Hilfer fractional evolution hemivariational inequalities with nonlocal initial conditions and optimal controls ∗

Abstract. In this paper, we mainly consider a control system governed by a Hilfer fractional evolution hemivariational inequality with a nonlocal initial condition. We first establish sufficient conditions for the existence of mild solutions to the addressed control system via properties of generalized Clarke subdifferential and a fixed point theorem for condensing multivalued maps. Then we present the existence of optimal state-control pairs of the limited Lagrange optimal systems governed by a Hilfer fractional evolution hemivariational inequality with a nonlocal initial condition. The optimal control results are derived without uniqueness of solutions for the control system.


Introduction
As a generalization of the ordinary differentiation and integration to arbitrary noninteger order, fractional calculus has been recognized as one of the most powerful tools to describe long-memory processes in the last decades.Many phenomena from viscoelasticity, electrochemistry, nonlinear oscillation in mechanics et al. can be modelled by ordinary and partial differential equations involving fractional derivatives; see, for instance, [1,2,5,7,8,15,24,28,29] and references therein.In [13], Hilfer proposed the Hilfer fractional derivative, which covers Riemann-Liouville fractional derivative and Caputo fractional derivative as special cases and appears in theoretical simulation of dielectric relaxation in glass forming materials.In [11], Gu and Trujillo studied existence of mild solutions to an evolution equation with Hilfer fractional derivative.In [12], Harrat et al. investigated solvability and optimal controls of impulsive Hilfer fractional delay evolution inclusions with Clarke subdifferential.As indicated in [9], the nonlocal initial condition can be more natural and more precise in describing some phenomena than the classical initial condition since some additional information is taken into account.There are some interesting results involved in nonlocal initial conditions for Hilfer fractional evolution equations.For example, in [26], Yang and Wang investigated approximate controllability of a Hilfer fractional differential inclusion with nonlocal initial conditions.In [27], Yang and Wang considered existence of mild solutions for a Hilfer fractional differential equation with nonlocal initial conditions.
The optimal control is one of the important and fundamental topics in the field of mathematical control theory, which plays a key role in control systems [17].In recent years, solvability and optimal control governed by fractional evolution equations has attracted great interest.For instance, the existence and optimal control for semilinear Caputo fractional finite time delay evolution systems of the order (0, 1) was concerned in [28,Chapter 4].Agarwal et al. investigated a survey on fuzzy fractional differential and optimal control nonlocal evolution equations in [3].Kumar considered the existence of optimal control for the system governed by semilinear Caputo fractional evolution equation of order (0, 1) with fixed delay in [16].Liu and Wang in [18] dealt with optimal controls of systems governed by semilinear Caputo fractional differential equations of order (0, 1) with not instantaneous impulses.Yan and Jia in [25] discussed optimal controls for Caputo fractional impulsive neutral stochastic integro-differential equations of order (1,2).On the other hand, hemivariational inequality finds its important applications to models in mechanics with nonsmooth and nonconvex energy superpotentials [22].Much attention has been paid to fractional evolution hemivariational inequalities recently.For example, Lu and Liu [19] studied the existence and controllability for a stochastic evolution hemivariational inequality in Caputo fractional derivative of order (0, 1).Lu, Liu et al. [20] investigated solvability and optimal controls for a semilinear fractional evolution hemivariational inequality in Caputo sense of order (0, 1).
Motivated by above mentioned work, the main objective of this paper is to consider the following Hilfer fractional evolution hemivariational inequality with a nonlocal initial condition: where D β,γ 0 + denotes the Hilfer fractional derivative, t ∈ I := (0, b], β ∈ [0, 1], γ ∈ (0, 1), •, • denotes the inner product (induced by the duality paring) of a separable reflexible Banach space X.The notation F 0 (t, •; •) represents the generalized (Clarke) directional derivative of a locally Lipschitz function F(t, •) : X → R. The state x takes values in the separable reflexible Banach space X.The control u takes its value from a separable reflexive Banach Y , and is given in a suitable admissible control set U ad .The operator B : I → B(Y, X), where B(Y, X) denotes the space of all bounded linear operators from Y into X.The operator A is the infinitesimal generator of a strongly continuous semigroup of a bounded linear operator family {S(t)} t 0 on Banach space X.Let I = [0, b], the function g : C(I, X) → X is continuous and compact.
In this paper, we shall establish sufficient conditions for the existence of mild solutions to system (1) and present the existence of optimal state-control pairs of the limited https://www.mii.vu.lt/NALagrange optimal systems governed by system (1).We note that the solvability for system (1) has a relation to a suitable fractional evolution inclusion with a nonlocal initial condition here.Thus the uniqueness of solutions to system (1) cannot be guaranteed by the usual condition (see condition (C3), Section 3).We shall establish optimal control results based upon the compactness result of a certain operator (see Lemma 9, Section 4).
The rest of this paper is organized as follows.Section 2 is preliminaries.Section 3 is devoted to solvability for system (1).Section 4 is involved in the existence of optimal state-control pairs of the limited Lagrange optimal control problems governed by system (1).

Preliminaries
In this section, we introduce some notations, definitions, and lemmas on fractional calculus, multivalued analysis and the generalized directional derivative.We can refer to [2, 6, 10, 13-15, 21, 28, 29] for detailed results and topics.
Denote by B(X) the space of all bounded linear operators from X into itself.Let C(I, X), C(I , X) denote the spaces of all continuous functions from I or I to X, respectively.Denote ν Thus, C(I, X) is a Banach space.Let L p (I, X) (1 p < +∞) be the Banach space of all X-valued Bochner-integrable functions defined on I with the norm x L p = ( I x(t) p dt) 1/p .For γ > 0, we define where Γ(•) is the gamma function.We also define g 0 ≡ δ 0 , the Dirac delta.
A multivalued map G : The multivalued map G : Z → P(Z) is called upper semicontinuous (u.s.c.) on Z if for each z 0 ∈ Z, the set G(z 0 ) is a nonempty, closed subset of Z, and if for each open set O of Z containing G(z 0 ), there exists an open neighborhood V of z 0 such that G(V) ⊆ O. Also, G is said to be completely continuous if G(B) is relatively compact for every B ∈ P b (Z).G has a fixed point if there exists z ∈ Z such that z ∈ G(Z).
If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c.if and only if G has a closed graph, i.e., A multivalued map : The generalized directional derivative (in the sense of Clarke) of a locally Lipshitz function h : Z → R at x in the direction d is denoted by h 0 (x, d), which is given by The Clarke subdifferential or the generalized gradient of h at x is denoted by ∂h(x), which is a subset of Z * defined by We have the following facts, which can be referred to [6] for more details.(ii) For each x ∈ O, the generalized gradient ∂h(x) is a nonempty, convex, weak *compact subset of Z * , and y Z * L for each y ∈ ∂h(x) (where L > 0 is the Lipschitz constant of h near y).(iii) The graph of the generalized gradient ∂h is close in Z × Z * w * topology, i.e., if {x n } ⊂ O and {y n } ⊂ Z * are sequences such that y n ∈ ∂h(x n ) and x n → x in Z, y n → y weak * in Z * , then y ∈ ∂h(x) (where Z * w * denotes the Banach space Z * equipped with the w * -topology).
We list the following results, which can be found in [10].In what follows, we introduce the admissible control set as [28, p. 141].Let Y be another separable reflexive Banach space from which the control u takes values.Let 1 < p < +∞ and L p (I, Y ) denote the usual Banach space of all Y -valued Bochner integrable functions having pth power summable norms.We assume that the multivalued map U : where Ω is a bounded set of Y .The admissible control set is defined as Then U ad = ∅, which can be found in [14].

Existence results
In order to investigate system (1), we can consider the following fractional evolution inclusion: We see that each solution of system ( 2) is also a solution of system (1).In fact, if x(t) ∈ C(I, X) is a solution of system (1), then there exists a function f (t) ∈ ∂F (t, x(t)), a.e.t ∈ I, and satisfies the following equation: In view of above equation, we obtain Owing to the facts that f (t) ∈ ∂F(t, x(t)) and f (t), d) X F 0 (t, x(t); d), we have It is shown that we can investigate system (1) by the corresponding evolution inclusion system (2).
In order to define the mild solution to system (2), we now introduce the following Wright function: Definition 2. (See [11].)A function x ∈ C(I, X) is said to be a mild solution to system (2) if there exists a function f ∈ L p (I, X) such that f (t) ∈ ∂F(t, x(t)), a.e.t ∈ I, and the following equation holds: where We now list the following conditions: (C1) The function t → S(t) is continuous in B(X) for all t > 0, and there exists a constant M > 1 such that S(t) M .(C2) The operator S(t) is compact for t > 0. (C5) There exists a constant L g such that for every Define the operator N : L q (I, X) → P(L p (I, X)) (1/p + 1/q = 1) as Now we have the following basic results.Lemma 6. (See [11].)Under condition (C2), the following results hold true: (i) The operator P γ (t) is continuous in the uniform operator topology for t > 0.
Theorem 1. Assume that conditions (C1)-(C5) are satisfied.Then system (2) admits at least one mild solution in a suitable ball B r on I, provided that Proof.Now, we define the multivalued map Φ : C(I, X) → P(C(I, X)) as where u(t) ∈ U ad .Clearly, the fixed points of Φ are mild solutions to system (2).We show that Φ admits a fixed point.The proof will be given in several steps.
Step 1.We show that there exists r > 0 such that Φ(B r ) ⊆ B r .
In fact, for any φ 1 , φ 2 ∈ Φ(x), there exist f 1 , f 2 ∈ N (x) such that for t ∈ I , Let ϑ ∈ [0, 1], then for each t ∈ I , we have Step 3. Φ is closed for each x ∈ B r .Let {φ n } n 1 ∈ Φ(x) such that φ n → φ in C(I, X).Then there exists f n ∈ N (x) such that for each t ∈ I ,

From (C3) and
Step 1 we know that {f n } n 1 ⊆ L p (I, X) is bounded.In view of Lemma 7, N (x) is weakly compact, and we may assume, passing to a subsequence if necessary, that f n → f , weakly in L p (I, X).Then for each t ∈ I ,
For terms I 4 , I 8 , taking into account Lemmas 2 and 6, Hölder inequality, and conditions (C3)-(C4), we have as ε → 0, The right-hand side of above inequalities tends to zero independently of x ∈ B r , and thus For t = 0, the conclusion obviously holds.Let 0 < t b be fixed.Taking into account that https://www.mii.vu.lt/NAFor each ∈ (0, t), t ∈ (0, b], x ∈ B r , and any δ > 0, we define an operator Φ ,δ 2 on B r by Φ ,δ 2 (x) the set of φ ,δ 2 such that From the compactness of S( γ δ) ( γ δ > 0) we deduce that the set )} is relatively compact in X for any ∈ (0, t) and any δ > 0. Furthermore, we have , the last inequality tends to zero as → 0 and δ → 0, i.e., there are relatively compact sets arbitrarily close to the set V (t) (t > 0).Hence, V (t) is relatively compact in X for all t ∈ (0, b].As a consequence of above steps (i)-(iii) and the Arzela-Ascoli theorem, we can deduce that Φ 2 is completely continuous.
Proof.Note that for each x ∈ B r , and define Analogously to Steps 4(i)-(iii) in the proof of Theorem 1, we can show that (Φ 1 x)(t) is completely continuous.
For any u ∈ U ad , let S(u) denote all mild solutions to systems (1) in B r defined in Theorem 1. Denote x u ∈ B r by the mild solution of system (1) corresponding to the control u ∈ U ad , we consider the following limited Lagrange problem: Problem.Find x 0 ∈ B r ⊆ C(I, X) and u 0 ∈ U ad such that for all u ∈ U ad , J (x 0 , u 0 ) J (x u , u), where and x 0 ∈ B r denotes the mild solution to system (1) related to the control u 0 ∈ U ad .
We remark that under the conditions of Theorem 1, a pair (x(•), u(•)) is feasible if it verifies system (1) for x(•) ∈ B r , and if In order to deal with the existence of optimal state-control pairs for problem (10), we further impose the following condition: Proof.For any given u ∈ U ad , we define If the set S(u) admits only finitely many elements, there exists some xu ∈ S(u) such that J (x u , u) = inf x u ∈S(u) J (x u , u) = J (u).It is trivial if the set S(u) admits infinitely many elements and inf x u ∈S(u) J (x u , u) = +∞.Now, we assume that J (u) = inf x u ∈S(u) J (x u , u) < +∞.By condition (C6) we have J (u) > −∞.For the sake of convenience, we divide the proof into the following several steps.
Step 1.Based upon the definition of infimum, there exists a sequence {x u n } ⊆ S(u) satisfying J(x u n , u) → J(u) as n → ∞.Taking into account that {x u n , u} is a sequence of feasible pairs, we have Step 2. It is shown that there exists some xu ∈ S(u) such that J (x u , u) = inf x u ∈S(u) J (x u , u) = J (u).
To achieve this aim, we first prove that for each u ∈ U ad , {x u n } is relatively compact in C(I, X).From Step 1 we have In view of Lemma 10 and Steps 4(i)-(iii) in the proof of Theorem 1, we can conclude that {I 1 x u n }, {I 2 x u n }, {I 3 x u n } are all relatively compact subsets of C(I, X).In consequence, the set {x u n } is relatively compact in C(I, X) for u ∈ U ad .Without loss of generality, we may assume that x u n → xu in CI, X) for u ∈ U ad as n → ∞.
Moreover, by conditions (C3), (C5 ), we have f u n (t) → f (t), a.e.t ∈ I, and f u n (t) ψ(t) + r , g(x u n ) → g(x u ).Let n → ∞ on both sides of (11), by the Lebesgue dominated convergence theorem, we obtain that which implies that xu ∈ S(u).Thus, through the definition of a feasible pair, condition (C6) and Balder theorem [4], we have i.e., J (x u , u) = J (u).This implies that J (u) admits its minimum at xu ∈ C(I, X) for each u ∈ U ad .
Step 3. It is shown that there exists u 0 ∈ U ad such that J (u 0 ) J (u) for all u ∈ U ad .
If inf u∈U ad J (u) = +∞, it is trivial.Assume that inf u∈U ad J (u) < +∞.By condition (C6) again, we can prove that inf u∈U ad J (u) > −∞, and similarly to Step 1, there exists a sequence {u n } ⊆ U ad such that J (u n ) → inf u∈U ad J (u) as n → ∞.Since {u n } ⊆ U ad , {u n } is bounded in L p (I, Y ) and L p (I, Y ) is a reflexive Banach space for 1/γ < p < +∞, there exists a subsequence still denoted by {u n } weakly converges to some u 0 ∈ L p (I, Y ) as n → ∞.Note that U ad is closed and convex, by Lemma 4 it follows that u 0 ∈ U ad .
Let xun be the mild solution to system (1) related to u n , where J (u n ) attains its minimum.Then (x un , u n ) is a feasible pair and verifies the following integral equation xun } are all relatively compact subsets of C(I, X).Additionally, by the fact {u n } weakly converges to some u 0 ∈ L p (I, Y ) and Lemma 9, Λ 3 is compact and Λ 3 u n → Λ 3 u 0 as n → ∞.Thus, the set {x un } ⊂ C(I, X) is relatively compact, and there exists a subsequence still denoted by {x un }, xu 0 ∈ C(I, X) such that xun → xu 0 in C(I, X) as n → ∞.Furthermore, by conditions (C3), (C5 ), we have fn (t) → f * (t), a.e.t ∈ I, and fn (t) ψ(t) + r , g(x un ) → g(x u 0 ).Let n → ∞ in both sides of (12), by the Lebesgue dominated convergence theorem, we have Therefore, J (x u 0 , u 0 ) = J(u 0 ) = inf x u 0 ∈S(u 0 ) J x u 0 , u 0 .Furthermore, J (u 0 ) = inf u∈U ad J (u), i.e., J admits its minimum at u 0 ∈ U ad .This finishes the proof.
Let operator A : D(A) ⊂ X → X be defined by Av = v with the domain D(A) := {v ∈ X: v ∈ H 2 ([0, π]), v(0) = v(π) = 0}.Then A generates a strongly continuous semigroup {S(t)} t 0 , which is compact for t > 0, analytic and self-adjoint.It is known that A has discrete spectrum with eigenvalues of the form −n 2 , n ∈ N, and the corresponding normalized eigenvectors are given by e n (s) := 2/π sin(ns).Moreover, {e n : n ∈ N} is an orthonormal basis for X, and thus A can be written as Az = ∞ n=1 n 2 z, e n e n , z ∈ D(A).Particularly, S(t) e −t (see [23] for details).Let g(x)(y) = m i=0 π 0 k(y, z)x(t i )(z) dz = m i=0 π 0 k(y, z)x(t i , z) dz, thus g satisfies condition (C5 )(see [26]).Note that problem (13) can be rewritten in the abstract form (2). According to Theorems 1-2, Eq. ( 13) has a mild solution for , L g properly small, and its corresponding limited Lagrange problem admits at least one optimal feasible pair.

Lemma 3 .
Let h : O → R be a locally Lipschitz function on an open set O of Z. Then the following results hold: (i) For each d ∈ Z, one has h 0 (x; d) = max{ y, d , y ∈ ∂h(x)}.

Lemma 4 .Lemma 5 .
The closure and weak closure of a convex subset of a normed space are the same.Let D be a nonempty bounded and convex subset of a Banach space Z. Suppose that Υ : D → P(D) is an u.s.c., condensing multivalued map.If for each x ∈ D, Υ (x) is a closed convex set in D. Then Υ has a fixed point.
and w n w in L p (I, X), then we have w ∈ N (z).Remark 1. (See[28, p. 141]) According to condition (C4) and the definition of the admissible set U ad , it is concluded that Bu ∈ L p (I, X) with 1/γ < p < ∞ for all u ∈ U ad .