ON EVOLUTION QUASI-VARIATIONAL INEQUALITIES AND IMPLICIT STATE-DEPENDENT SWEEPING PROCESSES

. In this paper, we study a variant of the state-dependent sweeping process with velocity constraint. The constraint C ( · ,u ) depends upon the unknown state u , which causes one of the main diﬃculties in the mathematical treatment of quasi-variational inequalities. Our aim is to show how a ﬁxed point approach can lead to an existence theorem for this implicit diﬀerential inclusion. By using Schauder’s ﬁxed point theorem combined with a recent existence and uniqueness theorem in the case where the moving set C does not depend explicitly on the state u (i.e. C := C ( t )) given in [3], we prove a new existence result of solutions of the quasi-variational sweeping process in the inﬁnite dimensional Hilbert spaces with a velocity constraint. Contrary to the classical state-dependent sweeping process, no conditions on the size of the Lipschitz constant of the moving set, with respect to the state, is required.

(Communicated by the associate editor name) Abstract. In this paper, we study a variant of the state-dependent sweeping process with velocity constraint. The constraint C(·, u) depends upon the unknown state u, which causes one of the main difficulties in the mathematical treatment of quasi-variational inequalities. Our aim is to show how a fixed point approach can lead to an existence theorem for this implicit differential inclusion. By using Schauder's fixed point theorem combined with a recent existence and uniqueness theorem in the case where the moving set C does not depend explicitly on the state u (i.e. C := C(t)) given in [3], we prove a new existence result of solutions of the quasi-variational sweeping process in the infinite dimensional Hilbert spaces with a velocity constraint. Contrary to the classical state-dependent sweeping process, no conditions on the size of the Lipschitz constant of the moving set, with respect to the state, is required.
1. Introduction. In [11], J.J Moreau introduced and studied thoroughly the following differential inclusioṅ u(t) ∈ −N C(t) (u(t)) a.e. on [0, T ], u(0) = u 0 ∈ C(0), where C : [0, T ] ⇒ H is a set-valued mapping defined from [0, T ] (T > 0) to a Hilbert space H and takes convex and closed values, and N C(t) (u(t)) denotes the outward normal cone, in the sense of convex analysis, to the set C(t) at u(t).
Translating the dynamic (1) to a mechanical language, we obtain the following interpretation: -if the position u(t) of a material point lies in the interior of the moving set C(t), thenu(t) = 0, which means that the material point remains at rest; -when the boundary of the moving set C(t) catches up the material point, then this latter is pushed in an inward normal direction by the boundary of C(t) to stay inside the moving set and satisfies the constraint. This mechanical visualization leads J.J. Moreau to call this problem the sweeping process: the material point is swept by the moving set. The original concrete motivations of the sweeping process by J.J. Moreau are the study of quasi-static evolution in elastoplasticity [11]. Nowadays, many models in nonsmooth mechanics, mathematical economics, simulation of switched electrical circuits and crowd motion can be formulated by the sweeping process approach (for more details, we refer to [1,2,3,4,7,8,10,12] and references therein). Problem (1) is equivalent to the following evolution variational inequality with a moving constraint subset: As a special case, the sweeping process includes the following time-dependent variational inequality: Several extensions of the sweeping process in diverse ways have been studied in the literature (see for instance [2,3,9] and references therein). In this paper, we are interested in a new variant of the state-dependent sweeping process with velocity constraint with the following assumptions: (H 1 ) A 0 : H −→ H is a bounded symmetric linear operator which is coercive, that is, for all x ∈ H, A 0 x, x ≥ α x 2 , for some real constant α > 0; and A 1 : H → H is a bounded symmetric linear operator which is positive semi-definite in the standard sense, that is, x) ⊂ H is a nonempty closed and convex set, C(t, x) ⊂ K ⊂ rB H for some compact set K in H and some r > 0; (H 3 ) The set C(t, x) varies in a Lipschitz continuous way with respect to the Hausdorff metric d H , that is, there exist a constants L 1 , L 2 ≥ 0 such that The nonsmooth dynamic (2) can be rewritten as the the following evolution quasi-variational inequality:    Find u : [0, T ] −→ H, with u(0) = u 0 ∈ H, such thatu(t) ∈ C(t, u(t)) a.e. t ∈ [0, T ] and a 1 (u(t), v −u(t)) + a 0 (u(t), v −u(t)) ≥ f (t), v −u(t) , ∀v ∈ C(t, u(t)).

EVOLUTION QUASI-VARIATIONAL INEQUALITIES 3
Here a 0 (·, ·) and a 1 (·, ·) are real bilinear, bounded and symmetric forms associated to the operators A 1 and A 0 defined in (H 1 ). The above type of evolution quasi-variational inequalities are widely used in applied mathematics, unilateral mechanics and various fields of sciences and engineering such as for instance traffic networks, energy market, transportation, elastoplasticity etc . . . (see e.g. [6]). In [9], the authors proved the existence of Lipschitz continuous solutions to the classical state dependent sweeping process where the moving set C(t, u) is supposed to move in a Lipschitz continuous way with the constants 0 ≤ L 1 and 0 ≤ L 2 < 1. In the case L 2 > 1, the differential inclusion (4) may have no absolutely continuous solutions. In fact, a counter-example is given in [9] (see Example 3.1 in [9]) . Now, unlike problem (4), the following simple example shows that the existence of solutions of problem (2) can be expected even for L 2 > 1.
Consequently, u is a solution of (2) with the initial value u(0) = 0.
The moving set C(u) of Example 1.
The aim of the present paper is to show how a fixed point approach can lead to the existence theorem for the implicit differential inclusion described by (2). By using Schauder's fixed point theorem and the existence and uniqueness theorem for the problem (2) with time-dependent C (i.e. C := C(t)) proved in [3], we give a new proof of the quasi-variational sweeping process (2) in the infinite dimensional Hilbert spaces. The paper is organized as follows. In section 2 we introduce some notations and state some preliminary results needed throughout the paper. In section 3, we present an existence theorem related to problem (2). A uniqueness result for the particular case of a shifted moving set is discussed in section 4. Section 5 is devoted to some concluding remarks and open questions.
2. Notation and preliminaries. Let H be a real Hilbert space with an inner product ·, · . For any x ∈ H and r ≥ 0, the closed ball centered at x with radius r will be denoted by B(x, r). For x = 0 and r = 1, we set B instead of B(0, 1). For a closed convex subset C of H the set x ∈ C, denotes the normal cone to C at x. Let d H (C 1 , C 2 ) denotes the Hausdorff distance between two subsets C 1 and C 2 of H defined as follows Let us recall the following classical Schauder's fixed point lemmas that will be used in the proof of our main result, for a reference we refer e.g. to [13].
Lemma 2.1. Let K be a closed convex subset of the Banach space X. Suppose f : K → K a compact map (i.e. bounded sets in K are mapped into relatively compact sets). Then, f has a fixed point in K.
We recall the following existence and uniqueness theorem proved in [3]. Then, for any initial point u 0 ∈ H, the evolution variational inequality with velocity constraint • Step 1.Construction of approximants u n i : for n ∈ N, let 0 = t n 0 < t n 1 < . . . < t n i < . . . < t n n = T, be a finite partition of the interval [0, T ]. We denote by η n i = t n i+1 − t n i = T n := η n , i = 0, 1, . . . , n the length of the time step. f (t n i ) f n i , I n i :=]t n i , t n i+1 ]. Algorithm. Fix n ≥ 2 and set η n = T n , u n 0 = u 0 and f n 0 = f (t n 0 ). For i = 0, 1, . . . , n − 1 -Compute f n i+1 = f (t n i+1 ) -Solve for z n i+1 the following variational inequalities where rht stands for the right hand term. -Update u n i+1 = u n i + η n z n i+1 . • Step 2.Construction of a sequence of functions (u n (.)) via an interpolation method: • Step 3.Convergence of (u n (.)): Step 4.We show that u(.) is a solution of the initial problem.
• Step 5.Uniqueness of the solution: monotonicity and coercivity arguments.
Remark 2. The coercivity assumption on A 0 in Lemma 2.2 was relaxed in [5] by assuming that A 0 is a semi-coercive operator. An existence result is proved in [5], however the uniqueness of solution, which is an important issue for applying Schauder's fixed point Theorem, is not guaranteed and the technique used in this paper can not be adapted.
3. Main result. The following theorem establishes our main existence result in this paper.
Proof. Let us apply Schauder's fixed point argument (Lemma 2.1) to deduce the existence result.
It is easy to check that K is a closed convex subset of C([0, T ], H).

SAMIR ADLY AND TAHAR HADDAD
For each t ∈ [0, T ] and each v ∈ K let us set Obviously, for each t ∈ [0, T ], the set valued map D(·) is convex and closed.
Let now x ∈ H and t, s ∈ [0, T ] such that s < t.
Therefore, for x ∈ H and t, s ∈ [0, T ] one has and w(t) ≤ u 0 + rT.
Step 3. Let us define the mapping Ψ : K → K by, Ψ(v) = w such that w is the unique solution for the problem (P v ). It's clear that Ψ(K) ⊂ K. To apply the classical Schauder's fixed point theorem (see Lemma 2.1), it is sufficient to show that Ψ is continuous with respect to · C([0,T ],H) and that Ψ(K) is relatively compact in the Banach space C([0, T ], H]). We prove the latter by the classical Ascoli-Arzela theorem. Indeed, let v ∈ K and t, s ∈ [0, T ] This implies that {Ψ(v)} v∈K is equi-Lipschitz and therefore equicontinuous.

SAMIR ADLY AND TAHAR HADDAD
By (H 3 ), we have By adding both inequalities (10) and (12) we obtain Since A 1 is monotone, we deduce Consequently, for almost every t ∈ [0, T ], (14) implies By integrating, using the equalities w n (0) = w(0) = u 0 and the coerciveness of A 0 , we have for every t ∈ [0, T ] The last inequality gives Therefore, Ψ is continuous. This completes the proof. 4. Uniqueness in the case of a shifted moving set. Due to the dependence of the constraint C(·, u) upon the unknown state u, the uniqueness of a solution to problem (2) can not be expected in general. However, the uniqueness of a solution could be proved in the particular case where the operator A 1 ≡ 0 and the statedependent moving set C(·, u) has the following structure: where K(t) ⊂ K 0 is a fixed compact convex set of H for every t ∈ [0, T ] and m : H → H is a single-valued and L 2 -Lipschitz operator such that m(u) ∈ K 1 for some fixed compact convex set of H and all u ∈ H. Let us consider the following problem with K(t) and m(u) defined in ( H 2 ) and ( H 3 ). It is clear that in this case the Lipschitz condition (3) is satisfied. Indeed, by ( Hence, d H (C(t, u), C(s, v)) ≤ L 1 |t − s| + L 2 u − v , for all t, s ∈ [0, T ], u, v ∈ H. All assumptions of Theorem 3.1 are satisfied. Hence, problem (16) has at least one solution. Suppose that (u 1 , u 2 ) are two solutions for (16) such that u 1 (0) = u 2 (0) = u 0 . Then, for almost every t ∈ [0, T ], we have and Using the fact thatu 2 (t) ∈ C(t, u 2 (t)) = K(t) + m(u 2 (t)), we replace in (17) x :=u 2 (t) − m(u 2 (t)) + m(u 1 (t)) ∈ C(t, u 1 (t)), we obtain for a.e. t ∈ [0, T ], On the other hand, asu 1 (t) ∈ C(t, u 1 (t)) = K(t) + m(u 1 (t)), by taking in (18) x :=u 1 (t) − m(u 1 (t)) + m(u 2 (t)) ∈ C(t, u 2 (t)), we get for a.e. t ∈ [0, T ], Adding the inequalities (19) and (20), we obtain Since m(·) is L 2 Lipschitz, we deduce Consequently, for almost every t ∈ [0, T ], By integrating, using the equalities u 1 (0) = u 2 (0) = u 0 and the coerciveness of A 0 , we have Gronwall's Lemma permits us to deduce that u 1 (t) − u 2 (t) 2 = 0. Therefore u 1 = u 2 . Which completes the proof.
5. Concluding remarks. In this note, using a fixed point argument, we give a new existence result for a variant of the state-dependent sweeping process with velocity constraint. The fact that the constraint C(·, u) depends upon the unknown state u, makes the study of the evolution quasi-variational inequalities more complicate. The smallness condition of the Lipschitz constant of the moving set with respect to the state, used in the classical state-dependent sweeping process (4), is not required here. We handled the situation using the Schauder fixed point Theorem which requires a compactness assumption of the moving set. Many issues remain open and need further investigations. It includes the relaxation of the compactness assumption or the use of an other technique. The extension of this result beyond the convex setting could be also of great interest. In fact, the class of prox-regular sets play an important role in optimization and control theory with concrete applications and constitutes an alternative assumption to the convexity used in this paper. This out of the scope of the current paper and will be the subject of a future research project.