Unbounded Second-Order State-Dependent Moreau’s Sweeping Processes in Hilbert Spaces

In this paper, an existence and uniqueness result of a class of second-order sweeping processes, with velocity in the moving set under perturbation in infinite-dimensional Hilbert spaces, is studied by using an implicit discretization scheme. It is assumed that the moving set depends on the time, the state and is possibly unbounded. The assumptions on the Lipschitz continuity and the compactness of the moving set, and the linear growth boundedness of the perturbation force are weaker than the ones used in previous papers.


Introduction
In 1971, the sweeping process was introduced and deeply studied by J. J. Moreau in a series of papers (see, e.g., [1][2][3][4]). This kind of problems plays an important role in elasto-plasticity, quasi-statics and nonsmooth dynamics with unilateral con-  straints. Roughly speaking, a point is swept by a moving closed and convex set, which depends on time in a Hilbert space and can be formulated in the form of first-order differential inclusion involving normal cone operators. Sweeping processes represent a nice and powerful mathematical framework for many nonsmooth dynamical systems, including Lagrangian systems. There are plenty of existence and uniqueness results (see, e.g., [5][6][7][8][9][10]) for variants of first-order sweeping processes in the literature. The second-order sweeping processes have been also considered by many authors (see, e.g., [7,[10][11][12][13][14]). In [11], Castaing studied for the first time the second-order sweeping processes, where the moving set depends on the state with convex, compact values. Let us note that the boundedness assumption on the moving set for the second-order case is essential in most previous works: see, for example, some recent papers [7,10,13,14]. In [15] , Castaing et al. considered the possibly unbounded moving set satisfying the classical Lipschitz continuity assumption with respect to Hausdorff distance. However, it is difficult for unbounded set to hold this assumption since the Hausdorff distance of two unbounded sets may equal the infinity, for example, the case of rotating hyperplane. In this paper, we propose an implicit discretization scheme based on the Moreau's catching-up algorithm [3] with different techniques to analyze the second-order sweeping processes under perturbation in Hilbert spaces. The moving set depends on the time, the state and is possibly unbounded. The set is supposed to be closed, convex and to have a Lipschitz variation of intersection with some particular ball (with a Lipschitz constant depending on the radius of the ball). It is obvious that this kind of Lipschitz continuity assumption is more feasible than the classical one for the unbounded moving set. The perturbation force is supposed to be upper semicontinuous with convex and weakly compact values and only need to satisfy the weak linear growth condition (i.e., the intersection between the perturbation force and the ball with linear growth is nonempty). In addition, the compactness assumption on the moving set is weaker than the one used in previous works [7,10,[13][14][15], since it only requires to check the Kuratowski measure of noncompactness for a fixed ball. We also consider the case when the moving set is anti-monotone (which replaces the compactness assumption) as in [14] under the current settings. Our methodology is based on convex and variational analysis [16,17].
The paper is organized as follows. In Sect. 2, we recall some basic notations, definitions and useful results which are used throughout the paper. The existence and uniqueness of solutions are thoroughly analyzed in Sect. 3. Some conclusions end up the paper in Sect. 4.

Notation and Preliminaries
We begin with some notations used in the paper. Let H be a real Hilbert space. Denote by ·, · , · the scalar product and the corresponding norm in H . Denote by I the identity operator, by B the unit ball in H . The distance from a point x to a set K is denoted by d(x, K ). If K is closed and convex, then for each x ∈ H , there exists uniquely a point y ∈ K which is nearest to x and set y := proj(K ; x). The normal cone of K is given by The support function of K is defined as follows It is not difficult to see that The Hausdorff distance between the sets A and B is given by for some L C > 0. Set Then, Φ is weakly lower semicontinuous, i.e., The following lemma is a discrete version of Gronwall's inequality. Then, for all n, we have Finally, we recall the Kuratowski measure of noncompactness for a bounded set B in H , which is defined as follows One has the following lemma (see, e.g., [ for some x 0 ∈ H and r > 0.

Main Result
In this section, the existence and uniqueness of solutions of the following second-order sweeping processes are analyzed thoroughly under weaker assumptions, by using an implicit discretization scheme and techniques different from previous works. The moving set C is supposed to be nonempty, closed, convex and to have a Lipschitz variation of intersection with some particular ball. The perturbation force F is upper semicontinuous with convex, weakly compact values and satisfies the weak linear growth condition. For details, let us make the assumptions below. Let H be a real Hilbert space and let be given Remark 1 Let us consider the simple case when C(·) depends only on t and is expressed in the form of unilateral inequality constraints, i.e., where function g k : [0, T ] × R n → R is supposed to be of class C 1 for each k = 1, 2, . . . , m. Clearly, the convexity of the functions g k implies the convexity of the set C(t). In order to go beyond the convexity assumption of the set C(·), the class of prox-regular sets is more appropriate. However, the sublevels of prox-regular functions and levels of differentiable mappings with Lipschitz derivatives may fail to be prox-regular. We need some qualification conditions on the functions g k to ensure the prox-regularity of the set C(t) (see [20] for more details). In the present paper, we will content ourselves with the convexity assumption.
The following assumptions will be useful. and where γ is the Kuratowski measure of noncompactness.

Assumption 2
The set-valued mapping F : gph(C) ⇒ H is upper semicontinuous with convex, weakly compact values in H and satisfies the weak linear growth condition, i.e., there exists L F > 0 such that, for all t ∈ [0, T ], x ∈ H and y ∈ C(t, x), then Here gph(C) denotes the graph of C.
Now we are ready for the main result.

Theorem 3.1 (Existence) Let H be a Hilbert space and let Assumptions 1, 2 hold.
Then, for given initial condition u 0 ∈ H, v 0 ∈ C(0, u 0 ), there exists a solution u in the following sense Proof We choose some positive integer n such that M 1 T /n < 1 and set h n := T /n, where Clearly u n i+1 is defined uniquely in terms of u n i and v n i . The first line of (9) can be rewritten as We have the algorithm to construct the sequences Iteration. One has current points u n Then The algorithm is well defined thanks to (i) of Assumption 1. Now we prove that , generated by the algorithm above, are uniformly bounded. Particularly, we show that It is obviously true for i = 0. Suppose that (12) holds for up to some i ∈ {0, 1, . . . , n − 1}, we will prove that (12) also holds for i + 1. Indeed, one has max{ u n Consequently On the other hand From (15) and (16), one has Consequently, by induction we have On the other hand, one has Furthermore, from (13) In conclusion, the sequences (u n i ) n i=0 , (v n i ) n i=0 are uniformly bounded by M 1 (more precisely by are uniformly bounded by M 2 . We construct the sequences of functions (u n (·)) n , (v n (·)) n , ( f n (·)) n , (θ n (·)) n , (η n (·)) n from [0, T ] to H as follows: Then, for all t ∈]t n i , t n i+1 [ and max{ sup The sequence v n (·) n is equi-Lipschitz with ratio Next we prove that the set (t) = {v n (t)} is relatively compact for all t ∈ [0, T ].
In next step, we prove that for every t ∈ [0, T ],u(t) ∈ C(t, u(t)). From the fact that v n i ∈ C(t n i , u n i ) for all i, we deduce for every t ∈ [0, T ] that v n (θ n (t)) ∈ C θ n (t), u n (θ n (t)) ∩ M 1 B ⊂ C(t, u(t)) It is easy to see that for every t ∈ [0, T ], v n (θ n (t)) → v(t) =u(t) and |θ n (t) − t| + u n (θ n (t)) − u(t) → 0 as n → +∞ because of (22) and the strongly convergence of v n (·) to v(t), u n (·) to u(·) in C([0, T ]; H ). Since C(t, u(t)) is closed, we obtain thatu(t) ∈ C(t, u(t)) for every t ∈ [0, T ]. It remains to prove thaẗ Let us define From (9) we have, for almost every t ∈ [0, T ], thaṫ v n (t) + f n (t) ∈ −N C η n (t),u n (η n (t)) v n (η n (t)) Then, for all c ∈ D η n (t) , we get Let us begin by estimating the second term in (29). First we prove that Note that we also have as n → +∞. So we get (30).
Since f n (t) ≤ M 2 for all t ∈ [0, T ], the sequence ( f n ) is bounded in L ∞ ([0, T ]; H ). Therefore we can extract a subsequence, without relabeling for simplicity, converging weakly to some mapping f (·) in L ∞ ([0, T ]; H ). On the other hand, v n (η n (·)) converges strongly tou(·) in L 1 ([0, T ]; H ), so one has From (30) and (31), one deduces that For the first term in (29), we will show that Let us recall that (Lemma 2.1) the convex function respectively. Consequently, one implies that On the other hand It leads to the following inequality lim inf From (34) and (35), we get the desired result (33). From (29), (32) and (33), we deduce that Note thatu(t) ∈ D(t) for every t ∈ [0, T ], we have From (36) and (37), one infers that or equivalently, On the other hand, one has f n (t) ∈ F θ n (t), u n (θ n (t)), v n (θ n (t)) for all t ∈ [0, T ] and F is upper semicontinuous with convex and weakly compact values in H . Classically, we obtain that f (t) ∈ F t, u(t),u(t) for almost all t ∈ [0, T ] (see, e.g., [21,). Thus The result has been proved.
Remark 2 (i) It is obvious that the result is still true if we replace Assumption 1-(i) by the classical Lipschitz continuity assumption (see, e.g., [15]): However, it is difficult for unbounded set to hold this kind of assumption, since the Hausdorff distance of two unbounded sets may equal the infinity. For example, the rotating hyperplane never satisfies (41), but satisfies Assumption 1-(i) with suitable parameters. This observation was also stated in [22], when the author studied the first-order sweeping processes with the convex moving set depending on the time. Note that in our paper, the local Lipschitz variation of the moving set is assumed in a fixed ball, while in [22], it is necessary to consider in any ball. Particularly, if F ≡ 0 and 0 ∈ C(t, x) for all t ∈ [0, T ] and x ∈ H , then (5) can be replaced by where M 1 := u 0 + v 0 T . Indeed, from (11) and (ii) The compactness assumption on the moving set C is also weaker than the one used in previous works [7,10,[13][14][15] since it only requires to check the Kuratowski measure of noncompactness for a fixed ball. Furthermore, the perturbation force F only needs to satisfy the weak linear growth condition. (iii) In many applications, in practice, the set C(·) could be unbounded. This is the case, e.g., when C(·) coincides with a moving convex and closed cone. Such systems are called nonlinear complementarity systems and are of great interest in the modeling of nonregular electrical systems (see, e.g., Section 3.4 in [5]). (iv) The compactness assumption can be replaced by the anti-monotonicity of C as in [14]. Again, one does not need the boundedness and the classical Lipschitz continuity of C.
Theorem 3.2 Let Assumptions 1-(i), 2 hold and suppose that −C(t, ·) is monotone for each t ∈ [0, T ]. Furthermore, assume that F is monotone with respect to the third variable on gph(C), i.e., for all (t i , Then, for each initial condition, there exists a solution in the sense of Theorem 3.1. Proof We construct the sequences (u n i ) n i=0 , (v n i ) n i=0 , ( f n i ) n i=0 and the sequences of functions (u n (·)) n , (v n (·)) n , ( f n (·)) n , (θ n (·)) n , (η n (·)) n as in Theorem 3.1. From the proof of Theorem 3.1, it is sufficient to prove the strong convergence of sequence v n (·) in C([0, T ]; H ). First we prove the convergence of u n (·). For all positive integers m ≥ n, let due to the M 1 -Lipschitz continuity of u m (·), u n (·) and the boundedness by M 1 of u m i , u n j . Consequently, which implies that (u n (·)) n is a Cauchy sequence in C([0, T ]; H ). Thus, there exists a M 1 -Lipschitz function u(·) such that u n (·) converges to u(·) uniformly and Next we show the uniform convergence of (v n (·)) n . By using (22), (43) and the Lipschitz continuity of C, u n (·), v m (·), one has the following estimation In particular, we imply that d C η n (t),u n (η n (t)) (v m (t)) → 0 as m, n → +∞. From (27) and the fact that v n (t) + f n (t) ≤ 2M 2 , one has −v n (t) − f n (t) ∈ N C η n (t),u n (η n (t)) v n (η n (t)) = 2M 2 ∂d C(η n (t),u n (η n (t))) (v n (η n (t))).
It deduces that (v n (·)) n is a Cauchy sequence in C([0, T ]; H ), which leads to the uniform convergence of (v n (·)) n . Thus, the result has followed.
Then, the result follows by using Gronwall's inequality.

Conclusions
In this paper, by using tools from convex and variational analysis, the existence and uniqueness result of a class of second-order state-dependent sweeping processes in Hilbert space, has been studied carefully. It is remarkable that the moving set is possibly unbounded and all main assumptions (the Lipschitz continuity and the compactness of the moving set, the linear growth boundedness of the perturbation force) are weaker than the ones used in previous works, which allows more applications in practice.