Approximate controllability for second order nonlinear evolution hemivariational inequalities

The goal of this paper is to study approximate controllability for control systems driven by abstract second order nonlinear evolution hemivariational inequalities in Hilbert spaces. First, the concept of a mild solution of our problem is defined by using the cosine operator theory and the generalized Clarke subdifferential. Next, the existence and the approximate controllability of mild solutions are formulated and proved by means of the fixed points strategy. Finally, an example is provided to illustrate our main results.


Introduction
Let H be a separable Hilbert space and J = [0, b] be a real interval, where b > 0. The purpose of this paper is to consider the solvability and approximate controllability of a system governed by the following hemivariational inequality −x (t) + Ax(t) + Bu(t), v H + F 0 (t, x(t); v) ≥ 0, t ∈ J, x(0) = x 0 , x (0) = y 0 , (1.1) where A : D(A) ⊂ H → H is a closed, linear and densely defined operator which generates a strongly continuous cosine family {C(t) | t ∈ J} on H.The notation F 0 (t, •; •) stands for the generalized Clarke directional derivative (cf.[5]) of a locally Lipschitz function F(t, •) : H → R, u ∈ L 2 (J, U) is a control function, the admissible control set U is also a Hilbert space, and B is a bounded linear operator from U into H.
The variational inequalities in the elliptic and parabolic cases were introduced by Lions and Stampacchia in [20], while the hyperbolic variational inequalities were introduced by Lions in [14].Many important physical applications were given in the monographs [8] and [16], whose introductions contain many historical details, and a good survey is given in the paper [17].Concerning the control-theoretical aspects, we would also like to mention the books [15] and [18,19], considered to be the most important references on control of linear partial differential equations.They also contain numerous results closely related to the subject of the present paper.
It is well known that many problems from mechanics (elasticity theory, semipermeability, electrostatics, hydraulics, fluid flow), economics and so on can be modeled by subdifferential inclusions or hemivariational inequalities, and we refer to [32] for more applications of hemivariational inequalities.Recently, the existence of solutions for hemivariational inequalities has been proved by many authors.For instance, the authors of [12,22] considered the problems with elliptic hemivariational inequalities, and in [4,21,[27][28][29][30], the researchers discussed the problems of parabolic hemivariational inequalities.However, to our knowledge, only a few results on existence of solutions were obtained for hyperbolic hemivariational inequalities, and this is one of our motivation for the study of such hemivariational inequality (1.1) in the present work.
It is well know that the importance of the wave equation is not only because it is the most relevant hyperbolic partial differential equation but also relies on the fact that it models a large class of vibrating phenomena such as vibrations of elastic bodies and the sound propagation.For example, let Ω ⊂ R n and ∆ be the Laplace operator, the wave equation (here x = x(t, θ) is the displacement function and f is an arbitrary external forcing function) provides a good application for the amplitude vibrations of an elastic string or a flexible membrane occupying the region Ω under a force acting on the vibrating structure.In applications of mathematical methods to physics, one is often concerned with the study of more complicated nonlinear wave equations such as the Klein-Gordon equation of the form where µ, η ∈ R. A large number of work has been devoted to the study of the Cauchy problem for the nonlinear Klein-Gordon equation.One of the nonlinear equations of greatest interest in the development of theoretical physics is the following where f (t, x, x t ) depends nonlinearly on x and x t and is, in a sense, close to a "monotone" function.Equations of this type arise naturally in many contexts, for instance, in classical mechanics, fluid dynamics, quantum field theory (cf.[37]) and have been extensively studied in the last decade (cf.[10,13,37,38,40]).Motivated by the aforementioned contributions, we are interested in the following model which is met in contact mechanics and can be described by the Clarke subdifferential of a nonconvex function It is supposed that F is a known function of the external force of the form Here ∂j(t, θ, η) denotes the generalized gradient of Clarke (cf.[5]) with respect to the third variable of a function j : (0, b) × Ω × R → R which is assumed to be locally Lipschitz in η.The multivalued function ∂j(t, θ, •) : R → 2 R is generally nonmonotone and it includes the vertical jumps.In a physicist's language, it means that the law is characterized by the generalized gradient of a nonsmooth potential j.The system (1.2), (1.3) can serve as a prototype of a dynamic model of nonmonotone skin friction in plane elasticity.For a description of such problems which include beams in adhesive contact, Kirchhoff plates, and other engineering applications, we refer to [26,[31][32][33].We underline that due to the lack of convexity of the function j(t, θ, •), the above problem cannot be formulated as a variational inequality.Its variational formulation leads to a hyperbolic hemivariational inequality, a simple version of the problem (1.1) with Ay = ∆y.
The first goal of our work is to study the existence of solutions for the system (1.1).Secondly, we are also curious about the fact whether or not the system (1.1) can get a good behavior as desired under the proper action of the law of supply.That is, the main properties of hyperbolic hemivariational inequalities such as time-reversibility and the lack of regularizing effects, have some very interesting and important consequences in control problems, too.At present, optimal control problems for hemivariational inequalities have been examined in a number of publications.In particular, we refer to Haslinger and Panagiotopoulos [11] on the existence of optimal control pairs for a class of coercive hemivariational inequalities, Migórski and Ochal [28] about the optimal control problems with the parabolic hemivariational inequalities, J. Park and S. Park [35] on the optimal control problems for the hyperbolic linear systems, and to Tolstonogov [41,42] about the optimal control problems for subdifferential type differential inclusions.Very recently, Liu and Li [23] studied the approximate controllability for a class of first order hemivariational inequalities.However, there is still little information available on the approximate controllability of hyperbolic hemivariational inequalities like (1.1).Therefore, it is worth to extend the main results of our previous paper in [23] to the control system (1.1).
The paper is organized as follows.In Section 2 we recall the notation and some basic definitions, and preliminary facts, we use throughout the paper.In Section 3, we are concerned with the existence of mild solutions of the system (1.1).The approximate controllability of our problems is analyzed in Section 4, while Section 5 is devoted to a concrete application of our main results.

Preliminaries
For a Banach space E with the norm • E , E * denotes its dual and •, • E the duality pairing between E * and E. The symbol L(X, Y) denotes the space of bounded linear operators from a Banach space X to a Banach space Y. C(J, E) is the Banach space of all continuous functions from , where x stands for the generalized derivative of x, i.e., Let P (E) be the set of all nonempty subsets of E. We will use the following notation Moreover, B r (0) and B r (0) denote, respectively, the open and the closed balls in a Banach space E centered at origin and of radius r > 0.
The key tool in our main results is the following fixed point theorem stated in [7].Then, either (i) the operator inclusion x ∈ Ax + Bx has a solution in B r (0), or (ii) there exist an element u ∈ E with u = r such that λu ∈ Au + Bu for some λ > 1.
Next, we recall some concepts of nonsmooth analysis (see [5] for more details).Let j : E → R be a locally Lipschitz function on a Banach space E. The Clarke generalized directional derivative j 0 (x; v) of j at the point x ∈ E in the direction v ∈ E is defined by The generalized gradient of j at x ∈ E is the subset of E * given by In the sequel, we shall study the existence of mild solutions and approximate controllability of the following second order evolution inclusion where H is a separable Hilbert space, A : Therefore, if we want to prove the solvability of the hemivariational inequality (1.1), we only need to deal with the inclusion (2.1).Similarly to [23], we say that x ∈ W 1,2 (J, H) is a solution of (2.1), if there exists f ∈ L 2 (J, H) such that f (t) ∈ ∂F(t, x(t)) for a.e.t ∈ J and From now on, in order to obtain the solution to problem (2.1), let us consider the following abstract second order initial value problem The following basic results concerning strongly continuous cosine operator can be found in the books [1,9] and in the papers [2,34,36,39].Definition 2.2.A strongly continuous operator C : R → L(H) is called a cosine operator, if C(0) = I (identity operator) and The linear operator A defined by is the generator of the strongly continuous cosine operator C, D(A) is the domain of A.
It is known that the generator A is a linear, closed, and densely defined operator on H. Next, let C be a cosine operator on H with generator A. The sine operator S : R → L(H) associated with the strongly continuous cosine operator C is defined by In the sequel, we collect some further properties of a cosine operator C and its relations with both the generator A and the associated sine operator S.

Proposition 2.3. The following assertions hold.
(i) There exist M A ≥ 1 and ω ≥ 0 such that C(t) ≤ M A e ω|t| and S(t) ≤ M A e ω|t| .
The uniform boundedness principle, together with Proposition 2.3 (i), implies that C(t) and S(t) are uniformly bounded on J by some positive constants M C and M S , respectively.Now, from the argument above, we may introduce the following concept.Definition 2.4.For each u ∈ L 2 (J, U), a function x ∈ C(J, H) is called a mild solution of the system (2.1) if there exists f ∈ L 2 (J, H) such that f (t) ∈ ∂F(t, x(t)) for a.e.t ∈ J and Throughout this paper, by a solution of system (2.1), we mean the mild solution.

Existence of mild solutions
The purpose of this section is to state the existence of mild solutions for problem (2.1).
We list the following standing hypotheses of this paper.
H(F 1 ) : the function x → F(t, x) is locally Lipschitz for all t ∈ J, H(F 2 ) : the function t → F(t, x) is measurable for all x ∈ H, H(F 3 ) : there exist a function k ∈ L 2 (J, R + ) and a constant l > 0 such that and all x ∈ H, H(S) : the sine operator S(t) associated with the operator A is compact for all t ≥ 0.
Next, we define a multivalued operator N : From [31, Lemma 5.3], we know that the multifunction N has nonempty, convex and weakly compact values for each x ∈ L 2 (J, H).Moreover, we have the following lemma.Lemma 3.1 ([30, Lemma 11]).If H(F 1 )-H(F 3 ) hold, then the operator N satisfies: if x n → x in L 2 (J, H), w n → w weakly in L 2 (J, H) and w n ∈ N (x n ), then w ∈ N (x).
In the sequel, for any x ∈ C(J, H) ⊂ L 2 (J, H), we can define a multivalued operator B : C(J, H) → 2 C(J,H) as follows The following property of the multivalued operator B is an essential result for proving the existence of mild solutions for system (2.1).Lemma 3.2.For each u ∈ L 2 (J, U), under the hypotheses H(F 1 )-H(F 3 ) and H(S), the multivalued operator B : C(J, H) → 2 C(J,H) is completely continuous, u.s.c., and has compact and convex values.
Proof.Firstly, for all x ∈ C(J, H), the convexity of values of the operator B(x) is obvious by the convexity of N (x).Next, for convenience, we divide the proof into two steps.
Step 1: We show that the multivalued operator B is completely continuous and has compact values.
First, we show that the operator B is bounded, i.e., for all x ∈ B r (0) with r > 0, B(B r (0)) is bounded in C(J, H).In fact, for all x ∈ B r (0), ϕ ∈ B(x), by using H(F 3 ) and the Hölder inequality, we obtain, for all t ∈ J Therefore, B(B r (0)) is a bounded subset in C(J, H).
Next, we prove that {B(x) | x ∈ B r (0)} is equicontinuous.To this end, let 0 < τ 1 < τ 2 ≤ b and δ > 0 be small enough.Then, we obtain Since Proposition 2.3 (iii) implies the continuity of S(t) in the uniform operator topology, it can be easily seen that the right-hand side of the above inequality is independent of x ∈ B r (0) and tends to zero, as τ 2 → τ 1 .Hence, we obtain that {B(x) | x ∈ B r (0)} is an equicontinuous subset of C(J, H).
Finally, from the assumption H(S) and by the definition of a relatively compact set, it is not difficult to check that {ϕ(t) | ϕ ∈ B(B r (0))} is relatively compact in H. Thus, by the generalized Ascoli-Arzelà Theorem, we get that B is a multivalued compact map.
Step 2: The operator B has a closed graph.
Let x n ∈ C(J, H), y n ∈ B(x n ) be such that x n → x and y n → y.We will prove that y ∈ B(x).Thus y n ∈ B(x n ) implies that there exists f n ∈ N (x n ) such that for all t ∈ J, we have

Define the linear continuous operator
Since x n → x, it follows from Lemma 3.1 and the compactness of the operator for some f ∈ N (x), i.e., B has a closed graph.Therefore, since B takes compact values, by Proposition 3.3.12(2) of [31], we know that B is u.s.c.The proof of the lemma is complete.
We are now in a position to obtain the main result of this section.
Theorem 3.3.For each u ∈ L 2 (J, U), if the hypotheses H(F 1 )-H(F 3 ) and H(S) are satisfied, then the system (2.1) has a mild solution on J.
Proof.It is clear that the multivalued map : C(J, H) → 2 C(J,H) defined by has nonempty values, since N (x) = ∅.In view of the definition of , the problem of finding mild solutions of (2.1) is equivalent to obtaining fixed points of .To prove this, we set = A + B, where B is defined by (3.2) and A is given by According to Theorem 2.1, it is sufficient to show that there exists no element x ∈ C(J, H) with x = r such that λx ∈ Ax + Bx for some λ > 1.Indeed, let λx ∈ A(x) + B(x) with λ > 1, and suppose that there exists f ∈ N (x) such that Then, by the assumptions, we obtain Applying the Gronwall inequality, from the last expression, we obtain Clearly, K r is an open subset of C(J, H).As an immediate consequence of Lemma 3.2, B : K r → P cv,cp (H) is u.s.c. and compact and it is not difficult to get that A : K r → H is a single-valued contraction with k < 1 2 .Furthermore, from the choice of K r , there is no x ∈ C(J, H) with x = r such that λx ∈ Ax + Bx for some λ > 1.
Thus, by Theorem 2.1, we obtain that the operator inclusion x ∈ Ax + Bx has a solution in K r which is a mild solution of system (2.1).The proof of the theorem is complete.

Approximate controllability results
Controllability is one of the fundamental concepts in mathematical control theory.This is a qualitative property of dynamical control systems and it is of particular importance in control theory.In recent years, controllability problems for various types of nonlinear dynamical systems in infinite dimensional spaces by using different kinds of approaches have been considered in many publications, see [2, 3, 23-25, 34, 36] and the references therein.In this section, we turn our attention to approximate controllability of second order evolution inclusion (2.1).Following [6], we recall the following notions.(a) Control system (2.1) is said to be exactly controllable on J if, for all x 0 , x 1 ∈ H, there exists u ∈ L 2 (J, U) such that the mild solution to system (2.1) satisfies x(0; u) = x 0 and x(b; u) = x 1 .
(b) Control system (2.1) is approximately controllable on J if, for every x 0 , x 1 ∈ H, and for every ε > 0, there exists a control u ∈ L 2 (J, U) such that the mild solution x of the problem (2.1) satisfies x(0) = x 0 and x(b) − x 1 < ε.Equivalently, we may say that the attainable set of system (2.1) with the initial value x 0 at the terminal time b, i.e., K b (F) = {x(b) ∈ H | x(•) is a mild solution of system (2.1) corresponding to u ∈ L 2 (J, U)} is dense in H.
In order to analyze the approximate controllability of problem (2.1), we shall consider the linear system which is associated with (2.1), namely x (t) = Ax(t) + Bu(t) for t ∈ J, x(0) = x 0 , x (0) = y 0 . (4.1) The controllability map corresponding to the linear system (4.1) is defined by which is a nonnegative, bounded, and linear operator on H.Here B * denotes the adjoint of B and S * (t − s) is the adjoint of S(t − s).Therefore, the inverse of εI + Γ b 0 exists, for any ε > 0, so the resolvent R(ε, −Γ b 0 ) = (εI + Γ b 0 ) −1 is well defined.The resolvent is useful in the study of the controllability properties of system (4.1).In this respect, we state a useful characterization of the approximate controllability for (4.1) in terms of the resolvent.

Lemma 4.2 ([3, Theorem 2]
).The linear system (4.1) is approximately controllable on J if and only if εR(ε, −Γ b 0 ) → 0, as ε → 0 + in the strong operator topology.At this point, we develop our fixed point approach.For any x ∈ C(J, H) ⊂ L 2 (J, H), there holds N (x) = ∅, where N is defined by (3.1).Hence, for every ε > 0, we can define the multivalued map ε : C(J, H) → 2 C(J,H) as follows Similarly, as in the proof of Theorem 3.3, we decompose ε in the following way for x ∈ C(J, H).The following property of the multivalued operator B ε is an essential result for obtaining the main results of this section.Then for all ε > 0 the operator B ε : C(J, H) → 2 C(J,H) is completely continuous, u.s.c., and has compact and convex values.
Proof.The proof is similar to the one of Lemma 3.2 and for this reason we omit it here.
Theorem 4.4.Suppose all of the hypotheses of Lemma 4.3.Then for all ε > 0 the map ε has a fixed point on J provided the system (4.1) is approximately controllable.
Proof.From the decomposition of ε , our problem is reduced to find the solutions of the operator inclusion x ∈ A ε x + B ε x.According to Theorem 2.1, it is sufficient to show that there is no element x ∈ C(J, H) with x = R such that λx ∈ A ε x + B ε x for some λ > 1.Indeed, let λx ∈ A ε x + B ε x with λ > 1, and assume that there exists f ∈ N (x) such that where From the assumption H(F 4 ), we know that there exists g ∈ L 2 (J, R + ) such that ∂F(t, x) H = sup{ f H | f ∈ ∂F(t, x)} ≤ g(t) for a.e.t ∈ J.Then, we obtain Clearly, K R is an open subset of C(J, H).As an immediate consequence of Lemma 4.3, B ε : K R → P cv,cp (H) is u.s.c. and compact and it is also easy to see that A ε : K R → H is a single-valued contraction with contraction constant k < 1 2 .Moreover, by the choice of K R , there is no x ∈ C(J, H) with x = R such that λx ∈ A ε x + B ε x for some λ > 1.
Thus, by Theorem 2.1, the operator inclusion x ∈ A ε x + B ε x has a solution in K R which is also a fixed point of ε on J.The proof is complete.
From now on, with the aforementioned theorems in mind, we deliver the second main result of this paper.Theorem 4.5.Assume the hypotheses of Lemma 4.3.Then system (2.1) is approximately controllable on J if the system (4.1) is approximately controllable on J.
Proof.From Theorem 4.4, we know that the operator ε has a fixed point in C(J, H) for all ε > 0. Let x ε be a fixed point of ε in C(J, H).It is easy to see that any fixed point of ε is a mild solution of (2.1) for control u ε .Therefore, there exists f ε ∈ N (x ε ) such that Then, by the property This fact, combined with hypothesis H(F 4 ) and the Dunford-Pettis Theorem, guarantees that the set { f ε (•)} is weakly compact in L 2 (J, H).Thus, there is a subsequence, still denoted by { f ε (•)} that converges weakly in L 2 (J, H) to f (•).We set Moreover, from the last inequality, we get which implies that system (2.1) is approximately controllable on J.The proof is complete.

Theorem 2 . 1 .
Let A : B r (0) → E and B : B r (0) → P cv,cp (E) be two operators such that (a) A is a single-valued contraction with contraction constant k < 1 2 , and (b) B u.s.c. and compact.
The control function u ∈ L 2 (J, U) where U is a Hilbert space of admissible controls.We remark that, by the definition of the generalized Clarke subdifferential, problem (2.1) is equivalent to the hemivariational inequality(1.1) [5]ear and densely defined operator which generates a strongly continuous cosine family {C(t) | t ∈ J} on H.The notation ∂F stands for the generalized Clarke subdifferential (cf.[5]) of a locally Lipschitz function F(t, •) : H → R and B ∈ L(U, H).
Lemma 4.3.Suppose that hypotheses H(S), H(F 1 ) and H(F 2 ) are satisfied.Moreover, we assume that H(F 4 ) : there exists a multivalued function G : J → 2 H with weakly compact values which is square integrable such that ∂F(t, x) ⊂ G(t) for a.e.t ∈ J, all x ∈ H.