Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model

The existence and uniqueness/multiplicity of phase locked solution for continuum Kuramoto model was studied in [ 12 , 29 ]. However, its asymptotic behavior is still unknown. In this paper we concern the asymptotic property of classic solutions to continuum Kuramoto model. In particular, we prove the convergence towards a phase locked state and its stability, provided suitable initial data and coupling strength. The main strategy is the quasi-gradient flow approach based on Łojasiewicz inequality. For this aim, we establish a Łojasiewicz type inequality in infinite dimensions for continuum Kuramoto model which is a nonlocal integro-differential equation. General theorems for convergence and stability of (generalized) quasi-gradient system in an abstract setting are also provided based on Łojasiewicz inequality.


1.1.
Motivation. The Kuramoto model [17,18], given by the following coupled oscillators has been a successful model for describing the synchronization process of large populations of weakly coupled oscillators which often appears in natural and engineering systems. Here, θ i is the phase of the ith oscillator, ω i is its natural frequency, and the oscillators are all-to-all coupled through the sinusoidal function with strength α > 0. This type of all-to-all coupling has played an important role in the analysis of (1). For example, for finite population of oscillators, there have been some studies on the complete synchronization of (1), where the analysis was built on the all-to-all coupling scheme [6,11,13]. In the limit as N → ∞, there are two ways to recast the Kuramoto model. The first one is to use the one-oscillator probability density to derive a kinetic Kuramoto equation in the mean-field limit, see for example, [1,2,3,8,9,26,27,28]. For the stability issue, in [4] Carrilo et al. studied the complete synchronization of kinetic Kuramoto model and derive some estimate for stability of measure-valued solutions in the Wasserstein distance. In [9], Dietert et al. discussed the stability of partially phase-locked states in kinetic Kuramoto model. Another way to reformulate the Kuramoto model as N → ∞ is to take a continuum limit in (1) so that we can derive a single integro-differential equation: This equation, referred as continuum Kuramoto model (CKM), can be formally constructed by interpreting the coupling term in (1) as a Riemann sum and sending N → ∞, where θ(x, t) now describes a continuum of dynamical systems distributed along I := [0, 1] and ω(x) describes the distributed natural frequencies. On the other hand, this model can be derived from (1) by applying Strong Law of Large Numbers with N → ∞ when natural frequencies {ω i } are subject to probability distributions that are independent and identically distributed [12]. A connection between the kinetic and continuum Kuramoto models is that the CKM can be regarded as a subclass of solutions of the kinetic Kuramoto model when the kinetic distribution δ θ=θ(t,x) dx is taken. Moreover, this class is invariant under the evolution of the kinetic Kuramoto model. In [12], Ermentrout derives a nonlinear equation and its equivalent formulation which gives a criterion for existence of phase-locked solutions of (2). Following this approach, Troy [29] investigated uniqueness or exact multiplicity of these solutions. He derived a criteria, depending on a parameter, to guarantee the uniqueness of phase-locked solution or coexistence of exactly two solutions. In [24,25], Medvedev derived the continuum limit of Kuramoto model on networks with increasing number of nodes by using the ideas from graph limits [23], and discussed the relation between solutions of discrete model and the continuum limit. We should note that the dynamic evolution of (2) has not been touched. In [12,29] they concerned only the static problems regarding its phase-locked solutions; in other words, the equilibrium of (2) was considered through the static equation As pointed in [29], the convergence of system (2) and the stability of its equilibrium (3) are interesting problems. To the best of our knowledge, the only related work should be directed to the stability of the measure-valued solution of kinetic Kuramoto model in the Wasserstein distance [4], in the sense that CKM describes a subclass of solutions of the kinetic Kuramoto model. However, the dynamic properties of the system (2) in itself have not been studied. In this paper, we will study the convergence of solutions to system (2) in C[0, 1] or L 2 [0, 1] (see Theorem 5.5 and Remark 4). We also consider the stability of its equilibrium (3) in the above topologies. Motivated by this problem, we will establish a Lojasiewicz inequality in an infinite-dimensional Hilbert space for (2), and develop abstract theorems for the convergence and stability of generalized gradient systems in a Banach space which is continuously embedded into a Hilbert space (Theorems 4.1 and 4.4). In the following subsection, we briefly introduce our strategy.
1.2. Our strategy. The classic Lojasiewicz inequality was established in finite dimensions [22], which reveals a fundamental relation between a potential function E : R N → R and its gradient, more precisely, where N (x 0 ) is a neighborhood of x 0 , ρ ∈ 0, 1 2 and C > 0 are constants independent of x ∈ N (x 0 ). The exponent ρ that makes (4) hold is called Lojasiewicz exponent of E at x 0 . This inequality provides a powerful tool for proving the convergence of trajectories of gradient systemẋ = −∇E(x) towards a single equilibrium. A celebrated result was given in [22] which claims that Lojasiewicz inequality holds for any real analytic function E : R N → R, although the computation of Lojasiewicz exponent ρ can be a hard problem. It is worthy to mention that the computation of exponent is relevant to the convergence rate; precisely, ρ = 1 2 implies exponential decay and ρ < 1 2 implies algebraic decay. In [30] it was noticed that the discrete Kuramoto model (1) has a Lyapunov function so that it can be reformulated as a standard gradient system with an analytic potential. Based on this reformulation, in [10,14,20] the Lojasiewicz inequality in finite dimensions was used to prove the convergence towards a phase-locked state for system (1). Two exponents show up for (1), ρ = 1 2 or 1 3 , depending on the configurations of phase-locked states [20]. For other studies on the second-order Kuramoto model with inertia by this inequality we refer to [7,19,21].
The Lojasiewicz inequality can be extended to abstract settings in infinite dimensions, for example, the so-called Lojasiewicz-Simon inequality [5]. However, unlike the case of finite dimensions, in infinite dimensions the Lojasiewicz type inequality may fail for a real analytic functional on a Hilbert space H, see [15,Proposition 3.5]. Actually, this example also indicates that Lojasiewicz inequality can fail even when we confine the argument x in a compact subset of the neighborhood N (x 0 ). Therefore, to verify a Lojasiewicz inequality for a real analytic functional in infinite dimensions is not a trivial work, even when we confine the argument in a compact subset of a neighborhood. In order to borrow the idea of Lojasiewicz inequality in finite dimensions to study the convergence and stability of CKM, we first need to prove a Lojasiewicz inequality for the specific system (2).

ZHUCHUN LI, YI LIU AND XIAOPING XUE
Define a functional E : L 2 [0, 1] → R as follows: Then (2) can be regarded as a quasi-gradient flow with potential E (see Remark 3). Two abstract theorems for the convergence and stability of quasi-gradient systems are established in a Banach space which is continuously embedded into a Hilbert space (Theorems 4.1 and 4.4). We prove a Lojasiewicz inequality for E when we confine the argument θ in a relatively compact subset, i.e, M in Theorem 3.2.
Using the Lojasiewicz inequality and abstract theorems we can obtain the desired results for the asymptotic behaviors of (2).

1.3.
Contribution. The contribution of this paper is threefold. First, we prove the convergence and stability for CKM which was noticed as an open question in [29]. Second, we prove a Lojasiewicz type inequality with an exponent ρ = 1 4 for infinite dimensional CKM which is a nonlocal integro-differential equation. As far as we know, this is the first result for this issue. Third, we present a novel setting for convergence and stability of generalized gradient systems in Banach spaces.
The rest of this paper is organized as follows. In Section 2, we show the existence and uniqueness of solution for (2). In Section 3, we prove a Lojasiewicz inequality for CKM. In Section 4, the abstract theorems are presented. Finally in Section 5 we give the main results in this paper for the convergence and stability of CKM. Then the conclusion is given in Section 6.

Preliminaries.
2.1. On the model. We consider the following integro-differential equation Let θ(x, t) be a solution of (7), we introduce natural frequency and phase fluctuations:ω then by (7) we have Note that the integral of right hand side goes to zero, so we have Without loss of any generality, let's assume 1 0θ (x, 0)dx = 0, then we have 1 0θ (x, t) dx = 0, ∀t ≥ 0. We now drop the hat to simplify the notation and obtain the following system: Obviously, along the solution of (8) the mean phase is conserved, i.e., Instead of (7), hereafter we will work on (8).
In this paper, we will study the convergence of classic solutions of (8), or more precisely, whether there exists an equilibrium θ * of (8) such that In the remaining part of this section, we address the existence and uniqueness of solution to (8).
then X is a Banach space and F is a mapping from X to itself. Let θ 1 and θ 2 be arbitrarily chosen in X , we have Hence, the mapping F : X → X is locally Lipschitz continuous. Consider the abstract differential equation in X :

ZHUCHUN LI, YI LIU AND XIAOPING XUE
By the standard Cauchy's theory, we see that the system (9) admits a unique solution θ ∈ C 1 ([0, T ), X ). Next, we show that the local solution can be extended so that it is global in time. We note that Let θ(t) be the solution of (9) on [0, T ), T < +∞, then we have and further

This surely implies that sup
Therefore, lim t→T − θ(t) exists in X . Then we can extend the solution to [0, +∞), and we conclude the existence and uniqueness of global solution for (9), and then (8).
Next, we discuss the time evolution of the x-derivative of classic solution to (8). First we give the following Lemma. Lemma 2.3. Let g ∈ C[0, 1]. The following differential equation admits a unique solution in C[0, 1].
Proof. For notational simplicity let's denote V = C[0, 1]. We consider the differential equation in space V: Note that Thus, the mapping G : V → V is locally Lipschitz. Moreover, we have for some constant M . We use the similar argument as in Proposition 2.2 to see that the integro-differential equation (11) admits a unique solution on [0, +∞).
Note that the solution of (8) satisfies By Lemma 2.3, there is a unique solution to the equation (12) with initial data a(x, 0) = θ 0 (x). Therefore, we see the following remark.
Remark 1. Let θ(x, t) be the unique solution to (8), which is guaranteed by Proposition 2.2, then its derivative ∂ ∂x θ(x, t) is the unique solution to (12).
3. Lojasiewicz inequality. In this section, we will establish a Lojasiewicz inequality for potential (6). For convenience, we drop the negative sign in (6) and consider the functional E 1 = −E, i.e., Here, the gradient is defined through the standard inner product in Hilbert space Hence, as h → 0. Since the function sin(·) is odd, we find that Hence, This implies (13).
Throughout this paper, for θ ∈ C 1 [0, 1] we will denote the phase diameter as For given constants > 0 andD ∈ (0, π 2 ), we let Theorem 3.2. Let and * be positive constants, and let θ * ∈ M * be an equilibrium of (8). Then there exist constants δ > 0 and β > 0 such that It is obvious that for θ ∈ M with θ − θ * C[0,1] < δ, we have η ≤ 2δ. Since θ * is an equilibrium of (7), we have Then we see that We estimate the terms I 1 and I 2 as follows: and for I 2 , Thus, Then we find • Step 2. Estimate of ∇E 1 (θ). Note that

ZHUCHUN LI, YI LIU AND XIAOPING XUE
We denote the integrand by q(x), that is, We notice that and denote Then we find Then Since there exists a constant β 2 > 0 such that Without loss of any generality, let's say q 2 (x) ≥ β 2 η 2 . In order to find a lower bound for ∇E 1 (θ) 2 L 2 [0,1] , we consider the x-derivative of q(x) and find that If |x −x| ≤ |q(x)| 2( + * ) , we have |q(x)| ≥ |q(x)| 2 and q 2 (x) ≥ q 2 (x) 4 . Then we recall (15) to see that Finally, we combine Step 1 and Step 2, i.e., the relations (14) and (16), to obtain the desired inequality.
Remark 2. Unlike the usual Lojasiewicz inequality in either finite or infinite dimensions, for the Lojasiewicz inequality in Theorem 3.2, we need to confine the argument θ in a relatively compact subset, i.e, M . The reason for this lies in the case that the equation (8) and potential (6) are nonlocal. The rationality for using such a subset lies in the fact that the trajectories of (2) do enter such a set and stay therein when suitable parameters are provided (see Section 5).
Next, we show that the equilibrium θ * is an extremal point of the functional E 1 .
We then choose sufficiently small δ with δ < cosD 2c in Theorem 3.2 such that η < cosD c , then Therefore, θ * is a local maximal point for the functional E 1 .

4.
Convergence and stability of generalized gradient systems. In this section, we present two results for the asymptotic property of an abstract differential equation based on Lojasiewicz inequality. Let V be a Banach space and H be a Hilbert space such that V is continuously embedded into H (we write V → H). Let M be a relatively compact subset in V. We consider the following differential equation in V: where f : V → V is locally Lipschitz continuous. Let A = {u ∈ V | f (u) = 0} and we denote the ω-limit set of u 0 by Ω(u 0 ), i.e., Ω(u 0 ) consists of all accumulation points in V of the trajectory {u(t)} t≥0 initialed at u 0 .

Convergence.
Theorem 4.1. Assume that there exists a functional E ∈ C 1 (H, R) such that: Let u : R + → V be a solution to (17) with u(t) ∈ M for all t ∈ R + . Assume that for any v ∈ Ω(u 0 ), there exists a neighborhood N V (v) of v in V, and two constants ρ ∈ 0, 1 2 and C > 0, such that Then, there exists u * ∈ A such that Remark 3. The abstract differential equation (17) which admits a functional E satisfying (i) and (ii) is refereed as a (generalized) quasi-gradient system with potential E.
Proof of Theorem 4.1.
So E(u(t)) is monotonically decreasing. On the other hand, it is bounded, so we know that E(u(t)) converges as t → ∞. Without loss of any generality, we assume E(u(t)) ≥ 0 and lim t→+∞ E(u(t)) = 0 (then E(u * ) = 0). If E(u(t 1 )) = 0 for some t 1 , we must have E(u(t)) = 0 for all t ≥ t 1 . Then (18) implies that u(t 1 ) ∈ A and u(t) = u(t 1 ) for all t ≥ t 1 . This proves the convergence. In the following, we assume that E(u(t)) > 0 for all t ∈ R + . We first recall that the gradient inequality holds at u * , i.e., for some constants ρ * ∈ 0, 1 2 and σ * > 0. We can choose t N such that u(t N ) − u * V < σ * and let Then T = ∅ is open. We now claim that: Proof of the Claim (20). If the open set T consists of finite number of open intervals, then as a result of t n ∈ T and t n → +∞, there must be an interval of the form (t * , +∞) such that (t * , +∞) ⊂ T . So, (20) is obviously true. In the following, we assume that the open set T consists of infinitely many open intervals, and none of them has the form of (t * , +∞). In this case, we can find a sequence of open intervals, say (α k , β k ), and a subsequence {t n k } of {t n } such that t n k ∈ (α k , β k ), (k = 1, 2, . . . ).
Next we show that σ 1 > 0. Otherwise, there must exist {a l } ⊂ A such that a l H → 0. As A is relatively compact in V, we can find a subsequence {a lp } of {a l } and a * ∈ V such that a lp − a * V → 0. Note that a lp V = σ * , so a * V = σ * . On the other hand, the relation a lp − a * H → 0 implies a * H = 0, and hence a * = 0. This contradicts (21). This proves that σ 1 > 0. We now choose a t n k 0 ∈ (α k0 , β k0 ) such that Obviously, we have u(t) − u * V < σ * for all t ∈ (t n k 0 , β k0 ). By assumption (ii) and relation (19) we can derive that So, we have which, together with (23), implies that We combine this relation with (23) to derive However, the definition of σ 1 in (22) implies which leads to a contradiction. The claim (20) is proved. We recall the relation (24) to find that By the Cauchy principle for convergence, we have lim t→+∞ u(t) − u * H = 0.
Next we show that lim t→+∞ u(t) − u * V = 0. Suppose not, i.e., As {u(s n )} is relatively compact, without loss of generality we can assume u(s n ) → v * in V for some v * . Then u * − v * V > 0. On the other hand, u(s n ) → v * in H, while (26) implies that u(s n ) → u * in H, so we have u * = v * . This is a contradiction. Therefore, we have lim t→+∞ u(t) − u * V = 0. (19), then u(t) → u * in H exponentially fast. In other words, there exist positive constants C, λ and T such that (19), then u(t) → u * in H algebraically slow. More precisely, there exist positive constants C and T such that Proof. We recall (4.1) and take into account of Claim (20) to see that for all t > t * , Then, we invoke (19) and the assumption that f (u) = ∇E(u) in H, to see that +∞ t u(s) Hds = +∞ t f (u(s)) Hds < C Then Sinceμ(t) = − u(t) H , we use (27) to find that This implies that µ(t) satisfies that following differential inequality: for some constant L > 0. Consider the differential equatioṅ By the principle of comparison, we have µ(t) ≤ y(t), ∀ t > t * . Then the desired estimates in (i) and (ii) follows from (28) and solving the differential equation (29).

4.2.
Stability. We assume that the differential equation (17) admits a unique global solution on [0, +∞). We will consider the stability of equilibrium of (17) in the following sense. Definition 4.3. We say the equilibrium u * ∈ A is V − H stable with respect to M, if for any ε > 0, there exists δ > 0 such that for all u 0 ∈ M with u 0 − u * V < δ we have where u(t) is the solution to (17) with initial value u 0 .
Theorem 4.4. Let u * ∈ A. Assume that there exists a functional E ∈ C 1 (H, R) such that: (i) ∇E(u * ) = 0; (i) there exists a neighborhood N V (u * ) of u * , and constants ρ ∈ (0, 1 2 ] and C 1 , C 2 > 0, such that (i) the relatively compact set M ⊂ V is positively invariant for (17), and u * is a local minimal point of E(·) on N V (u * ) ∩ M.
Then, u * is V − H stable with respect to M.
Proof. Without loss of any generality, we assume E(u * ) = 0. By the hypothesis (ii) and (iii), there exists a constant σ > 0 such that for all u ∈ M with u − u * V < σ we have We may assume u H ≤ u V for all u ∈ V. Let S σ = u ∈ M u − u * V = σ , and let r σ = inf u∈Sσ u−u * H . As S σ is relatively compact, we have r σ > 0. By the continuity of E(·) and E(u * ) = 0, for any ε ∈ (0, r σ ), there exists δ ∈ (0, ε 2 ) ∩ (0, σ) such that For initial data u 0 ∈ M, we have u(t) ∈ M. Now, we let u 0 ∈ M with u 0 −u * V < δ, and let By the continuity we see that T is well-defined. Next we show that sup T = +∞. Assume the opposite, say sup T = t 1 < +∞, then we should have By (17), (30) 1 and (31), for t ∈ (0, t 1 ) we have We integrate this inequality to find

5.
Convergence and stability of the Kuramoto model. In this section we will apply Theorems 4.1, 4.4 and 3.2 to prove the convergence and stability of CKM (8).
In order to apply the above theorems for this model, we should find some framework to guarantee that the solution to (8) is confined in M 0 for some constant 0 > 0, i.e., θ(·, t) ∈ M 0 . In order to do this we first present several lemmas as a prior estimates.

5.1.
A prior estimates. f (x, t), then g 1 (t), g 2 (t) are locally absolutely continuous, and where Proof. We give a proof only for g 1 (t), and it is similar for g 2 (t).
On the other hand, for anyx ∈ M (t), Taking the limit as h → 0, we find that Due to the compactness of M (t), we have For |ω (x)|.