180, 57
+Whipple, F. L. 1973, in NASA Special Publication, ed. C. L. Hemenway, P. M.
+Millman, & A. F. Cook, Vol. 319, 355
+Windmark, F., Birnstiel, T., Güttler, C., et al. 2012, A&A, 540, A73
+Youdin, A. N. & Lithwick, Y. 2007, Icarus, 192, 588
+Article number, page 8 of 8
+

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+arXiv:2301.01477v1  [stat.ME]  4 Jan 2023
+Reliability Analysis of Load-sharing Systems using a
+Flexible Model with Piecewise Linear Functions
+Shilpi Biswas ∗, Ayon Ganguly †, and Debanjan Mitra ‡
+Abstract
+Aiming for accurate estimation of system reliability of load-sharing systems, a flex-
+ible model for such systems is constructed by approximating the cumulative hazard
+functions of component lifetimes using piecewise linear functions. The advantages of
+the resulting model are that it is data-driven and it does not use prohibitive assump-
+tions on the underlying component lifetimes. Due to its flexible nature, the model is
+capable of providing a good fit to data obtained from load-sharing systems in general,
+thus resulting in an accurate estimation of important reliability characteristics. Es-
+timates of reliability at a mission time, quantile function, mean time to failure, and
+mean residual time for load-sharing systems are developed under the proposed model
+involving piecewise linear functions. Maximum likelihood estimation and construction
+of confidence intervals for the proposed model are discussed in detail. The performance
+of the proposed model is observed to be quite satisfactory through a detailed Monte
+Carlo simulation study. Analysis of a load-sharing data pertaining to the lives of a
+two-motor load-sharing system is provided as an illustrative example. In summary,
+this article presents a comprehensive discussion on a flexible model that can be used
+for load-sharing systems under minimal assumptions.
+Keywords: Load-sharing systems; Cumulative hazard function; Baseline hazard; Piecewise
+linear approximation; Maximum likelihood estimation; Fisher information; Bootstrap; Con-
+fidence interval; Quantile function; Mean time to failure; Reliability at a mission time; Mean
+residual time.
+1
+Introduction
+1.1
+Background
+Dynamic models are suitable for reliability systems where failure or degradation of one or
+more components affects the performance of the surviving or operating components. Load-
+sharing systems are appropriate examples where such models can be used. The total load
+on a load-sharing system is shared between its components; when a component fails within
+∗Indian Institute of Technology Guwahati, Assam 781039, India; Email: [email protected]
+†Indian Institute of Technology Guwahati, Assam 781039, India; Email: [email protected]
+‡Indian Institute of Management Udaipur, Rajasthan 313001, India; Email: [email protected]
+1
+
+the system, the total load gets redistributed over the remaining operating components. As a
+result of a higher stress due to this extra load, the failure rates of the operating components
+increase.
+Common examples of load-sharing systems are those where components are connected
+in parallel, such as central processing units (CPUs) of multi-processor computers, cables
+of a suspension bridge, valves or pumps in hydraulic systems, electrical generator systems
+etc. Load-sharing systems are found in other spheres as well, such as the kidney system in
+humans. When one of the kidneys fails or deteriorates, the other kidney experiences elevated
+stress and has an increased chance of failure.
+The load-share rule among the operating components depends on the physical charac-
+teristics of the system involved. In an equal load-share rule, the extra load caused by the
+failed components is shared equally by the operating components. On the other hand, a
+local load-share rule implies that the extra load is shared by the neighboring components of
+the failed ones. A monotone load-sharing rule more generally assumes that the load on the
+operating components is non-decreasing with respect to the failure of other components in
+the system [18].
+1.2
+Literature review
+One of the early major contributions to the literature on load-sharing systems was by
+Daniels [9], describing the increasing stress on yarn fibres with successive breakings of indi-
+vidual fibres within a bundle. In the same context of the textile industry, the early-period
+literature saw developments by Coleman [4, 5], Rosen [29], and Harlow and Phoenix [13, 14],
+among others. In general, the topic attracted the attentions of several researchers, and sig-
+nificant theoretical contributions were made, for example, by Birnbaum and Saunders [6],
+Freund [12], Ross [30], Schechner [31], Lee et al. [19], Hollander and Pena [15], and Lynch [22].
+While most studies on load-sharing systems in the early-period were based on a known
+load-share rule, Kim and Kvam [16] presented a statistical methodology for multicompo-
+nent load-sharing systems with an unknown load-share rule. In fact, the work of Kim and
+Kvam [16] was also important for another reason: they used the hypothetical latent variable
+approach for modelling the component lifetimes. The latent variable approach was later
+adapted by Park [27, 28] for developing an inferential framework for load-sharing systems
+assuming the component lifetimes to be exponential, Weibull, and lognormally distributed
+random variables.
+The use of parametric models has a long history in the literature on load-sharing models.
+Exponential distribution has been extensively used for modelling lifetimes of components of
+load-sharing systems [32, 20, 24]. However, the property of a constant hazard rate of the
+exponential distribution is not practical for most applications. The tampered failure rate
+model for load-sharing systems, proposed by Suprasad et al. [33], was thus developed to
+accommodate a wide range of failure-time distributions for the components. In this connec-
+tion, the use of accelerated life testing models for load-sharing systems may be mentioned;
+see Mettas and Vassiliou [23], Amari and Bergman [1], and Kong and Ye [17]. A family
+of parametric distributions was used for modelling the lives of two-component load-sharing
+systems by Deshpande et al. [10]. Asha et al. [2] used a frailty-based model to this effect. A
+recent contribution in this direction is by Franco et al. [11] who used generalized Freund’s
+2
+
+bivariate exponential model for two-component load-sharing systems. See also the references
+cited in these articles.
+Recently, several authors have explored diverse areas concerning load-sharing systems.
+The damage accumulation of load-sharing systems was modelled by M¨uller and Meyer [25].
+Luo et al.[21] developed a model for correlated lifetimes in dynamic environments incorpo-
+rating the load-sharing criterion. Brown et al. [7] explored a spatial model for load-sharing
+where the extra load due to failure of a component is shared more by the operating com-
+ponents that are in close proximity of the failed component than those that are distant.
+Nezakati and Ramzakh [26], and Zhao et al. [36] connected degradation of components to
+load-sharing phenomena.
+In an interesting development, Che et al. [8] considered man-
+machine units (MMUs) as units of analysis where load-sharing was possible due to machine
+issues as well as human issues. They studied the load-sharing of the MMUs, attempting to
+capture the complex dependence between machines and their operators. A general model,
+called the load-strength model, was studied by Zhang et al. [35]. It is to be noted that most
+of the studies on load-sharing systems have used parametric models for analysis so far, thus
+heavily relying on the modelling assumptions for suitability of their analyses.
+1.3
+Aim and Motivation
+Our aim in this paper is to develop an appropriate estimate for the system reliability or
+reliability at mission time (RMT) of load-sharing systems. The aim, also, is to accurately
+estimate quantile function of the underlying system lifetime distribution, mean time to failure
+(MTTF), and mean residual time (MRT) of load-sharing systems.
+These quantities are
+important to fully understand the characteristics of a load-sharing system; also, they are of
+practical importance for making various strategies and plans.
+Naturally, the quality of estimation of RMT, quantile function, MTTF, and MRT of a
+load-sharing system depends on the suitability of the model that is fitted to the lifetimes
+of its components capturing the load-share rule.
+To this effect, we develop a model for
+the component lifetimes involving piecewise linear approximations (PLAs) of the cumulative
+hazard functions, capturing the unknown load-share rule at each of the successive stages of
+component failures. The model is data-driven, and does not require prohibitive parametric
+assumptions for component lifetime distributions. Due to this flexibility, the PLA-based
+model is capable of providing a good fit to load-sharing data. An example, elaborated in a
+later section, is as follows.
+Data pertaining to a load-sharing system where each system was a parallel combination
+of two motors were analysed by Asha et al. [2] and Franco et al. [11].
+Asha et al. [2]
+assumed Weibull distributions for the component lifetimes, although data for one of the two
+component motors showed clear empirical evidence that the assumption was not satisfied. A
+generalized bivariate Freund distribution was assumed for the component lifetimes by Franco
+et al. [11]. To this data, we have fitted our proposed PLA-based model, and have observed
+according to the Akaike’s information criterion (AIC) for model selection, the PLA-based
+model is a much better fit compared to the Weibull model of Asha et al. [2] and generalized
+bivariate Freund model of Franco et al. [11]. The immediate and obvious result of this is
+a much more accurate estimation of the RMT, quantile function, MTTF, and MRT of the
+system lifetimes. The details of this analysis are given in a later section.
+3
+
+The main contributions of this paper are as follows:
+• We develop a flexible, data-driven model based on PLA for modelling component
+lifetimes of a load-sharing system. The model does not require prohibitive parametric
+assumptions on the underlying component lifetimes.
+• We develop inference for the proposed PLA-based model based on data from multi-
+component load-sharing systems.
+• Under the proposed PLA-based model, we develop methods to accurately estimate im-
+portant reliability characteristics such as system reliability or RMT, quantile function,
+MTTF, and MRT of load-sharing systems.
+The rest of this article is structured as follows. In Section 2, the proposed PLA-based
+model for load-sharing systems is presented.
+Section 3 contains likelihood inference for
+the model based on data from multi-component load-sharing systems, including relevant
+details of derivation of MLEs, construction of confidence intervals, and a general guidance
+on selection of cut-points for the piecewise linear functions. Estimation of system reliability,
+quantile function, MTTF, and MRT of load-sharing systems in this setting are given in
+Section 4. Based on component lifetime data from a two-component load-sharing system,
+an illustrative example of application of the PLA-based model and estimation of various
+important reliability characteristics are presented in Section 5. In Section 6, results of a
+detailed Monte Carlo simulation experiment investigating the efficacy and robustness of the
+PLA-based model are presented.
+Finally, the paper is concluded with some remarks in
+Section 7.
+2
+The Piecewise Linear Approximation Model for
+Cumulative Hazard
+In general, a PLA is a helpful tool for modelling data, avoiding strong parametric assump-
+tions. In survival analysis, piecewise linear functions are used extensively. Recently, Bal-
+akrishnan et al. [3] proposed a PLA-based model for the hazard rate of a population with
+a cured proportion; see also the references therein. In this article, we develop a PLA-based
+model for load-sharing systems with unknown load-share rules. Specifically, we model the
+cumulative hazard functions of the component lifetime distributions using PLAs. At each
+of the successive stages of component failures, as the lifetime distributions of the remaining
+operating components change, a new PLA for the cumulative hazard is used. The model
+can be suitably tuned by choosing the number of linear pieces for the PLA at each stage of
+failure. The principal advantage of the proposed PLA-based modelling approach is that it
+uses minimal model assumptions.
+Consider a J-component load-sharing system. Here, a J-component load-sharing system
+means a load-sharing system with J components that are connected in parallel. Assume that
+the failed components of the system are not replaced or repaired. When the components fail
+one by one, after each failure the total load on the system gets redistributed over the remain-
+ing operational components. As a result the operational components experience a higher load
+4
+
+than before. At the beginning when all components are operational, let U(0)
+1 , U(0)
+2 , . . . , U(0)
+J
+denote the latent lifetimes of the components, and Y (0) denote the system lifetime till the
+first component failure. Obviously,
+Y (0) = min
+�
+U(0)
+1 , U(0)
+2 , . . . , U(0)
+J
+�
+.
+Similarly, for j = 1, 2, . . . , J − 1, let Y (j) denote the system lifetime between j-th and
+(j + 1)-st component failures. Then,
+Y (j) = min
+�
+U(j)
+1 , U(j)
+2 , . . . , U(j)
+J−j
+�
+,
+where U(j)
+1 , U(j)
+2 , . . . U(j)
+J−j denote the latent lifetimes of the operational components after the
+j-th component failure, j = 1, 2, . . . , J − 1. For all values of j, U(j)
+1 , . . . , U(j)
+J−j are assumed
+to be independent and identically distributed random variables. It is further assumed that
+�
+U(j)
+ℓ , ℓ = 1, 2, . . . , J − j; j = 0, 1, . . . , J − 1
+�
+are independent random variables.
+Let h(j)(·) and H(j)(·) denote the hazard rate (HR) and cumulative hazard function
+(CHF), respectively, of the distribution of U(j)
+1 , j = 0, 1, 2, . . . , J −1. Here, we assume that
+the HR h(j) (·) is a non-decreasing function for all j. For y > 0, the survival function (SF)
+of Y (j) is given by
+P
+�
+Y (j) > y
+�
+= P
+�
+min
+�
+U(j)
+1 , U(j)
+2 , . . . , U(j)
+J−j
+�
+> y
+�
+= e−(J−j)H(j)(y).
+Hence, for y > 0, the cumulative distribution function (CDF) and probability density func-
+tion (PDF) of Y (j) are given by
+F (j)(y) = 1 − e−(J−j)H(j)(y)
+and
+f (j)(y) = (J − j)h(j)(y) e−(J−j)H(j)(y),
+respectively.
+Now, suppose there are n J-component load-sharing systems, and let Y (j)
+i
+denote the
+system lifetime between j-th and (j + 1)-st component failures for the i-th system, i =
+1, 2, . . . , n, j = 0, 1, . . . , J − 1. Suppose the observed values of Y (j)
+1 , Y (j)
+2 , . . . , Y (j)
+n
+are
+y(j)
+1 , y(j)
+2 , . . . , y(j)
+n , respectively. Let, for j = 0, 1, . . . , J − 1, ξ(j) =
+�
+τ (j)
+0 , τ (j)
+1 , . . . , τ (j)
+N
+�
+denote a set of N + 1 cut-points over the time scale y(j)
+1 , . . . , y(j)
+n , with the restrictions that
+τ (j)
+0
+< τ (j)
+1
+< τ (j)
+2
+< . . . < τ (j)
+N ,
+τ (j)
+0
+≤ min
+�
+y(j)
+1 , . . . , y(j)
+n
+�
+and τ (j)
+N ≥ max
+�
+y(j)
+1 , . . . , y(j)
+n
+�
+.
+Initially, ξ(j) is taken to be fixed and known. We discuss how to choose ��(j) in a later section.
+The proposed model approximates the CHF H(j)(·) by a piecewise linear function defined
+over intervals [τ (j)
+k−1, τ (j)
+k ), k = 1, 2, . . . , N, constructed by the consecutive cut points in ξ(j).
+Therefore, over the range [τ (0)
+0 , τ (0)
+N ), the CHF H(0)(·) is approximated by Λ(0)(·), where
+Λ(0)(t) =
+N
+�
+k=1
+(ak + bkt) 1[τ (0)
+k−1, τ (0)
+k
+)(t),
+(2.1)
+5
+
+with ak’s and bk’s as real constants and
+1A(t) =
+�
+1
+if t ∈ A
+0
+if t ̸∈ A.
+One of the possible ways to extend the PLA beyond τ (0)
+N
+would be to extend the last line
+segment aN + bNt to [τ (0)
+N , ∞). Therefore, the CHF corresponding to PLA over the range
+[τ (0)
+0 , ∞) is
+Λ(0)(t) =
+N
+�
+k=1
+(ak + bkt) 1[τ (0)
+k−1, τ (0)
+k
+)(t) + (aN + bNt)1[τ (0)
+N , ∞)(t),
+with Λ(0)(τ (0)
+0 ) = 0. We also assume that Λ(0)(·) is a continuous function. As Λ(0)(τ (0)
+0 ) = 0,
+using the assumption of continuity, ai’s can be expressed in terms of bi’s as follows:
+a1 = −b1τ (0)
+0
+and
+ak =
+k−1
+�
+ℓ=1
+(bℓ − bℓ+1) τ (0)
+ℓ
++ a1 =
+k−1
+�
+ℓ=1
+bℓ
+�
+τ (0)
+ℓ
+− τ (0)
+ℓ−1
+�
+− bkτ (0)
+k−1,
+for k = 1, 2, 3, . . . , N.
+Note that the above model can be equivalently described in terms of HRs.
+In this
+approach, h(0)(·) over the range [τ (0)
+0 , τ (0)
+N ) is approximated by a piecewise constant function
+λ(0)(·), where
+λ(0)(t) =
+N
+�
+i=1
+bk1[τ (0)
+k−1, τ (0)
+k
+) (t) .
+(2.2)
+After failure of one or more components within the system, the direct impact of the
+increased load will be an increased HR for the operational components. To incorporate this
+information, after the failure of j components of the system, we approximate h(j)(·) over
+[τ (j)
+0 , τ (j)
+N ), j = 1, 2, . . . , J − 1, using the piecewise constant function λ(j)(·), where
+λ(j)(t) = γj
+N
+�
+k=1
+bk1[τ (j)
+k−1, τ (j)
+k
+) (t) ,
+(2.3)
+with
+1 < γ1 < γ2 < . . . < γJ−1.
+The PLAs to the CHFs, corresponding to the PLAs of the HRs given in Eq.(2.3) are given
+by
+Λ(j)(t) = γj
+N
+�
+k=1
+�k−1
+�
+ℓ=1
+bℓ
+�
+τ (j)
+ℓ
+− τ (j)
+ℓ−1
+�
++ bk
+�
+t − τ (j)
+k−1
+��
+1[τ (j)
+k−1, τ (j)
+k
+) (t) .
+(2.4)
+To meet the non-decreasing nature of the HR, we assume that 0 < b1 < b2 < . . . < bN. Note
+that the parameters γ1, γ2, . . . , γJ−1 reflect the load-share rule of increased HRs. We treat
+γ1, γ2, . . . , γJ−1 as unknown parameters, and estimate them from component failure data.
+It may be mentioned here that the PLA model can be interpreted as an approximation
+of the underlying lifetime distribution by several exponential models (with different rate
+parameters) over the ranges specified by the cut-points.
+6
+
+3
+Likelihood Inference
+The parameters involved in the PLA-based model are estimated from the component failure
+data obtained from a set of load-sharing systems. The available data on component failures
+from n J-component load-sharing systems is of the form
+Data =
+�
+y(j)
+i
+: i = 1, 2, . . . , n; j = 0, 1, . . . , J − 1
+�
+,
+where y(j)
+i
+is the observed system lifetime between j-th and (j + 1)-st component failures for
+the i-th system. For j = 0, 1, 2, . . . , J − 1, and k = 1, 2, . . . , N, define
+I(j)
+k
+=
+�
+i : y(j)
+i
+∈
+�
+τ (j)
+k−1, τ (j)
+k
+��
+and
+n(j)
+k
+= |I(j)
+k |.
+Obviously, �N
+k=1 n(j)
+k
+= n. The likelihood function for the PLA model is then given by
+L (θ) =
+n
+�
+i=1
+J−1
+�
+j=0
+�
+(J − j)γj
+N
+�
+k=1
+bk1[τ (j)
+k−1, τ (j)
+k
+)
+�
+y(j)
+i
+�
+e
+−(J−j)γj
+��k−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
++bk
+�
+y(j)
+i
+−τ (j)
+k−1
+���
+,
+(3.1)
+where γ0 = 1 and θ = (γ1, γ2, . . . , γJ−1, b1, b2, . . . , bN)′ is the vector of parameters. The
+corresponding log-likelihood function, ignoring additive constant, can be expressed as
+l (θ) =
+N
+�
+k=1
+��J−1
+�
+j=0
+n(j)
+k
+�
+ln bk −
+�J−1
+�
+j=0
+(J − j)γjT (j)
+k
+�
+bk
+�
++ n
+J−1
+�
+j=0
+ln γj,
+(3.2)
+where
+T (j)
+k
+=
+�
+i∈I(j)
+k
+�
+y(j)
+i
+− τ (j)
+k−1
+�
++
+�
+n −
+k
+�
+ℓ=1
+n(j)
+ℓ
+� �
+τ (j)
+k
+− τ (j)
+k−1
+�
+,
+for k = 1, 2, . . . , N; j = 0, 1, . . . , J − 1. Equating partial derivative of the log-likelihood
+function in Eq.(3.2) with respect to bk to zero, we can express bk in terms of the load-share
+parameters γ = (γ1, γ2, . . . , γJ−1) as
+bk = bk (γ) =
+J−1
+�
+j=0
+n(j)
+k
+J−1
+�
+j=0
+(J − j)γjT (j)
+k
+,
+k = 1, ..., N.
+(3.3)
+Substituting bk(γ) from Eq.(3.3) in Eq.(3.2), the profile log-likelihood in γ, ignoring additive
+constant, is obtained as
+˜l (γ) =
+N
+�
+k=1
+��J−1
+�
+j=0
+n(j)
+k
+� �
+ln
+�J−1
+�
+j=0
+n(j)
+k
+�
+− ln
+�J−1
+�
+j=0
+(J − j)γjT (j)
+k
+���
++ n
+J−1
+�
+j=0
+ln γj.
+(3.4)
+7
+
+For optimizing the profile log-likelihood ˜l (γ) in γ, any routine maximizer of a standard
+statistical software may be used. Once the MLEs �γ1, �γ2, . . . , �γJ−1 of γ1, γ2, . . . , γJ−1 are
+obtained by numerical optimization of ˜l (γ), they can be plugged into Eq.(3.3) to get MLEs
+of bk as
+�bk = bk (�γ1, . . . , �γJ−1) ,
+k = 1, 2, . . . , N.
+3.1
+A special case: two-component load-sharing systems
+For analysing data from two-component load-sharing systems, if two linear pieces are used
+in the PLA-based model, MLEs can be derived analytically and explicitly. Consider the case
+when J = 2 and N = 2. In this case, the log-likelihood function simplifies to
+l (θ) =
+2
+�
+k=1
+��
+1
+�
+j=0
+n(j)
+k
+�
+ln bk −
+�
+1
+�
+j=0
+(2 − j)γjT (j)
+k
+�
+bk
+�
++ n
+1
+�
+j=0
+ln γj,
+(3.5)
+with
+T (j)
+k
+=
+�
+i∈I(j)
+k
+�
+y(j)
+i
+− τ (j)
+k−1
+�
++
+�
+n −
+k
+�
+ℓ=1
+n(j)
+ℓ
+� �
+τ (j)
+k
+− τ (j)
+k−1
+�
+,
+for k = 1, 2, j = 0, 1 and γ0 = 1. Here, θ = (γ1, b1, b2).
+Equating ∂l(θ)
+∂b1 and ∂l(θ)
+∂b2 to zero, we get
+b1 =
+n(0)
+1
++ n(1)
+1
+2T (0)
+1
++ γ1T (1)
+1
+(3.6)
+b2 =
+n(0)
+2
++ n(1)
+2
+2T (0)
+2
++ γ1T (1)
+2
+.
+(3.7)
+Equating ∂l(θ)
+∂γ1 to zero gives
+γ1 = T (1)
+1 b1 + T (1)
+2 b2,
+(3.8)
+in which, substituting b1 and b2 from Eqs.(3.6) and (3.7), a quadratic equation in γ1 is
+obtained as follows
+Q(γ1) = nγ2
+1B0,12 + 2γ1
+��
+n(0)
+1
++ n(1)
+1
+− n
+�
+B2,1 +
+�
+n(0)
+2
++ n(1)
+2
+− n
+�
+B1,2
+�
+− 4nB12,0 = 0,
+(3.9)
+with B0,12 = T (1)
+1 T (1)
+2 , B1,2 = T (0)
+1 T (1)
+2 , B2,1 = T (0)
+2 T (1)
+1
+and B12,0 = T (0)
+1 T (0)
+2 . Solving Q(γ1) =
+0, we have two values of γ1 from which we choose the suitable one, and then from equations
+(3.6) and (3.7) we get the MLEs of b1 and b2, respectively.
+8
+
+3.2
+Confidence Intervals
+As discussed above, the MLEs for the parameters of the PLA-based model are not available
+in explicit form in general, except for the special case of two-component load-sharing systems
+considered in Section 3.1. As a result, exact confidence intervals for the model parameters
+cannot be obtained. Asymptotic confidence intervals may be constructed in two possible
+ways: by using the Fisher information matrix, and by applying a bootstrap-based technique.
+3.2.1
+CIs using Fisher information matrix
+Using the asymptotic properties of the MLEs, it can be shown that for large sample size
+n, the distribution of √n(�θ − θ) is approximated by a multi-variate normal distribution
+N(0, I−1(�θ)), where the dimension of the multi-variate normal distribution is same as that
+of the parameter vector θ, and the asymptotic variance-covariance matrix I−1(θ) is the in-
+verse of the Fisher information matrix I(θ), evaluated at the MLE �θ. The Fisher information
+matrix I(θ) is defined as the expected value of the observed information matrix J(θ) which
+is calculated from the negative of the second-order derivatives of the log-likelihood function.
+That is, I(θ) = E(J(θ)), where J(θ) = −∇2(log L(θ)). In situations where analytical calcu-
+lation of the Fisher information is difficult or intractable, it may be either replaced by the
+observed information matrix, or may be calculated by simulations.
+From the asymptotic variance-covariance matrix I−1(θ), individual asymptotic variances
+of the MLEs can be pulled out, and asymptotic confidence intervals can be constructed. For
+example, corresponding to the MLE �γ1 using the asymptotic variance
+�
+V ar(ˆγ1) obtained from
+I−1(θ), asymptotic confidence intervals for γ1 can be constructed as:
+�
+�γ1 − zα/2
+�
+�
+V ar(ˆγ1), �γ1 + zα/2
+�
+�
+V ar(ˆγ1)
+�
+,
+where zα is the 100(1 − α)% point of the standard normal distribution.
+Special case: two-component load-sharing systems
+For the special case of two-component load-sharing systems considered in Section 3.1, the
+Fisher information matrix can be worked out explicitly. In this case,
+J(θ) = −
+
+
+
+
+
+
+
+
+∂2l(θ)
+∂γ2
+1
+∂2l(θ)
+∂γ1∂b1
+∂2l(θ)
+∂γ1∂b2
+∂2l(θ)
+∂b1∂γ1
+∂2l(θ)
+∂b2
+1
+∂2l(θ)
+∂b1∂b2
+∂2l(θ)
+∂b2∂γ1
+∂2l(θ)
+∂b2∂b1
+∂2l(θ)
+∂b2
+2
+
+
+
+
+
+
+
+
+= −
+
+
+
+
+− n
+γ2
+1
+−T (1)
+1
+−T (1)
+2
+−T (1)
+1
+−n(0)
+1 +n(1)
+1
+b12
+0
+−T (1)
+2
+0
+−n(0)
+2 +n(1)
+2
+b22
+
+
+
+ .
+9
+
+Hence, the Fisher information matrix is
+I(θ) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+n
+γ2
+1
+E
+
+
+
+�
+i∈I(1)
+1
+Y (1)
+i
+
+
+ + E
+�
+N(1)
+2
+�
+τ (1)
+1
+E
+
+
+
+�
+i∈I(1)
+1
+Y (1)
+i
+
+
+ − E
+�
+N(1)
+2
+�
+τ (1)
+1
+E
+
+
+
+�
+i∈I(1)
+1
+Y (1)
+i
+
+
+ + E
+�
+N(1)
+2
+�
+τ (1)
+1
+E
+�
+N(0)
+1
+�
++E
+�
+N(1)
+1
+�
+b2
+1
+0
+E
+
+
+
+�
+i∈I(1)
+1
+Y (1)
+i
+
+
+ − E
+�
+N(1)
+2
+�
+τ (1)
+1
+0
+E
+�
+N(0)
+2
+�
++E
+�
+N(1)
+2
+�
+b2
+2
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+,
+where N(j)
+k
+is the number of Y (j)
+i
+in [τ (j)
+k−1, τ (j)
+k ), k = 1, 2, j = 0, 1, i = 1, ..., n. An outline
+of calculations of the relevant expectations for the Fisher information matrix is given in
+Appendix A. The inverse of the Fisher information matrix is obtained as
+�
+I−1(θ)
+�
+=
+1
+|I(θ)|
+
+
+A11(θ)
+−A12(θ)
+A13(θ)
+−A21(θ)
+A22(θ)
+−A23(θ)
+A31(θ)
+−A32(θ)
+A33(θ)
+
+ ,
+where the determinant of I(θ) is
+|I(θ)|
+=
+n
+�
+2 −
+�
+e−2b1τ (0)
+1
++ e−γ1b1τ (1)
+1
+�� �
+e−2b1τ (0)
+1
++ e−γ1b1τ (1)
+1
+�
+γ2
+1b2
+1b2
+2
+−
+e−2γ1b1τ (1)
+1
+�
+1
+γ1b2
+�2 �
+2 −
+�
+e−2b1τ (0)
+1
++ e−γ1b1τ (1)
+1
+��
+b2
+1
+−
+�
+1
+γ1b1
+�
+1 − (1 + γ1b1τ (1)
+1 )e−γ1b1τ (1)
+1
+�
++ τ (1)
+1 e−γ1b1τ (1)
+1
+�2 �
+e−2b1τ (0)
+1
++ e−γ1b1τ (1)
+1
+�
+b2
+2
+,
+A11(θ) =
+�
+2 −
+�
+e−2b1τ (0)
+1
++ e−γ1b1τ (1)
+1
+�� �
+e−2b1τ (0)
+1
++ e−γ1b1τ (1)
+1
+�
+b2
+1b2
+2
+,
+A22(θ) =
+n
+�
+e−2b1τ (0)
+1
++ e−γ1b1τ (1)
+1
+�
+γ2
+1b2
+2
+− e−2γ1b1τ (1)
+1
+� 1
+γ1b2
+�2
+,
+A33(θ) =
+n
+�
+2 −
+�
+e−2b1τ (0)
+1
++ e−γ1b1τ (1)
+1
+��
+γ2
+1b2
+1
+−
+� 1
+γ1b1
+�
+1 − (1 + γ1b1τ (1)
+1 )e−γ1b1τ (1)
+1
+�
++ τ (1)
+1 e−γ1b1τ (1)
+1
+�2
+,
+A12(θ) = A21(θ) =
+�
+1
+γ1b1
+�
+1 − (1 + γ1b1τ (1)
+1 )e−γ1b1τ (1)
+1
+�
++ τ (1)
+1 e−γ1b1τ (1)
+1
+� �
+e−2b1τ (0)
+1
++ e−γ1b1τ (1)
+1
+�
+b2
+2
+,
+10
+
+A13(θ) = A31(θ) = −
+e−γ1b1τ (1)
+1
+�
+1
+γ1b2
+� �
+2 −
+�
+e−2b1τ (0)
+1
++ e−γ1b1τ (1)
+1
+��
+b2
+1
+,
+A23(θ) = A32(θ) = −
+��
+1 − (1 + γ1b1τ (1)
+1 )e−γ1b1τ (1)
+1
+�
++ γ1b1τ (1)
+1 e−γ1b1τ (1)
+1
+�
+e−γ1b1τ (1)
+1
+γ2
+1b1b2
+.
+Evaluating I−1(θ) at the MLE �θ, the asymptotic variance-covariance matrix of the MLEs
+is obtained.
+Hence, 100(1 − α)% asymptotic confidence intervals for γ1, b1, and b2 are
+obtained as
+�
+�γ1 −zα/2
+�
+A11(ˆθ)
+|I(ˆθ)| , �γ1 +zα/2
+�
+A11(ˆθ)
+|I(ˆθ)|
+�
+,
+�
+�b1 −zα/2
+�
+A22(ˆθ)
+|I(ˆθ)| , �b1 +zα/2
+�
+A22(ˆθ)
+|I(ˆθ)|
+�
+, and
+�
+�b2 − zα/2
+�
+A33(ˆθ)
+|I(ˆθ)| , �b2 + zα/2
+�
+A33(ˆθ)
+|I(ˆθ)|
+�
+, respectively.
+3.2.2
+Bootstrap confidence intervals
+Using the MLE �θ, B bootstrap samples can be obtained in the same sampling framework;
+let �θ
+∗
+s =
+�
+�γ∗
+1s,�b∗
+1s,�b∗
+2s
+�
+denote the bootstrap estimates, s = 1, ..., B. Bootstrap bias and
+standard error are defined as
+biasb(�γ1) = �γ∗
+1 − �γ1,
+biasb(�b1) = �b∗
+1 −�b1,
+biasb(�b2) = �b∗
+2 −�b2
+and
+SEb(�γ1) =
+�
+�
+�
+�
+1
+B − 1
+B
+�
+s=1
+�
+�γ∗
+1s − �
+γ∗
+1
+�2
+, SEb(�b1) =
+�
+�
+�
+�
+1
+B − 1
+B
+�
+s=1
+�
+�b∗
+1s − �b∗
+1
+�2
+, SEb(�b2) =
+�
+�
+�
+�
+1
+B − 1
+B
+�
+s=1
+�
+�b∗
+2s − �b∗
+2
+�2
+,
+where
+�γ∗
+1 = 1
+B
+B
+�
+s=1
+�γ∗
+1s,
+�b∗
+1 = 1
+B
+B
+�
+s=1
+�b∗
+1s,
+�b∗
+2 = 1
+B
+B
+�
+s=1
+�b∗
+2s.
+Finally, a 100(1 − α)% bootstrap confidence interval for γ1 can be calculated as
+�
+�γ1 − biasb(�γ1) − zα/2SEb(�γ1), �γ1 − biasb(�γ1) + zα/2SEb(�γ1)
+�
+.
+Bootstrap confidence intervals for b1 and b2 can be calculated similarly.
+For percentile bootstrap confidence intervals for, say γ1, the bootstrap estimates of �γ1
+are first ordered in terms of magnitude:
+�γ∗
+1(1) < �γ∗
+1(2) < ... < �γ∗
+1(B).
+Then, a 100(1−α)% percentile bootstrap confidence interval for γ1 is
+�
+�γ∗
+1([ αB
+2 ]), �γ∗
+1([(1− α
+2 )B])
+�
+.
+Similarly, percentile bootstrap confidence intervals can be calculated for b1 and b2.
+11
+
+3.3
+Choice of Cut Points
+The number and position of the cut-points for constructing the PLA-based model need to
+be suitably chosen, so that the model can closely approximate the underlying CHF, but
+avoid overfitting. A large number of cut points would provide a close local approximation
+to the underlying CHF. However, apart from being computationally expensive, a close local
+approximation may also lead to overfitting in which case it would be difficult to use the
+PLA-based model to predict future failures of components or systems.
+One of the possible ways to choose the number and position of the cut-points is by looking
+at the plot of the nonparametric estimator of CHF. From such a plot, observing the areas
+where the nonparametric estimate changes significantly, one can determine the positions and
+number of cut-points.
+More objectively, one can choose the positions of a given number of cut-points by max-
+imizing the log-likelihood function. For example, for three cut-points (N = 2), the natural
+choice for τ (j)
+0
+is min
+�
+y(j)
+1 , . . . , y(j)
+n
+�
+and τ (j)
+2
+is max
+�
+y(j)
+1 , . . . , y(j)
+n
+�
+. Now to choose the
+position of τ (j)
+1 , one may take τ (j)
+1
+equal to different sample quantiles of
+�
+y(j)
+1 , . . . , y(j)
+n
+�
+and
+choose one that provides the maximum value of log-likelihood function evaluated at MLE.
+This process can be expressed as an algorithm as follows.
+Algorithm:
+• Step 1: Fix 0 < p1 < p2 < 1.
+• Step 2: Find the number of y(j)
+1 , . . . , y(j)
+n
+that are between p1-th and p2-th sample
+quantiles of
+�
+y(j)
+1 , . . . , y(j)
+n
+�
+. Denote this number by l. Note that l does not depend
+on j = 0, 1, . . . , J − 1.
+• Step 3: Set aj1 = p1-th quantile of
+�
+y(j)
+1 , . . . , y(j)
+n
+�
+, j = 0, 1, . . . , J − 1.
+• Step 4: Set LL1= the value of log-likelihood function evaluated at MLE taking τ (j)
+1
+=
+aj1, j = 0, 1, . . . , J − 1.
+• Step 5: Set aj2 = min
+�
+y(j)
+i
+> aj1; i = 1, 2, . . . , n
+�
+, j = 0, 1, . . . , J − 1.
+• Step 6: Set LL2= the value of log-likelihood function evaluated at MLE taking τ (j)
+1
+=
+aj2, j = 0, 1, . . . , J − 1.
+• Step 7: Repeat the steps 5 and 6 to obtain LL1, LL2, . . . , LLl.
+• Step 8: Set k∗ = arg max
+1≤k≤l
+LLk.
+• Step 9: The final cut points are τ (j)
+1
+= ajk∗, j = 0, 1, . . . , J − 1.
+12
+
+4
+Estimation of various reliability characteristics
+The final goal of fitting a model to load-sharing data, naturally, is accurate estimation of
+reliability characteristics of load-sharing systems. As the PLA-based model provides a good
+fit to load-sharing data due to the model’s flexible nature, it is natural that the important
+reliability characteristics of load-sharing systems can also be estimated quite accurately
+under this model. In this section, we develop estimates of reliability characteristics such as
+the quantile function, MTTF, RMT, and MRT of load-sharing systems under the PLA-based
+model. Details of these derivations are given in Appendix B for interested readers.
+Under the PLA-based model, the quantile function of Y (j) which is the system lifetime
+between the j-th and (j + 1)-st component failures, j = 0, ..., J − 1, is given by
+η(p) = inf
+�
+y ∈ R : G(j)(y) ≥ p
+�
+,
+0 < p < 1,
+where G(j)(y) = 1 − e−(J−j)Λ(j)(y). Using the expression of Λ(j)(y) given in Section 2, it is
+possible to work out an explicit formula for the quantile function η(p), as follows:
+η(p) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+τ (j)
+k−1 −
+log(1−p)
+(J−j)γjbk − 1
+bk ·
+k−1
+�
+ℓ=1
+bℓ(τ (j)
+ℓ
+− τ (j)
+ℓ−1), if p ∈
+�
+G(j)(τ (j)
+k−1), G(j)(τ (j)
+k )
+�
+,
+for k = 1, 2, . . . , N.
+τ (j)
+N−1 −
+log(1−p)
+(J−j)γjbN −
+1
+bN ·
+N−1
+�
+ℓ=1
+bℓ(τ (j)
+ℓ
+− τ (j)
+ℓ−1), if p ∈
+�
+G(j)(τ (j)
+N ), 1
+�
+.
+The mean time to failure or MTTF of a load-sharing system is the expected time the
+system operates till its failure. Let T denote the system failure time; then, T = �J−1
+j=0 Y (j).
+The MTTF of a load-sharing system under the PLA-based model is given by
+E(T) =
+J−1
+�
+j=0
+N
+�
+s=1
+�e−κj,s−1 − e−κj,s
+(J − j)γjbℓ
+�
+,
+where
+κj,s = (J − j)γj
+s
+�
+ℓ=1
+bℓ
+�
+τ (j)
+ℓ
+− τ (j)
+ℓ−1
+�
+.
+Reliability at a mission time or RMT of a system is the probability that the system will
+operate till a desired time t0; it is calculated as the survival probability of the system at
+time t0, i.e., S(t0) = P(T > t0) = P
+�J−1
+�
+j=0
+Y (j) > t0
+�
+. An explicit expression for RMT may
+be derived by using the distribution of the system lifetime T.
+However, as Y (j)s, j = 0, ..., J − 1 are independent but not identically distributed, it is
+difficult to obtain an explicit expression for the distribution of the system lifetime T, where
+T = �J−1
+j=0 Y (j). It is evident from the moment generating function φT(t) of T, which, under
+the PLA-based model, is given by
+φT(t) =
+J−1
+�
+j=0
+N
+�
+s=1
+(J − j)bsγj
+(J − j)bsγj − t
+�
+etτ (j)
+s−1−κj,s−1 − etτ (j)
+s
+−κj,s
+�
+if t < γ1bN,
+13
+
+where
+κj,s = (J − j)γj
+s
+�
+ℓ=1
+bℓ
+�
+τ (j)
+ℓ
+− τ (j)
+ℓ−1
+�
+.
+From here, it is clear that it is difficult to find the RMT analytically under this model.
+However, for this model, RMT can be estimated using Monte Carlo simulations.
+For a
+Monte Carlo estimate of the RMT at a pre-specified time t0, one needs to generate R data
+points ti, i = 1, 2, . . . , R, as realisations of the system lifetime T, and find R(t0)
+R , where R(t0)
+is the number of realisations of the system lifetime that exceed t0. For a reasonably good
+estimate of RMT, a large value of R should be used.
+The mean residual time or MRT of a system is the expected additional time the system
+will survive if it has already survived a given time t. That is,
+MRT(t) = E(T − t|T > t) =
+� ∞
+t
+sfT|T>t(s)ds − t.
+Therefore, analytical derivation of MRT requires the truncated distribution of the system
+lifetime T, and it is difficult to obtain the truncated distribution of T in this case. Instead,
+an estimate of the MRT can be given using Monte Carlo simulations. We generate R data
+points t∗
+i , i = 1, 2, . . . , R, as realisations of the truncated lifetime T|T > t, and a Monte
+Carlo estimate of the MRT for load-sharing systems under the PLA-based model is then
+given by
+�
+MRT(t) =
+R
+�
+i=1
+t∗
+i
+R
+− t.
+5
+Data Analysis
+In this section, we present an illustrative example using data from load-sharing systems
+comprising of two components. Very recently, this data have been analysed by Sutar and
+Naik-Nimbalkar [34], Asha et al. [2] and Franco et al. [11]. The data consist of information
+on component lifetimes of 18 two-component load-sharing systems. Each system is a parallel
+combination of two motors - “A” and “B”. When both motors A and B are in working
+condition, the total load on the system is shared between them. When one of the motors
+fails, the entire load goes to the operational motor.
+Sutar and Naik-Nimbalkar [34] observed that the load-sharing phenomenon existed for
+the systems considered in this dataset. Asha et al. [2] assumed Weibull lifetimes for the
+components. From the Weibull Q-Q plots for the lifetimes of motor A and B reported in
+Asha et al. [2], it was observed that although the Weibull model assumption for the lifetimes
+of motor B was reasonable, the lifetimes of motor A did not follow a Weibull distribution.
+This motivated us to consider the PLA-based modelling approach for the lifetimes of the
+load-sharing systems in this case.
+The dataset is reproduced in Table 1 for ready reference of the readers. The average
+and standard deviation of first component failure times are 178.61 and 62.75, respectively,
+14
+
+0
+100
+200
+300
+0
+100
+200
+300
+Sample quantile
+Population quantile
+(a) Q-Q plot for Y (0)
+0
+25
+50
+75
+100
+125
+0
+25
+50
+75
+100
+125
+Sample quantile
+Population quantile
+(b) Q-Q plot for Y (1)
+Figure 1: Q-Q plots
+0.25
+0.50
+0.75
+1.00
+100
+150
+200
+250
+300
+Time
+SF
+(a) Plot of SF for Y (0)
+0.00
+0.25
+0.50
+0.75
+1.00
+0
+40
+80
+120
+Time
+SF
+(b) Plot of SF for Y (1)
+Figure 2: Plots of SFs
+while those of the lifetime between first and second component failures are 49.72 and 29.45,
+respectively. We consider three cut points for the PLA-based model (i.e., N = 2). The
+estimates of the model parameters are reported in Table 2. The Q-Q plots for Y (0) and Y (1)
+are given in Figures 1a and 1b, respectively. The plots of the estimated SF and CHF are
+given in Figures 2 and 3, respectively. These figures indicate that the PLA-based model fits
+the data quite adequately.
+15
+
+0.0
+0.5
+1.0
+1.5
+100
+150
+200
+250
+300
+Time
+CHF
+(a) Plot of CHF for Y (0)
+0
+2
+4
+6
+0
+40
+80
+120
+Time
+CHF
+(b) Plot of CHF for Y (1)
+Figure 3: Plots of CHFs
+Table 1: Time to failure (in days) data set for two motors in a load-sharing configuration
+System
+Time to failure of motor A
+Time to failure of motor B
+Event description
+1
+102
+65
+B failed first
+2
+84
+148
+A failed first
+3
+88
+202
+A failed first
+4
+156
+121
+B failed first
+5
+148
+123
+B failed first
+6
+139
+150
+A failed first
+7
+245
+156
+B failed first
+8
+235
+172
+B failed first
+9
+220
+192
+B failed first
+10
+207
+214
+A failed first
+11
+250
+212
+B failed first
+12
+212
+220
+A failed first
+13
+213
+265
+A failed first
+14
+220
+275
+A failed first
+15
+243
+300
+A failed first
+16
+300
+248
+B failed first
+17
+257
+330
+A failed first
+18
+263
+350
+A failed first
+A Kolmogorov-Smirnov type test has been performed to test the following hypotheses:
+H0 : True model is specified by Eqs. (2.2) and (2.3)
+against
+H1 : True model is not specified by Eqs. (2.2) and (2.3)
+16
+
+Table 2: Point and interval estimates of parameters of the PLA-based model when applied
+to the two-motor load-sharing data
+Parameter
+MLE
+Std. Error
+Asymptotic
+Percentile bootstrap
+Bootstrap
+γ1
+4.2712
+1.1901
+(1.9386, 6.6038)
+(3.0754, 8.0279)
+(0.8456, 5.8172)
+b1
+0.0034
+0.0008
+(0.0019, 0.0048)
+(0.0021, 0.0062)
+(0.0008, 0.0052)
+b2
+0.0134
+0.0039
+(0.0056, 0.0212)
+(0.0061, 0.0209)
+(0.0083, 0.0232)
+Table 3: Mean residual time and reliability in mission time
+t0
+MRTt0
+RMTt0
+102.00
+124.223
+0.963
+167.50
+88.678
+0.706
+227.50
+60.646
+0.466
+272.50
+42.794
+0.271
+350.00
+36.919
+0.044
+based on the test statistics
+Tn = max
+1≤i≤n
+���� �G(0) �
+Y (0)
+i:n
+�
+− i
+n
+���� + max
+1≤i≤n
+���� �G(1) �
+Y (1)
+i:n
+�
+− i
+n
+���� ,
+where �G(j)(·) is the estimated cumulative distribution function corresponding to PLA-based
+model, and Y (j)
+i:n is the i-th order statistics corresponding to Y (j)
+i
+, j = 0, 1, i = 1, 2, . . . , n.
+The observed value of the test statistics Tn is found to be 0.414 based on this data. The
+Monte Carlo estimate of the corresponding p-value is 0.71. Therefore, the null hypothesis
+cannot be rejected at significance level 0.05, and we conclude that it is quite reasonable to
+use the PLA-based model for this data.
+It may also be noted here that for this data, the value of the Akaike’s information
+criterion (AIC) for the model considered by Asha et al. [2] is 480.50, and that for the best
+model considered by Franco et al. [11] is 409.65. In contrast, the AIC value for the PLA-
+based model turns out to be 369.34, implying that the PLA-based model is more suitable
+for the two-motor load-sharing systems data considered here.
+For the PLA-based model, the estimated value of γ1 is 4.2712, which empirically implies
+that the load-sharing model is quite appropriate in this case. The same comment can also
+be made from the plots, by noting that the plot of the SF of the distribution of time between
+first and second failure component times diminishes to zero more quickly compared to that
+of first component failure times in Figure 2.
+The reliability characteristics of the two-motor load-sharing systems are also estimated
+by using the expressions and techniques described in Section 4. The MTTF is calculated
+to be 221.36 days. Monte Carlo estimates of the MRT and RMT are calculated at different
+sample percentile points of the system failure times and are presented in Table 3.
+17
+
+6
+Simulation Study
+The accuracy of the proposed PLA-based model in fitting data from load-sharing systems is
+of utmost importance as the subsequent estimation of reliability characteristics depends on
+the PLA-based model. In this section, we present results of a Monte Carlo simulation study
+that examines the performance of the proposed PLA-based model in two directions. First,
+based on samples generated from a parent process with piecewise linear CHF, we assess the
+performance of the proposed estimation method that is presented in Section 3. Then, the
+efficacy of the PLA-based model in fitting data generated from a parent process represented
+by some parametric models is also assessed. The simulations are carried out by using R
+software. For the simulations, we consider two-component load-sharing systems.
+6.1
+Assessing performance of the estimation method
+To assess the performance of the estimation methods, we consider an underlying cumulative
+hazard that is made up of two linear pieces. To this effect, we generate samples from the
+model specified by Eqs.(2.2) and (2.3) with J = 2 and N = 2. The true parameter values
+are taken to be b1 = 0.01, 0.05; b2 = 0.1, 0.5; γ1 = 5; τ (0)
+1
+= ln 2
+2b1 ; τ (1)
+1
+=
+ln 2
+γ1b1. The estimation
+is performed based on samples of size n = 100 and 200. The average estimates (AE), mean
+square errors (MSE), variance (VAR) of the MLEs based on 5000 Monte Carlo replications
+are reported in Tables 4, 5, and 6. The coverage percentage (CP) and average lengths (AL)
+of 95% confidence intervals are also reported in the same tables.
+From the Tables 4, 5 and 6, we observe that the average estimates of γ1, b1 and b2 are
+very close to the true values, and the MSEs as well as VARs are quite small as desired. It is
+also noticed that the performance of all the constructed confidence intervals is satisfactory.
+These results demonstrate that the proposed inferential techniques can accurately estimate
+the parameters of the PLA-based model.
+Table 4: Performance measures for estimates of γ1
+n
+b1
+b2
+AE
+MSE
+VAR
+Asymptotic
+Percentile bootstrap
+Bootstrap
+CP
+AL
+CP
+AL
+CP
+AL
+0.01 0.1
+5.0231
+0.3388811
+0.3384155
+94.38
+2.2566
+99.94
+2.2012
+83.58
+2.2156
+0.5
+5.0178
+0.2880872
+0.2878297
+95.84
+2.2509
+99.94
+2.1278
+86.68
+2.1993
+100
+0.05 0.1
+5.0209
+0.3556721
+0.3553026
+93.98
+2.2711
+98.52
+2.3093
+88.60
+2.3226
+0.5
+5.0231
+0.3388811
+0.3384155
+94.38
+2.2566
+99.94
+2.2012
+83.58
+2.2156
+0.01 0.1
+5.0144
+0.1472563
+0.1470773
+96.20
+1.5963
+99.90
+1.5000
+85.06
+1.5093
+0.5
+5.0127
+0.1373738
+0.1372388
+96.64
+1.5949
+99.86
+1.4395
+84.42
+1.4476
+200
+0.05 0.1
+5.0145
+0.1698002
+0.1696241
+94.84
+1.6037
+98.56
+1.6165
+89.56
+1.6246
+0.5
+5.0144
+0.1472563
+0.1470773
+96.20
+1.5963
+99.90
+1.5000
+85.06
+1.5093
+18
+
+Table 5: Performance measures for estimates of b1
+n
+b1
+b2
+AE
+MSE
+VAR
+Asymptotic
+Percentile bootstrap
+Bootstrap
+CP
+AL
+CP
+AL
+CP
+AL
+0.01 0.1
+0.0108
+0.0000022
+0.0000016
+87.24
+0.0041
+81.66
+0.0052
+92.88
+0.0052
+0.5
+0.0110
+0.0000025
+0.0000016
+84.56
+0.0042
+68.70
+0.0053
+93.96
+0.0054
+100
+0.05 0.1
+0.0513
+0.0000389
+0.0000371
+90.10
+0.0201
+95.96
+0.0245
+93.70
+0.0246
+0.5
+0.0528
+0.0000457
+0.0000376
+89.18
+0.0203
+89.88
+0.0252
+92.70
+0.0253
+0.01 0.1
+0.0105
+0.0000010
+0.0000008
+86.42
+0.0029
+81.74
+0.0035
+93.48
+0.0036
+0.5
+0.0106
+0.0000011
+0.0000008
+84.74
+0.0029
+74.08
+0.0036
+94.56
+0.0036
+200
+0.05 0.1
+0.0508
+0.0000186
+0.0000180
+90.06
+0.0140
+95.14
+0.0169
+93.08
+0.0169
+0.5
+0.0525
+0.0000255
+0.0000193
+86.42
+0.0143
+81.74
+0.0177
+93.48
+0.0178
+Table 6: Performance measures for estimates of b2
+n
+b1
+b2
+AE
+MSE
+VAR
+Asymptotic
+Percentile bootstrap
+Bootstrap
+CP
+AL
+CP
+AL
+CP
+AL
+0.01 0.1
+0.1006
+0.0001691
+0.0001688
+92.70
+0.0464
+96.60
+0.0534
+94.00
+0.0530
+0.5
+0.5067
+0.0038339
+0.0037895
+94.52
+0.2344
+97.12
+0.2675
+95.12
+0.2676
+100
+0.05 0.1
+0.1030
+0.0001794
+0.0001705
+93.76
+0.0472
+95.46
+0.0529
+94.70
+0.0532
+0.5
+0.5028
+0.0042337
+0.0042268
+92.68
+0.2315
+96.34
+0.2650
+94.00
+0.2633
+0.01 0.1
+0.1002
+0.0000725
+0.0000725
+94.22
+0.0325
+96.72
+0.0343
+93.76
+0.0345
+0.5
+0.5030
+0.0017348
+0.0017261
+95.06
+0.1632
+96.52
+0.1665
+93.60
+0.1674
+200
+0.05 0.1
+0.1011
+0.0000780
+0.0000768
+93.84
+0.0326
+96.10
+0.0351
+94.08
+0.0352
+0.5
+0.5010
+0.0018134
+0.0018128
+94.22
+0.1623
+96.72
+0.1717
+93.76
+0.1725
+6.2
+Assessing efficacy of the PLA-based model in fitting data
+from other models
+Now, we examine the robustness of the PLA-based model in the following manner.
+We
+generate load-sharing data from parametric models, and then fit the PLA-based model to
+the data.
+The model fit is then assessed with respect to an integrated measure that is
+suitably defined to reflect the quality of approximation provided by the PLA-based model.
+The measure, which we call the Absolute Integrated Error (AIE), is as follows. For j = 0, 1,
+let S(j)
+TGP(·) and H(j)
+TGP(·) denote the SF and CHF of the lifetimes between j-th and (j + 1)-
+st failures. Also, assume that the estimated SF and CHF based on PLA-based model are
+denoted by �S(j)
+P LA(·) and �H(j)
+P LA(·), respectively. Then the AIE, based on the SF and CHF,
+respectively, are defined as
+AIE(j)
+SF = 1
+R
+R
+�
+k=1
+1
+y(j)
+max − y(j)
+min
+� y(j)
+max
+y(j)
+min
+���S(j)
+TGP(t) − �S(j)
+P CA(t)
+��� dt,
+19
+
+Table 7: AIE based on SF and CHF for Weibull distribution with k = 3, β = 1.
+n
+α
+AIE(0)
+SF
+AIE(1)
+SF
+AIE(0)
+CHF
+AIE(1)
+CHF
+50
+1.0
+0.0379
+0.0291
+0.1503
+0.2981
+1.5
+0.0436
+0.0434
+0.1329
+0.2633
+100
+1.0
+0.0266
+0.0183
+0.1282
+0.2541
+1.5
+0.0326
+0.0301
+0.1231
+0.2440
+Table 8: AIE of the survival and cumulative hazard function of quadratic distribution for
+κ1 = 0.5, ˜κ1 = 2κ1 = 1, ˜κ2 > 2κ2.
+n
+κ2
+˜κ2
+AIE(0)
+SF
+AIE(1)
+SF
+AIE(0)
+CHF
+AIE(1)
+CHF
+50
+0.50
+1.50
+0.0380
+0.0368
+0.1262
+0.2536
+2.00
+0.0380
+0.0389
+0.1261
+0.2555
+0.70
+1.50
+0.0389
+0.0363
+0.1261
+0.2524
+2.00
+0.0388
+0.0383
+0.1258
+0.2539
+100
+0.50
+1.50
+0.0289
+0.0262
+0.1185
+0.2506
+2.00
+0.0289
+0.0281
+0.1178
+0.2575
+0.70
+1.50
+0.0301
+0.0257
+0.1217
+0.2465
+2.00
+0.0299
+0.0274
+0.1206
+0.2528
+AIE(j)
+CHF = 1
+R
+R
+�
+k=1
+1
+y(j)
+max − y(j)
+min
+� y(j)
+max
+y(j)
+min
+���H(j)
+TGP(t) − �H(j)
+P CA(t)
+��� dt,
+where y(j)
+min = min
+�
+y(j)
+1 , y(j)
+2 , . . . , y(j)
+n
+�
+, y(j)
+max = max
+�
+y(j)
+1 , y(j)
+2 , . . . , y(j)
+n
+�
+, j = 0, 1.
+For generating load-sharing data from parametric models, two scenarios are considered:
+(a) Case - 1: It is assumed that the lifetimes of each components of a two-component load-
+sharing system are independent and identically distributed as Weibull distribution with shape
+parameter α and scale parameter β when both the components are working. After the first
+failure, the lifetime of the surviving component is assumed to follow a Weibull distribution
+with same shape parameter α, but a different scale parameter kβ, where k > 2 is to ensure
+the increase of load on the surviving component. For β = 1, k = 3, we take α = 1 and 1.5.
+(b) Case - 2: In the second scenario, the component lifetimes are assumed to be independent
+and identically distributed according to a distribution with quadratic CHF κ1t + κ2t2 when
+both components are working. After the first failure, the lifetime of the surviving component
+is assumed to follow a quadratic CHF with different parameters ˜κ1 and ˜κ2. We take several
+values of the parameters κ1, κ2, ˜κ1, and ˜κ2 ensuring the fact that the CHF increases after
+one component fails in the system.
+The numerical results are reported in Tables 7, 8, and 9. For all cases, it is observed
+that the values of AIE based on SF and CHF are reasonably small, indicating that the
+20
+
+Table 9: AIE of the survival and cumulative hazard function of quadratic distribution for
+˜κ1 > 2κ1, κ2 = 0.5, ˜κ2 = 2κ2 = 1.
+n
+κ1
+˜κ1
+AIE(0)
+SF
+AIE(1)
+SF
+AIE(0)
+CHF
+AIE(1)
+CHF
+50
+0.50
+1.50
+0.0388
+0.0313
+0.1283
+0.2570
+2.00
+0.0397
+0.0307
+0.1309
+0.2672
+0.70
+1.50
+0.0372
+0.0314
+0.1284
+0.2567
+2.00
+0.0377
+0.0304
+0.1301
+0.2644
+100
+0.50
+1.50
+0.0306
+0.0210
+0.1265
+0.2290
+2.00
+0.0319
+0.0206
+0.1325
+0.2285
+0.70
+1.50
+0.0278
+0.0210
+0.1184
+0.2331
+2.00
+0.0285
+0.0198
+0.1227
+0.2271
+PLA-based model provides quite a satisfactory approximation to the data generated from
+different parent populations.
+7
+Concluding Remarks
+In this article, a PLA-based model for the CHF is proposed for data from load-sharing sys-
+tems and then important reliability characteristics such as quantile function, RMT, MTTF,
+and MRT of load-sharing systems are estimated under the proposed model. The principal
+advantages of the model are that it is data-driven, and does not use strong parametric as-
+sumptions for the underlying lifetime variable. Likelihood inference for the proposed model is
+discussed in detail. It is observed that for two-component load-sharing systems, it is possible
+to obtain explicit expressions for the MLEs of parameters of the PLA-based model. Construc-
+tion of confidence intervals using the Fisher information matrix and bootstrap approaches
+are also discussed. Derivations of the important reliability characteristics are provided in
+this setting.
+A Monte Carlo simulation study is performed to examine (a) the performance of the
+methods of inference, and (b) the efficacy of the PLA-based model to fit load-sharing data
+in general.
+It is shown that the PLA-based model performs quite satisfactorily in both
+cases.
+Analysis of data pertaining to components lifetimes of a two-motor load-sharing
+system is provided as an illustration. It is illustrated that the PLA-based model is supe-
+rior to the models that have been considered for this data in the literature of load-sharing
+systems. In summary, in this paper, an efficient PLA-based modelling framework using mini-
+mal assumptions for load-sharing systems is discussed, and estimates of important reliability
+characteristics for load-sharing systems in this setting are developed.
+21
+
+Funding information
+• The research of Ayon Ganguly is supported by the Mathematical Research Impact Cen-
+tric Support (File no. MTR/2017/000700) from the Science and Engineering Research
+Board, Department of Science and Technology, Government of India.
+• The research of Debanjan Mitra is supported by the Mathematical Research Impact
+Centric Support (File no. MTR/2021/000533) from the Science and Engineering Re-
+search Board, Department of Science and Technology, Government of India.
+References
+[1] S.V. Amari and R. Bergman. Reliability analysis of k-out-of-n load-sharing systems. In
+Annual Reliability and Maintainability Symposium, pages 440–445, 2008.
+[2] G Asha, AV Raja, and N. Ravishanker. Reliability modelling incorporating load share
+and frailty. Applied Stochastic Models in Business and Industry, 34:206–223, 2018.
+[3] N. Balakrishnan, M. Koutras, F. Milienos, and S. Pal. Piecewise linear approximations
+for cure rate models. Methodology and Computing in Applied Probability, 18(4):937–966,
+2016.
+[4] C.D. Bernard. Time dependence of mechanical breakdown in bundles of fibers. I. con-
+stant total load. Journal of Applied Physics, 28(9):1058–1064, 1957.
+[5] C.D. Bernard. Statistics and time dependence of mechanical breakdown in fibers. Jour-
+nal of Applied Physics, 29(6):968–983, 1958.
+[6] Z. W. Birnbaum and S. C. Saunders. A statistical model for life-length of materials.
+Journal of the American Statistical Association, 53(281):151–160, 1958.
+[7] B. Brown, B. Liu, S. McIntyre, and M. Revie.
+Reliability analysis of load-sharing
+systems with spatial dependence and proximity effects.
+Reliability Engineering and
+System Safety, 2022.
+[8] H. Che, S. Zeng, K. Li, and J. Guo. Reliability analysis of load-sharing man-machine
+systems subject to machine degradation, human errors, and random shocks. Reliability
+Engineering and System Safety, 2022.
+[9] H.E. Daniels. The statistical theory of the strength of bundles of threads I. Proceedings
+of the Royal Society of London, 83:405–435, 1945.
+[10] JV Deshpande, I. Dewan, and UV Naik-Nimbalkar. A family of distributions to model
+load sharing systems.
+Journal of Statistical Planning and Inference, 140:1441–1451,
+2010.
+[11] M Franco, JM Vivo, and D Kundu. A generalized Freund bivariate model for a two-
+component load sharing system. Reliability Engineering and System Safety, 2020.
+22
+
+[12] JE Freund. A bivariate extension of the exponential distribution. Journal of American
+Statistical Association, 56:971–976, 1961.
+[13] D. G. Harlow and S. L. Phoenix.
+The chain-of-bundles probability model for the
+strength of fibrous materials I: analysis and conjectures. Journal of composite materials,
+12(2):195–214, 1978.
+[14] D. G. Harlow and S. L. Phoenix. Probability distributions for the strength of fibrous
+materials under local load sharing I: two-level failure and edge effects. Advances in
+Applied Probability, pages 68–94, 1982.
+[15] M. Hollander and E. A. Pena. Dynamic reliability models with conditional proportional
+hazards. Lifetime Data Analysis, 1(4):377–401, 1995.
+[16] H. Kim and P.H. Kvam. Reliability estimation based on system data with an unknown
+load share rule. Lifetime Data Analysis, 10:83–94, 2004.
+[17] Y. Kong and Z. S. Ye. A cumulative-exposure-based algorithm for failure data from a
+load-sharing system. IEEE Transactions on Reliability, 65(2):1001–1012, 2016.
+[18] H. P. Kvam and J. C. Lu. Load-sharing models. Math and Computer Science Faculty
+Publications, 2008.
+[19] S. Lee, S. Durham, and J. Lynch. On the calculation of the reliability of general load
+sharing system. Journal of Applied Probability, 32:777–792, 1995.
+[20] H.H. Lin, K.H. Chen, and R.T. Wang. A multivariant exponential shared-load model.
+IEEE Transactions on Reliability, 42(1):165–171, 1993.
+[21] C. Luo, L. Chen, and A. Xu. Modelling and estimation of system reliability under dy-
+namic operating environments and lifetime ordering constraints. Reliability Engineering
+and System Safety, 2022.
+[22] J D Lynch. On the joint distribution of component failures for monotone load-sharing
+systems. Journal of Statistical Planning and Inference, 78:13–21, 1999.
+[23] A. Mettas and P. Vassiliou. Application of quantitative accelerated life models on load
+sharing redundancy. In Annual Symposium Reliability and Maintainability, 2004-RAMS,
+pages 293–296. IEEE, 2004.
+[24] R. Mohammad, A. Kalam, and S.V. Amari. Reliability evaluation of phased-mission
+systems with load-sharing components. In Proceedings Annual Reliability and Main-
+tainability Symposium, pages 1–6. IEEE, 2012.
+[25] C. H. Muller and R. Meyer. Inference of intensity-based models for load-sharing systems
+with damage accumulation. IEEE Transactions on Reliability, 71:539–554, 2012.
+[26] E. Nezakati and M. Ramzakh. Reliability analysis of a load sharing k-out-of-n:f degra-
+dation system with dependent competing failures. Reliability Engineering and System
+Safety, 2020.
+23
+
+[27] C. Park. Parameter estimation for the reliability of load-sharing systems. IIE Transac-
+tions, 42:753–765, 2010.
+[28] C. Park. Parameter estimation from load-sharing system data using the expectation-
+maximization algorithm. IIE Transactions, 45:147–163, 2013.
+[29] B. W. Rosen. Tensile failure of fibrous composites. AIAA journal, 2(11):1985–1991,
+1964.
+[30] S.M. Ross. A model in which component failure rates depend on the working set. Naval
+research logistics quarterly, 31(2):297–300, 1984.
+[31] Z. Schechner. A load-sharing model: The linear breakdown rule. Naval research logistics
+quarterly, 31(1):137–144, 1984.
+[32] J. Shao and L.R. Lamberson. Modeling a shared-load k-out-of-n: G system. IEEE
+Transactions on Reliability, 40(2):205–209, 1991.
+[33] A.V. Suprasad, M.B. Krishna, and P. Hoang. Tampered failure rate load-sharing sys-
+tems: Status and perspectives. In Handbook of performability engineering, pages 291–
+308. Springer, 2008.
+[34] S S Sutar and U V Naik-Nimbalkar. Accelerated failure time models for load sharing
+systems. IEEE Transactions on Reliability, 63:706–714, 2014.
+[35] J. Zhang, Y. Zhao, and X. Ma. Reliability modeling methods for load-sharing k-out-of-
+n system subject to discrete external load. Reliability Engineering and System Safety,
+2020.
+[36] X. Zhao, B. Liu, and Y. Liu. Reliability modeling and analysis of load-sharing systems
+with continuously degrading components. IEEE Transactions on Reliability, 67:1096–
+1110, 2018.
+Appendix A: Calculation of Fisher information ma-
+trix for two-component load sharing systems
+For calculating I(θ), the required expectations are E
+�
+N(0)
+1
+�
+, E
+�
+N(0)
+2
+�
+, E
+�
+N(1)
+1
+�
+, E
+�
+N(1)
+2
+�
+,
+E
+
+
+
+�
+i∈I(1)
+1
+Y (1)
+i
+
+
+ and E
+
+
+
+�
+i∈I(1)
+2
+Y (1)
+i
+
+
+.
+Note that
+N(0)
+k
+∼ Bin(n, p(0)
+k ),
+N(1)
+k
+∼ Bin(n, p(1)
+k ),
+with
+p(0)
+k
+= P
+�
+Y (0)
+i
+∈ [τ (0)
+k−1, τ (0)
+k )
+�
+,
+p(1)
+k
+= P
+�
+Y (1)
+i
+∈ [τ (1)
+k−1, τ (1)
+k )
+�
+,
+k = 1, 2.
+24
+
+In case of a two-component load-sharing system, PDF of Y (j)
+i
+, j = 1, 2, is given by
+gY (j)
+i (y) = (2 − j)λ(j)(y)e−(2−j)
+� y
+0 λ(j)(u)du.
+Hence,
+p(0)
+1
+=
+� τ (0)
+1
+0
+gY (0)
+i
+(y)dy = 1 − e−2b1τ (0)
+1 ,
+p(1)
+1
+=
+� τ (1)
+1
+0
+gY (1)
+i
+(y)dy = 1 − e−γ1b1τ (1)
+1 .
+Then, p(0)
+2
+= 1 − p(0)
+1
+= e−2b1τ (0)
+1
+and p(1)
+2
+= 1 − p(1)
+1
+= e−γ1b1τ (1)
+1 . Therefore,
+E(N(0)
+1 ) = 1−e−2b1τ (0)
+1 ,
+E(N(0)
+2 ) = e−2b1τ (0)
+1 ,
+E(N(1)
+1 ) = 1−e−γ1b1τ (1)
+1 ,
+E(N(1)
+2 ) = e−γ1b1τ (1)
+1 .
+Now,
+E
+
+
+
+�
+i∈I(1)
+1
+Y (1)
+i
+
+
+ = E
+
+
+E
+
+
+
+�
+i∈I(1)
+1
+Y (1)
+i
+|N(1)
+1
+= n(1)
+1
+
+
+
+
+
+ .
+For i ∈ I(1)
+1 , Y (1)
+i
+follows a right truncated exponential distribution with PDF
+γ1b1e−γ1b1y
+1−e−γ1b1τ(1)
+1
+for
+0 < y < τ (1)
+1 . Hence, for i ∈ I(1)
+1 ,
+E
+�
+Y (1)
+i
+�
+=
+� τ (1)
+1
+0
+y γ1b1e−γ1b1y
+1 − e−γ1b1τ (1)
+1
+dy =
+1
+γ1b1
+�
+1 − (1 + γ1b1τ (1)
+1 )e−γ1b1τ (1)
+1
+1 − e−γ1b1τ (1)
+1
+�
+.
+Therefore,
+E
+
+
+
+�
+i∈I(1)
+1
+Y (1)
+i
+
+
+ =
+1
+γ1b1
+�
+1 − (1 + γ1b1τ (1)
+1 )e−γ1b1τ (1)
+1
+1 − e−γ1b1τ (1)
+1
+�
+E(N(1)
+1 )
+=
+1
+γ1b1
+�
+1 − (1 + γ1b1τ (1)
+1 )e−γ1b1τ (1)
+1
+1 − e−γ1b1τ (1)
+1
+� �
+1 − e−γ1b1τ (1)
+1
+�
+=
+1
+γ1b1
+�
+1 − (1 + γ1b1τ (1)
+1 )e−γ1b1τ (1)
+1
+�
+.
+Similarly,
+E
+
+
+
+�
+i∈I(1)
+2
+Y (1)
+i
+
+
+ = E
+
+
+E
+
+
+��
+�
+i∈I(1)
+2
+Y (1)
+i
+|N(1)
+2
+= n(1)
+2
+
+
+
+
+
+ .
+For i ∈ I(1)
+2 , Y (1)
+i
+follows a left truncated exponential distribution with PDF γ1b2e−γ1b2y
+e−γ1b2τ(1)
+1
+for
+y > τ (1)
+1 . Hence,
+E
+�
+Y (1)
+i
+�
+=
+� ∞
+τ (1)
+1
+yγ1b2e−γ1b2y
+e−γ1b2τ (1)
+1
+dy =
+1
+γ1b2
++ τ (1)
+1 .
+Therefore,
+E
+
+
+
+�
+i∈I(1)
+2
+Y (1)
+i
+
+
+ =
+� 1
+γ1b2
++ τ (1)
+1
+�
+E(N(1)
+2 ) =
+� 1
+γ1b2
++ τ ′
+1
+�
+e−γ1b1τ (1)
+1 .
+25
+
+Appendix B: Calculations of some important relia-
+bility characteristics
+Derivation of the quantile function:
+Denote p = G(j)(y) for y ∈
+�
+τ (j)
+k−1, τ (j)
+k
+�
+; then, y = η(p) for p ∈
+�
+G(j)(τ (j)
+k−1), G(j)(τ (j)
+k )
+�
+,
+k = 1, 2, . . . , N. Now,
+p = 1 − e
+−(J−j)γj
+��k−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
++bk
+�
+y−τ (j)
+k−1
+��
+=⇒ bk
+�
+y − τ (j)
+k−1
+�
+= −log(1 − p)
+(J − j)γj
+−
+k−1
+�
+ℓ=1
+bℓ
+�
+τ (j)
+ℓ
+− τ (j)
+ℓ−1
+�
+=⇒ y = τ (j)
+k−1 − log(1 − p)
+(J − j)γjbk
+− 1
+bk
+k−1
+�
+ℓ=1
+bℓ
+�
+τ (j)
+ℓ
+− τ (j)
+ℓ−1
+�
+, if p ∈
+�
+G(j)(τ (j)
+k−1), G(j)(τ (j)
+k )
+�
+,
+k = 1, 2, . . . , N.
+If y ∈
+�
+τ (j)
+N , ∞
+�
+, then y = η(p) for p ∈
+�
+G(j)(τ (j)
+N ), 1
+�
+.
+Therefore,
+p = 1 − e
+−(J−j)γj
+��N−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
++bN
+�
+y−τ (j)
+N−1
+��
+=⇒ y = τ (j)
+N−1 − log(1 − p)
+(J − j)γjbN
+− 1
+bN
+N−1
+�
+ℓ=1
+bℓ
+�
+τ (j)
+ℓ
+− τ (j)
+ℓ−1
+�
+, if p ∈
+�
+G(j)(τ (j)
+N ), 1
+�
+.
+Derivation of MTTF:
+MTTF of the system lifetime T is given by E(T) = E
+�J−1
+�
+j=0
+Y (j)
+�
+=
+J−1
+�
+j=0
+E(Y (j)), where
+E(Y (j)) =
+� ∞
+0
+P(Y (j) > y)dy =
+� τ (j)
+N−1
+0
+e−(J−j)Λ(j)(y)dy+
+� ∞
+τ (j)
+N−1
+e−(J−j)Λ(j)(y)dy = I1+I2 (say).
+Here,
+I1 =
+� τ (j)
+N−1
+0
+e
+−(J−j)γj
+�N
+k=1
+��k−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
++bk
+�
+y−τ (j)
+k−1
+��
+1
+[τ(0)
+k−1, τ(0)
+k
+)(y)
+dy
+=
+N−1
+�
+s=1
+� τ (j)
+s
+τ (j)
+s−1
+e
+−(J−j)γj
+��s−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
++bs
+�
+y−τ (j)
+s−1
+��
+dy
+=
+N−1
+�
+s=1
+�
+e
+−(J−j)γj
+�s−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+� � τ (j)
+s
+τ (j)
+s−1
+e
+−(J−j)γjbs
+�
+y−τ (j)
+s−1
+�
+dy
+�
+=
+N−1
+�
+s=1
+
+
+e
+−(J−j)γj
+�s−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+� 
+1 − e
+−(J−j)γjbs
+�
+τ (j)
+s
+−τ (j)
+s−1
+�
+(J − j)γjbs
+
+
+
+
+
+26
+
+=
+N−1
+�
+s=1
+
+
+
+e
+−(J−j)γj
+�s−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
+− e
+−(J−j)γj
+�s
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
+(J − j)γjbs
+
+
+
+and
+I2 =
+� ∞
+τ (j)
+N−1
+e
+−(J−j)γj
+�N
+k=1
+��k−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
++bk
+�
+y−τ (j)
+k−1
+��
+1
+[τ(0)
+k−1, τ(0)
+k
+)(y)
+dy
+=
+� ∞
+τ (j)
+N−1
+e
+−(J−j)γj
+��N−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
++bN
+�
+y−τ (j)
+N−1
+��
+dy
+= e
+−(J−j)γj
+�N−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+� � ∞
+τ (j)
+N−1
+e
+−(J−j)γjbN
+�
+y−τ (j)
+N−1
+�
+dy
+= e
+−(J−j)γj
+�N−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+� �
+1
+(J − j)γjbN
+�
+= e
+−(J−j)γj
+�N−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
+(J − j)γjbN
+.
+Therefore,
+E(Y (j)) =
+N
+�
+s=1
+
+
+
+e
+−(J−j)γj
+�s−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
+− e
+−(J−j)γj
+�s
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
+(J − j)γjbs
+
+
+ .
+From here, the results follows immediately.
+Derivation of moment generating function of system lifetime:
+Note that the system lifetime MGF of T is T =
+J−1
+�
+j=0
+Y (j), where Y (j)’s are independent
+for j = 0, 1, . . . , (J − 1). Therefore, the MGF of T is φT(t) =
+J−1
+�
+j=0
+φY (j)(t). Now,
+φY (j)(t)
+= E(etY (j)) =
+� ∞
+0
+etygY (j)(y)dy
+=
+� τ (j)
+N−1
+0
+ety(J − j)λ(j)(y)e−(J−j)Λ(j)(y)dy +
+� ∞
+τ (j)
+N−1
+ety(J − j)λ(j)(y)e−(J−j)Λ(j)(y)dy
+= I1 + I2 (say) ,
+where gY (j)(y) = (J − j)λ(j)(y)e−(J−j)Λ(j)(y). For t ∈ R,
+I1 =
+� τ (j)
+N−1
+0
+ety(J − j)γj
+N
+�
+k=1
+bk1[τ (j)
+k−1, τ (j)
+k
+) (y) e
+−(J−j)γj
+�N
+k=1
+��k−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
++bk
+�
+y−τ (j)
+k−1
+��
+dy
+=
+N−1
+�
+s=1
+(J − j)bsγj
+� τ (j)
+s
+τ (j)
+s−1
+e
+−
+�
+(J−j)γj
+��s−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
++bs
+�
+y−τ (j)
+s−1
+��
+−ty
+�
+dy
+27
+
+=
+N−1
+�
+s=1
+�
+(J − j)bsγje
+−(J−j)γj
+�s−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+� � τ (j)
+s
+τ (j)
+s−1
+e
+−
+�
+(J−j)γjbs
+�
+y−τ (j)
+s−1
+�
+−ty
+�
+dy
+�
+=
+N−1
+�
+s=1
+
+
+(J − j)bsγje
+−(J−j)γj
+�s−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+� 
+etτ (j)
+s−1 − e
+−
+�
+(J−j)γjbs
+�
+τ (j)
+s
+−τ (j)
+s−1
+�
+−tτ (j)
+s
+�
+(J − j)γjbs − t
+
+
+
+
+
+=
+N−1
+�
+s=1
+(J − j)bsγj
+(J − j)bsγj − t
+�
+e
+−
+�
+(J−j)γj
+�s−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
+−tτ (j)
+s−1
+�
+−e
+−
+�
+(J−j)γj
+�s
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
+−tτ (j)
+s
+��
+.
+For t < (J − j)γjbN,
+I2 =
+� ∞
+τ (j)
+N−1
+ety(J − j)γj
+N
+�
+k=1
+bk1[τ (j)
+k−1, τ (j)
+k
+) (y) e
+−(J−j)γj
+�N
+k=1
+��k−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
++bk
+�
+y−τ (j)
+k−1
+��
+dy
+= (J − j)bNγj
+� ∞
+τ (j)
+N−1
+e
+ty−(J−j)γj
+��N−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
++bN
+�
+y−τ (j)
+N−1
+��
+dy
+= (J − j)bNγje
+−(J−j)γj
+�N−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+� � ∞
+τ (j)
+N−1
+e
+−
+�
+(J−j)γjbN
+�
+y−τ (j)
+N−1
+�
+−ty
+�
+dy
+= (J − j)bNγje
+−(J−j)γj
+�N−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+� �
+etτ (j)
+N−1
+(J − j)γjbN − t
+�
+= (J − j)bNγj · e
+tτ (j)
+N−1−(J−j)γj
+�N−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
+(J − j)bNγj − t
+.
+Therefore, for t < (J − j)γjbN,
+φY (j)(t) =
+N
+�
+s=1
+(J − j)bsγj
+(J − j)bsγj − t
+�
+e
+−
+�
+(J−j)γj
+�s−1
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
+−tτ (j)
+s−1
+�
+−e
+−
+�
+(J−j)γj
+�s
+ℓ=1 bℓ
+�
+τ (j)
+ℓ
+−τ (j)
+ℓ−1
+�
+−tτ (j)
+s
+��
+.
+From here the result follows immediately.
+28
+

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+1
+Synchronization of Josephson junctions in series
+array
+Abhijit Bhattacharyya
+Abstract—Multi-qubit quantum processors coupled to net-
+working provides the state-of-the-art quantum computing plat-
+form. However, each qubit has unique eigenfrequency even
+though fabricated in the same process. To continue quantum
+gate operations besides the detection and correction of errors it
+is required that the qubits must be synchronized in the same
+frequency. This study uses Kuramoto model which is a link
+between statistical mean-field technique and non-linear dynamics
+to synchronize the qubits applying small noise in the system. This
+noise could be any externally applied noise function or just noise
+from the difference of frequencies of qubits. The Kuramoto model
+tunes the coupled oscillators adjusting the coupling strength
+between the oscillators to evolve from the state of incoherence to
+the synchronized state.
+Index Terms—Josephson junction, Kuramoto Model, synchro-
+nization, oscillators
+I. INTRODUCTION
+J
+Osephson junction controls the flow of magnetic flux
+quanta through frequency and voltage. Modern instruments
+require measurement of voltage with a reproducible capability
+exceeding the uncertainty of realization of the SI volt (cur-
+rently 0.4 parts on 106). Before 1972, SI volt was represented
+by using carefully stabilised Weston cell banks [1]. Drift and
+transportability problems with these electrochemical artifact
+standards limited the uniformity of voltage standards to about
+1 part in 106. These uniformity was drastically improved by
+the usage of Josephson junction [1].
+Josephson equation for supercurrent through a supercon-
+ducting tunnel junction, called as DC Josephson Effect, is
+defined as [2]–[4]
+I = Ic sin
+�4πe
+h
+�
+V dt
+�
+,
+(1)
+where Ic is critical current, h is Planck’s constant and e is
+electron charge. When a dc voltage is applied in equation
+(1), the phase will vary linearly with time and current will
+be sinusoidal with amplitude Ic and frequency fJ = 2eV/h.
+The magnetic flux threading a superconducting loop or hole is
+quantized [5]. The superconducting magnetic flux quantum Φ0
+= h/(2e) is 2.0678×10−15 Wb. The inverse of flux quantum
+1/Φ0 is called Josephson constant KJ defined as 2e/h has a
+value of 483.597 GHz/mV . During each oscillation, a single
+quantum of magnetic flux h/(2e) passes through the junction
+which is very difficult to measure. However, if an alternating
+current with frequency f is applied across the junction, there
+Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400
+094, India
+[email protected]; [email protected]
+is a range of bias current for which flow of flux quanta
+will phaselock to the applied frequency. Under this phase
+locked condition, the average voltage across the junction is
+precisely (h/2e)f. This effect is known as ac Josephson effect
+observed as a constant voltage step at V =(h/2e)f in the I −V
+characteristic curve. This means a Josephson junction can act
+as a “Voltage to frequency converter”. It is also possible for
+the junction to phaselock with the harmonics of fJ resulting
+in a series of steps at voltages V =nf(h/2e), where n is an
+integer denoting step number. This accuracy was limited to
+the condition that a Josephson voltage higher than 10mV was
+never used [6]. Therefore, if one obtain Josephson voltage
+over 100 mV , the accuracy could be remarkably improved
+besides the ability to vary the Jsephson voltage with the
+frequency and step number could be utilized as potentiometer.
+Series array of Josephson junction [6] has been effectively
+used in development of a potentiometer system to produce (1-
+10)V [1], [6] with uncertainty about 2.5 × 10−9 [6]. Larger
+series arrays were initially considered as impractical due to
+junction nonuniformity. The nonuniformity demanded each
+junction to be biased separately. In 1977, Levinsen et al [7]
+stated the important of the parameter βc=4πeIcR2C/h in
+determining the characteristics of RF induced Josephson steps.
+This βc is measure of the damping of Josephson oscillations
+by the junction shunting resistance R.
+The Josephson junction is also a natural choice for sub-
+millimeter local oscillator [8], [9] as one may capitalize the
+voltage controlled oscillator property. However, the disadvan-
+tage, in this case lies in very low power output. The Josephson
+constant clearly indicates that with dc voltage bias at 1 mV
+at 483.6 GHz, the junction may accept 100 µA current
+keeping under the limitation of Ic which limits the maximum
+output RF power at about 100 nW. This requirement indicates
+series array of junctions with a common current bias demands
+keeping all the junctions in phase.
+However, the issue with series array of junctions operated
+with common current bias arises with nonuniformity of each
+junction due to fabrication processes [1], [6]. When junctions
+are connected in series, the system behaves as a coupled os-
+cillator and understanding the periodic solutions is important.
+Two special types of periodic solutions exist [10], namely, in-
+phase state and splay state.
+An in-phase state with period T is a state where all the
+oscillators always possess the same phase at all times, i.e.
+θi(t) = θj(t), and θi(t + T) = θi(t) + 2π.
+The splay-phase or anti-phase or rotating wave state with
+period T is a solution where the oscillators can be labeled
+so that θi(t) = Θ(t + jT/N) for all j for some function
+arXiv:2301.03787v1  [quant-ph]  10 Jan 2023
+
+2
+Θ(t + T) = Θ(t) + 2π. Thus, this state indicates that all the
+oscillators have the same waveform Θ(t) except for a shift in
+time. As per [10], one may imagine that each oscillator “fires”
+when it reaches a certain angle. For an in-phase solution, all
+the oscillators fire simultaneously at every instant T, while
+splay-phase state has a single oscillator firing every T/N
+instant. Therefore, for splay-phase state, oscillators nearly
+coincide or coincide when ˙θ is small where as for large
+values, oscillators are not coherent. The definition of splay-
+phase does not imply that the phases of the oscillators are
+equi-spaced around he circle. The oscillators bunch up for
+smaller ˙θ while spread out for large ˙θ. Therefore, splay-state
+shows non-uniformity in the distribution of oscillators as they
+are coherent for smaller ˙θ. It has been shown that [10], [11],
+the non-uniformity can be removed by determining a set of
+“natural” angles ϕj, so that the splay-phase solution satisfies
+ϕj(t) = 2πj/N + 2πt/T + const. The “natural” angle based
+dynamical system gets locked. This provides an idea of phase-
+locking N oscillators, like N Josephson junctions, having
+eigenfrequencies with smaller spread which may get locked
+to some resonating frequency.
+Kuramoto model provides an exactly solvable mean-field
+model of coupled nonlinear oscillators connecting a large of
+them having distributed natural frequencies. This model links
+mean-field techniques and nonlinear dynamics together and
+also provides precise technique to tune the synchronization.
+Section II discusses the theory of the Kuramoto model,
+Section III discusses on the reduction of the equations for
+the Josephson junctions connected in series to the Kuramoto
+Model framework and section IV discusses on the numerical
+analysis of the results for the generalised Kuramoto Model
+theory and Kuramoto model for Josephson junctions.
+II. KURAMOTO MODEL
+Let us consider a system of N globally coupled differential
+equations with the stable limits cycles. Yoshiki Kuramoto de-
+veloped a mathematical model for coupled oscillators (n ⩾ 2)
+to synchronize which is known as “Kuramoto model” [12].
+In this model, each jth oscillator is represented by a phase
+variable θj(t), possessing its own natural frequency ωj ∈ R.
+The dynamics of the system of coupled N oscillators becomes
+˙θj(t) = ωj +
+N
+�
+i=1,j̸=i
+Kji sin (θj(t) − θi(t)) , j ∈ {1, . . . , N} ,
+(2)
+where Kji is coupling coefficient of the jth oscillator with all
+other oscillators in the system. Kuramoto assumed mean field
+coupling among phase oscillators such that Kji ≈ K/N ⩾ 0
+where K is mean coupling strength which changes (2) as
+˙θj(t) = ωj + K
+N
+N
+�
+i=1,j̸=i
+sin (θj(t) − θi(t)) , j ∈ {1, . . . , N} ,
+(3)
+where, K ⩾ 0 is the coupling strength among the oscillators
+whose frequencies are distributed with a probability density
+g(ω). One may find a suitable rotating frame like θj → θj −
+Ωt transforming the system so that natural frequencies of the
+oscillators may have zero mean, where Ω is the first moment of
+the distribution function of natural frequencies g(ω). Therfore,
+one may consider the normal form calculation for the system
+such that one may define the system of equations as
+˙θj = fj(θj) + K
+N
+N
+�
+i=1,i̸=j
+g (θi, θj) , θj ∈ Rd, j = 1, . . . , N,
+(4)
+where, function fj(θj) are eigenfrequencies defining the nat-
+ural dynamics in the system. Here coupling parameter K
+has been added with coupling strength K/N, g is the phase
+response curve defining the interaction of the system. In the
+following section, we are not discussing with the stability of
+the dynamical system, bifurcation etc while one may consult
+other references like [13].
+In the original paper [12], Kuramoto considered the proba-
+bility density g(ω) to be uni-modal and symmetric centered at
+mean frequency ω so that, without loss of generality, one can
+assume that the mean frequency ω = 0 after a shift leading to
+g(ω) = g(−ω) for the even and symmetric distribution g(ω).
+To diagnose the feasibility of synchronization, Kuramoto
+introduced the order parameter R(t) projecting the oscillation
+on unit circle where R(t) : 0 ⩽ R(t) ⩽ 1 is a measure of the
+coherence of oscillators as
+R(t)eȷψ(t) = 1
+N
+N
+�
+i=1
+eȷθi(t),
+(5)
+where R(t) = 0 for asynchronised oscillators,
+and R(t) > 0 for synchronization.
+The quantity ψ(t) refers to average phase of all the oscillators
+at an instant t. Physically, this order parameter R(t) is the
+centroid of a set of N points eȷθi distributed in the unit circle in
+the complex plane at the instant t. If the phases are uniformly
+spread in the range [−π, π], then R → 0 indicates that the
+oscillators are not synchronized. All the oscillators become
+synchronized with the same average phase ψ(t) for R(t) ≈ 1.
+If the dynamics show stability of R(t) at 1, then the oscillators
+are synchronized and phaselocked. Eq. (3) may be re-written
+by multiplying Ke−ȷθj on both sides of (5) and equating the
+imaginary parts of the both sides to reduce (3) to
+˙θj(t) = ωj + KR(t) sin (ψ(t) − θj(t)) = vj(θ, ω, t) (say).
+(6)
+Here, vj(θ, ω, t) is the angular velocity of a given oscillator
+with phase θ and natural frequency ω at the instant t. The
+equation (6) reveals that the interaction is set through R(t)
+and ψ(t) while the phases θj seem to evolve independently
+from each other. Also the effective coupling is proportional to
+the order parameter R(t) creating a feedback relation between
+coupling and synchronization. In the limit K → 0, (6) reduces
+to
+θj(t) ≈ ωjt + θ(0),
+(7)
+where, θj(0) denotes initial phase of the jth oscillator and
+(7) suggests that each oscillator oscillates with own natural
+frequencies in the absence of coupling.
+
+3
+In the limit of infinite number of oscillators having a
+distribution of frequency, phase over time, Kuramoto de-
+scribed the system by the probability density ρ (θ, ω, t) so
+that ρ (θ, ω, t) dθ gives the fraction of oscillators with phase
+between θ(t) and θ(t) + dθ(t) at the instant t for a given
+natural frequency ω. Since ρ is non-negative and 2π-periodic
+in θ satisfying the normalization condition
+� π
+−π
+ρ (θ, ω, t) dθ = 1.
+(8)
+The probability density function g must also obey the equation
+of continuity using the angular velocity v(θ, ω, t) as
+∂ρ(θ, ω, t)
+∂t
++ ∂
+∂θ {ρ(θ, ω, t).v} = 0,
+∂ρ(θ, ω, t)
+∂t
++
+∂
+∂θ [ρ(θ, ω, t) {ω + KR(t) sin (ψ(t) − θ(t))}] = 0.(9)
+In the limit R(t) → 0, the dynamics provides stationary
+solution for ρ(θ, ω, t) = 1/(2π).
+In the continuum limit, (5) gets re-defined by the order
+parameter R(t) and the average phase ψ(t) incorporating
+previously described frequency distribution as
+R(t)eȷψ(t) =
+� π
+−π
+� ∞
+−∞
+eȷθρ (θ, ω, t) g(ω)dωdθ.
+(10)
+In the strong coupling limit where K → ∞ indicate K ≫
+Kc where Kc is critical coupling strength and (6) reduces to
+system having phases reduced to the average phase as θ(t) =
+ωt + θ(0) = ψ(t).
+From (6), if oscillators get into phaselocked condition,
+vi(t) → 0 which provides
+ωj = KR(t) sin (θj(t) − ψ(t)) , −π
+2 ⩽ (θj(t) − ψ(t)) ⩽ π
+2 .
+(11)
+From (9), partially synchronized state leading to a locked
+system can be described as
+∂
+∂t(ρ(θ, ω, t)) = 0 which also
+means
+∂
+∂θ (ρ(θ, ω, t).v(t)) = 0. Eq. (11), in this partial
+synchronized state for vj(t) → 0 and
+∂
+∂t (ρ(θ, ω, t)) = 0,
+reduces to
+ω
+KR(t) → sin(θj(t) − ψ(t)),
+which means
+ρ(θ, ω, t) = δ
+�
+θj(t) − ψ(t) − sin−1
+�
+ω
+KR(t)
+��
+H(cos θ),
+(12)
+such that |ω| ⩽ KR(t) and
+H(x) =
+1,
+x > 0,
+0,
+elsewhere..
+(13)
+Now, for the other condition
+∂
+∂θ (ρ(θ, ω, t)v(t)) = 0 using
+(6),
+ρ(θ, ω, t)v(t) = C(say) = constant,
+or,
+ρ(θ, ω, t) =
+C
+|ω + KR(t) sin(θj(t) − ψ(t))|,
+|ω| � KR(t).
+(14)
+The constant C can be determined from (8) such that (14)
+reduces to
+ρ(θ, ω, t) =
+�
+ω2 − K2R2(t)
+2π|ω − KR(t) sin(θj(t) − ψ(t))|,
+|ω| � KR(t).
+(15)
+Therefore, the constraint on the probablity density of the
+oscillators may be
+ρ(θ, ω, t) = δ
+�
+θj(t) − ψ(t) − sin−1
+�
+ω
+KR(t)
+��
+H(cos θ),
+for |ω| ⩽ KR(t)
+(16)
+and
+ρ(θ, ω, t) =
+�
+ω2 − K2R2(t)
+2π|ω − KR(t) sin(θj(t) − ψ(t))|, elsewhere .
+(17)
+Here δ is the Dirac delta function. Eqs. (16) and (17) indicate
+that partial synchronized states are divided into two groups
+depending on the natural frequencies. Oscillators having con-
+straint |ω| ⩽ KR(t) operate in mean-field resulting in locking
+in a common average phase ψ(t) = Ωt where Ω is the average
+frequency of the ensemble of the oscillators in this regime. On
+the other side, the second group of oscillators having constraint
+|ω| > KR(t) rotate incoherently which are called as drifting
+oscillators.
+Inserting (16) and (17) in (10) we get
+R(t)
+=
+� π
+−π
+� ∞
+−∞
+eȷ(φ(t)−ψ(t))
+δ
+�
+θ(t) − ψ(t) − sin−1
+�
+ω
+KR(t)
+��
+g(ω)dθdω
++
+� π
+−π
+�
+|ω|⩽KR(t)
+�
+ω2 − K2R2(t)g(ω)dθdω
+2π|ω − KR(t) sin(θ(t) − ψ(t))|.
+(18)
+Since g(ω) is even and symmetric, g(ω) = g(−ω) and
+ρ(θ + π, −ω) = ρ(θ, ω). The even function condition makes
+the second term of (18) vanish which physically means all
+the incoherent oscillator solutions vanish resulting in order
+parameter R(t) only for coherent synchronized oscillators that
+reform as
+R(t)
+=
+�
+|ω|⩽KR(t)
+cos
+�
+sin−1
+�
+ω
+KR(t)
+��
+g(ω)dωdθ,
+=
+�
+π
+2
+− π
+2
+cos θg (KR(t) sin θ) KR(t) cos θdθ,
+=
+KR(t)
+�
+π
+2
+− π
+2
+cos2 θg (KR(t) sin θ) dθ.
+(19)
+Here, (19) shows a trivial solution for which order parameter
+R(t) = 0 which actually shows incoherence as discussed
+earlier for ρ (θ, ω, t) = 1/(2π). However, (19) also suggests
+1 = K
+�
+π
+2
+− π
+2
+cos2 θ g (KR(t) sin θ) dθ.
+
+4
+Setting R(t) = 0, considering K = Kc - the critical coupling
+strength we get,
+Kc =
+2
+πg(0),
+(20)
+that triggers the synchronization. In general, expanding the
+right hand side of (19) in terms of powers of KR(t) and
+considering g′′(0) < 0 the order parameter can be written as
+R(t) ∼
+�
+−8 (K − Kc)
+K3c g′′(0) ,
+(21)
+which shows that near the transition point, the order parameter
+[12], [14] yields the form R(t) ∼ (K − Kc)β with β = 1/2
+like second order phase transition.
+The Kuramoto model can be generalized for a complex
+network including the connectivity parameter in the coupling
+term as
+˙θj = ωj +
+N
+�
+i=1
+KjiAji sin(θj − θi),
+(22)
+where, Kji is the coupling strength between nodes j and i.
+Aji is the element of the adjacency matrix A (Aji = 1 if there
+is a connection between j and i else Aji = 0 otherwise).
+Any real system may have noise. Let us discuss on the
+effect of the noise for the Kuramoto model. The noise may
+arise from the variation of frequency of incoherent oscillators
+as they may not be identical or there may either be an external
+white noise or white noise inherent to the system. Therefore
+the model (3) could be reframed as
+˙θj
+=
+σωj + K
+N
+N
+�
+i=1
+sin (θj(t) − θi(t)) +
+√
+Γηj(t),
+:
+j ∈ {1, . . . , N} ,
+(23)
+where, both ωj and ηj(t) are Gaussian distributions having
+zero mean and unit variance while σ and Γ behave as am-
+plitudes of the noise. Here last term refers to white noise in
+the system. Therefore (23) physically indicates locally coupled
+oscillators having natural frequencies of oscillators derived
+from Gaussian distribution in presence of stochastic effects
+like white noise due to fluctuations in the system. The reason
+for stochastic behavior may vary for different systems while
+any natural process exhibit stochastic behavior. .
+The situation of lim σ → 0 refers to the Kuramoto model
+having identical oscillators in presence of gaussian white
+noise. The system behaves as if the system is in contact with
+a heat source and the dynamics is evolving in the statistical
+equilibrium.
+The situation for lim Γ → 0 indicates that the Kuramoto
+model has been constructed with oscillators having distributed
+natural frequencies in absence of gaussian white noise. The
+system behaves as nonlinear dynamical system relaxing to the
+non-equilibrium stationary state.
+Beside this brief summary, one may also consult articles
+like [15].
+Next, let us transform the Josephson equations for series
+array of junctions to Kuramoto model.
+III. KURAMOTO MODEL FOR JOSEPHSON JUNCTION
+SERIES
+The Josephson junction array can be constructed using
+Kirchhoff’s laws considering each Josephson junction as a
+parallel circuit of two elements: an ideal resistance ρ carrying
+ideal current Iρ and a junction carrying critical current Ic.
+Actual Josephson junction also contains a capacitor in parallel
+to the nonlinear inductor which we have neglected due to
+its very small value. Let each of N junctions be connected
+serially and then coupled to external load having inductance
+L, resistance R and capacitance C.
+C
+R
+L
+Ib
+ρ1
+I1
+ρ2
+I2
+ρN
+IN
+Fig. 1. Schematic circuit of qubits connected in series parallel to a Load.
+Let us consider Josephson junction in the series array, say
+jth junction and following Josephson equation; we express the
+circuit shown in the Fig.1 as
+V (t)
+ρj
++ Ij sin φj + dQ
+dt = Ib,
+which can be written as,
+dφj
+dt = 2πρj
+Φ0
+�
+Ib − Ij sin φj − dQ
+dt
+�
+.
+(24)
+Further,
+L ¨Q + R ˙Q + Q
+C =
+N
+�
+k=1
+Vk,
+or,
+L ¨Q +
+�
+R +
+N
+�
+k=1
+ρk
+�
+dQ
+dt + Q
+C = −
+N
+�
+k=1
+Ikρk sin φk,
+(25)
+where Q is the charge on load capacitor, Φ0 = h/(2e) is
+magnetic flux quantum, h is Planck’s constant, e being the
+charge of an electron. Here, junction resistance ρk for any
+junction k is very small compared to the load variable Q/C
+such that one may consider, Q/C − �
+k ρkIb ≈ Q/C. To
+understand the effect of external parameters like L, C and R
+on each junction, one may consider a scaled version of those
+parameters by choosing
+l = L
+N , r = R
+N , c = NC.
+(26)
+
+5
+Here, it is to be noted that
+Φ0
+Ij
+=
+1
+2πf 2
+j Cj
+where fj is the frequency and Cj is the capacitance of jth
+junction.
+Let us now consider transformation of time t and charge Q
+so that (24) and (25) become dimensionless. From (24)
+Φ0
+2πρjIj
+dφj
+dt + sin φj + 1
+Ij
+dQ
+dt = Ib
+Ij
+= αj.
+Let us consider the following transformation relation to
+transform time t to dimensionless form τ as
+Φ0
+2πρjIj
+d
+dt ≡ d
+dτ .
+(27)
+such that we may write
+dφj
+dτ + sin φj + 2πρj
+Φ0
+dQ
+dτ = αj.
+(28)
+Substituting dimensionless time τ and scaled parameters as
+in (26) in (25) we get,
+L
+N
+�2πρjIj
+Φ0
+�2 d2Q
+dτ 2
++
+(R + �N
+k=1 ρk)
+N
+�2πρjIj
+Φ0
+� dQ
+dτ
++
+Q
+NC = 1
+N
+N
+�
+k=1
+−Ikρk sin φk,
+or, l
+�2πρjIj
+Φ0
+�2 d2Q
+dτ 2
++
+�
+r +
+�N
+k=1 ρk
+N
+� �2πρjIj
+Φ0
+� dQ
+dτ
++
+Q
+c = − 1
+N
+N
+�
+k=1
+Ikρk sin φk.
+(29)
+Let us also consider the following transformation to transform
+charge Q to dimensionless form q as
+2πρjIj
+Φ0
+Q ≡ qj.
+(30)
+Therefore, through (29), (25) transforms as
+d2qj
+dτ 2 + γj
+dqj
+dτ + ω2
+0jqj = −δj
+N
+N
+�
+k=1
+Ikρk sin φk.
+(31)
+Eq. (30) can be used to rewrite (28) as
+dφj
+dτ + sin φj + ϵj
+dqj
+dτ = αj,
+(32)
+where coefficients may be written as
+γj
+=
+�
+Φ0
+2πρjIj
+� �1
+l
+� �
+r +
+�N
+k=1 ρk
+N
+�
+,
+(33)
+ω2
+0j
+=
+�
+Φ0
+2πρjIj
+�2 1
+lc,
+(34)
+δj
+=
+�
+Φ0
+2πρjIj
+� 1
+l ,
+(35)
+and ϵj
+=
+1
+Ij
+.
+(36)
+Let us write the equation (32) in the uncoupled form for
+ϵj → 0 or ˙Q → 0 such that we get,
+dφj
+dτ = αj − sin φj.
+(37)
+As discussed in the Section I, the splay-state shows that
+transforming the dynamical system equations make a rigid sys-
+tem with coherent frequencies in weak coupling or uncoupled
+limit. Hence, let us transform φj in (24) into ‘natural’ angle
+ψj such that dψj
+dt = constant. Eq. (37) can be transformed in
+terms of the ‘natural’ angle ψj such that dψj/dt − c, where c
+is constant to be determined, i.e. transformation as φj → ψj as
+uniform rotation with first derivative remaining constant. The
+constant ‘c’ may be determined with the fact that the time to
+complete one cycle by these two sets of coordinates must be
+same. Thus,
+T
+=
+� T
+0
+dτ
+=
+� 2π
+0
+dψj
+c
+=
+� 2π
+0
+dψj
+ωj
+=
+� 2π
+0
+dφj
+(αj − sin φj).
+or, 2π
+ωj
+=
+2π
+��
+α2
+j − 1
+�, for αj ⩾ 0 i.e. Ib ⩾ Ij,
+which shows
+ωj =
+�
+α2
+j − 1.
+(38)
+Then the transformation to the natural angles satisfies
+dψj =
+�
+α2
+j − 1
+αj − sin φj
+dφj,
+(39)
+which on integration yields
+ψj = 2 tan−1
+��
+αj − 1
+αj + 1 tan
+�φj
+2 + π
+4
+��
+.
+(40)
+At this point, one may construct a transformation function
+ψ(φj) to translate any angle φj to its natural angle ψj while
+another transformation function φ(ψj) may be used to invert
+as
+ψ (φ) = 2 tan−1
+��
+α − 1
+α + 1 tan
+�φ
+2 + π
+4
+��
+, (41)
+φ (ψ) = 2 tan−1
+��
+α + 1
+α − 1 tan
+�ψ
+2
+��
+− π
+2 . (42)
+Here, we use the shorthand: ψj ≡ ψ(φj) and φj ≡ φ(ψj).
+From (40),
+sin φj = 1 − αj cos ψj
+αj − cos ψj
+= αj −
+α2
+j − 1
+αj − cos ψj
+.
+(43)
+Detailed derivation of (43) from (40) is shown in appendix A.
+Therefore, one may rewrite (32) using (39) and (43) as
+dψj
+dτ
+=
+dψj
+dφj
+dφj
+dτ =
+�
+α2
+j − 1
+αj − sin φj
+.
+�
+αj − sin φj − ϵj
+dqj
+dτ
+�
+,
+=
+�
+α2
+j − 1 −
+ϵj
+�
+α2
+j − 1
+αj − sin φj
+dqj
+dτ .
+(44)
+
+6
+Let us rescale non-dimensional quantity τ as ˜τ such that
+τ =
+˜τ
+�
+α2
+j − 1
+=⇒
+d
+d˜τ ≡
+1
+�
+α2
+j − 1
+d
+dτ
+=⇒
+d2
+dτ 2 ≡
+�
+α2
+j − 1
+� d2
+d˜τ 2 .
+(45)
+Eq. (44), using (45), transforms as
+dψj
+d˜τ = 1 −
+ϵj
+�
+α2
+j − 1
+αj − sin φj
+dqj
+d˜τ ,
+(46)
+The weak-coupling solution of (44) may be written as
+ψj(τ) ≡
+��
+α2
+j − 1
+�
+τ + cj = ˜τ + ψj0,
+(47)
+where cj is the integration constant. Initially, at τ
+= 0,
+one may assume initial phase as ψj0 such that cj=ψj0. The
+reference [11] discusses about the importance of the weak
+coupling condition for the Josephson junction arrays and drift
+in ψj may be obtained by averaging (46) over one cycle as
+�dψj
+d˜τ
+�
+= 1 − 1
+2π
+� 2π
+0
+ϵj
+�
+α2
+j − 1
+αj − sin φj
+�dqj
+d˜τ
+�
+d˜τ.
+(48)
+Similarly, one may rewrite non-dimensional charge equation
+(31) in terms of ˜τ as
+�
+α2
+j − 1
+� d2qj
+d˜τ 2
++
+γj
+�
+α2
+j − 1dqj
+d˜τ + ω2
+0jqj
+=
+−δj
+N
+N
+�
+k=1
+Ikρk sin φk.
+(49)
+It is usually convenient to write sin(φj)=sin(φ(ψj)) in terms
+of its Fourier series as
+sin φ(ψk)
+=
+∞
+�
+n=0
+Akn cos (nψkn)
+=
+∞
+�
+n=0
+Akn cos {n (˜τ + ck)} .
+(50)
+Then (49) reduces to
+�
+α2
+j − 1
+� d2qj
+d˜τ 2
++
+γj
+�
+α2
+j − 1dqj
+d˜τ + ω2
+0jqj
+=
+−δj
+N
+N
+�
+k=1
+∞
+�
+n=0
+IkρkAkn cos {n (˜τ + ck)} .
+(51)
+One may obtain the steady-state solution of (51) as
+qj(˜τ)
+=
+−δj
+N IkρkBkn cos {n (˜τ + ck) + βkn} ,(52)
+dqj(˜τ)
+d˜τ
+=
+δj
+N nIkρkBkn sin {n (˜τ + ck) + βkn} , (53)
+d2qj(˜τ)
+d˜τ 2
+=
+δj
+N n2IkρkBkn cos {n (˜τ + ck) + βkn} ,(54)
+where
+B2
+kn
+=
+A2
+kn
+n2γ2
+j
+�
+α2
+j − 1
+�
++
+�
+n2 �
+α2
+j − 1
+�
+− ω2
+0j
+�2 , (55)
+βkn
+=
+tan−1
+�
+�
+nγj
+�
+α2
+j − 1
+n2 �
+α2
+j − 1
+�
+− ω2
+0j
+�
+� = βn.
+(56)
+Using the expression (43), one may derive Akn and obtain
+Ak0
+=
+1
+π
+� π
+−π
+1 − αk cos ψk
+αk − cos ψk
+dψk,
+(57)
+Akn
+=
+1
+π
+� π
+−π
+1 − αk cos ψk
+αk − cos ψk
+cos
+�nπψk
+π
+�
+dψk (58)
+where n ̸= 0.
+Bkn denotes the amplitude of the linear damped oscillator
+while βkn denotes its phase. Therefore, Bkn must be chosen
+to be positive.
+Now, (48) may be re-written as
+�dψj
+d˜τ
+�
+=
+1 −
+ϵjδj
+�
+α2
+j − 1
+2πN
+� 2π
+0
+�
+1
+αj − sin φj
+×
+N
+�
+k=1
+∞
+�
+n=0
+nIkρkBkn sin {n (˜τ + ck) + βkn}
+�
+d˜τ.
+(59)
+Using (43),
+sin φj = αj −
+α2
+j − 1
+αj − cos ψj
+,
+or, αj − sin φj =
+α2
+j − 1
+αj − cos ψj
+.
+(60)
+With this (59) may be modified using (60) to
+or,
+�dψj
+d˜τ
+�
+= 1 + Kj
+N
+N
+�
+k=1
+Ak sin (cj − ck − ζj) ,
+(61)
+where,
+Kj
+=
+ϵjδj
+�
+α2
+j − 1
+�
+γ2
+j
+�
+α2
+j − 1
+�2 +
+�
+ω2
+0j −
+�
+α2
+j − 1
+�2�2 ,
+(62)
+AK
+=
+Ikρk
+�
+1 − α2
+k + αk
+�
+α2
+k − 1
+�
+,
+(63)
+ζj
+=
+tan−1
+�
+�
+γj
+�
+α2
+j − 1
+α2
+j − 1 − ω2
+0j
+�
+� = β1j.
+(64)
+Reader may check detailed description of the derivation in the
+appendix B.
+In the final step, one may replace the ‘initial values’ of
+phases by their slowly evolving components like ⟨ψj(˜τ)⟩ and
+⟨ψk(˜τ)⟩. Also one may get firstorder averaged equation by
+dropping the angular brackets so that (61) transforms to
+dψj
+d˜τ = 1 + Kj
+N
+N
+�
+k=1
+Ak sin (ψj(˜τ) − ψk(˜τ) − δ) .
+(65)
+
+7
+Eq. (65) resembles the Kuramoto model in a generalized
+form. For the sake of mathematical formalities, it is important
+to note that except terms corresponding to n = 1 terms for
+other values of n becomes zero.
+To arrive at (65), it was assumed that the fabrication process
+may not guarantee exactly same values of parameters for each
+junction and hence one may consider that each junction has
+different internal resistance and different critical current. The
+difference may be very small for junctions prepared in the
+same batch. If the fabrication process is done in very skilled
+sequence (65) may turn into special form for assuming ρ1 =
+ρ2 = . . . = ρN = ρ (say) and I1 = I2 = . . . = IN = Ic(say)
+so that each junction has nearly same frequency f (say). This
+case of identical junctions has been studied extensively in may
+literatures.
+The transformation (27) for time leads to
+Φ0
+2πρIc
+d
+dt ≡ d
+dτ .
+(66)
+while (30) entails
+2πρIc
+Φ0
+Q ≡ q.
+(67)
+Consequently, (31) reduces to
+d2q
+dτ 2 + γ dq
+dτ + ω2
+0q = − β
+N
+N
+�
+k=1
+sin φk,
+(68)
+where
+γ
+=
+� Φ0
+2πρIc
+� � 1
+lρ
+�
+(r + ρ) ,
+(69)
+ω2
+0
+=
+� Φ0
+2πρIc
+�2 1
+lc,
+(70)
+β
+=
+� Φ0
+2πρIc
+� 1
+l ,
+(71)
+(72)
+Eqs. (55) and (56) become
+B2
+n =
+A2
+n
+n2γ2 (α2 − 1) + {n2 (α2 − 1) − ω2
+0}2 , (73)
+βn = tan−1
+�
+nγ
+√
+α2 − 1
+ω2
+0 − n2 (α2 − 1)
+�
+.
+(74)
+Repeating the earlier exercise, one may obtain the final phase
+equation (59) as
+�dψj
+d˜τ
+�
+=
+1 −
+β
+2πN
+√
+α2 − 1
+� 2π
+0
+(α − cos (τ + cj))
+×
+N
+�
+k=1
+∞
+�
+n=0
+nBn sin {n (˜τ + ck) + βn} d˜τ.
+(75)
+In the case of identical junctions, computation shows that
+only B1 exists while others are evaluated to zero. Thus, (75)
+becomes
+�dψj
+d˜τ
+�
+=
+1 −
+B1β
+2πN
+√
+α2 − 1
+� 2π
+0
+(α − cos (τ + cj))
+×
+N
+�
+k=1
+∞
+�
+n=0
+n sin {n (˜τ + ck) + βn} d˜τ.
+Integrating, we get
+dψj
+d˜τ = 1 + K
+N
+N
+�
+k=1
+sin (ψj(˜τ) − ψk(˜τ) − β1) ,
+(76)
+where
+K =
+πB1β
+2π
+√
+α2 − 1
+.
+(77)
+Eq. (76) exactly resembles as the Kuramoto model.
+In the following section, let us try to understand general
+characteristics of the Kuramoto model in general and in the
+context of Josephson junction array.
+IV. ANALYSIS
+A C + + code has been developed alongwith DISLIN
+code to analyse the equations. DISLIN [16] is a freely
+available graph plotting routine that plots during runtime and
+can be stored. In this section, let us first investigate basic
+Fig. 2.
+Kuramoto model in arbitrary unit for 100 oscillators with K=4
+showing synchronization after a certain settling time within a band of
+frequency range.
+Kuramoto model as discussed in (3) including the effect of
+coupling strength (K). If K is properly tuned, one may expect
+synchronization as shown in Fig.2.
+Here we consider that the oscillators are oscillating possess-
+ing a frequency distribution g(ω). One may control width of
+the distribution while keeping the zero mean.
+We consider Logistic and Lorentzian fuctions having
+width β. Oscillators tend get to be synchronized if K is equal
+to or more than some threshold value Kc as discussed in (20).
+g(ω) =
+exp (−ω/β)
+β [1 + exp (−ω/β)]2
+(78)
+The Logistic function is described as (78) which shows
+g(0)=1/(4β) where, β is the width. Likewise, one may define
+the Lorentzian function as (79)
+g(ω) =
+b
+(ω2 + b2),
+(79)
+
+Kuramoto model for 1oo oscillators having K=4 without any noise
+0.5
+a
+sin
+0
+-0.5
+0
+2
+4
+6
+8
+10
+time8
+Fig. 3.
+100 oscillators with K=0.1 having Logistic distribution of width
+0.001.
+Fig. 4. 100 oscillators with K=0.1 having Lorentzian distribution of width
+0.001.
+so that one get g(0)=2/(πb). This g(0) estimates thresh-
+old value of the coupling strength as Kc=2/πg(0).
+One
+may compare Fig.3 with Fig.5 where the latter is operating
+with threshold coupling. The synchronization for the latter
+shows phase space of order parameter as a dot denoting
+synchronization. Figs.4 and 6 also show similar observation
+of synchronization.
+This theoretical study clearly heps us to understand the sig-
+nificance of coupling strength and the treatment of frequency
+range of oscillators to start with.
+Next one may apply this understanding in the case of
+Josephson junction. The situation is very much different hereas
+the definition of K is complex for both non-identical and
+Fig. 5. 100 oscillators oscillating with k=Kc=0.509 with Logistic function
+of width 0.2.
+Fig. 6. 100 oscillators oscillating with k=Kc=0.4 with Lorentzian function
+of width 0.2.
+identical junction arrays as evident from either (65) or (76)
+respectively. Let us consider mean frequency may be around
+5 GHz. Figs.7 and 8 show simulated results of systems
+of 100 Josephson junctions in non-identical and identical
+configurations respectively operated for ˜τ = 25. The interesting
+part is that synchronization is not pulling the oscillators to a
+certain unique frequency. Rather, oscillators tend to cool down
+to a narrow band of frequencies resulting in an arc in phase
+space diagram which resembles as if oscillators have a certain
+‘viscosity’ in the combined system. For the non identical case,
+the spread of Ic is considered very small like 0.1% while
+variation in ρj is about 0.05 % as fabrication is much better
+and junctions fabricated in the same substrate will not vary
+
+Phase graph of 10o oscillators forK=0.100 dt=0.001simulation time=10
+(a) Phase graph of 100 oscillators
+(b) Order Paraneter of acillators
+1.0
+10
+sin
+P
+-1.心
+0.0
+2.0
+4.0
+B.0
+08
+10.0
+0.0
+2.0
+4.0
+08
+10.0
+time (T)
+time (r)
+(o)orderParameterof oscillatorsatT=0
+(d) Order Parameter of oncillatore at T =io
+1.1
+1.1
+40
+40
+90
+90
+0.3
+UTS
+0.1
+sin
+0.1
+0.1
+0.1
+0.8
+0.3
+0.5
+0.5
+0.7
+0.7
+0.9
+8'0-
+-L1
+11
+0.9
+20
+0.6
+0.3
+心1
+2
+6'0-
+0
+0.5
+0.3
+01
+0.5
+COSPhase graph of 10o oscillators forK=0.100 dt=0.001simulation time=10
+(a) Phase graph of 100 oscillators
+(b) Order Paraneter of acillators
+1.0
+10
+sin
+P
+1.心
+0.0
+2.0
+4.0
+B.0
+08
+10.0
+0.0
+2.0
+4.0
+08
+10.0
+time (r)
+time (r)
+(o)orderParameterof oscillatorsatT=0
+(d) Order Parameter of oncillatore at T =io
+1.1
+1.1
+40
+40
+90
+90
+0
+UTS
+0.1
+++
+sin
+0.1
+0.1
+0.1
+0.8
+0.3
+0.5
+0.5
+0.7
+0.7
+8'0-
+-L1
+11
+6'0
+0.3
+心1
+0.8
+0
+0.5
+0.3
+01
+0.5
+F'T
+COSPhasegraph of 100oscillatorsforK=0.509dt=0.100simulation time=30
+(a) Phase graph of 100 oscillators
+[b) Order Paraneter of acillators
+1.0
+20
+P 0.5
+一
+1.心
+0.0
+6.0
+12.0
+18.0
+24.0
+30.0
+0.0
+6.0
+12.0
+18.0
+24.
+30.0
+time (-)
+time (-)
+(o)Order Parameter of oscillators atT=0
+(d) Order Parameter of oBcillatorB at T =90
+1.1
+1.1
+B0
+2'0
+90
+90
+sin
+UTS
+0.1 
+++++
+0.1 
+-0.1
+0.1
+0.8
+0.3
+0.5
+-0.5
+0.7
+0.7
+0.9
+-0.8
+-L1
+-L1
+0.E
+0.3
+心1
+0.9
+0.7
+0.5
+0.3
+01
+0.5
+COSPhasegraphof100oscillatorsforK=0.400dt=0.100simulationtime=30
+(a) Phase graph of 1o0 oscillators
+(b) Order Paraneter of oacillators
+1.0
+二
+0.5
+>
+1.心
+0.0
+6.0
+12.0
+18.0
+24.0
+30.0
+0.0
+6.0
+12.0
+18.0
+24.
+30.0
+time (-)
+time (-)
+(o)orderParameterof oscillators atT=0
+(d) Order Parameter of oBcillatore at T=go
+1.1
+1.1
+10
+2'0
+90
+9'0
+sin
+sin
+0.1
++++ +.
+0.1 
+-0.1
+0.1
+0.8
+0.3
+0.5
+-0.5
+0.7
+0.7
+0.9
+-0.8
+-L1
+11
+0
+0.E
+0.3
+心1
+0.9
+0.7
+0.5
+0.3
+01
+0.5
+COS9
+Fig. 7. 100 non-identical Josephson junctions operating with mean frequency
+of 5 GHz having mean Ic = 10 µA, mean internal resistance ρj = 4.2 kΩ
+connected in series array to external load with parameters L=1 nH, C = 1
+µF and R = 2 Ω treated with bias current Ib = 12 µA synchronizes within
+a narrow band of distribution. The final phase space is not a dot!
+Fig. 8. 100 identical Josephson junctions operating at mean frequency of 5
+GHz having mean Ic = 10 µA, internal resistance ρ = 4.2 kΩ connected in
+series array to external load with parameters L=1 nH, C = 1 µF and R = 2
+Ω treated with bias current Ib = 12 µA synchronizes within a narrow band
+of distribution. The final phase space is not a dot!
+too much. Another point to note is that the oscillators in the
+non-identical case tend to syncronize faster and better than the
+other case, possibly due to the noisy environment.
+It has already been discussed that Kuramoto model stands
+on the assumption that a large number of oscillators have
+been considered. In our experimental regime, one may need
+to use smaller number of oscillators say 5 or 10 oscillators as
+shown in Fig.9 as asynchronized. The order parameter R is
+Fig. 9. 5 identical oscillators having Ic = 10 µA and ρ = 4.2 kΩ operating
+with 5 GHz frequency.
+also shown to be oscillating at a lower value. The observation
+was made for ˜τ = 25. The circuit parameters were kept same
+as those for 100 oscillators. Evidently oscillators were not
+syhronized. The case for the 5 non-identical oscillators is same
+as 9. Now, to tune the circuit, let us select Ic as 10 µA and ρj
+Fig. 10.
+5 non identical Josephson junctions are partially syncronized
+changing Ib to 10.8785 µA.
+= 4.2 kΩ as before as we wish to experiment with the same
+junctions while we change Ib - the bias current. In the Fig.10,
+the synchronization is observed where one oscillator is out of
+sync while the rest 4 oscillators come closer to lie in a band
+very fast ˜τ ≈ 1.
+
+Phase graph of 100 oseillators for K=12205.95dt=0.001 simulation time=25
+(a) Phase graph of 100 oscillators
+[b) Order Paraneter of acillators
+1.0
+B'0
+0.5
+一
+-1.心
+0.1
+0.0
+6.0
+10.0
+15.0
+20.心
+25.0
+0.0
+5.0
+10.0
+15.0
+20.心
+25.0
+time (r)
+time (r)
+(o) Order Parameter of oscillators at T=0
+(d) Order Parameter of oBcillatorB at T=z5
+1.1
+1.1
++ +
+2'0
+2'0
+90
+90
+us
+0
+sin
+0.1
+0.1 
+0.1
+0.1
+0.8
+0.3
+0.5
+0.5
+0.7
+0.7
+0.9
+-0.9 
+-L1
+11
+0.9
+20
+0.E
+0.3
+心1
+0.9
+0.7
+0.5
+0.3
+心1
+cosu
+cosyPhase graph of 100 oseillators for K=12205.95dt=0.001 simulation time=25
+(a) Phase graph of 1oo oscillators
+(b) Order Paraneter of oacillators
+1.0
+10
+TTTTT
+0.8
+sin
+一
+0.4 
+0.2
+-1.心
+0.0
+0.0
+6.0
+10.0
+15.0
+20.心
+25.0
+0.0
+5.0
+10.0
+15.0
+20.心
+25.0
+time (r)
+time (r)
+(o)orderParameterof oscillators atT=0
+(d) Order Parameter of oBcillatorB at T=z5
+1.1
+1.1
+2'0
+2'0
+90
+90
+sin
+us
+0
+0.1
+0.1 
+-0.1
+0.1
+0.8
+0.3
+0.5
+0.5
+0.7
+0.7
+0.9
+-0.9 
+-L1
+11
+0.9
+20
+0.E
+0.3
+心1
+0.9
+0.7
+0.5
+0.3
+心1
+cosu
+cosyPhasegraph of 5oseillators forK=4307.83 dt=0.001 simulationtime=25
+(a) Phase graph of 5 oscillators
+(b) Order Paraneter of acillators
+1.0
+0.9
+0.7
+sin
+9'0
+1.心
+0.1
+0.0
+6.0
+10.0
+15.0
+20.0
+25.0
+0.0
+5.0
+10.0
+15.0
+20.0
+25.0
+time (r)
+time (r)
+(o)OrderParameter of oscillators atT=0
+(d) Order Parameter of oBcillator at T =25
+1.1
+1.1
+2'0
+2'0
+90
+90
+us
+0
+0.1
+0.1
+-0.1 
+0.1
+0.8
+0.3
+0.5
+-0.5
+0.7
+0.7
+0.9
+-0.9 
+-L1
+11
+0.E
+0.3
+TO-
+0.9
+0.7
+0.5
+0.3
+心1
+0.5
+cosyPhase graph of 5 oscillators for K=30333.88dt=0.00lsimulation time=15
+(a) Phase graph of 5oscillators
+(b) Order Paraneter of oacillators
+1.0
+80
+P 0.5
+0.1 -
+-1.心
+0.0
+12.心
+15.0
+0.0
+6.0
+12.心
+15.0
+time (r)
+time (r)
+(o)orderParameterof oscillators atT=0
+(d) Order Parameter of oBcillatore at T =15
+1.1
+1.1
+2'0
+++
+20
+90
+90
+uIs
+0.1 
+0.1
+0.1
+-0.1
+0.8
+0.3
+0.5
+0.5
+0.7
+0.7
+-0.9 
+-L1
+1L1
+0.E
+0.3
+心1
+0.7
+0.5
+E'O
+心1
+cosW
+cosu10
+Fig. 11. 5 identical Josephson junctions are partially syncronized changing
+Ib to 10.877 µA.
+V. CONCLUSION
+The exercises demonstrated in Figs.10 and 11 show the
+possibility of synchronization for few oscillators following
+Kuramoto model. However, order parameter show in-course
+instability which later settles down.
+This study helps to understand applicability of junctions in
+series array and steps to control the level of synchronization.
+The process is easier and synchronization is performed well
+for larger number of junctions while partial synchronization is
+also possible following the Kuramoto model. However, this
+study does not state any conclusive equation for threshold
+coupling for Josephson junction as it discussed in case of
+general oscillators. This aspect will be discussed in future.
+APPENDIX A
+From (40),
+tan
+�φj
+2 + π
+4
+�
+=
+�
+αj + 1
+αj − 1 tan ψj
+2 ,
+or,
+�
+tan
+�φj
+2 + π
+4
+�
++ 1
+� �
+tan
+�φj
+2 + π
+4
+�
+− 1
+�
+=
+��
+αj + 1
+αj − 1 tan ψj
+2 + 1
+� ��
+αj + 1
+αj − 1 tan ψj
+2 − 1
+�
+,
+or,
+�
+1 + tan φj
+2
+1 − tan φj
+2
++ 1
+� �
+1 + tan φj
+2
+1 − tan φj
+2
+− 1
+�
+= αj + 1
+αj − 1 tan2 ψj
+2 − 1,
+or,
+�
+cos φj
+2 + sin φj
+2
+cos φj
+2 − sin φj
+2
++ 1
+� �
+cos φj
+2 + sin φj
+2
+cos φj
+2 − sin φj
+2
+− 1
+�
+= αj + 1
+αj − 1 tan2 ψj
+2 − 1,
+or,
+2 cos φj
+2 × 2 sin φj
+2
+�
+cos φj
+2 − sin φj
+2
+�2 =
+2 sin φj
+�
+cos φj
+2 − sin φj
+2
+�2
+= αj + 1
+αj − 1 tan2 ψj
+2 − 1,
+or,
+2 sin φj
+1 − sin φj
+= αj tan2 ψj
+2 + tan2 ψj
+2 − αj + 1
+αj − 1
+,
+or,
+1 − sin φj
+2 sin φj
+=
+1 − αj cos ψj
+(αj − 1) cos2 ψj
+2
+,
+or,
+1
+2 sin φj
+= 1
+2 +
+1 − αj cos ψj
+(αj − 1) cos2 ψj
+2
+,
+or,
+sin φj = 1 − αj cos ψj
+αj − cos ψj
+,
+or,
+sin φj ≡ sin φ(ψj) = αj −
+�
+α2
+j − 1
+�
+αj − cos ψj
+.
+
+Phasegraph of 5oscillators forK=30453.72dt=0.00l simulationtime=15
+(a) Phase graph of 5 oscillators
+(b) Order Paraneter of oacillators
+1.0
+E0
+sin
+ 0.5
+1.心
+0.0
+12.心
+15.0
+0.0
+6.0
+12.心
+15.0
+time (T)
+time (r)
+(o)orderParameterof oscillators at T=Q
+(d) Order Parameter of oBcillatore at T =15
+1.1
+1.1
+2'0
+20
+90
+90
+sin
+uIs
+0.1 
+0.1
+-0.1
+0.1
+0.8
+0.3
+-0.8
+0.5
+0.7
+0.7
+0.9
+-L1
+-11
+0.3
+心1
+0.7
+0.5
+心1
+cos
+cosu11
+APPENDIX B
+�dψj
+d˜τ
+�
+= 1 −
+ϵjδj
+�
+α2
+j − 1
+2πN
+� 2π
+0
+�
+αj − cos ψj
+α2
+j − 1
+×
+N
+�
+k=1
+∞
+�
+n=0
+nIkρkBkn sin {n (˜τ + ck) + βkn}
+�
+d˜τ.
+or,
+�dψj
+d˜τ
+�
+= 1 −
+ϵjδj
+2πN
+�
+α2
+j − 1
+� 2π
+0
+(αj − cos (˜τ + cj))
+×
+N
+�
+k=1
+∞
+�
+n=0
+nIkρkBkn sin {n (˜τ + ck) + βkn} d˜τ.
+or,
+�dψj
+d˜τ
+�
+= 1
++
+ϵjδj
+N
+�
+α2
+j − 1
+�
+γ2
+j
+�
+α2
+j − 1
+�2 +
+�
+ω2
+0j −
+�
+α2
+j − 1
+�2�2
+×
+N
+�
+k=1
+Ikρk
+�
+1 − α2
+k + αk
+�
+α2
+k − 1
+�
+sin (cj − ck − ζj) .
+or,
+�dψj
+d˜τ
+�
+= 1 + Kj
+N
+N
+�
+k=1
+Ikρk
+�
+1 − α2
+k + αk
+�
+α2
+k − 1
+�
+× sin (cj − ck − ζj) ,
+or,
+�dψj
+d˜τ
+�
+= 1 + Kj
+N
+N
+�
+k=1
+Ak sin (cj − ck − ζj) .
+ACKNOWLEDGMENT
+The author would like to Sudhir R Jain, for his ideas, inspi-
+ration and continuous support to conceptualize, understand and
+formulate the problem. The author also expresses gratitude to
+Susmita Bhattacharyya and Tilottoma Bhattacharyya for their
+guidance.
+REFERENCES
+[1] S. P. Benz and C. A. Hamilton, “Application of josephson effect to
+voltage metrology,” Proceedings of the IEEE, vol. 92, no. 10, pp. 1617–
+1629, 2004.
+[2] B. D. Josephson, “Possible new effects in supercondictive tunneling,”
+Physics Letters, vol. 1, no. 7, pp. 251–253, 1962.
+[3] ——, “Coupled superconductors,” Reiew of Modern Physics, vol. 36,
+no. 1, pp. 216–220, 1964.
+[4] ——, “The discovery of tunneling supercurrents,” Nobel Lectures, 1973.
+[5] B. S. D. Jr. and W. M. Fairbank, “Experimental evidence for quantized
+flux in superconducting cylinders,” Physical Review Letters, vol. 7, no. 2,
+pp. 43–46, 1961.
+[6] T. Endo, masao Koyanagi, and A. Nakamura, “High accuracy josephson
+potentiometer,” IEEE Transactions on Instrumentation and Measure-
+ment, vol. IM-32, no. 1, pp. 267–271, 1983.
+[7] M. T. Levinsen, R. Y. Chiao, M. J. Feldman, and B. A. Tucker, “Applied
+physics letters,” Applied Physics Letters, vol. 31, p. 776, 1977.
+[8] S. P. Benz and C. J. Burroughs, “Coherent emission from two dimen-
+sional josephson junction arrays,” Applied Physics Letters, vol. 58(19),
+pp. 2162–2164, 1991.
+[9] K. Wan, A. K. Jain, and J. E. Lukens, “Submillimeter wave generation
+using josephson junction arrays,” Applied Physics Letters, vol. 54, pp.
+1805–1807, 1989.
+[10] J. W. Swift, S. H. Strogatz, and K. Wisenfield, “Averaging of globally
+coupled oscillators,” Physica D, vol. 55, pp. 239–250, 1992.
+[11] K. Wisenfeld and J. W. Swift, “Averaged equations for josephson
+junction series arrays,” Physical Review E, vol. 51, pp. 1020–1025, 1995.
+[12] H. Sakaguchi and Y. Kuramoto, “Kuramoto order parameters and phase
+concentration for the kuramoto-sakaguchi equation with frustration,”
+Progress of Theoretical Physics, vol. 76(3), pp. 576–581, 1986.
+[13] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical
+Systems, and Bifurcations of Vector fields.
+Springer Verlag New York,
+2002.
+[14] S.-Y. Ha, J. Morales, and Y. Zhang, “A soluble active rotator model
+showing phase transitions via mutual entrainment,” Communications on
+pure and applied analysis, vol. 20(7 & 8), pp. 2579–2612, 2021.
+[15] D. Florian and F. Bullo, “On the critical coupling for kuramoto oscilla-
+tors,” SIAM Journal of Apllied Dynamical Systems, vol. 10, 2011.
+[16] H. Michels, “Dislin software.” [Online]. Available: http://www.dislin.de/
+

DNE3T4oBgHgl3EQfUwpD/content/tmp_files/2301.04453v1.pdf.txt

@@ -0,0 +1,927 @@
+Nakayama et al.: Preparation of Papers for IEEE Access
+.
+.
+VOLUME 4, 2016
+1
+arXiv:2301.04453v1  [eess.SY]  11 Jan 2023
+
+TEEEAccesSDate of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
+Digital Object Identifier 10.1109/ACCESS.2017.DOI
+Trajectory Tracking Control of
+The Second-order Chained Form System
+by Using State Transitions
+MAYU NAKAYAMA1, MASAHIDE ITO1, (Member, IEEE)
+1School of Information Science and Technology, Aichi Prefectural University, Nagakute, Aichi, Japan
+Corresponding author: Masahide Ito (e-mail: [email protected]).
+ABSTRACT This paper proposes a novel control approach composed of sinusoidal reference trajectories
+and trajectory tracking controller for the second-order chained form system. The system is well-known as
+a canonical form for a class of second-order nonholonomic systems obtained by appropriate transformation
+of the generalized coordinates and control inputs. The system is decomposed into three subsystems, two of
+them are the so-called double integrators and the other subsystem is a nonlinear system depending on one of
+the double integrators. The double integrators are linearly controllable, which enables to transit the value of
+the position state in order to modify the nature of the nonlinear system that depends on them. Transiting the
+value to “one” corresponds to modifying the nonlinear subsystem into the double integrator; transiting the
+value to “zero” corresponds to modifying the nonlinear subsystem into an uncontrollable linear autonomous
+system. Focusing on this nature, this paper proposes a feedforward control strategy. Furthermore, from the
+perspective of practical usefulness, the control strategy is extended into trajectory tracking control by using
+proportional-derivative feedback. The effectiveness of the proposed method is demonstrated through several
+numerical experiments including an application to an underactuated manipulator.
+INDEX TERMS
+nonholonomic systems; state transitions; the second-order chained form; trajectory
+tracking control
+I. INTRODUCTION
+N
+ONHOLONOMIC systems are nonlinear dynamical
+systems with non-integrable differential constraints,
+whose control problems have been attracting many re-
+searchers and engineers for the last three decades. The main
+reason is that the nonholonomic systems do not satisfy
+Brockett’s theorem [1]. The challenging and negative fact
+means that there is not any smooth time-invariant feedback
+control law to be able to stabilize them. The applications
+include various types of robotic vehicles and manipulation.
+Some of them have been often used as a kind of bench-
+mark platform to demonstrate the performance of a proposed
+controller for not only a control problem of a single robotic
+system and also a distributed control problem of multiagent
+robotic systems.
+The class subject to acceleration constraints—called
+second-order nonholonomic systems—includes real exam-
+ples such as a V/STOL aircraft [2], an underactuated manip-
+ulator [3], an underactuated hovercraft [4], and a crane [5].
+These systems can be represented in a canonical system
+called the second-order chained form by coordinate and in-
+put transformations. The second-order chained form system
+is also affected by Brockett’s theorem [1]. To avoid this
+difficulty, there are several ingenious control approaches.
+The stabilizing controllers proposed in [4], [6]–[8] exploit
+discontinuity or time-variance; [3], [9] and [10] reduce the
+control problem into a trajectory tracking problem. Other
+than those, [11] and [12] consider a motion planning problem
+(in other words, a feedforward control problem).
+For the second-order chained form system, this paper
+presents a novel control approach composed of sinusoidal
+reference trajectories and a simple trajectory tracking con-
+troller. The second-order chained form system is decomposed
+into three subsystems. Two of them are the so-called dou-
+ble integrators; the other subsystem is a nonlinear system
+depending on one of the double integrators. The double
+integrator is linearly controllable, which enables to transit the
+value of the position state in order to modify the nature of the
+nonlinear subsystem. Transiting the value into “one” corre-
+sponds to modifying the nonlinear subsystem into the double
+2
+VOLUME 4, 2016
+
+IEEEAccesS
+Multidisciplinary  Rapid Review  Open Access JournalNakayama et al.: Preparation of Papers for IEEE Access
+integrator; transiting the value into “zero” corresponds to
+modifying the nonlinear subsystem into a linear autonomous
+system. Focusing on this nature, this paper proposes a feed-
+forward control strategy. Furthermore, from the perspective
+of practical usefulness, the control strategy is extended into
+trajectory tracking control by using proportional-derivative
+(PD) feedback.
+The remainder of this paper is organized as follows: Sec-
+tion II presents that the second-order chained form system
+can be decomposed to linear subsystems by using state
+transitions. On the basis of such system nature, Section III
+proposes a feedforward control strategy and also a trajectory
+tracking controller of PD feedback. Section IV applies the
+proposed control approach to an underactuated manipulator
+and evaluates it through numerical experiments. The last
+section concludes the paper with a summary and future work.
+II. SUBSYSTEM DECOMPOSITION OF THE
+SECOND-ORDER CHAINED FORM SYSTEM BY USING
+STATE TRANSITIONS
+Consider the following second-order chained form system:
+d2
+dt2 ξ =
+�
+�
+1
+0
+0
+1
+ξ2
+0
+�
+� u,
+(1)
+where ξ = [ξ1, ξ2, ξ3]⊤ and u = [u1, u2]⊤ are the gen-
+eralized coordinate vector and the generalized input vector,
+respectively. This system is well-known as a canonical form
+for a class of second-order nonholonomic systems, which
+can be resulted from the original dynamical model via an
+appropriate transformation of the generalized coordinates
+and control inputs. Representing the system (1) as an affine
+nonlinear system:
+d
+dt
+�
+�������
+ξ1
+ξ2
+ξ3
+˙ξ1
+˙ξ2
+˙ξ3
+�
+�������
+=
+�
+�������
+˙ξ1
+˙ξ2
+˙ξ3
+0
+0
+0
+�
+�������
++
+�
+�������
+0
+0
+0
+1
+0
+ξ2
+�
+�������
+u1 +
+�
+�������
+0
+0
+0
+0
+1
+0
+�
+�������
+u2,
+(2)
+we
+can
+easily
+confirm
+that
+the
+equilibrium
+points
+(ξ⋆
+1, ξ⋆
+2, ξ⋆
+3, 0, 0, 0), ξ⋆
+1, ξ⋆
+2, ξ⋆
+3
+∈
+R are small-time local
+controllable (STLC) via Sussmann’s theorem [13].
+By focusing on the control inputs, the system (1) can be
+decomposed into the following two subsystems:
+d
+dt
+�
+���
+ξ1
+ξ3
+˙ξ1
+˙ξ3
+�
+��� =
+�
+���
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+�
+���
+�
+���
+ξ1
+ξ3
+˙ξ1
+˙ξ3
+�
+��� +
+�
+���
+0
+0
+1
+ξ2
+�
+��� u1,
+(3a)
+d
+dt
+� ξ2
+˙ξ2
+�
+=
+� 0
+1
+0
+0
+� � ξ2
+˙ξ2
+�
++
+� 0
+1
+�
+u2.
+(3b)
+The subsystem (3b) with respect to the control input u2 is
+a linear and controllable system represented by the double
+integrator. On the other hand, the subsystem (3a) with respect
+to the input u1 is a four-dimensional nonlinear system whose
+input matrix depends on the state variable ξ2. The subsys-
+tem (3a) can be further decomposed as follows:
+d
+dt
+� ξ1
+˙ξ1
+�
+=
+� 0
+1
+0
+0
+� � ξ1
+˙ξ1
+�
++
+� 0
+1
+�
+u1,
+(4a)
+d
+dt
+� ξ3
+˙ξ3
+�
+=
+� 0
+1
+0
+0
+� � ξ3
+˙ξ3
+�
++
+� 0
+ξ2
+�
+u1.
+(4b)
+The subsystem (4a) of the double integrator is linear and
+controllable; the subsystem (4b) inherits the nonlinearity of
+the system (3a).
+Fig. 1 shows a block diagram describing the above-
+mentioned subsystem decomposition explicitly. The state of
+the subsystem (3b) can be transited to be a constant value
+because of the linear controllability. For example, by setting
+time intervals where ξ2 is “zero” and also ξ2 is “one”, the
+nonlinear subsystem (4b) can be treated as a linear system.
+During the time interval of ξ2 = 1, the subsystems (4a)
+and (4b) are linear which have the same double integrator
+structure and control input u1. On the other hand, during
+the time interval of ξ2 = 0, the subsystem (3a) becomes
+a linear autonomous (i.e., uncontrollable) system and the
+subsystem (4a) can be controlled independently from sub-
+system (4b) by the control input u1.
+Remark 1. Some conventional approaches such as in [14],
+[10] and [15] exploit a different subsystem decomposition
+that can decompose the system (1) as follows:
+d
+dt
+� ξ1
+˙ξ1
+�
+=
+� 0
+1
+0
+0
+� � ξ1
+˙ξ1
+�
++
+� 0
+1
+�
+u1,
+(5a)
+d
+dt
+�
+���
+ξ2
+ξ3
+˙ξ2
+˙ξ3
+�
+��� =
+�
+���
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+0
+0
+u1
+0
+0
+0
+�
+���
+�
+���
+ξ2
+ξ3
+˙ξ2
+˙ξ3
+�
+��� +
+�
+���
+0
+0
+1
+0
+�
+��� u2.
+(5b)
+The subsystem (5a) is the same with (4a); the subsystem (5b)
+has a variable structure depending on u1. The subsystem (5b)
+is linear when u1 is a non-zero constant, which reduces a
+control problem of the second-order chained form system into
+a simultaneous stabilizing problem of the two subsystems (5a)
+and (5b). When u1 becomes zero before the end of control,
+however, the subsystem (5b) will be uncontrollable with a
+pole at the origin and then the whole of the subsystem loses
+the controllability. This subsystem decomposition, therefore,
+needs control in consideration with u1.
+III. PROPOSED CONTROL APPROACH
+In this paper, a control task of a rest-to-rest motion is ad-
+dressed. For this task, the authors propose a control approach
+composed of sinusoidal reference trajectories and a trajec-
+tory tracking controller. In particular, a feedforward control
+strategy that generates the reference trajectories exploits the
+system decomposition based on state transition described in
+the previous section.
+The feedforward control strategy using system switching
+based on state transitions in ξ2 is as follows:
+VOLUME 4, 2016
+3
+
+TEEEAccesSNakayama et al.: Preparation of Papers for IEEE Access
+�
+�
+u1
+˙ξ1
+ξ1
+×
+�
+�
+˙ξ3
+ξ3
+�
+�
+u2
+˙ξ2
+ξ2
+�
+�
+u1
+˙ξ1
+ξ1
+�
+�
+˙ξ3
+ξ3
+�
+�
+u1
+˙ξ1
+ξ1
+�
+�
+˙ξ3
+ξ3
+⇐⇒
+when ξ2 = 1
+when ξ2 = 0
+FIGURE 1. Subsystem decomposition of the second-order chained form by using ξ2’s state transitions between 0 and 1.
+Step 1
+Transit ξ2 from any initial value to 1 by using
+u1(t) = 0, u2(t) = q2(t);
+Step 2
+Transit ξ3 from any initial value to any desired
+value (in conjunction with it, ξ1 is also driven)
+by using u1(t) = q3(t), u2(t) = 0;
+Step 3
+Transit ξ2 from 1 to 0 by using u1(t)
+=
+0, u2(t) = q2(t);
+Step 4
+Transit ξ1 from any value in Step 2 to any de-
+sired value by using u1(t) = q1(t), u2(t) = 0;
+Step 5
+Transit ξ2 from 0 to any desired value by using
+u1(t) = 0, u2(t) = q2(t).
+A control input in Step k (k = 1, 2, . . . , 5) is designed
+by an appropriate sinusoidal function qi(t) (i = 1, 2, 3)
+without any feedback. This control strategy is namely mo-
+tion planning, which naturally cannot deal with disturbance.
+Therefore, we provide a trajectory tracking controller that
+follow the reference trajectory.
+Consider to drive the state variables ξi(t), ˙ξi(t) of the
+system (1) by the following sinusoidal functions with pe-
+riod T = 2π/ω and amplitude ak:
+qi(t) = akω2 sin ωt.
+(6)
+Then, at time t (≤ kT), trajectories of a subsystem with non-
+zero input are derived as
+˙ξi(t) = ˙ξi((k − 1)T) − akω cos ωt + akω,
+(7)
+ξi(t) = ξi((k − 1)T) + ˙ξi((k − 1)T)t
+− ˙ξi((k − 1)T)(k − 1)T
+− ak sin ωt + akωt − ak(k − 1)ωT,
+(8)
+respectively, where ξi((k − 1)T) and ˙ξi((k − 1)T) are initial
+values of the state variables in Step k. Thus, at the end of
+k-th period (t = kT), the state transitions are represented as
+˙ξi(kT) = ˙ξi((k − 1)T),
+(9)
+ξi(kT) = ξi((k − 1)T) + ˙ξi((k − 1)T)T + 2πak,
+(10)
+which means that a displacement of 2πak on ξi is obtained.
+This can be seen that the desired displacement is extracted by
+using the amplitude ak as a tuning parameter.
+By setting the trajectories (6), (7), (8) as reference trajec-
+tories qref
+i (t), ξref
+i (t), ˙ξref
+i (t), a PD feedback control system
+can be designed for trajectory tracking. A linear system of a
+double integrator can be represented in the following state-
+space form with the state zi = [ξi, ˙ξi]⊤ and control input qi:
+˙zi =
+� 0
+1
+0
+0
+�
+� �� �
+A
+zi +
+� 0
+1
+�
+����
+b
+qi(t, zi).
+(11)
+In Step k, a feedback controller for trajectory tracking to zref
+i
+is given as follows:
+qi(t, zi) = qref
+i (t) + k ei,
+(12)
+where ei := zref
+i
+− zi and k = [kp, kd] is a feedback gain
+matrix. The system (11) yields the closed-loop system ˙ei =
+(A − bk)ei. By choosing the feedback gain k so that (A −
+bk) is Hurwitz-stable, the closed-loop system is stabilized,
+that is, zi tracks zref
+i .
+IV. NUMERICAL EXPERIMENTS
+In this section, we evaluate the effectiveness of the proposed
+control approach through numerical experiments.
+Firstly, we validate the proposed controller for the second-
+order chained form system. A numerical experiment was per-
+formed with T = 1 s, ξ(0) = [3, 0.5, 1]⊤, ˙ξ(0) = 03, ξ⋆ =
+[1, 1, 0]⊤, and ˙ξ⋆ = 03. Fig. 2 shows the simulation results
+when choosing a1 = 1/(4π), a2 = a3 = a4 = −1/(2π),
+and a5 = 1/(2π). The ordinary differential equations was
+numerically solved by ODE45 of MATLAB [16] with a rela-
+tive tolerance of 1×10−3. The results indicate that each state
+reached to the target value ξ⋆ with the remaining errors at t =
+5T: ξ(5T)−ξ⋆ = [−2.7×10−8, 1.0×10−10, −4.7×10−8]⊤
+and ˙ξ(5T)− ˙ξ⋆ = [1.3×10−8, −8.9×10−9, −2.1×10−8]⊤,
+which means that the desired control is achieved.
+Secondly, the proposed controller is applied to an underac-
+tuated manipulator—a typical example of second-order non-
+holonomic systems—as shown in Fig. 3. This manipulator
+has first two joints being actuated and the last joint being
+unactuated. The system representation can be converted to
+the second-order chained form system. Even if the third joint
+cannot be driven due to no actuator, the acceleration (α1, α2)
+acting on the center of percussion of the third link can be
+treated equivalently as a control input owing to dynamic cou-
+pling effect—the rotational actuation of the first and second
+4
+VOLUME 4, 2016
+
+TEEEAccesSNakayama et al.: Preparation of Papers for IEEE Access
+TABLE 1. Definition of variables and parameters
+(x, y) : position of the center of percussion of the third link in the frame O-XY ;
+θ
+: angle of the third link relative to X-axis;
+d3
+: distance between the third joint and the center of mass of the third link;
+m3
+: mass of the third link;
+I3
+: moment of inertia mass of the third link;
+LCoP : distance between the third joint and the center of percussion of the third link
+�
+LCoP := (I3 + m3d2
+3)/(m3d3)
+�
+;
+α1
+: translational acceleration along the third link;
+α2
+: angular acceleration around the center of percussion of the third link.
+FIGURE 2. Simulation results of trajectory tracking control
+joints propagates through the links. For simplicity, assume
+that there is no disturbance such as load, friction, linear and
+nonlinear damping, etc. The main variables are defined as in
+Table 1.
+Let χ := [x, y, θ]⊤ and α = [α1, α2]⊤. Yoshikawa, et
+al. [11] provided a set of coordinate and input transforma-
+tions to convert the manipulator dynamics derived from the
+Lagrange’s equation of motion into the following system
+𝜃
+1st revolute joint
+(actuated)
+𝑥
+𝑑!
+2nd revolute joint
+(actuated)
+3rd revolute joint
+(unactuated)
+𝑦
+𝑋
+𝑌
+𝑂
+center of percussion 
+of 3rd link
+:
+𝛼"
+𝛼#
+FIGURE 3. A three-joint manipulator with passive third joint
+representation:
+¨χ =
+�
+�
+cos θ
+0
+sin θ
+0
+0
+1
+�
+� α.
+(13)
+Using the coordinate transformation
+�
+�
+ξ1
+ξ2
+ξ3
+�
+� =
+�
+�
+x − LCoP
+tan θ
+y
+�
+� ,
+�
+�
+˙ξ1
+˙ξ2
+˙ξ3
+�
+� =
+�
+�
+˙x
+˙θ sec2 θ
+˙y
+�
+�
+(14)
+and the input transformation
+� α1
+α2
+�
+=
+�
+u1 sec θ
+u2 cos2 θ − 2 ˙θ2 tan θ
+�
+,
+(15)
+the system (13) can be transformed into the second-order
+chained form system (1). Note that both transformation are
+singular point at θ = ±π/2.
+For the third joint of the underactuated manipulator with
+m3 = 0.6 kg, d3 = 0.3 m, and I3 = 4.5 × 10−3 kg · m2,
+steer from initial values χ(0)
+=
+[3.33 m, 1 m, 4.6 ×
+10−1 rad]⊤, ˙χ(0)
+=
+03 to the desired ones χ⋆
+=
+[1 m, 0 m, 0 rad]⊤, ˙χ⋆ = 03.
+Fig. 4 shows a simulation result with the period T = 1 s
+and the feedback gain kp = kd = 1. In this case, from (14),
+we have ξ(0) = [3, 0.5, 1]⊤ and ξ⋆ = [0.67, 0, 0]⊤. It can
+be confirmed that each state converges to the desired value in
+the both system representation.
+VOLUME 4, 2016
+5
+
+TEEEAccesSNakayama et al.: Preparation of Papers for IEEE Access
+(a) States and inputs of the second-order chained form system
+(b) Status and inputs of a three-joint underactuated manipulator
+FIGURE 4. Numerical results
+Furthermore, to verify the effect of feedback control, an-
+other case with an initial value error was simulated. For
+a rest-to-rest motion from χ(0)
+=
+[3.33 m, 1 m, 4.6 ×
+10−1 rad]⊤ to χ⋆ = [1.33 m, 0 m, 7.8 × 10−1 rad]⊤ with
+the zero velocities, the initial value error of +10% is given to
+θ, i.e., χ(0) = [3.33 m, 1 m, 5.1 × 10−1 rad]⊤. The result
+is shown in Fig. 5. The dashed lines indicate the target
+trajectories. It can be observed that tracking error due to the
+initial value error is alleviated over time.
+Similarly, when initial value errors of ±1%, ±10%, and
+±30% on θ are given the tracking errors at the end of control
+at t = 5T are summarized in Table 2. The terminal values
+of the tracking errors do not increase greatly even if the
+magnitude of the initial value error increases. Consequently,
+it is confirmed that the feedback of trajectory tracking has a
+sufficient effect on initial value errors. Note that the terminal
+error on x is relatively larger than the one on θ. The proposed
+control method attempts to settle the system by focusing on a
+single state every step. In addition, the state in which the con-
+trol step ends has no chance to be controlled directly. For such
+a state, there can be a secondary state transition that yields
+in control steps that focus on the other states. Therefore, if
+a state fails to converge into its reference trajectory within
+the control step due to initial value error or disturbance, it
+behaves unexpectedly until the end of the control strategy.
+In particular, ξ2—the state used for switching the systems—
+has a negative effect on the other states because the reference
+trajectory is not computed correctly. Furthermore, the error
+remaining in the velocity state (ξ4, ξ5, ξ6) causes a drift in
+the position state (ξ1, ξ2, ξ3) even if the input is zero in
+the following control steps. This is explained by numerical
+experiments shown in Fig. 5. Note that θ is related to ξ2
+as specified in (14). This means that θ affects the other
+states (x, y) when not converging completely. On the other
+hand, since ξ2 is settled in the final step (i.e., Step 5), the
+propagation from the error in the velocity state is small.
+Therefore, the error remaining in θ is considered to be smaller
+than in x.
+V. CONCLUSION
+In this paper, a novel control approach composed of sinu-
+soidal reference trajectories and a simple trajectory tracking
+6
+VOLUME 4, 2016
+
+TEEEAccesSNakayama et al.: Preparation of Papers for IEEE Access
+FIGURE 5. Given an initial value error(+10%)
+controller for the second-order chained form system was
+proposed. The key idea is a subsystem decomposition of the
+second-order chained form system by using state transitions.
+The effectiveness of the proposed algorithm was demon-
+strated by numerical results including an application to a
+three-joint underactuated manipulator. In particular, it can be
+confirmed that the feedback control works well against the
+initial value error.
+The future work of this research is to verify the proposed
+approach via experiments on an actual robot.
+REFERENCES
+[1] R. W. Brockett: “Asymptotic stability and feedback stabilization,” in
+Differential Geometric Control Theory (Eds. by R. W. Brockett, R. S.
+Millmann and H. J. Sussmann), Birkhauser, Boston, pp. 181–191, 1983.
+[2] J. Hauser, S. Sastry, and G. Meyer: “Nonlinear control design for slightly
+TABLE 2. Error from target value by the initial value error
+Case
+χ(0) − χ⋆
+χ(5T) − χ⋆
+w/o init. err.
+�
+�
+0 m
+0 m
+0 rad
+�
+�
+�
+�
+4.5 × 10−8 m
+4.4 × 10−7 m
+4.9 × 10−9 rad
+�
+�
+w/ +1 % init. err.
+�
+�
+0 m
+0 m
+4.6 × 10−3 rad
+�
+�
+�
+�
+2.9 × 10−3 m
+−6.9 × 10−4 m
+−8.5 × 10−4 rad
+�
+�
+w/ −1 % init. err.
+�
+�
+0 m
+0 m
+−4.6 × 10−3 rad
+�
+�
+�
+�
+−2.9 × 10−3 m
+7.4 × 10−4 m
+8.5 × 10−4 rad
+�
+�
+w/ +10 % init. err.
+�
+�
+0 m
+0 m
+4.6 × 10−2 rad
+�
+�
+�
+�
+3.0 × 10−2 m
+−4.8 × 10−3 m
+−8.8 × 10−3 rad
+�
+�
+w/ −10 % init. err.
+�
+�
+0 m
+0 m
+−4.6 × 10−2 rad
+�
+�
+�
+�
+−2.9 × 10−2 m
+9.9 × 10−3 m
+8.3 × 10−3 rad
+�
+�
+w/ +30 % init. err.
+�
+�
+0 m
+0 m
+1.4 × 10−1 rad
+�
+�
+�
+�
+9.1 × 10−2 m
+3.3 × 10−3 m
+−2.8 × 10−2 rad
+�
+�
+w/ −30 % init. err.
+�
+�
+0 m
+0 m
+−1.4 × 10−1 rad
+�
+�
+�
+�
+−8.5 × 10−2 m
+4.3 × 10−2 m
+2.3 × 10−2 rad
+�
+�
+non-minimum phase systems: application to V/STOL aircraft,” Automat-
+ica, Vol. 28, No. 4, pp. 665–679, 1992.
+[3] H. Arai, K. Tanie, and N. Shiroma: “Nonholonomic control of a three-
+DOF planar underactuated manipulator,” IEEE Transactions on Robotics
+Automation, Vol. 14, No. 5, pp. 681–695, 1998.
+[4] G. He, C. Zhang, W. Sun, and Z. Geng: “Stabilizing the second-order
+nonholonomic systems with chained form by finite-time stabilizing con-
+trollers,” Robotica, Vol. 34, pp. 2344–2367, 2016.
+[5] M. Nowicki, W. Respondek, J. Piasek, and K. Kozłowski: “Geometry and
+flatness of m-crane systems,” Bulletin of The Polish Academy of Sciences,
+Technical Sciences, Vol. 67, No. 5, pp. 893–903, 2019.
+[6] S.S. Ge, Z. Sun, T.H. Lee, and M.W. Spong: “Feedback linearization and
+stabilization of second-order nonholonomic chained systems,” Interna-
+tional Journal of Control, Vol. 74, pp. 1383–1392, 2001.
+[7] K. Pettersen and O. Egeland: “Exponential stabilization of an underac-
+tuated surface vessel,” in Proceedings of the 35th IEEE International
+Conference on Decision and Control (CDC’96), Vol. 1, pp. 967–972, 1996.
+[8] K. Pettersen and O. Egeland: “Position and attitude control of an au-
+tonomous underwater vehicle,” in Proceedings of the 35th IEEE Interna-
+tional Conference on Decision and Control (CDC’96), pp. 987–991, 1996.
+[9] A. De Luca and G. Oriolo: “Trajectory planning and control for planar
+robots with passive last joint,” International Journal of Robotics Research,
+Vol. 21, No. 5–6, pp. 575–590, 2002.
+[10] N.P.I. Aneke, H. Nijmeijer, and A.G. de Jager: “Tracking control of
+second-order chained form systems by cascaded backstepping,” Interna-
+tional Journal of Robust and Nonlinear Control, Vol. 13, No. 2, pp. 95–
+115.
+[11] T. Yoshikawa, K. Kobayashi, and T. Watanabe, “Design of a desirable
+trajectory and convergent control for 3-D.O.F manipulator with a nonholo-
+nomic constraint,” in Proceedings of the IEEE International Conference
+on Robotics and Automation (ICRA’00), San Francisco, CA, USA, Vol. 2,
+pp. 1805–1810, 2000.
+[12] M. Ito: “Motion planning of a second-order nonholonomic chained form
+system based on holonomy extraction,” Electronics, Vol. 8, No. 11,
+pp. 1337, 2019.
+[13] H. Sussmann: “A general theorem on local controllability,” SIAM Journal
+on Control and Optimization, Vol. 25, No. 1, pp. 158–194, 1987.
+[14] T. Nam, T. Tamura, T. Mita, and Y. Kim: “Control of the high-order
+VOLUME 4, 2016
+7
+
+TEEEAccesSNakayama et al.: Preparation of Papers for IEEE Access
+chained form system,” in Proceedings of the 41st SICE Annual Confer-
+ence, Vol. 4, pp. 2196–2201, 2002.
+[15] A. Hably and N. Marchand: “Bounded control of a general extended
+chained form systems,” in Proceedings of the 53rd IEEE Conference on
+Decision and Control (CDC’14), pp. 6342–6347, Los Angeles, CA, USA,
+2014.
+[16] MathWorks, “ode45: Solve nonstiff differential equations—medium order
+method,” Documentation for MATLAB R2022b, 2022. [Online]. Avail-
+able: https://www.mathworks.com/help/matlab/ref/ode45.html. Accessed
+on: Jan 11, 2023.
+MAYU NAKAYAMA was born in Kiyosu, Aichi,
+Japan in 1997. She received the B.S. and M.S. de-
+grees in information science and technology from
+Aichi Prefectural University (APU), Nagakute,
+Aichi, Japan, in 2020 and 2022.
+She is currently with DENSO Corporation. Her
+research interests include nonlinear control for
+underactuated systems.
+MASAHIDE ITO (M’10) was born in Nagoya,
+Aichi, Japan in 1979. He received the B.S., M.S.,
+and Ph.D. degrees in information science and tech-
+nology from Aichi Prefectural University (APU),
+Nagakute, Aichi, Japan, in 2002, 2004, and 2008.
+He is currently an Associate Professor with
+the School of Information Science and Technol-
+ogy, APU. His research interests include visual
+feedback control of robotic systems and nonlinear
+control for underactuated systems.
+8
+VOLUME 4, 2016
+
+TEEEAccesS

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+page_content='SY]  11 Jan 2023  TEEEAccesSDate of publication xxxx 00, 0000, date of current version xxxx 00, 0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Digital Object Identifier 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='1109/ACCESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='DOI  Trajectory Tracking Control of  The Second-order Chained Form System  by Using State Transitions  MAYU NAKAYAMA1, MASAHIDE ITO1, (Member, IEEE)  1School of Information Science and Technology, Aichi Prefectural University, Nagakute, Aichi, Japan  Corresponding author: Masahide Ito (e-mail: masa-ito@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='aichi-pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='jp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  ABSTRACT This paper proposes a novel control approach composed of sinusoidal reference trajectories  and trajectory tracking controller for the second-order chained form system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The system is well-known as  a canonical form for a class of second-order nonholonomic systems obtained by appropriate transformation  of the generalized coordinates and control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The system is decomposed into three subsystems, two of  them are the so-called double integrators and the other subsystem is a nonlinear system depending on one of  the double integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The double integrators are linearly controllable, which enables to transit the value of  the position state in order to modify the nature of the nonlinear system that depends on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Transiting the  value to “one” corresponds to modifying the nonlinear subsystem into the double integrator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' transiting the  value to “zero” corresponds to modifying the nonlinear subsystem into an uncontrollable linear autonomous  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Focusing on this nature, this paper proposes a feedforward control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Furthermore, from the  perspective of practical usefulness, the control strategy is extended into trajectory tracking control by using  proportional-derivative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The effectiveness of the proposed method is demonstrated through several  numerical experiments including an application to an underactuated manipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  INDEX TERMS  nonholonomic systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' state transitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' the second-order chained form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' trajectory  tracking control  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' INTRODUCTION  N  ONHOLONOMIC systems are nonlinear dynamical  systems with non-integrable differential constraints,  whose control problems have been attracting many re-  searchers and engineers for the last three decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The main  reason is that the nonholonomic systems do not satisfy  Brockett’s theorem [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The challenging and negative fact  means that there is not any smooth time-invariant feedback  control law to be able to stabilize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The applications  include various types of robotic vehicles and manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Some of them have been often used as a kind of bench-  mark platform to demonstrate the performance of a proposed  controller for not only a control problem of a single robotic  system and also a distributed control problem of multiagent  robotic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  The class subject to acceleration constraints—called  second-order nonholonomic systems—includes real exam-  ples such as a V/STOL aircraft [2], an underactuated manip-  ulator [3], an underactuated hovercraft [4], and a crane [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  These systems can be represented in a canonical system  called the second-order chained form by coordinate and in-  put transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The second-order chained form system  is also affected by Brockett’s theorem [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' To avoid this  difficulty, there are several ingenious control approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  The stabilizing controllers proposed in [4], [6]–[8] exploit  discontinuity or time-variance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' [3], [9] and [10] reduce the  control problem into a trajectory tracking problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Other  than those, [11] and [12] consider a motion planning problem  (in other words, a feedforward control problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  For the second-order chained form system, this paper  presents a novel control approach composed of sinusoidal  reference trajectories and a simple trajectory tracking con-  troller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The second-order chained form system is decomposed  into three subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Two of them are the so-called dou-  ble integrators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' the other subsystem is a nonlinear system  depending on one of the double integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The double  integrator is linearly controllable, which enables to transit the  value of the position state in order to modify the nature of the  nonlinear subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Transiting the value into “one” corre-  sponds to modifying the nonlinear subsystem into the double  2  VOLUME 4, 2016  IEEEAccesS  Multidisciplinary  Rapid Review  Open Access JournalNakayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' : Preparation of Papers for IEEE Access  integrator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' transiting the value into “zero” corresponds to  modifying the nonlinear subsystem into a linear autonomous  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Focusing on this nature, this paper proposes a feed-  forward control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Furthermore, from the perspective  of practical usefulness, the control strategy is extended into  trajectory tracking control by using proportional-derivative  (PD) feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  The remainder of this paper is organized as follows: Sec-  tion II presents that the second-order chained form system  can be decomposed to linear subsystems by using state  transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' On the basis of such system nature, Section III  proposes a feedforward control strategy and also a trajectory  tracking controller of PD feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Section IV applies the  proposed control approach to an underactuated manipulator  and evaluates it through numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The last  section concludes the paper with a summary and future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' SUBSYSTEM DECOMPOSITION OF THE  SECOND-ORDER CHAINED FORM SYSTEM BY USING  STATE TRANSITIONS  Consider the following second-order chained form system:  d2  dt2 ξ =  �  �  1  0  0  1  ξ2  0  �  � u,  (1)  where ξ = [ξ1, ξ2, ξ3]⊤ and u = [u1, u2]⊤ are the gen-  eralized coordinate vector and the generalized input vector,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' This system is well-known as a canonical form  for a class of second-order nonholonomic systems, which  can be resulted from the original dynamical model via an  appropriate transformation of the generalized coordinates  and control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Representing the system (1) as an affine  nonlinear system:  d  dt  �  �������  ξ1  ξ2  ξ3  ˙ξ1  ˙ξ2  ˙ξ3  �  �������  =  �  �������  ˙ξ1  ˙ξ2  ˙ξ3  0  0  0  �  �������  +  �  �������  0  0  0  1  0  ξ2  �  �������  u1 +  �  �������  0  0  0  0  1  0  �  �������  u2,  (2)  we  can  easily  confirm  that  the  equilibrium  points  (ξ⋆  1, ξ⋆  2, ξ⋆  3, 0, 0, 0), ξ⋆  1, ξ⋆  2, ξ⋆  3  ∈  R are small-time local  controllable (STLC) via Sussmann’s theorem [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  By focusing on the control inputs, the system (1) can be  decomposed into the following two subsystems:  d  dt  �  ���  ξ1  ξ3  ˙ξ1  ˙ξ3  �  ��� =  �  ���  0  0  1  0  0  0  0  1  0  0  0  0  0  0  0  0  �  ���  �  ���  ξ1  ξ3  ˙ξ1  ˙ξ3  �  ��� +  �  ���  0  0  1  ξ2  �  ��� u1,  (3a)  d  dt  � ξ2  ˙ξ2  �  =  � 0  1  0  0  � � ξ2  ˙ξ2  �  +  � 0  1  �  u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  (3b)  The subsystem (3b) with respect to the control input u2 is  a linear and controllable system represented by the double  integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' On the other hand, the subsystem (3a) with respect  to the input u1 is a four-dimensional nonlinear system whose  input matrix depends on the state variable ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The subsys-  tem (3a) can be further decomposed as follows:  d  dt  � ξ1  ˙ξ1  �  =  � 0  1  0  0  � � ξ1  ˙ξ1  �  +  � 0  1  �  u1,  (4a)  d  dt  � ξ3  ˙ξ3  �  =  � 0  1  0  0  � � ξ3  ˙ξ3  �  +  � 0  ξ2  �  u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  (4b)  The subsystem (4a) of the double integrator is linear and  controllable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' the subsystem (4b) inherits the nonlinearity of  the system (3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 1 shows a block diagram describing the above-  mentioned subsystem decomposition explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The state of  the subsystem (3b) can be transited to be a constant value  because of the linear controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' For example, by setting  time intervals where ξ2 is “zero” and also ξ2 is “one”, the  nonlinear subsystem (4b) can be treated as a linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  During the time interval of ξ2 = 1, the subsystems (4a)  and (4b) are linear which have the same double integrator  structure and control input u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' On the other hand, during  the time interval of ξ2 = 0, the subsystem (3a) becomes  a linear autonomous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=', uncontrollable) system and the  subsystem (4a) can be controlled independently from sub-  system (4b) by the control input u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Some conventional approaches such as in [14],  [10] and [15] exploit a different subsystem decomposition  that can decompose the system (1) as follows:  d  dt  � ξ1  ˙ξ1  �  =  � 0  1  0  0  � � ξ1  ˙ξ1  �  +  � 0  1  �  u1,  (5a)  d  dt  �  ���  ξ2  ξ3  ˙ξ2  ˙ξ3  �  ��� =  �  ���  0  0  1  0  0  0  0  1  0  0  0  0  u1  0  0  0  �  ���  �  ���  ξ2  ξ3  ˙ξ2  ˙ξ3  �  ��� +  �  ���  0  0  1  0  �  ��� u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  (5b)  The subsystem (5a) is the same with (4a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' the subsystem (5b)  has a variable structure depending on u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The subsystem (5b)  is linear when u1 is a non-zero constant, which reduces a  control problem of the second-order chained form system into  a simultaneous stabilizing problem of the two subsystems (5a)  and (5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' When u1 becomes zero before the end of control,  however, the subsystem (5b) will be uncontrollable with a  pole at the origin and then the whole of the subsystem loses  the controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' This subsystem decomposition, therefore,  needs control in consideration with u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' PROPOSED CONTROL APPROACH  In this paper, a control task of a rest-to-rest motion is ad-  dressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' For this task, the authors propose a control approach  composed of sinusoidal reference trajectories and a trajec-  tory tracking controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' In particular, a feedforward control  strategy that generates the reference trajectories exploits the  system decomposition based on state transition described in  the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  The feedforward control strategy using system switching  based on state transitions in ξ2 is as follows:  VOLUME 4, 2016  3  TEEEAccesSNakayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' : Preparation of Papers for IEEE Access  �  �  u1  ˙ξ1  ξ1  ×  �  �  ˙ξ3  ξ3  �  �  u2  ˙ξ2  ξ2  �  �  u1  ˙ξ1  ξ1  �  �  ˙ξ3  ξ3  �  �  u1  ˙ξ1  ξ1  �  �  ˙ξ3  ξ3  ⇐⇒  when ξ2 = 1  when ξ2 = 0  FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Subsystem decomposition of the second-order chained form by using ξ2’s state transitions between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Step 1  Transit ξ2 from any initial value to 1 by using  u1(t) = 0, u2(t) = q2(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Step 2  Transit ξ3 from any initial value to any desired  value (in conjunction with it, ξ1 is also driven)  by using u1(t) = q3(t), u2(t) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Step 3  Transit ξ2 from 1 to 0 by using u1(t)  =  0, u2(t) = q2(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Step 4  Transit ξ1 from any value in Step 2 to any de-  sired value by using u1(t) = q1(t), u2(t) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Step 5  Transit ξ2 from 0 to any desired value by using  u1(t) = 0, u2(t) = q2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  A control input in Step k (k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' , 5) is designed  by an appropriate sinusoidal function qi(t) (i = 1, 2, 3)  without any feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' This control strategy is namely mo-  tion planning, which naturally cannot deal with disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Therefore, we provide a trajectory tracking controller that  follow the reference trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Consider to drive the state variables ξi(t), ˙ξi(t) of the  system (1) by the following sinusoidal functions with pe-  riod T = 2π/ω and amplitude ak:  qi(t) = akω2 sin ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  (6)  Then, at time t (≤ kT), trajectories of a subsystem with non-  zero input are derived as  ˙ξi(t) = ˙ξi((k − 1)T) − akω cos ωt + akω,  (7)  ξi(t) = ξi((k − 1)T) + ˙ξi((k − 1)T)t  − ˙ξi((k − 1)T)(k − 1)T  − ak sin ωt + akωt − ak(k − 1)ωT,  (8)  respectively, where ξi((k − 1)T) and ˙ξi((k − 1)T) are initial  values of the state variables in Step k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Thus, at the end of  k-th period (t = kT), the state transitions are represented as  ˙ξi(kT) = ˙ξi((k − 1)T),  (9)  ξi(kT) = ξi((k − 1)T) + ˙ξi((k − 1)T)T + 2πak,  (10)  which means that a displacement of 2πak on ξi is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  This can be seen that the desired displacement is extracted by  using the amplitude ak as a tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  By setting the trajectories (6), (7), (8) as reference trajec-  tories qref  i (t), ξref  i (t), ˙ξref  i (t), a PD feedback control system  can be designed for trajectory tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' A linear system of a  double integrator can be represented in the following state-  space form with the state zi = [ξi, ˙ξi]⊤ and control input qi:  ˙zi =  � 0  1  0  0  �  � �� �  A  zi +  � 0  1  �  ����  b  qi(t, zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  (11)  In Step k, a feedback controller for trajectory tracking to zref  i  is given as follows:  qi(t, zi) = qref  i (t) + k ei,  (12)  where ei := zref  i  − zi and k = [kp, kd] is a feedback gain  matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The system (11) yields the closed-loop system ˙ei =  (A − bk)ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' By choosing the feedback gain k so that (A −  bk) is Hurwitz-stable, the closed-loop system is stabilized,  that is, zi tracks zref  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' NUMERICAL EXPERIMENTS  In this section, we evaluate the effectiveness of the proposed  control approach through numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Firstly, we validate the proposed controller for the second-  order chained form system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' A numerical experiment was per-  formed with T = 1 s, ξ(0) = [3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='5, 1]⊤, ˙ξ(0) = 03, ξ⋆ =  [1, 1, 0]⊤, and ˙ξ⋆ = 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 2 shows the simulation results  when choosing a1 = 1/(4π), a2 = a3 = a4 = −1/(2π),  and a5 = 1/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The ordinary differential equations was  numerically solved by ODE45 of MATLAB [16] with a rela-  tive tolerance of 1×10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The results indicate that each state  reached to the target value ξ⋆ with the remaining errors at t =  5T: ξ(5T)−ξ⋆ = [−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='7×10−8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='0×10−10, −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='7×10−8]⊤  and ˙ξ(5T)− ˙ξ⋆ = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='3×10−8, −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='9×10−9, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='1×10−8]⊤,  which means that the desired control is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Secondly, the proposed controller is applied to an underac-  tuated manipulator—a typical example of second-order non-  holonomic systems—as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' This manipulator  has first two joints being actuated and the last joint being  unactuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The system representation can be converted to  the second-order chained form system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Even if the third joint  cannot be driven due to no actuator, the acceleration (α1, α2)  acting on the center of percussion of the third link can be  treated equivalently as a control input owing to dynamic cou-  pling effect—the rotational actuation of the first and second  4  VOLUME 4, 2016  TEEEAccesSNakayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' : Preparation of Papers for IEEE Access  TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Definition of variables and parameters  (x, y) : position of the center of percussion of the third link in the frame O-XY ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  θ  : angle of the third link relative to X-axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  d3  : distance between the third joint and the center of mass of the third link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  m3  : mass of the third link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  I3  : moment of inertia mass of the third link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  LCoP : distance between the third joint and the center of percussion of the third link  �  LCoP := (I3 + m3d2  3)/(m3d3)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  α1  : translational acceleration along the third link;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  α2  : angular acceleration around the center of percussion of the third link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Simulation results of trajectory tracking control  joints propagates through the links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' For simplicity, assume  that there is no disturbance such as load, friction, linear and  nonlinear damping, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The main variables are defined as in  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Let χ := [x, y, θ]⊤ and α = [α1, α2]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Yoshikawa, et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' [11] provided a set of coordinate and input transforma-  tions to convert the manipulator dynamics derived from the  Lagrange’s equation of motion into the following system  𝜃  1st revolute joint  (actuated)  𝑥  𝑑!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  2nd revolute joint  (actuated)  3rd revolute joint  (unactuated)  𝑦  𝑋  𝑌  𝑂  center of percussion  of 3rd link  :  𝛼"  𝛼#  FIGURE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' A three-joint manipulator with passive third joint  representation:  ¨χ =  �  �  cos θ  0  sin θ  0  0  1  �  � α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  (13)  Using the coordinate transformation  �  �  ξ1  ξ2  ξ3  �  � =  �  �  x − LCoP  tan θ  y  �  � ,  �  �  ˙ξ1  ˙ξ2  ˙ξ3  �  � =  �  �  ˙x  ˙θ sec2 θ  ˙y  �  �  (14)  and the input transformation  � α1  α2  �  =  �  u1 sec θ  u2 cos2 θ − 2 ˙θ2 tan θ  ��  ,  (15)  the system (13) can be transformed into the second-order  chained form system (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Note that both transformation are  singular point at θ = ±π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  For the third joint of the underactuated manipulator with  m3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='6 kg, d3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='3 m, and I3 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='5 × 10−3 kg · m2,  steer from initial values χ(0)  =  [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='33 m, 1 m, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='6 ×  10−1 rad]⊤, ˙χ(0)  =  03 to the desired ones χ⋆  =  [1 m, 0 m, 0 rad]⊤, ˙χ⋆ = 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 4 shows a simulation result with the period T = 1 s  and the feedback gain kp = kd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' In this case, from (14),  we have ξ(0) = [3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='5, 1]⊤ and ξ⋆ = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='67, 0, 0]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' It can  be confirmed that each state converges to the desired value in  the both system representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  VOLUME 4, 2016  5  TEEEAccesSNakayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' : Preparation of Papers for IEEE Access  (a) States and inputs of the second-order chained form system  (b) Status and inputs of a three-joint underactuated manipulator  FIGURE 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Numerical results  Furthermore, to verify the effect of feedback control, an-  other case with an initial value error was simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' For  a rest-to-rest motion from χ(0)  =  [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='33 m, 1 m, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='6 ×  10−1 rad]⊤ to χ⋆ = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='33 m, 0 m, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='8 × 10−1 rad]⊤ with  the zero velocities, the initial value error of +10% is given to  θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=', χ(0) = [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='33 m, 1 m, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='1 × 10−1 rad]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The result  is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The dashed lines indicate the target  trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' It can be observed that tracking error due to the  initial value error is alleviated over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Similarly, when initial value errors of ±1%, ±10%, and  ±30% on θ are given the tracking errors at the end of control  at t = 5T are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The terminal values  of the tracking errors do not increase greatly even if the  magnitude of the initial value error increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Consequently,  it is confirmed that the feedback of trajectory tracking has a  sufficient effect on initial value errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Note that the terminal  error on x is relatively larger than the one on θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The proposed  control method attempts to settle the system by focusing on a  single state every step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' In addition, the state in which the con-  trol step ends has no chance to be controlled directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' For such  a state, there can be a secondary state transition that yields  in control steps that focus on the other states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Therefore, if  a state fails to converge into its reference trajectory within  the control step due to initial value error or disturbance, it  behaves unexpectedly until the end of the control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  In particular, ξ2—the state used for switching the systems—  has a negative effect on the other states because the reference  trajectory is not computed correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Furthermore, the error  remaining in the velocity state (ξ4, ξ5, ξ6) causes a drift in  the position state (ξ1, ξ2, ξ3) even if the input is zero in  the following control steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' This is explained by numerical  experiments shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Note that θ is related to ξ2  as specified in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' This means that θ affects the other  states (x, y) when not converging completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' On the other  hand, since ξ2 is settled in the final step (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=', Step 5), the  propagation from the error in the velocity state is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Therefore, the error remaining in θ is considered to be smaller  than in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' CONCLUSION  In this paper, a novel control approach composed of sinu-  soidal reference trajectories and a simple trajectory tracking  6  VOLUME 4, 2016  TEEEAccesSNakayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' : Preparation of Papers for IEEE Access  FIGURE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Given an initial value error(+10%)  controller for the second-order chained form system was  proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' The key idea is a subsystem decomposition of the  second-order chained form system by using state transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  The effectiveness of the proposed algorithm was demon-  strated by numerical results including an application to a  three-joint underactuated manipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' In particular, it can be  confirmed that the feedback control works well against the  initial value error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  The future work of this research is to verify the proposed  approach via experiments on an actual robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  REFERENCES  [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Brockett: “Asymptotic stability and feedback stabilization,” in  Differential Geometric Control Theory (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Brockett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  Millmann and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Sussmann), Birkhauser, Boston, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 181–191, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Hauser, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Sastry, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Meyer: “Nonlinear control design for slightly  TABLE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Error from target value by the initial value error  Case  χ(0) − χ⋆  χ(5T) − χ⋆  w/o init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  �  �  0 m  0 m  0 rad  �  �  �  �  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='5 × 10−8 m  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='4 × 10−7 m  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='9 × 10−9 rad  �  �  w/ +1 % init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  �  �  0 m  0 m  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='6 × 10−3 rad  �  �  �  �  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='9 × 10−3 m  −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='9 × 10−4 m  −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='5 × 10−4 rad  �  �  w/ −1 % init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  �  �  0 m  0 m  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='6 × 10−3 rad  �  �  �  �  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='9 × 10−3 m  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='4 × 10−4 m  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='5 × 10−4 rad  �  �  w/ +10 % init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  �  �  0 m  0 m  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='6 × 10−2 rad  �  �  �  �  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='0 × 10−2 m  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='8 × 10−3 m  −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
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+page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  �  �  0 m  0 m  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
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+page_content='9 × 10−2 m  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='9 × 10−3 m  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='3 × 10−3 rad  �  �  w/ +30 % init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  �  �  0 m  0 m  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='4 × 10−1 rad  �  �  �  �  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='1 × 10−2 m  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='3 × 10−3 m  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='8 × 10−2 rad  �  �  w/ −30 % init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  �  �  0 m  0 m  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='4 × 10−1 rad  �  �  �  �  −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='5 × 10−2 m  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='3 × 10−2 m  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='3 × 10−2 rad  �  �  non-minimum phase systems: application to V/STOL aircraft,” Automat-  ica, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 28, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 665–679, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [3] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Arai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Tanie, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Shiroma: “Nonholonomic control of a three-  DOF planar underactuated manipulator,” IEEE Transactions on Robotics  Automation, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 14, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 681–695, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' He, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Sun, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Geng: “Stabilizing the second-order  nonholonomic systems with chained form by finite-time stabilizing con-  trollers,” Robotica, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 34, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 2344–2367, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Nowicki, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Respondek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Piasek, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Kozłowski: “Geometry and  flatness of m-crane systems,” Bulletin of The Polish Academy of Sciences,  Technical Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 67, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 893–903, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Ge, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Sun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Lee, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Spong: “Feedback linearization and  stabilization of second-order nonholonomic chained systems,” Interna-  tional Journal of Control, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 74, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 1383–1392, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [7] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Pettersen and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Egeland: “Exponential stabilization of an underac-  tuated surface vessel,” in Proceedings of the 35th IEEE International  Conference on Decision and Control (CDC’96), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 967–972, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [8] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Pettersen and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Egeland: “Position and attitude control of an au-  tonomous underwater vehicle,” in Proceedings of the 35th IEEE Interna-  tional Conference on Decision and Control (CDC’96), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 987–991, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' De Luca and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Oriolo: “Trajectory planning and control for planar  robots with passive last joint,” International Journal of Robotics Research,  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 21, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 5–6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 575–590, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [10] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Aneke, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Nijmeijer, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' de Jager: “Tracking control of  second-order chained form systems by cascaded backstepping,” Interna-  tional Journal of Robust and Nonlinear Control, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 95–  115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [11] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Yoshikawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Kobayashi, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Watanabe, “Design of a desirable  trajectory and convergent control for 3-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='F manipulator with a nonholo-  nomic constraint,” in Proceedings of the IEEE International Conference  on Robotics and Automation (ICRA’00), San Francisco, CA, USA, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 2,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 1805–1810, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Ito: “Motion planning of a second-order nonholonomic chained form  system based on holonomy extraction,” Electronics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 8, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 11,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 1337, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Sussmann: “A general theorem on local controllability,” SIAM Journal  on Control and Optimization, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 25, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 158–194, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [14] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Nam, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Tamura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Mita, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Kim: “Control of the high-order  VOLUME 4, 2016  7  TEEEAccesSNakayama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' : Preparation of Papers for IEEE Access  chained form system,” in Proceedings of the 41st SICE Annual Confer-  ence, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 2196–2201, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Hably and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Marchand: “Bounded control of a general extended  chained form systems,” in Proceedings of the 53rd IEEE Conference on  Decision and Control (CDC’14), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' 6342–6347, Los Angeles, CA, USA,  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  [16] MathWorks, “ode45: Solve nonstiff differential equations—medium order  method,” Documentation for MATLAB R2022b, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Avail-  able: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='mathworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='com/help/matlab/ref/ode45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Accessed  on: Jan 11, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  MAYU NAKAYAMA was born in Kiyosu, Aichi,  Japan in 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' She received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' de-  grees in information science and technology from  Aichi Prefectural University (APU), Nagakute,  Aichi, Japan, in 2020 and 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  She is currently with DENSO Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' Her  research interests include nonlinear control for  underactuated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  MASAHIDE ITO (M’10) was born in Nagoya,  Aichi, Japan in 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' He received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=',  and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' degrees in information science and tech-  nology from Aichi Prefectural University (APU),  Nagakute, Aichi, Japan, in 2002, 2004, and 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  He is currently an Associate Professor with  the School of Information Science and Technol-  ogy, APU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content=' His research interests include visual  feedback control of robotic systems and nonlinear  control for underactuated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}
+page_content='  8  VOLUME 4, 2016  TEEEAccesS' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE3T4oBgHgl3EQfUwpD/content/2301.04453v1.pdf'}

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+A Variant Prescribed Curvature Flow on Closed Surfaces with
+Negative Euler Characteristic
+Franziska Borer∗
+Peter Elbau†
+Tobias Weth‡
+Abstract
+On a closed Riemannian surface (M, ¯g) with negative Euler characteristic, we study the problem of
+finding conformal metrics with prescribed volume A > 0 and the property that their Gauss curvatures
+fλ = f + λ are given as the sum of a prescribed function f ∈ C∞(M) and an additive constant λ. Our
+main tool in this study is a new variant of the prescribed Gauss curvature flow, for which we establish local
+well-posedness and global compactness results. In contrast to previous work, our approach does not require
+any sign conditions on f. Moreover, we exhibit conditions under which the function fλ is sign changing and
+the standard prescribed Gauss curvature flow is not applicable.
+Acknowledgment
+This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project
+408275461 (Smoothing and Non-Smoothing via Ricci Flow).
+We would like to thank Esther Cabezas–Rivas for helpful discussions.
+1. Introduction
+Let (M, ¯g) be a two-dimensional, smooth, closed, connected, oriented Riemann manifold endowed with a smooth
+background metric ¯g. A classical problem raised by Kazdan and Warner in [11] and [10] is the question which
+smooth functions f : M → R arise as the Gauss curvature Kg of a conformal metric g(x) = e2u(x)¯g(x) on M
+and to characterise the set of all such metrics.
+For a constant function f, this prescribed Gauss curvature problem is exactly the statement of the Uni-
+formisation Theorem (see e.g. [16], [12]):
+There exists a metric g which is pointwise conformal to ¯g and has constant Gauss curvature Kg ≡ ¯K ∈ R.
+We now use this statement to assume in the following without loss of generality that the background metric
+¯g itself has constant Gauss curvature K¯g ≡ ¯K ∈ R. Furthermore we can normalise the volume of (M, ¯g) to
+one. We recall that the Gauss curvature of a conformal metric g(x) = e2u(x)¯g(x) on M is given by the Gauss
+equation
+Kg(x) = e−2u(x)(−∆¯gu(x) + ¯K).
+(1.1)
+Therefore the problem reduces to the question for which functions f there exists a conformal factor u solving
+the equation
+− ∆¯gu(x) + ¯K = f(x)e2u(x)
+in M.
+(1.2)
+Given a solution u, we may integrate (1.2) with respect to the measure µ¯g on M induced by the Riemannian
+volume form. Using the Gauss–Bonnet Theorem, we then obtain the identity
+�
+M
+f(x)dµg(x) =
+�
+M
+¯Kdµ¯g(x) = ¯K vol¯g = ¯K = 2πχ(M),
+(1.3)
+where dµg(x) = e2u(x)dµ¯g(x) is the element of area in the metric g(x) = e2u(x)¯g(x).
+We note that (1.3)
+immediately yields necessary conditions on f for the solvability of the prescribed Gauss curvature problem. In
+particular, if ±χ(M) > 0, then ±f must be positive somewhere. Moreover, if χ(M) = 0, then f must change
+sign or must be identically zero.
+∗Technical University of Berlin, Faculty II—Mathematics and Natural Sciences, Straße des 17. Juni 136, 10623 Berlin, Germany
+email: [email protected]
+†Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
+email: [email protected]
+‡Goethe University Frankfurt, Institut f¨ur Mathematik, Robert-Mayer-Straße 10, 60629 Frankfurt, Germany
+email: [email protected]
+1
+arXiv:2301.12015v1  [math.AP]  27 Jan 2023
+
+2
+Franziska Borer, Peter Elbau, Tobias Weth
+In the present paper we focus on the case χ(M) < 0, so M is a surface of genus greater than one and
+¯K < 0. The complementary cases χ(M) ≥ 0—i.e., the cases where M = S2 or M = T, the 2-torus—will be
+discussed briefly at the end of this introduction, and we also refer the reader to [18, 19, 2, 8] and the references
+therein. Multiplying equation (1.2) with the factor e−2u and integrating over M with respect to the measure µ¯g,
+we get the following necessary condition—already mentioned by Kazdan and Warner in [11]—for the average
+¯f :=
+1
+vol¯g
+�
+M f(x)dµ¯g(x), with vol¯g :=
+�
+M dµ¯g(x):
+¯f =
+1
+vol¯g
+�
+M
+f(x)dµ¯g(x) =
+�
+M
+(−∆¯gu(x) + ¯K)e−2u(x)dµ¯g(x)
+=
+�
+M
+(−2|∇¯gu(x)|2
+¯g + ¯K)e−2u(x)dµ¯g(x) < 0.
+(1.4)
+This condition is not sufficient. Indeed, it has already been pointed out in [11, Theorem 10.5] that in the case
+χ(M) < 0 there always exist functions f ∈ C∞(M) with ¯f < 0 and the property that (1.2) has no solution.
+We recall that solutions of (1.2) can be characterised as critical points of the functional
+Ef : H1(M, ¯g) → R;
+Ef(u) := 1
+2
+�
+M
+�
+|∇¯gu(x)|2
+¯g + 2 ¯Ku(x) − f(x)e2u(x)�
+dµ¯g(x).
+(1.5)
+Under the assumption χ(M) < 0, i.e., ¯K < 0, the functional Ef is strictly convex and coercive on H1(M, ¯g)
+if f ≤ 0 and f does not vanish identically. Hence, as noted in [7], the functional Ef admits a unique critical
+point uf ∈ H1(M, ¯g) in this case, which is a strict absolute minimiser of Ef and a (weak) solution of (1.2).
+The situation is more delicate in the case where fλ = f0 + λ, where f0 ≤ 0 is a smooth, nonconstant function
+on M with maxx∈M f0(x) = 0, and λ > 0. In the case where λ > 0 sufficiently small (depending on f0), it was
+shown in [7] and [1] that the corresponding functional Efλ admits a local minimiser uλ and a further critical
+point uλ ̸= uλ of mountain pass type.
+These results motivate our present work, where we suggest a new flow approach to the prescribed Gausss
+curvature problem in the case χ(M) < 0. It is important to note here that there is an intrinsic motivation to
+formulate the static problem in a flow context. Typically, elliptic theories are regarded as the static case of the
+corresponding parabolic problem; in that sense, many times the better-understood elliptic theory has been a
+source of intuition to generalise the corresponding results in the parabolic case. Examples of this feedback are
+minimal surfaces/mean curvature flow, harmonic maps/solutions of the heat equation, and the uniformisation
+theorem/the two-dimensional normalised Ricci flow.
+In this spirit, a flow approach to (1.2), the so-called prescribed Gauss curvature flow, was first introduced
+by Struwe in [18] (and [2]) for the case M = S2 with the standard background metric and a positive function
+f ∈ C2(M). More precisely, he considers a family of metrics (g(t, ·))t≥0 which fulfils the initial value problem
+∂tg(t, x) = 2(α(t)f(x) − Kg(t,·)(x))g(t, x)
+in (0, T) × M;
+(1.6)
+g(0, x) = g0(x)
+on {0} × M,
+(1.7)
+with
+α(t) =
+�
+M Kg(t,·)(x)dµg(t,·)(x)
+�
+M f(x)dµg(t,·)(x)
+=
+2πχ(M)
+�
+M f(x)dµg(t,·)(x).
+(1.8)
+This choice of α(t) ensures that the volume of (M, g(t, ·)) remains constant throughout the deformation, i.e.,
+�
+M
+dµg(t,·)(x) =
+�
+M
+e2u(t,x)dµ¯g(x) ≡ volg0
+for all t ≥ 0,
+where g0 denotes the initial metric on M.
+Equivalently one may consider the evolution equation for the
+associated conformal factor u given by g(t, x) = e2u(t,x)¯g(x):
+∂tu(t, x) = α(t)f(x) − Kg(t,·)(x)
+in (0, T) × M;
+(1.9)
+u(0, x) = u0(x)
+on {0} × M.
+(1.10)
+Here the initial value u0 is given by g0(x) = e2u0(x)¯g(x). The flow associated to this parabolic equation is
+usually called the prescribed Gauss curvature flow. With the help of this flow, Struwe [18] provided a new proof
+of a result by Chang and Yang [6] on sufficient criteria for a function f to be the Gauss curvature of a metric
+g(x) = e2u(x)gS2(x) on S2. He also proved the sharpness of these criteria.
+In the case of surfaces with genus greater than one, i.e., with negative Euler characteristic, the prescribed
+Gauss curvature flow was used by Ho in [9] to prove that any smooth, strictly negative function on a surface
+with negative Euler characteristic can be realised as the Gaussian curvature of some metric. More precisely,
+
+Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic
+3
+assuming that χ(M) < 0 and that f ∈ C∞(M) is a strictly negative function, he proves that equation (1.9) has
+a solution which is defined for all times and converges to a metric g∞ with Gaussian curvature Kg∞ satisfying
+Kg∞(x) = α∞f(x)
+for some constant α∞.
+While the prescribed Gauss curvature flow is a higly useful tool in the cases where f is of fixed sign, it
+cannot be used in the case where f is sign-changing. Indeed, in this case we may have
+�
+M f(x)dµg(t,·)(x) = 0
+along the flow and then the normalising factor α(t) is not well-defined by (1.8). As a consequence, a long-time
+solution of (1.9) might not exist. In particular, the static existence results of [7] and [1] can not be recovered
+and reinterpreted with the standard prescribed Gauss curvature flow.
+In this paper we develop a new flow approach to (1.2) in the case χ(M) < 0 for general f ∈ C∞(M), which
+sheds new light on the results in [7], [1] and [9]. The main idea is to replace the multiplicative normalisation in
+(1.9) by an additive normalisation, as will be described in details in the next chapter.
+At this point, it should be noted that the normalisation factor α(t) in the prescribed Gauss curvature flow
+given by (1.8) is also not the appropriate choice in the case of the torus, where, as noted before, f has to
+change sign or be identically zero in order to arise as the Gauss curvature of a conformal metric. The case of
+the torus was considered by Struwe in [19], where, in particular, he used to a flow approach to reprove and
+partially improve a result by Galimberti [8] on the static problem. In this approach, the normalisation in (1.8)
+is replaced by
+α(t) =
+�
+M f(x)Kg(t,·)(x)dµg(t,·)(x)
+�
+M f 2(x)dµg(t,·)(x)
+.
+(1.11)
+With this choice, Struwe shows that for any smooth
+u0 ∈ C∗ :=
+�
+u ∈ H1(M, ¯g) |
+�
+M
+f(x)e2u(x)dµ¯g(x) = 0,
+�
+M
+e2u(x)dµ¯g(x) = 1
+�
+there exists a unique, global smooth solution u of (1.9) satisfying u(t, ·) ∈ C∗ for all t > 0. Moreover, u(t, ·) →
+u∞(·) in H2(M, ¯g) (and smoothly) as t → ∞ suitably, where u∞ + c∞ is a smooth solution of (1.2) for some
+c∞ ∈ R.
+In principle, the normalisation (1.11) could also be considered in the case χ(M) < 0, but then the flow is not
+volume-preserving anymore, which results in a failure of uniform estimates for solutions of (1.9). Consequently,
+we were not able to make use of the associated flow in this case.
+The paper is organised as follows. In Section 2 we set up the framework for the new variant of the prescribed
+Gauss curvature flow with additive normalisation, and we collect basic properties of it. In Section 3, we then
+present our main result on the long-time existence and convergence of the flow (for suitable times tk → ∞) to
+solutions of the corresponding static problem. In particular, our results show how sign changing functions of
+the form fλ = f0 + λ arise depending on various assumptions on the shape of f0 and on the fixed volume A of
+M with respect to the metric g(t). Before proving our results on the time-dependent problem, we first derive,
+in Section 4, some results on the static problem with volume constraint. Most of these results will then be used
+in Section 5, where the parabolic problem is studied in detail and the main results of the paper are proved. In
+the appendix, we provide some regularity estimates and a variant of a maximum princple for a class of linear
+evolution problems with H¨older continuous coefficients.
+In the remainder of the paper, we will use the short form f, g(t), u(t), Kg(t), volg(t) :=
+�
+M dµg(t) =
+�
+M e2u(t)dµ¯g, and so on instead of f(x), g(t, x), u(t, x), Kg(t,·)(x),
+�
+M dµg(t,·)(x) =
+�
+M e2u(t,x)dµ¯g(x), et cetera.
+2. A New Flow Approach and Some of its Properties
+Let f ∈ C∞(M) be a smooth function. We consider now the additive rescaled prescribed Gauss curvature flow
+given by
+∂tu(t) = f − Kg(t) − α(t) = f − e−2u(t)(∆¯gu(t) − ¯K) − α(t)
+in (0, T) × M,
+(2.1)
+where α(t) is chosen such that the volume volg(t) of M with respect to g(t) = e2u(t)¯g remains constant along
+the flow, that is, we require the condition
+1
+2
+d
+dt volg(t) =
+�
+M
+∂tu(t)dµg(t) =
+�
+M
+(f − Kg(t) − α(t))dµg(t) = 0.
+(2.2)
+Solving for α(t) then we find
+α(t) =
+1
+volg(t)
+��
+M
+fdµg(t) − ¯K
+�
+.
+
+4
+Franziska Borer, Peter Elbau, Tobias Weth
+So, starting with
+u0 ∈ Cp,A :=
+�
+v ∈ W 2,p(M, ¯g) |
+�
+M
+e2vdµ¯g = A
+�
+,
+p > 2,
+for a given A > 0, we have
+volg(t) = volg(0) = volg0 = A,
+for all t ≥ 0,
+hence we can define
+αA(t) = 1
+A
+��
+M
+fdµg(t) − ¯K
+�
+.
+(2.3)
+Therefore in the following we consider the flow
+∂tu(t) = f − Kg(t) − αA(t)
+in (0, T) × M;
+(2.4)
+u(0) = u0 ∈ Cp,A
+on {0} × M,
+(2.5)
+with αA(t) is chosen like in (2.3). We can now state some first properties of the flow.
+Proposition 2.1. Let u be a (sufficiently smooth) solution of (2.4), (2.5). Then
+1. the volume volg(t) of (M, g(t)) is preserved along the flow, i.e., volg(t) ≡ volg0 = A for all t ≥ 0;
+2. along this trajectory, we have a uniform bound for α given by
+α(t) ≥ min
+x∈M f(x) + | ¯K|
+A =: α1 > −∞
+(2.6)
+and
+α(t) ≤ max
+x∈M f(x) + | ¯K|
+A =: α2 < ∞;
+(2.7)
+3. the flow is invariant under adding or subtracting a constant C > 0 to the function f;
+4. and the energy Ef, defined in (1.5), is decreasing in time along the flow, so
+Ef(u(t)) ≤ Ef(u0)
+for all t ≥ 0.
+Proof. The first statement directly follows by (2.2) and the choice of α in (2.3).
+The second one we get since f is smooth and volg(t) = A.
+To show the invariance of the flow, let C > 0 be a constant. We then replace f by f ± C in (2.4) and see that
+f ± C − Kg(t) − 1
+A
+��
+M
+(f ± C)dµg(t) − ¯K
+�
+= f − Kg(t) − 1
+A
+��
+M
+fdµg(t) − ¯K
+�
+= ∂tu(t).
+So, the flow (2.4) is left unchanged if we replace f by f ± C for a constant C > 0.
+To see that the energy Ef is decreasing along the flow, we use (2.2) and get
+d
+dtEf(u(t)) =
+�
+M
+(−∆¯gu(t) + ¯K − fe2u(t))∂tu(t)dµ¯g
+=
+�
+M
+((−∆¯gu(t) + ¯K)e−2u(t) − f)e2u(t)∂tu(t)dµ¯g
+=
+�
+M
+((−∆¯gu(t) + ¯K)e−2u(t) − f)∂tu(t)dµg(t)
+=
+�
+M
+(Kg(t) − f)∂tu(t)dµg(t) =
+�
+M
+(Kg(t) − f + α(t))∂tu(t)dµg(t)
+= −
+�
+M
+|∂tu(t)|2dµg(t) ≤ 0.
+(2.8)
+Therefore on an interval [0, T], we have the uniform a-priori bound
+Ef(u(T)) +
+� T
+0
+�
+M
+|∂tu(t)|2dµg(t)dt = Ef(u(0))
+(2.9)
+for any T > 0.
+
+Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic
+5
+3. Main Results
+The following is our first main result.
+Theorem 3.1. Let f ∈ C∞(M), p > 2, and u0 ∈ Cp,A for a given A > 0. Then the initial value problem (2.4),
+(2.5) admits a unique global solution u ∈ C([0, ∞); C(M)) ∩ C([0, ∞); H1(M, ¯g)) ∩ C∞((0, ∞) × M) satisfying
+the energy bound Ef(u(t)) ≤ Ef(u0) for all t.
+Moreover, u is uniformly bounded in the sense that
+sup
+t>0
+∥u(t)∥L∞(M,¯g) < ∞.
+Furthermore, as t → ∞ suitably, u converges to a function u∞ in H2(M, ¯g) solving the equation
+− ∆¯gu + ¯K = fλe2u
+in M,
+(3.1)
+where fλ := f + λ with
+λ = 1
+A
+�
+¯K −
+�
+M
+fe2u∞dµ¯g
+�
+.
+(3.2)
+In other words, u∞ induces a metric g∞ with Gauss curvature Kg∞ satisfying
+Kg∞(x) = fλ(x) = f(x) + λ
+for
+x ∈ M.
+(3.3)
+Remark 3.2. For functions f < 0, the convergence of the flow (1.9) is shown in [9]. For the additive rescaled
+flow (2.4) with initial data (2.5) we get convergence for arbitrary functions f ∈ C∞(M). In general we do not
+have any information about λ and therefore no information about the sign of fλ in Theorem 3.1. On the other
+hand, more information can be derived for certain functions f ∈ C∞(M) and certain values of A > 0.
+(i) In the case where A ≤ −
+¯
+K
+∥f∥L∞(M,¯g) , it follows that
+λ = 1
+A
+�
+¯K −
+�
+M
+fe2udµ¯g
+�
+≤
+¯K
+A + ∥f∥L∞(M,¯g)
+A
+�
+M
+e2udµ¯g =
+¯K
+A + ∥f∥L∞(M,¯g) ≤ 0
+for every solution u ∈ C2,A :=
+�
+v ∈ H2(M, ¯g) |
+�
+M e2vdµ¯g = 0
+�
+of the static problem (3.1), and therefore
+this also applies to λ in Theorem 3.1 in this case.
+(ii) The following theorems show that fλ in Theorem 3.1 may change sign if A > −
+¯
+K
+∥f∥L∞(M,¯g) , so in this case
+we get a solution of the static problem (1.2) for sign-changing functions f ∈ C∞(M) by using the additive
+rescaled prescribed Gauss curvature flow (2.4).
+Theorem 3.3. Let p > 2. For every A > 0 and c > −
+¯
+K
+A there exists ε = ε(c, A, ¯K) > 0 with the following
+property.
+If u0 ≡ 1
+2 log(A) ∈ Cp,A and f ∈ C∞(M) with −c ≤ f ≤ 0 and ∥f + c∥L1(M,¯g) < ε is chosen in Theorem 3.1,
+then the value λ defined in (3.2) is positive.
+In particular, if f has zeros on M, then fλ in (3.3) is sign changing.
+Under fairly general assumptions on f, we can prove that λ > 0 if A is sufficiently large and u0 ∈ Cp,A is
+chosen suitably.
+Theorem 3.4. Let f ∈ C∞(M) be nonconstant with maxx∈M f(x) = 0. Then there exists κ > 0 with the
+property that for every A ≥ κ there exists u0 ∈ Cp,A such that the value λ defined in (3.2) is positive.
+In fact we have even more information on the associated limit u∞ in this case, see Corollary 4.8 below.
+It remains open how large λ can be depending on A and f. The only upper bound we have is
+λ < −
+�
+M
+fdµ¯g,
+(3.4)
+since we must have
+¯fλ =
+1
+vol¯g
+�
+M
+fλdµ¯g =
+�
+M
+fdµ¯g + λ
+!< 0,
+so that fλ fulfills the necessary condition (1.4) provided by Kazdan and Warner in [11].
+
+6
+Franziska Borer, Peter Elbau, Tobias Weth
+4. The static Minimisation Problem with Volume Constraint
+To obtain additional information on the limiting function u∞ and the value λ ∈ R associated to it by (3.2) and
+(3.3), we need to consider the associated static setting for the prescribed Gauss curvature problem with the
+additional condition of prescribed volume.
+Before going into the details of this static problem, we recall an important and highly useful estimate.
+The following lemma (see e.g. [5, Corollary 1.7]) is a consequence of the Trudinger’s inequality [20] which was
+improved by Moser in [15] (for more details see e.g. [19, Theorem 2.1 and Theorem 2.2]):
+Lemma 4.1. For a two-dimensional, closed Riemannian manifold (M, ¯g) there are constants η > 0 and CMT >
+0 such that
+�
+M
+e(u−¯u)dµ¯g ≤ CMT exp
+�
+η∥∇¯gu∥2
+L2(M,¯g)
+�
+(4.1)
+for all u ∈ H1(M, ¯g) where
+¯u :=
+1
+vol¯g
+�
+M
+u dµ¯g =
+�
+M
+u dµ¯g,
+in view of our assumption that vol¯g = 1.
+As a consequence of Lemma 4.1, we have
+�
+M
+epudµ¯g = ep¯u
+�
+M
+e(pu− ¯
+pu)dµ¯g ≤ ep¯uCMT exp
+�
+η∥∇¯g(pu)∥2
+L2(M,¯g)
+�
+< ∞
+for every u ∈ H1(M, ¯g) and p > 0. Consequently, for a given A > 0, the set
+C1,A :=
+�
+u ∈ H1(M, ¯g) | V (u) :=
+�
+M
+e2udµ¯g = A
+�
+(4.2)
+is well defined and coincides with the closure of C2,A with respect to the H1-norm. We also note that
+¯u ≤ 1
+2 log(A)
+for u ∈ C1,A,
+(4.3)
+since by Jensen’s inequality and our assumption that vol¯g = 1 we have
+2¯u = −
+�
+M
+2udµ¯g =
+�
+M
+2udµ¯g ≤ log
+�
+−
+�
+e2udµ¯g
+�
+= log(A)
+for u ∈ C1,A.
+Furthermore we want to recall the Gagliardo–Nirenberg–Ladyˇzhenskaya interpolation, see e.g. [4].
+Lemma 4.2 (Gagliardo–Nirenberg–Ladyˇzhenskaya inequality). There exists a constant CGNL > 0 such that
+we have for every ζ ∈ H1(M, ¯g) the inequality
+∥ζ∥4
+L4(M,¯g) ≤ CGNL∥ζ∥2
+L2(M,¯g)∥ζ∥2
+H1(M,¯g).
+Now we enter the details of the static prescribed Gauss curvature problem with volume constraint. In this
+problem, we wish to find, for given f ∈ C∞(M) and A > 0, critical points of the restriction of the functional
+Ef defined in (1.5) to the set C1,A. A critical point u ∈ C1,A of this restriction is a solution of (3.1) for some
+λ ∈ R, where, here and in the following, we put again fλ := f + λ ∈ C∞(M). In other words, such a critical
+point induces, similarly as the limit u∞ in Theorem 3.1, a metric gu with Gauss curvature Kgu satisfying
+Kgu(x) = fλ(x) = f(x) + λ. The unknown λ ∈ R arises in this context as a Lagrangian multiplier and is a
+posteriori characterised again by
+λ = 1
+A
+�
+¯K −
+�
+M
+fe2udµ¯g
+�
+.
+In the study of critical points of the restriction of Ef to C1,A, it is natural to consider the minimisation
+problem first. For this we set
+mf,A =
+inf
+u∈C1,A Ef(u).
+We have the following estimates for mf,A:
+Lemma 4.3. Let f ∈ C∞(M), A > 0. Then we have
+mf,A ≤ 1
+2
+�
+¯K log(A) − A
+�
+M
+fdµ¯g
+�
+.
+(4.4)
+Moreover, if max f ≥ 0, then we have
+lim sup
+A→∞
+mf,A
+A
+≤ 0.
+(4.5)
+
+Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic
+7
+Proof. Let u0(A) ≡ 1
+2 log(A), so that
+�
+M e2u0(A)dµ¯g = A. Hence u0(A) is the (unique) constant function in
+C1,A, and
+mf,A ≤ Ef(u0(A)) = 1
+2
+�
+M
+(|∇¯gu0(A)|2
+¯g + 2 ¯Ku0(A) − fe2u0(A))dµ¯g
+= 1
+2
+�
+M
+( ¯K log(A) − fA)dµ¯g
+= 1
+2
+�
+¯K log(A) − A
+�
+M
+fdµ¯g
+�
+.
+This shows (4.4). To show (4.5), we let ε > 0. Since f ∈ C∞(M) and max f ≥ 0 by assumption, there exists
+an open set Ω ⊂ M with f ≥ −ε on Ω. Next, let ψ ∈ C∞(M), ψ ≥ 0, be a function supported in Ω and with
+∥ψ∥L∞(M,¯g) = 2. Consequently, the set Ω′ := {x ∈ M | ψ > 1} is a nonempty open subset of Ω, and therefore
+µ¯g(Ω′) > 0.
+Next we consider the continuous function
+h : [0, ∞) → [0, ∞);
+h(τ) =
+�
+M
+e2τψdµ¯g
+and we note that h(0) =
+�
+M dµ¯g = 1, and that
+h(τ) ≥
+�
+Ω′ e2τψdµ¯g ≥ e2τµ¯g(Ω′)
+for τ ≥ 0.
+Hence for every A ≥ 1 there exists
+0 ≤ τA ≤ 1
+2
+�
+log(A) − log(µ¯g(Ω′))
+�
+(4.6)
+with h(τA) = A and therefore τAψ ∈ C1,A. Consequently,
+mf,A ≤ Ef(τAψ) = 1
+2
+�
+M
+(|∇¯gτAψ|2
+¯g + 2 ¯KτAψ − fe2τAψ)dµ¯g
+= τ 2
+Ac1 − τAc2 − c3 − 1
+2
+�
+Ω
+fe2τAψdµ¯g
+with
+c1 = 1
+2
+�
+M
+|∇¯gψ|2
+¯gdµ¯g,
+c2 = − ¯K
+�
+M
+ψdµ¯g
+and
+c3 = 1
+2
+�
+M\Ω
+fdµ¯g.
+Since f ≥ −ε on Ω, we thus deduce that
+mf,A ≤ τ 2
+Ac1 − 2τAc2 + c3 + ε
+2
+�
+Ω
+e2τAψdµ¯g ≤ τ 2
+Ac1 − 2τAc2 + c3 + εA
+2 .
+Since τA
+A → 0 as A → ∞ by (4.6), we conclude that
+lim sup
+A→∞
+mf,A
+A
+≤ ε
+2.
+Since ε > 0 was chosen arbitrarily, (4.5) follows.
+Lemma 4.4. Let f ∈ C∞(M) nonconstant with maxx∈M f(x) = 0. For every ε > 0 there exists κ0 > 0 with
+the following property. If A ≥ κ0 and u ∈ C1,A is a solution of
+− ∆¯gu + ¯K = (f + λ)e2u
+(4.7)
+for some λ ∈ R with Ef(u) < εA
+2 , then we have λ < ε.
+Proof. For given ε > 0, we may choose κ0 > 0 sufficiently large so that | ¯
+K|
+2
+log(A)
+|A|
+< ε
+2 for A ≥ κ0.
+Now, let A ≥ κ0, and let u ∈ C1,A be a solution of (4.7) satisfying Ef(u) < εA
+2 . Integrating (4.7) over M
+with respect to µ¯g and using that vol¯g(M) = 1 and
+�
+M e2udµ¯g = A, we obtain
+λ = 1
+A
+�
+¯K −
+�
+M
+fe2udµ¯g
+�
+≤ − 1
+A
+�
+M
+fe2udµ¯g
+= 1
+A
+�
+Ef(u) − 1
+2
+�
+M
+(|∇¯gu|2
+¯g + 2 ¯Ku)dµ¯g
+�
+≤ 1
+A
+�
+Ef(u) + | ¯K|¯u
+�
+≤ ε
+2 + | ¯K|
+2
+log(A)
+A
+< ε,
+as claimed. Here we used (4.3) to estimate ¯u.
+
+8
+Franziska Borer, Peter Elbau, Tobias Weth
+Proposition 4.5. Let f ∈ C∞(M) be a nonconstant function with maxx∈M f(x) = 0. Moreover, let λn → 0+
+for n → ∞, and let (un)n∈N be a sequence of solutions of
+− ∆¯gun + ¯K = (f + λn)e2un
+in M
+(4.8)
+which are weakly stable in the sense that
+�
+M
+(|∇¯gh|2
+¯g − 2(f + λn)e2unh2)dµ¯g ≥ 0
+for all h ∈ H1(M).
+(4.9)
+Then un → u0 in C2(M), where u0 is the unique solution of
+− ∆¯gu0 + ¯K = fe2u0
+in M.
+(4.10)
+Proof. We only need to show that
+(un)n∈N is bounded in C2,α(M) for some α > 0.
+(4.11)
+Indeed, assuming this for the moment, we may complete the argument as follows. Suppose by contradiction
+that there exists ε > 0 and a subsequence, also denoted by (un)n∈N, with the property that
+∥un − u0∥C2(M) ≥ ε
+for all n ∈ N.
+(4.12)
+By (4.11) and the compactness of the embedding C2,α(M) �→ C2(M), we may then pass to a subsequence, still
+denoted by (un)n∈N, with un → u∗ in C2(M) for some u∗ ∈ C2(M). Passing to the limit in (4.8), we then see
+that u∗ is a solution of (4.10), which by uniqueness implies that u∗ = u0. This contradicts (4.12), and thus the
+claim follows.
+The proof of (4.11) follows by similar arguments as in [7, p. 1063 f.]. Since the framework is slightly different,
+we sketch the main steps here for the convenience of the reader. We first note that, by the same argument as
+in [7, p. 1063 f.], there exists a constant C0 > 0 with
+un ≥ −C0
+for all n.
+(4.13)
+Since {f < 0} is a nonempty open subset of M by assumption, we may fix a nonempty open subdomain
+Ω ⊂⊂ {f < 0}. By [1, Appendix], there exists a constant C1 > 0 with
+∥u+
+n ∥H1(Ω,¯g) ≤ C1
+for all n
+and therefore
+�
+Ω
+e2undµ¯g ≤
+�
+Ω
+e2u+
+n dµ¯g ≤ C2
+for all n
+(4.14)
+for some C2 > 0 by the Moser–Trudinger inequality.
+Next, we consider a nontrivial, nonpositive function
+h ∈ C∞
+c (Ω) ⊂ C∞(M) and the unique solution w ∈ C∞(M) of the equation
+−∆¯gw + ¯K = he2w
+in M.
+Moreover, we let wn := un − w, and we note that wn satisfies
+−∆¯gwn + he2w = (f + λn)e2un
+in M.
+Multiplying this equation by e2wn and integrating by parts, we obtain
+�
+M
+(f + λn)e2(un+wn)dµ¯g =
+�
+M
+�
+−∆¯gwn + he2w�
+e2wndµ¯g =
+�
+M
+�
+2e2wn|∇¯gwn|2
+¯g + he2(w+wn)�
+dµ¯g
+= 2
+�
+M
+|∇¯gewn|2
+¯gdµ¯g +
+�
+Ω
+he2undµ¯g.
+(4.15)
+Moreover, applying (4.9) to h = ewn gives
+�
+M
+(f + λn)e2(un+wn)dµ¯g ≤ 1
+2
+�
+M
+|∇¯gewn|2
+¯gdµ¯g.
+(4.16)
+Combining (4.14), (4.15) and (4.16) yields
+∥∇¯gewn∥2
+L2(M,¯g) ≤ −2
+3
+�
+Ω
+he2undµ¯g ≤ 2
+3∥h∥L∞(M,¯g)C2
+for all n.
+(4.17)
+
+Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic
+9
+Next we claim that also ∥ewn∥L2(M,¯g) remains uniformly bounded. Suppose by contradiction that
+∥ewn∥L2(M,¯g) → ∞
+as n → ∞.
+(4.18)
+We then set vn :=
+ewn
+∥ewn∥L2(M,¯g) , and we note that
+∥vn∥L2(M,¯g) = 1
+for all n
+and
+∥∇¯gvn∥2
+L2(M,¯g) → 0
+as n → ∞
+(4.19)
+by (4.17). Consequently, we may pass to a subsequence satisfying vn ⇀ v in H1(M, ¯g), where v is a constant
+function with
+∥v∥L2(M,¯g) = 1.
+(4.20)
+However, since
+∥ewn∥L2(Ω,¯g) ≤ ∥eun∥L2(Ω,¯g)∥e−w∥L∞(Ω,¯g) ≤
+�
+C2∥e−w∥L∞(Ω,¯g)
+for all n ∈ N
+by (4.14) and therefore
+∥v∥L2(Ω,¯g) = lim
+n→∞ ∥vn∥L2(Ω,¯g) = lim
+n→∞
+∥ewn∥L2(Ω,¯g)
+∥ewn∥L2(M,¯g)
+= 0
+by (4.18), we conclude that the constant function v must vanish identically, contradicting (4.20).
+Consequently, ∥ewn∥L2(M,¯g) remains uniformly bounded, which by (4.17) implies that ewn remains bounded
+in H1(M, ¯g) and therefore in Lp(M, ¯g) for any p < ∞. Since eun ≤ ∥ew∥L∞(M,¯g)ewn on M for all n ∈ N, it thus
+follows that also eun remains bounded in Lp(M, ¯g) for any p < ∞. Moreover, by (4.13), the same applies to
+the sequence un itself. Therefore, applying successively elliptic Lp and Schauder estimates to (4.8), we deduce
+(4.11), as required.
+Proposition 4.6. Let f ∈ C∞(M) be a nonconstant function with maxx∈M f(x) = 0. Then there exists λ♯ and
+a C1-curve (−∞, λ♯] → C2(M);
+λ �→ uλ with the following properties.
+(i) If λ ≤ 0, then uλ is the unique solution of
+− ∆¯gu + ¯K = fλe2u
+in M
+(4.21)
+and a global minimum of Efλ.
+(ii) If λ ∈ (0, λ♯], then uλ is the unique weakly stable solution of (4.21) in the sense of (4.9), and it is a local
+minimum of Efλ.
+(iii) The curve of functions λ �→ uλ is pointwisely strictly increasing on M, and so the volume function
+(−∞, λ♯] → [0, ∞);
+λ �→ V (λ) :=
+�
+M
+e2uλdµ¯g
+(4.22)
+is continuous and strictly increasing.
+Proof. We already know that, for λ ≤ 0, the energy Efλ admits a strict global minimiser uλ which depends
+smoothly on λ. Moreover, by [1, Proposition 2.4], the curve λ �→ uλ can be extended as a C1-curve to an
+interval (−∞, λ♯] for some λ♯ > 0. We also know from [1, Proposition 2.4] that, for λ ∈ (−∞, λ♯], the solution
+uλ is strongly stable in the sense that
+Cλ :=
+inf
+h∈H1(M,¯g)
+1
+∥h∥2
+H1(M,¯g)
+�
+M
+�
+|∇¯gh|2
+¯g − 2fλe2uλh2�
+dµ¯g > 0.
+(4.23)
+Here we note that the function λ �→ Cλ is continuous since uλ depends continuously on λ with respect to the
+C2-norm. Next we prove that, after making λ♯ > 0 smaller if necessary, the function uλ is the unique weakly
+stable solution of (4.21) for λ ∈ (0, λ♯]. Arguing by contradiction, we assume that there exists a sequence
+λn → 0+ and corresponding weakly stable solutions (un)n∈N of
+− ∆¯gun + ¯K = (f + λn)e2un
+in M
+(4.24)
+with the property that un ̸= uλn for every n ∈ N. By Proposition 4.5, we know that un → u0 in C2(M).
+Consequently, vn := un − uλn → 0 in C2(M) as n → ∞, whereas the functions vn solve
+− ∆¯gvn = (f + λn)
+�
+e2un − e2uλn �
+= (f + λn)e2uλn �
+e2vn − 1
+�
+in M
+for every n ∈ N.
+(4.25)
+
+10
+Franziska Borer, Peter Elbau, Tobias Weth
+Combining this fact with (4.23), we deduce that
+∥vn∥2
+H1(M,¯g) ≤ 1
+Cλ
+�
+M
+�
+|∇¯gvn|2
+¯g − 2(f + λn)e2uλn v2
+n
+�
+dµ¯g
+= 1
+Cλ
+�
+M
+(f + λn)e2uλn �
+e2vn − 1 − 2vn
+�
+vndµ¯g.
+Since vn → 0 in C2(M), there exists a constant C > 0 with |(e2vn − 1 − 2vn)vn| ≤ C|vn|3 on M for all n ∈ N,
+which then implies with H¨older’s inequality and Lemma 4.2 that
+∥vn∥2
+H1(M,¯g) ≤ C∥(f + λn)e2uλn ∥L∞(M,¯g)∥vn∥3
+L3(M,¯g)
+≤ C
+��
+M
+|vn|3· 4
+3 dµ¯g
+� 3
+4
+= C∥vn∥3
+L4(M,¯g) ≤ C∥vn∥3
+H1(M,¯g)
+with a constant C > 0 independent on M. This contradicts the fact that vn → 0 in H1(M) as n → ∞. The
+claim thus follows.
+It remains to prove that the curve of functions λ �→ uλ is pointwisely strictly increasing on M. This is a
+consequence of the uniqueness of weakly stable solutions stated in (ii) and the fact that, as noted in [7], if uλ0
+is a solution for some λ0 ∈ (−∞, λ♯], it is possible to construct, via the method of sub- and supersolutions, for
+every λ < λ0, a weakly stable solution uλ with uλ < uλ0 everywhere in M.
+Corollary 4.7. Let f ∈ C∞(M) be nonconstant with maxx∈M f(x) = 0, and let λ♯ > 0 be given as in
+Proposition 4.6. Then there exists κ1 > 0 with the following property.
+If A ≥ κ1 and u ∈ C1,A is a solution of
+− ∆¯gu + ¯K = (f + λ)e2u
+(4.26)
+for some λ ∈ R with Ef(u) < λ♯A
+2 , then 0 < λ < λ♯, and u is not a weakly stable solution of (4.26), so u ̸= uλ.
+Proof. Let κ0 > 0 be given as in Lemma 4.4 for ε = λ♯ > 0. Moreover, let
+κ1 := max
+�
+κ0, V (uλ♯)
+�
+with V defined in (4.22). Next, let u ∈ C1,A be a solution of (4.26) for some λ ∈ R with Ef(u) < λ♯A
+2 . From
+Lemma 4.4, we then deduce that 0 < λ < λ♯, and by Proposition 4.6 (iii) we have u ̸= uλ. Since uλ is the
+unique weakly stable solution of (4.26), it follows that u is not weakly stable.
+Corollary 4.8. Let p > 2, f ∈ C∞(M) be nonconstant with maxx∈M f(x) = 0, and let λ♯ > 0 be given as in
+Proposition 4.6. Then there exists κ > 0 with the property that for every A ≥ κ the set
+˜C :=
+�
+u0 ∈ C1,A ∩ W 2,p(M, ¯g) | Ef(u0) < λ♯A
+2
+�
+is nonempty, and for every u0 ∈ ˜C the global solution u ∈ C([0, ∞); C(M))∩C([0, ∞); H1(M, ¯g))∩C∞((0, ∞)×
+M) of the initial value problem (2.4), (2.5) converges, as t → ∞ suitably, to a solution u∞ of the static problem
+(4.26) for some λ ∈ (0, λ♯) which is not weakly stable and hence no local minimiser of Efλ.
+Proof. Let κ1 > 0 be given by Corollary 4.7. By (4.5), there exists κ ≥ κ1 > 0 with mf,A < λ♯A
+4
+for fixed
+A > κ. Consequently, there exists u0 ∈ C1,A ∩ W 2,p(M, ¯g) with Ef(u0) < λ♯A
+2 . By Theorem 3.1, the global
+solution u ∈ C([0, ∞); C(M)) ∩ C([0, ∞); H1(M, ¯g)) ∩ C∞((0, ∞) × M) of the initial value problem (2.4), (2.5)
+converges, as t → ∞ suitably, to a solution u∞ ∈ C1,A of the static problem (4.26) for some λ ∈ R, whereas
+Ef(u∞) ≤ Ef(u0) < λ♯A
+2 . Consequently, λ ∈ (0, λ♯) by Corollary 4.7, and u∞ is not weakly stable.
+5. Proof of the Main Results
+5.1. Notation and Some Regularity Results. In this chapter we summarise different kind of estimates
+which will be useful later. In the following, for T > 0 we use the notation
+Lp
+t Lr
+x := Lp([0, T]; Lr(M, ¯g))
+and
+Lp
+t Hq
+x := Lp([0, T]; Hq(M, ¯g)).
+A first regularity result is therefore given by Lemma 4.2.
+
+Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic
+11
+Remark 5.1. We have
+∥θ∥4
+Lp
+t L4x ≤ CGNL∥θ∥2
+Lp
+t L2x∥θ∥2
+Lp
+t H1x
+for θ ∈ Lp
+t H1
+x with p ∈ [1, ∞].
+Lemma 5.2 (Sobolev inequality). There exists a constant CS > 0 such that for every ρ ∈ L∞
+t H1
+x, T ≤ 1, we
+have
+∥ρ∥2
+L4
+t L4x ≤ CS(∥ρ∥2
+L∞
+t L2x + ∥∇¯gρ∥2
+L2
+t L2x) < ∞.
+(5.1)
+Proof. With Lemma 4.2 there exists a constant CGNL > 0 such that we have for all T ≤ 1
+∥ρ∥4
+L4
+t L4x =
+� T
+0
+∥ρ(t)∥4
+L4(M,¯g)dt ≤ CGNL
+� T
+0
+∥ρ(t)∥2
+L2(M,¯g)∥ρ(t)∥2
+H1(M,¯g)dt
+≤ CGNL∥ρ∥2
+L∞
+t L2x
+� T
+0
+(∥ρ(t)∥2
+L2(M,¯g) + ∥∇¯gρ(t)∥2
+L2(M,¯g))dt
+≤ CGNL · T ∥ρ∥4
+L∞
+t L2x + CGNL∥ρ∥2
+L∞
+t L2x∥∇¯gρ∥2
+L2
+t L2x
+≤ CGNL
+�
+∥ρ∥4
+L∞
+t L2x + ∥ρ∥2
+L∞
+t L2x∥∇¯gρ∥2
+L2
+t L2x
+�
+.
+By using Young’s inequality we have
+∥ρ∥L∞
+t L2x∥∇¯gρ∥L2
+t L2x ≤ 1
+2
+�
+∥ρ∥2
+L∞
+t L2x + ∥∇¯gρ∥2
+L2
+t L2x
+�
+and therefore
+∥ρ∥2
+L4
+t L4x ≤ C
+1
+2
+GNL
+�
+∥ρ∥4
+L∞
+t L2x + 1
+4(∥ρ∥2
+L∞
+t L2x + ∥∇¯gρ∥2
+L2
+t L2x)2
+≤ C
+1
+2
+GNL(∥ρ∥2
+L∞
+t L2x + 1
+2∥ρ∥2
+L∞
+t L2x + 1
+2∥∇¯gρ∥2
+L2
+t L2x)
+≤ 3
+2C
+1
+2
+GNL(∥ρ∥2
+L∞
+t L2x + ∥∇¯gρ∥2
+L2
+t L2x)
+=: CS(∥ρ∥2
+L∞
+t L2x + ∥∇¯gρ∥2
+L2
+t L2x).
+Since T is finite, ρ ∈ L∞
+t H1
+x implies that ρ ∈ Lp
+t H1
+x for all p ∈ [1, ∞] which shows that the upper bound is
+finite.
+Furthermore, since T < ∞ and vol¯g = 1, with Lemma 4.1 we also have for every p, s ∈ [1, ∞] that Lq
+tLr
+x ⊂
+Ls
+tLp
+x for q ≥ s, r ≥ p.
+Since we will often use it in the following, we recall that for v ∈ CtCx := C([0, T], C(M)) we have
+∥1 − ev∥2
+L∞
+t L∞
+x ≤ e2∥v∥L∞
+t
+L∞
+x ∥v∥2
+L∞
+t L∞
+x
+(5.2)
+since for x ∈ R we get with the Taylor expansion
+|ex − 1| = |1 − ex| ≤ |x|e|x|.
+(5.3)
+Lemma 5.3. With Lemma 4.1 we get the following statements:
+1. For a (sufficiently smooth) solution u of (2.4), (2.5) we have
+¯u(t) ≥ 1
+2 log
+� A
+Cup
+�
+=: m0(A, Ef(u0), f, CMT, η1),
+(5.4)
+with Cup = CMT exp(4η1(2Ef(u0) + | ¯K| log(A) + A maxx∈M f(x))) where η1 is a number determined by
+Lemma 4.1. So, especially for a solution u of (2.4), (2.5) we have the uniform bound
+m0 ≤ ¯u(t) ≤ 1
+2 log(A),
+(5.5)
+where we used (4.3) and the volume preserving property to get the upper bound of ¯u(t).
+2. For a solution u of (2.4), (2.5) we have for all p ∈ R that
+�
+M
+e2pu(t)dµ¯g ≤ Cint(A, CMT, Ef(u0), f, ¯K, η1, η2, p),
+(5.6)
+where again, η1, η2 are numbers determined by Lemma 4.1.
+
+12
+Franziska Borer, Peter Elbau, Tobias Weth
+3. For this part we choose f = f0 where f0 ≤ 0 is a nonconstant, smooth function with maxx∈M f0(x) = 0.
+Then there exists a constant Clow = Clow(Cint, f0) > 0 such that
+�
+M
+|f0|dµg(t) ≥ Clow.
+(5.7)
+Proof.
+1. Let u be a solution of (2.4), (2.5). We then know that u(t) ∈ CA. So, with (2.9) we have for all
+t ≥ 0 that
+∥∇¯gu(t)∥2
+L2(M,¯g) = 2Ef(u(t)) −
+�
+M
+(2 ¯Ku(t) − fe2u(t))dµ¯g
+= 2Ef(u(t)) +
+�
+M
+(2| ¯K|u(t) + fe2u(t))dµ¯g
+≤ 2Ef(u0) + | ¯K| log(A) + A max
+x∈M f(x),
+(5.8)
+where we used the fact that
+�
+M 2u(t)dµ¯g ≤ log(A) by (4.3) and since
+�
+M e2u(t)dµ¯g ≡ A. With this and
+Lemma 4.1 we can now estimate
+A =
+�
+M
+e2u(t)dµ¯g = e2¯u(t)
+�
+M
+e2(u(t)−¯u(t))dµ¯g
+≤ e2¯u(t)CMT exp(η1∥∇¯g(2u(t))∥2
+L2(M,¯g))
+≤ e2¯u(t)CMT exp(4η1(2Ef(u0) + | ¯K| log(A) + A max
+x∈M f(x)))
+=: Cupe2¯u(t),
+with Cup = Cup(A, CMT, Ef(u0), f, ¯K, η1) > 0 and therefore
+¯u(t) ≥ 1
+2 log
+� A
+Cup
+�
+=: m0(A, CMT, Ef(u0), f, ¯K, η1) ∈ R.
+So, for a solution u(t) ∈ CA of (2.4), (2.5) we get the uniform bound
+m0 ≤ ¯u(t) ≤ 1
+2 log(A).
+2. Let u be a solution of (2.4), (2.5). So, u(t) ∈ CA. With Lemma 4.1, (5.5), and (5.8) we directly get for
+any p ∈ R that
+�
+M
+e2pu(t)dµ¯g = e2p¯u(t)
+�
+M
+e2p(u(t)−¯u(t))dµ¯g
+≤ e2p¯u(t)CMT exp(4η2p2∥∇¯gu(t)∥2
+L2(M,¯g))
+≤ Cint,
+(5.9)
+where Cint = Cint(A, CMT, Ef(u0), f, ¯K, η1, η2, p) > 0.
+3. Similar to [19, Lemma 2.3] we see by the choice of f0, H¨older’s inequality, and (5.9) that
+0 <
+����
+�
+M
+�
+|f0|dµ¯g
+����
+2
+≤
+�
+M
+|f0|e2u(t)dµ¯g
+�
+M
+e−2u(t)dµ¯g ≤ Cint
+�
+M
+|f0|e2u(t)dµ¯g
+(5.10)
+which shows the claim.
+So, Lemma 5.3 is proven.
+Now we can turn to the proofs of the main results.
+5.2. Short-Time Existence. Let A > 0. We are looking for a short-time solution of (2.4) with initial data
+(2.5). Using the Gauss equation (1.1) we can rewrite (2.4), (2.5) in the following way:
+∂tu(t) = f − Kg(t) − αA(t)
+= e−2u(t)∆¯gu(t) + ¯K
+� 1
+A − e−2u(t)
+�
++ f − 1
+A
+�
+M
+fe2u(t)dµ¯g;
+(5.11)
+u(0) = u0 ∈ Cp,A :=
+�
+u ∈ W 2,p(M, ¯g) |
+�
+M
+e2u = A
+�
+,
+(5.12)
+
+Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic
+13
+with p > 2, where
+αA(t) = 1
+A
+��
+M
+fdµg(t) − ¯K
+�
+.
+To find a solution of (5.11), (5.12), we consider the linear equation
+∂tu(t) = e−2v(t)∆¯gu(t) + ¯K
+� 1
+A − e−2v(t)
+�
++ f − 1
+A
+�
+M
+fe2v(t)dµ¯g;
+(5.13)
+u(0) = u0 ∈ Cp,A,
+(5.14)
+and use a fixed point argument in the space (X, ∥ · ∥X) := (CtCx, ∥ · ∥CtCx). First we observe that for v ∈ CtCx,
+equation (5.13) is strongly parabolic. Furthermoren, with p > 2 and the fact that M is compact, we have
+u0 ∈ Cp,A ⊂ H2(M, ¯g), and therefore u0 ∈ L∞(M, ¯g).
+For the fixed point argument we fix R = R(u0) := ∥u0∥L∞(M,¯g) + 1. For fixed T > 0, let
+X = CtCx = C([0, T], C(M, ¯g)) �→ L∞
+t L∞
+x
+with
+∥u∥X =
+max
+t∈[0,T ], x∈M |u(x, t)|.
+For v ∈ X, by [14, Theorem 7.32] and the appendix, we get a unique solution uv ∈ W 2,1
+p
+= W 1,p
+t
+Lp
+x ∩Lp
+t W 2,p
+x
+of (5.13), (5.14) for t ∈ [0, T], x ∈ M. On XR = {U ∈ X | ∥U∥X ≤ R}, we now define the function Φ as follows:
+for v ∈ XR, let Φ(v) =: uv be the unique solution of (5.13), (5.14). First, we want to show that Φ : XR → XR
+if T > 0 is chosen small enough.
+Lemma 5.4. If T > 0 is fixed with
+T ≤
+�
+| ¯K|e2(∥u0∥L∞(M,¯g)+1) + ∥f∥L∞(M,¯g)
+�
+1 + e2(∥u0∥L∞(M,¯g)+1)
+A
+��−1
+(5.15)
+and v ∈ XR, then Φ(v) ∈ XR.
+Proof. With Proposition 6.3 (ii) we directly get
+∥Φ(v)∥L∞
+t L∞
+x = ∥uv∥L∞
+t L∞
+x ≤ ∥u+
+0 ∥L∞(M,¯g) + TdT
+(5.16)
+where
+dT ≤ | ¯K|e2∥v∥L∞
+t
+L∞
+x + ∥f∥L∞(M,¯g) + ∥f∥L∞(M,¯g)e2∥v∥L∞
+t
+L∞
+x
+A
+≤ | ¯K|e2R + ∥f∥L∞(M,¯g)
+�
+1 + e2R
+A
+�
+,
+hence
+∥Φ(v)∥L∞
+t L∞
+x ≤ T
+�
+| ¯K|e2R + ∥f∥L∞(M,¯g)
+�
+1 + e2R
+A
+��
++ ∥u+
+0 ∥L∞(M,¯g)
+≤ 1 + ∥u0∥L∞(M,¯g) = R,
+by (5.15) and since R = ∥u0∥L∞(M,¯g) + 1, which shows the claim.
+We now use Schauder’s fixed point Theorem [17] to show the following proposition.
+Proposition 5.5. If u0 ∈ Cp,A ⊂ W 2,p(M, ¯g) and T > 0 is fixed with (5.15), then there exists a short-time
+solution u ∈ X ∩ C∞(M × (0, T)) of (5.11), (5.12).
+Moreover, any such solution satisfies u ∈ C([0, T), H1(M, ¯g)).
+Proof. Step 1: First we recall Schauder’s Theorem: It asserts that if H is a nonempty, convex, and closed
+subset of a Banach space B and F is a continuous mapping of H into itself such that F(H) is a relatively
+compact subset of H, then F has a fixed point.
+In our case, B ˆ=X = C([0, T]; C(M, ¯g)), H ˆ=XR = {u ∈ X | ∥u∥X = ∥u∥CtCx ≤ R}, and F ˆ=Φ. So to show
+the existence of a fixed point of Φ in XR, it remains to show that
+1. Φ : XR → XR ist continuous and
+
+14
+Franziska Borer, Peter Elbau, Tobias Weth
+2. Φ(XR) ⊂ XR is relatively compact.
+In a first step we show that Φ : XR → XR ist continuous. For this, let (vn)n∈N ⊂ XR be a sequence with
+∥vn − v∥X → 0 for n → ∞ with v ∈ XR. With Proposition 6.1 we know that for all vn there exists un ∈ W 2,1
+p
+,
+p > 2, which is the unique solution of (5.13), (5.14) such that
+∥un∥W 2,1
+p
+≤ C(∥u0∥W 2,p(M,¯g) + ∥dn∥Lp
+t Lp
+x)
+with
+dn(t) := ¯K
+� 1
+A − e−2vn(t)
+�
++ f − 1
+A
+�
+M
+fe2vn(t)dµ¯g.
+Since vn → v in CtCx and therefore vn → v in L∞
+t L∞
+x , we know that vn → v in Lp
+t Lp
+x for all p. Furthermore,
+since the exponential map is continuous, we have e±2vn → e±2v in Lp
+t Lp
+x for all p, and therefore dn → d in Lp
+t Lp
+x
+for all p.
+Hence, for every ε > 0 there exist NV , Nd ∈ N such that
+∥vn − v∥Lp
+t Lp
+x < ε
+for all n ≥ N
+and
+∥dn − d∥Lp
+t Lp
+x < ε
+for all n ≥ N,
+with N := max{NV , Nd}.
+Furthermore we have the estimate
+∥e2vn − e2v∥L∞
+t L∞
+x = ∥(e2vn−2v − 1)e2v∥L∞
+t L∞
+x ≤ ∥e2vn−2v − 1∥L∞
+t L∞
+x ∥e2v∥L∞
+t L∞
+x
+≤ ∥2vn − 2v∥e∥2Vn−2V ∥L∞
+t
+L∞
+x ∥e2v∥L∞
+t L∞
+x < 2εe2εe2R,
+and similarly ∥e−2vn − e−2v∥L∞
+t L∞
+x < 2εe2εe2R.
+Considering now the difference un − u, where un = Φ(vn) and u = Φ(v), we see that un − u fulfils the
+equation
+∂t(un − u)(t) = e−2vn(t)∆¯gun(t) + dn(t) − e−2v(t)∆¯gu(t) − d(t)
+= e−2vn(t)∆¯g(un − u)(t) + (e−2vn(t) − e−2v(t))∆¯gu(t) + dn(t) − d(t)
+with
+∥un − u∥W 2,1
+p
+≤ C∥(e−2vn − e−2v)∆¯gu + dn − d∥Lp
+t Lp
+x
+≤ C
+�
+∥e−2vn − e−2v∥L∞
+t L∞
+x ∥∆¯gu∥Lp
+t Lp
+x + ∥dn − d∥Lp
+t Lp
+x
+�
+≤ C(2εe2εe2R∥∆¯gu∥Lp
+t Lp
+x + ε)
+for n ≥ N.
+Since ∥∆¯gu∥Lp
+t Lp
+x is finite and ε > 0 was arbitrary, we see that ∥Φ(vn) − Φ(v)∥W 2,1
+p
+→ 0 for n → ∞. So, we
+get
+∥Φ(vn) − Φ(v)∥X ≤ C∥Φ(vn) − Φ(v)∥Cα ≤ C∥Φ(vn) − Φ(v)∥W 2,1
+p
+→ 0
+for n → ∞
+which shows the continuity of Φ : XR → XR.
+In a second step we show that Φ(XR) is relatively compact. For this let (un)n∈N ⊂ Φ(XR) be an arbitrary
+sequence in the image of Φ. So, again with Proposition 6.1, we see that for every un ∈ Φ(XR) there exists a
+vn ∈ XR with Φ(vn) = un such that
+∥un∥W 2,1
+p
+≤ C(∥u0∥W 2,p(M,¯g) + ∥dn∥Lp
+t Lp
+x)
+≤ C
+�
+∥u0∥W 2,p(M,¯g) + T| ¯K|
+A
++ ∥ ¯Ke−2vn∥Lp
+t Lp
+x + ∥f∥Lp
+t Lp
+x +
+����
+1
+A
+�
+M
+fe2vndµ¯g
+����
+Lp
+t Lp
+x
+�
+≤ C
+�
+∥u0∥W 2,p(M,¯g) + T| ¯K|
+A
++ | ¯K|e2R + T∥f∥L∞(M,¯g) + T
+A∥f∥L∞(M,¯g)e2R
+�
+≤ C(A, f, ¯K, R, T, u0) =: Cd.
+So, (un)n∈N is uniformly bounded in W 2,1
+p
+((0, T) × M). Using now that W 2,1
+p
+((0, T) × M) is continuously
+embedded in Cα([0, T] × M) for some 0 < α < 1 and this on the other hand is compactly embedded in
+Cβ([0, T] × M) for some 0 < β < α < 1 we can conclude the claim.
+We have thus proved that Φ has a fixed point u in XR, which then is a (strong) solution u ∈ W 2,1
+p
+((0, T) × M)
+of (5.11), (5.12).
+Step 2: We now show that u ∈ C∞(M × (0, T)).
+To see this, we first note the trivial fact that u ∈
+W 2,1
+p
+((0, T)×M) is a strong solution of (5.13), (5.14) with v = u. Since then v ∈ W 2,1
+p
+((0, T)×M) ⊂ Cα([0, T]×
+
+Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic
+15
+M), [14, Theorems 5.9 and 5.10] imply the existence of a classical solution ˜u ∈ X ∩ C2+α′,1+α′
+loc
+((0, T) × M)
+of (5.13), (5.14) with v = u for some α′ > 0. Here C2+α′,1+α′
+loc
+((0, T) × M) denotes the space of functions
+f ∈ C2,1((0, T) × M) with the property that ∂tf and all derivatives up to second order of f with respect to
+x ∈ M are locally α′-H¨older continuous. In particular, ˜u ∈ W 2,1
+p
+((ε, T − ε) × M) for ε ∈ (0, T). The function
+w := u − ˜u ∈ W 2,1
+p
+((ε, T − ε) × M) is then a strong solution of the initial value problem
+∂tw(t) = e−2v(t)∆¯gw(t)
+for t ∈ (ε, T − ε),
+w(ε) = u(ε, ·) − ˜u(ε, ·).
+By Proposition 6.3 (ii) we then have |w| ≤ ∥u(ε, ·) − ˜u(ε, ·)∥L∞(M,¯g) on (ε, T − ε) × M, whereas ∥u(ε, ·) −
+˜u(ε, ·)∥L∞(M,¯g) → 0 as ε → 0 by the continuity of u and ˜u. It thus follows that u ≡ ˜u on (0, T) × M), and
+therefore u ∈ C2+α′,1+α′
+loc
+((0, T) × M). Since u solves (5.13), (5.14) with v = u ∈ C2+α′,1+α′
+loc
+((0, T) × M),
+we can apply [14, Theorems 5.9] and the above argument again to get u ∈ C4+α′′,2+α′′
+loc
+((0, T) × M) for some
+α′′ > 0.
+Repeating this argument inductively, we get u ∈ C
+k, k
+2
+loc ((0, T) × M) for every k > 0, and hence
+u ∈ C∞(M × (0, T)).
+Step 3: It remains to show that any solution u ∈ X ∩ C∞((0, T) × M) of (5.11), (5.12) also satisfies u ∈
+C([0, T), H1(M, ¯g)). Since u ∈ C∞((0, T) × M), only the continuity in t = 0 needs to be proved. Setting
+φ(t) = ∥u(t)∥2
+H1(M,¯g) for t ∈ (0, T), we see that
+1
+2(φ(t2) − φ(t1)) = 1
+2
+� t2
+t1
+∂t∥u(t)∥2
+H1(M,¯g) dt =
+� t2
+t1
+�
+M
+�
+u(t)∂tu(t) + ∇u(t)∇∂tu(t)
+�
+dµ¯gdt
+=
+� t2
+t1
+�
+M
+�
+u(t)∂tu(t) − [∆u(t)]∂tu(t)
+�
+dµ¯gdt
+and therefore, by H¨older’s inequality,
+1
+2|φ(t2) − φ(t1)| ≤
+� t2
+t1
+�
+M
+�
+|u||∂tu| + |∆u||∂tu|
+�
+dµ¯gdt
+≤ C∥∂tu∥Lp((0,T )×M)
+�
+∥u∥Lp((0,T )×M) + ∥∆u∥Lp((0,T )×M)
+�
+(t2 − t1)β
+≤ C∥u∥W 1,2
+p
+((0,T )×M)(t2 − t1)β,
+for 0 < t1 < t2 < T with some β > 0 depending on p > 2, which implies that the function φ is uniformly
+continuous and therefore bounded on (0, T).
+We now assume by contradiction that u is not continuous at t = 0 with respect to the H1(M, ¯g)-norm. Then
+there exists a sequence (tn)n∈N in (0, T) and ε > 0 with tn → 0+ as n → ∞ and
+∥u(tn) − u0∥H1(M,¯g) ≥ ε
+for all n ∈ N.
+(5.17)
+Since ∥u(tn)∥2
+H1(M,¯g) = φ(tn) remains bounded as n → ∞, we conclude that, passing to a subsequence, the
+sequence u(tn) converges weakly in H1(M, ¯g) and therefore strongly in L2(M, ¯g). Since the strong L2-limit
+of u(tn) must be u0 = u(0) as a consequence of the fact that u ∈ X, we deduce that u(tn) ⇀ u0 weakly in
+H1(M, ¯g) as n → ∞. Combining this information with Proposition 6.1 from the appendix, we deduce that
+lim sup
+n→∞ ∥u(tn)∥2
+H1(M,¯g) ≤ ∥u0∥2
+H1(M,¯g) ≤ lim inf
+n→∞ ∥u(tn)∥2
+H1(M,¯g)
+(5.18)
+and therefore ∥u(tn)∥H1(M,¯g) → ∥u0∥H1(M,¯g). Note here that this part of Proposition 6.1 applies since u solves
+(5.13), (5.14) with v = u ∈ W 2,1
+p
+((0, T) × M) ⊂ Cα([0, T] × M) for some α > 0. From (5.18) and the uniform
+convexity of the Hilbert space H1(M, ¯g), we conclude that u(tn) → u0 strongly in H1(M, ¯g), contrary to
+(5.17).
+5.3. Uniqueness. We now show that the solution from Proposition 5.5 is unique.
+Lemma 5.6. Let u0 ∈ W 2,p(M, ¯g), p > 2, and T > 0 be fixed with (5.15). Then the short-time solution of
+u ∈ X ∩ C∞(M × (0, T)) of (5.11), (5.12) given by Proposition 5.5 is unique.
+Proof. Let u1, u2 ∈ X ∩ C∞(M × (0, T)) be two solutions of (5.11), (5.12). The difference u := u1 − u2 ∈
+X ∩ C∞(M × (0, T)) then fulfils
+∂tu(t) = e−2u1(t)∆¯gu1(t) − e−2u2(t)∆¯gu2(t)
+− ¯K(e−2u1(t) − e−2u2(t)) − 1
+A
+�
+M
+f(e2u1(t) − e2u2(t))dµ¯g
+= e−2u1(t)∆¯gu(t) + ∆¯gu2(t)
+�
+e−2u1(t) − e−2u2(t)�
+− ¯K(e−2u1(t) − e−2u2(t)) − 1
+A
+�
+M
+f(e2u1(t) − e2u2(t))dµ¯g
+for t ∈ (0, T).
+(5.19)
+
+16
+Franziska Borer, Peter Elbau, Tobias Weth
+In the following, the letter C denotes different positive constants. Multiplying (5.19) with 2u and integrating
+over M gives
+d
+dt∥u(t)∥2
+L2(M,¯g) = 2
+�
+M
+u(t)∂tu(t)dµ¯g
+= 2
+�
+M
+e−2u1(t)u(t)∆¯gu(t)dµ¯g + 2
+�
+M
+u(t)∆¯gu2(t)
+�
+e−2u1(t) − e−2u2(t)�
+dµ¯g
+(5.20)
+− 2
+�
+M
+¯Ku(t)(e−2u1(t) − e−2u2(t))dµ¯g − 2
+A
+�
+M
+f(e2u1(t) − e2u2(t))dµ¯g
+�
+M
+u(t)dµ¯g
+≤ 2
+�
+M
+e−2u1(t)u(t)∆¯gu(t) + 2
+�
+M
+V (t, x)u2(t) + 2ρ(t)∥u(t)∥L2(M,¯g)
+�
+M
+|u(t)|dµ¯g
+≤ 2
+�
+−
+�
+M
+e−2u1(t)|∇¯gu(t)|2
+¯g + 2
+�
+M
+e−2u1(t)u(t)⟨∇¯gu1(t), ∇¯gu(t)⟩¯gdµ¯g
+�
++ 2∥V (t, ·)∥Lp(M,¯g)∥u(t)∥2
+L2p′(M,¯g) + C∥u(t)∥2
+L2(M,¯g)
+≤ C∥∇¯gu1(t)∥L4(M,¯g)∥u(t)∥L4(M,¯g)∥∇¯gu(t)∥L2(M,¯g)
++ 2∥V (t, ·)∥Lp(M,¯g)∥u(t)∥2
+L2p′(M,¯g) + C∥u(t)∥2
+L2(M,¯g)
+≤ C
+�
+∥u1(t)∥H2(M,¯g)∥u(t)∥2
+H1(M,¯g) + 2∥V (t, ·)∥Lp(M,¯g)∥u(t)∥2
+H1(M,¯g) + ∥u(t)∥2
+L2(M,¯g)
+�
+≤ C
+�
+∥u1(t)∥H2(M,¯g) + 2∥V (t, ·)∥Lp(M,¯g) + 1
+�
+∥u∥2
+H1(M,¯g),
+(5.21)
+with functions V ∈ Lp((0, T) × M) ∩ C∞((0, T) × M) and ρ ∈ L∞(0, T). Here we used the Sobolev embeddings
+H1(M, ¯g) �→ Lρ(M) for ρ ∈ [1, ∞). Multiplying (5.19) with −2∆u and integrating over M yields
+d
+dt∥∇gu(t)∥2
+L2(M,¯g) = 2
+�
+M
+∇u(t)∇∂tu(t)dµ¯g = −2
+�
+M
+∆gu(t)∂tu(t)dµ¯g
+≤ −2
+�
+M
+e−2u1(t)|∆¯gu(t)|2dµ¯g + 2
+�
+M
+V (x, t)|u(t)||∆u(t)|dµ¯g
+≤ −κ∥∆¯gu(t)∥2
+L2(M,¯g) + 2∥V (t, ·)∥Lp(M,¯g)∥u∥Lα(M,¯g)∥∆gu∥L2(M,¯g)
+≤ −κ∥∆¯gu(t)∥2
+L2(M,¯g) + 1
+κ∥V (t, ·)∥2
+Lp(M,¯g)∥u∥2
+Lα(M,¯g) + κ∥∆gu∥2
+L2(M,¯g)
+= 1
+κ∥V (t, ·)∥2
+Lp(M,¯g)∥u∥2
+Lα(M,¯g) ≤ C∥V (t, ·)∥2
+Lp(M,¯g)∥u∥2
+H1(M,¯g),
+(5.22)
+where we used first H¨older’s inequality with α =
+2p
+p−2, then Young’s inequality and finally Sobolev embeddings
+again. Here we note that, by making C > 0 larger if necessary, we may assume that the constants are the same
+in (5.21) and (5.22). Combining these estimates gives
+d
+dt∥u(t)∥2
+H1(M,¯g) ≤ g(t)∥u(t)∥2
+H1(M,¯g)
+for t ∈ (0, T)
+(5.23)
+with the function g ∈ L1(0, T) given by g1(t) = C
+�
+∥u1(t)∥H2(M,¯g) + 3∥V (t, ·)∥Lp(M,¯g) + 1
+�
+. Integrating and
+using the fact that u ∈ C([0, T), H1(M, ¯g)) by Proposition 5.5 with u(0) = u1(0) − u2(0) = 0, we see that
+∥u(t)∥2
+H1(M,¯g) ≤
+� t
+0
+g(s)∥u(s)∥2
+H1(M,¯g) ds
+for t ∈ [0, T).
+It then follows from Gronwall’s inequality [3] that ∥u(t)∥2
+H1(M,¯g) ≡ 0 on [0, T), hence u1 ≡ u2.
+5.4. Global Existence. From Section 5.2 and Section 5.3 we know that there exists a unique solution
+u ∈ C([0, T], C(M)) ∩ C([0, 1], H1(M, ¯g)) ∩ C∞((0, T) × M),
+of the initial value problem (5.11), (5.12). In particular we know that u ∈ L∞
+t L∞
+x for t ∈ [0, T], where T > 0
+is given by (5.15). In this section we want to show that u posses an L∞-a-priori bound on any time interval
+[0, T], T < ∞, and therefore, u is the unique global solution of (5.11), (5.12). For this we partially follow the
+idea of [2, Chapter 6].
+Lemma 5.7. For every T > 0, there exists M(T) > 0 such that we have
+sup
+t∈[0,T ]
+∥u(t)∥L∞(M,¯g) ≤ M(T).
+
+Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic
+17
+Proof. Let
+I :=
+�
+t ≥ 0
+��� u is a solution of (5.11) on (0, t] × M
+with initial data u(0) ∈ Cp,A
+�
+,
+Tmax := sup I, and Tk ⊂ I a sequence in I such that Tk → Tmax for k → ∞.
+For any t ∈ [0, Tk] and any xmax(t) ∈ M where
+u(t, xmax(t)) = max
+x∈M u(t, x) ≥ 0
+we have with ∂tu(t) = ∆g(t)u(t) − e−2u(t) ¯K + f − α(t) and the upper bound for |α| which is given by
+α0 := max{|α1|, |α2|},
+(5.24)
+that
+d
+dt [u(t, xmax(t))] = ∂tu(t, xmax(t)) ≤ | ¯K|e−2u(t,xmax(t)) + f(xmax(t)) + α0
+≤ | ¯K| + ∥f∥L∞(M,¯g) + α0,
+(5.25)
+where we used that ∇¯gu(t, xmax(t)) = 0 and therefore
+d
+dt [u(t, xmax(t)] = ∂tu(t, xmax(t)) + ∇¯gu(t, xmax(t)) ˙xmax(t) = ∂tu(t, xmax(t)).
+Integrating (5.25) on both side with respect to t and taking the supremum over t yields (together with the
+fact that u(0) = u0 ∈ Cp,A)
+sup
+t∈[0,Tk]
+x∈M
+u(t, x) ≤ Tk(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup
+x∈M
+u0(x)
+→ Tmax(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup
+x∈M
+u0(x) =: M1(Tmax) < ∞
+(5.26)
+for k → ∞ which shows the upper bound for u.
+Analogously, at any point xmin(t) ∈ M where
+u(t, xmin(t)) = min
+x∈M u(t, x) ≤ 0
+we have with ∂tu(t) = ∆g(t)u(t) − e−2u(t) ¯K + f − α(t), the fact that ¯K < 0, and the upper bound for |α| given
+by α0 that
+d
+dt [u(t, xmin(t))] = ∂tu(t, xmin(t)) ≥ −∥f∥L∞(M,¯g) − α0.
+(5.27)
+Integrating (5.27) on both side with respect to t and taking the infimum over t yields (together with the fact
+that u(0) = u0 ∈ Cp,A)
+inf
+t∈[0,Tk]
+x∈M
+u(t, x) ≥ −Tk(∥f∥L∞(M,¯g) + α0) + inf
+x∈M u0(x)
+→ −Tmax(∥f∥L∞(M,¯g) + α0) + inf
+x∈M u0(x) =: M2(Tmax) > −∞
+(5.28)
+for k → ∞ which shows the lower bound for u.
+So, we get
+sup
+t∈[0,T ]
+x∈M
+|u(t, x)| ≤ max{|M1(T)|, |M2(T)|}
+≤ T(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup
+x∈M
+|u0(x)| =: M(T)
+(5.29)
+which shows the claim.
+In fact, with the help of (2.9) we can turn (5.29) into a uniform estimate for all time.
+Lemma 5.8. Let u be the global, smooth solution of (5.11) with u(0) = u0 ∈ Cp,A.
+Then we have that
+supt>0 ∥u(t)∥L∞(M,¯g) ≤ Cuni < ∞.
+
+18
+Franziska Borer, Peter Elbau, Tobias Weth
+Proof. We follow the proof of [19, Lemma 2.5].
+By using the fact that u(t) is a volume preserving solution of (5.11) with u(0) = u0 ∈ Cp,A and therefore
+�
+M e2u(t)dµ¯g ≡ A, we get with (4.3) and the fact that ¯K < 0 that
+Ef(u(t)) = 1
+2∥∇¯gu(t)∥2
+L2(M,¯g) +
+�
+M
+¯Ku(t)dµ¯g − 1
+2
+�
+M
+fe2u(t)dµ¯g
+≥
+¯K
+2
+�
+M
+2u(t)dµ¯g − 1
+2
+�
+M
+fe2u(t)dµ¯g
+≥
+¯K
+2 log(A) − A
+2 ∥f∥L∞(M,¯g) > −∞.
+(5.30)
+Defining
+F(t) :=
+�
+M
+|∂tu(t)|2dµg(t) =
+�
+M
+|∂tu(t)|2e2u(t)dµ¯g
+and using the uniform lower bound of Ef given by (5.30), we then get from (2.8) or (2.9), respectively, the
+estimate
+� ∞
+0
+F(t)dt =
+� ∞
+0
+�
+M
+|∂tu(t)|2dµg(t)dt ≤ Ef(u0) + | ¯K|
+2 | log(A)| + A
+2 ∥f∥L∞(M,¯g).
+(5.31)
+Hence, for any T > 0 we find tT ∈ [T, T + 1] such that
+F(tT ) =
+inf
+t∈(T,T +1) F(t) ≤ Ef(u0) + | ¯K|
+2 | log(A)| + A
+2 ∥f∥L∞(M,¯g).
+(5.32)
+So, at time tT we get with (2.1), H¨olders inequality, (5.6), and (5.32) that
+∥∆¯gu(tT )∥L
+3
+2 (M,¯g)
+≤ ∥e2u(tT )∂tu(tT )∥L
+3
+2 (M,¯g) + ∥ ¯K∥L
+3
+2 (M,¯g) + ∥e2u(tT )f∥L
+3
+2 (M,¯g) + ∥e2u(tT )α(tT )∥L
+3
+2 (M,¯g)
+≤ ∥eu(tT )∥L6(M,¯g)F(tT )
+1
+2 + | ¯K| +
+��
+M
+e3u(tT )|f|
+3
+2 dµ¯g
+� 2
+3
++
+��
+M
+e3u(tT )|α(tT )|
+3
+2 dµ¯g
+� 2
+3
+≤ C
+1
+6
+int(A, Ef(u0), f, ¯K, η1, η2, 3)
+�
+Ef(u0) + | ¯K|
+2 | log(A)| + A
+2 ∥f∥L∞(M,¯g)
+� 1
+2
++ | ¯K|
++ C
+2
+3
+int
+�
+A, Ef(u0), f, ¯K, η1, η2, 3
+2
+�
+(∥f∥L∞(M,¯g) + α0)
+=: C10
+�
+A, Ef(u0), f, ¯K, η1, η2, 3
+2, 3
+�
+.
+(5.33)
+Furthermore, with Sobolev’s embedding theorem we have W 2, 3
+2 ⊂ C0, 2
+3 . Therefore we get with Poincar´e’s
+inequality, the Calder´on–Zygmund inequality for closed surfaces, and with (5.33) that
+∥u(tT ) − ¯u(tT )∥
+3
+2
+L∞(M,¯g) ≤ C∥u(tT ) − ¯u(tT )∥
+3
+2
+W 2, 3
+2 (M,¯g) ≤ C∥∇2
+¯gu(tT )∥
+3
+2
+L
+3
+2 (M,¯g)
+≤ C∥∆¯gu(tT )∥
+3
+2
+L
+3
+2 (M,¯g) ≤ CC
+3
+2
+10,
+(5.34)
+and therefore with (5.5) we obtain the uniform bound
+∥u(tT )∥L∞(M,¯g) ≤ CC10 + max
+�
+|m0|, 1
+2| log(A)|
+�
+.
+(5.35)
+Upon shifting time by tT , from (5.29) we now get
+sup
+s∈[T +1,T +2]
+∥u(s)∥L∞(M,¯g) ≤
+sup
+s∈[tT ,T +2]
+∥u(s)∥L∞(M,¯g)
+≤ 2(| ¯K| + ∥f∥L∞(M,¯g) + α0) + sup
+x∈M
+|u(tT , x)|
+≤ 2(| ¯K| + ∥f∥L∞(M,¯g) + α0) + CC10 + max
+�
+|m0|, 1
+2| log(A)|
+�
+.
+(5.36)
+Since T > 0 is arbitrary, the claim follows.
+
+Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic
+19
+5.5. Convergence of the Flow. Let f0 ≤ 0 be a smooth, nonconstant function withmaxx∈M f0(x) = 0.
+Following here the argumentation of [19], and using (5.31), we know that for a suitable sequence tl → ∞,
+l → ∞, with associated metrics gl = g(tl) we obtain convergence
+�
+M
+|∂tu(tl)|2dµg(tl) =
+�
+M
+|f0 − Kgl − α(tl)|2dµg(tl) → 0
+for l → ∞.
+(5.37)
+Provided that we can also show convergence of the associated sequence of metrics g(tl) to a limit metric
+g∞
+A = e2u∞
+A ¯g with Gauss curvature Kg∞
+A , it then follows that Kg∞
+A = f0 − α∞
+A for a constant α∞
+A . Later we will
+have a closer look at this constant α∞
+A .
+Lemma 5.9. For F(t) =
+�
+M |∂tu(t)|2dµg(t) as above, we have F(t) → 0 for t → ∞.
+Proof. First we consider the evolution equation of the curvature Kg(t) and of α(t). From the Gauss equation
+(1.1) we get for the curvature that
+∂tKg(t) = ∂t(−e−2u(t)∆¯gu(t) + e−2u(t) ¯K)
+= −2∂tu(t)Kg(t) − ∆g(t)∂tu(t)
+= 2Kg(t)(Kg(t) − f0 + α(t)) + ∆g(t)(Kg(t) − f0 + α(t))
+= 2(Kg(t) − f0 + α(t))2 + 2(f0 − α(t))(Kg(t) − f0 + α(t)) + ∆g(t)(Kg(t) − f0 + α(t)).
+(5.38)
+With (2.3) we get for the evolution equation for α(t):
+d
+dtα(t) = 2
+A
+�
+M
+f0e2u(t)∂tu(t)dµ¯g = 2
+A
+�
+M
+f0(f0 − Kg(t) − α(t))dµg(t).
+(5.39)
+So, with (5.38) and (5.39) we arrive at
+∂t(Kg(t) − f0 − α(t)) − ∆g(t)(Kg(t) − f0 + α(t))
+= 2(Kg(t) − f0 + α(t))2 + 2(f0 − α(t))(Kg(t) − f0 + α(t))
++ 2
+A
+�
+M
+f0(Kg(t) − f0 + α(t))dµg(t).
+(5.40)
+Following the proof of Lemma 3.1 in [19] we therefore get
+1
+2
+d
+dt
+�
+M
+|f0 − Kg(t) − α(t)|2dµg(t)
+=
+�
+M
+��
+∂tKg(t) +
+� d
+dtα(t)
+��
+(Kg(t) − f0 + α(t)) − (Kg(t) − f0 − α(t))3
+�
+dµg(t)
+= −
+�
+M
+|∇g(t)(Kg(t) − f0 + α(t))|2
+g(t)dµg(t) + 2
+�
+M
+(f0 − α(t))(Kg(t) − f0 + α(t))2dµg(t)
++
+�
+M
+(Kg(t) − f0 + α(t))3dµg(t),
+(5.41)
+where we used in the second step the fact that
+� d
+dtα(t)
+� �
+M
+(Kg(t) − f0 + α(t))dµg(t) = 0
+by (2.2).
+With H¨older’s inequality we can estimate
+�
+M
+(Kg(t) − f0 + α(t))3dµg(t) ≤ ∥∂tu(t)∥3
+L3(M,g(t)) ≤ ∥∂tu(t)∥L2(M,g(t))∥∂tu(t)∥2
+L4(M,g(t))
+(5.42)
+
+20
+Franziska Borer, Peter Elbau, Tobias Weth
+and by Lemma 4.2 we further get with the uniform bound for u ∈ CtCx that
+∥∂tu(t)∥2
+L4(M,g(t))
+=
+��
+M
+|∂tu(t)|4e2u(t)dµ¯g
+� 1
+2
+≤ e∥u∥L∞
+t
+L∞
+x ∥∂tu(t)∥2
+L4(M,¯g)
+≤ e∥u∥L∞
+t
+L∞
+x �
+CGNL∥∂tu(t)∥L2(M,¯g)∥∂tu(t)∥H1(M,¯g)
+= e∥u∥L∞
+t
+L∞
+x �
+CGNL
+��
+M
+|∂tu(t)|2e2u(t)e−2u(t)dµ¯g
+� 1
+2 ��
+M
+|∂tu(t)|2e2u(t)e−2u(t)dµ¯g +
+�
+M
+|∇¯g∂tu(t)|2
+¯gdµ¯g
+� 1
+2
+= e∥u∥L∞
+t
+L∞
+x �
+CGNL
+��
+M
+|∂tu(t)|2e−2u(t)dµg(t)
+� 1
+2 ��
+M
+|∂tu(t)|2e−2u(t)dµg(t) +
+�
+M
+|∇g(t)∂tu(t)|2
+g(t)dµg(t)
+� 1
+2
+≤ e∥u∥L∞
+t
+L∞
+x max{e∥u∥L∞
+t
+L∞
+x , e2∥u∥L∞
+t
+L∞
+x }
+�
+CGNL∥∂tu(t)∥L2(M,g(t))∥∂tu(t)∥H1(M,g(t))
+=: ˜C2∥∂tu(t)∥L2(M,g(t))∥∂tu(t)∥H1(M,g(t)),
+(5.43)
+where we used the fact that
+�
+M
+|∇¯g∂tu(t)|2
+¯gdµ¯g =
+�
+M
+|∇g(t)∂tu(t)|2
+g(t)dµg(t) =: G(t).
+Plugging in (5.43) into (5.42) we arrive at
+�
+M
+(Kg(t) − f0 + α(t))3dµg(t) ≤ ˜C2∥∂tu(t)∥2
+L2(M,g(t))∥∂tu(t)∥H1(M,g(t))
+≤
+˜C2
+2
+2 ∥∂tu(t)∥4
+L2(M,g(t)) + 1
+2∥∂tu(t)∥2
+H1(M,g(t))
+≤ ˜C2
+2F 2(t) + 1
+2(F(t) + G(t)),
+(5.44)
+where we used Young’s inequality in the second step.
+With α0 = max{|α1|, |α2|} > 0 we furthermore have that
+2
+�
+M
+(f0 − α(t))(Kg(t) − f0 + α(t))2dµg(t) ≤ 2(∥f0∥L∞(M,¯g) + α0)F(t) =: ˜C3(α0, f0)F(t)
+So, (5.41) yields
+d
+dtF(t) + G(t) ≤ 2
+�
+˜C3F(t) + ˜C2
+2F 2(t) + 1
+2F(t)
+�
+= (2 ˜C3 + 1)F(t) + 2 ˜C2
+2F 2(t)
+=: ˜C4F(t) + 2 ˜C2
+2F 2(t).
+(5.45)
+We recall that with (5.31) we have lim inft→∞ F(t) = 0 and therefore we know that there exist tl → ∞ with
+F(tl) → 0 as l → ∞, see (5.37).
+By integrating (5.45) over (tl, t) ⊂ (tl, T) and taking the supremum over (tl, T) we get with
+� T
+tl G(t)dt ≥ 0
+that
+sup
+t∈(tl,T )
+F(t) ≤ F(tl) + ˜C4
+� T
+tl
+F(t)dt + 2 ˜C2
+2
+� T
+tl
+F 2(t)dt
+≤ F(tl) + ˜C4
+� T
+tl
+F(t)dt + 2 ˜C2
+2
+sup
+t∈(tl,T )
+F(t)
+� T
+tl
+F(t)dt
+≤ F(tl) + ˜C4
+� T
+tl
+F(t)dt + 2 ˜C2
+2
+sup
+t∈(tl,T )
+F(t)
+� ∞
+tl
+F(t)dt.
+With (5.31) we also have
+� ∞
+tl F(t)dt → 0 for l → ∞. So, for T > 0 big enough such that for tl < T big
+enough we have that 2 ˜C2
+2
+� ∞
+tl F(t)dt is small enough to guarantee that 1 − 2 ˜C2
+2
+� ∞
+tl F(t)dt > 0 and therefore the
+term 2 ˜C2
+2 supt∈(tl,T ) F(t)
+� ∞
+tl F(t)dt can be absorbed on the left hand side. So, we get
+sup
+t∈(tl,T )
+F(t) ≤
+1
+�
+1 − 2 ˜C2
+2
+� ∞
+tl F(t)dt
+�
+�
+F(tl) + ˜C4
+� T
+tl
+F(t)dt
+�
+.
+
+Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic
+21
+Letting T → ∞ yields
+sup
+t∈(tl,∞)
+F(t) ≤
+1
+�
+1 − ˜C2
+2
+� ∞
+tl F(t)dt
+�
+�
+F(tl) + ˜C4
+� ∞
+tl
+F(t)dt
+�
+→ 0
+as l → ∞
+which shows the claim.
+To prove now the convergence of the flow, let A > 0 and u0 ∈ Cp,A, p > 2. Furthermore let f ∈ C∞(M) be
+a smooth, nonconstant function, and (f0, λ) ∈ C∞(M) × R the unique pair such that
+f = f0 + λ
+with f0 ≤ 0, f0 nonconstant, and maxx∈M f0(x) = 0. Since by Proposition 2.1 the additive rescaled prescribed
+Gauss curvature flow (2.4) is invariant under adding or subtracting a constant C > 0 to the function f, for all
+functions
+f ∈ {f0 + λ | λ ∈ R}
+we consider the same flow given by
+∂tu(t) = f0 − Kg(t) − αA(t)
+in (0, T) × M,
+(5.46)
+which is (2.4) with f replaced by f0.
+With (2.9) we know that
+1
+2
+�
+M
+(|∇¯gu(T)|2
+¯g + 2 ¯Ku(T) − f0e2u(T ))dµ¯g = Ef0(u(T)) ≤ Ef0(u(0)).
+So, we get with (4.3) that
+1
+2
+�
+M
+|∇¯gu(T)|2
+¯gdµ¯g = Ef0(u(T)) −
+�
+M
+¯Ku(T)dµ¯g + 1
+2
+�
+M
+f0e2u(T )dµ¯g
+≤ Ef0(u(T)) + | ¯K|
+�
+M
+u(T)dµ¯g
+≤ Ef0(u(0)) + | ¯K|
+2 | log(A)|.
+So, u is uniformly (in T) bounded in H1(M, ¯g), i.e., ∥u∥L∞
+t H1x ≤ C.
+We now consider ul := u(tl) for a suitable sequence tl → ∞. By the Theorem of Banach-Alao˘glu we know
+that (ul)l is weak∗ relatively compact in H1(M, ¯g) and therefore (since H1 is reflexive) also weak relatively
+compact. This means that that there exists a subsequence ulk which we again call ul such that ul → u∞
+A weakly
+in H1(M, ¯g) and therefore strongly in L2(M, ¯g) (by a direct consequence of the Rellich–Kondrachov embedding
+Theorem). Furthermore with (2.6) and (2.7) we know that αl := α(tl) → α∞
+A as l → ∞. Moreover we have
+e±ul → e±u∞
+A (as l → ∞) in Lp(M, ¯g) for any 2 ≤ p < ∞. Indeed, with Lemma 5.8 and (5.3) we have
+∥eul − eu∞
+A ∥p
+Lp(M,¯g) =
+�
+M
+epul|1 − eu∞
+A −ul|pdµ¯g ≤ epCuni
+�
+M
+|1 − eu∞
+A −ul|pdµ¯g
+≤ epCuni
+�
+M
+|u∞
+A − ul|pep|u∞
+A −ul||dµ¯g
+≤ epCunie2pCuni
+�
+M
+|u∞
+A − ul|p−2|u∞
+A − ul|2dµ¯g
+≤ e3pCuni(2Cuni)p−2∥u∞
+A − ul∥2
+L2(M,¯g) → 0
+as l → ∞.
+Replacing ul by −ul we get also e−ul → e−u∞
+A in Lp(M, ¯g) as l → ∞ for any p < ∞. Moreover, with Lemma 5.8
+and Lemma 5.9 we also have e2ul∂tul → 0 in L2(M, ¯g) as l → ∞. Furthermore we have
+∥e2ulαl − e2u∞
+A α∞
+A ∥L2(M,¯g) ≤ ∥e2ul(αl − α∞
+A )∥L2(M,¯g) + ∥α∞
+A (e2ul − e2u∞
+A )∥L2(M,¯g)
+≤ ∥e2ul∥L∞(M,¯g)|αl − α∞
+A |A
+1
+2 + |α∞
+A |∥e2ul − e2u∞
+A ∥L2(M,¯g)
+→ 0
+for l → ∞.
+So, considering our evolution equation (5.11), we therefore get
+∆¯gul = e2ul∂tul + ¯K − e2ulf0 + e2ulαl
+→ ¯K − e2u∞
+A f0 + e2u∞
+A α∞
+A =: (∆¯gu)∞
+A
+
+22
+Franziska Borer, Peter Elbau, Tobias Weth
+in L2(M, ¯g).
+Since the Laplace operator ∆¯g is closed we know that (∆¯gu)∞
+A = ∆¯gu∞
+A .
+Hence ∥∆¯g(ul −
+u∞
+A )∥L2(M,¯g) → 0 as l → ∞. So, we even have strong convergence ul → u∞
+A in H2(M, ¯g) and uniformly.
+Thus, passing to the limit l → ∞ in the equation
+e2ul∂tul − ∆¯gul = − ¯K + e2ulf0 − e2ulαl
+we get the identity
+−∆¯gu∞
+A = − ¯K + e2u∞
+A f0 − e2u∞
+A α∞
+A
+and therefore
+Kg∞
+A = f0 − α∞
+A = f0 + 1
+A
+�
+¯K +
+�
+M
+|f0|dµg∞
+A
+�
+which shows the convergence of the flow.
+5.6. The Sign of the Constant α∞
+A . In this subsection we prove Theorem 3.3 and Theorem 3.4, with other
+words, under certain assumptions we can now further estimate the expression
+1
+A
+�
+¯K +
+�
+M
+|f0|dµg∞
+A
+�
+to show that it is positive.
+The proof of Theorem 3.4 is already covered by the proof of Corollary 4.8. So we can turn to Theorem 3.3.
+Proof of Theorem 3.3. We have seen in Lemma 5.7 that in the case where u0 ≡ 1
+2 log(A) ∈ Cp,A, the uniform
+L∞-bound on the global solution of the initial value problem (5.11), (5.12) only depends on A and an upper
+bound on ∥f∥L∞(M,¯g). In other words, if A > 0 and c > 0 are fixed, then there exists τ > 0 with the property
+that
+sup
+t>0
+∥u(t)∥L∞(M,¯g) ≤ τ
+for every f ∈ C∞(M) with ∥f∥L∞(M,¯g) ≤ c and the corresponding solution u of the initial value problem (5.11),
+(5.12) with u0 ≡ 1
+2 log(A) ∈ Cp,A. Consequently, we also have ∥u∞∥L∞(M,¯g) ≤ τ under the current assumptions
+on f, which implies that
+λ = 1
+A
+�
+¯K −
+�
+M
+fe2u∞dµ¯g
+�
+= 1
+A
+�
+¯K + cA −
+�
+M
+(f + c)e2u∞dµ¯g
+�
+≥ c +
+¯K
+A − ∥f + c∥L1(M,¯g)∥e2u∞∥L∞(M,¯g) ≥ c +
+¯K
+A − ∥f + c∥L1(M,¯g)e2τ.
+Hence, if ∥f + c∥L1(M,¯g) < ε := c+ ¯
+K
+A
+e2τ , we have λ > 0.
+6. Appendix
+As before, let (M, ¯g) be a two-dimensional, smooth, closed, connected, oriented Riemann manifold endowed
+with a smooth background metric ¯g. For a domain Ω ⊂ M × R and p ≥ 1, we let W 2,1
+p
+(Ω) denote the space of
+functions u ∈ Lp(Ω) which have weak derivatives Du, D2u and ∂tu in Lp(Ω). In the following, we fix p > 2,
+which implies that
+W 2,1
+p
+(Ω) is continuously embedded in Cα(Ω) for some α = α(p) > 0,
+(6.1)
+see e.g. [13, Lemma 3.3]. We consider the linear parabolic problem
+∂tu(x, t) = a(x, t)∆¯gu(x, t) + c(x, t)u(x, t) + d(x, t),
+(6.2)
+with a, c, d ∈ C(Ω) and d ∈ Lp(Ω). We say that a function u ∈ W 2,1
+p
+(Ω) is a (strong) solution of (6.2) in Ω if
+(6.2) holds almost everywhere in Ω. Specifically, we consider (6.2) on the cylindrical domains ΩT = M × (0, T)
+and �ΩT = M × (−∞, T) in the following.
+In particular, we consider strong solutions of (6.2) together with the initial condition
+u(0, x) = u0(x)
+in M
+(6.3)
+with u0 ∈ W 2,p(M, ¯g), which is supposed to hold in the (initial) trace sense.
+
+Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic
+23
+Proposition 6.1. Let T > 0, a, c ∈ C(ΩT ) with aT :=
+min
+(x,t)∈ΩT
+a(x, t) > 0, let d ∈ Lp(ΩT ) for some p > 2, and
+let u0 ∈ W 2,p(M, ¯g).
+Then the initial value problem (6.2), (6.3) has a unique strong solution u ∈ W 2,1
+p
+(ΩT ). Moreover, u satisfies
+the estimate
+∥u∥W 2,1
+p
+(ΩT ) ≤ C
+�
+∥u0∥W 2,p(M,¯g) + ∥d∥Lp(ΩT )
+�
+(6.4)
+with a constant C > 0 depending only on ∥a∥L∞(ΩT ), ∥c∥L∞(ΩT ) and aT . Moreover, C does not increase after
+making T smaller.
+If, moreover, a, c, d ∈ Cα(ΩT ) for some α > 0, then u ∈ C(ΩT )∩C2,1(ΩT ) is a classical solution of (6.2), (6.3),
+and we have the inequality
+∥u0∥H1(M,¯g) ≥ lim sup
+t→0+ ∥u(t)∥H1(M,¯g)
+(6.5)
+Proof. In the following, the letter C stands for various positive constants depending only on ∥a∥L∞(ΩT ),
+∥c∥L∞(ΩT ), and aT , and which do not increase after making T smaller.
+Step 1: We first assume that we are given a strong solution u ∈ W 2,1
+p
+(ΩT ) of (6.2), (6.3) with u0 ≡ 0 ∈
+W 2,p(M, ¯g). We then define v : �ΩT → R by
+v(x, t) =
+�
+u(x, t),
+for t > 0;
+0,
+for t ≤ 0.
+Then v ∈ W 2,1
+p
+(�ΩT ) solves (6.2) with a, c, d replaced by suitable extensions ˜a, ˜c, ∈ L∞(�ΩT ), ˜d ∈ Lp(�ΩT ) satisfying
+˜a(x, t) = a(x, 0), ˜c(x, t) = c(x, 0) and ˜d(x, t) = 0 for t ≤ 0, x ∈ M.
+Therefore, [14, Theorem 7.22] gives rise to the uniform bound
+∥D2v∥Lp(�ΩT ) + ∥∂tv∥Lp(�ΩT ) ≤ C
+�
+∥ ˜d∥Lp(�ΩT ) + ∥v∥Lp(�ΩT )
+�
+.
+(6.6)
+This translates into the estimate
+∥D2u∥Lp(ΩT ) + ∥∂tu∥Lp(ΩT ) ≤ C
+�
+∥d∥Lp(ΩT ) + ∥u∥Lp(ΩT )
+�
+.
+(6.7)
+Moreover, setting V (t) := ∥u(t)∥p
+Lp(M,¯g) for t ∈ R, we have V (0) = 0 and
+˙V (t) = p
+�
+M
+|u(t)|p−2u(t)∂tu(t)dµ¯g ≤ pV (t)
+1
+p′ ∥∂tu(t)∥Lp(M,¯g)
+≤ p
+�
+V (t)
+p′
++
+∥∂tu(t)∥p
+Lp(M,¯g)
+p
+�
+= p
+p′ V (t) + ∥∂tu(t)∥p
+Lp(M,¯g)
+for t ∈ (0, T), therefore
+V (t) =
+� t
+0
+˙V (s) ds ≤ p
+p′
+� t
+0
+V (s) ds + ∥∂tu∥p
+Lp(Ωt)
+≤ p
+p′
+� t
+0
+V (s) ds + C
+�
+∥d∥p
+Lp(Ωt) + ∥u∥p
+Lp(Ωt)
+�
+≤ C
+�� t
+0
+V (s) ds + ∥d∥p
+Lp(Ωt)
+�
+.
+By Gronwall’s inequality we get V (t) ≤ C∥d∥p
+Lp(Ωt) and thus
+∥u(t)∥Lp(M,¯g) ≤ C∥d∥Lp(Ωt)
+for t ∈ [0, T].
+(6.8)
+This already implies the uniqueness of strong solutions of (6.2), (6.3), since the difference u of two solutions
+u1, u2 ∈ W 2,1
+p
+(ΩT ) of (6.2), (6.3) satisfies (6.2), (6.3) with u0 = 0 and d = 0. Moreover, if u ∈ W 2,1
+p
+(ΩT ) is a
+strong solution of (6.2), (6.3), then the function ˆu ∈ W 2,1
+p
+(ΩT ) given by ˆu(x, t) := u(x, t) − u0(x) safisfies (6.2),
+(6.3) with u0 = 0 and d replaced by ˆd given by
+ˆd(x, t) = d(x, t) + a(x, t)∆¯gu0(x) + c(x, t)u0(x).
+Consequently, combining (6.7) and (6.8), and using an interpolation estimate for Du, we find that
+∥u∥W 2,1
+p
+(ΩT ) ≤ ∥ˆu∥W 2,1
+p
+(ΩT ) + ∥u0∥W 2,p(M,¯g) ≤ C
+�
+∥ ˆd∥Lp(ΩT ) + ∥ˆu∥Lp(ΩT )
+�
++ ∥u0∥W 2,p(M,¯g)
+≤ C∥ ˆd∥Lp(ΩT ) + ∥u0∥W 2,p(M,¯g) ≤ C
+�
+∥d∥Lp(ΩT ) + ∥u0∥W 2,p(M,¯g)
+�
+,
+
+24
+Franziska Borer, Peter Elbau, Tobias Weth
+as claimed in (6.4).
+Step 2 (Existence): In the case where a, c, d ∈ Cα(ΩT ) and u0 ∈ C2+α(M), the existence of a classical
+solution u ∈ C(ΩT ) ∩ C2,1(ΩT ) of (6.2), (6.3) follows as in [14, Theorem 5.14].
+In the general case we consider (6.2), (6.3) with coefficients an, cn, dn ∈ Cα(ΩT ), u0,n ∈ C2+α(M), in place
+of a, c, d, u0 with the property that an → a, cn → c in L∞(ΩT ), dn → d ∈ Lp(ΩT ) as well as u0,n → u0 in
+W 2,p. The associated unique solutions un ∈ C(ΩT ) ∩ C2,1(ΩT ) are uniformly bounded in W 2,1
+p
+(ΩT ) by (6.4),
+and therefore we have un ⇀ u in W 2,1
+p
+(ΩT ) after passing to a subsequence. For every φ ∈ C∞
+c (ΩT ), we then
+have
+�
+ΩT
+�
+∂tu(x, t) − a(x, t)∆¯gu(t.x) − c(x, t)u(x, t) − d(x, t)
+�
+φ(x, t)dµ¯g(x)dt
+= lim
+n→∞
+�
+ΩT
+�
+∂tun(x, t) − an(x, t)∆¯gun(x, t) − cn(x, t)un(x, t) − dn(x, t)
+�
+φ(x, t)dµ¯g(x)dt = 0,
+and from this we deduce that ∂tu(x, t) − a(x, t)∆¯gu(x, t) − c(x, t)u(x, t) − d(x, t) = 0 almost everywhere in ΩT ,
+so u is a strong solution of (6.2).
+Step 3: It remains to show the inequality (6.5) in the case where a, c, d ∈ Cα(ΩT ) for some α > 0. Since
+u ∈ C(ΩT ) ∩ C2,1(ΩT ) in this case and therefore
+∥u0∥L2(M,¯g) = lim
+t→0+ ∥u(t)∥L2(M,¯g),
+it suffices to show that
+∥∇u0∥L2(M,¯g) ≥ lim sup
+t→0+ ∥∇u(t)∥L2(M,¯g).
+(6.9)
+If u0 ∈ C2+α(M) for some α > 0, this follows by [14, Theorem 5.14] with lim in place of lim sup, since the
+function t �→ u(t) is continuous from [0, T) → C2+α(M) in this case. Moreover, in this case we have, by H¨older’s
+and Young’s inequality,
+d
+dt∥∇u(t)∥2
+L2(M,¯g) = −
+�
+M
+∂tu(t)∆u(t)dµ¯g
+= −
+�
+M
+�
+a(t)|∆u(t)|2 + c(t)u(t)∆u(t) + d(t)∆u(t)
+�
+dµ¯g
+≤ −aT ∥∆¯gu(t)∥2
+L2(M,¯g) + ∥c(t)u(t) + d(t)∥L2(M,¯g)∥∆¯gu(t)∥L2(M,¯g)
+≤ −aT ∥∆¯gu(t)∥2
+L2(M,¯g) + aT ∥∆¯gu(t)∥2
+L2(M,¯g) +
+1
+4aT
+∥c(t)u(t) + d(t)∥2
+L2(M,¯g)
+=
+1
+4aT
+∥c(t)u(t) + d(t)∥2
+L2(M,¯g),
+and therefore
+∥∇u(t)∥2
+L2(M,¯g) ≤ ∥∇u(0)∥2
+L2(M,¯g) +
+1
+4aT
+� t
+0
+∥c(s)u(s) + d(s)∥2
+L2(M,¯g) ds
+for t > 0.
+(6.10)
+In the general case, we consider (6.2), (6.3) with a sequence of initial conditions un,0 in place of u0, where
+un,0 → u0 in H2(M).
+The associated unique solutions un ∈ C(ΩT ) ∩ C2,1(ΩT ) are uniformly bounded in
+W 2,1
+p
+(ΩT ) by (6.4), and they are also uniformly bounded in C2,1([ε, T] × M) by [14, Theorem 5.15] for every
+ε ∈ (0, T). Fix t ∈ (0, T). Passing to a subsequence, we may assume that un ⇀ u in W 2,1
+p
+(ΩT ), un → u
+strongly in C0(ΩT ) and un(t) → u(t) strongly in C1(M). As in Step 2, we see, by testing with φ ∈ C∞
+c (ΩT ),
+that ∂tu(x, t) − a(x, t)∆¯gu(x, t) − c(x, t)u(x, t) − d(x, t) = 0 almost everywhere in ΩT , so u is the unique strong
+solution of (6.2), (6.3). Moreover, by (6.10) we have
+∥∇u(t)∥2
+L2(M,¯g) = lim
+n→∞ ∥∇un(t)∥2
+L2(M,¯g)
+≤ lim
+n→∞
+�
+∥∇un(0)∥2
+L2(M) +
+1
+4aT
+� t
+0
+∥c(s)un(s) + d(s)∥2
+L2(M,¯g) ds
+�
+= ∥∇u(0)∥2
+L2(M,¯g) +
+1
+4aT
+� t
+0
+∥c(s)u(s) + d(s)∥2
+L2(M,¯g) ds.
+It thus follows that
+∥∇u(t)∥2
+L2(M,¯g) − ∥∇u(0)∥2
+L2(M,¯g) ≤
+1
+4aT
+� t
+0
+∥c(s)u(s) + d(s)∥2
+L2(M,¯g) ds
+
+Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic
+25
+and therefore
+lim sup
+t→0
+�
+∥∇u(t)∥2
+L2(M,¯g) − ∥∇u(0)∥2
+L2(M,¯g)
+�
+≤
+1
+4aT
+lim
+t→0+
+� t
+0
+∥c(s)u(s) + d(s)∥2
+L2(M,¯g) ds = 0,
+as claimed in (6.9).
+Next we prove a maximum principle for solutions of (6.2), (6.3). We need the following preliminary lemma.
+Lemma 6.2. Let T > 0.
+(i) For any function u ∈ C2(M) we have
+�
+{x∈M|u(x)>0}
+∆¯gudµ¯g ≤ 0.
+(ii) Let u, ρ ∈ C1([0, T]) be functions with u(0) ≤ 0 and ρ(T) ≥ 0. Then
+�
+{t∈[0,T ]|u(t)>0}
+�
+ρ(t)∂tu(t) + κu(t)
+�
+dt ≥ 0
+with
+κ :=
+sup
+s∈(0,T )
+∂tρ(s).
+(6.11)
+(iii) Let u ∈ C2,1(ΩT ) ∩ C0,1(ΩT ), ρ ∈ C0,1(ΩT ) be functions with u ≤ 0 on M × {0} and ρ ≥ 0 on M × {T}.
+Then we have
+�
+{(x,t)∈M×[0,T ]|u(x,t)>0}
+(ρ(x, t)∂tu(x, t) + κu(x, t) − ∆¯gu(x, t))dµ¯g(x)dt ≥ 0
+with
+κ :=
+sup
+(s,x)∈M×(0,T )
+∂tρ(s, x).
+(6.12)
+Proof. (i) By Lebesgue’s theorem, it suffices to prove
+�
+{x∈M|u(x)>εn}
+∆¯gudµ¯g ≤ 0
+(6.13)
+for a sequence εn → 0+. By Sard’s Lemma, we may choose this sequence such that Ωε := {x ∈ M | u(x) > εn}
+is an open set of class C1, whereas the outer unit vector field of Ωε is given by (x, t) �→ −
+∇¯gu(x,t)
+|∇¯gu(x,t)|¯g . Hence
+(6.13) follows from the divergence theorem.
+(ii) The set {t ∈ [0, T] | u(t) > 0} is a union of at most countably many open intervals Ij, j ∈ N. For any such
+interval, partial integration gives
+�
+Ij
+�
+ρ(t)∂tu(t) + ∂tρ(t)u(t)
+�
+dt =
+�
+0,
+if T ̸∈ Ij;
+ρ(T)u(T) ≥ 0,
+if T ∈ Ij.
+Consequently,
+�
+{t∈[0,T ]|u(t)>0}
+ρ(t)∂tu(t) dt ≥ −
+�
+{t∈[0,T ]|u(t)>0}
+∂tρ(t)u(t) dt ≥ −
+�
+{t∈[0,T ]|u(t)>0}
+κu(t) dt
+with κ given in (6.11). This shows the claim.
+(iii) This is a direct consequence of (i), (ii) and Fubini’s theorem.
+Proposition 6.3. (Maximum principle)
+Let T > 0, a, c ∈ C(ΩT ) with aT :=
+min
+(x,t)∈ΩT
+a(x, t) > 0, let d ∈ Lp(ΩT ) for some p > 2 with dT :=
+sup(x,t)∈ΩT d(x, t) < ∞, and let u0 ∈ W 2,p(M, ¯g).
+Moreover, let u ∈ W 2,1
+p
+(ΩT ) be the unique solution of
+(6.2), (6.3).
+(i) If u0 ≤ 0 on M and dT ≤ 0, then u ≤ 0 on ΩT .
+(ii) If c ≡ 0 on ΩT , then
+u(x, t) ≤ ∥u+
+0 ∥L∞(M,¯g) + tdT
+for t ∈ [0, T], x ∈ M.
+(6.14)
+
+26
+Franziska Borer, Peter Elbau, Tobias Weth
+Proof. (i) Step 1: We consider the special case a ∈ C0,1(ΩT ), u0 ≤ 0 and dT ≤ −ε for some ε > 0. We put
+ρ := 1
+a ∈ C0,1(ΩT ) and κ :=
+sup
+(s,x)∈M×(0,T )
+∂tρ(s, x) as in (6.12). Moreover, we consider the function
+˘u ∈ W 2,1
+p
+(ΩT ),
+˘u(x, t) = e−˘κtu(x, t)
+with ˘κ =
+|κ|
+min(x,t)∈ΩT ρ(x,t) + ∥c∥L∞(ΩT ), noting that ˘u satisfies
+ρ(x, t)∂t˘u(x, t) − ∆¯g˘u(x, t) + κ˘u(x, t)
+= e−˘κt�
+u(x, t)(ρ(x, t)c(x, t) − ρ(x, t)˘κ + κ) + ρ(x, t)d(x, t)
+�
+≤ −ρ(x, t)εe−˘κt
+almost everywhere in {(x, t) ∈ ΩT | ˘u(x, t) > 0}.
+(6.15)
+We now let (un)n∈N be a sequence in C2,1(ΩT ) ∩ C0,1(ΩT ) with un(x, 0) ≤ 0 and un → ˘u in W 2,1
+p
+(ΩT ).
+Since the functions gn := 1{(x,t)∈M×[0,T ]|un(x,t)>0} are bounded in Lp′(ΩT ), we may pass to a subsequence such
+that gn ⇀ g in Lp′(ΩT ), where g ≥ 0 and g ≡ 1 in {(x, t) ∈ M × [0, T] | ˘u(x, t) > 0}, since un → ˘u uniformly
+as a consequence of (6.1) and therefore gn → 1 pointwisely on {(x, t) ∈ M × [0, T] | ˘u(x, t) > 0}. Applying
+Lemma 6.2 (iii) to un, we find that
+0 ≤
+�
+{(x,t)∈M×[0,T ]|un(x,t)>0}
+�
+ρ(x, t)∂tun(t) − ∆¯gun(x, t) + κun(x, t)
+�
+dµ¯g(x)dt
+=
+�
+M×(0,T )
+gn(x, t)
+�
+ρ(x, t)∂tun(x, t) − ∆¯gun(x, t) + κun(x, t)
+�
+dµ¯g(x)dt
+for all n ∈ N and therefore
+0 ≤ lim
+n→∞
+�
+M×(0,T )
+gn(x, t)
+�
+ρ(x, t)∂tun(x, t) − ∆¯gun(x, t) + κun(x, t)
+�
+dµ¯g(x)dt
+=
+�
+M×(0,T )
+g(x, t)
+�
+ρ(x, t)∂t˘u(x, t) − ∆¯g˘u(x, t) + κ˘u(x, t)
+�
+dµ¯gdt
+≤ −
+�
+M×(0,T )
+g(x, t)ρ(x, t)εe−˘κtdµ¯g(x)dt ≤ −
+�
+{(x,t)∈M×(0,T )|˘u(x,t)>0}
+ρ(x, t)εe−˘κtdµ¯g(x)dt.
+We thus conclude that {(x, t) ∈ M × (0, T) | ˘u(x, t) > 0} = {(x, t) ∈ M × (0, T) | u(x, t) > 0} = ∅ and therefore
+u ≤ 0 in M × (0, T).
+Step 2: In the special case where a ∈ C0,1(ΩT ), u0 ≤ 0 and dT ≤ 0, we may apply Step 1 to the functions
+uε ∈ W 2,1
+p
+(ΩT ) defined by uε(x, t) = u(x, t) − εt, which yields that uε ≤ 0 for every ε > 0 and therefore u ≤ 0
+in ΩT .
+Step 3: In the general case, we consider a sequence an ∈ C0,1(ΩT ) with an → a in C(ΩT ), and we let un denote
+the associated solutions of (6.2), (6.3) with a replaced by an. As in the end of the proof of Proposition 6.1, we
+then find that, after passing to a subsequence, un ⇀ ˜u in W 2,1
+p
+(ΩT ), where ˜u is a solution of (6.2), (6.3). By
+uniqueness, we have u = ˜u. Moreover, since un ≤ 0 for all n by Step 3, we have u = ˜u ≤ 0, as required.
+(ii) We consider the function v ∈ W 2,1
+p
+(ΩT ) given by v(x, t) = u(x, t)−∥u+
+0 ∥L∞(M) −tdT , which, by assumption,
+satisfies (6.2), (6.3) with c ≡ 0, d − dT in place of d and u0 − ∥u+
+0 ∥L∞(M) in place of u0. Then (i) yields v ≤ 0
+in ΩT , and therefore u satisfies (6.14).
+References
+[1]
+F. Borer, L. Galimberti, and M. Struwe. “Large” conformal Metrics of prescribed Gauss Curvature on
+Surfaces of higher Genus. Commentarii Mathematici Helvetici 90.2 (2015), pp. 407–428. doi: 10.4171
+/CMH/358.
+[2]
+R. Buzzano, M. Schulz, and M. Struwe. Variational Methods in Geometric Analysis. (to appear).
+[3]
+T. Cazenave, A. Haraux, and Y. Martel. An Introduction to Semilinear Evolution Equations. Oxford
+Lecture Series in Mathematics and its Application 13. The Clarendon Press, Oxford University Press,
+1999.
+[4]
+J. Ceccon and M. Montenegro. Optimal Lp-Riemannian Gagliardo-Nirenberg inequalities. Mathematische
+Zeitschrift 258.4 (2008), pp. 851–873. issn: 0025-5874. doi: 10.1007/s00209-007-0202-8.
+[5]
+S.-Y. A. Chang. Non-linear elliptic equations in conformal geometry. Zurich Lectures in Advanced Math-
+ematics 2. European Mathematical Society, 2004.
+
+Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic
+27
+[6]
+S.-Y. A. Chang and P. C. Yang. Prescribing Gaussian curvature on S2. Acta Mathematica 159.1 (1987),
+pp. 215–259. issn: 1871-2509. doi: 10.1007/BF02392560.
+[7]
+W.-Y. Ding and J.-Q. Liu. A Note on the Problem of Prescribing Gaussian Curvature on Surfaces. Trans-
+actions of the American Mathematical Society 347.3 (1995), pp. 1059–1066. issn: 00029947. url: http:
+//www.jstor.org/stable/2154889.
+[8]
+L. Galimberti. Compactness issues and bubbling phenomena for the prescribed Gaussian curvature equation
+on the torus. Calculus of Variations and Partial Differential Equations 54.3 (2015), pp. 2483–2501. doi:
+10.1007/s00526-015-0872-8.
+[9]
+P. T. Ho. Prescribed Curvature Flow on Surfaces. Indiana University Mathematics Journal 60.5 (2011),
+pp. 1517–1541. issn: 00222518, 19435258. url: http://www.jstor.org/stable/24903835.
+[10]
+J. L. Kazdan and F. W. Warner. Curvature Functions for Open 2-Manifolds. Annals of Mathematics 99.2
+(1974), pp. 203–219. issn: 0003486X. url: http://www.jstor.org/stable/1970898.
+[11]
+J. L. Kazdan and F. W. Warner. Curvature Functions for Compact 2-Manifolds. Annals of Mathematics
+99.1 (1974), pp. 14–47. issn: 0003486X. url: http://www.jstor.org/stable/1971012.
+[12]
+P. Koebe. ¨Uber die Uniformisierung beliebiger analytischer Kurven (Dritte Mitteilung). Nachrichten von
+der Gesellschaft der Wissenschaften zu G¨ottingen, Mathematisch-Physikalische Klasse 1 (1908), pp. 337–
+358.
+[13]
+O. Ladyˇzenskaja, V. Solonnikov, and N. Ural’ceva. Linear and Quasilinear Equations of Parabolic Type.
+Vol. 23. Translations of mathematical monographs. Providence, RI: American Mathematical Society, 1968.
+[14]
+G. M. Lieberman. Second Order Parabolic Differential Equations. World Scientific, 1996. doi: 10.1142/3
+302.
+[15]
+J. Moser. A Sharp Form of an Inequality by N. Trudinger. Indiana University Mathematics Journal 20.11
+(1971), pp. 1077–1092. issn: 00222518, 19435258. url: http://www.jstor.org/stable/24890183.
+[16]
+H. Poincar´e. Sur l’uniformisation des fonctions analytiques. Acta Mathematica 31 (1908), pp. 1–64.
+[17]
+J. Schauder. Der Fixpunktsatz in Funktionalra¨umen. Studia Mathematica 2.1 (1930), pp. 171–180. url:
+http://eudml.org/doc/217247.
+[18]
+M. Struwe. A flow approach to Nirenberg’s problem. Duke Math. J. 128.1 (2005), pp. 19–64. doi: 10.121
+5/S0012-7094-04-12812-X.
+[19]
+M. Struwe. “Bubbling” of the prescribed curvature flow on the torus. Journal of the European Mathematical
+Society 22.10 (2020), pp. 3223–3262.
+[20]
+N. S. Trudinger. On embeddings into Orlicz spaces and some applications. J. Math. Mech. 17 (1967),
+pp. 473–484.
+

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+Impact of Charge Conversion on NV-Center Relaxometry
+Isabel Cardoso Barbosa, Jonas Gutsche, and Artur Widera∗
+Department of Physics and State Research Center OPTIMAS,
+University of Kaiserslautern-Landau,
+Erwin-Schroedinger-Str. 46, 67663 Kaiserslautern, Germany
+(Dated: January 4, 2023)
+Relaxometry schemes employing nitrogen-vacancy (NV) centers in diamonds are essential in bi-
+ology and physics to detect a reduction of the color centers’ characteristic spin relaxation (T1) time
+caused by, e.g., paramagnetic molecules in proximity. However, while only the negatively-charged
+NV center is to be probed in these pulsed-laser measurements, an inevitable consequence of the
+laser excitation is the conversion to the neutrally-charged NV state, interfering with the result for
+the negatively-charged NV centers’ T1 time or even dominating the response signal. In this work, we
+perform relaxometry measurements on an NV ensemble in nanodiamond combining a 520 nm excita-
+tion laser and microwave excitation while simultaneously recording the fluorescence signals of both
+charge states via independent beam paths. Correlating the fluorescence intensity ratios to the fluo-
+rescence spectra at each laser power, we monitor the ratios of both charge states during the T1-time
+measurement and systematically disclose the excitation-power-dependent charge conversion. Even
+at laser intensities below saturation, we observe charge conversion, while at higher intensities, charge
+conversion outweighs spin relaxation. These results underline the necessity of low excitation power
+and fluorescence normalization before the relaxation time to accurately determine the T1 time and
+characterize paramagnetic species close to the sensing diamond.
+I.
+INTRODUCTION
+The negatively-charged nitrogen-vacancy (NV) cen-
+ter in diamond constitutes a versatile tool for the detec-
+tion of magnetic [1–9] and electric [10] fields with high
+sensitivity and spatial resolution. Measurement of the
+NV centers’ spin relaxation (T1) time is widely applied
+in different fields of science to detect magnetic noise
+[11, 12].
+Various so-called relaxometry measurement
+schemes employ a reduction of the NV centers’ T1 time
+with the host nanodiamond exposed to paramagnetic
+molecules fluctuating at the NV centers’ resonance fre-
+quency [13–15]. Thus, relaxometry schemes have been
+used to detect a superparamagnetic nanoparticle [16], or
+paramagnetic Gd3+ ions [15, 17–20]. Further, relaxome-
+try with NV− centers has been utilized to trace chemical
+reactions involving radicals [21, 22]. Also, the NV cen-
+ters’ T1 time as a measure for the presence of paramag-
+netic noise gains momentum in biological applications
+[7, 12]. Individual ferritin proteins have been detected
+[23] and relaxometry has been applied to detect radicals
+even inside cells [24–27].
+Especially in the field of biology, T1 measurement
+schemes are often conducted only with optical excita-
+tion of the NV− centers, while the readout of their spin
+states is realized by detection of the ensemble’s fluores-
+cence intensity. However, recent results indicate that a
+second process impeding the NV− centers’ fluorescence
+signal is present in relaxometry measurements [20, 28–
+31]. The laser pulse that is fundamental for preparation
+∗ Author
+to
+whom
+correspondence
+should
+be
+addressed:
+[email protected]
+of the NV− centers’ spin state can additionally ionize the
+NV− center to its neutrally-charged state, NV0. Conver-
+sion under illumination and back-conversion in the dark
+influence the NV− centers’ fluorescence signal, compli-
+cating a seemingly simple measurement. A quantitative
+determination of the unwanted contribution of the NV0
+state to the NV− relaxometry data is, however, elusive.
+In this work, we compare the results of two relaxometry
+schemes well-known in literature for the same nanodi-
+amond at varying laser powers. Additionally, we intro-
+duce a novel method to extract the ratio of the two NV
+charge states from the NV centers’ fluorescence spectra
+throughout the entire measurement sequence to give an
+insight into the vivid NV charge dynamics we observe
+in our data.
+A level scheme of the NV center in diamond is de-
+picted in Fig. 1, including the negatively-charged NV−
+[32–34], the neutrally-charged NV0 and transitions from
+NV− to NV0 under green illumination [35, 36]. We in-
+clude transitions independent of excitation power from
+the NV0’s ground state to NV−, reflecting the observa-
+tion of recharging processes in the dark in [28, 29] and
+in this work.
+Using a 520 nm laser, we non-resonantly excite the
+NV− centers from their triplet ground state 3A2 to the
+electronically-excited state 3E. Because 3E’s states mS =
+±1 are preferentially depopulated via the NV− centers’
+singlet states 1A1 and 1E, illumination with a green laser
+will spin polarize the NV− centers into their ground
+spin state mS = 0 [34].
+The T1 time describes how
+long this spin polarization persists until the spin popu-
+lation decays to a thermally mixed state [34]. It can reach
+up to 6 ms in bulk diamonds at room temperature [37]
+and is influenced by paramagnetic centers within the
+arXiv:2301.01063v1  [quant-ph]  3 Jan 2023
+
+2
+host diamond or on its surface [38, 39]. In the simplest
+T1 measurement scheme, spin polarization is achieved
+by a laser pulse, followed by a second readout-laser
+pulse after a variable relaxation time τ. Besides differ-
+ent durations, the two laser pulses are identical. There-
+fore, the readout pulse is capable of spin-polarizing and
+ionizing the NV-center ensemble as well as the initial-
+ization pulse. Additionally, the spin-polarization pulse
+provides information about the charge-conversion pro-
+cesses during laser excitation.
+To determine the T1 time of NV− centers of a spe-
+cific orientation in the diamond crystal, coherent spin
+manipulation is introduced in these measurements [38].
+Here, a resonant microwave π pulse transfers the pop-
+ulation of these NV− centers from mS = 0 to mS = +1
+or mS = −1 after the spin-polarization pulse. A sec-
+ond laser pulse is used for the readout of the spin state.
+Repetition of the sequence with the π pulse omitted and
+subtracting the readout signals from each other yields a
+spin-polarization signal as a function of τ that is robust
+against background fluorescence [38, 40].
+In the following, we present our experimental sys-
+tem in Section II. Our results are divided into two main
+parts. We first analyze fluorescence spectra of NV cen-
+ters in a single nanodiamond to assign concentration ra-
+tios to count ratios measured with SPCMs in Section III.
+This knowledge allows us to quantify the NV0 contribu-
+tion during the spin-relaxation dynamics in Section IV.
+FIG. 1. Level scheme of the NV center in diamond. Depicted
+are levels of the negatively-charged NV− and the neutrally-
+charged NV0 and transitions between the two charge states.
+Gray arrows show transitions between NV−’s triplet and sin-
+glet states, mediated via intersystem crossing (ISC). Green ar-
+rows denote transitions driven by a green laser, red and or-
+ange arrows mark the fluorescence of the NV charge states.
+Light-green dashed arrows between mS states are transitions
+driven by microwave radiation at 2.87 GHz at zero magnetic
+field. Additionally, the light-green dashed arrows represent
+the relaxation of the spin-polarized state to a thermally mixed
+state without illumination (T1). The purple dashed arrows de-
+note charge transfer processes in the dark.
+II.
+EXPERIMENTAL SYSTEM
+We
+perform
+our
+studies
+on
+a
+single
+nanodia-
+mond
+crystal
+of
+size
+750 nm
+commercially
+avail-
+able
+from
+Adamas
+Nano
+as
+water
+suspension
+(NDNV/NVN700nm2mg).
+As specified by the man-
+ufacturer,
+the nanodiamonds’ NV concentration is
+[NV] ≈ 0.5 ppm, which is about 2 × 104 NV centers per
+diamond.
+For sample preparation, the suspension is
+treated in an ultrasonic bath to prevent the formation of
+crystal agglomerates. We spin-coat the nanodiamonds
+to a glass substrate and subsequently remove the sol-
+vent by evaporating the residual water on a hot contact
+plate.
+To probe the NV centers in a single nanodiamond, we
+use a microscope consisting of an optical excitation and
+detection section and a microwave setup, as shown in
+Fig. 2. A CW-laser source of wavelength λ = 520 nm
+is used to optically excite the NV centers with a max-
+imum laser power of 4.9 mW.
+The laser beam is fo-
+50:50 NPBS
+NF 514 nm
+DM 550 nm
+laser 
+520 nm
+AOM
+objective
+sample
+permanent magnet
+MW antenna
+LP 550 nm
+ND1
+ND2
+SP 625 nm
+LP 665 nm
+to SPCM2
+> 665 nm
+to SPCM1
+    < 600 nm
+to spectrometer
+spectrometer
+camera
+tube lens
+grating
+(a)
+(b)
+FIG. 2.
+Experimental setup for recording NV fluorescence
+spectra and relaxometry data. In both setups, the excitation
+is the same, but the detection sections are different for the re-
+spective application. (a) NV centers in a single crystal nanodi-
+amond are excited by a 520 nm-laser in combination with an
+acousto-optic modulator (AOM). The light stemming from the
+sample is filtered by a dichroic mirror (DM), a longpass filter
+(LP) and a notch filter (NF) with given wavelenghts and passes
+a non-polarizing beamsplitter (NPBS). The remaining fluores-
+cence is spectrally resolved on a camera chip. This setup is
+used for the measurement of the NV fluorescence spectra. (b)
+The NV fluorescence is split into two arms of a beamsplitter,
+additionally filtered with an LP or a shortpass filter (SP) and
+detected with fiber-coupled SPCMs. The SP is tilted to only
+transmit fluorescence below 600 nm.
+To keep the detectors
+below saturation, neutral-density (ND) filters are used. Lu-
+minescence above 665 nm (NV− fluorescence) is detected in
+SPCM2, while light below 600 nm (NV0 fluorescence) is de-
+tected in SPCM1. Transitions of the NV− centers’ spin states
+mS are driven with a microwave (MW) antenna. This setup is
+used for the measurement of the charge-state dependent relax-
+ometry.
+
+3
+cused to a spot-size diameter of 700 nm (1/e2 diame-
+ter), reaching a maximum intensity of ∼ 2500 kW cm−2.
+Pulses are generated by an AOM with an edge width of
+about 120 ns. Laser light is guided through an objective
+(NA = 0.5, WD = 2.1 mm) and focused at the position
+of the nanodiamond. Fluorescent light stemming from
+the sample is guided back through the objective and fil-
+tered by a dichroic mirror with a cut-on wavelength of
+550 nm. Next, the fluorescence light is filtered by an ad-
+ditional 550 nm-longpass filter and a 514 nm-notch filter
+to prevent detection of reflected laser light. The filtered
+fluorescence light is branched at a 50:50 non-polarizing
+beamsplitter, giving the possibility to further filter the
+luminescence and collect it in two separate detectors. In
+particular, our setup allows for tailoring the transmitted
+wavelengths to the spectral regions, where either pho-
+ton emission from the neutral or the negative NV charge
+state dominates in each beam path individually. Thus,
+we can easily discriminate between the emission of both
+charge states in our measurements. In this work, we
+make use of different detectors. While for spectral anal-
+ysis of the NV centers’ fluorescence, we use a spectrom-
+eter (Fig. 2 (a)), we employ two single-photon counting
+modules (SPCMs) as detectors for our spin-relaxation
+measurements (Fig. 2 (b)) in combination with a time-
+to-digital converter.
+Microwave signals are generated, amplified, and
+brought close to the nanodiamond using a microwave
+antenna structure written on a glass substrate.
+All
+experiments are carried out under ambient conditions
+and in an external magnetic field in the order of 12 mT
+caused by a permanent magnet to split the NV centers’
+ODMR resonances. In our ODMR spectrum, eight reso-
+nances appear because of the four existing orientations
+of NV centers in the single diamond crystal. We select
+one resonance to drive Rabi oscillations, from which we
+determine a π-pulse length of 170 ns.
+III.
+FLUORESCENCE SPECTRA
+A.
+Setup
+To spectrally resolve the NV centers’ fluorescence, we
+use a spectrometer. The incoming fluorescence light is
+dispersed at a grating (600 grooves/mm), and an achro-
+matic tube lens translates the angle dispersion into a
+spatial dispersion. Thus, the detection of light of dif-
+ferent wavelengths at different positions of a camera’s
+chip is facilitated, and spectra are obtained from 500 nm
+to 760 nm. With this setup, we achieve a resolution of
+∆λ ≈ 0.19 nm/pixel. Each spectrum consists of a mean
+of at least 20 spectra recorded at each laser power. We
+correct the spectra for the wavelength-dependent prop-
+erties of optical elements in the beam path and subtract
+a background.
+B.
+Concentration ratio assignment
+Corrected fluorescence spectra of a monocrystalline
+nanodiamond for excitation laser powers from 1 % to
+100 % are depicted in Fig. 3 (a). Two features, the NV0s’
+ZPL at ∼ 575 nm [41] and the NV−s’ ZPL at ∼ 639 nm
+[42] are clearly visible.
+The overlapping fluorescence
+spectra of both NV charge states show phonon broad-
+ening. Conform with the observation in [1], but con-
+trary to the results in [31], the NV0s’ ZPL intensity
+increases with higher laser power with respect to the
+NV−s’ ZPL in our sample. These results indicate a lower
+[NV−]/[NV0] ratio at higher laser powers and thus an
+increasing charge conversion for higher powers.
+We obtain area-normalized extracted spectra for NV−
+and for NV0 from our recorded data as shown in
+Fig. 3 (b). We conduct the spectra decomposition anal-
+ysis of our spectra according to Alsid et al. and follow
+the nomenclature given in reference [43]. The fraction of
+[NV−] of the total NV concentration [NVtotal] is defined
+(a)
+(b)
+FIG. 3. NV fluorescence spectra. (a) Spectra recorded at laser
+powers from 1 % to 100 %. The NV0s’ ZPL at ∼ 575 nm and the
+NV−s’ ZPL at ∼ 639 nm are evident and marked in the spec-
+trum. For better visibility, spectra were normalized to the sum
+of the NV charge states’ ZPL intensities. (b) Area-normalized
+decomposed basis functions for NV0 and NV−.
+
+4
+by
+[NV−]
+[NVtotal] =
+[NV−]
+[NV−] + [NV0] =
+c−
+c− + κ520c0
+.
+(1)
+Thus, the concentration ratio between NV charge
+states [NV−]/[NV0] can be described with
+[NV−]
+[NV0] = c−
+c0
+1
+κ520
+.
+(2)
+Here, c− and c0 describe the coefficients of the ba-
+sis functions of NV− and NV0 used to assemble an
+area-normalized composed spectrum at arbitrary laser
+power with the condition c− + c0 = 1.
+The correc-
+tion factor κ520 translates this fluorescence ratio c−/c0
+to the ratio of NV concentrations [NV−]/[NV0], taking
+into account the different lifetimes and the absorption
+cross sections of the two NV charge states [43]. Note
+the different subscript in our work for the excitation
+(a)
+(b)
+FIG. 4.
+(a) NV fractions as a function of the laser power we
+derived from spectral analysis. (b) NV ratios as a function of
+the fluorescence count ratio in the two SPCMs applied as de-
+tectors. Using the fit curve, we map the fluorescence count
+ratio to an NV ratio during relaxometry measurements. For
+fitting the [NV−]/[NV0] concentration ratio with f (x) = axn,
+we obtain a = 0.0135 ± 0.0001 and n = 1.334 ± 0.004. The re-
+ciprocal ratio [NV0]/[NV−] was not fit separately, displayed is
+the function g(x) = a−1x−n.
+wavelength of 520 nm compared to κ532 in [43].
+Us-
+ing ten spectra recorded at laser powers below the sat-
+uration intensity and the deviations from the linearity
+of the charge states’ fluorescence intensity with the ap-
+plied laser power, we find κ520 = 2.03 ± 0.07.
+The
+error denotes the statistical error from a weighted fit
+we performed on our measurement data.
+For a de-
+tailed description of the determination of κ520, see Ap-
+pendix B. This value is within the reported value for
+κ532 = 2.5 ± 0.5 for an excitation wavelength of 532 nm
+[43].
+We use our value for κ520 to calculate the frac-
+tions of [NV−] and [NV0] and the concentration ratio
+[NV−]/[NV0] as a function of the laser power.
+As shown in Fig. 4 (a), the fraction of [NV−] is high for
+low laser powers and decreases with higher laser pow-
+ers. At the lowest laser power of 0.1 %, about 73 % of
+the total NV concentration is [NV−], while at the high-
+est laser power, only about 21 % [NV−] remain. Already
+at laser powers of 2 % (∼ 51 kW cm−2), which is below
+saturation intensity (≈ 100 kW cm−2) [44], [NV0] out-
+weighs [NV−].
+Therefore, a significant influence due
+to charge conversion is to be considered in relaxometry
+measurements.
+Together with the recorded fluorescence-count-rate
+ratio of both SPCMs for each laser power, we assign each
+count-rate ratio ρSPCM2/ρSPCM1 a ratio [NV−]/[NV0].
+The results are shown in Fig. 4 (b).
+With an increas-
+ing ratio of ρSPCM2/ρSPCM1, the ratio [NV−]/[NV0] in-
+creases. We fit a power law (inverse-variance-weighted
+fit) to the ratio [NV−]/[NV0] to be able to trace the NV-
+concentration ratio over a broad range of count-rate ra-
+tios during the spin-relaxation measurements. Thereby
+we are able to quantitatively trace the contribution of
+NV0 during the spin-relaxation dynamics of the NV−
+centers in the following.
+IV.
+SPIN-RELAXATION MEASUREMENTS
+A.
+Setup and measurement sequences
+To separately detect the fluorescence of NV− and NV0
+throughout our measurement, different filters are used
+in the optical beam path as depicted in Fig. 2 (b). Af-
+ter passing a 50:50 non-polarizing beamsplitter, the sam-
+ple’s transmitted fluorescence light is guided through a
+665 nm-longpass filter, and mainly NV− fluorescence is
+detected. For the luminescence reflected by the beam-
+splitter, we use a tilted 625 nm-shortpass filter to collect
+NV0 fluorescence below 600 nm. Neutral-density filters
+are added in front of the beamsplitter and within its
+transmitted beam path to keep the SPCMs below sat-
+uration.
+For determining the longitudinal spin relaxation time
+T1, we conduct and compare two different and fre-
+quently used pulsed-measurement schemes, which we
+
+5
+term P1 and P2 in the following. These two pulse se-
+quences are depicted in Fig. 5.
+In the pulsed sequence P1, we choose an initialization
+pulse of 200 µs duration to spin polarize the NV-center
+ensemble to their spin states mS = 0. We apply a 5 µs
+normalization pulse 1 µs after the initialization pulse to
+probe the fluorescence intensity before a variable relax-
+ation time τ in gate Cπ
+1 . To assure a depopulation of the
+NV− centers’ singlet states, we choose the time between
+the two pulses to be longer than the singlet lifetimes of
+τmeta ≈ 150 ns at room temperature [45]. Approximately
+1.5 µs into τ, a resonant π pulse is applied. After τ, a
+readout pulse of duration 5 µs probes the fluorescence of
+both NV centers’ charge states in gate Rπ. Subsequently,
+the sequence is repeated with the π pulse omitted, ob-
+taining fluorescence intensities in gates C1 and R. The
+spin polarization as a function of τ for NV− is obtained
+by subtracting the fluorescence counts in Rπ from the
+counts in gate R. Details on measurement sequence P1
+can be found in [38, 40]. Sequence P1 provides a tech-
+nique for determination of the NV− centers’ T1 time ro-
+bust against background fluorescence [38, 40] and is be-
+lieved to be unaffected by charge-state conversion [29].
+Analysis of the second half of P1 represents an all-
+optical T1 measurement scheme as often applied in bi-
+ology [7, 24, 27]. Further, using only the second half
+of this sequence, we are able to obtain the fluorescence
+evolution as a function of τ for NV− and NV0, includ-
+ing effects caused by the charge-state conversion. Only
+taking into account the signal without the π pulse ap-
+laser
+laser
+MW
+MW
+detection
+detection
+(a) Sequence P1
+(b) Sequence P2
+FIG. 5.
+Longitudinal spin relaxation time (T1) measurement
+schemes applied in this work. The beginning of the second
+half of each sequence is indicated by a dashed line. (a) Se-
+quence P1.
+The NV ensemble is spin polarized by a laser
+pulse. Next, the fluorescence is detected by a control pulse
+(orange). Within the variable relaxation time τ, a resonant π
+pulse is applied (light-green). The spin state is read out by a
+third laser pulse (purple). The sequence is repeated with the
+π pulse omitted after a pause time tp. (b) Sequence P2. As
+opposed to P1, the readout pulse has the same length as the
+spin-polarization pulse. The control gates C1 within the ini-
+tialization pulse and C2 within the readout pulse will be com-
+pared in this work.
+plied, we obtain the fluorescence evolution by dividing
+the fluorescence counts in gate R by the counts in gate
+C1. Charge conversion during the relaxometry measure-
+ment has an effect on the NV− fluorescence as well as
+on the NV0 fluorescence during the relaxation time τ.
+Therefore, by only evaluating P1’s second half, we gain
+information about the charge conversion taking place
+alongside the spin relaxation. However, to obtain the
+NV− centers’ T1 time, the full sequence P1 is evaluated.
+As opposed to P1, P2 uses a normalization probe after
+the readout of the NV centers’ fluorescence [19, 21, 46].
+We choose the laser readout pulse to have the same du-
+ration as the initialization pulse (200 µs) and carry out
+the readout gates Rπ and R in the first 5 µs and the nor-
+malization probes Cπ
+2 and C2 in the last 5 µs of the read-
+out pulse. Scheme P2 assumes the NV centers to have
+the same fluorescence intensity at the end of the second
+pulse as at the end of the first pulse. To test this notion,
+we apply second normalization gates, Cπ
+1 and C1, within
+the last 5 µs of the initialization pulse and compare the
+results for both normalized data.
+Between readout and the upcoming initialization
+pulse, we insert a pause time tp between the sequences
+of 1 ms, which is in the order of T1, to minimize build-up
+effects from spin polarization during the cycle for both
+sequences. Each cycle is repeated 50 000 times, and the
+whole measurement is swept multiple times. The se-
+quences are repeated for different laser powers, ranging
+from 0.1 % to 11 % of the maximum laser power.
+B.
+Results and discussion
+In this section, we present and compare the experi-
+mental results for the spin-relaxation measurements for
+both sequences, P1 and P2.
+Using our experimental
+setup as described in Section IV A, we observe laser-
+power-dependent dynamics in the NV− and NV0 flu-
+orescence throughout our measurement. Fig. 6 depicts
+an example for the fluorescence as a function of τ for
+the NV0 fluorescence recorded at a laser power of 11 %
+with sequence P1. These results show the normalized
+fluorescence as a function of τ obtained from the second
+part of the measurement sequence without a microwave
+π pulse, dividing the fluorescence counts in gate R by
+the counts in gate C1. The normalized fluorescence as
+a function of τ decays exponentially. Different from the
+dynamics of the NV− center, we observe similar behav-
+ior for the NV0 fluorescence at all laser powers. We fit a
+biexponential function of type
+f 0(τ) = A e−τ/TR,1 + B e−τ/TR,2 + d
+(3)
+to our measurement data and obtain two recharge times
+in the order of TR,1 = 100 µs and TR,2 = 2.0 ms for
+all laser powers. We assign these time constants TR to
+an electron-recapturing process of NV0 during the dark
+
+6
+time τ, after an ionization from NV− to NV0 has pre-
+viously taken place in the initializing laser pulse. Re-
+markably, this process occurs even at the lowest laser
+power.
+Presumably, the presence of two components
+of TR is due to the different environments of NV cen-
+ters concerning charge transfer sites. Vacancies or elec-
+tronegative surface groups on the diamond surface are
+known to promote a charge conversion of NV− to NV0
+[47, 48]. We assume that the NV environment similarly
+affects the recharging process in the dark. Therefore, we
+attribute one component of TR to NV centers closer to
+the nanodiamond surface and the other to NV centers
+more proximate to the center of the crystal. We empha-
+size that both TR,1 and TR,2 we report match previously
+reported values for TR of 100 µs [28] and (3 ± 1) ms [20]
+and underline that they simultaneously appear as two
+components in our sample. We find that neither TR,1
+nor TR,2 changes as a function of the laser power. The
+coefficients of the exponential functions A and B do not
+change significantly from 1 % to 11 % laser power. How-
+ever, for the lowest laser power of 0.1 % A and B are
+smaller. We attribute this to little NV0 fluorescence ob-
+served at this low laser power due to less charge conver-
+sion, resulting in a lower signal-to-noise ratio (SNR) for
+the NV0 fluorescence.
+Further, we present the results for the normalized
+NV− fluorescence as a function of τ in Fig. 7 (a) for
+ascending laser powers. We conducted the experiment
+with sequence P1, and this data refers to the results with
+the π pulse omitted.
+The laser-power-dependent dy-
+namics of NV− and NV0 result in a drastic change of
+shape of the normalized fluorescence as a function of τ.
+While we observe an exponential decay in the lowest
+laser power, we find an inverted exponential profile of
+the NV− fluorescence at 11 % laser power. In-between
+laser powers show both an exponential decay and an in-
+0
+2
+4
+6
+8
+10
+τ (ms)
+0.7
+0.8
+0.9
+normalized fluorescence
+R/C1
+biexponential fit
+FIG. 6.
+NV0 fluorescence as a function of τ as recorded
+with sequence P1 (second half) by division of the fluorescence
+counts in R by the counts in C1 for 11 % laser power. With
+a biexponential fit function, we find TR,1 = (109 ± 7) µs and
+TR,2 = (2.1 ± 0.1) ms.
+crease, present in the fluorescence. This phenomenon of
+inverted exponential components in the recorded nor-
+malized fluorescence during a T1 measurement has been
+reported by [29] and attributed to a recharging process
+of NV0 to NV− during τ. However, a complete flip of
+the fluorescence alone by a laser power increase has not
+been reported so far. Remarkably, this behavior indi-
+cates that NV0 to NV− charge dynamics outweigh the
+NV− ensemble’s spin relaxation at high laser powers in
+our sample.
+To better understand the NV− power-dependent be-
+havior, we use the results from the spectral analysis to
+map the ratios of [NV−]/[NV0] to our relaxometry mea-
+surement data of sequence P1. The ratio [NV−]/[NV0]
+as a function of τ for all laser powers is summarized
+in Fig. 8 (a) and displayed in more detail in Fig. C1 (a).
+For all laser powers, even for the lowest, which lies well
+below saturation intensity, we observe an increase of
+[NV−]/[NV0] as a function of τ in the readout pulse R.
+We conclude that during τ a re-conversion from NV0 to
+NV− takes place in the dark, after ionization of NV−
+had occurred in the initialization pulse. While for the
+lowest power, the ratio [NV−]/[NV0] increases from 4.0
+to 7.6 over the variation of τ, [NV0] outweighs [NV−] at
+11 % laser power throughout the entire relaxation mea-
+surement, see Fig. C1 (a). As shown in Fig. 8 (a), the ra-
+tios increase by a factor of ∼ 2 from shortest to longest
+τ for all laser powers.
+The ratios [NV−]/[NV0] we find in control pulse C1
+as a function of τ also show a power-dependent be-
+havior. While the ratio increases as a function of τ in
+the lowest power, it is constant in the control pulse for
+the highest power.
+These power-dependent recharge
+processes in the control pulse we observe appear most
+likely due to build-up effects during the measurement
+cycle, as we explain in the following. At low powers,
+the initializing laser pulse spin polarizes the NV− cen-
+ters but does not ionize to a steady state of NV− and
+NV0. For short τ, the re-conversion in the dark of NV0
+to NV− has not completed, and the following laser pulse
+continues to ionize the NV− centers. However, at the
+highest power, each initialization pulse efficiently ion-
+izes to a steady state of the two NV charge states, reach-
+ing [NV−]/[NV0] ≈ 0.44. These results clearly show
+that the normalization in the sequence we perform is
+mandatory to only detect the change in the relative flu-
+orescence during the relaxation time τ and minimize in-
+fluences due to charges passed through cycles.
+At the lowest laser power of 0.1 %, we observe the
+highest ratio of [NV−]/[NV0] and therefore expect the
+most negligible influences of charge conversion on the
+NV−s’ spin relaxation. Thus, we fit a monoexponential
+function to the relative fluorescence as a function of τ
+and obtain T1 = (1.42 ± 0.06) ms for the NV− ensemble
+in the nanodiamond. To further underline the necessity
+of a normalization of the fluorescence intensity, we fit a
+
+7
+(a)
+(b)
+(c)
+FIG. 7. NV− fluorescence as a function of τ, recorded with sequence P1 for different laser powers. Insets show the characteristics
+of the sequences applied. (a) NV− fluorescence as obtained from the second half of P1, dividing the fluorescence counts in R by
+the counts in C1. We observe a transition from an exponential decay of the fluorescence to an inverted exponential profile with
+rising laser powers. For 0.1 % laser power, we perform a monoexponential fit and obtain T1 = (1.42 ± 0.06) ms. For laser powers
+from 1 % to 11 %, we fit a sum of three exponential functions as explained in the text. (b) Spin polarization of the NV− ensemble
+as obtained from the full sequence P1, subtracting the fluorescence counts in Rπ from the counts in R. Unlike (a), we observe an
+exponential decay at all laser powers in this measurement data. However, with increasing laser power, we observe a decrease
+in the amplitude of the exponential function. Fitting a monoexponential function to the data at 0.1 % laser power, we obtain
+T1 = (1.5 ± 0.1) ms, consistent with the T1 time we find in (a) at the same laser power. (c) NV− fluorescence as obtained from
+the second half of P2, dividing the fluorescence counts in R by the counts in C1 or C2. Fitting monoexponential functions to the
+fluorescence normalized by the counts in C1 or C2 yields T1 = (1.54 ± 0.06) ms or T1 = (1.50 ± 0.07) ms, respectively, consistent
+with the results named above. The NV− fluorescence qualitatively behaves as in sequence P1, see (a), and is fitted accordingly.
+On the contrary, the shape of the fluorescence depends on the position of the normalization gate, C1 or C2, especially visible at
+4 % laser power.
+monoexponential function to the non-normalized bare
+NV− fluorescence detected in R at 0.1 % laser power.
+We obtain a T1 time of (0.94 ± 0.05) ms, see Fig. C2 (a),
+which is drastically lower than the T1 time retrieved
+with normalization by the fluorescence counts in C1.
+For higher laser powers, we fit the normalized data
+with a function of type
+f −(τ) = −A e−τ/TR,1 − B e−τ/TR,2 + C e−τ/T1 + d
+(4)
+and restrict the time constants to TR,1 = 100 µs, TR,2 =
+2.0 ms and T1 = 1.4 ms. With this, we assume that the
+decay of [NV0] causes an increase of [NV−] and, there-
+fore, their fluorescence. Thus, the NV− fluorescence is
+best described by a sum of an exponential decay due
+to the loss of spin polarization and a biexponential in-
+verted component due to the recharging process of NV0
+to NV− in the dark. As shown in Fig. 7 (a), our fit func-
+tion Eq. (4) describes the measurement data from 1 % to
+11 % laser power very well. We emphasize that the mea-
+surement data for 1 % laser power does not visibly ap-
+pear to show this triexponential behavior. Fitting a mo-
+noexponential function to the NV− fluorescence at 1 %
+laser power, however, results in T1 = (1.28 ± 0.06) ms,
+see Fig. C2 (b), which deviates significantly from the
+value obtained at lower laser power.
+Measurement
+sequence
+P1
+is
+a
+well-established
+method to accurately measure the T1 time of the NV−
+
+8
+centers excited by a resonant π pulse [38].
+Since the
+π pulse only acts on the negatively-charged NV cen-
+ters, it is said to be independent of charge conversion
+processes alongside the spin polarization [29]. We com-
+pare the results we obtain in the complete measurement
+sequence P1, subtracting fluorescence intensities in Rπ
+from the counts in R, to the result we gave for the T1 time
+above without the π pulse taken into account. Remark-
+ably, although in Fig. 7 (a) we observe vivid dynamics
+ranging from exponential decay to an inverted exponen-
+tial profile in the NV− fluorescence as a function of τ,
+the complete sequence P1 yields a monoexponential de-
+crease for all laser powers, see Fig. 7 (b). For the lowest
+laser power, we obtain T1 = (1.5 ± 0.1) ms for sequence
+P1 comparing the fluorescence intensity with and with-
+out the resonant π pulse. This value matches the pre-
+viously determined T1 time when only considering the
+normalized signal without the π pulse for the lowest
+laser power. It does not match the T1 time obtained from
+the monoexponential fit we performed on the measure-
+ment data for 1 % laser power, stressing the effects of
+(a)
+(b)
+FIG. 8. Changes of the NV-charge-state ratio during the relax-
+ometry measurement from lowest to highest τ. (a) Sequence
+P1. The change of the ratio [NV−]/[NV0] is constant as a func-
+tion of the laser power in the readout pulse R, while it decays
+in the control pulse C1. (b) Sequence P2. The change of the
+ratio [NV−]/[NV0] in readout and control pulses show qual-
+itatively the same behavior as in sequence P1. However, the
+changes in the NV-charge-state ratio in C1 and C2 as a func-
+tion of the laser power differ.
+NV charge conversion within this measurement and the
+necessity for consideration of the two components TR,1
+and TR,2 in a triexponential fit function.
+However, the measurement sequence P1 is not en-
+tirely unaffected by the charge conversion process. Al-
+though the resonant π pulse does not directly act on
+the NV0 center (we observe no difference in the signals
+with and without the π pulse), the fluorescence contrast
+in the measurement decreases because of NV0 to NV−
+conversion. This lower contrast becomes noticeable in
+Fig. 7 (b) due to the decaying amplitude of the mono-
+exponential function with increasing laser power. The
+effect of NV− spin depolarization due to charge conver-
+sion has been previously investigated in [28, 35]. As a
+measure for the reliability of our measurement result,
+we use the area under the curves showing spin polar-
+ization as a function of τ for each laser power as a fluo-
+rescence contrast in the respective measurement. We di-
+vide this value by the Root mean squared error (RMSE)
+value we obtain from the fit result to account for fluc-
+tuations in our measurement data and define this value
+contrast/RMSE as the SNR. In Fig. 9, we show the SNR
+as a function of the laser power. In addition, we dis-
+play the value for T1 we obtain in the same graph. With
+the SNR decreasing, we observe a decrease in T1, accom-
+panied by a larger standard deviation with higher laser
+power. We conclude that the T1 time we measured at
+the lowest laser power is the most reliable one due to the
+highest SNR. In addition, we note that T1 seems to decay
+as a function of the laser power, although T1 should be
+independent of the excitation power. We attribute this
+decay of T1 to the lower SNR in the measurements at
+higher laser power due to increased charge conversion.
+From the results of sequence P1, we conclude that the
+normalization in the T1 measurement is essential to re-
+0
+5
+10
+laser power (%)
+5
+10
+15
+20
+25
+30
+SNR (arb. units)
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+T1 (ms)
+SNR
+T1
+FIG. 9. SNR and T1 time as a function of the laser power, ob-
+tained from measurements performed in sequence P1. With
+higher laser power, the SNR decreases, and so does the T1
+time.
+The standard error of T1 that we obtain from mono-
+exponential fits increases with higher laser power. The data
+is extracted from equal numbers of repetitions of relaxometry
+measurements for each laser power.
+
+9
+flect the charge-state processes alongside the NV− en-
+semble’s spin relaxation. Besides P1, sequence P2 is used
+in literature to determine a single NV center’s [46] or
+an ensemble’s [19] T1 time. While the π pulse is often
+omitted in these sequences, we chose to implement it
+for low laser powers for better comparison to the results
+obtained in P1. For laser powers starting from 3 %, we
+repeated the sequence without the π pulse and calcu-
+lated the mean values of the control and readout data
+taken. The results for sequence P2 with a π pulse in-
+cluded for 0.1 % laser power are shown in Fig. C2 (c).
+Using the data for 0.1 % laser power and subtracting Rπ
+from R, we obtain T1 = (1.45 ± 0.09) ms, which is the
+same result as in sequence P1. Since both sequences are
+used in the literature to measure an NV− ensemble’s T1
+time, we expect them to produce the same result for our
+NV ensemble when neglecting additional effects due
+to charge conversion. At this low laser power, charge
+conversion is inferior to spin relaxation. Therefore, the
+T1 times we obtain from both sequences do not differ.
+However, with higher laser power, charge conversion
+prevails, and both NV charge states’ fluorescence sig-
+nals are greatly affected by recharge in the dark.
+In order to evaluate the result of sequence P2 without
+the π pulse applied, we normalize the fluorescence in-
+tensities. To this end, we divide the counts in R obtained
+by the counts measured during the two control gates C1
+or C2, yielding two normalized fluorescence signals for
+each NV charge state. This way, we obtain two normal-
+ized fluorescence signals as a function of τ. If no charge
+conversion effects were present in this measurement,
+both signals for the normalized fluorescence should be
+equal. However, as pointed out, charge conversion is
+prominent in our sample, not only for high laser pow-
+ers. We show the NV− fluorescence as a function of τ
+we obtain from sequence P2 in Fig. 7 (c). Qualitatively
+similar to sequence P1, we see a smooth transition from
+an exponential decay at low laser powers to an inverted
+exponential profile at high laser powers. Similarly as in
+P1, we derive T1 = (1.54 ± 0.06) ms for normalization
+with C1 and T1 = (1.50 ± 0.07) ms for normalization
+with C2 for the lowest laser power. We emphasize that
+all T1 times we derive from the normalized NV− fluo-
+rescence in both sequences are equal within their stan-
+dard errors. In addition, the values for TR,1 and TR,2 we
+obtain from the NV0 fluorescence with sequence P2 are
+the same as in sequence P1. We fit the NV− fluorescence
+for laser powers from 1 % to 11 % in the same manner as
+for P1 using Eq. (4) and the aforementioned values for
+T1, TR,1 and TR,2. This triexponential fit function models
+our data well, regardless of the normalization we use.
+However, the amplitudes of the respective exponen-
+tial functions differ depending on the normalization, C1
+or C2, employed. Thus, the shapes of the fluorescence as
+a function of τ differ with the gates used for normaliza-
+tion, which is especially visible at 4 % laser power. To
+understand the difference in the measurement results
+that the positions of the normalization gate cause, we
+take the ratios [NV−]/[NV0] into account. In Fig. C1 (b)
+and (c), [NV−]/[NV0] as a function of τ for sequence P2
+is displayed and summarized as a change from shortest
+to longest τ in Fig. 8 (b) for each laser power. The ra-
+tios as a function of τ behave similarly to as observed
+with sequence P1 discussed above. We note that the ra-
+tios we obtain in our measurement for the two control
+gates C1 and C2 are different. For τ ≲ 1 ms the ratio
+[NV−]/[NV0] is smaller for C2 than for C1, while for val-
+ues τ ≳ 1 ms the opposite is the case, see Fig. C1 (c). For
+the same reasons discussed in sequence P1, this effect is
+prominent in laser powers up to 4 %. In contrast, for the
+highest laser power, the ratios in the control gates are
+approximately constant with τ and do not differ signifi-
+cantly. As pointed out in the discussion of P1, the results
+indicate that the first laser pulse does not ionize into a
+steady state of [NV−]/[NV0], and the second laser pulse
+continues to ionize NV− into NV0. Therefore, especially
+for small values of τ, the ratio [NV−]/[NV0] is smaller in
+C2 than in C1. For larger values of τ, recharge dynamics
+of NV0 to NV− in the dark add to the different ratios of
+[NV−]/[NV0] for both control gates. We do not exclude
+additional effects due to continued spin polarization of
+NV− in the second laser pulse, especially for low laser
+powers.
+It is for these reasons that in Fig. 8 (b) the changes of
+[NV−]/[NV0] as a function of the laser power are higher
+for C2 than for C1 for low powers and converge to the
+same value for higher laser powers. We therefore at-
+tribute the difference in the normalized fluorescences in
+Fig. 7 (c) when normalizing to C1 or C2 to the differences
+in [NV−]/[NV0] for C1 and C2, respectively.
+Both the results from measurement sequences P1 and
+P2 and the simultaneous mapping of [NV−]/[NV0] in-
+dicate that a charge conversion from NV− to NV0 dur-
+ing the spin-polarization pulse of a spin-relaxation mea-
+surement is inevitable. We emphasize that a normal-
+ization gate is mandatory to correctly display the fluo-
+rescence dynamics of NV0 and NV− as a function of τ.
+Comparison of the two control gates C1 and C2 shows
+that the normalized fluorescence signal depends on the
+positions of the gate used for normalization because of
+charge conversion processes that take place alongside
+the NV− ensemble’s spin relaxation.
+V.
+CONCLUSIONS
+This work examines laser-power-dependent dynam-
+ics of NV charge conversion within spin-relaxation mea-
+surements of the negatively-charged NV centers in a sin-
+gle nanodiamond. We present a new method of trac-
+ing the ratio of [NV−] to [NV0] during our sequence, in
+which we extract the relative concentrations of NV− to
+NV0 from their fluorescence spectra and perform a map-
+ping to fluorescence count ratios in two separate detec-
+
+10
+tors. From the analysis of low-excitation intensity spec-
+tra, we find κ520 = 2.03 ± 0.07. This correction factor
+κ520 allows us to translate the fluorescence ratio of NV−
+to NV0 to a concentration ratio, taking into account dif-
+ferent lifetimes and absorption cross sections for the two
+charge states. Combining our results, we conclude that
+ionization of NV− to NV0 during the optical initializa-
+tion and readout is inevitable and occurs even at low
+laser powers. A recharge process in the dark of NV0 to
+NV− significantly affects the NV− ensemble’s fluores-
+cence during the spin-relaxation measurement. We find
+the recharging in the dark to be biexponential with com-
+ponents TR,1 = 100 µs and TR,2 = 2.0 ms. At high laser
+powers, the effect of charge conversion outweighs spin
+relaxation, making it impossible to accurately measure a
+T1 time, even with a scheme involving a π pulse for two
+reasons. Firstly, recharging effects of NV0 to NV− in the
+dark dominate the NV− fluorescence signal. Secondly,
+the measurement of T1 is crucially impeded by a dimin-
+ished fluorescence contrast due to charge conversion. To
+determine the NV− centers’ T1 time at low laser powers,
+we find it necessary to conduct a pulsed sequence with
+a normalization gate included. We prove the normal-
+ization mandatory to accurately reflect the charge-state
+dynamics as a function of τ and mitigate additional ef-
+fects due to charge-state accumulation during the mea-
+surement cycle. Additionally, comparing two pulsed se-
+quences often used in the literature, we find that the po-
+sition of the normalization gates plays an essential role
+due to charge conversion during the measurement. We
+emphasize that including a normalization gate directly
+after the spin polarization before the relaxation time τ is
+a simple method to accurately display the fluorescence
+dynamics during the relaxation time. This way, com-
+paring the fluorescence counts in the readout gate to the
+counts in the control gate reliably reflects the spin relax-
+ation and the charge dynamics in the relaxometry mea-
+surement.
+Overall, we emphasize that the results presented in
+this work impact relaxometry schemes widely used in
+biology, chemistry, and physics.
+To further extend
+this work, the effects of different duration of the spin-
+polarization pulse and the readout pulse can be exam-
+ined and give insight into the steady-state dynamics of
+the NV centers. Further, the excitation of NV− can be
+conducted at longer wavelengths, changing the charge-
+state dynamics [49] and impacting the spin relaxation
+results. The influence of different NV and nitrogen con-
+centrations in diamonds of different sizes on the charge
+dynamics can be considered to unravel the mechanisms
+of charge conversion in the dark.
+ACKNOWLEDGMENTS
+We acknowledge support by the nano-structuring
+center NSC. This project was funded by the Deutsche
+Forschungsgemeinschaft
+(DFG,
+German
+Research
+Foundation)—Project-ID No.
+454931666.
+Further,
+I. C. B. thanks the Studienstiftung des deutschen Volkes
+for financial support.
+We thank Oliver Opaluch and
+Elke Neu-Ruffing for providing the microwave antenna
+in our experimental setup. Furthermore, we thank Sian
+Barbosa, Stefan Dix, and Dennis L¨onard for fruitful
+discussions and experimental support.
+Appendix A: Methods
+To understand the NV centers’ fluorescence evolu-
+tion as a function of τ in terms of charge conversion,
+we map the fluorescence count ratio detected in both
+SPCMs to a ratio of NV− and NV0 throughout the
+spin-relaxation measurement.
+For this, we combine
+the results of recorded NV spectra and spin-relaxation
+measurements. We choose a single nanodiamond and
+record fluorescence spectra at different laser powers us-
+ing the setup in the configuration shown in Fig. 2 (a).
+Both charge states, NV− and NV0, contribute to the
+recorded spectra between 500 nm and 750 nm because
+of the charge states’ overlapping phononic sidebands.
+For further analysis, we decompose the obtained spec-
+tra into NV− and NV0 basis functions as described by
+[43] using the spectra we recorded at the highest and
+lowest laser power. Employing our extracted basis func-
+tions, we obtain the fluorescence ratio of both NV charge
+states for all other laser powers with the help of MAT-
+LAB’s function nlinfit. We access the NV-charge-state
+ratio from the fluorescence ratio after determining the
+necessary correction factor κ520 [43]. A detailed descrip-
+tion of κ520’s derivation is given in Appendix B.
+Next, we assign the concentration ratio to a count ra-
+tio in our SPCM detectors. We alter the setup accord-
+ing to Fig. 2 (b). We illuminate the nanodiamond for
+1 s with a given laser power and record the fluorescence
+counts in both SPCMs. Using the data for each laser
+power, we map the NV concentration ratio to a count
+ratio in both SPCMs. At this point, we stress that we
+do not obtain the NV concentration ratio through fluo-
+rescence count ratios in SPCMs, but by analysis of the
+NV centers’ fluorescence spectra. This method provides
+the advantage that any influence of NV0 fluorescence
+in SPCM2 (> 665 nm) can be neglected because only a
+count ratio is considered in our analysis and a mapping
+to previously-assigned concentration ratios performed.
+Appendix B: Determination of κ520
+This section describes how we retrieve the correction
+factor κ520 from our measurement data. We derive κ520
+similarly to as described in [43].
+We recorded fluorescence spectra of the single dia-
+mond crystal with laser powers well below saturation
+intensity with our setup shown in Fig. 2 (a). To achieve
+
+11
+these laser powers, an additional ND filter was used in
+our laser-beam path. We correct the spectra for different
+exposure times we set in our camera due to the differ-
+ent NV luminescence intensities at different laser pow-
+ers. We show the spectra we obtain for different laser
+powers in Fig. B1. As can be seen, the overall fluores-
+cence counts increase with increasing laser power. We
+perform the spectra analysis as described in the main
+text to derive the coefficients c− and c0.
+Below saturation intensity, the luminescence of NV−
+and NV0 should scale linearly with the laser power [43].
+However, due to charge conversion, we observe devia-
+tions from this linearity. The coefficients c− and c0 we
+obtain directly represent the amount of NV− and NV0
+fluorescence in the given spectra. We scale these factors
+with the total integration value of the spectra in Fig. B1
+for each laser power and obtain measured fluorescence
+counts for both NV charge states at each laser power.
+Further, we take the fluorescence counts for NV− and
+NV0 of the lowest-intensity spectrum recorded and scale
+it with the laser power. This way, we obtain calculated
+fluorescence counts for each NV charge state that strictly
+increase linearly with the laser power.
+These fluorescence counts for NV− and NV0, mea-
+sured and calculated, are shown in Fig. B2 as a func-
+tion of the laser power. We note that the measured NV−
+fluorescence is lower than the calculated linear integra-
+tion value, while the NV0 fluorescence is higher. We per-
+form a weighted linear fit (inverse-variance weighting)
+for each data set and compare the slopes to one another
+for each NV charge state. We divide the two slope ratios
+by each other and obtain κ520 = 2.03 ± 0.07, while we
+derive the error from the statistical error of the fits we
+performed.
+550
+600
+650
+700
+750
+wavelength (nm)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+fluorescence counts
+×103
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+laser power (%)
+FIG. B1.
+NV fluorescence spectra recorded at laser pow-
+ers from 0.1 % to 1 %.
+The laser power is kept well below
+saturation intensity with a maximum laser power of ∼ 1 %
+(∼ 25 kW cm−2). The spectra were corrected for different cam-
+era exposure times used. An overall increase in the fluores-
+cence counts is observed with increasing laser power.
+(a)
+(b)
+FIG. B2.
+Determination of κ520. (a) Fluorescence counts for
+NV− as a function of the laser power. The red curve displays
+the fluorescence counts obtained from scaling the counts at
+the lowest laser power with the laser power. The blue curve
+depicts the fluorescence counts for NV− as a function of the
+laser power as we obtain it from the spectra. (b) Calculated
+and measured fluorescence counts for NV0 as a function of the
+laser power. Error bars are derived from the statistical errors
+for c− and c0 and are smaller than the data points shown in
+this graph.
+Appendix C: Supporting relaxometry data
+In Fig. C1, the ratios [NV−]/[NV0] are shown as a
+function of τ, recorded with sequences P1 and P2. We
+obtained the data as described in the main text. Fig. C2
+shows further supporting relaxometry data recorded
+with sequences P1 and P2.
+
+12
+(a)
+(b)
+(c)
+FIG. C1.
+NV-charge-state ratios as a function of τ, obtained from relaxometry measurements. The ratios in R, C1, and C2 are
+derived from the count-rate ratios of both SPCMs in the respective gates. (a) Sequence P1. The ratio [NV−]/[NV0] increases in the
+readout gate R for all laser powers as a function of τ. For lower laser powers, the ratio [NV−]/[NV0] increases in the control gate
+C1, while for the highest laser power, it is constant. (b) Sequence P2. The ratio [NV−]/[NV0] as a function of τ behaves similarly
+as in sequence P1. However, the NV-charge-state ratios as a function of τ are different in C1 and C2, indicating charge-conversion
+processes during the measurement. (c) Sequence P2. For better visibility, the ratios [NV−]/[NV0] for C1 and C2 as a function of
+τ are displayed from panel (b). At laser powers up to 4 %, the ratio [NV−]/[NV0] is smaller for C2 than for C1 for τ ≲ 1 ms. For
+values τ ≳ 1 ms, the opposite is the case. For visualization, τ = 1 ms is marked with a dashed line. At 11 % laser power, the ratios
+[NV−]/[NV0] are equal in C1 and C2.
+
+13
+(a)
+(b)
+(c)
+FIG. C2. Supporting results from relaxometry measurements.
+(a) NV− fluorescence obtained from sequence P1 at 0.1 % laser
+power in gate R. The data shown was not normalized by di-
+vision by the fluorescence counts in gate C1. Fitting a mono-
+exponential function to the data yields T1 = (0.94 ± 0.05) ms,
+which deviates drastically from the T1 time obtained in the full
+sequence P1 and in the case of normalization with C1. (b) NV−
+fluorescence obtained from sequence P1 at 1 % laser power
+by division of the fluorescence counts in R by the counts in
+C1. Fitting a monoexponential function instead of the triexpo-
+nential function yields T1 = (1.28 ± 0.06) ms, which does not
+match the value determined for T1 in the full sequence P1. (c)
+NV− spin polarization as obtained from the full sequence P2
+at 0.1 % laser power by subtracting the counts in Rπ from the
+counts in R. Fitting a monoexponential function to the data
+yields T1 = (1.45 ± 0.09) ms, which matches the previously
+determined values for T1 in sequence P1.
+
+14
+[1] V. M. Acosta, E. Bauch, M. P. Ledbetter, C. Santori, K.-
+M. C. Fu, P. E. Barclay, R. G. Beausoleil, H. Linget, J. F.
+Roch, F. Treussart, S. Chemerisov, W. Gawlik, and D. Bud-
+ker, Diamonds with a high density of nitrogen-vacancy
+centers for magnetometry applications, Physical Review
+B 80, 115202 (2009).
+[2] G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-
+Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hem-
+mer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Brats-
+chitsch, F. Jelezko, and J. Wrachtrup, Nanoscale imaging
+magnetometry with diamond spins under ambient condi-
+tions, Nature 455, 648 (2008).
+[3] J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M.
+Taylor, P. Cappellaro, L. Jiang, M. V. G. Dutt, E. Togan,
+A. S. Zibrov, A. Yacoby, R. L. Walsworth, and M. D. Lukin,
+Nanoscale magnetic sensing with an individual electronic
+spin in diamond, Nature 455, 644 (2008).
+[4] C. L. Degen, Scanning magnetic field microscope with a
+diamond single-spin sensor, Applied Physics Letters 92,
+243111 (2008).
+[5] J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Bud-
+ker, P. R. Hemmer, A. Yacoby, R. Walsworth, and M. D.
+Lukin, High-sensitivity diamond magnetometer with
+nanoscale resolution, Nature Physics 4, 810 (2008).
+[6] A. Laraoui, J. S. Hodges, and C. A. Meriles, Magnetome-
+try of random ac magnetic fields using a single nitrogen-
+vacancy center, Applied Physics Letters 97, 143104 (2010).
+[7] R. Schirhagl, K. Chang, M. Loretz, and C. L. Degen,
+Nitrogen-Vacancy Centers in Diamond: Nanoscale Sen-
+sors for Physics and Biology, Annual Review of Physical
+Chemistry 65, 83 (2014).
+[8] L. Thiel, Z. Wang, M. A. Tschudin, D. Rohner, I. Guti´errez-
+Lezama, N. Ubrig, M. Gibertini, E. Giannini, A. F. Mor-
+purgo, and P. Maletinsky, Probing magnetism in 2D mate-
+rials at the nanoscale with single-spin microscopy, Science
+(New York, N.Y.) 364, 973 (2019).
+[9] S. Dix, J. Gutsche, E. Waller, G. von Freymann, and
+A. Widera, Fiber-tip endoscope for optical and mi-
+crowave control, The Review of Scientific Instruments 93,
+095104 (2022).
+[10] F. Dolde, H. Fedder, M. W. Doherty, T. N¨obauer, F. Rempp,
+G. Balasubramanian, T. Wolf, F. Reinhard, L. C. L. Hollen-
+berg, F. Jelezko, and J. Wrachtrup, Electric-field sensing
+using single diamond spins, Nature Physics 7, 459 (2011).
+[11] M. Rollo, A. Finco, R. Tanos, F. Fabre, T. Devolder,
+I. Robert-Philip, and V. Jacques, Quantitative study of the
+response of a single NV defect in diamond to magnetic
+noise, Physical Review B 103, 235418 (2021).
+[12] A. Sigaeva, N. Norouzi, and R. Schirhagl, Intracellular Re-
+laxometry, Challenges, and Future Directions, ACS Cen-
+tral Science 8, 1484 (2022).
+[13] J. H. Cole and L. C. L. Hollenberg, Scanning quantum de-
+coherence microscopy, Nanotechnology 20, 495401 (2009).
+[14] L. T. Hall, J. H. Cole, C. D. Hill, and L. C. L. Hollenberg,
+Sensing of Fluctuating Nanoscale Magnetic Fields Using
+Nitrogen-Vacancy Centers in Diamond, Physical Review
+Letters 103, 220802 (2009).
+[15] S. Steinert, F. Ziem, L. T. Hall, A. Zappe, M. Schweikert,
+N. G¨otz, A. Aird, G. Balasubramanian, L. Hollenberg, and
+J. Wrachtrup, Magnetic spin imaging under ambient con-
+ditions with sub-cellular resolution, Nature Communica-
+tions 4, 1607 (2013).
+[16] D. Schmid-Lorch, T. H¨aberle, F. Reinhard, A. Zappe,
+M. Slota, L. Bogani, A. Finkler, and J. Wrachtrup, Re-
+laxometry and Dephasing Imaging of Superparamagnetic
+Magnetite Nanoparticles Using a Single Qubit, Nano Let-
+ters 15, 4942 (2015).
+[17] J.-P. Tetienne,
+T. Hingant,
+L. Rondin,
+A. Cavaill`es,
+L. Mayer, G. Dantelle, T. Gacoin, J. Wrachtrup, J.-F.
+Roch, and V. Jacques, Spin relaxometry of single nitrogen-
+vacancy defects in diamond nanocrystals for magnetic
+noise sensing, Physical Review B 87, 235436 (2013).
+[18] A. O. Sushkov, N. Chisholm, I. Lovchinsky, M. Kubo, P. K.
+Lo, S. D. Bennett, D. Hunger, A. Akimov, R. L. Walsworth,
+H. Park, and M. D. Lukin, All-Optical Sensing of a Single-
+Molecule Electron Spin, Nano Letters 14, 6443 (2014).
+[19] M. Pelliccione, B. A. Myers, L. M. A. Pascal, A. Das,
+and A. C. Bleszynski Jayich, Two-Dimensional Nanoscale
+Imaging of Gadolinium Spins via Scanning Probe Relax-
+ometry with a Single Spin in Diamond, Physical Review
+Applied 2, 054014 (2014).
+[20] F. Gorrini, R. Giri, C. E. Avalos, S. Tambalo, S. Man-
+nucci, L. Basso, N. Bazzanella, C. Dorigoni, M. Cazzanelli,
+P. Marzola, A. Miotello, and A. Bifone, Fast and Sensi-
+tive Detection of Paramagnetic Species Using Coupled
+Charge and Spin Dynamics in Strongly Fluorescent Nan-
+odiamonds, ACS Applied Materials & Interfaces 11, 24412
+(2019).
+[21] J. Barton, M. Gulka, J. Tarabek, Y. Mindarava, Z. Wang,
+J. Schimer, H. Raabova, J. Bednar, M. B. Plenio, F. Jelezko,
+M. Nesladek, and P. Cigler, Nanoscale Dynamic Read-
+out of a Chemical Redox Process Using Radicals Coupled
+with Nitrogen-Vacancy Centers in Nanodiamonds, ACS
+Nano 14, 12938 (2020).
+[22] F. Perona Mart´ınez, A. C. Nusantara, M. Chipaux, S. K.
+Padamati, and R. Schirhagl, Nanodiamond Relaxometry-
+Based Detection of Free-Radical Species When Produced
+in Chemical Reactions in Biologically Relevant Condi-
+tions, ACS Sensors 5, 3862 (2020).
+[23] E. Sch¨afer-Nolte, L. Schlipf, M. Ternes, F. Reinhard,
+K. Kern, and J. Wrachtrup, Tracking Temperature-
+Dependent Relaxation Times of Ferritin Nanomagnets
+with a Wideband Quantum Spectrometer, Physical Re-
+view Letters 113, 217204 (2014).
+[24] L. Nie, A. C. Nusantara, V. G. Damle, R. Sharmin, E. P. P.
+Evans, S. R. Hemelaar, K. J. van der Laan, R. Li, F. P. Per-
+ona Martinez, T. Vedelaar, M. Chipaux, and R. Schirhagl,
+Quantum monitoring of cellular metabolic activities in
+single mitochondria, Science Advances 7, eabf0573 (2021).
+[25] R. Sharmin, T. Hamoh, A. Sigaeva, A. Mzyk, V. G. Damle,
+A. Morita, T. Vedelaar, and R. Schirhagl, Fluorescent Nan-
+odiamonds for Detecting Free-Radical Generation in Real
+Time during Shear Stress in Human Umbilical Vein En-
+dothelial Cells, ACS Sensors 6, 4349 (2021).
+[26] A. Sigaeva, H. Shirzad, F. P. Martinez, A. C. Nusantara,
+N. Mougios, M. Chipaux, and R. Schirhagl, Diamond-
+Based Nanoscale Quantum Relaxometry for Sensing Free
+Radical Production in Cells, Small (Weinheim an der
+Bergstrasse, Germany) 18, e2105750 (2022).
+[27] N. Norouzi, A. C. Nusantara, Y. Ong, T. Hamoh, L. Nie,
+A. Morita, Y. Zhang, A. Mzyk, and R. Schirhagl, Relaxom-
+etry for detecting free radical generation during Bacteria’s
+
+15
+response to antibiotics, Carbon 199, 444 (2022).
+[28] J. Choi, S. Choi, G. Kucsko, P. C. Maurer, B. J. Shields,
+H. Sumiya, S. Onoda, J. Isoya, E. Demler, F. Jelezko,
+N. Y. Yao, and M. D. Lukin, Depolarization Dynamics in a
+Strongly Interacting Solid-State Spin Ensemble, Physical
+Review Letters 118, 093601 (2017).
+[29] R. Giri, F. Gorrini, C. Dorigoni, C. E. Avalos, M. Caz-
+zanelli, S. Tambalo, and A. Bifone, Coupled charge and
+spin dynamics in high-density ensembles of nitrogen-
+vacancy centers in diamond, Physical Review B 98,
+045401 (2018).
+[30] R. Giri, C. Dorigoni, S. Tambalo, F. Gorrini, and A. Bifone,
+Selective measurement of charge dynamics in an ensem-
+ble of nitrogen-vacancy centers in nanodiamond and bulk
+diamond, Physical Review B 99, 155426 (2019).
+[31] F. Gorrini, C. Dorigoni, D. Olivares-Postigo, R. Giri,
+P. Apr`a, F. Picollo, and A. Bifone, Long-Lived Ensembles
+of Shallow NV- Centers in Flat and Nanostructured Di-
+amonds by Photoconversion, ACS Applied Materials &
+Interfaces 13, 43221 (2021).
+[32] M. W. Doherty, N. B. Manson, P. Delaney, and L. C. L. Hol-
+lenberg, The negatively charged nitrogen-vacancy cen-
+tre in diamond: the electronic solution, New Journal of
+Physics 13, 025019 (2011).
+[33] S. Felton, A. M. Edmonds, M. E. Newton, P. M. Martineau,
+D. Fisher, and D. J. Twitchen, Electron paramagnetic reso-
+nance studies of the neutral nitrogen vacancy in diamond,
+Physical Review B 77, 081201(R) (2008).
+[34] E. V. Levine, M. J. Turner, P. Kehayias, C. A. Hart, N. Lan-
+gellier, R. Trubko, D. R. Glenn, R. R. Fu, and R. L.
+Walsworth, Principles and techniques of the quantum di-
+amond microscope, Nanophotonics 8, 1945 (2019).
+[35] X.-D. Chen, L.-M. Zhou, C.-L. Zou, C.-C. Li, Y. Dong, F.-W.
+Sun, and G.-C. Guo, Spin depolarization effect induced
+by charge state conversion of nitrogen vacancy center in
+diamond, Physical Review B 92, 104301 (2015).
+[36] I. Meirzada, Y. Hovav, S. A. Wolf, and N. Bar-Gill, Neg-
+ative charge enhancement of near-surface nitrogen va-
+cancy centers by multicolor excitation, Physical Review
+B 98, 245411 (2018).
+[37] B. Naydenov, F. Dolde, L. T. Hall, C. Shin, H. Fedder,
+L. C. L. Hollenberg, F. Jelezko, and J. Wrachtrup, Dynam-
+ical decoupling of a single-electron spin at room temper-
+ature, Physical Review B 83, 081201(R) (2011).
+[38] A. Jarmola, V. M. Acosta, K. Jensen, S. Chemerisov, and
+D. Budker, Temperature- and Magnetic-Field-Dependent
+Longitudinal Spin Relaxation in Nitrogen-Vacancy En-
+sembles in Diamond, Physical Review Letters 108, 197601
+(2012).
+[39] Y. Romach, C. M¨uller, T. Unden, L. J. Rogers, T. Isoda,
+K. M. Itoh, M. Markham, A. Stacey, J. Meijer, S. Pezza-
+gna, B. Naydenov, L. P. McGuinness, N. Bar-Gill, and
+F. Jelezko, Spectroscopy of Surface-Induced Noise Using
+Shallow Spins in Diamond, Physical Review Letters 114,
+017601 (2015).
+[40] M. Mr´ozek, D. Rudnicki, P. Kehayias, A. Jarmola, D. Bud-
+ker, and W. Gawlik, Longitudinal spin relaxation in
+nitrogen-vacancy ensembles in diamond, EPJ Quantum
+Technology 2, 10.1140/epjqt/s40507-015-0035-z (2015).
+[41] N. B. Manson, M. Hedges, M. S. J. Barson, R. Ahlefeldt,
+M. W. Doherty, H. Abe, T. Ohshima, and M. J. Sellars, NV
+− –N + pair centre in 1b diamond, New Journal of Physics
+20, 113037 (2018).
+[42] M. L. Juan, C. Bradac, B. Besga, M. Johnsson, G. Brennen,
+G. Molina-Terriza, and T. Volz, Cooperatively enhanced
+dipole forces from artificial atoms in trapped nanodia-
+monds, Nature Physics 13, 241 (2017).
+[43] S. T. Alsid, J. F. Barry, L. M. Pham, J. M. Schloss, M. F.
+O’Keeffe, P. Cappellaro, and D. A. Braje, Photolumines-
+cence Decomposition Analysis: A Technique to Charac-
+terize N - V Creation in Diamond, Physical Review Ap-
+plied 12, 044003 (2019).
+[44] T. Wolf,
+P. Neumann,
+K. Nakamura,
+H. Sumiya,
+T. Ohshima, J. Isoya, and J. Wrachtrup, Subpicotesla Dia-
+mond Magnetometry, Physical Review X 5, 041001 (2015).
+[45] L. Robledo, H. Bernien, T. van der Sar, and R. Hanson,
+Spin dynamics in the optical cycle of single nitrogen-
+vacancy centres in diamond, New Journal of Physics 13,
+025013 (2011).
+[46] T. de Guillebon, B. Vindolet, J.-F. Roch, V. Jacques, and
+L. Rondin, Temperature dependence of the longitudinal
+spin relaxation time T1 of single nitrogen-vacancy centers
+in nanodiamonds, Physical Review B 102, 165427 (2020).
+[47] L. Rondin,
+G. Dantelle,
+A. Slablab,
+F. Grosshans,
+F. Treussart,
+P. Bergonzo,
+S. Perruchas,
+T. Gacoin,
+M. Chaigneau, H.-C. Chang, V. Jacques, and J.-F. Roch,
+Surface-induced charge state conversion of nitrogen-
+vacancy defects in nanodiamonds, Physical Review B 82,
+115449 (2010).
+[48] E. R. Wilson, L. M. Parker, A. Orth, N. Nunn, M. Torelli,
+O. Shenderova, B. C. Gibson, and P. Reineck, The effect
+of particle size on nanodiamond fluorescence and col-
+loidal properties in biological media, Nanotechnology 30,
+385704 (2019).
+[49] S. Dhomkar, H. Jayakumar, P. R. Zangara, and C. A. Mer-
+iles, Charge Dynamics in near-Surface, Variable-Density
+Ensembles of Nitrogen-Vacancy Centers in Diamond,
+Nano Letters 18, 4046 (2018).
+