Charles Castaing
email: charles.castaing@gmail.com

Christiane Godet-Thobie
email: christiane.godet-thobie@univ-brest.fr

Manuel D P Monteiro Marques

Lionel Thibault
email: lionel.thibault@umontpellier.fr

Well-Posedness of evolution variational inequalities with applications

We are concerned in the present work with the existence and uniqueness of absolutely continuous solutions to a class of evolution problems governed by time-dependent subdifferential operators of the form

with various applications.

Introduction

In this work we are concerned with the existence and uniqueness of absolutely continuous solution to an evolution inclusion in a separable Hilbert where A(t) is a time dependent maximal monotone operator, B(t, u) defined for (t, u) ∈ [0, T ]×H is Lipschitz with respect to u. Then the existence and uniqueness of absolutely continuous solution to (1.2) is known in some particular cases in the literature, see e.g [START_REF] Azzam-Laouir | Perturbed evolution problems with continuous bounded variation in time and applications[END_REF][START_REF] Castaing | Evolution problems involving time dependent subdifferential operators[END_REF] and Barbu and Rascanu in [START_REF] Barbu | Parabolic variational inequalities with singular inputs[END_REF] dealing with existence of generalized solutions for parabolic variational inequalities with singular inputs and operators of the form

f (t) + dM dt (t) ∈ du dt (t) + Au(t) + ∂ϕ(u(t)),
where A is a linear coercive operator and ϕ is a lower semicontinuous convex function. There is an increasing activity around problem (1.2) since it contains several new applications such as sweeping process, relaxed problem and Skorohod problem etc. In this framework, problem (1.1) constitutes a new variational evolution inequality with the velocity inside the subdifferential in constrast to problem (1.2). Likewise problem (1.2), the study of (1.1) leads to several applications in a new setting such as the sweeping process, Skorohod problem, second order evolution and fractional differential equation [START_REF] Castaing | Evolution problems involving time dependent subdifferential operators[END_REF]. Although (1.1) deals with the deterministic case, it is a step towards the Skorohod problem in the stochastic setting, see the recent articles by Castaing-Raynaud de Fitte [START_REF] Castaing | A Skorohod problem governed by a closed convex moving set[END_REF][START_REF] Castaing | Sweeping Process Perturbed by Rough Signal[END_REF], Rascanu [START_REF] Rascanu | Deterministic and Stochastic Differential Equations in Hilbert Spaces Involving Multivalued Maximal Monotone Operators[END_REF], and L.Maticiuc, A. Rascanu, L. Slominski and M.Topolewski [START_REF] Maticiuc | Cadlag Skorohod problem driven by Maximal monotone operator[END_REF] for references on this stochastic subject. Let us mention the current situation of problem (1.1) in the literature. In [START_REF] Castaing | Evolution problems involving time dependent subdifferential operators[END_REF] it was dealt with the existence of absolutely continuous solutions to variational evolution inequalities in separable Hilbert space H of the forms f (t) -Au(t) ∈ ∂ϕ(t, du dt (t)) (1.3)

f (t) -Au(t) ∈ N C(t) ( du dt (t)), (1.4)
where f : [0, T ] → H is a continuous mapping, A : H → H is a linear continuous coercive symmetric operator, ϕ : [0, T ] × H →] -∞, +∞] is a normal convex integrand, and N C(t) (x) denotes the normal cone to a closed convex moving set C(t) ⊂ H. Some related variants of problem (1.4) dealing with two positive operators A and B are given in a series of papers by Adly et al [START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF][START_REF] Adly | An implicit sweeping process Approach to Quasistatic Evolution Variational inequalities[END_REF][START_REF] Adly | State -dependent Implicit Sweeping process in the frame work of Quasi static Evolutions Quasi-Variational Inequalities[END_REF]. We note that there is a new variant of problem (1.1) in a recent work by Bacho, Emmrich and Mielke [START_REF] Bacho | An existence result and evolutionary ?-convergence for perturbed gradient systems[END_REF] dealing with the following inclusion where B is a continuous mapping. In Mielke's paper, in order to solve the problem, it is proposed an algorithm due to De Giorgi combined with regularization of subdifferentials. Consult also a recent article by Migorski, Sofonea and Zeng [START_REF] Migorski | Well-Posedness of historoy dependent sweeping process[END_REF] dealing with the inclusion of the form

B(
- du dt (t) ∈ N C(t) (A du dt (t) + Bu(t))
where B : H → H is Lipschitz continuous. Note that in [START_REF] Migorski | Well-Posedness of historoy dependent sweeping process[END_REF] B is not assumed to be a positive operator in contrast to the results obtained by Adly et al [START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF][START_REF] Adly | An implicit sweeping process Approach to Quasistatic Evolution Variational inequalities[END_REF][START_REF] Adly | State -dependent Implicit Sweeping process in the frame work of Quasi static Evolutions Quasi-Variational Inequalities[END_REF].

Our aim in the present paper is to develop several variants of problem (1.1) along with diverse applications via some related variational limits.

Preliminaries

Throughout the paper, H is a real separable Hilbert space and E is a separable Banach space with B E as its closed unit ball centered at zero.

If I is an interval of R the spaces L 1 (I, E, dt) (L 1 E (I), for short) and L ∞ (I, E, dt) (L ∞ E (I), for short) denote the usual spaces with respect to the Lebesgue measure endowed with their canonical norms • 1 and • ∞ respectively. Similarly, C(I, E) or C E (I) will be the space of continuous mappings from I into E. For any subset Q ⊂ I the function 1 Q is defined by 1 Q (t) = 1 if t ∈ Q and 1 Q (t) = 0 otherwise. Given a convex function ϕ : H → R ∪ {-∞, +∞}, its effective domain dom ϕ is the set dom ϕ := {x ∈ H : ϕ(x) < +∞}, so the function ϕ is proper whenever dom ϕ = ∅ and ϕ does not take the value -∞. At any x ∈ H where ϕ is finite its subdifferential ∂ϕ(x) is defined by ∂ϕ(x) := {ζ ∈ H : ζ, y -x -ϕ(x) ≤ ϕ(y), ∀y ∈ H}.

If f (x) is not finite ∂ϕ(x) = ∅. Considering the Legendre-Fenchel conjugate ϕ * : H → R ∪ {-∞, +∞} with ϕ * (y) := sup{ y, x -f (x) : x ∈ H}, it is known that, when ϕ is a proper lower semicontinuous convex function, ϕ * is also proper lower semicontinuous and y ∈ ∂ϕ(x) ⇔ y, x = ϕ(x) + ϕ * (y).

Given a nonempty closed convex subset S of H, its indicator function δ(•, S) : H → R ∪ {+∞} is defined by δ(x, S) = 0 if x ∈ S and δ(x, S) = +∞ if x ∈ H \ S. Clearly, δ(•, S) is a proper lower semicontinuous convex function. Its subdifferential is called the normal cone of S, and obviously and it is known that (see, e.g., [START_REF] Castaing | Convex Analysis and Measurable Multifunctions[END_REF][START_REF] Thibault | Unilateral Variational Analysis[END_REF])

N S (x) = {ζ ∈ H : ζ, y -x ≤ 0, ∀y ∈ S} if x ∈ S ( 2
haus (S, S ) = sup u ≤1 |δ * (u, S) -δ * (u, S )|, (2.2)
which entails |δ * (y, S) -δ * (y, S )| ≤ y haus(S, S ) for all y ∈ H.

(2.3)

We recall and summarize two useful results, see for example [6, Corollary 2.9, Corollary 2.10]. Remind that a linear operator A : H → H is coercive if there is a real ω > 0 such that Ax, x ≥ ω x 2 for all x ∈ H.

(2.4) Proposition 2.1. Let A : H → H be a linear continuous and coercive operator. (a) If ϕ : H → [0, ∞] is a proper lower semicontinuous convex function, then for each f ∈ H the problem f ∈ Ay + ∂ϕ(y) admits a unique solution y. (b) If K is a closed convex subset in H, then for each f ∈ H the problem f ∈ Ay + N K (y) admits a unique solution y.

3 Preparatory variational limit theorems Proposition 3.1. Let H be a separable Hilbert space and I = [0, T ]. Let C : I ⇒ H be a closed convex valued scalarly measurable multimapping for which there is some real r > 0 such that C(t) ⊂ rB H for all t ∈ I. Let A be a continuous symmetric coercive linear operator on H and let B be a continuous compact linear operator on H. Let (θ n ) n∈N be a sequence of measurable functions from I into I such that for each t ∈ I one has θ n (t) → t and haus C(θ n (t)), C(t) → 0 as n → ∞.

Let (f n , f ) n∈N be a bounded sequence in L ∞ H (I) with f n (t) ≤ β, f (t) ≤ β (β > 0) for all n ∈ N such that (f n (t)) n converges to f (t) for each t ∈ I. Let (v n , v) n∈N be a bounded sequence in L ∞ H (I) with v n (t) ≤ γ, v(t) ≤ γ (γ > 0) for all n ∈ N such that (v n (t)) n converges weakly to v(t) for each t ∈ I. Let (ζ n ) n∈N be an integrable sequence in L Proof. We first verify that ζ(t) ∈ C(t) a.e.t ∈ I. Indeed, for every measurable set Z ⊂ I and for any x ∈ H, the function 1 Z x ∈ L ∞ H (I). By the inequality x, ζ n (t) ≤ δ * (x, C(θ n (t))) integrating on Z gives

I 1 Z x, ζ n (t), dt = Z x, ζ n (t) dt ≤ Z δ * (x, C(θ n (t)))dt.
Passing to the upper limit in this inequality we obtain This being true for any Lebesgue measurable set Z ⊂ I we deduce that for every x ∈ H x, u(t) ≤ δ * (x, C(t)) a.e. t ∈ I.

By the separability of H and the weak compactness and convexity of C(t) (see, e.g., ), we get the desired inclusion ζ(t) ∈ C(t) a.e. t ∈ I For each t ∈ I keeping in mind that v n (t) → v(t) weakly in H and B is a continuous compact linear operator, we see that Bv n (t) → Bv(t) strongly in H, so that Bv n (.) → Bv(.) weakly in L 1 H (I). Indeed, let any h ∈ L ∞ H (I). Then we have

T 0 h(t), Bv n (t) dt - T 0 h(t), Bv(t) dt ≤ T 0 | h(t), Bv n (t) -Bv(t) |dt ≤ |h| ∞ T 0 Bv n (t) -Bv(t) dt.
As T 0 Bv n (δ n (t))-Bv(t) dt → 0 as n → ∞ (by Lebesgue dominated convergence theorem), our assertion follows. Similarly as A is symmetric, we note that Aζ n → Aζ weakly in L 1 H (I). As a main consequence f n +Bv n -Aζ n → f +Bu-Aζ weakly in L 1 H (I). Then given any Lebesgue measurable subset Z ⊂ I we may apply the lower semicontinuity of convex integral functional ( [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF], Theorem 8. 1.16) to deduce that Z δ * (f (t)+Bv(t)-Aζ(t), C(t))dt ≤ lim inf n Z δ * (f n (t)+Bv n (t)-Aζ n (t), C(t))dt.

(3.1) This need a careful look. Indeed, we note that (t, x) → δ * (x, C(t)) is a normal lower semicontinuous convex integrand defined on [0, T ] × H and δ * (f n (t) + Bv n (t) -Aζ n (t), C(t)) is measurable and bounded by an integrable function independent of n ∈ N since taking some real constant α with f n (t) + Bv n (t) -Aζ n (t) ≤ α for all n ∈ N and t ∈ I we have

|δ * (f n (t) + Bv n (t) -Aζ n (t), C(t))| ≤ r f n (t) + Bv n (t) -Aζ n (t) ≤ αr.
Then writing An easy adaptation of the arguments in the above proposition furnishes the following variant. Proposition 3.2. Let H be a separable Hilbert space and I = [0, T ]. Let C : I ⇒ H be a closed convex valued scalarly measurable multimapping for which there is some real r > 0 such that C(t) ⊂ rB H for all t ∈ I. Let A be a continuous symmetric linear operator coercive on H and let B : H → H be a Lipschitz mapping. Let (θ n ) n∈N be a sequence of measurable functions from I into I such that for each t ∈ I one has θ n (t) → t and haus C(θ n (t)), C(t) → 0 as n → ∞.

δ * (f n (t) + Bv n (t) -Aζ n (t), C(t)) -δ * (f n (t) + Bv n (t) -Aζ n (t), C(θ n (t))) ≤ f n (t) + Bv n (t) -Aζ n (t)) haus C(t), C(θ n (t)) ≤ α haus C(t
Let (f n , f ) n∈N be a bounded sequence in L ∞ H (I) with f n (t) ≤ β, f (t) ≤ β (β > 0) for all n ∈ N such that (f n (t)) n converges to f (t) for each t ∈ I.

Let (v n , v) n∈N be a bounded sequence in L ∞ H (I) with v n (t) ≤ γ, v(t) ≤ γ (γ > 0) for all n ∈ N such that (v n (t)) n converges to v(t) for each t ∈ I. Let (ζ n ) n∈N be an integrable sequence in L 1 H (I) such that ζ n (t) ∈ C(θ n (t)) for all t ∈ I and such that (ζ n ) n∈N σ(L 1 H , L ∞ H ) converges in L 1 H (I) to ζ. Assume that f n (t) + Bv n (t) -A ζ n (t) ∈ N C(θn(t)) (ζ n (t)) for all n ∈ N, t ∈ I.
Then for a.e. t ∈ I one has

ζ(t) ∈ C(t) and f (t) + Bv(t) -A ζ(t) ∈ N C(t) (ζ(t)).
Another variant of Proposition 3.1 is available with the gradient ∇g of a convex function g in place of the mapping B. Proposition 3.3. Let H be a separable Hilbert space and I = [0, T ]. Let C : I ⇒ H be a closed convex valued scalarly measurable multimapping for which there is some real r > 0 such that C(t) ⊂ rB H for all t ∈ I. Let A be a continuous symmetric linear coercive operator on H and let g : H → R be a convex function Lipschitz on bounded sets and continuously differentiable on H. Let (θ n ) n∈N be a sequence of measurable functions from I into I such that for each t ∈ I one has θ n (t) → t and haus C(θ n (t)), C(t) → 0 as n → ∞.

Let (f n , f ) n∈N be a bounded sequence in L ∞ H (I) with f n (t) ≤ β, f (t) ≤ β (β > 0) for all n ∈ N such that (f n (t)) n converges to f (t) for each t ∈ I. Let (u n , u) n∈N be a sequence of absolutely continuous mappings

u n (t) = u 0 + t 0 un (s)ds, un (t) ∈ C(t) u(t) = u 0 + t 0 u(s)ds, u(t) ∈ C(t)
such that (u n ) n converges to u uniformly on I and ( un ) n conveges to u weakly in

L 1 H (I). Assume that for every n ∈ N f n (t) + ∇g(u n (t)) -A un (t) ∈ N C(θn(t)) ( un (t)) a.e. t ∈ I.
Then for a.e. t ∈ I one has

u(t) ∈ C(t) and f (t) + ∇g(u(t)) -A u(t) ∈ N C(t) ( u(t)).
Proof. First, we justify that u(t) ∈ C(t) a.e. t ∈ I. We proceed as in the proof of Proposition 3.1. Take any measurable Lebesgue set Z ⊂ I and any x ∈ H. The function

1 Z x ∈ L ∞ H (I). Writing x, un (t) ≤ δ * (x, C(θ n (t))),
we see that

I 1 Z x, un (t), dt = Z x, un (t) dt ≤ Z δ * (x, C(θ n (t)))dt.
Passing to the upper limit we obtain

Z 1 Z x, u(t) dt ≤ lim sup n Z δ * (x, C(θ n (t)))dt ≤ Z lim sup n δ * (x, C(θ n (t))) ≤ Z δ * (x, C(t))dt.
This being true for any Lebesgue measurable set Z ⊂ I we deduce that for every

x ∈ H x, u(t) ≤ δ * (x, C(t)) t exta.e. t ∈ I.
By the separability of H and the weak compactness and convexity of C(t), we get the desired inclusion u(t) ∈ C(t) a.e. t ∈ I. Now let γ > 0 be a Lipschitz constant of g on ( u 0 + r T )B H for some r > r. Take any Lebesgue measurable set Z ⊂ I. Since (f n ) n is uniformly bounded and pointwise converges to f , we have

lim n Z f n (t), un (t) dt = Z f (t), u(t) dt. (3.6)
Also by integrating on Z (we are ensured that the functions given are measurable) the inequality

δ * f n (t) + ∇g(u n (t))-A un (t), C(θ n (t)) + A un (t)-∇g(u n (t))-f n (t), un (t) ≤ 0 it ensues that with q n (t) := f n (t) + ∇g(u n (t)) -A un (t) Z δ * (q n (t), C(θ n (t)))dt + Z A un (t)-f n (t), un (t) dt- Z ∇g(u n (t)), un (t) dt ≤ 0. (3.7) We claim that Au n (•) → Au(•) weakly in L 1 H (I) and ∇g(u n (•)) → ∇g(u(•)) weakly in L 1 H (I), so q n (•) = f n (•) + ∇g(u n (•)) -A un (•) → q(•) := f (•) + ∇g(u(•)) -A u(•) weakly in L 1 H (I). Indeed, for any h ∈ L ∞ H (I) the weak convergence in L 1 H (I) of ( un ) n to u says that T 0 Ah(t), un (t) dt → T 0 Ah(t), u(t) dt, which means T 0 h(t), A un (t) dt → T 0 h(t), A u(t) dt.

This property for every

h ∈ L ∞ H (I) translates the weak convergence in L 1 H (I) of (A un ) n to A u. Concerning (∇g(u n (•))) n it converges strongly in L 1 H (I) to ∇g(u(•))
(by Lebesgue convergence theorem) since ∇g(u n (t)) → ∇g(u(t)) for each t ∈ I and ∇g(u n (t)) ≤ γ (by the γ-Lipschitz property of g on ( u 0 + r T )B H ). The desired weak convergence in L 1 H (I) of (q n ) n to q is justified. Let us consider now the term ∇g(u n (t), un (t) . The Lipschitz property of g on ( u 0 + rT )B H assures us that g • u n and g • u are absolutely continuous on I and at any t ∈ I where u and all u n are derivable and are derivable (by classical chain rule)

∇g(u(t)), u(t) = d dt g(u(t)) and ∇g(u n (t)), un (t) = d dt g(u n (t)).
From this we deduce that

T 0 ∇g(u(t)), u(t) dt = T 0 d dt g(u(t))dt = g(u(T )) -g(u(0)) = lim n g(u n (T )) -g(u n (0)) = lim n T 0 d dt g(u n (t))dt = lim n T 0 ∇g(u n (t)), un (t) dt. (3.8) Regarding T 0 A un (t), un (t) dt let us set ψ A (t, x) = Ax, x for all x ∈ H.
It is clear that ψ A is a positive lower semicontinuous convex normal integrand. By the lower semicontinuity of convex integral functional (see [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF]Theorem 8.1.6]) we obtain lim inf

n T 0 ψ A (t, un (t))dt ≥ T 0 ψ A (t, u(t))dt, that is, lim inf n T 0 A un (t), un (t) dt ≥ T 0 A u(t), u(t) dt.
(3.9)

The last step is concerned with T 0 δ * (q n (t), C(t))dt. Note that q n (t) ≤ β + γ + r A , hence δ * (q n (t), C(t)) ≥ -r(β + γ + r A ). Using this and the lower semicontinuous convex normal integrand (t, x) → δ * (x, C(t)) we obtain by [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF]Theorem 8.1.6]

again that lim inf n T 0 δ * (q n (t), C(t))dt ≥ T 0 δ * (g(t), C(t))dt. Since δ * q n (t), C(θ n (t) -δ * q n (t), C(t) ≤ (β+γ+r A ) haus C(θ n (t)), C(t) =: ε n (t) with T 0 ε n (t) → 0 as n → ∞, we are ensured that lim inf n T 0 δ * q n (t), C(t) dt ≥ T 0 δ * g(t), C(t) dt.
(3.10)

Putting together (3.37)-(3.39) yields

T 0 δ * (f (t)+∇g(u(t))-A u(t), C(t))dt+ T 0 -f (t)+∇g(u(t))+A u(t), u(t) dt ≤ 0.
(3.11) On the other hand, the inclusion u(t) ∈ C(t) says that for a.e.

t ∈ I δ * (f (t) + ∇g(u(t)) -A u(t), C(t)) ≥ f (t) + ∇g(u(t)) -A u(t), u(t) .
Taking this into account in (3.40) we deduce for a.e.

t ∈ I δ * f (t) + ∇g(u(t)) -A u(t), C(t) -f (t + ∇g(u(t)) -A u(t), u(t) ≤ 0
The latter inequality and the inclusion u(t) ∈ C(t) guarantees that for a.e.

t ∈ I f (t) + ∇g(u(t)) -A u(t) ∈ N C(t) ( u(t))
according to the description (2.1) of the normal cone.

Remark 3.1. Concerning the chain rule for g • u and g • u n above, one can more in Moreau-Valadier [START_REF] Moreau | A chain rule involving vector functions of bounded variations[END_REF] under weaker assumptions for the convex function g.

When θ n (t) = t ( i.e., C(θ n (t)) = C(t)) in Proposition 3.
) n∈N σ(L 1 H , L ∞ H ) converges in L 1 H (I) to ζ. Then ζ(t) ∈ C(t) for a.e. t ∈ I.
Proof. Let (e p ) p∈N be a dense sequence in H. Take any measurable set Z ⊂ I and any p ∈ N, and note that the mapping 1 Z e p ∈ L ∞ H (I). Considering the inequality (due to the inclusion

ζ n (t) ∈ C(t)) e p , ζ n (t) ≤ δ * (e p , C(t))
and integrating on Z ensure that

I 1 Z e p , ζ n (t), dt = Z e p , ζ n (t) dt ≤ Z δ * (e p , C(t))dt.
Passing to the limit in the latter inequality assures us that

Z 1 Z e p , ζ(t) dt ≤ Z δ * (e p , C(t))dt.
(f n , f ) n∈N be a bounded sequence in L ∞ H (I) with f n (t) ≤ β, f (t) ≤ β (β > 0) for all n ∈ N such that (f n (t)) n converges to f (t) for each t ∈ I. Let (v n , v) n∈N be a bounded sequence in L ∞ H (I) with v n (t) ≤ γ, v(t) ≤ γ (γ > 0) for all n ∈ N such that (v n (t)) n converges weakly to v(t) for each t ∈ I. Let (ζ n ) n∈N be an equi-integrable sequence in L 1 H (I) such that ζ n (t) ∈ C(t) for all t ∈ I and n ∈ N and such that (ζ n ) n∈N σ(L 1 H , L ∞ H ) converges in L 1 H (I) to ζ. Assume that f n (t) + Bv n (t) -A ζ n (t) ∈ N C(t) (ζ n (t)) for all n ∈ N, a.e. t ∈ I.
Then for a.e. t ∈ I one has

ζ(t) ∈ C(t) and f (t) + Bv(t) -A ζ(t) ∈ N C(t) (ζ(t)).
Proof. First, Lemma 3.2 justifies that ζ(t) ∈ C(t) for a.e. t ∈ I.

For each t ∈ I keeping in mind that v n (t) → v(t) weakly in H and B is a continuous compact linear operator, we see that Bv n (t) → Bv(t) strongly in H. The Lebesgue dominated convergence theorem says that

T 0 Bv n (t)) -Bv(t) dt → 0, i.e., Bv n (•) → Bv(•) strongly in L 1 H (I). We can also see that Aζ n → Aζ weakly in L 1 H (I) since for any h ∈ L ∞ H (I) T 0 h(t), Aζ n (t) dt = T 0 Ah(t), ζ n (t) dt → T 0 Ah(t), ζ(t) dt = T 0 h(t), Aζ(t) dt.
As a main consequence

f n + Bv n -Aζ n → f + Bu -Aζ weakly in L 1 H (I).
Given any Lebesgue measurable subset Z ⊂ I we may apply the lower semicontinuity of convex integral functional in [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF]Theorem 8.1.16] to derive that

Z δ * (f (t)+Bv(t)-Aζ(t), C(t))dt ≤ lim inf n Z δ * (f n (t)+Bv n (t)-Aζ n (t), C(t))dt.
(3.12) This needs a careful look. Choose a measurable selection s of t → C(t) ∩ r(t)B H , so s ∈ L ∞ H (I). We note that (t, x) → δ * (x, C(t)) is a normal lower semicontinuous convex integrand defined on I × H and δ * (f n (t) + Bv n (t) -Aζ n (t), C(t)) is measurable and minored by ρ n (t) := s(t), f n (t) + Bv n (t) -Aζ n (t) . Further, it is easy to check that (ρ n (•)) is equi-integrable in L1 R (I). Then by [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF]Theorem 8.1.6] we deduce that lim inf

n Z δ * (f n (t) + Bv n (t) -Aζ n (t), C(t))dt ≥ Z δ * (f (t) + Bv(t) -Aζ(t), C(t))dt, which confirms (3.32). Let us set ψ A (x) = Ax, x if x ∈ C(t) and ψ A (x) = +∞ if x / ∈ C(t).
Then it is clear that ψ A is a positive lower semicontinuous convex integrand. Apply again the lower semicontinuity of the convex integral functional ( [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF], Theorem 8.1.6) associated with the positive normal convex integrand ψ A we obtain On the other hand, the inclusion

lim inf n Z ψ A (ζ n (t)) dt ≥ Z ψ A (ζ(t)) dt, that is, lim inf n Z Aζ n (t), ζ n (t))dt ≥ Z Aζ(t), ζ(t)dt. ( 3
f n (t) = Bv n (t) -Aζ n (t) ∈ N C(t) (ζ n (t)) for a.e. t ∈ I ensures that δ * (f n (t) + Bv n (t) -Aζ n (t), C(t)) -f n (t) + Bv n (t) -Aζ n (t), ζ n (t) ≤ 0.
Integrating this inequality on I gives

T 0 δ * (f n (t) + Bv n (t) -Aζ n (t), C(t))dt + T 0 Aζ n (t), ζ n (t) dt + T 0 -f n (t) -Bv n (t), ζ n (t) dt ≤ 0.
Passing to the limit inferior as n → ∞ and using (3.32) we obtain

T 0 δ * (f (t) + Bv(t) -Aζ(t), C(t)) + Aζ(t) -Bv(t) -f (t), ζ(t) dt ≤ 0.
This and the inclusion ζ(t) ∈ C(t) a.e. t ∈ I allow us to conclude that

f (t) + Bv(t) -Aζ(t)) ∈ N C(t) (ζ(t)) a.e.
Let (f n , f ) n∈N be a bounded sequence in L ∞ H (I) with f n (t) ≤ β, f (t) ≤ β (β > 0) for all n ∈ N such that (f n (t)) n converges to f (t) for each t ∈ I.
Let (u n , u) n∈N be a sequence of absolutely continuous mappings From this we deduce that

u n (t) = u 0 + t 0 un (s)ds, un (t) ∈ C(t) u(t) = u 0 + t 0 u(s)ds, u(t) ∈ C(t) such that (u n ) n converges
T 0 ∇g(u(t)), u(t) dt = T 0 d dt g(u(t))dt = g(u(T )) -g(u(0)) = lim n g(u n (T )) -g(u n (0)) = lim n T 0 d dt g(u n (t))dt = lim n T 0 ∇g(u n (t)),
(•)) → ∇g(u(•)) weakly in L 1 H (I), so q n (•) = f n (•) + ∇g(u n (•)) -A un (•) → q(•) := f (•) + ∇g(u(•)) -A u(•)
weakly in L 1 H (I). Indeed, for any h ∈ L ∞ H (I) the weak convergence in L 1 H (I) of ( un ) n to u says that

T 0 Ah(t), un (t) dt → T 0 Ah(t), u(t) dt, which means T 0 h(t), A un (t) dt → T 0 h(t), A u(t) dt.
This property for every h ∈ L ∞ H (I) translates the weak convergence in L 1 H (I) of (A un ) n to A u. Concerning (∇g(u n (•))) n it converges strongly in L 1 H (I) to ∇g(u(•)) (by Lebesgue convergence theorem) since ∇g(u n (t)) → ∇g(u(t)) for each t ∈ I and ∇g(u n (t)) ≤ γ (by the γ-Lipschitz property of g on ( u 0 + r T )B H ). Altogether, the weak convergence in L 1 H (I) of (q n ) n to q is justified. Regarding T 0 A un (t), un (t) dt let us set ψ A (t, x) = Ax, x for all x ∈ C(t) and ψ A (t, x) = +∞ for x / ∈ C(t). It is clear that ψ A is a positive lower semicontinuous convex normal integrand. By the lower semicontinuity of convex integral functional (see [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF]Theorem 8.1.6]) we obtain lim inf

n T 0 ψ A (t, un (t))dt ≥ T 0 ψ A (t, u(t))dt, that is, lim inf n T 0 A un (t), un (t) dt ≥ T 0 A u(t), u(t) dt. (3.19)
The last step deals with

T 0 δ * (q n (t), C(t))dt. Fix a measurable selection s(•) of t → C(t) ∩ r(t)B H , set ρ n (t) := s(t), q n (t) , and observe that δ * (q n (t), C(t)) ≥ ρ n (t). It is not difficult to check that (ρ n ) n is equi-integrable in L 1 R (I)
. From this and the lower semicontinuous convex normal integrand (t, x) → δ * (x, C(t)) it follows by [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF]Theorem 8.1.6]

* (f (t) + ∇g(u(t)) -A u(t), C(t)) ≥ f (t) + ∇g(u(t)) -A u(t), u(t) .
Taking this into account in (3.40) we deduce for a.e.

t ∈ I δ * f (t) + ∇g(u(t)) -A u(t), C(t) -f (t + ∇g(u(t)) -A u(t), u(t) ≤ 0.
The latter inequality and the inclusion u(t) ∈ C(t) guarantees that for a.e.

t ∈ I f (t) + ∇g(u(t)) -A u(t) ∈ N C(t) ( u(t))
according to the description (2.1) of the normal cone.

We pass now to the situation of a subdifferential ∂ϕ(t, •) in place of the normal cone N C(t) (•). Proposition 3.6. Let H be a separable Hilbert space and I = [0, T ] Let ϕ : [0, T ] × H →] -∞, +∞] be a normal lower semicontinuous convex integrand for which there exists a convex weakly compact set Γ such that: (i) for all t ∈ I, domϕ t := Γ; (ii {ϕ(., u(.)), u ∈ S 1 Γ } is uniformly integrable; (iii) ϕ(t, x) ≤ ϕ(τ, x)+|v(t)-v(τ )| for all t, τ ∈ [0, T ], x ∈ Γ where v : [0, T ] → R + is a positive nondecreasing continuous function with v(0) = 0. Let A : H → H be a continuous symmetric linear coerciveoperator and let g : H → R be a convex function Lipschitz on bounded sets and continuously differentiable on H. Let (θ n ) n∈N be a sequence of measurable functions from I into I such that for each t ∈ I one has θ n (t) → t.

Let (f n , f ) n∈N be a bounded sequence in L ∞ H (I) with f n (t) ≤ β, f (t) ≤ β (β > 0) for all n ∈ N such that (f n (t)) n converges to f (t) for each t ∈ I. Let (v n , v) n∈N be a bounded sequence in L ∞ H (I) with v n (t) ≤ γ, v(t) ≤ γ (γ > 0) for all n ∈ N such that (v n (t)
) n converges strongly to v(t) for each t ∈ I. Let (u n , u) n∈N be a sequence of absolutely continuous mappings

u n (t) = u 0 + t 0 un (s)ds, un (t) ∈ Γ u(t) = u 0 + t 0 u(s)
q n (t) := f n (t) + ∇g(v n (t)) -A un (t) ∈ ∂ϕ(θ n (t), un (t))
so that by the normality of ϕ * , the function t → ϕ * (θ n (t), q n (t)) is measurable and integrable, the measurability and integrability of ∇g(v n (t)) is already ensured as above. We also note that q n , un is measurable and integrable and the sequence ( q n , un ) is uniformly integrable. Further, by (3.42) and condition (ii) we have

-ϕ(t, un (t)) + un (t), q n (t) ≤ ϕ * (t, q n (t)) ≤ ϕ * (θ n (t), q n (t)) + |v(t) -v(θ n (t))|, (3.23
) so that t → -ϕ(t, un (t)) + un (t), q n (t) is uniformly integrable thanks to (ii). We note that, for h n (t) := f n (t) + ∇g(v n (t)), (h n ) n is uniformly bounded and pointwise converges to h given by h This fact has been justified in the proof of Proposition 3.4. As A is symmetric, we also show that A un (•) → A u(•) weakly in L 1 H (I). As consequence q n = f n + ∇g(v n (.)) -A un (.) → q := f + ∇g(v(.)) -A u weakly in L 1 H (I). Further, let us set

(t) = f (t) + ∇g(v(t) in H. Hence h n → h strongly in L 1 H (I).
ψ A (x) = Ax, x if x ∈ Γ and ψ(x) = +∞ if x / ∈ Γ.
Then it is clear ψ is a positive lower semicontinuous convex integrand. Apply again the lower semicontinuity of the integral convex functional ( [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF], Theorem 8.1.6) associated with the positive normal lower semi continuous convex integrand ψ A we obtain

lim inf n Z ψ A (u n (t))dt ≥ Z ψ A (u(t)) d, t that is, lim inf n Z A un (t), un (t)) dt ≥ Z A u(t), u(t) dt. (3.25)
Now, we deduce using (3.43) and the lower semicontinuity of integral convex functional (see [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF]Theorem 8.1.6]) applied to ϕ * ,

Z ϕ * (t, q(t)) dt ≤ lim inf n Z ϕ * (t, q n (t)) dt ≤ lim inf n Z
ϕ * (θ n (t), q n (t))dt (3.26) This fact is justified because ϕ * (t, q n (t)) ≥ -ϕ(t, un (t)) + un (t), q n (t) , and the sequence (-ϕ(t, un (t)) + un (t), q n (t) ) is uniformly integrable. By

ϕ
ϕ(t, u(t)) dt + Z ϕ * (t, q(t)) dt ≤ Z u(t), q(t) dt.
By the measurability of the non negative function t → ϕ(t, u(t)) + ϕ * (t, q(t))u(t), q(t) , we deduce that for almost every t ∈ I ϕ(t, u(t)) + ϕ * (t, q(t)) -u(t), q(t) ≤ 0 along with u(t) ∈ Γ. So, it follows for almost every t ∈ I that ϕ(t, u(t)) + ϕ * (t, q(t)) = u(t), q(t) , or equivalently

q(t) = f (t) + ∇g(u(t)) -A u(t) ∈ ∂ϕ(t, u(t)).
The proof is finished.

Remark Proposition 3.6 hold true if we replace the gradient ∇g by a linear continuous operator B : H → H.

We finish this section with some new variants which have also some importance in further developments.

n Z δ * (-un (t), C(θ n (t))dt ≥ lim inf n Z δ * (-un (t), C(t))dt
To finish the proof we apply the lower semicontinuity of the integral convex functional ( [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF], Theorem 8.1.6) associated with the normal convex integrand

(t, x) → δ * (x, C(t)) by noting that (j) δ * (-un (t), C(t)) is minored by -un (t), h(t) with h(t) = A u(t) + ∇g(u(t)), (jj
) the minored sequence ( -un (t), h(t) ) is uniformly integrable. Then we are ensured by (3.7.4)

lim inf n Z δ * (-un (t), C(θ n (t))dt ≥ lim inf n Z δ * (-un (t), C(t))dt ≥ Z δ * (-u(t), C(t))dt
By combining (3.7.1) --(3.7.4) we get (3.7.5)

T 0 δ * (-u(t), C(t))dt + u(t), A u(t) + ∇g(u(t)) dt ≤ 0. as δ * (-u(t), C(t)) ≥ -u(t), Au(t) + ∇g(u(t))
a.e. that implies

T 0 [δ * (-u(t), C(t)) + u(t), Au(t) + ∇g(u(t)) ]dt = 0 so we conclude that δ * (-u(t), C(t)) + u(t), Au(t) + ∇g(u(t)) = 0 a.e. with A u(t) + ∇g(u(t)) ∈ C(t), just proving that -u(t) ∈ N C(t) (A u(t) + ∇g(u(t))) a.e.

Application We give an example of the existence of absolutelty continuous solution to the evolution inclusion of the form

-u(t) ∈ N C(t) (A u(t) + ∇g(u(t)))
where C is a convex compact valued continuous mapping and A is a linear continuous coercive: Ax, x ≥ γ||x|| 2 symmetric operator in H = R e .

Further variational limit involving time dependent maximal monotone operator

We introduce in the following the definition and some properties of maximal monotone operators needed in the proofs of our results, Let A : D(A) ⊂ H ⇒ H be a set-valued operator. The domain, the range and the graph of A are the following sets

D(A) = {x ∈ H : Ax = ∅}, R(A) = {y ∈ H : ∃ x ∈ D(A), y ∈ Ax} = ∪{Ax : x ∈ D(A)}, gph(A) = {(x, y) ∈ H × H : x ∈ D(A), y ∈ Ax}.
We say that A :

D(A) ⊂ H ⇒ H is monotone, if y 1 -y 2 , x 1 -x 2 ≥ 0 whenever (x i , y i ) ∈ gph(A), i = 1, 2.
It is maximal monotone, if its graph could not be contained strictly in the graph of any other monotone operator, in this case, for all λ > 0, R(I H + λA) = H, where I H stands for the identity mapping of H.

If A is a maximal monotone operator then, for every x ∈ D(A), Ax is non-empty, closed and convex. So that, the projection of the origin into Ax, A 0 (x), exists and is unique.

For λ > 0, we define the resolvent and the Yosida approximation of A respectively by, J A λ = (I H + λA) -1 and A λ = 1 λ I H -J A λ . These operators are both singlevalued and defined on the whole space H, and we have

J A λ x ∈ D(A) and A λ (x) ∈ A(J A λ x), for every x ∈ H, (3.28)
||A λ (x)|| ≤ ||A 0 x|| ∀x ∈ D(A).
(3.29)

Let A : D(A) ⊂ H ⇒ H and B : D(B) ⊂ H ⇒ H be two maximal monotone operators, then we denote by dis(A, B) the pseudo-distance between A and B defined by

dis(A, B) = sup y -y , x -x 1 + ||y|| + ||y || : (x, y) ∈ gph(A), (x , y ) ∈ gph(B) . (3.30) Clearly, dis(A, B) ∈ [0, +∞], dis(A, B) = dis(B, A) and dis(A, B) = 0 iff A = B. Lemma 3.3. Let A n (n ∈ N), A be maximal monotone operators of H such that dis(A n , A) → 0. Suppose also that x n ∈ D(A n ) with x n → x and y n ∈ A n (x n )
with y n → y weakly for some x, y ∈ H. Then x ∈ D(A) and y ∈ Ax.

Let I := [0, T .] Let t → A t : D(A t ) → 2 H be a time dependent maximal monotone operator. We say that A t : D(A t ) → 2 H is continuous in variation if there exist nondecreasing continuous function ρ : I → R + with ρ(0) = 0 such that dist(A t , A τ ) ≤ |ρ(t) -ρ(τ )| for all t, τ ∈ I.
Proposition 3.8. Let H be a separable Hilbert space and I = [0, T ]. Let C : I ⇒ H be a convex weakly compact valued multimapping for which there is some nondecreasing continuous function r :

I → R + such that haus(C(t), C(τ )) ≤ |r(t)-r(τ )| for all τ, t ∈ I. Let t → A t : D(A t ) → 2 H be
a continuous in variation time dependent maximal monotone operator: dist(A t , A τ ) ≤ |ρ(t) -ρ(τ )| for all t, τ ∈ I. Let B = ∇g where ∇g is the gradient of a convex continuous Gateaux differentiable function g : H → R + such that g(v(t)) is absolutely continuous for v : [0, T ] → H absolutely continuous. Let (θ n ) n∈N be a sequence of measurable functions from I into I such that for each t ∈ I one has θ n (t) → t Let (u n , u) n∈N be a bounded sequence of absolutely continuous mapping on H such that (u n (t)) n converges pointwise strongly to u and such that ( un

) n∈N σ(L 1 H , L ∞ H ) converges in L 1 H (I) to u. Let (v n
) be a sequence of bounded measurable mapping pointwise converging to a bounded measurable mapping v. Assume that

-un (t) ∈ N C(θn(t)) (v n (t) + ∇g(u n (t))) for all n ∈ N, a.e. t ∈ I. v n (t) ∈ A θn(t) u n (t) Then for a.e. t ∈ I one has a.e. t ∈ [0, T ] -u(t) ∈ N C(t) (v(t) + ∇g(u(t))) v(t) ∈ A t u(t) Proof. As -un (t) ∈ N C(θn(t)) (v n (t) + ∇g(u n (t))) then δ * (-un (t), C(θ n (t))) + un (t), v n (t) + ∇g(u n (t)) ≤ 0
By integrating on Z ∈ L(I) (we are ensured that the functions given are measurable) this inequality we get (3.7.1)

Z δ * (-un (t), C(θ n (t)))dt + Z un (t), v n (t) + ∇g(u n (t)) dt ≤ 0.
Now we have to consider first the term ∇g(u n (t), un (t) by using the special property of ∇g. In fact u n is absolutely continuous with derivative un and g(u n ) is absolutely continuous , so that by Moreau-Valadier [START_REF] Moreau | A chain rule involving vector functions of bounded variations[END_REF] ,

un (t), ∇g(u n (t)) = d dt g(u n (t))
From this fact, it is easy to deduce that lim inf

n T 0 ∇g(u n (t)), un (t) dt = lim inf n T 0 d dt g(u n (t)) dt (3.7.2) = lim inf n (g(u n (T ) -u n (0)) ≥ g(u(T ) -u(0)) = T 0 d dt g(u(t)) dt = T 0 ∇g(u(t)), u(t) dt Further we have (3.7.3) lim n Z un (t), v n (t) dt = Z u(t), v(t) dt.
This need a careful look. Indeed, since v n is uniformly bounded and pointwise strongly converge to v and un is uniformly integrable and un → u weakly in L 1 H (I) we ensure that v n -v → 0 uniformly on any uniformly integrable subset of L 1 H ([0, T ]), in other terms it converges to 0 with respect to the Mackey topology τ (L ∞ H (I), L 1 H (I)), as consequence (3.7.3) hold. If H = R e , here one may invoke a classical fact that on bounded subsets of L ∞ H the topology of convergence in measure coincides with the topology of uniform convergence on uniformly integrable sets, i.e. on relatively weakly compact subsets, alias the Mackey topology. This is a lemma due to Grothendieck [START_REF] Grothendieck | Espaces Vectoriels Topologiques Mat[END_REF] [Ch.5 §4 no 1 Prop. 1 and exercice]. By (ii) using the Hormander formula we have the estimation

Z δ * (-un (t), C(t))dt ≤ Z δ * (-un (t), C(θ n (t)dt + Z || un (t)|||r(θ n (t) -r(t)|dt with Z || un (t)|||r(θ n (t) -r(t)|dt → 0 using the fact that ρ n (t) := |r(θ n (t) - r(t)| → 0 and || un (t)|| is uniformly integrable.
Here the measurability of the functions given these integrals is granted; in fact it is easily seen that C is scalarly continuous. So that

lim inf n Z δ * (-un (t), C(θ n (t))dt ≥ lim inf n Z δ * (-un (t), C(t))dt
To finish the proof we apply the lower semicontinuity of the integral convex functional ( [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF], Theorem 8.1.6) associated with the normal convex integrand

(t, x) → δ * (x, C(t)) by noting that (j) δ * (-un (t), C(t)) is minored by h(t), -un (t) ; with h(t) = v(t) + ∇g(u(t)) ∈ C(t) for all t ∈ I. (jj) the minored sequence ( h(t), -un (t) ) is uniformly integrable with h(t), -un (t) ≤ δ * (-un (t), C(t)).
In fact the last inclusion is easily ensured by noting that

v n (t) + ∇g(u n (t)) ∈ C(θ n (t)) and v n (t) + ∇g(u n (t)) → v(t) + ∇g(u(t))
and the multimapping C is scalarly continuous. Then we are ensured (3.7.4)

lim inf n Z δ * (-un (t), C(θ n (t))dt ≥ lim inf n Z δ * (-un (t), C(t))dt ≥ Z δ * (-u(t), C(t))dt By combining (3.7.1) --(3.7.4) we get (3.7.5) T 0 δ * (-u(t), C(t))dt + u(t), v(t) + ∇g(u(t)) dt ≤ 0. as we have δ * (-u(t), C(t)) ≥ -u(t), v(t) + ∇g(u(t))
a.e. that implies

T 0 [δ * (-u(t), C(t)) + u(t), v(t) + ∇g(u(t)) ]dt = 0 so we conclude that δ * (-u(t), C(t)) + u(t), v(t) + ∇g(u(t)) = 0 a.e. with v(t) + ∇g(u(t)) ∈ C(t), just proving that -u(t) ∈ N C(t) (v(t) + ∇g(u(t))) a.e. It remain to check that v(t) ∈ A t u(t). Indeed, as θ n (t) -t → 0 and v n (t) ∈ A θn(t) u n (t), by Lemma 3.5, we get u(t) ∈ D(A t ) and v(t) ∈ A t u(t).
As further application we consider a class of time measurable maximal monotone perturbation t → A t : H → H satisfying the conditions.

(A 1 ) t → J λ (t, x) is L(I)-measurable for every λ > 0 and for every x ∈ H (A 2 ) |A 0 (t, x)| ≤ c(1 + ||x||) for all (t, x) ∈ I × H. (A 3 ) For λ ∈]0, 1], λ → 0 + , for fixed r > 0, for fixed t ∈ I, sup |z|≤r |A λn (t, z) - A 0 (t, z)| → 0. (A 4 ) For fixed t ∈ I, A 0 (t, .) is continuous on H.
It is important to note a useful consequence of these properties. Let (u n ) a uniformly bounded sequence of Lebesgue measurable mappings pointwise converging to measurable mapping u. Let us consider the measurable mapping v n (t) = A λn (t, u n (t)). Then (v n (t)) pointwise converge to a bounded measurable mapping v :

I → H. Indeed, let v n (t) = A λn (t, u n (t)), v m (t) = A λm (t, u m (t)), then let us write for every t ∈ I |v m (t) -v n (t)| ≤ |A λm (t, u m (t)) -A 0 (t, u m (t))| + |A 0 (t, u m (t)) -A 0 (t, u n (t))| +|A 0 (t, u n (t)) -A λn (t, u n (t))| → 0 when m, n → ∞ . Then (v n (t)
) is a Cauchy sequence which converges uniformly to v(t). Here the measurability of these mappings is ensured thank to (A 1 ). Note that (A 1 ). is not necessary when dealing with fixed maximal monotone A :

H → H. If t → A t : H → H is a measurable single valued maximal monotone operator satisfying (A 1 ) (A 2 ), then A λn (t, u n (t))) = A(t, J λ (t, u n (t))) → A(t, u(t)) := v(t)
As an example, consider A = ∇g the gradient of a C 1 convex Lipschitz. At this point, in further application,we only need (A 1 ) (A 2 ) and the condition (A C ): For any uniformly bounded sequence of Lebesgue measurable mappings (u n ) pointwise converging to measurable mapping u, for λ ∈]0, 1], λ n → 0 + , then A λn (t, u n (t)) pointwise converges to a bounded measurable mapping v(t). Theorem 3.4. Assume that H = R e , I = [0, T ]. Let C : I × H ⇒ H be a closed convex valued multimapping for which there is some nondecreasing continuous function r : I → R + such that haus(C(t, x), C(τ, y)) ≤ |r(t) -r(τ )| + a||x -y|| for all τ, t ∈ I and for all x, y ∈ H with r(0) = 0. Let B be a linear operator continuous coercive: Bx, x ≥ γ||x|| 2 ≥ 0 for all x ∈ H, symmetric operator on H.

Let A : H → 2 H be a maximal monotone operator satisfying (A) 1 |A 0 (x)| ≤ c(1 + ||x||) for all x ∈ H. (A) 2 :If λ n → 0 + , if (u n , u) is a bounded sequence of measurable mappings on H converging pointwise to u, then t → A λn u n (t) pointwise converge to v ∈ L ∞ H (I) . (H) 3 : A λn x ∈ C(0, x)
for all n ∈ N and for all x ∈ H. Then, for any u 0 ∈ H, there exist v ∈ L ∞ H (I) and an absolutely continuous mapping u :

I → H satisfying      -du dt (t) ∈ N C(t,u(t)) (Bu(t) + v(t))a.e. u(0) = u 0 ∈ H v(t) ∈ Au(t)a.e.
Further, one has u(t) ≤ ρ := r(T ) γ . Proof. Put I := [0, T ].We note that by Hormander formula (see e.g. Castaing-

Valadier [24]) (1.1.1) |δ * (e, C(t, x))-δ * (e, C(τ, y))| ≤ ||e||d H (C(t, x), C(τ, y)) ≤ ||e||(|r(t)-r(τ )|+a||x-y||)
so that C is scalarly continuous.

Step I. Construction of a sequence (u n ) n We will use the Moreau 's catching-up algorithm [START_REF] Moreau | Evolution problem asssociated with a moving convex set in a Hilbert space[END_REF]. We consider for each n ∈ N the following partition of the interval ]0, T ] given by

t n i = i T n := iη n for 0 ≤ i ≤ n, I n i :=]t n i , t n i+1 ] for 0 ≤ i ≤ n -1. Put u n 0 = u 0 ∈ H. By (H 2 ). There is z n 1 ∈ H such that -z n 1 ∈ N C(t n 1 ,u n 0 ) (Bz n 1 + A λn u n 0 ). with A λn u n 0 ∈ C(0, u n 0 ). Put u n 1 = u n 0 + η n z n 1 .
We have

γ||z n 1 || 2 ≤ Bz n 1 , z n 1 = [Bz n 1 + A λn u n 0 -v] + [v -A λn u n 0 ], z n 1 ≤ v -A λn u n 0 , z n 1 ≤ ||v -A λn u n 0 ||||z n 1 || ≤ d H (C(0, u n 0 ), C(t n 1 , u n 0 )||z n 1 || for all v ∈ C(t n 1 , u n 0 ). So ||z n 1 || ≤ r(T ) γ Suppose that u n 0 , u n 1 , .., u n i , z n 1 , z n 2 , ..z n i are constructed. As above by condition (H2) -z n i+1 ∈ N C(t n i+1 ,u n i ) (Bz n i+1 + A λn u n i ), with A λn u n i ∈ C(0, u n i ), and we set u n i+1 = u n i + η n z n i+1 .
Then by induction there are finite sequences (u n i ) n i=0 and (

z n i ) n i=1 such that -z n i+1 ∈ N C(t n i+1 ,u n i ) (Bz n i+1 + A λn u n i ) (3.31) u n i+1 = u n i + η n z n i+1 . From (u n i ) n i=0 , (z n i ) n i=1 , we construct a sequence of mappings (u n ) n∈N from I to H, u n (0) = u n 0 and for each i = 0, .., ..n -1 u n (t) = u n i + t -t n i η n (u n i+1 -u n i ) for t ∈]t n i , t n i+1 ].
Furthermore we have the estimate

γ||z n i+1 || 2 ≤ Bz n i+1 , z n i+1 = [Bz n i+1 + A λn u n i -v] + [v -A λn u n i ], z n i+1 ≤ v -A λn u n i , z n i+1 ≤ ||v -A λn u n i ||, ||z n i+1 || ≤ d H (C(0, u n i ), C(t n i+1 , u n i ))||z n i+1 || for all v ∈ C(t n i+1 , u n i ). Whence ||z n i+1 || ≤ 1 γ d H (C(0, u n i ), C(t n i+1 , u n i )) ≤ r(t n i+1 ) -r(0) γ ≤ r(T ) γ Keeping in mind that 1 η n (u n i+1 -u n i ) = z n i+1 ≤ ρ := r(T ) γ
From this we construct sequence u n (•) is Lipschitz continuous on I with ρ as a Lipschitz constant. This Lipschitz property of u n (•) ensures that u n (t) ≤ u 0 + ρT and u n (t) = u 0 + t t 0 un (s)ds for every t ∈ I with for every

t ∈]t n i , t n i+1 ] un (t) = u n i+1 -u n i η n = z n i+1 ∈ ρB H . Furthermore, for every t ∈ [0, T ], one has u n (t) = u 0 + t 0 un (s)ds, hence ||u n (t)|| ≤ ||u 0 || + ρT .
Now, let us define the step functions θ n , δ n : I -→ I by θ n (0) = δ n (0) = 0 and

θ n (t) = t n i+1 , δ n (t) = t n i if t ∈]t n i , t n i+1 ],
so the inclusion (4. We also note that the latter inclusion above yields

δ * (-un (t), C(θ n (t), u n (δ n (t)))) + un (t), B un (t) + A λn u n (δ n (t)) ≤ 0 with un (t) ∈ ρB H a.e. t ∈ I, so that un ∈ S 1 ρB H where S 1 ρB H = {ξ ∈ L 1 H (I) : ξ(t) ∈ ρB H a.e. t ∈ I }. a.e. t ∈ [0, T ] . We note that ||u n (t)|| ≤ ||u 0 || + ρT , for all t ∈ [0, T ] and u n (t) = u 0 + t 0 un (s)ds for all t ∈ [0, T ] so that |A λn u n (δ n (t))| ≤ |A 0 (u n (δ n (t))| ≤ c(1 + ||u n (δ n (t))|| ≤ c(1 + ||u 0 || + ρT )
Step II. Convergence to a solution. We note that S 1 L is a weakly compact convex set of L 1 H (I). Set

X := {ξ : I → H : ξ(t) = u 0 + t 0 ξ(s)ds, t ∈ [0, T ]; v ∈ S 1 L }.
It is clear that X is convex, equicontinuous and compact. As u n ∈ X , one can extract from (u n ) n a (not relabeled) subsequence which converges uniformly to u : I → H such that u(t) = u 0 + t 0 u(s)ds for all t ∈ I and such that ( un

) n σ(L 1 H (I), L ∞ H (I))-converges to u ∈ S 1 L .
Further, the inequality

u n (δ n (t)) -u n (t) ≤ ρ|δ n (t) -t|
assures us that u n (δ n (t)) n converges to u(t) to H for each t ∈ I. This and the σ(L 1 and integrating on Z gives

H (I), L ∞ H (I)) convergence of ( un ) n to u in L 1 H (I)
I 1 Z e, B un (t) + A λn u n (δ n (t)) dt ≤ Z δ * (x, C(θ n (t), u n (δ n (t))))dt.
Passing to the limit in this inequality and using the scalar continuity of C(•) and the fact that

B un → B u weakly in L 1 H (I), A λn (u n (δ n (.))) pointwise converge to v ∈ L ∞ H (I)
un (t), A λn u n (δ n (t)) dt = T 0 u(t), v(t) dt. because A λn u n (δ n (.)) τ (L ∞ H (I), L 1 H (I)) (Mackey ) converges to v in L ∞ H using Grothendieck Lemma, recalling that |A λn u n (δ n (.))| ≤ c(1 + ||u 0 || + ρT ).
As above we may apply the lower semicontinuity of integral convex functional ( [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF], Theorem 8.1.16) to deduce that

Z δ * (-u(t), C(t, u(t)))dt ≤ lim inf n Z δ * (-un (t), C(t, u(t)))dt 29 (1.1.2) ≤ lim inf n Z δ * (-un (t), C(θ n (t), u n (δ n (t))))dt by noting that B u(t) + v(t) ∈ C(t) with B u + v ∈ L ∞ H (I). Let us set ψ B (x) = Bx, x if x ∈ ρB H and ψ(x) = +∞ if x /
∈ ρB H . Then it is clear ψ is a positive lower semicontinuous convex integrand. Apply again the lower semicontinuity of the integral convex functional ( [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF], Theorem 8.1.6) associated with the positive normal convex integrand ψ B we obtain,

lim inf n Z ψ B ( un (t))dt ≥ Z ψ B ( u(t))dt that is (1.1.3) lim inf n Z B un (t), un (t))dt ≥ Z B u(t), u(t)dt.
By integrating on measurable subset Z ⊂ [0, T ] the inequality ( here measurability and integrability are guaranted)

δ * (-un (t), C(θ n (t), u n (δ n (t)))) + un (t), B un (t) + A λn u n (δ n (t)) dt ≤ 0 gives (1.1.4) Z δ * (-un (t), C(θ n (t), u n (δ n (t))))dt+ Z B un (t), un (t) dt+ Z un (t), A λn u n (δ n (t)
) dt ≤ 0 by passing to the limit when n goes to ∞ in this equality using (1.1.1)-(1.1.7) gives

(1.1.5) T 0 [δ * (-u(t), C(t, u(t))) + u(t), B u(t) + v(t) ]dt ≤ 0. As t → δ * (-u(t), C(t, u(t))) + u(t), B u(t) + v(t) is integrable, by (1.1.5) and B u(t) + v(t) ∈ C(t, u(t)), we have δ * (-u(t), C(t, u(t))) ≥ -u(t), B u(t) + v(t) that implies (1.1.6) T 0 [δ * (-u(t), C(t, u(t))) + u(t), B u(t) + v(t) ]dt = 0.
By

||J λn u n (δ n (t)) -u(t)|| ≤ ||J λn u n (δ n (t)) -J λn u(t)|| + ||J λn u(t) -u(t)|| ≤ ||u n (δ n (t)) -u(t)|| + ||J λn u(t) -u(t)|| → 0 From v n (t) = A λn u n (δ n (t)) ∈ AJ λn u n (δ n (t)) we have that (v n (t), J λn u n (δ n (t))) ∈ graphA
As graphA is closed that implies v(t) ∈ Au(t) a.e.

(H) 1 |A 0 (x)| ≤ c(1 + ||x||) for all x ∈ H. Let (f n , f ) n∈N be a bounded sequence in L ∞ H (I) with f n (t) ≤ β, f (t) ≤ β (β > 0) for all n ∈ N such that (f n (t)) n converges to f (t) for each t ∈ I. Let (v n , v) n∈N be a bounded sequence in L ∞ H (I) with v n (t) ≤ γ, v(t) ≤ γ (γ > 0) for all n ∈ N such that (v n (t)) n converges to v(t) for each t ∈ I. Let (ζ n ) n∈N be an equi-integrable sequence in L 1 H (I) such that ζ n (t) ∈ C(t) for all t ∈ I and n ∈ N and such that (ζ n ) n∈N σ(L 1 H , L ∞ H ) converges in L 1 H (I) to ζ. Assume that (i)f n (t) + A λn v n (t) -B ζ n (t) ∈ N C(t) (ζ n (t)) for all n ∈ N, a.e. t ∈ I.
and (ii) λ n → 0, lim n A λn v n (t) → w(t) pointwise, where w is a measurable mapping. Then for a.e. t ∈ I one has

ζ(t) ∈ C(t) and f (t) + w(t) -B ζ(t) ∈ N C(t) (ζ(t)). w(t) ∈ Av(t) Proof. We first verify that ζ(t) ∈ C(t)
A λn v n (t), ζ n (t)dt = lim n T 0 w(t), ζ(t)dt
We can also see that

Bζ n → Bζ weakly in L 1 H (I) since for any h ∈ L ∞ H (I) T 0 h(t), Bζ n (t) dt = T 0 Bh(t), ζ n (t) dt → T 0 Bh(t), ζ(t) dt = T 0 h(t), Bζ(t) dt.
As a main consequence f n + w n -Bζ n → f + w -Bζ weakly in L 1 H (I). Given any Lebesgue measurable subset Z ⊂ I we may apply the lower semicontinuity of convex integral functional in [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF]Theorem 8.1.16] to derive that

Z δ * (f (t) + w(t) -Bζ(t), C(t))dt ≤ lim inf n Z δ * (f n (t) + w n (t) -Bζ n (t), C(t))dt.
(3.32) This needs a careful look. Choose a bounded measurable selection

s of t → C(t) ∩ r(t)B H , so s ∈ L ∞ H (I). We note that (t, x) → δ * (x, C(t)) is a normal lower semicontinuous convex integrand defined on I × H and δ * (f n (t) + w n (t) - Bζ n (t), C(t)) is measurable and minored by ρ n (t) := s(t), f n (t)+w n (t))-Bζ n (t) . Further, it is easy to check that (ρ n (•)) is equi-integrable in L 1 R (I). Then by [22, Theorem 8.1.6] we deduce that lim inf n Z δ * (f n (t) + w n (t) -Bζ n (t), C(t))dt ≥ Z δ * (f (t) + w(t) -Bζ(t), C(t))dt, which confirms (3.32). Let us set ψ B (x) = Bx, x if x ∈ C(t) and ψ B (x) = +∞ if x / ∈ C(t).
Then it is clear that ψ B is a positive lower semicontinuous convex integrand. Apply again the lower semicontinuity of the convex integral functional ( [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF], Theorem 8.1.6) associated with the positive normal convex integrand ψ B we obtain

lim inf n Z ψ B (ζ n (t)) dt ≥ Z ψ B (ζ(t)) dt, that is, lim inf n Z Bζ n (t), ζ n (t))dt ≥ Z Bζ(t), ζ(t)dt. (3.33)
Further, we have

lim n Z w n (t)), ζ n (t) dt = Z w(t), ζ(t) dt, (3.34) lim n Z f n (t), ζ n (t) dt = Z f (t), ζ(t) dt. (3.35)
The two latter equality features require a careful justification. using the fact that a uniformly bounded sequence of measurable mapping (g n ) that pointwise converge to a bounded measurable mapping g,then (g n ) converge to g with respect to the Mackey topology τ (L ∞ H (I), L

δ * (f n (t) + w n (t) -Bζ n (t), C(t)) -f n (t) + w n (t) -Bζ n (t), ζ n (t) ≤ 0.
Integrating this inequality on I gives

T 0 δ * (f n (t) + w n (t) -Bζ n (t), C(t))dt + T 0 Bζ n (t), ζ n (t) dt + T 0 -f n (t) -w n (t), ζ n (t) dt ≤ 0.
Passing to the limit inferior as n → ∞ and using (3.32) we obtain

T 0 δ * (f (t) + w(t) -Aζ(t), C(t)) + Bζ(t) -w(t) -f (t), ζ(t) dt ≤ 0.
This and the inclusion ζ(t) ∈ C(t) a.e. t ∈ I allow us to conclude that

f (t) + w(t) -Bζ(t)) ∈ N C(t) (ζ(t)) a.e. t ∈ I
according to the description (2.1) of the normal cone. It remain to check that w(t) ∈ Av(t) a.e. Indeed J λn v n (t) → v(t) by writing

||J λn v n (t) -v(t)|| ≤ ||J λn v n (t)) -J λn v(t)|| + ||J λn v(t) -v(t)|| ≤ ||v n (t)) -v(t)|| + ||J λn v(t) -v(t)|| → 0 From w n (t) = A λn v n (t) ∈ AJ λn v n (t) we have that (w n (t), J λn v n (t)) ∈ graphA
As graphA is weakly strongly sequentially closed that implies w(t) ∈ Av(t) a.e. Proposition 3.10. Let H be a separable Hilbert space and I = [0, T ]. Let C : I ⇒ H be a closed convex valued scalarly measurable multimapping for which there is some real r > 0 such that C(t) ⊂ rB H for all t ∈ I. Let B be a continuous symmetric linear coercive operator on H and let A : H → 2 H be a maximal monotone operator satisfying

(H) 1 |A 0 (x)| ≤ c(1 + ||x||) for all x ∈ H. Let (θ n ) n∈N be a sequence of mea- surable functions from I into I such that for each t ∈ I one has θ n (t) → t and haus C(θ n (t)), C(t) → 0 as n → ∞. Let (f n , f ) n∈N be a bounded sequence in L ∞ H (I) with f n (t) ≤ β, f (t) ≤ β (β > 0) for all n ∈ N such that (f n (t)) n converges to f (t) for each t ∈ I.
Let (u n , u) n∈N be a sequence of absolutely continuous mappings we see that

u n (t) = u 0 + t 0 un (s)ds, un (t) ∈ C(t) u(t) = u 0 + t 0 u(s)ds, such that (u n ) n converges
I 1 Z x, un (t), dt = Z x, un (t) dt ≤ Z δ * (x, C(θ n (t)))dt.
Passing to the upper limit we obtain

Z 1 Z x, u(t) dt ≤ lim sup n Z δ * (x, C(θ n (t)))dt ≤ Z lim sup n δ * (x, C(θ n (t))) ≤ Z δ * (x, C(t))dt.
This being true for any Lebesgue measurable set Z ⊂ I we deduce that for every

x ∈ H x, u(t) ≤ δ * (x,
(I), L 1 H (I)in L 1 H (I), so q n (•) = f n (•) + A λn (u n (•)) -B un (•) → q(•) := f (•) + w(.) -B u(•) weakly in L 1 H (I). Indeed, for any h ∈ L ∞ H (I) the weak convergence in L 1 H (I) of ( un ) n to u says that T 0 Bh(t), un (t) dt → T 0 Bh(t), u(t) dt, which means T 0 h(t), B un (t) dt → T 0 h(t), B u(t) dt.

This property for every

h ∈ L ∞ H (I) translates the weak convergence in L 1 H (I) of (B un ) n to B u. Concerning (A λn (u n (•))) n it converges τ (L ∞ H (I), L 1 H (I) to w as have already seen.So weak convergence in L 1 H (I) of (q n ) n to q is justified. Regarding T 0 B un (t), un (t) dt let us set ψ B (t, x) = Bx, x if x ∈ C(t) and ψ B (t, x) = 0 if x / ∈ C(t)
. It is clear that ψ B is a positive lower semicontinuous convex normal integrand. By the lower semicontinuity of convex integral functional (see [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF]Theorem 8.1.6]) we obtain lim inf

n T 0 ψ B (t, un (t))dt ≥ T 0 ψ B (t, u(t))dt, that is, lim inf n T 0 B un (t), un (t) dt ≥ T 0 B u(t), u(t) dt. (3.38)
The last step is concerned with T 0 δ * (q n (t), C(t))dt. Note that q n (t) ≤ β + γ + r B , hence δ * (q n (t), C(t)) ≥ -r(β + γ + r B ). Using this and the lower semicontinuous convex normal integrand (t, x) → δ * (x, C(t)) we obtain by [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF]Theorem 8.1.6]

again that lim inf n T 0 δ * (q n (t), C(t))dt ≥ T 0 δ * (g(t), C(t))dt. Since δ * q n (t), C(θ n (t) -δ * q n (t), C(t) ≤ (β+γ+r B ) haus C(θ n (t)), C(t) =: ε n (t) with T 0 ε n (t) → 0 as n → ∞, we are ensured that lim inf n T 0 δ * q n (t), C(t) dt ≥ T 0 δ * g(t), C(t) dt. ( 3
||J λn u n (t) -u(t)|| ≤ ||J λn u n (t)) -J λn u(t)|| + ||J λn u(t) -u(t)|| ≤ ||u n (t)) -u(t)|| + ||J λn u(t) -u(t)|| → 0 From w n (t) = A λn u n (t) ∈ AJ λn u n (t) we have that (w n (t), J λn u n (t)) ∈ graphA
As graphA is weakly strongly sequentially closed that implies w(t) ∈ Au(t) a.e. Let B : H → H be a linear continuous coercive symmetric operator and let A : H → 2 H be a maximal monotone operator satisfying

(A) 1 |A 0 (x)| ≤ c(1 + ||x||) for all x ∈ H. (A) 2 :If λ n → 0 + , if (u n , u
) is a bounded sequence of measurable mappings on H converging pointwise to u, then t → A λn u n (t) pointwise converge to v ∈ L ∞ H (I) . Then, for any u 0 ∈ H, there exists an W 1,∞ H ([0, T ]) solution u : [0, T ] → H and a bounded measurable mapping w :

I → H such that      f (t) + w(t) -B du dt (t) ∈ N C(t) ( du dt (t)) u(0) = u 0 w(t) ∈ Au(t)
Further, one has u(t) ≤ ρ, where ρ := sup{ y : y ∈ C([0, T ])}.

Proof. Put I := [0, T ] and denote β := max{ f (t) : t ∈ I} (by continuity of f ). Noticing that the multimapping C(•) is upper semicontinuous from I into H the set C(I) is compact, and hence ρ := sup{ y : y ∈ C(I)} is finite and L := ρB H is compact and convex.

Step I. Construction of a sequence (u n ) n We will use the Moreau 's catching-up algorithm [START_REF] Moreau | Evolution problem asssociated with a moving convex set in a Hilbert space[END_REF]. We consider for each n ∈ N A λn the Yosida approxiamtion of A and the following partition of the interval ]0, T ] given by

t n i = i T n := iη n for 0 ≤ i ≤ n, I n i :=]t n i , t n i+1 ] for 0 ≤ i ≤ n -1.
Put u n 0 = u 0 and f n i = f (t n i ) for all i = 1, .., n. By Proposition 2.1(b), there is

z n 1 ∈ C(t n 1 ) ⊂ L such that f n 1 + A λn u n 0 -Bz n 1 ∈ N C(t n 1 ) (z n 1 ). Put u n 1 = u n 0 + η n z n 1 . Suppose that u n 0 , u n 1 , .., u n i , z n 1 , z n 2 , ..z n i are constructed. As above by Proposition 2.1(b) there exists z n i+1 ∈ C(t n i+1 ) ⊂ L such that f n i+1 + A λn u n i -Bz n i+1 ∈ N C(t n i+1 ) (z n i+1 ),
and we set u n i+1 = u n i + η n z n i+1 . Then by induction there are finite sequences (u n i ) n i=0 and (z n i ) n i=1 such that

f n i+1 + A λn u n i -Bz n i+1 ∈ N C(t n i+1 ) (z n i+1 ) (3.41) u n i+1 = u n i + η n z n i+1 . From (u n i ) n i=0 , (z n i ) n i=1 (f n i ) n i=0
, we construct two sequences of mappings (u n ) n∈N and (f n ) n∈N from I to H, by setting f n (0) = f n 1 , u n (0) = u n 0 and for each i = 0, .., ..n -1 we set f n (t) = f n i+1 and

f n (t) = f n i+1 and u n (t) = u n i + t -t n i η n (u n i+1 -u n i ) for t ∈]t n i , t n i+1 ].
Keeping in mind that

C(t) ⊂ L = ρB H we have u i ∈ C(t n i+1 ) ⊂ ρB H , so 1 η n (u n i+1 -u n i ) = z n i+1 ≤ ρ.
From this it is clear that u n (•) is Lipschitz continuous on I with ρ as a Lipschitz constant. This Lipschitz property of u n (•) ensures that u n (t) ≤ u 0 + ρT and u n (t) = u 0 + t t 0 un (s)ds for every t ∈ I. We also note that f n (t) ≤ β for all n ∈ N and t ∈ I. Now, let us define the step functions θ n , δ n : I -→ I by θ n (0) = δ n (0) = 0 and

θ n (t) = t n i+1 , δ n (t) = t n i if t ∈]t n i , t n i+1 ],
so the inclusion (4.1) becomes (0. We also note that the latter inclusion above yields

δ * (f n (t)+A λn u n (δ n (t))-B un (t), C(θ n (t)))+ -f n (t)-A λn u n (δ n (t))-Bdotu n (t), un (t) ≤ 0 with un (t) ∈ C(θ n (t)) ⊂ L a.e. t ∈ I, so that un ∈ S 1 L where S 1 L := {ξ ∈ L 1 H (I) : ξ(t) ∈ L a.e. t ∈ I }.
Step II. Convergence to a solution. We note that S 1 L is a weakly compact convex set of L 1 H (I) (see, e.g., [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF] and the references therein). Set

X := {ξ : I → H : ξ(t) = u 0 + t 0 ξ(s)ds, t ∈ [0, T ]; v ∈ S 1 L }.
It is clear that X is convex, equicontinuous and compact in C H ([0, T ]) As u n ∈ X , one can extract from (u n ) n a (not relabeled) subsequence which pointwise converges to u : I → H (i.e., u n (t) → u(t) in H for each t ∈ I) such that u(t) = u 0 + t 0 u(s)ds for all t ∈ I and such that ( un

) n σ(L 1 H (I), L ∞ H (I))-converges to u ∈ S 1
L . Further, the inequality

u n (δ n (t)) -u n (t) ≤ ρ|δ n (t) -t|
assures us that u n (δ n (t)) n converges in H for each t ∈ I. This and the σ(L 1 H (I), L ∞ H (I)) convergence of ( un ) n to u in L 1 H (I) along with the inclusion (4.1) allow us (according to the pointwise convergence of (f n ) n to f and the estimates from the hypotheses) to obtain that for a.e. t ∈ I the inclusions u(t) ∈ C(t)

f (t) + w(t) -B u(t) ∈ N C(t) ( u(t)) and w(t) ∈ Au(t)
Our first task is to prove the inclusion u(t) ∈ C(t) a.e. t ∈ I. Indeed, for every Lebesgue measurable set Z ⊂ I and for any x ∈ H , the function

1 Z x ∈ L ∞ H . Considering the inequality x, un (t) ≤ δ * (x, C(θ n (t)))
and integrating on Z gives

I 1 Z x, un (t), dt = Z x, un (t) dt ≤ Z δ * (x, C(θ n (t)))dt.
Passing to the limit in this inequality and using the scalar upper semicontinuity of C(•) we obtain

I 1 Z x, u(t) dt ≤ lim sup n Z δ * (x, C(θ n (t)))dt ≤ Z lim sup n δ * (x, C(θ n (t))) ≤ Z δ * (x, C(t))dt, which is equivalent to x, u(t) ≤ δ * (x, C(t
|δ * (f n (t)+A λn u n (δ n (t))-B un (t), C(t))| ≤ ||f n (t)+A λn u n (δ n (t))-B un (t)|||L| ≤ Constant furthermore δ * (f n (t) + A λn u n (δ n (t)) -B un (t), C(t)) ≥ f n (t) + A λn u n (δ n (t)) -B un (t), u(t)
where u(t) is a measurable selection of C (note that C is scalarly upper semicontinuous multimapping from [0, T ] to L). Further we have by (1.1.1) H ([0, T ]) by noting that a bounded sequence is L ∞ H (I) which pointwise converges to 0, converges to 0 uniformly on any uniformly integrable subset of L 1

|δ * (f n (t)+A λn u n (δ n (t))-B un (t), C(t))-δ * (f n (t)+A λn u n (δ n (t))-B un (t), C(θ n (t)))| ≤ ||f n (t) + A λn u n (δ n (t)) -B un (t))||d H (C(t), C(θ n (t))) ≤ ||f n (t) + A λn u n (δ n (t)) -B un (t)|||v(t) -v(θ n (t))| ≤ Constant|v(t) -v(θ n (t))| so that lim inf n Z δ * (f n (t) + A λn u n (δ n (t)) -B un (t), C(θ n (t)))dt ≥ lim inf n Z δ * (f n (t) + A λn u n (δ n (t)) -B un (t), C(t))dt (1.1.3) ≥ Z δ * (f (t) + w(t) -B u(t), C(t))dt. Let us set ψ B (x) = Bx, x if x ∈ L and ψ(x) = +∞ if x / ∈ L. Then it is clear ψ B is
H ([0, T ]), in other terms it converges to 0 with respect to the Mackey topology [START_REF] Castaing | Topologie de la convergence uniforme sur les parties uniformément intégrables de L 1 E et théorèmes de compacité faible dans certains espaces du type Köthe-Orlicz[END_REF]). 3 Now integrating on Z ⊂ [0, T ] the inequality ( here measurability and integrability are guaranted)

τ (L ∞ H ([0, T ]), L 1 H ([0, T ])) (see
δ * (f n (t) + A λn u n (δ n (t)) -B un (t), C(θ n (t))) + -f n (t) -A λn u n (δ n (t)), un (t) + B un (t), un (t) ≤ 0 gives Z δ * (f n (t) + A λn u n (δ n (t)) -B un (t), C(θ n (t))))dt
Z [δ * (f (t) + w(t) -B u(t), C(t)) + B u(t) -w(t) -f (t), u(t) ]dt ≤ 0. As t → δ * (f (t) + w(t) -B u(t), C(t)) + B u(t) -w(t) -f (t), u(t) is integrable, by (1.1.8) it follow (1.1.9) δ * (f (t) + w(t) -B u(t), C(t)) + B u(t) -w(t) -f (t), u(t) ≤ 0, a.e.
As u(t) ∈ C(t), we have

δ * (f (t) + w(t) -B u(t), C(t)) ≥ f (t) + w(t) -B u(t), u (
||J λn u n (δ n (t)) -u(t)|| ≤ ||J λn u n (δ n (t)) -J λn u(t)|| + ||J λn u(t) -u(t)|| ≤ ||u n (δ n (t)) -u(t)|| + ||J λn u(t) -u(t)|| → 0 From w n (t) = A λn u n (δ n (t)) ∈ AJ λn u n (δ n (t)) we have that (w n (t), J λn u n (δ n (t))) ∈ graphA
As graphA is weakly strongly sequentially closed that implies w(t) ∈ Au(t) a.e.

Remark

We cannot expect to have uniqueness of solution. In case when H is separable Hilbert space the result hold true if we replace the operator A is replaced by a linear compact operator, uniqueness of solutions hold true using the coerciveness of the operator B, then result is read as : there exist an absolutely continuous mapping u : I → H such tthat for a.e.

f (t) + Au(t) -B du dt (t) ∈ N C(t) ( du dt (t)) u(0) = u 0
If A : I → H is a continuous mapping with the growth condition ||Ax|| ≤ c(1 + ||x||), the result is read as : there exist an absolutely continuous mapping u : I → H such that for a.e.

f (t) + Au(t) -B du dt (t) ∈ N C(t) ( du dt (t)) u(0) = u 0 Proposition 3.11.
Let H be a separable Hilbert space and I = [0, T ] Let ϕ : [0, T ] × H →] -∞, +∞] be a normal lower semicontinuous convex integrand for which there exists a convex weakly compact set Γ such that: (i) for all t ∈ I, domϕ t := Γ; (ii {ϕ(., u(.)), u ∈ S ∞ Γ } is uniformly integrable; (iii) ϕ(t, x) ≤ ϕ(τ, x)+|v(t)-v(τ )| for all t, τ ∈ [0, T ], x ∈ Γ where v : [0, T ] → R + is a positive nondecreasing continuous function with v(0) = 0. Let B : H → H be a continuous symmetric linear coerciveoperator, let A t : H → 2 H be a time measurable maximal monotone operator satisfying (A) 1 |A 0 (t, x)| ≤ c(1 + ||x||) for all (t, x) ∈ I × H. Let (θ n ) n∈N be a sequence of measurable functions from I into I such that for each t ∈ I one has θ n (t) → t.

Let (f n , f ) n∈N be a bounded sequence in L ∞ H (I) with f n (t) ≤ β, f (t) ≤ β (β > 0) for all n ∈ N such that (f n (t)) n converges to f (t) for each t ∈ I.
Let (u n , u) n∈N be a sequence of absolutely continuous mappings

u n (t) = u 0 + t 0 un (s)ds, un (t) ∈ Γ u(t) = u 0 + t 0 u(s)
q n (t) := f n (t) + A λn (t, u n (t)) -B un (t) ∈ ∂ϕ(θ n (t), un (t))
so that by the normality of ϕ * , the function t → ϕ * (θ n (t), q n (t)) is measurable and integrable, the measurability and integrability of A λn (t, u n (t)) is already ensured as above. We also note that q n , un is measurable and integrable and the sequence ( q n , un ) is uniformly integrable. Further, by (3.42) and condition (ii) we have -ϕ(t, un (t)) + un (t), q n (t) ≤ ϕ * (t, q n (t)) ≤ ϕ * (θ n (t), q n (t)) + |v(t) -v(θ n (t))|, (3.44) so that t → -ϕ(t, un (t)) + un (t), q n (t) is uniformly integrable thanks to (ii). We note that, for h n (t) := f n (t) + A λn (t, u n (t)), (h n ) n is uniformly bounded and pointwise converges to h given by h(t) = f (t) + v(t) in H. As consequence, for every measurable set Z in I, we have

lim n→∞ Z h n (t), un (t) dt = Z h(t), u(t) dt.
(3.45)

This fact has been justified in the proof of Proposition 3.4. As B is symmetric, we also show that B un (

•) → A u(•) weakly in L 1 H (I). As consequence q n = f n + A λn (t, u n (t)) -B un (.) → q := f + v(t) -B u weakly in L 1 H (I). Further, let us set ψ B (x) = Bx, x if x ∈ Γ and ψ(x) = +∞ if x / ∈ Γ.
ϕ(t, u(t)) dt + Z ϕ * (t, q(t)) dt ≤ Z u(t), q(t) dt.
By the measurability of the non negative function t → ϕ(t, u(t)) + ϕ * (t, q(t))u(t), q(t) , we deduce that for almost every t ∈ I ϕ(t, u(t)) + ϕ * (t, q(t)) -u(t), q(t) ≤ 0 along with u(t) ∈ Γ. So, it follows for almost every t ∈ I that ϕ(t, u(t)) + ϕ * (t, q(t)) = u(t), q(t) , or equivalently

q(t) = f (t) + v(t) -A u(t) ∈ ∂ϕ(t,
ϕ : [0, T ] × K →] -∞, +∞] be a normal lower semicontinuous convex integrand such that (i {ϕ(., u(.)), u ∈ S 1 K } is uniformly integrable. (ii) ϕ(t, x) ≤ ϕ(τ, x)+|v(t)-v(τ )| for all t, τ ∈ [0, T ], x ∈ K where v : [0, T ] → R + is a positive nondecreasing continuous function with v(0) = 0.
Let B : H → H be a linear continuous coercive symmetric operator and let A : H → 2 H be a maximal monotone operator satisfying the properties

(A) 1 |A 0 x)| ≤ c(1 + ||x||) for all (t, x) ∈ I × H. (A) 2 :If λ n → 0 + , if (u n , u
) is a bounded sequence of measurable mappings on H converging pointwise to u, then A λn u n (.) pointwise converge to v ∈ L ∞ H (I) Let f : I → H be a continuous mapping.

Then, for any u 0 ∈ H, there is an absolutely continuous mapping u : i → H and a bounded measurable mapping v : I → H such that for a.e.,

f (t) + v(t) -B u(t) ∈ ∂ϕ(t, du dt (t)) u(0) = u 0 v(t) ∈ Au(t) Further the solution set {(u, v)} is compact in C H (I) × w-L 2 H (I).
Proof. We will use again the Moreau 's catching-up algorithm. We consider for each n ∈ N the following partition of the interval I = [0, T ].

t n i = i T n := iη n for 0 ≤ i ≤ n. I n i :=]t n i , t n i+1 ] for 0 ≤ i ≤ n -1. Put u n 0 = u 0 and f n i = f (t n i
) for all i = 1, .., n. By Proposition 1.1 1) , there is

z n 1 ∈ K such that f n 1 + A λn u n 0 -Bz n 1 ∈ ∂ϕ(t n 1 , z n 1 ). Put u n 1 = u n 0 + η n z n 1 .
Suppose that u n 0 , u n 1 , .., u n i , z n 1 , z n 2 , ..z n i are constructed. As above by Proposition 1.1 1) there exists z n i+1 ∈ K such that

f n i+1 + A λn u n i -Bz n i+1 ∈ ∂ϕ(t n i+1 , z n i+1 ).
and we set u n i+1 = u n i + η n z n i+1 . Then by induction there are finite sequences (u n i ) n i=0 and (z n i ) n i=1 such that

f n i+1 + A λn u n i -Bz n i+1 ∈ ∂ϕ(t n i+1 , z n i+1 ) u n i+1 = u n i + η n z n i+1 From (u n i ) n i=0 , (z n i ) n i=1 (f n i ) n i=0 , we construct two sequences u n from [0, T ] to H, f n from [0, T ] to H, by setting f n (0) = f n 1 , u n (0) = u n 0 and for each i = 0, .., ..n -1 we set f n (t) = f n i+1 and u n (t) = u n i + t -t n i η n (u n i+1 -u n i ) for t ∈]t n i , t n i+1 ].
Clearly, the mapping u n (.) is Lipschitz continuous on [0, T ], and ρ is a Lipschitz constant of u n (.) on [0, T ] since for every

t ∈]t n i , t n i+1 ] un (t) = u n i+1 -u n i η n = z n i+1 ∈ K ⊂ ρB H .
Furthermore, for every t ∈ [0, T ], one has u n (t) = u 0 + t 0 un (s)ds, hence ||u n (t)|| ≤ ||u 0 || + ρT . We have

f n i+1 + A λn u n i -Bz n i+1 ∈ ∂ϕ(t n i+1 , z n i+1 ).
Now, let us define the step functions θ n , δ n : I -→ I by

θ n (t) = t n i+1 , δ n (t) = t n i if t ∈]t n i , t n i+1 ] and θ n (0) = δ n (0) = 0,
X := {v : [0, T ] → H : v(t) = u 0 + t 0 v(s)ds, t ∈ [0, T ]; v ∈ S 1 K }.
Then it is clear that S 1 K is convex and weakly compact in L 1 H ([0, T ]) (see e.g. [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF] and the references therein) and that X is convex, equicontinuous and compact in C H ([0, T ]). As (u n ) ⊂ X , one can extract from (u n ) a subsequence not relabelled which pointwise converges to u : [0, T ] → H such that u(t) = u 0 + t 0 u(s)ds, for all t ∈ [0, T ] and ( un ) σ(L 1 H ([0, T ]), L ∞ H ([0, T ]))-converges to u ∈ S 1 K . By using the normality of ϕ, the mappings t → ϕ(θ n (t), un (t)) and t → ϕ(t, un (t)) are measurable and integrable. By construction we have

g n (t) := f n (t) + A λn u n (δ n (t)) -B un (t) ∈ ∂ϕ(θ n (t), un (t)).
by (A) 2 :let λ n → 0 + , then A λn u n (δ n (t)) pointwise converges to a bounded measurable mapping v : I → H. For simplicity set

g(t) := f (t) + v(t) -B u(t).
As f n → f pointwise strongly, u n (δ n (.)) → u(.)) pointwise strongly, and un → u weakly in L 1 E ([0, T ]), A λn u n (δ n (t)) pointwise converges to v, a direct application of Proposition 3.11 gives

g(t) := f (t) + v(t) -B u(t) ∈ ∂ϕ(t, u(t))
a.e It remain to check v(t) ∈ Au(t) a.e. Indeed J λn (t, u n (t))-→ u(t) by writing

||J λn u n (δ n (t)) -u(t)|| ≤ ||J λn u n (δ n (t)) -J λn u(t))|| + ||J λn u(t)) -u(t)|| ≤ ||u n (δ n (t)) -u(t)|| + ||J λn (t, u(t)) -u(t)|| → 0 From v n (t) = A λn u n (δ n (t)) ∈ AJ λn u n (δ n (t)) we show that (v n , w n ) ∈ graphA with w n (t) = J λn u n (δ n (t))
). As graphA is sequentially strong weakly closed, with v n → v strongly pointwise in H and w n → u strongly pointwise in H this gives (v, u) ∈ graphA, that implies v(t) ∈ Au(t) a.e. and finish the proof. The compactness of solution set follows from the arguments given in the variational limit theorem, cf Proposition 0.7. Theorem 3.7. Let H be a separable Hilbert space. Let K be a convex compact subset of

H. Let ϕ : [0, T ] × K →] -∞, +∞] be a normal lower semicontinuous convex integrand such that (i {ϕ(., u(.)), u ∈ S 1 K } is uniformly integrable. (ii) ϕ(t, x) ≤ ϕ(τ, x)+|v(t)-v(τ )| for all t, τ ∈ [0, T ], x ∈ K where v : [0, T ] → R + is a positive nondecreasing continuous function with v(0) = 0.
Let B : H → H be a linear continuous coercive symmetric operator. Let f : I → H be a continuous mapping. Let g : H → R be a convex function Lipschitz on bounded sets and continuously differentiable on H. Then, for any u 0 ∈ H, there is an absolutely continuous mapping u : I → H such that for a.e.,

f (t) + ∇g(u(t)) -B u(t) ∈ ∂ϕ(t, du dt (t)) u(0) = u 0
Further the solution set is compact in C H (I).

Proof. We will use again the Moreau 's catching-up algorithm. We consider for each n ∈ N the following partition of the interval I = [0, T ].

t n i = i T n := iη n for 0 ≤ i ≤ n. I n i :=]t n i , t n i+1 ] for 0 ≤ i ≤ n -1. Put u n 0 = u 0 and f n i = f (t n i
) for all i = 1, .., n. By Proposition 1.1 1) , there is

z n 1 ∈ K such that f n 1 + ∇g(u n 0 ) -Bz n 1 ∈ ∂ϕ(t n 1 , z n 1 ). Put u n 1 = u n 0 + η n z n 1 . Suppose that u n 0 , u n 1 , .., u n i , z n 1 , z n 2 , .
.z n i are constructed. As above by Proposition 1.1 1) there exists z n i+1 ∈ K such that

f n i+1 + ∇g(u n i ) -Bz n i+1 ∈ ∂ϕ(t n i+1 , z n i+1 ).
and we set u n i+1 = u n i + η n z n i+1 . Then by induction there are finite sequences (u n i ) n i=0 and (z n i ) n i=1 such that

f n i+1 + ∇g(u n i ) -Bz n i+1 ∈ ∂ϕ(t n i+1 , z n i+1 ) u n i+1 = u n i + η n z n i+1 From (u n i ) n i=0 , (z n i ) n i=1 (f n i ) n i=0 , we construct two sequences u n from [0, T ] to H, f n from [0, T ] to H, by setting f n (0) = f n 1 , u n (0) = u n 0 and for each i = 0, .., ..n -1 we set f n (t) = f n i+1 and u n (t) = u n i + t -t n i η n (u n i+1 -u n i ) for t ∈]t n i , t n i+1 ].
Clearly, the mapping u n (.) is Lipschitz continuous on [0, T ], and ρ is a Lipschitz constant of u n (.) on [0, T ] since for every

t ∈]t n i , t n i+1 ] un (t) = u n i+1 -u n i η n = z n i+1 ∈ K ⊂ ρB H .
Furthermore, for every t ∈ [0, T ], one has u n (t) = u 0 + t 0 un (s)ds, hence ||u n (t)|| ≤ ||u 0 || + ρT . We have

f n i+1 + ∇g(u n i ) -Bz n i+1 ∈ ∂ϕ(t n i+1 , z n i+1 ).
Now, let us define the step functions θ n , δ n : I -→ I by

θ n (t) = t n i+1 , δ n (t) = t n i if t ∈]t n i , t n i+1 ] and θ n (0) = δ n (0) = 0,
X := {v : [0, T ] → H : v(t) = u 0 + t 0 v(s)ds, t ∈ [0, T ]; v ∈ S 1 K }.
Then it is clear that S 1 K is convex and weakly compact in L 1 H ([0, T ]) (see e.g. [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF] and the references therein) and that X is convex, equicontinuous and compact in C H ([0, T ]). As (u n ) ⊂ X , one can extract from (u n ) a subsequence not relabelled which pointwise converges to u : [0, T ] → H such that u(t) = u 0 + t 0 u(s)ds, for all t ∈ [0, T ] and ( un )

σ(L 1 H ([0, T ]), L ∞ H ([0, T ]))-converges to u ∈ S 1 K .
By using the normality of ϕ, the mappings t → ϕ(θ n (t), un (t)) and t → ϕ(t, un (t)) are measurable and integrable. By construction we have

g n (t) := f n (t) + ∇g(u n (δ n (t))) -B un (t) ∈ ∂ϕ(θ n (t), un (t)).
For simplicity set g(t) := f (t) + ∇g(u(t)) -B u(t).

As f n → f pointwise strongly, u n (δ n (.)) → u(.)) pointwise strongly, and un → u weakly in L 1 E ([0, T ]), ∇g(u n (δ n (t))) pointwise converges to ∇g(u(t)), a direct application of Proposition 3.6 gives

g(t) := f (t) + ∇g(u(t)) -B u(t) ∈ ∂ϕ(t, u(t))
a.e and finish the proof. Lemma 3.8. Let a time measurable maximal monotone operator t → A t : H → H satisfying the conditions. (A 1 ) t → J λ (t, x) is L(I)-measurable for every λ > 0 and for every x ∈ H

(A 2 ) |A 0 (t, x)| ≤ c(1 + ||x||) for all (t, x) ∈ I × H. If (u n ) is a uniformly bounded sequence of Lebesgue measurable mappings point- wise converging to measurable mapping u, for λ ∈]0, 1], λ n → 0 + , A λn (t, u n (t)) weakly converge in L 2 H (I) to a bounded measurable mapping v, then v(t) ∈ A(t, u(t)) a.e. Proof. Let us set v n (t) := A λn (t, u n (t)). From Lemma we are ensured that (v n ) is bounded measurable and (v n ) wealy converge to v in L 2 H (I). We have v n (t) := A λn (t, u n (t)) ∈ A(t, J λ (t, u n (t))) with w n (t) := J λ (t, u n (t))) → u(t)) pointwise. Indeed J λn (t, u n (t))-→ u(t) by writing ||J λn (t, u n (t)) -u(t)|| ≤ ||J λn (t, u(t)) -J λn (t, u(t))|| + ||J λn (t, u(t)) -u(t)|| ≤ ||u n (t) -u(t)|| + ||J λn (t, u(t)) -u(t)|| → 0 As ||u n (t) -J λn (t, u n (t))|| = λ n ||A λn (t, u n (t))|| ≤ λ n |A 0 (t, u n (t)) ≤ c(1 + ||u n (t)||) with λ n < 1, J λn (t, u n (t)
) is uniformly bounded and pointwise converge to u(t), so that t → J λn (t, u n (t)) converge to u in L 2 H (I). Then from v n (t) ∈ A(t, w n (t)) with v n → v weakly in L 2 H (I) and w n → u strongly in L 2 H (I) and (v n , w n ) ∈ A * . by Lemma 0.1, (v, u) ∈ A * i.e v(t) ∈ A(t, u(t)) a.e.

Well-posedness of inclusion (1.1)

Our main proofs in this section are build upon the variational inequalities in Proposition 2.1 and the variational limits in Section 3 as well as upon an explicit catching-up algorithm (alias Moreau's algorithm). We stress the fact that our algorithm and tools are self contained apart from the use of the mentioned variational inequalities.

haus (C(t), C(τ )) ≤ |v(t) -v(τ )| for all t, τ ∈ [0, T ].
Let A : H → H be a linear continuous coercive symmetric operator and let B : H → H be a linear continuous compact operator. Then, for any u 0 ∈ H, the evolution inclusion

f (t) + Bu(t) -A du dt (t) ∈ N C(t) ( du dt (t)) u(0) = u 0 admits a unique W 1,∞ H ([0, T ]) solution u : [0, T ] → H.
Further, one has u(t) ≤ ρ, where ρ := sup{ y : y ∈ C([0, T ])}.

Proof. Put I := [0, T ] and denote β := max{ f (t) : t ∈ I} (by continuity of f ). Noticing that the multimapping C(•) is upper semicontinuous from I into H endowed with the weak topology, the set C(I) is weakly compact, and hence ρ := sup{ y : y ∈ C(I)} is finite and L := ρB H is weakly compact and convex.

Step I. Construction of a sequence (u n ) n We will use the Moreau 's catching-up algorithm [START_REF] Moreau | Evolution problem asssociated with a moving convex set in a Hilbert space[END_REF]. We consider for each n ∈ N the following partition of the interval ]0, T ] given by

t n i = i T n := iη n for 0 ≤ i ≤ n, I n i :=]t n i , t n i+1 ] for 0 ≤ i ≤ n -1.
Put u n 0 = u 0 and f n i = f (t n i ) for all i = 1, .., n. By Proposition 2.1(b), there is

z n 1 ∈ C(t n 1 ) ⊂ L such that f n 1 + Bu n 0 -Az n 1 ∈ N C(t n 1 ) (z n 1 ). Put u n 1 = u n 0 + η n z n 1 . Suppose that u n 0 , u n 1 , .., u n i , z n 1 , z n 2 , .
.z n i are constructed. As above by Proposition 2.1(b) there exists z n i+1 ∈ C(t n i+1 ) ⊂ L such that

f n i+1 + Bu n i -Az n i+1 ∈ N C(t n i+1 ) (z n i+1 ),
and we set u n i+1 = u n i + η n z n i+1 . Then by induction there are finite sequences (u n i ) n i=0 and (z n i ) n i=1 such that

f n i+1 + Bu n i -Az n i+1 ∈ N C(t n i+1 ) (z n i+1 ) (4.1)
u n i+1 = u n i + η n z n i+1 . From (u n i ) n i=0 , (z n i ) n i=1 (f n i ) n i=0 , we construct two sequences of mappings (u n ) n∈N and (f n ) n∈N from I to H, by setting f n (0) = f n 1 , u n (0) = u n 0 and for each i = 0, .., ..n -1 we set f n (t) = f n i+1 and f n (t) = f n i+1 and u n (t) = u n i + t -t n i η n (u n i+1 -u n i ) for t ∈]t n i , t n i+1 ].
Keeping in mind that C(t) ⊂ L = ρB H we have

u i ∈ C(t n i+1 ) ⊂ ρB H , so 1 η n (u n i+1 -u n i ) = z n i+1 ≤ ρ.
From this it is clear that u n (•) is Lipschitz continuous on I with ρ as a Lipschitz constant. This Lipschitz property of u n (•) ensures that u n (t) ≤ u 0 + ρT and u n (t) = u 0 + t t 0 un (s)ds for every t ∈ I. We also note that f n (t) ≤ β for all n ∈ N and t ∈ I. Now, let us define the step functions θ n , δ n : I -→ I by θ n (0) = δ n (0) = 0 and

θ n (t) = t n i+1 , δ n (t) = t n i if t ∈]t n i , t n i+1 ],
so the inclusion (4.1) becomes

f n (t) + Bu n (δ n (t)) -A un (t) ∈ N C(θn(t)) ( un (t)) a.e. t ∈ I.
For each t ∈ I we observe that there is some i ∈ {0, ..., n-1} such that t ∈ [t n i , t n i+1 [, and then

|θ n (t) -t| → 0 and |δ n (t) -t| → 0 as n → +∞.

We also note that the latter inclusion above yields

δ * (f n (t)+Bu n (δ n (t))-A un (t), C(θ n (t)))+ -f n (t)-Bu n (δ n (t))+A un (t), un (t) ≤ 0 with un (t) ∈ C(θ n (t)) ⊂ L a.e. t ∈ I, so that un ∈ S 1 L where S 1 L := {ξ ∈ L 1 H (I) : ξ(t) ∈ L a.e. t ∈ I }.
Step II. Convergence to a solution. We note that S 1 L is a weakly compact convex set of L 1 H (I) (see, e.g., [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF] and the references therein). Set

X := {ξ : I → H : ξ(t) = u 0 + t 0 ξ(s)ds, t ∈ [0, T ]; v ∈ S 1 L }.
It is clear that X is convex, equicontinuous and weakly compact [START_REF] O'regan | Fixed point theorem for weakly sequentially closed maps[END_REF] in C H ([0, T ]) (see [START_REF] O'regan | Fixed point theorem for weakly sequentially closed maps[END_REF]). As u n ∈ X , one can extract from (u n ) n a (not relabeled) subsequence which pointwise weakly converges to u : I → H (i.e., u n (t) → u(t) weakly in H for each t ∈ I) such that u(t) = u 0 + t 0 u(s)ds for all t ∈ I and such that ( un

) n σ(L 1 H (I), L ∞ H (I))-converges to u ∈ S 1 L . Further, the inequality u n (δ n (t)) -u n (t) ≤ ρ|δ n (t) -t|
assures us that u n (δ n (t)) n converges weakly in H for each t ∈ I. This and the σ(L 1 H (I), L ∞ H (I)) convergence of ( un ) n to u in L 1 H (I) along with the inclusion (4.1) allow us (according to the pointwise convergence of (f n ) n to f and the estimates from the hypotheses) to apply Proposition 3.1, with v n = u n • δ n and ζ n = un , to obtain that for a.e. t ∈ I the inclusions u(t) ∈ C(t) and

f (t) + Bu(t) -A u(t) ∈ N C(t) ( u(t))
hold true. This says that u(•) is solution of the inclusion of the theorem.

STEP III. Uniqueness.

The uniqueness of solutions follows easily from the coerciveness of the operator A. Indeed let u 1 and u 2 be two solutions. An easy computation gives

A u2 (t) -A u1 (t), u2 (t) -u1 (t) + Bu 2 (t) -Bu 1 (t), u2 (t) -u1 (t) ≤ 0, so that A u2 (t) -A u1 (t), u2 (t) -u1 (t) ≤ |B| u 2 (t) -u 1 (t) u2 (t) -u1 (t) .
By coerciveness of A we deduce that

ω u2 (t) -u1 (t) 2 ≤ |B| u 2 (t) -u 1 (t) u2 (t) -u1 (t) . This entails that u2 (t) -u1 (t) ≤ |B| ω u 2 (t) -u 1 (t) ≤ |B| ω t 0 u2 (
∇g(v(t)), v(t) = d dt g(v(t))
Hence by using this fact and the tool developed in Theorem 1.1, we obtain a variant of Theorem 1.1 by noting that lim inf

n T 0 ∇g(u n (t)), un (t)dt = lim inf n T 0 d dt g(u n (t))dt ≥ T 0 d dt g(u(t))dt = T 0 ∇g(u(t)), u(t)dt
It is obvious that a linear continuous operator and a gradient do not enjoy similar properties, showing the interest of the new variant we give further. This remark has some importance in further developments. Now we present a variant dealing with the existence and uniqueness of absolutely continuous solution to the evolution inclusion of the form

f (t) + Bu(t) -A u(t) ∈ N C(t) (t, du dt (t))
where f is a continuous mapping f : I → H, A is a coercive symmetric operator, and B : H → H is a Lipschitz mapping.

Theorem 4.3. Let f : [0, T ] → H be a continuous mapping and let v : [0, T ] → R + be a non-negative nondecreasing continuous function with v(0) = 0. Let C : [0, T ] → H be a strongly compact convex valued multimapping such that haus(C(t), C(τ )) ≤ |v(t) -v(τ )| for all t, τ ∈ [0, T ].
Let A : H → H be a linear continuous coercive symmetric operator and let B : H → H be a Lipschitz mapping, that is, for some real constant M > 0, Bx -By ≤ M x -y for all x, y ∈ H for some positive constant M . Then, for any u 0 ∈ H, the evolution inclusion

f (t) + Bu(t) -A du dt (t) ∈ N C(t) ( du dt (t)) u(0) = u 0 admits a unique W 1,∞ H ([0, T ]) solution u : [0, T ] → H. Further, u(t) ≤ ρ a.e. t ∈ [0, T ], where ρ := max{ y : y ∈ C([0, T ])}.
Proof. Put I := [0, T ] and denote β := max{ f (t) : t ∈ I} (by continuity of f ). Noticing that the multimapping C(•) is upper semicontinuous from I into H endowed with the norm topology, the set C(I) is norm compact, and hence ρ := sup{ y : y ∈ C(I)} is finite and L := co C(I) is convex and norm compact.

Step I. The sequence (u n ) n is constructed as in Theorem 4.1.

Step II. With the strongly compact set L = co C(I) at hands, we see that the set X in the proof of Theorem 4.1 is strongly compact in C H (I). Since u n ∈ X we can extract from (u n ) n a (not relabeled) sequence which pointwise converges to u : I → H (i.e., u n (t) → u(t) strongly in H for each t ∈ I) such that u(t) = u 0 + t 0 u(s)ds, for all t ∈ I and ( un

) n σ(L 1 H ([0, T ]), L ∞ H ([0, T ]))-converges to u ∈ S 1
L . The inequality

u n (δ n (t)) -u n (t) ≤ ρ|δ n (t) -t|
ensures that the sequence u n (δ n (t)) n strongly converges to u(t) for each t ∈ I.

Consequently, we can follow

Step II in the proof of Theorem 4.1 by applying Proposition 3.2 in place of Proposition 3.1, to arrive to the fact that u(•) is a solution of the inclusion in the present theorem.

Step III. The arguments for the uniqueness are the same as for Theorem 4.1.

Similarly, in the proof of Theorem 4.1 employing Proposition 3.10 instead of Proposition 3.1 we easily obtain the following case with the gradient ∇g of a convex function g in place of B.

Theorem 4.4. Let f : [0, T ] → H be a continuous mapping and let v : [0, T ] → R + be a positive nondecreasing continuous function with v(0) = 0. Let C : [0, T ] → H be a strongly compact convex valued multimapping such that

haus (C(t), C(τ )) ≤ |v(t) -v(τ )| for all t, τ ∈ [0, T ].
Let A : H → H be a linear continuous coercive symmetric operator and let g be a convex function on H Lipschitz on bounded sets and continuously differentiable on H. Then, for any u 0 ∈ H, the evolution inclusion

f (t) + ∇g(u(t)) -A du dt (t) ∈ N C(t) ( du dt (t)) u(0) = u 0 admits at least a W 1,∞ H ([0, T ]) solution u : [0, T ] → H.
Further, one has u(t) ≤ ρ, where ρ := sup{ y : y ∈ C([0, T ])}.

We present another variant dealing with the existence and uniqueness of absolutely continuous solution to the evolution inclusion of the form

f (t) + Bu(t) -A u(t) ∈ ∂ϕ(t, du dt (t))
where f is a bounded continuous mapping f : I → H, A is a coercive symmetric operator, and B : H → H be a linear continuous mapping ∂ϕ is the subdifferential of a normal lower semicontinuous convex integrand ϕ.

Theorem 4.5. Let H be a separable Hilbert space.Let K be a convex compact subset of

H. Let ϕ : [0, T ] × K →] -∞, +∞] be a normal lower semicontinuous convex integrand such that (i {ϕ(., u(.)), u ∈ S 1 K } is uniformly integrable. (ii) ϕ(t, x) ≤ ϕ(τ, x)+|v(t)-v(τ )| for all t, τ ∈ [0, T ], x ∈ K where v : [0, T ] → R + is a positive nondecreasing continuous function with v(0) = 0.
Let A : H → H be a linear continuous coercive symmetric operator and B : H → H be a linear continuous mapping.

Then, for any u 0 ∈ H, the evolution inclusion

f (t) + Bu(t) -A u(t) ∈ ∂ϕ(t, du dt (t)) u(0) = u 0 admits a unique W 1,∞ H ([0, T ]) solution u : [0, T ] → H.
Proof. We will use again the Moreau 's catching-up algorithm. We consider for each n ∈ N the following partition of the interval I = [0, T ].

t n i = i T n := iη n for 0 ≤ i ≤ n. I n i :=]t n i , t n i+1 ] for 0 ≤ i ≤ n -1. Put u n 0 = u 0 and f n i = f (t n i
) for all i = 1, .., n. By Proposition 1.1 1) , there is

z n 1 ∈ K such that f n 1 + Bu n 0 -Az n 1 ∈ ∂ϕ(t n 1 , z n 1 ). Put u n 1 = u n 0 + η n z n 1 . Suppose that u n 0 , u n 1 , .., u n i , z n 1 , z n 2 , .
.z n i are constructed. As above by Proposition 1.1 1) there exists z n i+1 ∈ K such that

f n i+1 + Bu n i -Az n i+1 ∈ ∂ϕ(t n i+1 , z n i+1 ).
and we set u n i+1 = u n i + η n z n i+1 . Then by induction there are finite sequences (u n i ) n i=0 and (z n i ) n i=1 such that

f n i+1 + Bu n i -Az n i+1 ∈ ∂ϕ(t n i+1 , z n i+1 ) u n i+1 = u n i + η n z n i+1 56 From (u n i ) n i=0 , (z n i ) n i=1 (f n i ) n i=0 , we construct two sequences u n from [0, T ] to H, f n from [0, T ] to H, by setting f n (0) = f n 1 , u n (0) = u n 0 and for each i = 0, .., ..n -1 we set f n (t) = f n i+1 and u n (t) = u n i + t -t n i η n (u n i+1 -u n i ) for t ∈]t n i , t n i+1 ].
Clearly, the mapping u n (.) is Lipschitz continuous on [0, T ], and ρ is a Lipschitz constant of u n (.) on [0, T ] since for every

t ∈]t n i , t n i+1 ] un (t) = u n i+1 -u n i η n = z n i+1 ∈ K ⊂ ρB H .
Furthermore, for every t ∈ [0, T ], one has u n (t) = u 0 + t 0 un (s)ds, hence ||u n (t)|| ≤ ||u 0 || + ρT . We have

f n i+1 + Bu n i -Az n i+1 ∈ ∂ϕ(t n i+1 , z n i+1 ).
Now, let us define the step functions θ n , δ n : I -→ I by

θ n (t) = t n i+1 , δ n (t) = t n i if t ∈]t n i , t n i+1 ] and θ n (0) = δ n (0) = 0,
:= {h ∈ L 1 H ([0, T ]) : h(t) ∈ K a.e.} and let X := {v : [0, T ] → H : v(t) = u 0 + t 0 v(s)ds, t ∈ [0, T ]; v ∈ S 1 K }.
Then it is clear that S 1 K is convex and weakly compact in L 1 H ([0, T ]) (see e.g. [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF] and the references therein) and that X is convex, equicontinuous and compact in C H ([0, T ]). As (u n ) ⊂ X , one can extract from (u n ) a subsequence not relabelled which pointwise converges to u :

[0, T ] → H such that u(t) = u 0 + t 0 u(s)ds, for all t ∈ [0, T ] and ( un ) σ(L 1 H ([0, T ]), L ∞ H ([0, T ]))-converges to u ∈ S 1 K . As ϕ is normal lower semicontinuous convex, the conjugate function ϕ * : [0, T ] × H →] -∞, +∞] ϕ * (t, y) = sup x∈K [ x, y -ϕ(t, x)]
is normal, see e.g Castaing-Valadier [START_REF] Castaing | Convex Analysis and Measurable Multifunctions[END_REF] and satisfies

ϕ * (t, y) ≤ ϕ * (τ, y) + |v(t) -v(τ )|
for all t, τ ∈ [0, T ], y ∈ H using assumption (ii) ( [START_REF] Peralba | Équations d'évolution dans un espace de Hilbert, associées à des opérateurs sous-différentiels[END_REF], Proposition 27).

By using the normality of ϕ, the mappings t → ϕ(θ n (t), un (t)) and t → ϕ(t, un (t)) are measurable and integrable. By construction we have

g n (t) := f n (t) + Bu n (δ n (t)) -A un (t) ∈ ∂ϕ(θ n (t), un (t)).
For simplicity set

g(t) := f (t) + Bu(t)) -A u(t).
As f n → f pointwise strongly, u n (δ n (.)) → u(.)) pointwise strongly, and un → u weakly in L 1 E ([0, T ]), a direct application of Proposition 3.11 and its remark gives

g(t) := f (t) + Bu(t)) -A u(t) ∈ ∂ϕ(t, u(t))
a.e and finish the proof.

Remarks 1) The uniqueness of solutions follows easily from the coerciveness of the operator A. Indeed let u 1 and u 2 two solutions. then by an easy computation,

A u2 (t) -A u1 (t), u2 (t) -u1 (t) + Bu 2 (t) -Bu 1 (t), u2 (t) -u1 (t) ≤ 0 so that A u2 (t) -A u1 (t), u2 (t) -u1 (t) ≤ |B|||u 2 (t) -u 1 (t)|||| u2 (t) -u1 (t)||
By coerciveness, we deduce that

ω|| u2 (t) -u1 (t)|| 2 ≤ |B|||u 2 (t) -u 1 (t)|||| u2 (t) -u1 (t)|| Whence || u2 (t) -u1 (t)|| ≤ |B| ω ||u 2 (t) -u 1 (t)|| ≤ |B| ω t 0 || u2 (s-) u1 (s)||ds
By Gronwall lemma u1 (t) = u2 (t) a and so u

1 (t) = u2 (t) since u 1 (t) = u 0 + t 0 u1 (s)ds, ∀t ∈ [0, T ], u 2 (t) = u 0 + t 0 u2 (s)ds, ∀t ∈ [0, T ].
2) Theorem 4.5 holds if we replace the operator B by the gradient ∇g of a smooth function g.

3) Theorem 4.5 generalizes Theorem 6 in [START_REF] Castaing | Evolution problems involving time dependent subdifferential operators[END_REF] dealing with finite dimensional space.

Applications

A Skorokhod problem

We present at first a new version of the Skorokhod problem in Castaing et al [START_REF] Castaing | A Skorohod problem governed by a closed convex moving set[END_REF][START_REF] Castaing | Sweeping Process Perturbed by Rough Signal[END_REF] dealing with the sweeping process associated with an absolutely continuous (or continuous) closed convex moving set C(t) in H. Here the novelty is the velocity inside the nrmal cone operator. We will denote, as usual, by L(R d , R e ) the space of linear mappings Λ from R d to R e endowed with the operator norm

|Λ| := sup x∈R d , x R d =1 Λ(x) R e .
Given a mapping Q : (i) g(t, s, x) ≤ M g for all (t, s, x) ∈ I × I × R e ; (ii) g(t, s, x)-g(t, s, y) ≤ M g ||x-y|| for all t, s ∈ I, x, y ∈ R e with the perturbed Lebesgue integral t 0 g(t, s, x(s))ds defined on in C(I, R e ). Let a ∈ C(0). Then there exist a BVC mapping x : [0, 1] → R e and an absolutely continuous mapping u :

I → L(R d , R e )
[0, 1] → R e satisfying              x(0) = u(0) = a x(t) = h(t) + k(t) + Bu(t), ∀t ∈ I h(t) = t 0 b(τ, x(τ ))dz τ , ∀t ∈ I k(t) = t 0 g(t, s, x(s))ds, ∀t ∈ I t 0 b(s, x(s))dz s + t 0 g(t, s, x(s))ds + Bu(t) -A u(t) ∈ N C(t) ( u(t)) a.e. t ∈ I Proof. Let M := max{M b , M g }. Set for all t ∈ I = [0, 1] x 0 (t) = a, h 1 (t) =
h 1 (t) -h 1 (s) ≤ M |z| 1-var:[s,t]
for all 0 ≤ s ≤ t ≤ 1, and in particular

h 1 (t) ≤ M |z| 1-var:[0,t] ≤ M |z| 1-var:[0,1] for all t ∈ [0, 1]. Let us set for all t ∈ I = [0, 1] k 1 (t) = t 0
g(t, s, x 0 (s))ds for all t ∈ I = [0, 1] and note that k 1 is continuous with k 1 (t) ≤ M for all t ∈ I. By an easy computation, using conditions (i) and (ii) we have the estimate k 1 (t) -k 1 (τ ) ≤ M |t -τ |, for all τ, t ∈ I. By Theorem 4.1 there is a unique absolutely continuous mapping u 1 : I → H solution of the problem

u 1 (0) = a h 1 (t) + k 1 (t) + Bu 1 (t) -A u1 (t) ∈ N C(t) ( u1 (t)) a.e. t ∈ I with u 1 (t) = a + t 0 u1 (s)ds for all t ∈ I and u1 (t) ≤ M a.e. t ∈ I. Set x 1 (t) = h 1 (t) + k 1 (t) + Bu 1 (t) = t 0 b(τ, x 0 (τ )dz τ + t 0 g(t, s, x 0 (s))ds + Bu 1 (t).
Then x 1 is BVC with x 1 (0) = a. Now we construct x n by induction as follows. Let for all t ∈ I

h n (t) = t 0 b(τ, x n-1 (τ ))dz τ k n (t) = t 0 g t, s, x n-1 (s) ds. Then ||k n (t) -k n (τ )|| ≤ M |t -τ |,
for all τ, t ∈ I with k n (t) ≤ M for all t ∈ I. By Proposition 2.2 in Friz-Victoir [START_REF] Friz | Multidimensional Stochastic Processes as Rough Paths[END_REF] we have the estimate

h n (t) -h n (s) ≤ M |z| 1-var:[s,t] (5.1)
for all 0 ≤ s ≤ t ≤ 1, and in particular

h n (t) ≤ M |z| 1-var:[0,t] ≤ M |z| 1-var:[0,1] (5
(t) = h n (t)+k n (t)+Bu n (t) = t 0 b(τ, x n-1 (τ ))dz τ + t 0 g t, s, x n-1 (s) ds+Bu n (t) so that x n is BVC.
As (u n ) n is equi-Lipschitz continuous (with M as Lipschitz constant) we may suppose that (u n ) n converges uniformly to a Lipschitz continuous mapping u : I → H with u(t) = a + t 0 u(s)ds for all t ∈ I and with u(t) ≤ M for a.e. t ∈ I. We may also suppose that ( un ) n weakly converges in L 1 H (I) to u, and by Arzela-Ascoli theorem we may suppose that (k n ) n converges uniformly to a continuous mapping k : I → H. Thanks to (5.1) (h n ) n is bounded and equicontinuous. By Arzela-Ascoli theorem again, we may suppose that (h n ) n converge uniformly to a continuous mapping h. Similarly (k n ) n is bounded and equi-Lipschitz. By Arzeala-Ascoli theorem, we may suppose that (k n ) n converges uniformly to a continuous mapping k. Then putting x n (t) = h n (t)+k n (t)+Bu n (t) and x(t) := h(t)+k(t)+Bu(t), we see that (x n ) n converges uniformly to x, and b(., x n-1 (.)) n converges uniformly to b(., x(.)) according to the Lipschitz condition (b). Therefore, by Friz-Victoir [25, Proposition 2.7] t 0 b(τ, x n-1 (τ ))dz τ converge uniformly in t ∈ I to t 0 b(τ, x(τ ))dz τ as n → ∞. By hypothesis (ii), g(t, s, x n-1 (s)) converges to g(t, s, x(s)) for every (t, s) ∈ I × I, hence t 0 g(t, s, x n-1 (s))ds → t 0 g(t, s, x(s))ds for each t ∈ I by Lebesgue dominated convergence theorem. So we can write From the inclusion h n (t) + k n (t) + Bu n (t) -A un (t)) ∈ N C(t) ( un (t)) a.e. t ∈ I and the above convergence, applying Proposition 3.1 we obtain

x(t) = lim n→∞ x n (t) = lim n→∞ t 0 b(τ, x n-1 (τ ))dz τ + lim
h(t) + k(t) + Bu(t) -A u(t) ∈ N C(t) ( u(t)) a.e. t ∈ I.
The proof is therefore complete.

Our tools allow to state several variants of Theorem 5.1 according to the nature of the perturbation and the operator. Actually Theorem 5.1 holds true if B : H → H is a Lipschitz mapping, that is, there is some real constant M > 0 such that Bx -By ≤ M x -y for all x, y ∈ H. Theorem 5.1 is still valid if we replace B by the gradient ∇g of a positive convex function g : H → R Lipschitz on bounded sets and continuously differentiable.

Towards an application in Optimal Control problem

In the previous results we have developed the Skorokhod problem associated with the sweeping process with Riemann-Stieltjes integral perturbation. This leads to study the following optimal control problem.

|Λ| := sup x∈R d , x R d =1 Λ(x) R e .
Let us consider a continuous integrand operator b :

[0, 1] × R e → L(R d , R e ) satisfying (a) |b(t, x)| ≤ M for all (t, x) ∈ I × R e ; (b) |b(t, x) -b(t, y)| ≤ M x -y R e for all (t, x, y) ∈ I × R e × R e .
Let V : R d → R e be a bounded continuous mapping. Let L : [0, 1]×R e ×R e ×R e → [0, ∞[ be a lower semicontinuous integrand such that L(t, x, y, .) is convex on R e for every (t, x, y) ∈ [0, 1] × R e × R e . Then the problem of minimizing the cost function

1 0 L(t, x(t), y(t), ẏ(t))dt subject to        dx t = V (x t )dz t , t ∈ [0, 1] x 0 = ψ ∈ R e y(0) = y 0 ∈ C(0) t 0 b(τ, x(τ ))dz τ + By(t) -A ẏ(t) ∈ N C(t
(u,v)∈Y T 0 L(t, u(t), v(t), v(t))dt,
where Y is the set of solutions (x, y) to the above dynamical system. First by [START_REF] Friz | Multidimensional Stochastic Processes as Rough Paths[END_REF]Theorem 3.4] we assert that the C 1-var (I, R e )-solution set to We then conclude that (x, y) is an optimal solution.

dx t = V (x t )dz t , t ∈ I x 0 = ψ ∈ R e
Several variants of the preceding theorem are available using Theorems 4.3 and 4.4 along with Propositions 3.2 and 3.10.

In the following we will examine a Bolza problem and its relaxation associated with a Young integral perturbation of a sweeping process with a Lipschitzian moving compact set C(t), say haus R e (C(t), C(s)) ≤ α|t -s|. First, we need some notation and background on Young integral and Young measures in this special context. Young integral. Let z ∈ C 1-var ([0, T ], R d ), that is, z is a bounded variation continuous mapping defined on [0, T ] with values in R d . We remind that L(R d , R e ) denotes the space of linear mappings Λ from R d to R e endowed with the operator norm |Λ| := sup

x∈R d , x R d =1 |Λ(x)| R e .
Let us consider a continuous integrand operator b :

[0, T ] × R e → L(R d , R e ) satisfying (B 1 ) : |b(t, x)| ≤ M, ∀x ∈ R e (B 2 ) : |b(t, x) -b(τ, y)| ≤ ρ(t) -ρ(τ ) + M x -y R e , 0 ≤ τ ≤ t ≤ T, ∀x, y ∈ R e
where ρ : [0, T ] → R + is a positive nondecreasing continuous function and M is a positive constant. If a sequence (u n ) n of continuous mappings from [0, T ] into R e is uniformly bounded and uniformly bounded in variation, then the sequence (y n ) n , with y n (t) = b(t, u n (t)), is formed with mappings which are continuous, uniformly bounded and uniformly bounded in variation from [0, T ] to L(R d , R e ), shortly

y n ∈ C 1-var ([0, T ], L(R d , R e )
). Indeed we have

|y n (t) -y n (τ )| ≤ ρ(t) -ρ(τ ) + M u n (t) -u n (τ ) R e
for all τ ≤ t ≤ T , so that sup n |y n | 1-var;[s,t] < ∞ for all 0 ≤ s ≤ t ≤ T . As consequence the Young integral t 0 y n (s)dz s of y n against z is well-defined and belongs to C 1-var ([0, T ], R e ) according to Friz-Victoir [START_REF] Friz | Multidimensional Stochastic Processes as Rough Paths[END_REF], with the following estimates ) is a simple consequence of the Valadier result mentioned above since any BVRC mappin,g is Borelian. Take and ).

t s y n (τ )dz τ ≤ 1 1 -2 1-θ |z| 1-var;[s,t] |y n | 1-var;[s,t] + |y n (s)| z(t) -z(s) R d ≤ 1 1 -2 1-θ |z| 1-var;[s,t] |y n | 1-var;[s,t] + M z(t) -z(s) R d for all 0 ≤ s ≤ t ≤ T with θ = 2 and . 0 y n (τ )dz τ 1-var;[s,t] ≤ C(1, 1)|z| 1-var;[s,t] |y n | 1-var;[s,t] + |y n | ∞;[s,t] for all 0 ≤ s ≤ t ≤ T . As consequence t s y n dz ≤ 1 1 -2 1-θ |z| 1-var;[s,t] |y n | 1-var;[s,t] + |y n (s)| z(t) -z(s) R d ≤ 1 1 -2 1-θ |z| 1-var;[s,t] |y n | 1-var;[s,t] + M ||z(t) -z(s)|| R d ≤ 1 1 -2 1-θ |z| 1-var;[s,t]
(P S ∞ K ) t 0 ζ(s)ds + Bu(t) + A u(t) ∈ N C(t) ( u(t)), t ∈ I, ζ ∈ S ∞ K u(0) = a ∈ C(0).
any ζ ∈ V ∞ K . As S BV RC K is sequentially dense in V ∞ K with respect to the σ(L ∞ E , L 1 E ) topology, there exists a sequence (ζ n ) in S BV RC K such that (ζ n ) σ(L ∞ E , L 1 E ) converge to ζ. For simplicity set f n (t) =
This holds for every

ζ ∈ V ∞ K , hence inf (P V ∞ K ) ≥ inf (P S BV RC K
).

For simplicity we considered in Theorem 5.2 a simple perturbation control integral in the form Young measures. For the sake of completeness of the next development of Theorem 5.3, we summarize some useful facts concerning Young measures. Let (Ω, F, P ) be a complete probability space. Let X be a Polish space and let C b (X, R) be the space of all bounded continuous functions defined on X. Let M 1 + (X) be the set of all Borel probability measures on X equipped with the narrow topology. A Young measure ν : Ω → M 1 + (X) is, by definition, a scalarly measurable mapping from Ω into M 1 + (X), that is, for every f ∈ C b (X, R), the mapping ω → f, ν ω := X f (x) dν ω (x) is F-measurable. A sequence (ν n ) in the space of Young measures Y(Ω, F, P ; M 1 + (X)) stably converges to a Young measure ν ∈ Y(Ω, F, P ; M 1 + (X)) if the following holds:

lim n→∞ A X f (x) dν n ω (x) dP (ω) = A X f (x) dν ω (x) dP (ω)
for every A ∈ F and for every f ∈ C b (X, R).

In the remainder Z is a compact metric space, M 1 + (Z) is the space of all Radon probability measures on Z. We will endow M 1 + (Z) with the narrow topology so that M 1 + (Z) is a compact metrizable space. For I := [0, 1] let us denote by Y(I; M 1 + (Z)) the space of all Young measures defined on I endowed with the stable topology so that Y(I; M 1 + (Z)) is a compact metrizable space with respect to this topology. By the Portmanteau Theorem for Young measures [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF]Theorem 2

.1.3], a sequence (ν n ) in Y(I; M 1 + (Z)) stably converges to ν ∈ Y(I; M 1 + (Z)) if lim n→∞ T 0 Z h t (z)dν n t (z) dt = T 0 Z h t (z)dν t (z) dt
for all h ∈ L 1 (I, C R (Z)); here C R (Z) denotes the space of all continuous real valued functions defined on Z endowed with the norm of uniform convergence. Finally let Γ be a measurable multimapping defined on I with nonempty compact values in Z and let S Γ be the set of all Lebesgue measurable selections of Γ (alias original controls).

Let C : [0, T ] ⇒ R e be a compact valued Lipschitzian multimapping and let f : I × Z → R e be a mapping satisfying [START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF] for every fixed t ∈ I, f (t, •) is continuous on Z, (2) for every z ∈ Z, f (•, z) is Lebesgue-measurable on I;

(3) there is a constant M > 0 such that f (t, z) ≤ M for all (t, z) in I × Z.

We aim to present some relaxation problems in the framework of Optimal Control Theory. We consider the evolution inclusion (PO) associated with original controls

(PO) t 0 f (s, ζ(s))ds + Bu ζ (t) -A uζ (t) ∈ N C(t) ( uζ (t))), a.e. t ∈ I, u ζ (0) = u 0 ∈ C(0),
where ζ belongs to the set Z := S 1 Γ of all original controls, which means that ζ is a Lebesgue-measurable selection of Γ, and the evolution inclusion (PR) associated with relaxed controls

(PR) t 0 [ Z f (s, z)ν s (dz)]ds + Bu ν (t) -A uν (t) ∈ N C(t) ( uν (t))), a.e. t ∈ I u ν (0) = u 0 ∈ C(0),
where ν belongs to the set R := S Σ of all relaxed controls, which means that ν is a Lebesgue-measurable selection of the multimapping Σ defined by

Σ(t) := σ ∈ M 1 + (Z) : σ(Γ(t)) = 1
for all t ∈ I. Note that, for ν ∈ R, the mapping

h ν : (t, z) → Z f (t, z)ν t (dz)
inherits the properties (1) for every fixed t ∈ I, h ν (t, •) is continuous on Z;

(2) for every z ∈ Z, h ν (•, z) is Lebesgue-measurable on I;

(3) there is a constant M > 0 such that h ν (t, z) ≤ M for all (t, z) in I × Z. Consequently, for each ζ ∈ Z (resp. ν ∈ R), the evolution inclusion (PO) (resp. (PR)) has a unique Lipschitz continuous solution. Moreover, there is an a priori bound for the Lipschitz ratio of solutions which easily implies that the solution sets (SO) and (SR) (to (PO) and (PR)) are equi-Lipschitz.

We can now prove the following theorem establishing some topological properties of the solution sets (S O ) and (S R ), namely we obtain the typical relaxation result that the former is dense in the latter. Then the following hold: (a) the solution set (S R ) to

(PR) t 0 [ Z f (s, z)ν s (dz)]ds+Bu ν (t)-A uν (t) ∈ N C(t) ( uν (t))), a.e. t ∈ I, ν ∈ R u ν (0) = u 0 ∈ C(0) is nonempty and compact in C(I, R e ). (b) the solution set (S O ) to (PO) t 0 f (s, ζ(s))ds + Bu ζ (t) -A uζ (t) ∈ N C(t) ( uζ (t))), a.e. t ∈ I, ζ ∈ Z u ζ (0) = u 0 ∈ C(0)
is dense in (S R ) with respect to the topology of uniform convergence.

Proof. (a) By Theorem 4.1, the solution set (S R ) is bounded and equi-Lipschitz. Then (S R ) is relatively compact in C(I, R e ), by Arzela-Ascoli theorem. Therefore, for (ν n ) ⊂ R, there is a subsequence still denoted by (u ν n ) which converges uniformly to a Lipschitz continuous mapping u ∞ with u∞ (t) ≤ K a.e. t ∈ I and such that also ( uν n ) σ(L 1 (I, R e ; dt), L ∞ (I, R e ; dt))-converges to u∞ . As R is compact and metrizable for the stable topology, we may suppose that (ν n ) stably converges to ν ∞ ∈ R. Since the continuous functions g n and g, given for all t ∈ I by

g n (t) := t 0 [ Z f (s, z)ν n s (dz)]ds and
g(t) := t 0 [ Z f (s, z)ν ∞ s (dz)]ds,
are uniformly bounded, and since g n (t) → g(t) for every t ∈ I, from the inclusion

g n (t) + Bu n ν (t) -A uν (t) ∈ N C(t) ( un ν (t)))
and Proposition 3.2, we deduce that

g(t) + Bu ν ∞ (t) -A uν ∞ (t) ∈ N C(t) ( uν ∞ (t))).
This proves the first part of the theorem.

(P R ) t 0 [ Z f (s, z)ν s (dz)]ds + Bu ν (t) -A uν (t) ∈ N C(t) ( uν (t))), a.e. t ∈ I, ν ∈ R u ν (0) = u 0 ∈ C(0)
and the problem

(P O ) t 0 f (s, ζ(s))ds + Bu ζ (t) -A uζ (t) ∈ N C(t) ( uζ (t))), a.e. t ∈ I, ζ ∈ Z u(0) = a ∈ C(0)
Then one has inf (P R ) = inf (P O ) and inf

(P R ) T 0 t 0 b(s, u(s))dz s , t 0 [ Z h(s, z)ν s (dz)]ds dt has a minimizer. Proof. The inequality inf (P R ) ≥ inf (P O ) is clear. Fix any ν ∈ R. Let (ζ n ) n in Z with lim n t 0 f (s, ζ n (s))ds = t 0 [ Z f (s, z)ν s (dz)
]ds for all t ∈ I.

Let u n be the unique Lipschitz solution to

-un (t) ∈ N C(t) u n (t) + t 0 f (s, ζ n (s))ds, a.e. t ∈ I u n (0) = a ∈ C(0)
and let v be the unique Lipschitz solution to

-v(t) ∈ N C(t) v(t) + t 0 [ Z f (s, z)ν s (dz)]ds, a.e. t ∈ I v(0) = a ∈ C(0).
In view of the first step of the proof of Theorem 5.3 the sequence (u n ) n converges uniformly to v. For simplicity set g n (t) =

(t) = t 0 [ Z f (s, z)ν s (dz)]ds, so that lim n g n (t), k n (t) = g(t), k(t) . Since (g n ) n , g, (k n ) n , k are uniformly bounded, we deduce that lim n→∞ T 0 t 0 b(s, u n (s))dz s , t 0 h(s, ζ n (s))ds dt = lim n→∞ T 0 g n (t), k n (t) dt = T 0 g(t), k(t) dt = t 0 b(s, v(s)dz s , t 0 [ Z h(s, z)ν s (dz)]ds dt. As T 0 t 0 b(s, u n (s)dz s , t 0 h(s, ζ n (s))ds dt ≥ inf(P O )
for all n ∈ N, it follows by taking the limit that

T 0 t 0 b(s, v(s))dz s , t 0 [ Z h(s, z)ν s (dz)]ds dt ≥ inf(P O )
Since this holds for every ν ∈ R, we conclude that inf (P R ) ≥ inf (P O ).

Towards fractional inclusion coupled with EVI and sweeping process

Now given I = [0, 1] we investigate a class of boundary value problems governed by a fractional differential inclusion (FDI) (5.3) in the separable Hilbert space H coupled with the evolution inclusion governed by the (EVI) (5.4) and sweeping process (5.6) below.

D α h(t) + λD α-1 h(t) = u(t), t ∈ I, I β 0 + h(t) | t=0 := lim t→0 t 0 (t -s) β-1 Γ(β) h(s)ds = 0, h(1) = I γ 0 + h(1) = 1 0 (1 -s) γ-1 Γ(γ) h(s)ds, (5.3
) f (t, h(t)) + Bu(t) -A u(t) ∈ ∂ϕ(t, u(t)) a.e. t ∈ I and f (t, h(t)) + Bu(t) -A u(t) ∈ N C(t) ( u(t)) a.e. t ∈ I, where α ∈]1, 2], β ∈ [0, 2-α],
λ ≥ 0, γ > 0 are given constants, D α is the standard Riemann-Liouville fractional derivative , Γ is the Classical gamma function.

For the convenience of the reader, we begin with a few reminders of the concepts which will be used in this subsection. Definition 5.1 (Fractional Bochner integral). Let E be a separable Banach space and f : I = [0, 1] → E. The fractional Bochner-integral of order α > 0 of the function f is defined by

I α 0 + f (t) := t 0 (t -s) α-1 Γ(α) f (s)ds, t > 0.
In the above definition, the sign " " denotes the classical Bochner integral.

D α f (t) := D α 0 + f (t) = d n dt n I n-α 0 + f (t) = d n dt n t 0 (t -s) n-α-1 Γ(n -α) f (s)ds,
where n = [α] + 1 and [α] is the integer part of α.

We denote by W α,1 B,E (I) the space of all continuous functions in C E (I) such that their Riemann-Liouville fractional derivatives of order α -1 are continuous and their Riemann-Liouville fractional derivatives of order α are Bochner integrable. Green function and its properties. In all the rest of this subsection

α ∈]1, 2], β ∈ [0, 2 -α], λ ≥ 0, γ > 0. Let G : [0, 1] × [0, 1] → R be the Green function defined by G(t, s) = ϕ(s)I α-1 0 + (exp(-λt)) +    exp(λs)I α-1 s + (exp(-λt)), 0 ≤ s ≤ t ≤ 1, 0, 0 ≤ t ≤ s ≤ 1, (5.4
) where

ϕ(s) = exp(λs) µ 0 I α-1+γ s + (exp(-λt)) (1) -I α-1 s + (exp(-λt)) (1) (5.5) with µ 0 = I α-1 0 + (exp(-λt)) (1) -I α-1+γ 0 + (exp(-λt)) (1). ( 5.6)
We recall and summarize a useful result ( [START_REF] Castaing | On fractional differential inclusions with nonlocal boundary conditions[END_REF]).

Lemma 5.6. Let E be a separable Banach space and let G be the function defined by (5.4)-(5.6). For α ∈]1, 2] the following hold:

(a) G(•, •) satisfies the estimate

|G(t, s)| ≤ 1 Γ(α) 1 + Γ(γ + 1) |µ 0 |Γ(α)Γ(γ + 1) + 1 = M G . (b) If u ∈ W α,1 B,E ([0, 1]) satisfying boundary conditions (5.3), then u(t) = 1 0 G(t, s) D α u (s) + λD α-1 u(s) ds for every t ∈ [0, 1]. {u f : f ∈ S 1 X } is bounded, equicontinuous in C E (I).
To show first the relative compactness take any sequence (u fn ) n in X . We note that, for each n ∈ N, we have u fn ∈ W α,1

B,E (I) , and

u fn (t) = 1 0 G(t, s)f n (s)ds, t ∈ I,
with by Lemma 5.6

• I β 0 + u fn (t)| t=0 = 0, u fn (1) = I γ 0 + u(1), • D α-1 u fn (t) = t 0 exp(-λ(t -s))f n (s)ds + exp(-λt) 1 0 ϕ(s)f n (s)ds, t ∈ I, • (D α u fn ) (t) + λ D α-1 u fn (t) = f n (t), t ∈ I.
Consider any t 1 , t 2 ∈ I with t 1 < t 2 . Let us write

u fn (t 2 ) -u fn (t 1 ) = 1 0 G(t, s)(f n (s) -f n (s))ds = 1 0 ϕ(s)f n (s)ds t 2 0 e -λτ Γ(α -1) (t 2 -τ ) α-2 dτ - t 1 0 e -λτ Γ(α -1) (t 1 -τ ) α-2 dτ + t 2 0 e λs t 2 s (t 2 -τ ) α-2 Γ(α -1) e -λτ dτ f n (s)ds - t 1 0 e λs t 1 s e -λτ Γ(α -1) (t 1 -τ ) α-2 dτ f n (s)ds,
which gives with Φ(t, τ ) := (t -τ ) α-2 /Γ(α -1)

u fn (t 2 ) -u fn (t 1 ) = 1 0 ϕ(s)f n (s)ds t 1 0 e -λτ Φ(t 2 , τ ) -Φ(t 1 , τ ) dτ + t 2 t 1 e -λτ
e -λτ (t 1 -τ ) α-2 -(t 2 -τ ) α-2 Γ(α -1) dτ + 1 0 |ϕ(s)| + e λs |X(s)|ds t 2 t 1 e -λτ (t 2 -τ ) α-2 Γ(α -1) dτ + t 2 t 1 e λs |X(s)|ds t 2 t 1 e -λτ (t 2 -τ ) α-2 Γ(α -1) dτ.
It is easy to see, after an integration by parts, that

t 2 t 1 e -λτ (t 2 -τ ) α-2 Γ(α -1) dτ = e -λt 1 (t 2 -t 1 ) α-2 Γ(α) + λ t 2 t 1 e -λτ (t 2 -τ ) α-1 Γ(α) dτ ≤ 1 + λ Γ(α) (t 2 -t 1 ) α-1
and

t 1 0 e -λτ (t 1 -τ ) α-2 -(t 2 -τ ) α-2 Γ(α -1) dτ ≤ t 1 0 (t 1 -τ ) α-2 -(t 2 -τ ) α-2 Γ(α -1) dτ = (t 2 -t 1 ) α-1 + t α-1 1 -t α-1 2 Γ(α) .
Using for p ∈]0, 1] the inequality |a p -b p | ≤ |a -b| p for all a, b ≥ 0, we deduce that

t 1 0 e -λτ (t 2 -τ ) α-2 -(t 1 -τ ) α-2 Γ(α -1) dτ ≤ 2 Γ(α) (t 2 -t 1 ) α-1 .
Then, since α ∈]1, 2], we can estimate u fn (t 2 ) -u fn (t 1 ) by

u fn (t 2 ) -u fn (t 1 ) ≤ K|t 2 -t 1 | α-1 with K = 1 0 (3 + λ)|φ(s)| + (4 + 2λ
)e λs |X(s)|ds. This shows that {u fn : n ∈ N} is equicontinuous in C E (I). Moreover, for each t ∈ I the set {u fn (t) : n ∈ N} is contained in the compact convex set 1 0 G(t, s)X(s)ds [?, 24], so that X is relatively compact in C E (I) as claimed. It remains to justify that X is closed in [START_REF] Castaing | Convex Analysis and Measurable Multifunctions[END_REF]) we may suppose that (

C E (I). Let (u fn ) n in X converging to u ∞ in C E (I). As S 1 X is σ(L 1 E , L ∞ E * )-compact (see, e.g,
f n ) n σ(L 1 E , L ∞ E * )-converges to f ∞ ∈ S 1 X . Then (u fn ) n pointwise weakly converges to u f∞ , with u ∞ ∈ C E (I) given by u f∞ (t) = 1 0 G(t, s)f ∞ (s)ds. Therefore, for each t ∈ I u ∞ (t) = w-lim n→∞ u fn (t) = w-lim n→∞ 1 0 G(t, s)f n (s)ds = 1 0 G(t, s)f ∞ (s)ds = u f∞ (t),
so u ∞ = u f∞ , and the desired closedness of X in C E (I) is confirmed. The proof of the theorem is complete.

We can now state and prove the theorem concerned with a fractional inclusion coupled with an EVI. Theorem 5.9. Let I := [0, 1] and H be a separable Hilbert space. Let K be a compact convex equilibrated subset of H. Let ϕ : I × H →] -∞, +∞] be a normal lower semicontinuous convex integrand such that dom ϕ(t, •) = K for all t ∈ I and Theorem 5.16. Let I := [0, 1] and H be a separable Hilbert space. Let K be a convex compact equilibrated subset of H. Let ϕ : [0, 1] × K →] -∞, +∞] be a normal lower semicontinuous convex integrand such that (i {ϕ(., u(.)), u ∈ S 1 K } is uniformly integrable. where A(t) is a time dependent Lipschitz variation single valued maximal monotone operator and ∇ϕ is the gradient of a smooth Lipschitz function ϕ is stated, by constrast with classical cases dealing with some special fixed operators; cf. Attouch et al [START_REF] Attouch | The Dynamics of Elastic Shocks via Epigraphical Regularization of a Differential Inclusion . Barrier and Penalty Approximations[END_REF], Paoli [START_REF] Paoli | An existence result for non-smooth vibro-impact problem[END_REF] and Schatzman [START_REF] Schatzman | Problèmes unilatéraux d' évolution du second ordre en temps[END_REF]. Problems of second order have some importance in Mechanics [START_REF] Monteiro Marques | Differential inclusions nonsmooth mechanical problems, shocks and dry friction[END_REF], which may require a more general evolution inclusion of the form 0 ∈ ü(t) + A(t) u(t) + ∂ϕ(u(t)), t ∈ [0, T ] here ∂ϕ(u(t)) denotes the subdifferential of a proper lower semicontinuous convex function ϕ at the point u(t). Existence for this problem is stated in [START_REF] Castaing | Second order evolution problems with time dependent maximal monotone operator and applications[END_REF] via a variational approach. In this spirit, existence of solution for a second order problem dealing with time and state dependent maximal monotone A t,x and multivalued perturbation F (t, x, y) of the form -ü(t) ∈ A t,u(t) u(t) + F (t, u(t), u(t)) is given in [START_REF] Castaing | On a fractional differential inclusion with boundary conditions and application to subdifferential operators[END_REF]. Second order evolution inclusion under consideration require concise and original proofs. We will give below a main variational limit result which help to give a meaning of the variational limit solution to the second order evolution inclusion of the form f (t) ∈ ü(t) + A u(t) + ∂ϕ(u(t))

(ii) ϕ(t, x) ≤ ϕ(τ, x) + |v(t) -v(τ )| for all t, τ ∈ [0, 1], x ∈ K where v : [0, 1] → R + is
where A is a linear continuous coercive symmetric operator and ∂ϕ is the subdifferential of a lower convex lower semi continuous function and f ∈ L 2 H ([0, T ]). In this context our results contains novelties since the second velocity is in inside the subdifferential (resp. the normal cone). These variants are not comparable.

We recall below some notations and summarize some results which describe the limiting behavior of a bounded sequence in L 1 H ([0, T ]). See ( [START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF], Proposition 6.5.17). Let us recall a useful Gronwall type lemma [START_REF] Castaing | Functional evolution equations governed by nonconvex sweeping process[END_REF].

Lemma 5.17. (A Gronwall-like inequality.) Let p, q, r : [0, T ] → [0, ∞[ be three nonnegative Lebesgue integrable functions such that for almost all t ∈ [0, T ] r(t) ≤ p(t) + q(t) t 0 r(s) ds.

Then r(t) ≤ p(t) + q(t) Here is a main variational limit result which help to give a meaning of the variational limit solution to the second order evolution inclusion f (t) ∈ ü(t) + A u(t) + ∂ϕ(u(t))

where A is a linear continuous coercive symmetric operator and ∂ϕ is the subdifferential of a lower convex lower semi continuous function ϕ and f ∈ L Assume that (i) satisfies the inclusion ϕ n (u n (t)) < +∞.

f n σ(L 2 R d , L 2 R d )-converges to f ∞ ∈ L 2 R d ([0, T ]), (ii) ϕ n epi-converges to ϕ ∞ , (iii) lim n u n (0) = u ∞ 0 ∈ dom ϕ ∞ , lim n ϕ n (u n (0)) = ϕ ∞ (u ∞ 0 ),
f ∞ (t) -ζ ∞ (t) -A u∞ (t) ∈ ∂ϕ ∞ (u ∞ (t
Step 2 Estimation of ||ü n (.)||. As z n (t) := f n (t) -ün (t) -A un (t) ∈ ∂ϕ n (u n (t))

by the subdifferential inequality for convex lower semi continuous functions we have

ϕ n (x) ≥ ϕ n (u n (t)) + x -u n (t), z n (t)
for all x ∈ R d . Now let v ∈ B L ∞ R d ([0,T ]) , the closed unit ball of L ∞ R d [0, T ]). By taking x = w(t) := x 0 + r 0 v(t) in the preceding inequality we get ϕ n (w(t)) ≥ ϕ n (u n (t)) + w(t) -u n (t), z n (t) . which satisfies the property: for any nonnegative measure θ on [0, T ] with respect to which n s is absolutely continuous here h ϕ * ∞ denotes the recession function of ϕ * ∞ . Indeed, as n belongs to ∂J ϕ∞ (u ∞ ) by applying Theorem 5 in [START_REF] Rockafellar | Integrals which are convex functionals, II[END_REF] we have Recall that ]) -I ϕ∞ (u ∞ ) From (3.3.13) we deduce

Integrating the preceding inequality gives

dn a dt = f ∞ -ζ ∞ -A u∞ ∈ ∂I ϕ∞ (u ∞ ) that is (3.3.13) I ϕ * ∞ ( dn a dt ) = f ∞ -ζ ∞ -A u∞ , u ∞ L 1 R d ([0,T ]),L ∞ R d ([0,T
J * ϕ∞ (n) = u ∞ , n C R d ([0,T ]),C R d ([0,T ]) -J ϕ∞ (u ∞ ) = u ∞ , n C R d ([0,T ]),C R d ([0,T ]) -I ϕ∞ (u ∞ ) = T 0 u ∞ (t), f ∞ (t) -ζ ∞ (t) -A u∞ (t) dt + T 0 u ∞ (t

. 1 )

 1 and N S (x) = ∅ if x ∈ H \ S. The Legendre-Fenchel conjugate δ * (•, S) : H → R ∪ {+∞} coincides with the support function of S, that is, δ * (y, S) = sup x∈S y, x for all y ∈ H. If S, S are both nonempty closed bounded convex sets of H, the Hausdorff distance between S and S can be defined by haus (S, S ) = sup x∈H |d S (x) -d S (x)|,

Z 1 Z

 1 x, ζ(t) dt ≤ lim sup n Z δ * (x, C(θ n (t)))dt ≤ Z lim sup n δ * (x, C(θ n (t))) ≤ Z δ * (x, C(t))dt.

  t ∈ I according to the description (2.1) of the normal cone. Similarly to Proposition 3.4, when C(θ n (t) = C(t), the next result relaxes the condition C(t) ⊂ rB H in Proposition 3.10. Proposition 3.5. Let H be a separable Hilbert space and I = [0, T ]. Let C : I ⇒ H be a scalarly measurable multimapping with closed convex weakly locally compact values containing no line, for which there exist r ∈ L ∞ R + (I) such that C(t) ∩ r(t)B H = ∅ for all t ∈ I. Let A : H → H be a continuous symmetric linear coerciveoperator and let g : H → R be a convex function Lipschitz on bounded sets and continuously differentiable on H.

Theorem 3 . 5 .

 35 Let f : [0, T ] → H = R e be a continuous mapping and let v : [0, T ] → R + be a positive nondecreasing continuous function with v(0) = 0. Let C : [0, T ] → H be a compact convex valued multimapping such that haus (C(t), C(τ )) ≤ |v(t) -v(τ )| for all t, τ ∈ [0, T ].

B

  un (t), un (t) dt + Z -f n (t) -A λn u n (δ n (t)), un (t) dt ≤ 0 by passing to the limit when n goes to ∞ in this equality using (1.1.1)-(1.1.7) gives(1.1.8)

Theorem 4 . 1 .

 41 Let f : [0, T ] → H be a continuous mapping and let v : [0, T ] → R + be a positive nondecreasing continuous function with v(0) = 0. Let C : [0, T ] → H be a weakly compact convex valued multimapping such that

Theorem 5 . 1 .

 51 on a compact interval I, it will be convenient to write |Q(•)| ∞:I := sup t∈I |Q(t)|. Let I := [0, 1] and H = R e . Let v : I → R + be a positive nondecreasing continuous function with v(0) = 0. Let C : I ⇒ R e be a compact convex valued multimapping such that(i) there is a real constant M > 0 such that C(t) ⊂ M B H for all t ∈ I; (ii) haus (C(t), C(τ )) ≤ |v(t) -v(τ )| for all t, τ ∈ I.Let A : R e → R e be a coercive symmetric linear operator and let B : R e → R e be a linear operator. Let z ∈ C 1-var ([0, 1], R d ), the space of continuous mappings of bounded variation defined on [0, 1] with values in R d . Let b : I × R e → L(R d , R e ) be a continuous integrand operator satisfying for some real M b > 0 (a) |b(t, x)| ≤ M b for all (t, x) ∈ I × R e ; (b) |b(t, x) -b(t, y)| ≤ M b x -y R e for all (t, x, y) ∈ I × R e × R e with the perturbed Riemann-Stieljies integral t 0 b(τ, x(τ ))dz τ defined for x ∈ C(I, R e ). Let g : I × I × R e → R e be a continuous mapping satisfying for real M g > 0:

  , a)dz τ , so by Proposition 2.2 in Friz-Victoir [25], we have t 0 b(τ, a)dz τ ≤ |b(., a)| ∞:[0,1] |z| 1-var:[0,t] . , a)dz τ , we see by condition (a) that

  , s, x n-1 (s))ds + lim n→∞ Bu n (t) = t 0 b(τ, x(τ ))dz τ + t 0 g(t, s, x(s))ds + Bu(t).

Proposition 5 . 1 .

 51 Let I := [0, 1] and H = R e . Let v : I → R + be a positive nondecreasing continuous function with v(0) = 0. Let C : I ⇒ R e be a compact convex valued multimapping such that (i) C(t) ⊂ M B R e for all t ∈ I where M is a positive constant; (ii) haus R e (C(t), C(τ )) ≤ |v(t) -v(τ )| for all t, τ ∈ I. Let A : R e → R e be a coercive symmetric linear operator and let B : R e → R e be a clinear operator. Let z ∈ C 1-var ([0, 1], R d ), the space of continuous functions of bounded variation defined on [0, 1] with values in R d . Let L(R d , R e ) be the space of linear mappings Λ from R d to R e endowed with the operator norm

1 0L 1 0L

 11 is compact in C(I, R e ) and so is the W 1,∞ (I, R e )-solution set toy(0) = y 0 ∈ C(0) t 0 b(τ, x(τ ))dz τ + By(t) -A dy dt (t) ∈ N C(t) ( dy dt (t)), a.e. t ∈ I. Then (x n ) n converges uniformly to some x ∈ C 1-var (I, R e ) with x t = ψ + t 0 V (x s )dz s , and (y n ) n converges to y ∈ W 1,∞ (I, R e )and ( ẏn ) n converges to ẏ weakly in L 1 R e (I). Applying the lower semicontinuity of the integral functional ([22, Theorem 8.16]) gives lim inf n (t, x n (t), y n (t), ẏn (t))dt ≥ (t, x(t), y(t), ẏ(t))dt. From the inclusion t 0 b(τ, x n (τ ))dz τ + By n (t) -A ẏn (t) ∈ N C(t) ( dy n dt (t)) and the fact that lim n t 0 b(τ, x n (τ ))dz τ = t 0 b(τ, x(τ ))dz τ uniformly with respect to t ∈ I (cf the proof of Theorem 5.1) we obtain by using Proposition 3.2 that t 0 b(τ, x(τ ))dz τ + By(t) -A ẏ(t) ∈ N C(t) ( dy dt (t)) a.e. t ∈ I.

  sup n |y n | 1-var;[s,t] + sup s∈[0,T ] M ||z(t) -z(s)|| R d as well as the problem inf , u(s))dz s , ζ(t) dt associated with the dynamic system

t0

  ζ n (s)ds , and f (t) = t 0 ζ(s)ds for all t ∈ [0, T ].Then it is clear that f n and f continuous uniformly bounded with f n (t) → f (t) for every t ∈ [0, T ]. Let u n be the unique Lipschitz solution tot 0 ζ n (s)ds + Bu n (t) + A un (t) ∈ N C(t) ( un (t)), t ∈ I, u n (0) = a ∈ C(0)and let v be the unique Lipschitz solution tot 0 ζ(s)ds + Bv(t) + A v(t) ∈ N C(t) ( v(t)), t ∈ I, u(0) = a ∈ C(0)In view Theorem 4.1 and Proposition 3.1 the sequence (u n ) n is equi-Lipchitz and converges uniformly to v. For simplicity set g n (t) = t 0 b(s, u n (s))dz s for all t ∈ [0, T ]. Applying the foregoing estimates concerning the Young integral, the sequence (g n ) n is uniformly bounded, equicontinuous and uniformly bounded in variation. Indeed, by condition (B 2 ), the sequence (b(•, u n (•))) n converge uniformly to b(•, v(•)). As b(., u n (.)) and b(., v(.)) are bounded and uniformly bounded in variation, by Friz-Victoir ([25, Proposition 6.12]) (g n ) n converges uniformly to g defined by g(t) = t 0 b(s, v(s))dz s , so that lim , u n (s))dz s , ζ n (t) dt = T 0 t 0 b(s, v(s)dz s , ζ(t) dt. , u n (s)dz s , ζ n (t) dt ≥ inf , u(s))dz s , ζ(t) dt for all n ∈ N, it follows by taking the limit that , v(s))dz s , ζ(t) dt ≥ inf (P S BV RC K

  t 0 ζ(s)ds. This theorem is still valid with the perturbation control integral t 0 Dζ(s)ds where D : R e → R e is a linear operator. Several other variants of Theorem 5.2 are available using Theorems 4.1 and 4.3 along with Propositions 3.2 and 3.10.

Theorem 5 . 3 .

 53 Let I := [0, 1] and let C : I ⇒ R e be a compact convex valued Lipschitz multimapping. Let f : I × Z → R e be a mapping satisfying (1), (2), (3).

3 Theorem 5 . 4 .

 354 (b) The second part follows by continuity and density, since Z is dense in R with respect to the stable topology ([22, Lemma 7.1.1]). With notation and assumptions in Theorem 5.With notation and assumptions in Theorem 5.3 let us consider the problem inf (P R ) T 0 t 0 b(s, u(s))dz s , t 0 [ Z f (s, z)ν s (dz)]ds dt associated with the dynamic system

t

  0 b(s, u n (s))dz s for all t ∈ I. Apply the foregoing estimates related to the Young integral, the sequence (g n ) n is uniformly bounded, equicontinuous and uniformly bounded in variation. By condition (B 2 ) the sequence (b(•, u n (•))) n converges uniformly to b(•, v(•)). As b(•, u n (•)) and b(•, v(•)) are bounded and uniformly bounded in variation, by Friz-Victoir ([25, Proposition 6.12]) (g n ) n converges uniformly to g defined by g(t) = t 0 b(s, v(s))dz s . For simplicity set k n (t) = t 0 f (s, ζ n (s))ds and k

Lemma 5 . 5 (

 55 [START_REF] Phung | On a fractional differential inclusion with integral boundary conditions in Banach space[END_REF]). Let f ∈ L 1 (I, E, dt) with I = [0, 1].(a) If α ∈]0, 1[ then I α 0 + f exists almost everywhere on I andI α 0 + f ∈ L 1 (I, E, dt). (b) If α ∈ [1, ∞) then I α 0 + f ∈ C E (I).Definition 5.2. Let E be a separable Banach space and let f ∈ L 1 E (I, E, dt) with I := [0, 1]. One defines the Riemann-Liouville fractional derivative of order α > 0 of f by

Proposition 5 . 2 .

 52 Let H be a separable Hilbert space. Let (ζ n ) be a bounded sequence in L 1 H ([0, T ]). Then the following hold: 1) (ζ n ) (up to an extracted subsequence) stably converges to a Young measure ν that is, there exist a subsequence (ζ n ) of (ζ n ) and a Young measure ν belonging to the space of Young measure Y([0, T ]; H σ ) with t → bar(ν t ) ∈ L 1 H ([0, T ]) (here bar(ν t ) denotes the barycenter of ν t ) such thatlim n→∞ T 0 h(t, ζ n (t))) dt) = T 0 [ H h(t, x) ν t (dx)] dt for all bounded Carathéodory integrands h : [0, T ] × H σ → R, 2) (ζ n ) (upto an extracted subsequence) weakly biting converges to an integrable function f ∈ L 1 H ([0, T ]), which means that, there is a subsequence (ζ m ) of (ζ n ) and an increasing sequence of Lebesgue-measurable sets (A p ) with lim p λ(A p ) = 1 and f ∈ L 1 H ([0, T ]) such that, for each p,

  ), ζ m (t) dt = Ap h(t), f (t) dt for all h ∈ L ∞H ([0, T ]), 3) (ζ n ) (up to an extracted subsequence) Komlós converges to an integrable function g ∈ L 1H ([0, T ]), which means that, there is a subsequence (ζ β(m) ) and an integrable function g ∈ L 1H ([0, T ]), such thatlim n→∞ 1 n Σ n j=1 ζ γ(j) (t) = g(t), a.e. ∈ [0, T ],for every subsequence(f γ(n) ) of (f β(n) ).4) There is a filter U finer than the Fréchet filter such that U -lim n ζ n = l ∈ (L ∞ H ) weak where (L ∞ H ) weak is the second dual of L 1 H ([0, T ]). Let w la ∈ L 1H ([0, T ]) be the density of the absolutely continuous part l a of l in the decomposition l = l a + l s in absolutely continuous part l a and singular part l s . If we have considered the same extracted subsequence in 1), 2), 3), 4), then one hasf (t) = g(t) = bar(ν t ) = w la (t) a.e. t ∈ [0, T ] By W 2,1 R d ([0, T ]) (resp. W 2,2 R d ([0, T ])we denote the set of all continuous functions in C R d ([0, T ]) such that their first derivatives are continuous and their second derivatives belong to L 1 R d ([0, T ]) (resp. L 2 R d ([0, T ])) and by W 1,1 BV ([0, T ]) we denote the set of all continuous functions in C R d ([0, T ]) such that their first derivatives are of bounded variation (BV for short).

  ) dτ )] ds for all t ∈ [0, T ].

2 H

 2 ([0, T ]). Proposition 5.3. Assume that β ∈ L 2 R + ([0, T ]) and A is a linear continuous symmetric and coercive operator: Ax, x ≥ M ||x|| 2 for all x ∈ R d where M is a positive constant. Let n ∈ N and ϕ n : R d → R + be a C 1 , convex, Lipschitz function and let ϕ ∞ be a nonnegative l.s.c proper function defined on R d such that ϕ n (x) ≤ ϕ ∞ (x) for all n ∈ N and for all x ∈ R d . Letf n ∈ L 2 R d ([0, T ]) such that ||f n (t)|| ≤ β(t), ∀n ∈ N, ∀t ∈ [0, T ]. For each n ∈ N, let u n be a W 2,1 R d ([0, T ])-solution to the problem f n (t) ∈ ün (t) + A un (t) + ∂ϕ n (u n (t)), t ∈ [0, T ]u n (0) = u n 0 ; un (0) = un 0 .

ϕ 0 A

 0 and lim n un (0) = u∞ 0 , (iv) There exist r 0 > 0 and x 0 ∈ R d such thatsup ∞ (x 0 + r 0 v(t)) < +∞ here B L ∞ R d ([0,T ]) is the closed unit ball in L ∞ R d ([0, T ]). (a) Then up to extracted subsequences, (u n ) converges uniformly to an W 1,1BV ([0, T ])-function u ∞ and ( un ) pointwisely converges to a BV function v ∞ with v ∞ = u∞ , and(ü n ) biting converges to a function ζ ∞ ∈ L 1 R d ([0,T]) so that the limit function u ∞ , u∞ and the biting limit ζ ∞ satisfy the variational inclusionf ∞ ∈ ζ ∞ + A u∞ + ∂I ϕ∞ (u ∞ )here ∂I ϕ∞ denotes the subdifferential of the convex lower semicontinuous integral functionalI ϕ∞ defined on L ∞ R d ([0, T ]) I ϕ∞ (u) := T 0 ϕ ∞ (u(t)) dt, ∀u ∈ L ∞ R d ([0, T ]). Furthermore lim n ϕ n (u n (t)) = ϕ ∞ (u ∞ (t)) < ∞ a.e. and lim n T 0 ϕ n (u n (t))dt = T 0 ϕ ∞ (u ∞ (t))dt.Subsequently, the energy estimate holds true almost everywheret ∈ [0, T ], ϕ ∞ (u ∞ (t)) + 1 2 || u∞ (t)|| 2 = ϕ ∞ (u ∞ u∞ (s), u∞ (s) ds + t 0 u∞ (s), f ∞ (s) ds(c) Consequently the density f ∞ -ζ ∞ -A u∞ of the absolutely continuous part n a n a (h) := T 0 f ∞ (t) -ζ ∞ (t) -A u∞ (t), h(t) dt, ∀h ∈ C R d ([0, T ])

T 0 x 0 0 x 0

 0000 + r 0 v(t) -u n (t), z n (t) dt = T -u n (t), z n (t) dt + r 0 T 0 v(t), z n (t) dt ≤ T 0 ϕ n (x 0 + r 0 v(t))dt -), z n (t) dt ≤ T 0 ϕ n (x 0 + r 0 v(t))dt -T 0 ϕ n (u n (t))dt -T 0 x 0 -u n (t), z n (t) dt.We compute the last integral in the preceding inequality. For simplicity, let us set v n (t) = u n (t) -x 0 for all t ∈ [0, T ]. By integration by parts and taking account into (u n (t),z n (t) dt = -T 0 v n (t), vn (t) + A vn (t) -f n (t) dt = -[ v n (t), vn (t) + Av n (t)] T 0 + T 0 vn (t), vn (t) + Av n (t) dt + T 0 v n (t), f n (t) dt ≤ -v n (T ), vn (T ) + v n (0), vn (0) -Av n (T ), v n (T ) + Av n (0), v n (0) + T 0 || vn (t)|| 2 dt + T 0 vn (t), Av n (t) dt + T 0 v n (t), f n (t) dt.and the singular part B * l s satisfies the equationB * l s , h = l s , h , ∀h ∈ C R d ([0, T ]).As B * l = n, using (3.3.10)-(3.3.11), it turns out that n is the sum of the absolutely continuous measure n a withn a , h = T 0 f ∞ -ζ ∞ (t) -A u∞ (t), h(t) dt, ∀h ∈ C R d ([0, T ])and the singular part n s given by n s , h = l s , h , ∀h ∈ C R d ([0, T ]).

  t))dt, ∀v ∈ L 1 R d ([0, T ]).

  Problem (1.1) is interpreted as an evolution variational inequality (EVI) with the velocity inside the subdifferential. Generally, the model for parabolic evolution inclusion is a differential inclusion of the form

	B(t, u(t)) ∈	du dt	(t) + A(t)u(t) + ∂ϕ(t, u(t)), t ∈ [0, T ],	(1.2)

space H in the form f (t) + Bu(t) -A du dt (t) ∈ ∂ϕ(t, du dt (t)), t ∈ [0, T ]. (1.1) Above f : [0, T ] → H is a continuous mapping, B : H → H is an operator, A : H → H is a linear continuous coercive and symmetric operator, ϕ : [0, T ] × H → ] -∞, +∞] is a normal lower semicontinuous convex integrand, and ∂ϕ(t, .) is the subdifferential of ϕ(t, .).

  1 H (I) such that ζ n (t) ∈ C(θ n (t)) for all t ∈ I and such that (ζ n ) n∈N σ(L 1 H , L ∞ H ) converges in L 1 H (I) to ζ. Assume that f n (t) + Bv n (t) -A ζ n (t) ∈ N C(θn(t)) (ζ n (t)) for all n ∈ N,a.e. t ∈ I. Then for a.e. t ∈ I one has ζ(t) ∈ C(t) and f (t) + Bv(t) -A ζ(t) ∈ N C(t) (ζ(t)).

  ), C(θ n (t)) , ∈ rB H . Then it is clear that ψ A is a positive lower semicontinuous convex integrand. Apply again the lower semicontinuity of the convex integral functional ([START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF], Theorem 8.1.6) associated with the positive normal convex integrand ψ A we obtain

	f (t), ζ(t) dt
	we can see that that is, since ζ n → ζ weakly in L 1 lim inf n Z δ lim inf n lim inf n Z Aζ n (t), ζ n (t))dt ≥ Z ψ A (ζ n (t))dt ≥ H (I). It ensues that lim n Z f n (t), ζ n (t) dt = Z Z Z ψ A (ζ(t))dt, Aζ(t), ζ(t)dt. f (t), ζ(t) dt. Since we already saw that Bv Z Now putting q	(3.3) (3.5)

* (f n (t) + Bv n (t) -Aζ n (t), C(θ n (t)))dt ≥ lim inf n Z δ * (f n (t) + Bv n (t) -Aζ n (t), C(t))dt ≥ Z δ * (f (t) + Bv(t) -Aζ(t), C(t))dt. (3.2) Let us set ψ A (x) = Ax, x if x ∈ rB H and ψ A (x) = +∞ if x / n (t) → Bv(t) strongly in H for each t ∈ I, we also have by the Lebesgue dominated convergence theorem lim n Z Bv n (t), ζ n (t) dt = Z Bv(t), ζ(t) dt. (3.4) Further, by the Lebesgue dominated convergence theorem again the inequality

| f n (t), ζ n (t) -f (t), ζ n (t) | ≤ r f n (t) -f (t) gives Z f n (t), ζ n (t) -f (t), ζ n (t) dt → 0. Also as n → ∞ Z f (t), ζ n (t) dt → n (t) := f n (t) + Bv n (t) -Aζ n (t)

and integrating on Z ⊂ [0, T ] the inequality ( here measurability and integrability are guaranted)

δ * (q n (t), C(θ n (t))) + -f n (t) -Bv n (t), ζ n (t) + Aζ n (t), ζ n (t) ≤ 0 gives Z δ * (q n (t), C(θ n (t)))dt + Z Av n (t), ζ n (t) dt + Z -f n (t) -Bv n (t), ζ n (t) dt ≤ 0,

so passing to the limit as n → ∞ in this equality and using (3.1)-(3.5) yield

Z δ * (f (t) + Bv(t) -Aζ(t), C(t)) + Aζ(t) -Bv(t) -f (t), ζ(t) dt ≤ 0. As t → δ * (f (t) + Bv(t) -Aζ(t), C(t)) + Aζ(t) -Bv(t) -f (t), ζ(t)

is integrable and as the latter inequality holds true for any Lebesgue measurable set Z ⊂ I, it follows that for a.e. t ∈ I δ * (f (t) + Bv(t) -Aζ(t), C(t)) + Aζ(t) -Bv(t) -f (t), ζ(t) ≤ 0. This and the inclusion ζ(t) ∈ C(t) a.e. t ∈ I allow us to conclude that f (t) + Bv(t) -Aζ(t)) ∈ N C(t) (ζ(t)) a.e. t ∈ I according to the description (2.1) of the normal cone.

  1 the uniform boundedness condition C(t) ⊂ rB H can be relaxed as shown in the next proposition. Let us prove first a lemma.

Lemma 3.2. Let H be a separable Hilbert space and I = [0, T ]. Let C : I ⇒ H be a scalarly measurable multimapping with closed convex weakly locally compact values which contain no line. Let (ζ n ) n∈N be sequence in L 1 H (I) such that ζ n (t) ∈ C(t) for all t ∈ I and n ∈ N and such that (ζ n

  This being true for any Lebesgue measurable set Z ⊂ I, it follows that for every p ∈ N e Proposition 3.4. Let H be a separable Hilbert space and I = [0, T ]. Let C : I ⇒ H be a scalarly measurable multimapping with closed convex weakly locally compact values which contain no line and for which there exist r ∈ L ∞ R + (I) such that C(t) ∩ r(t)B H = ∅ for all t ∈ I. Let A : H → H be a continuous symmetric linear coercive operator and let B : H → H be a continuous compact linear operator. Let

p , ζ(t) ≤ δ * (e p , C(t)) a.e. t ∈ I. As H is separable and C(t) is closed convex weakly locally compact and contains no line, by ( Castaing-Valadier [24, Proposition III-35]), we get the desired inclusion ζ(t) ∈ C(t) a.e. t ∈ I.

  to u uniformly on I and ( un ) n converges to u weakly in The inclusion u(t) ∈ C(t) for a.e. t ∈ I is justified by Lemma 3.2. Take a real r > r and let γ > 0 be a Lipschitz constant of g on ( u 0 + r T )B H . Take any Lebesgue measurable set Z ⊂ I. Let us consider the term ∇g(u n (t)), un (t) . The Lipschitz property of g on ( u 0 + r T )B H assures us that g • u n and g • u are absolutely continuous on I and at any t ∈ I where u and all u n are derivable

	Proof. ∇g(u(t)), u(t) =	d dt	g(u(t)) and ∇g(u n (t)), un (t) =	d dt	g(u n (t)).
	L 1 H (I). Assume that for every n ∈ N
	f n (t) + ∇g(u n (t)) -A un (t) ∈ N C(t) ( un (t)) a.e. t ∈ I.
	Then for a.e. t ∈ I one has
	u(t) ∈ C(t) and f (t) + ∇g(u(t)) -A u(t) ∈ N C(t) ( u(t)).

  Lipschitz constant of g on γB H , so that ∇g is bounded on γB H and ∇g(v n ) → ∇g(v) strongly . Take any Lebesgue measurable set Z ⊂ I.

	is normal, see e.g Castaing-Valadier [24], and satisfies
		ϕ * (t, y) ≤ ϕ * (τ, y) + |v(t) -v(τ )|
	for all t, τ ∈ I, y ∈ H using assumption (ii) ([36], Proposition 27). By using
	the normality of ϕ, the functions t → ϕ(θ n (t), un (t)) and t → ϕ(t, un (t)) are
	measurable and integrable.By assumption we have
						ds,
	such that (u n ) n converges to u uniformly on I and ( un ) n converges to u weakly in
	L 1 H (I). Assume that for every n ∈ N
	f n (t) + ∇g(v n (t)) -A un (t) ∈ ∂ϕ(θ n (t), un (t)) a.e. t ∈ I.
	Then for a.e. t ∈ I one has
	u(t) ∈ Γ and f (t) + ∇g(v(t)) -A u(t) ∈ ∂ϕ(t, u(t)).
	Proof. First, Lemma 3.2 tells us that u(t) ∈ Γ for a.e. t ∈ I.
	Let κ > 0 be a Since (f n ) n is uniformly bounded and pointwise converges to f and ( un ) n weakly
	converges in L 1 H ([0, T ]) to u , we have
	lim n	Z	f n (t), un (t) dt =	Z	f (t), u(t) dt.	(3.21)
	This fact has been already justified in the proof of Proposition 3.4. As ϕ is normal
	lower semicontinuous convex integrand, the conjugate function ϕ * : I × H →
	] -∞, +∞]
		ϕ * (t, y) = sup	[ x, y -ϕ(t, x)]	(3.22)
			x∈H

  As consequence, for every measurable set Z in I, we have

	lim n→∞ Z	h n (t), un (t) dt
	= lim n→∞ Z	h n (t) -h(t), un (t) dt + lim n→∞ Z	h(t), un (t) dt
	= lim n→∞ Z	h(t), un (t) dt =	Z	h(t), u(t) dt.	(3.24)

  Proposition 3.7. Let H be a separable Hilbert space and I = [0, T ]. Let C : I ⇒ H be a convex weakly compact valued multimapping for which there is some nondecreasing continuous function r : I → R + such that haus(C(t), C(τ )) ≤ |r(t)-r(τ )| for all τ, t ∈ I Let A be a linear continuous coercive symmetric operator on H and let B = ∇g where ∇g is the gradient of a convex continuous Gateaux differentiable function g : H → R + such that g(v(t)) is absolutely continuous for v : [0, T ] → H absolutely continuous. Let (θ n ) n∈N be a sequence of measurable functions from I into I such that for each t ∈ I one has θ n (t) → t. Let (u n , u) n∈N be a bounded sequence of absolutely continuous mapping on H such that (u n (t)) n converges pointwise strongly to u and such that ( un ) n∈N σ(L 1

								T
							=	∇g(u(t)), u(t) dt
								0
	Further let Ψ(x) = x, Ax for all x ∈ H. Then Ψ is positive lower semicontinuous
	convex on H. As un weakly converges converges in L 1 H (I) to u, again by ([22],
	Theorem 8.1.6) we get we have
	(3.7.3)					lim inf n	Z	Ψ( un (t))dt ≥	Z	Ψ( u(t))dt
	As it is readly seen that A un weakly converge to A u and ∇g(u n (t))) → ∇g(u(t))),
	then let e ∈ H, we have
				e, A un (t) + ∇g(u n (t))) ≤ δ * (e, C(θ n (t)))
	Hence for Z ∈ L(I) converges in L 1 H (I) to u. Assume that e, A un (t) + ∇g(u n (t))) dt ≤	δ * (e, C(θ n (t)))dt	H , L ∞ H )
		-un (t) ∈ N C(θn(t)) (A un (t) + ∇g(u n (t)))for all n ∈ N, a.e. t ∈ I. Z Z
	Then for a.e. t ∈ I one has -u(t) ∈ N C(t) (A u(t) + ∇g(u(t))). Passing to the limit in this inequality we get
	Z Proof. As -un (t) ∈ N C(θn(t)) (A un (t) + ∇g(u n (t))) then Z e, A u(t) + ∇g(u(t))) dt ≤ δ * (e, C(t))dt
	δ * (-un (t), C(θ n (t))) + un (t), A un (t) + ∇g(u n (t)) ≤ 0 By [24] we conclude that A u(t) + ∇g(ut))) ∈ C(t) By (ii) using the Hormander
	By integrating on Z ∈ L(I) (we are ensured that the functions given are measur-formula we have the estimation
	able) this inequality we get
	(3.7.1)	δ * (-un (t), C(θ n (t)))dt +	un (t), A un (t) + ∇g(u n (t)) dt ≤ 0.
		Z						Z
	Now we have to consider first the term ∇g(u n (t), un (t) by using the special
	property of ∇g. In fact u n is absolutely continuous with derivative un and g(u n )
	is absolutely continuous , so that by Moreau-Valadier [35] ,
						un (t), ∇g(u n (t)) =	d dt	g(u n (t))
	From this fact, it is easy to deduce that
		lim inf n	0	T	∇g(u n (t)), un (t) dt = lim inf n	0	T	d dt	g(u n (t)) dt
	(3.7.2)	= lim inf n	(g(u n (T ) -u n (0)) ≥ g(u(T ) -u(0)) =	0	T	d dt	g(u(t)) dt

Z δ * (-un (t), C(t))dt ≤ Z δ * (-un (t), C(θ n (t)dt + Z || un (t)|||r(θ n (t) -r(t)|dt with Z || un (t)|||r(θ n (t) -r(t)|dt → 0 using the fact that ρ n (t) := |r(θ n (t)r(t)| → 0 and || un (t)|| is uniformly integrable. Here the measurability of the functions given these integrals is granted; in fact it is easily seen that C is scalarly continuous. So that lim inf

  along with the inclusion (4.1) allow us to obtain that for a.e. t ∈ I the required inclusion. This need a careful look. Indeed, for every Lebesgue measurable set Z ⊂ I and for any e ∈ H , the function 1 Z e ∈ L ∞

H (I). Considering the inequality e, B un (t) + A λn u n (δ n (t)) ≤ δ * (e, C(θ n (t), u n (δ n (t))))

  a.e. t ∈ I. Let (e p ) p∈N be a dense sequence in H. Take any measurable set Z ⊂ I and any p ∈ N, and note that the mapping 1 Z e p ∈ L ∞ H (I). Considering the inequality By the inequality (due to the inclusion ζ n (t) ∈ C(t)) e Passing to the limit in the latter inequality assures us that As H is separable and C(t) is closed convex weakly locally compact and contains no line, by ( Castaing-Valadier [24, Proposition III-35]), we get the desired inclusion ζ(t) ∈ C(t) a.e. t ∈ I. For each t ∈ I keeping in mind that v n (t) → v(t) strongly in H and A is a maximal montone operator satisfying (H) 1 we see that |A λn v n (t)| ≤ c(1 + γ) strongly in H so that by our assumption w(t) = lim n A λn v n (t) ∈ c(1+γ)B H . Then the uniformly bounded sequence w n := A λn v n (.) → w for the Mackey topology τ (L ∞

p , ζ n (t) ≤ δ * (e p , C(t)) and integrating on Z ensure that I 1 Z e p , ζ n (t), dt = Z e p , ζ n (t) dt ≤ Z δ * (e p , C(t))dt. Z 1 Z e p , ζ(t) dt ≤ Z δ * (e p , C(t))dt. This being true for any Lebesgue measurable set Z ⊂ I, it follows that for every p ∈ N e p , ζ(t) ≤ δ * (e p , C(t)) a.e. t ∈ I. H (I), L 1 H (I) .As consequence lim n T 0

  to u uniformly on I and ( un ) n converges to u weakly in L 1

H (I). Assume that for every n ∈ N

f n (t) + A λn u n (t) -B un (t) ∈ N C(θn(t)) ( un (t)) a.e. t ∈ I.

and λ n → 0, lim n A λn u n (t) → w(t) pointwise, where w is a measurable mapping. Then for a.e. t ∈ I one has

u(t) ∈ C(t) and f (t) + w(t) -B u(t) ∈ N C(t) ( u(t)). w(t) ∈ Au(t)

Proof. First, we justify that u(t) ∈ C(t) a.e. t ∈ I. Take any measurable Lebesgue set Z ⊂ I and any x ∈ H. The function 1 Z x ∈ L ∞ H (I). Writing

x, un (t) ≤ δ * (x, C(θ n (t))),

  C(t)) t exta.e. t ∈ I.

By the separability of H and the weak compactness and convexity of C(t), we get the desired inclusion u(t) ∈ C(t) a.e. t ∈ I.

Since (f n ) n is uniformly bounded and pointwise converges to f , we have

lim n Z f n (t), un (t) dt = Z f (t), u(t) dt.

(3.36)

this fact is explained. Also by integrating on Z (we are ensured that the functions given are measurable) the inequality it ensues that with q n (t

) := f n (t)+A λn u n (t)-B un (t) Z δ * (q n (t), C(θ n (t)))dt + Z B un (t)-f n (t)

, un (t) dt-Z A λn u n (t), un (t) dt ≤ 0. (3.37) We claim that Bu n (•) → Bu(•) weakly in L 1 H (I) and as above w n (t) : A λn u n (t) → w(t) for with respect to the Mackey topology τ (L ∞ H

  By the separability of H and the weak compactness of C(t) (se, e.g.,), we get the desired inclusion u(t) ∈ C(t) a.e. Let h ∈ L ∞ H (I). We first note that A λn u n (δ n (t)) → w(t) by A 2 so that (w n (t) :A λn u n (δ n (t))) is a uniformly bounded measurable sequence pointwise converging to w. As consequence, we have lim Similarly we note that B un → B u weakly in L 1 H (I). As a main consequencef n + A λn u n (δ n (t)) -B un → f + w -B u weakly in L 1H (I). Then we may apply the lower semicontinuity of integral convex functional ([START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF], Theorem 8.1.16) to (f n (t)+A λn u n (δ n (t))-B un (t), C(t))dt for every Lebesque measurable set Z ⊂ [0, T ]. This need a careful look. Indeed, we note that (t, x) → δ * (x, C(t)) is a normal lower semicontinuous convex integrand defined on [0, T ] × H and δ * (f n (t) + A λn u n (δ n (t)) -B un (t), C(t)) is measurable and integrable:

	deduce that
	(1.1.2)
	Z	δ * (f (t)+w(t)-B u(t), C(t))dt ≤ lim inf n	Z	δ

)) a.e. t ∈ I.

n T 0 un (t), w n (t) dt = T 0 u(t), w(t) dt. *

  a positive lower semicontinuous convex integrand. Apply again the lower semicontinuity of the integral convex functional ([START_REF] Castaing | Young measures on topological spaces with applications in control theory and probability theory[END_REF], Theorem 8.1.6) associated with the positive normal convex integrand ψ B we obtain,

	Whence we have
	(1.1.5)	lim n	Z	A λn u n (δ n (t)), un (t) dt =	Z	w(t)), u(t) dt.
	Similarly we have
	(1.1.6)		lim n	Z	f n (t), un (t) dt =	Z	f (t), u(t) dt
	because f n is uniformly bounded and pointwise strongly converge to f and un →	u
	weakly in L 1
			lim inf n	Z	ψ B ( un (t))dt ≥	Z	ψ B ( u(t))dt
	that is
	(1.1.4)	lim inf

n Z B un (t), un (t))dt ≥ Z B u(t), u(t)dt.

  ds, such that (u n ) n converges to u uniformly on I and ( un ) n converges to u weakly in L 1Since (f n ) n is uniformly bounded and pointwise converges to f and ( un ) n weakly converges in L 1

	H ([0, T ]) to u , we have
	lim n	Z	f n (t), un (t) dt =	Z	f (t), u(t) dt.	(3.42)
	This fact has been already justified in the proof of Proposition 3.4. As ϕ is normal
	lower semicontinuous convex integrand, the conjugate function ϕ * : I × H →
	] -∞, +∞]
		ϕ * (t, y) = sup	[ x, y -ϕ(t, x)]	(3.43)
			x∈H
	is normal, see e.g Castaing-Valadier [24], and satisfies
		ϕ

H (I). Assume that for every n ∈ N ( * )f n (t) + A λn (t, u n (t)) -B un (t) ∈ ∂ϕ(θ n (t), un (t)) a.e. t ∈ I.

( * * ) λ n ∈]0, 1] → 0 and A λn (t, u n (t)) → v(t)

pointwise strongly Then for a.e. t ∈ I one has u(t) ∈ Γ, f (t) + v(t) -B u(t) ∈ ∂ϕ(t, u(t)) and v(t) ∈ A(t, u(t)) Proof. First, Lemma 3.2 tells us that u(t) ∈ Γ for a.e. t ∈ I. First, we justify that u(t) ∈ Γ a.e. t ∈ I. Let (e p ) be a dense sequence in H. For every measurable set Z ⊂ I and for any e p ∈ H, the function 1 Z e p ∈ L ∞ H (I). By the inequality e p , un (t) ≤ δ * (e p , Γ) integrating on Z gives I 1 Z e p , un (t) dt = Z e p , un (t) dt ≤ Z δ * (e p , Γ)dt. Passing to the upper limit in this inequality we obtain Z 1 Z e p , u(t) dt ≤ Z δ * (e p , Γ)dt. This being true for any Lebesgue measurable set Z ⊂ I we deduce that for every e p ∈ H e p , u(t) ≤ δ * (e p , Γ) a.e. t ∈ I. As H is separable and Γ is closed convex weakly locally weakly compact which contain no line, by (Castaing-Valadier [24, Proposition III-35]), we get the desired inclusion u(t) ∈ Γ a.e. t ∈ I. Take any Lebesgue measurable set Z ⊂ I. * (t, y) ≤ ϕ * (τ, y) + |v(t) -v(τ )| for all t, τ ∈ I, y ∈ H using assumption (ii) ([36], Proposition 27). By using the normality of ϕ, the functions t → ϕ(θ n (t), un (t)) and t → ϕ(t, un (t)) are measurable and integrable.By assumption we have

  As ||u n (t) -J λn (t, u n (t))|| = λ n ||A λn (t, u n (t))|| ≤ λ n |A 0 (t, u n (t)) ≤ c(1 + ||u n (t)||) with λ n < 1, J λn (t, u n (t)) is uniformly bounded and pointwise converge to u(t),so that t → J λn (t, u n (t)) converge to u in L 2 H (I). From v n (t) = A λn (t, u n (t)) ∈ A(t, J λn (t, u n (t)) we show that (v n (t), J λn (t, u n (t)) ∈ graphA t , so that (v n , w n ) ∈ graphA * with w n (t) = J λn (t,u n (t)). As graphA * is sequentially strong weakly closed by Lemma 0.1, with v n → v strongly hence weakly in L 2 H (I) and w n → u strongly in L 2 H (I) applying Lemma 0.1 gives (v, u) ∈ graphA * , that implies v(t) ∈ A(t, u(t)) a.e. The proof is finished. Theorem 3.6. Let H be a separable Hilbert space.Let K be a convex compact subset of H. Let

u(t)).

It remain to check v(t) ∈ A(t, u(t)) a.e. Indeed J λn (t, u n (t))-→ u(t) by writing

||J λn (t, u n (t)) -u(t)|| ≤ ||J λn (t, u(t)) -J λn (t, u(t))|| + ||J λn (t, u(t)) -u(t)|| ≤ ||u n (t) -u(t)|| + ||J λn (t, u(t)) -u(t)|| → 0

  We note that ||u n (t)|| ≤ ||u 0 || + ρT , ||f n (.)|| ≤ β for all t ∈ [0, T ] and u n (t) = u 0 +

	and observe that for each t ∈ I, there is i+1 [, and then, i , t n i ∈ {0, ..., n -1} such that t ∈ [t n 0 un (s)ds for all t ∈ [0, T ] with un ∈ K a.e. Step 2 Convergence of the algorithm and final conclusion |θ t Let S 1 K := {h ∈ L 1 H ([0, T ]) : h(t) ∈ K a.e.} and let

n (t) -t| → 0 and |δ n (t) -t| → 0 as n → +∞, So, the last inclusion becomes

f n (t) + ∇g(u n (δ n (t))) -B un (t) ∈ ∂ϕ(θ n (t), un (t))

a.e. t ∈ [0, T ] .

  Remark 4.2. The tools developed above allow to obtain further variants. The fact that C(t) is weakly compact is required, and mainly the coerciveness of A and the compactness assumption for the operator B. An inspection of the proof of Theorem 4.1, shows that the compactness assumption on B is required to prove the Fatou property, So as a possible variant we may substitute the bounded operator B by the gradient ∇g of a positive convex continuous Gateaux differentiable function g : H → R such that g(v(t)) is absolutely continuous for v : [0, T ] → H absolutely continuous, so that by invoking the chain rule formula, see[START_REF] Moreau | A chain rule involving vector functions of bounded variations[END_REF], we have the equality

	lim inf n	Z	B(u n (δ n (t)), un (t) dt ≥	Z	Bu(t), u(t) dt

s) -u1 (s) ds.

By Gronwall lemma u1 (t) = u2 (t) a.e. t ∈ I, and so u 1 (t) = u 2 (t) for every t ∈ I since u 1 (t) = u 0 + t 0 u1 (s)ds and u 2 (t) = u 0 + t 0 u2 (s)ds.

  Step 2 Convergence of the algorithm and final conclusion Let S 1 K

and observe that for each t ∈ I, there is i ∈ {0, ..., n -1} such that t ∈ [t n i , t n i+1 [, and then, |θ n (t) -t| → 0 and |δ n (t) -t| → 0 as n → +∞, So, the last inclusion becomes f n (t) + Bu n (δ n (t)) -A un (t) ∈ ∂ϕ(θ n (t), un (t)) a.e. t ∈ [0, T ] . We note that ||u n (t)|| ≤ ||u 0 || + ρT , ||f n (.)|| ≤ β for all t ∈ [0, T ] and u n (t) = u 0 + t 0 un (s)ds for all t ∈ [0, T ] with un ∈ K a.e.

  .2) for all 0 ≤ t ≤ 1. By Theorem 1.1 there is a unique absolutely continuous mapping u n : I → H solution of the problemu n (0) = a, h n (t) + k n (t) + Bu n (t) -A un (t) ∈ N C(t) ( un (t)) a.e. ∈ t ∈ I

	with u n (t) = a +	t 0	un (s)ds for all t ∈ I and un (t) ≤ M a.e. t ∈ I. Set for all
	t ∈ I
	x n

  ) ( dy dt (t)), a.e. t ∈ [0, T ] has an optimal solution.Proof. Let us consider a minimizing sequence (x n , y n ) n in Y, that is,

		T
	lim n→∞	0	L(t, x n (t), y n (t), ẏn (t))dt = inf

  a positive nondecreasing continuous function with v(0) = 0. Let A : H → H be a linear continuous coercive symmetric operator and let g be a convex function on H Lipschitz on bounded sets and continuously differentiable on H whose gradient is locally Lipschitz. Let f : I ×H → H be a bounded mapping :||f (t, x)|| ≤ M for all (t, x) ∈ [0, 1]×H such that (i) f (•, x) is L(I) measurable for all x ∈ H, (ii) f (t, •) is continuous on H for all t ∈ I.Then given a, b ∈ H, there is an absolutely continuous mapping x : I → H and an absolutely continuous mapping u : I → H satisfying Kds ⊂ b+K∀t ∈ [0, 1] with v h uniformly bounded and equi-absolutely continuous. Now let us consider the set X defined byX := {ξ f : I → H : f ∈ S 1 b+K },each mapping ξ f being given for every t ∈ I byξ f (t) = a +We note that X is convex compact in C H (I). Now for each h ∈ X , by Theorem 4.5 again denote by u h the unique absolutely continuous solution of the differential inclusion where A(t) is a time dependent Lipschitz variation maximal monotone operator and the perturbation f (t, .) is boundedly Lipschitz is stated in[START_REF] Azzam-Laouir | Perturbed evolution problems with continuous bounded variation in time and applications[END_REF] . In particular, existence and uniqueness of solution to 0 = ü(t) + A(t) u(t) + ∇ϕ(u(t)), t ∈ [0, T ]

	 	x(t) = a + u(0) = b	t 0 u(s)ds, t ∈ I
	
			t	dv h
			0

t 0 f (s, x(s))ds + ∇g(u(t)) -A du dt (t) ∈ ∂ϕ(t, du dt (t)), a.e. t ∈ [0, 1] Proof. For any continuous mapping h : I → H, the mapping f h : I × H → H defined by f h (t) := f (t, h(t)) for all t ∈ I is L(I) measurable and satisfies |f h (t)| ≤ M for all t ∈ I. Then by Theorem 4.5 and the assumptions on f , there is a unique absolutely continuous solution v h to the inclusion v h (0) = b ∈ H t 0 f (s, h(s))ds + ∇g(v h (t)) -A dv h dt (t) ∈ ∂ϕ(t, dv h dt (t)), a.e. t ∈ [0, 1] with dv h dt (t) ∈ K a.e. so that v h (t) = b+ ds (s)ds ∈ b+ t 0 t 0 f (s) ds, t ∈ I, f ∈ S 1 b+K t 0 f (s, h(s))ds + ∇g(u h (t)) -A du h dt (t) ∈ ∂ϕ(t, du h dt (t)), a.e. t ∈ I

  )), a.e.. and for any nonnegative measure θ on [0, T ] with respect to which n s is absolutely continuous here h ϕ * ∞ denotes the recession function of ϕ * ∞ . Proof. Step 1 || un (.)|| and ϕ n (u n (.)) are uniformly bounded. Multiplying scalarly the inclusionf n (t) -ün (t) -A un (t) ∈ ∂ϕ n (u n (t))by un (t) and applying the chain rule theorem ([START_REF] Moreau | A chain rule involving vector functions of bounded variations[END_REF], Theorem 2) yieldsun (t), f n (t) -un (t), ün (t) -un (t), A un (t) = || un (s)|| 2 ds + ||f n || L 2 R d ([0,T ]) (Then from (iii), the preceding estimate and the Gronwall like inequality (Lemma 3.1), it is immediate that

	(3.3.2)	sup			sup	|| un (t)|| < +∞ and sup	sup
		n≥1	t∈[0,T ]		n≥1	t∈[0,T ]
					0	T	h ϕ * ∞ (	dn s dθ	(t))dθ(t) =	0	T	u ∞ (t),	dn s dθ	(t) dθ(t)
											d dt	[ϕ n (u n (t))]
	that is
	(3.3.1)	-A un (t), un (t) + un (t), f n (t) =	d dt	[ϕ n (u n (t)) +	1 2	|| un (t)|| 2 ].
	By integrating on [0, t] this equality we get
						ϕ n (u n (t)) +	1 2	|| un (t)|| 2 = ϕ n (u n (0)) +	1 2	|| un (0)|| 2
											t	t
									-	A un (s), un (s) ds +	un (s), f n (s) ds
											0	0
											≤ ϕ n (u n (0)) +	1 2	|| un (0)|| 2
		+M				t		t	|| un (s)|| 2 ds)	1 2
									0		0
											≤ ϕ n (u n (0)) +	1 2	|| un (0)|| 2
		+M	0	t	|| un (s)|| 2 ds +	1 2	||f n || L 2 R d ([0,T ]) (1 +	0	t	|| un (s)|| 2 ds)
											≤ ϕ n (u n (0)) +	1 2	|| un (0)|| 2
		+M		0	t	|| un (s)|| 2 ds +	1 2	||β|| L 2

R ([0,T ]) (1 + t 0 || un (s)|| 2 ds).

If H = R e , here one may invoke a classical fact that on bounded subsets of L ∞ H the topology of convergence in measure coincides with the topology of uniform convergence on uniformly integrable sets, i.e. on relatively weakly compact subsets, alias the Mackey topology. This is a lemma due to Grothendieck[START_REF] Grothendieck | Espaces Vectoriels Topologiques Mat[END_REF] [Ch.5 §4 no 1 Prop. 1 and exercice].

If H = R e , here one may invoke a classical fact that on bounded subsets of L ∞ H the topology of convergence in measure coincides with the topology of uniform convergence on uniformly integrable sets, i.e. on relatively weakly compact subsets, alias the Mackey topology. This is a lemma due to Grothendieck[START_REF] Grothendieck | Espaces Vectoriels Topologiques Mat[END_REF] [Ch.5 §4 no 1 Prop. 1 and exercice].

If H = R e , one may invoke a classical fact that on bounded subsets of L ∞ H the topology of convergence in measure coincides with the topology of uniform convergence on uniformly integrable sets, i.e. on relatively weakly compact subsets, alias the Mackey topology. This is a lemma due to Grothendieck[START_REF] Grothendieck | Espaces Vectoriels Topologiques Mat[END_REF] [Ch.5 §4 no 1 Prop.1 and exercice]

for all 0 ≤ s ≤ t ≤ T with θ = 2 and for all 0 ≤ s ≤ t ≤ T . Shortly the sequence g n (.) = . 0 y n dz of C 1-var ([0, T ], R e ) mappings is uniformly bounded, equicontinuous and uniformly bounded in variation.

Now let E be a separable reflexive Banach space. Let us consider a weakly compact convex valued multimapping K : [0, 1] ⇒ B E with bounded right continuous retraction in the sense, there is a bounded and right continuous function ρ : [0, 1] → R + such that haus(K(t), K(τ )) ≤ ρ(τ ) -ρ(t) for all t ≤ τ ∈ [0, [START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF] and such that its graph is Borelian, that is, gph (C) ∈ B([0, T ]) ⊗ B(E). We consider the control sets given by [START_REF] Adly | Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities[END_REF], E, dt), u(t) ∈ K(t), ∀t ∈ [0, T ].} By J.J. Moreau ([34], Prop.5 d, p. 198) and Valadier [START_REF] Valadier | Une proprieté de l'ensemble des selections a variation bornée d'une multiplication a retraction bornee[END_REF] these sets are non empty and cl S BV RC K = S ∞ K , here cl denotes the closure with respect to the σ(L ∞ E , L 1 E * )topology. Shortly, S BV RC K is dense in S ∞ K with respect to this topology. Our next theorem, say theorem 5.2, will present relaxation results for a Bolza optimal control problem governed by EVI sweeping processes of the types developed in the previous sections. The control will belong either in S BV RC K or S ∞ K . In the theorem A : R e → R e is a coercive symmetric linear operator while B is a linear operator on R e and b : [0, T ] × R e → L(R d , R e ) is a mapping satisfying (B 1 ) and (B 2 ).

B,E ([0, 1]) and one has for every t ∈ [0, 1]

exp(-λ(t -s))f (s)ds + exp(-λt)

From Lemma 5.6 we derive a crucial feature.

Lemma 5.7. Let E be a separable Banach space and let f ∈ L 1 (I, E, dt) with I := [0, 1]. Then the boundary value problem

has a unique W α,1 B,E (I)-solution defined by

G(t, s)f (s)ds, textf orall t ∈ I.

Theorem 5.8. Let E be a separable Banach space, α ∈]1, 2) and I := [0, 1].

Let X : I ⇒ E be a compact convex valued measurable multimapping such that X(t) ⊂ γB E for all t ∈ I, where γ is a positive constant. Let S 1 X be the set of all measurable selections of X. Then the W α, 1 B,E (I)-solutions set of problem

is compact in C E (I).

Proof. By virtue of Lemma 5.7 the W α,1 B,E (I)-solutions set X to the above inclusion is characterized by

(5.7)

Claim: X is bounded, convex, equicontinuous and compact in C E (I).

From the definition of the Green function G, it is not difficult to show that (i {ϕ(., u(.)), u ∈ S 1 K } is uniformly integrable; (ii) ϕ(t, x) ≤ ϕ(τ, x) + |v(t) -v(τ )| for all t, τ ∈ I, x ∈ K, where v : I → R + is a positive nondecreasing continuous function with v(0) = 0. Let A : H → H be a continuous coercive symmetric linear operator and B : H → H be a continuous linear operator. Let f : I × H → H be a bounded continuous mapping, say f (t, x) ≤ M for all (t, x) ∈ I × H.

Then for any u 0 ∈ H, there exist a W α,1 B,H ([0, 1]) mapping x : I → H and an absolutely continuous mapping u :

Proof. Let us consider the compact convex (cf Theorem 5.8 and (5.7)) subset X in the Banach space C H (I) defined by

where S 1 u 0 +K denotes the set of all integrable selections of the compact convex valued constant multimapping u 0 + K. For each h ∈ X , by Theorem 4.5 and the assumptions on f , there is a unique absolutely continuous solution v h to the inclusion

For each h ∈ X consider the mapping Φ(h) defined on I by

It is clear that Φ(h) ∈ X . Let us check that Φ is continuous on X . It is sufficient to show that, if (h n ) n uniformly converges to h in X , then for v hn denoting the absolutely continuous solution of the inclusion

dv hn dt (t)), a.e. t ∈ I, the sequence (v hn ) n uniformly converges to the absolutely continuous solution v h of the inclusion

As (v hn ) n is equi-absolutely continuous with v hn (t) ∈ u 0 + t 0 Kds ⊂ u 0 + K for all t ∈ I we may suppose that (v hn ) n uniformly converges to an absolutely continuous mapping u on I. Since v hn (t) = u 0 + ]0,t] dv hn ds (s)ds, for all t ∈ I and dv hn ds (s) ∈ K a.e. s ∈ I, we may also suppose that (

Identifying the limits yields for every t ∈ I u(t) = u 0 + t 0 w(s)ds, so u = w. Therefore, by applying the arguments in the above variational limit result Cf. Proposition 3.6we get

On the other hand, writing

we see by Lemma 5.6(a) that sup

This inequality and the uniform convergence of (v hn

So by putting x = h and u = v h we conclude that (x, u) solves the dynamic EVI

The proof is complete.

Further variants of the above results are available. For instance, we are able to state the existence of solution to the dynamic system

Our tools also allow to treat other variants by considering other class of FDI given in [START_REF] Castaing | On a fractional differential inclusion in Banach space under weak compactness condition[END_REF][START_REF] Castaing | Satco, Optimal control problems governed by a second order ordinary differential equation with m-point boundary condition[END_REF][START_REF] Castaing | Fractional order of evolution inclusion coupled with a time and state dependent maximal monotone operator[END_REF][START_REF] Castaing | On a fractional differential inclusion with boundary conditions and application to subdifferential operators[END_REF][START_REF] Castaing | On fractional differential inclusions with nonlocal boundary conditions[END_REF]. We study below an example of a Caputo fractional differential inclusion governed by an EVI. For the sake of completeness, we recall some needed properties for the fractional calculus and provide a series of lemmas on the fractional integral. Throughout we assume α ∈]1, 2]. Definition 5.3. The Caputo fractional derivative of order γ > 0 of a function

Here n = [γ] + 1 and [γ] denotes the integer part of γ.

Denote by

where c D α-1 u and c D α u are the fractional Caputo derivatives of order α -1 and α of u, respectively.

We summarize some properties of a Green function given in Lemma 2.1 of [START_REF] Castaing | On a fractional differential inclusion with boundary conditions and application to subdifferential operators[END_REF].

Let f ∈ L ∞ H (I). Then the system defined by

.

We recall and summarize a crucial lemma (Lemma 3.5 of [START_REF] Castaing | On a fractional differential inclusion with boundary conditions and application to subdifferential operators[END_REF]) for our next theorem.

Lemma 5.11. Let X : [0, T ] ⇒ H be a convex compact valued measurable mapping such that

Now comes an existence result with a Caputo fractional differential inclusion. Theorem 5.12. Let I := [0, 1] and H be a separable Hilbert space. Let K be a convex compact equilibrated subset of H. Let ϕ : [0, 1] × K →] -∞, +∞] be a normal lower semicontinuous convex integrand such that (i {ϕ(., u(.)), u ∈ S 1 K } is uniformly integrable. (ii) ϕ(t, x) ≤ ϕ(τ, x) + |v(t) -v(τ )| for all t, τ ∈ [0, 1], x ∈ K where v : [0, 1] → R + is a positive nondecreasing continuous function with v(0) = 0. Let A : H → H be a linear continuous coercive symmetric operator and B : H → H be a linear continuous mapping. Let f : I ×H → H be a bounded mapping :

Then given a ∈ H, there is a W α,∞ H (I) mapping x : I → H and an absolutely continuous mapping u :

Proof. For any continuous mapping h : I → H, the mapping f h :

Then by Theorem 4.4 and the assumptions on f , there is a unique absolutely continuous solution v h to the inclusion

with v h uniformly bounded and equi-absolutely continuous: dv h dt ∈ K. Now let us consider the set X defined by

each mapping ξ f being given for every t ∈ I by

where G is the Green function given in Lemma 5.10. We note that X is convex compact in C E (I) by Lemma 5.11. Now for each h ∈ X , by Theorem 4.4 again denote by u h the unique absolutely continuous solution of the differential inclusion

For each h ∈ X let us set (again with the above Green function G)

Then it is clear that Φ(h) ∈ X because u h (t) ∈ a + K for all t ∈ I. Hence Φ(X ) is equicontinuous and relatively compact in the Banach space C E (I) because Φ(X ) ⊂ X . Now we check that Φ is continuous relative to X . It is enough to show that, if (h n ) n converges uniformly to h in X , then the sequence (u hn ) n , where each u hn is the unique absolutely continuous solution of the differential inclusion

uniformly converges to the unique absolutely continuous solution u h of the differential inclusion

We note that (u hn ) n is equicontinuous since for every n ∈ N one has uhn (t) ∈ K for almost all t ∈ I. Further, {u hn (t) : n ∈ N} is included in the compact set a + K for every t ∈ I. The Arzelà-Ascoli theorem tells us that {u hn : n ∈ N} is relatively compact in C H (I). So by extracting a subsequence, we may suppose that (u hn ) n converges uniformly on I to some mapping ζ : I → H with

This combined with the variational limit theorem (Cf. Proposition 5.3) gives

So using the uniqueness of solution of the latter differential inclusion we obtain that ζ = u h . Now let us write by Lemma 5.10 and boundedness of the Green function G

Coming back to Lemma 5.10 and applying the above notations, this means that we have just shown that there exists a mapping h ∈ W α,∞ B,H (I) satisfying

The proof of the theorem is then complete

We finish this section with two variants. Theorem 5.13. Let I = [0, 1] and let v : [0, 1] → R + be a positive nondecreasing continuous function with v(0) = 0. Let C : [0, 1] → H be a convex compact valued multimapping such that

Let A : H → H be a linear continuous coercive symmetric operator and let B : H → H be a linear continuous compact operator. Let f : I ×H → H be a bounded mapping :

Then given a ∈ H, there is a W α,∞ H (I) mapping x : I → H and an absolutely continuous mapping u :

Then by Theorem 4.1 and the assumptions on f , there is a unique absolutely continuous solution v h to the inclusion

with v h uniformly bounded and equi-absolutely continuous. Now let us consider the set X defined by

each mapping ξ f being given for every t ∈ I by

where G is the Green function given in Lemma 5.10. We note that X is convex compact in C E (I) by Lemma 5.11. Now for each h ∈ X , by Theorem 4.1 again denote by u h the unique absolutely continuous solution of the differential inclusion

For each h ∈ X let us set (again with the above Green function G)

Then it is clear that Φ(h) ∈ X because u h (t) ∈ a + coK for all t ∈ I. Hence Φ(X ) is equicontinuous and relatively compact in the Banach space C E (I) because Φ(X ) ⊂ X . Now we check that Φ is continuous relative to X . It is enough to show that, if (h n ) n converges uniformly to h in X , then the sequence (u hn ) n , where each u hn is the unique absolutely continuous solution of the differential inclusion

uniformly converges to the unique absolutely continuous solution u h of the differential inclusion

which entails that Φ(h n ) → Φ(h) uniformly on I, as desired. Then Φ : X → X is continuous, hence Φ has a fixed point, say h = Φ(h) ∈ X . This means that for every t ∈ I

), a.e. t ∈ I. Coming back to Lemma 5.10 and applying the above notations, this means that we have just shown that there exists a mapping h ∈ W α,∞ B,H (I) satisfying Let A : H → H be a linear continuous coercive symmetric operator and let g be a convex function on H Lipschitz on bounded sets and continuously differentiable on H whose gradient is locally Lipschitz.

Let

Then given a ∈ H, there is a W α,∞ H (I) mapping x : I → H and an absolutely continuous mapping u :

Proof. The proof is omitted by repeating the arguments given in the proof of Theorem 5.13. Here we apply Theorem 4.4 with the variational limit given in Proposition 3.3.

An easy inspection of the tool developed above lead to a second order evolution like mechanical problem with dry friction. Let A : H → H be a linear continuous coercive symmetric operator and let g be a convex function on H Lipschitz on bounded sets and continuously differentiable on H whose gradient is locally Lipschitz. Let f : I ×H → H be a bounded mapping :

Then given a, b ∈ H, there is an absolutely continuous mapping x : I → H and an absolutely continuous mapping u :

Proof. For any continuous mapping h : I → H, the mapping f h : I × H → H defined by f h (t) := f (t, h(t)) for all t ∈ I is L(I) measurable and satisfies |f h (t)| ≤ M for all t ∈ I. Then by Theorem 4.4 and the assumptions on f , there is a unique absolutely continuous solution v h to the inclusion

with v h uniformly bounded and equi-absolutely continuous. Now let us consider the set X defined by

each mapping ξ f being given for every t ∈ I by

We note that X is convex compact in C H (I). Now for each h ∈ X , by Theorem 4.4 again denote by u h the unique absolutely continuous solution of the differential inclusion

For each h ∈ X let us set Φ(h)(t) = a + t 0 u h (s) ds, for all t ∈ I.

Then it is clear that Φ(h) ∈ X because u h (t) ∈ b + coK for all t ∈ I. Hence Φ(X ) is equicontinuous and relatively compact in the Banach space C E (I) because Φ(X ) ⊂ X . Now we check that Φ is continuous relative to X . It is enough to show that, if (h n ) n converges uniformly to h in X , then the sequence (u hn ) n , where each u hn is the unique absolutely continuous solution of the differential inclusion

uniformly converges to the unique absolutely continuous solution u h of the differential inclusion

which entails that Φ(h n ) → Φ(h) uniformly on I, as desired. Then Φ : X → X is continuous, hence Φ has a fixed point, say h = Φ(h) ∈ X . This means that for every t ∈ I

), a.e. t ∈ I. So we have just shown that there is an absolutely continuous mapping x :

Then it is clear that Φ(h) ∈ X because u h (t) ∈ b + K for all t ∈ I. Hence Φ(X ) is equicontinuous and relatively compact in the Banach space C E (I) because Φ(X ) ⊂ X . Now we check that Φ is continuous relative to X . It is enough to show that, if (h n ) n converges uniformly to h in X , then the sequence (u hn ) n , where each u hn is the unique absolutely continuous solution of the differential inclusion

du hn dt (t)), a.e. t ∈ I, uniformly converges to the unique absolutely continuous solution u h of the differential inclusion

), a.e. t ∈ I. We note that (u hn ) n is equicontinuous since for every n ∈ N one has uhn (t) ∈ K for almost all t ∈ I. This combined with the variational limit theorem (Cf. Proposition 3.6) gives

So using the uniqueness of solution of the latter differential inclusion we obtain that ζ = u h . Now let us write

which entails that Φ(h n ) → Φ(h) uniformly on I, as desired. Then Φ : X → X is continuous, hence Φ has a fixed point, say h = Φ(h) ∈ X . This means that for every t ∈ I

So we have just shown that there is an absolutely continuous mapping x : with given operator A and B and given gradient ∇g, perturbation f and time dependent subdifferential operator ∂ϕ t and also in the context of sweeping process

Existence and uniqueness of solution for a general second order evolution inclusion in a separable Hilbert space of the form

Further (ü n ) weakly converges to the vector measure m ∈ M b R d ([0, T ]) so that the limit functions u ∞ (.) and the limit measure m satisfy the following variational inequality:

In other words, the vector measure

. Let l a be the density of the absolutely continuous part l a of l in the decomposition l = l a +l s in absolutely continuous part l a and singular part l s . Then

Further n belongs to the subdifferential ∂J ϕ∞ (u ∞ ) of the convex lower semicontinuous integral functional J ϕ∞ defined on

) , here L is a generic positive constant independent of n ∈ N. By (iv) and (3.3.5) we conclude that (

), then so is (ü n ). It turns out that the sequence ( un ) is uniformly bounded by using (3.3.2) and is bounded in variation. By Helly theorem, we may assume that ( un ) pointwisely converges to a BV function v ∞ : [0, T ] → R d and the sequence (u n ) converges uniformly to an absolutely continuous function u ∞ with u∞ = v ∞ a.e. At this point, it is clear that ( un ) converges in

2) and the dominated convergence theorem. Hence (A un (.)) converges in L 1 R d ([0, T ]) to Av ∞ (.). Step 3. Young measure limit and biting limit of ün . As (ü n ) is bounded in

) (here bar (ν t ) denotes the barycenter of ν t ). Further by Proposition 3.1, we may assume that (ü n ) biting converges to a function ζ ∞ : t → bar (ν t ) that is, there exists a decreasing sequence of Lebesgue-measurable sets (B p ) with lim p λ(B p ) = 0 such that the restriction of (ü n ) on each B c p converges weakly in

for every B ∈ B c p ∩ L([0, T ]), and for every w ∈ L ∞ R d ([0, T ]), where W n (t) = M un (t) -f n (t) and W (t) = M u∞ -f ∞ . Indeed, we note that (w(t) -u n (t)) is a bounded sequence in L ∞ R d ([0, 1]) which pointwisely converges to w(t) -u ∞ (t), it converges uniformly on every uniformly integrable subset of L 1 R d ([0, T ]) by virtue of a Grothendieck Lemma [START_REF] Grothendieck | Espaces Vectoriels Topologiques Mat[END_REF], recalling here that the restriction of -ü n -W n on each B c p is uniformly integrable. Now, since ϕ n lower epiconverges to ϕ ∞ , for every Lebesgue-measurable set A in [0, T ], by virtue of Corollary 4.7 in [START_REF] Castaing | Some variational convergence results with application to evolution inclusions[END_REF], we have

This shows that t → -bar (ν t ) -W (t) is a subgradient at the point u ∞ of the convex integral functional

As this inclusion is true on each B c p and B c p ↑ [0, T ], we conclude that

, a.e. on [0, T ].

Step 4. Measure limit in

, we may assume that (ü n ) weakly converges to the vector measure m ∈ M b R d ([0, T ]) so that the limit functions u ∞ (.) and the limit measure m satisfy the following variational inequality:

. In other words, the vector measure -m + [-A u∞ + f ∞ ] dt = -m -W.dt belongs to the subdifferential ∂J ϕ∞ (u ∞ ) of the convex functional integral

and noting that ϕ ∞ (w(t)) ≥ ϕ n (w(t)) gives immediately

uniformly integrable, and weakly converges to W := A u∞ -f ∞ and the bounded sequence in w(t) -u n (t) pointwise converges to w -u ∞ so that it converges uniformly on uniformly integrable subsets by virtue of Grothendieck lemma. Whence follows

which shows that that the vector measure -m -W.dt is a subgradient at the point u ∞ of the of the convex integral functional

and subsequently, the energy estimate holds for a.e t ∈ [0, T ]:

With the above results and notations, applying the subdifferential inequality

with w = u ∞ , integrating on [0, T ], and passing to the limit when n goes to ∞, gives the inequality

By the estimate (3.4.2) and the boundedness in

Hence we get lim n ϕ n (u n (t)) = ϕ ∞ (u ∞ (t)) a.e. Subsequently, using (iii) the passage to the limit when n goes to ∞ in the equation

By noting that (f n ) is uniformly integrable and un is uniformly bounded and pointwise converges to u∞ , by virtue of Grothendieck lemma [START_REF] Grothendieck | Espaces Vectoriels Topologiques Mat[END_REF], it converges uniformly on uniformly integrable (= relatively weakly compact) subsets of

Step 6. Localization of further limits and final step.

. Furthermore, (z n ) can be viewed as a bounded sequence in C R d ([0, T ]) . Hence there are a filter U finer than the Fréchet filter Using the inclusion

that is

we see that

Coming back to the inclusion z n (t) ∈ ∂ϕ n (u n (t)), we have ϕ n (x) ≥ ϕ n (u n (t)) + x -u n (t), z n (t) for all x ∈ R d . By substituting x by h(t) in this inequality, here h ∈ L ∞ R d ([0, T ]), and by integrating Then we have B * l = B * l a + B * l s , l ∈ L ∞ R d ([0, T ]) being the limit of (z n = f n -ün -A un ) under the filter U given in section 4 and l = l a + l s being the decomposition of l in absolutely continuous part l a and singular part l s . It follows that B * l, h = B * l a , h + B * l s , h = l a , h + l s , h for all h ∈ C R d ([0, T ]). But it is already seen that

so that the measure B * l a is absolutely continuous