Optimal control of feedback control systems governed by hemivariational inequalities

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Optimal control of feedback control systems governed by hemivariational inequalities✩ Yong Huang a , Zhenhai Liu b,c,∗ , Biao Zeng d a

Department of Mathematics, Baise University, Baise 533000, Guangxi Province, PR China

b

Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation, Guangxi University for Nationalities, Nanning 530006, Guangxi Province, PR China c

College of Sciences, Guangxi University for Nationalities, Nanning 530006, Guangxi Province, PR China

d

Faculty of Mathematics and Computer Science, Jagiellonian University, Institute of Computer Science, ul. Łojasiewicza 6, 30-348 Krakow, Poland

article

info

Article history: Received 20 November 2014 Received in revised form 16 August 2015 Accepted 23 August 2015 Available online xxxx Keywords: Feedback optimal control Hemivariational inequalities Clarke’s subdifferential Feasible pairs

abstract This paper is mainly concerned with the feedback control systems governed by evolution hemivariational inequalities. By using the properties of multimaps and Clarke’s subdifferential, we formulate some sufficient conditions to guarantee the existence result of feasible pairs of the feedback control systems. We also present an existence result of optimal control pairs for an optimal control problem. We emphasize that our results cannot be obtained straightforwardly from the previous works. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Let H be a separable Hilbert space, ⟨·, ·⟩H the inner product of H. A : D(A) ⊆ H → H is the infinitesimal generator of a uniformly bounded compact C0 -semigroup {T (t )}t >0 on H. Let V be a reflexive Banach space, u : [0, T ] → V a control function and B : V → H a bounded linear operator. The notation F 0 (t , ·; ·) stands for the generalized Clarke’s directional derivative (cf. [1]) of a locally Lipschitz function F (t , ·) : H → R. In this paper, we firstly study the existence of solutions of the following evolution hemivariational inequalities:



⟨−x′ (t ) + Ax(t ) + Bu(t ), v⟩H + F 0 (t , x(t ); v) ≥ 0, x(0) = x0 ∈ H .

a.e. t ∈ [0, T ], ∀v ∈ H ,

(1.1)

Next, we shall be concerned with the existence of feasible pairs of the following feedback control systems:

 ⟨−x′ (t ) + Ax(t ) + Bu(t ), v⟩H + F 0 (t , x(t ); v) ≥ 0, u(t ) ∈ U (t , x(t )), x(0) = x ∈ H , 0

a.e. t ∈ [0, T ], ∀v ∈ H , (1.2)

where U : [0, T ] × H → P (V ) is a multimap. ✩ Project supported by NNSF of China Grant Nos. 11271087, 61263006, NSF of Guangxi Grant Nos. 2013GXNSFAA019022, 2014GXNSFDA118002, Scientific Research Fund of Guangxi Education Department Grant No. 2013YB243 and Special Funds of Guangxi Distinguished Experts Construction Engineering. ∗ Corresponding author. Tel.: +86 771 3265663, +86 771 3260370. E-mail addresses: [email protected] (Y. Huang), [email protected] (Z. Liu), [email protected] (B. Zeng).

http://dx.doi.org/10.1016/j.camwa.2015.08.029 0898-1221/© 2015 Elsevier Ltd. All rights reserved.

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Hemivariational inequalities have important applications in mechanics and engineering, especially in nonsmooth analysis and optimization (see [2–16]). With the development of the study of hemivariational inequalities, some scholars have begun to pay their attentions to the optimal control problems for hemivariational inequalities. In particular, Haslinger and Panagiotopoulos [17] obtained the existence of optimal control pairs for a class of coercive hemivariational inequalities. In [18], Migorski and Ochal investigated the optimal control problems for the parabolic hemivariational inequalities. J.Y. Park and S.H. Park [15,16] showed the existence of optimal control pairs to the hyperbolic linear systems. In [19,20], Tolstonogov paid his attention to the optimal control problems for differential inclusions with subdifferential type. Feedback control systems are ubiquitous around us, including trajectory planning of a robot manipulator, guidance of a tactical missile toward a moving target, regulation of room temperature, and control of string vibrations. Optimal feedback control of semilinear evolution equations in Banach spaces has been studied [21–24]. However, the study for the optimal control of feedback control systems described by evolution hemivariational inequalities is still untreated topic in the literature and this fact is the motivation of the present work. The aim of this paper is study the existence result of feasible pairs of feedback optimal control systems for evolution hemivariational inequalities. By using the properties of multimaps and Clarke’s subdifferential, a new set of sufficient conditions are formulated to guarantee our main results. The rest of this paper is organized as follows. In the next section, we will introduce some useful preliminaries and physical models. In Section 3, some sufficient conditions and techniques are established for the existence of feasible pairs of problem (1.2). We first study the existence of solutions of (1.1) by a fixed point theorem of multimaps. In Section 4, we will study the optimal control of problem (1.2). 2. Preliminaries and physical models In this section, we first introduce some basic preliminaries which are used throughout this paper. The norm of the Hilbert space H will be denoted by ∥ · ∥H . Let J = [0, T ]. For a uniformly bounded C0 -semigroup {T (t )}t ≥0 , there exists M > 0 such that supt ∈[0,∞) ∥T (t )∥ ≤ M [25]. Let C (J , H ) denote the Banach space of all continuous functions from J into H with the norm ∥x∥C = supt ∈J ∥x(t )∥H , L2 (J , H ) denote the Banach space of all Bochner L2 -integrable functions from J into H with the

 T

1

norm ∥x∥L2 = 0 ∥x(s)∥2 ds 2 . Let us recall some definitions and properties about multimaps. For more details we refer to [26–31]. Let X and Y be two topological spaces. Denote by P (Y ) [C (Y ), K (Y ), Kv (Y )] the collections of all nonempty [respectively, nonempty closed, nonempty compact, nonempty compact convex] subsets of Y . A multimap F : J → C (X ) is said to be measurable, if F −1 (Q ) := {x ∈ J |F (x) ∩ Q ̸= ∅} ∈ L for every closed set Q ⊂ X , where L denotes the σ -field of Lebesgue measurable sets on J. Every measurable multimap F admits a measurable selection f : J → X , i.e., f is measurable and f (t ) ∈ F (t ) for a.e. t ∈ J. A multimap F : X → C (Y ) is said to be upper semicontinuous (or u.s.c. for short), if for every −1 (D) = {x ∈ X : F (x) ⊂ D} is open in X ; weakly u.s.c., if F : X → C (Yw ) is u.s.c., where Yw is open subset D ⊂ Y the set F+ the space Y equipped with a weak topology. A multimap F : X → C (Y ) is said to be closed if its graph Gr (F ) := {(x, y) ∈ X × Y : x ∈ X , y ∈ F (x)} is a closed subset of X × Y ; compact, if F maps bounded sets of X into relatively compact sets in Y . We have the following important property for multimaps. Lemma 2.1 ([29, Theorem 1.1.12]). Let X and Y be metric spaces and F : X → K (Y ) a closed compact multimap. Then F is u.s.c. Definition 2.2 ([23]). Let X be a Banach space and Y be a metric space. Let F : X → P (Y ) be a multimap. We say F possesses the Cesari property at x0 ∈ X , if



coF (Oδ (x0 )) = F (x0 ),

δ>0

where coD is the closed convex hull of D, Oδ (x) is the δ -neighborhood of x. If F has the Cesari property at every point x ∈ Z ⊂ X , we simply say that F has the Cesari property on Z . Lemma 2.3 ([23, Proposition 4.2]). Let X be a Banach space and Y be a metric space. Let F : X → P (Y ) be u.s.c. with convex and closed valued. Then F has the Cesari property on X . Now, let us proceed to the definition of the Clarke’s subdifferential for a locally Lipschitz function j : X → R, where X is a Banach space and X ∗ is the dual space of X (one can see [1,32,14]). We denote by j0 (x; v) the Clarke’s generalized directional derivative of j at the point x ∈ X in the direction v ∈ X , that is j0 (x; v) := lim sup

λ→0+ , ζ →x

j(ζ + λv) − j(ζ )

λ

.

Recall also that the Clarke’s subdifferential or generalized gradient of j at x ∈ X , denoted by ∂ j(x), is a subset of X ∗ given by

∂ j(x) := {x∗ ∈ X ∗ : j0 (x; v) ≥ ⟨x∗ , v⟩, ∀v ∈ X }.

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Lemma 2.4 ([14, Proposition 3.23]). If j : X → R is locally Lipschitz function on a subset E of X , then (i) the function (x, v) → j0 (x; v) is u.s.c. from E × X into R, i.e., for all x ∈ E , v ∈ X , {xn } ⊂ E , {vn } ⊂ X such that xn → x in E and vn → v in X , we have lim sup j0 (xn ; vn ) ≤ j0 (x; v); (ii) for every x ∈ E the gradient ∂ j(x) is a nonempty, convex, and weakly∗ compact subset of X ∗ which is bounded by the Lipschitz constant Kx > 0 of j near x; (iii) the graph of ∂ j is closed in X × Xw∗ ∗ ; (iv) the multimap ∂ j is u.s.c. from E into Xw∗ ∗ , i.e., ∂ j is weakly u.s.c. from E into X ∗ . The key tool in one of our main results is the following fixed point theorem. Theorem 2.5 ([33], Nonlinear Alternative for Kakutani Maps). Let X be a Banach space, C a closed convex subset of X , D an open subset of C (relative to C ) and 0 ∈ D. Suppose that z : D → Kv (C ) is an u.s.c. and compact multimap. Then either (i) z has a fixed point in D, or (ii) there are x ∈ ∂ D and λ ∈ (0, 1) with x ∈ λzx. In the sequel, we shall study the existence of mild solutions (see, Definition 3.6) of the following semilinear inclusion:



x′ (t ) ∈ Ax(t ) + Bu(t ) + ∂ F (t , x(t )), x(0) = x0 ∈ H ,

a.e. t ∈ J ,

(2.1)

where A : D(A) ⊆ H → H is the infinitesimal generator of a C0 -semigroup T (t )(t ≥ 0) on a separable Hilbert space H. The notation ∂ F stands for the generalized Clarke subdifferential (cf. [1]) of a locally Lipschitz function F (t , ·) : H → R. The control function u takes value in L2 (J , V ) and the admissible controls set V is a Hilbert space, B is a bounded linear operator from V into H. We say that x ∈ W 1,2 (J , H ) is a solution of (2.1) if there exists f ∈ L2 (J , H ) such that f (t ) ∈ ∂ F (t , x(t )) and



x′ (t ) = Ax(t ) + Bu(t ) + f (t ), x(0) = x0 ∈ H ,

a.e. t ∈ J ,

which implies



⟨−x′ (t ) + Ax(t ) + Bu(t ), v⟩H + ⟨f (t ), v⟩H = 0 a.e. t ∈ J , ∀v ∈ H , x(0) = x0 ∈ H .

Since f (t ) ∈ ∂ F (t , x(t )) and ⟨f (t ), v⟩H ≤ F 0 (t , x(t ); v), we obtain

 ⟨−x′ (t ) + Ax(t ) + Bu(t ), v⟩H + F 0 (t , x(t ); v) ≥ 0, x(0) = x0 ∈ H .

a.e. t ∈ J , ∀v ∈ H ,

Hence, any solutions of the problem (2.1) are also solutions of the problem (1.1). Similarly, the feedback control systems (1.2) of hemivariational inequalities can be reduced to the following feedback control systems with Clarke’s subdifferential: x′ (t ) ∈ Ax(t ) + Bu(t ) + ∂ F (t , x(t )), u(t ) ∈ U (t , x(t )), x(0) = x0 ∈ H .



a.e. t ∈ J , (2.2)

Therefore, in order to study the hemivariational inequality (1.1) and feedback control system (1.2), we only need to deal with the semilinear inclusion (2.1) and feedback control system (2.2). We note that the problem (1.1) arises in many important models for distributed parameter control problems and that a large class of identification problems enter our formulation. Let us indicate a problem which is one of the motivations for the study of hemivariational inequality (1.1) (cf [18]). We consider the following initial value problem with heat equation

 2   ∂ y = ∂ y + Bu + f (y), (x, t ) ∈ (0, π ) × (0, 1), ∂t ∂ x2 y ( t , 0 ) = y(t , π ) = 0, t ∈ (0, 1),   y(0, x) = y0 (x), x ∈ (0, 1).

(2.3)

This problem represents the heat flow with a temperature dependent source. Here y = y(x, t ) represents the temperature at the point x ∈ (0, π ) and time t ∈ (0, 1). The temperature of boundaries is zero and the initial temperature is y0 (x) (x ∈ (0, π )). u is a control function. f (y) is a heat source dependent of temperature. It is supposed that the control u is a feedback control by the temperature y such that u ∈ U (t , y),

a.e. (x, t ) ∈ (0, π ) × (0, 1)

(2.4)

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f is a known function of the temperature of the form f (x, t ) ∈ ∂ F (x, t , y(x, t )) a.e. (x, t ) ∈ (0, π ) × (0, 1).

(2.5)

Here ∂ F (x, t , ξ ) denotes the Clarke’s generalized gradient with respect to the last variable of a function F : (0, π ) × (0, 1) × R → R which is assumed to be locally Lipschitz in ξ . The multivalued function ∂ F (x, t , ·) : R → 2R 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 F . Take H = L2 (0, π ), y(t )(·) = y(t , ·) and the operator A : D(A) ⊂ H → H is defined by Ay = y′′ , where the domain D(A) is given by

{y ∈ H : y′ , y′′ ∈ H , y(0) = y(π )}. Then, A can be written as Ay = −

∞ 

n2 (y, yn )yn ,

y ∈ D(A),

n =1

√

where yn (x) = 2/π sin nx (n = 1, 2, . . .) is an orthonormal basis of H. It is well known that A generates of a strongly continuous semigroup T (t )(t > 0) in H, which is compact and analytic (see [25]), given by T (t )y =

∞ 

2

e−n t (y, yn )yn ,

y ∈ H,

and

∥T (t )∥ ≤ e−1 < 1 = M .

n =1

Now we let the function F : (0, 1) × H → R be given by F (t , y) =

1



j(x, t , y(x))dx,

t ∈ (0, 1), y ∈ H ,

0

where j(x, t , z ) =

z



φ(x, t , θ )dθ ,

(x, t ) ∈ (0, π ) × (0, 1), z ∈ R.

0

Assume that φ : (0, π ) × (0, 1) × R → R be a function satisfying: (i) (ii) (iii) (iv)

for all x ∈ (0, π ), z ∈ R, φ(·, x, z ) : (0, 1) → R is measurable; for all t ∈ (0, 1), z ∈ R, φ(t , ·, z ) : (0, π ) → R is continuous; for all z ∈ R there exists a constant c1 > 0 such that |φ(·, ·, z )| ≤ c1 (1 + |z |) for z ∈ R; for every z ∈ R, φ(·, ·, z ± 0) exists.

If φ satisfies (iii), then we have that, ∂ j(z ) ⊂ [φ(z ), φ(z )] for z ∈ R (we omit (x, t ) here), where φ(z ) and φ(z ) denote the essential supremum and essential infimum of φ at z (see [1, p. 34]). If φ satisfies (i)–(iv), then the function j(·, ·, ·) defined above satisfies the following properties: (i) (ii) (iii) (iv) (v)

for all x ∈ (0, π ), z ∈ R, j(·, x, z ) is measurable and j(·, ·, 0) ∈ L2 ((0, π ) × (0, 1)); for all t ∈ (0, 1), z ∈ R, j(t , ·, z ) : (0, π ) → R is continuous; for all (x, t ) ∈ (0, π ) × (0, 1), j(x, t , ·) : R → R is locally Lipschitz; there exists a constant c2 > 0 such that |η| ≤ c2 (1 + |z |) for all η ∈ ∂ j(x, t , z ), (x, t ) ∈ (0, π ) × (0, 1); there exists a constant c3 > 0 such that j0 (x, t , z ; −z ) ≤ c3 (1 + |z |) for all (x, t ) ∈ (0, π ) × (0, 1).

Let V be a reflexive Banach space, u : (0, 1) → V a control function and B : V → R a bounded linear operator. Thus, combining (2.4)–(2.5), problem (2.3) turns to be problem (2.2). Therefore, the variational formulation of the above problem leads to the hemivariational inequality (1.1) and is met, for example, in the nonmonotone nonconvex interior semipermeability problems. For the latter, Panagiotopoulos [34] considered a temperature control problem in which they regulated the temperature to deviate as little as possible from a given interval. We remark that the monotone semipermeability problems, leading to variational inequalities, have been studied by Duvaut and Lions in [35] under the assumption that F (x, t , ·) is a proper, lower semicontinuous, convex function which means that ∂ F (x, t , ·) is a maximal monotone operator in R2 . From the above, our problems in this paper are valuable and it is worth to do further research on this subject. For more details, one can see [36,1,37,15]. 3. The existence of feasible pairs In this section we study the existence of feasible pairs for problem (2.2).

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At the first, we study the existence of solutions of problem (2.1). We will make the following conditions.

(HT ) : T (t ) is a compact operator for every t > 0. Let F : J × H → R be a function satisfying the following conditions: (F1) F (·, x)J → R is measurable for every x ∈ H; (F2) F (t , ·) : H → R is locally Lipschitz for a.e. t ∈ J; (F3) there exist a function φ ∈ L2 (J , R+ ) and a constant L > 0 such that ∥∂ F (t , x)∥ = sup{∥y∥H : y ∈ ∂ F (t , x)} ≤ φ(t ) + L∥x∥H for all x ∈ H and a.e. t ∈ J. By the symbol of SΨ2 we will denote the set of all Bochner L2 -integrable selections of the multimap Ψ : J → P (X ), i.e. SΨ2 = {ψ ∈ L2 (J , H ) : ψ(t ) ∈ Ψ (t ) for a.e. t ∈ J }. Define the superposition multioperator PF : C (J , H ) → P (L2 (J , H )) as

PF (x) = S∂2F (·,x(·)) . We have the following property for the operator PF . Lemma 3.1 ([14]). If conditions (F1)–(F3) are satisfied, then for every x ∈ C (J , H ), the set PF (x) has nonempty, convex and weakly compact values. Moreover, Operator PF is closed in C (J , H ) × L2w (J , H ). Lemma 3.2. If the condition (F1)–(F3) are satisfied, then for a.e. t ∈ J, the multimap ∂ F (t , ·) : H → P (H ) has the Cesari property, i.e.,



co∂ F (t , Oδ (x)) = ∂ F (t , x),

δ>0

for all x ∈ H. Proof. On one hand, it is clear that for any δ > 0,

∂ F (t , x) ⊂ co∂ F (t , Oδ (x)), for all x ∈ H, a.e. t ∈ J. Therefore,

∂ F ( t , x) ⊂



co∂ F (t , Oδ (x)),

δ>0

for all x ∈ H, a.e. t ∈ J. On the other hand, let t ∈ J , x ∈ H be fixed. For any neighborhood V ⊃ ∂ F (t , x) (in the sense of weak∗ topology), from (iv) of Lemma 2.4, there exists a δ > 0 such that ∂ F (t , Oδ (x)) ⊂ V . Since Xw∗ ∗ is locally convex, we can choose that V is convex. Therefore, co∂ F (t , Oδ (x)) ⊂ V  . V for all neighborhood V of ∂ F (t , x). To the contrary, there exists y ∈ V and Now, we show that ∂ F (t , x) = y ̸∈ ∂ F (t , x). Then there exists a closed set Uy ∋ y such that Uy ∩∂ F (t , x) = ∅. By the Separation Property [38, Theorem 1.10], there exist a neighborhood U ′ of Uy and a neighborhood V ′ of ∂ F (t , x) such that U ′ ∩ V ′ = ∅. This shows y ̸∈ V ′ which is a contradiction. Therefore,



co∂ F (t , Oδ (x)) ⊂



V = ∂ F (t , x).

δ>0

The proof is complete.



Now, we give the definition of mild solutions of (2.1). Definition 3.3 (See [10]). A function x ∈ C (J , H ) is said to be a mild solution of problem (2.1) on the interval J if x(t ) = T (t )x0 +

t



T (t − s)(Bu(s) + f (s))ds,

t ∈ J,

0

where f ∈ PF (x). Lemma 3.4 ([23, Lemma 3.5]). If condition (HT ) holds, then operator G : Lp (J , H ) → C (J , H ) for some p > 1, given by

(Gf )(·) =

·



T (· − s)f (s)ds, 0

is compact for f ∈ Lp (J , H ).

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Now we can obtain the following result. Theorem 3.5. If the conditions (HT ) and (F1)–(F3) are satisfied, then for any u ∈ L2 (J , V ), problem (2.1) has at least one mild solution in C (J , H ). Proof. Consider the multimap z : C (J , H ) → Kv (C (J , H )) defined by



zx = y ∈ C (J , H ) : y(t ) = T (t )x0 +

t





T (t − s)(Bu(s) + f (s))ds , 0

where f ∈ PF (x). We divide six steps to verify that z has a fixed point in C (J , H ). Step 1. zx is convex for each x ∈ C (J , H ). Indeed, if y1 , y2 ∈ zx, then there exist f1 , f2 ∈ PF (x) such that for every t ∈ J we have yi (t ) = T (t )x0 +



t

T (t − s)(Bu(s) + fi (s))ds,

i = 1, 2.

0

Let 0 ≤ d ≤ 1. Then for every t ∈ J we have

(dy1 + (1 − d)y2 )(t ) = T (t )x0 +

t



T (t − s)(Bu(s) + (df1 + (1 − d)f2 )(s))ds. 0

By Lemma 3.1, PF (x) is convex. Then df1 + (1 − d)f2 ∈ PF (x). Hence dy1 + (1 − d)y2 ∈ zx. Step 2. z maps bounded sets into bounded sets in C (J , H ). For ∀k0 > 0, let Bk0 = {x ∈ C (J , H ) : ∥x∥C ≤ k0 }. For ∀x ∈ Bk0 and t ∈ J, we have

∥(zx)(t )∥ ≤ ∥T (t )x0 ∥ +

t



∥T (t − s)(Bu(s) + f (s))∥ds 0 t



[φ(s) + L∥x(s)∥X + ∥B∥ ∥u(s)∥V ]ds

≤ M ∥x 0 ∥ + M 0

1

≤ M ∥x0 ∥ + M (∥φ∥L2 + ∥B∥ ∥u∥L2 )T 2 + MLk0 T . Therefore, {zx, x ∈ Bk0 } is bounded in C (J , H ). Step 3. z maps bounded sets into equicontinuous sets of C (J , H ). In the following, we will show that {zx, x ∈ Bk0 } is a family of equicontinuous functions. On one hand, for any x ∈ Bk0 , when t1 = 0, 0 < t2 ≤ δ0 and δ0 is small enough, we obtain

  ∥(zx)(t2 ) − (zx)(t1 )∥ ≤ ∥T (t2 )x0 − x0 ∥ +  

t2 0

 

T (t2 − s)(Bu(s) + f (s))ds  1

≤ ∥T (t2 )x0 − x0 ∥ + M (∥φ∥L2 + ∥B∥ ∥u∥L2 )δ02 + MLk0 δ0 . Then, we can easily see that ∥(zx)(t2 ) − (zx)(t1 )∥ tends to zero independently of x ∈ Bk0 as δ0 → 0. δ On the other hand, for any x ∈ Bk0 and 20 ≤ t1 < t2 ≤ T , we obtain

 t  2 ∥(zx)(t2 ) − (zx)(t1 )∥ ≤ ∥T (t2 )x0 − T (t1 )x0 ∥ +  T (t2 − s)(Bu(s) + f (s))ds  0   t1  − T (t1 − s)(Bu(s) + f (s))ds  0  t    2  ≤ ∥T (t2 )x0 − T (t1 )x0 ∥ +  T ( t − s )( Bu ( s ) + f ( s )) ds 2   t1  t   1  + [T (t2 − s) − T (t1 − s)](Bu(s) + f (s))ds   0

:= Q1 + Q2 + Q3 .

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By (F3) we have 1

Q2 ≤ M (∥φ∥L2 + ∥B∥ ∥u∥L2 )(t2 − t1 ) 2 + MLk0 (t2 − t1 ). δ0

> 0 and δ > 0 being small enough, we obtain 2   t   t1 −δ    1     Q3 ≤  [ T ( t − s ) − T ( t − s )]( Bu ( s ) + f ( s )) ds + [ T ( t − s ) − T ( t − s )]( Bu ( s ) + f ( s )) ds 2 1 2 1     0 t1 −δ   1 sup ∥T (t2 − s) − T (t1 − s)∥ ≤ (∥φ∥L2 + ∥B∥ ∥u∥L2 )(t1 − δ) 2 + Lk0 (t1 − δ)

For t1 ≥

s∈[0,t1 −δ]

1 2

+ 2M (∥φ∥L2 + ∥B∥ ∥u∥L2 )δ + 2MLk0 δ. Since (HT ) implies the continuous of T (t ) (t > 0) in t in the uniform operator topology, it is easily seen that Q3 tends to zero independently of x ∈ Bk0 as t2 → t1 , δ → 0. It is clear that Qi (i = 1, 2) tends to zero as t2 → t1 does not depend on particular choice of x. Thus, we get that ∥(zx)(t2 ) − (zx)(t1 )∥ tends to zero independently of x ∈ Bk0 as δ0 → 0, which means that {zx, x ∈ Bk0 } is equicontinuous. Step 4. z is a compact multivalued map. We prove that for any t ∈ J , Λ(t ) = {(zx)(t ), x ∈ Bk0 } is relatively compact in H. Clearly, Λ(0) = {(zx)(0), x ∈ Bk0 } = {0} is compact. Let 0 < t ≤ T be fixed. For ∀ε ∈ (0, t ), define an operator zε on Bk0 as follows:

(zε x)(t ) = T (t )x0 +

t −ε



T (t − s)(Bu(s) + f (s))ds 0

= T ( t ) x0 +

t −ε



T (ε)T (t − s − ε)(Bu(s) + f (s))ds 0

= T (ε)T (t − ε)x0 + T (ε)



t −ε

T (t − s − ε)(Bu(s) + f (s))ds

0

:= T (ε)z (t , ε), where x ∈ Bk0 , f ∈ PF (x). Then from the compactness of T (ε) (ε > 0), we obtain that the set Λε (t ) = {(zε x)(t ), x ∈ Bk0 } is relatively compact in H for ∀ε ∈ (0, t ). Moreover, for every x ∈ Bk0 , we have

∥(zx)(t ) − (zε x)(t )∥ =



t

T (t − s)(Bu(s) + f (s))ds t −ε 1

≤ M (∥φ∥L2 + ∥B∥ ∥u∥L2 )ε 2 + MLk0 ε. Therefore, there are relatively compact sets arbitrarily close to the set Λ(t ) (t > 0). Hence the set Λ(t ) (t > 0) is also relatively compact in H. Therefore, from Steps 2–3, {zx, x ∈ Bk0 } is relatively compact by the generalized Ascoli–Arzela theorem. Thus, z is a compact multivalued map. Step 5. z has a closed graph. Let xn ∈ C (J , H ), yn ∈ zxn such that xn → x and yn → y. We will prove that y ∈ zx. Now yn ∈ zxn implies that there exists fn ∈ PF (xn ) such that for each t ∈ J, yn = T (t )x0 +

t



T (t − s)(Bu(s) + fn (s))ds. 0

From (F3), we may assume that fn ⇀ f for some f ∈ L2 (J , H ). Define the continuous linear operator G : L2 (J , H ) → C (J , H ) as

(Gf )(·) =

·



T (· − s)f (s)ds, 0

for f ∈ L2 (J , H ). Since xn → x, it follows from Lemmas 3.1 and 3.4 that y = T (t )x0 +



t

T (t − s)(Bu(s) + f (s))ds

0

and f ∈ PF (x), i.e. z has a closed graph. Therefore, since z takes compact values, from Lemma 2.1 we deduce that z is u.s.c. Step 6. According to Theorem 2.5, it is sufficient to show that there exists an open set D ⊂ C (J , H ) such that there is no x ∈ ∂ D satisfying x ∈ λzx for ∀λ ∈ (0, 1). In fact, let x ∈ λzx for some λ ∈ (0, 1). Then, there exists f ∈ PF (x) such that for each t ∈ J, x(t ) = λT (t )x0 + λ

t



T (t − s)(Bu(s) + f (s))ds. 0

8

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We have



t

∥T (t − s)(Bu(s) + f (s))∥ds  t 1 ≤ M (∥φ∥L2 + ∥B∥ ∥u∥L2 )T 2 + ML ∥x(s)∥X ds.

∥x(t )∥X ≤ ∥T (t )x0 ∥ +

0

0

Let W (t ) = ∥x(t )∥X . Then we have W (t ) ≤ ρ + ML

t



W (s)ds, 0

where 1

ρ = M (∥φ∥L2 + ∥B∥ ∥u∥L2 )T 2 . It follows from the standard Gronwall inequality [39] that W (t ) ≤ ρ eMLT , which implies

∥x∥C ≤ ρ eMLT := l. Set D = {x ∈ C (J , H ) : ∥x∥C < l + 1}. Clearly, D is an open subset of C (J , H ), z : D → Kv (C (J , H )) is u.s.c and compact. From the choice of D, there is no x ∈ ∂ D satisfying x ∈ λzx for some λ ∈ (0, 1). Therefore, by Theorem 2.5 we deduce that z has a fixed point x∗ in C (J , H ). Consequently, the problem (2.1) has at least one mild solution on C (J , H ).  Now, we give the following definition. Definition 3.6. A pair (x, u) is said to be feasible if (x, u) satisfies (2.2) for t ∈ J. To the readers’ convenience, we denote V [0, T ] = {u : [0, T ] → V |u(·) is measurable}, H [0, T ] = {(x, u) ∈ C (J , H ) × V [0, T ]| (x, u) is feasible}. Now, we study the existence result of feasible pairs for problem (2.2). We assume that the feedback multimap U : J ×H → P (V ) satisfies the following conditions: (U1) there exist a function φ1 ∈ L2 (J , R+ ) and a constant L1 > 0, such that

∥U (t , x)∥ = sup ∥z ∥V ≤ φ1 (t ) + L1 ∥x∥H z ∈U (t ,x)

for all (t , x) ∈ J × H; (U2) for a.e. t ∈ J , x ∈ H, the set U (t , x) satisfies the following



coU (Oδ (t , x)) = U (t , x).

δ>0

Remark 3.7. By Lemma 2.1, condition (U2) is fulfilled if U is u.s.c. with convex and closed valued. Now, we are in the position to present the main result of this section. Theorem 3.8. If conditions (HT ), (F1)–(F3) and (U1), (U2) are satisfied, then the set H [0, T ] is nonempty. Proof. For any k > 0, let tj = k T , 0 ≤ j ≤ k − 1. We set j

uk ( t ) =

k−1 

uj χ[tj ,tj+1 ) (t ),

t ∈ J,

j =0

where χ[tj ,tj+1 ) is the character function of interval [tj , tj+1 ). The sequence {uj } is constructed as follows.

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Firstly, we take u0 ∈ U (0, x0 ). By Theorem 3.5, there exists xk (·) given by xk (t ) = T (t )x0 +

t





T (t − s)(Bu0 (s) + fk (s))ds,

t ∈ 0,

0

T



k

,

where fk ∈ PF (xk ). Then take u1 ∈ U ( Tk , xk ( Tk )). We can repeat this procedure to obtain xk on [ Tk , end up with the following:

2T k

], etc. By induction, we

 t    T (t − s)(Buk (s) + fk (s))ds, t ∈ J , xk (t ) = T (t )x0 + 0      jT jT jT (j + 1)T   uk ( t ) ∈ U , xk , t∈ , , 0 ≤ j ≤ k − 1, k

k

k

k

where fk ∈ PF (xk ). From the proof of Theorem 3.5, it is easy to prove that there exists r0 > 0 such that

∥x k ∥C ≤ r 0 . Moreover, it comes from (F3) and (U1) that there exist r1 , r2 > 0 such that

∥uk (·)∥L2 ≤ r1 ,

∥fk (·)∥L2 ≤ r2 .

Therefore, there are subsequences of {uk (·)} and {fk (·)}, denoted by {uk (·)} and {fk (·)} again, such that uk (·) ⇀ u(·) in L2 (J , V ),

fk (·) ⇀ f (·) in L2 (J , H ).

(3.1)

From (HT ), by Lemma 3.4 we have that for any t ∈ J, t



T (t − s)(Buk (s) + fk (s))ds → 0

t



T (t − s)(Bu(s) + f (s))ds. 0

Let x(t ) = T (t )x0 +

t



T (t − s)(Bu(s) + f (s))ds,

t ∈ J.

0

Then, xk (t ) → x(t ), uniformly in t ∈ J, i.e. xk (·) → x(·) in C (J , H ). Hence, for any δ > 0, there exists a k0 > 0 such that xk (t ) ∈ Oδ (x(t )),

t ∈ J , k ≥ k0 .

(3.2)

On the other hand, by the definition of uk (·) for k large enough, we have uk (t ) ∈ U (tj , xk (tj )) ⊂ U (Oδ (t , x(t ))),

 ∀t ∈

jT (j + 1)T k

,

k



, 0 ≤ j ≤ k − 1.

Secondly, by (3.1) and Mazur Theorem [23, Chapter 2, Corollary 2.8], let ail , bil ≥ 0 and

φl (·) =



ail ui+l (·) → u(·)

in L2 (J , V ),

ψl (·) =



(3.3)



Then, there are subsequences of {φl } and {ψl }, denoted by {φl } and {ψl } again, such that

φl (t ) → u(t ) in V ,

ψl (t ) → f (t ) in H , a.e. t ∈ J .

Hence, from (3.2) and (3.3), for l large enough,

φl (t ) ∈ coU (Oδ (t , x(t ))),

ψl (t ) ∈ co∂ F (t , Oδ (x(t ))),

a.e. t ∈ J .

Thus, for any δ > 0, u(t ) ∈ coU (Oδ (t , x(t ))),

f (t ) ∈ co∂ F (t , Oδ (x(t ))),

a.e. t ∈ J .

From (U2) and Lemma 3.2, we have u(t ) ∈ U (t , x(t )),

f (t ) ∈ ∂ F (t , x(t )),

a.e. t ∈ J .

Therefore, (x, u) is a feasible pair in J. The proof is complete.



ail =

bil fi+l (·) → f (·) in L2 (J , H ).

i≥1

i≥1

i≥1



i≥1

bil = 1 such that

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4. Existence of optimal state–control pairs In this section, we consider the optimal control problem stated as follows. Problem (ϕ): find a pair (x0 , u0 ) ∈ H [0, T ] such that

ϕ(x0 , u0 ) ≤ ϕ(x, u), for all (x, u) ∈ H [0, T ], T where ϕ(x, u) = 0 f0 (t , x(t ), u(t ))dt. We make the following assumptions on f0 :

(f0 1) the functional f0 : J × H × V → R ∪ {±∞} is Borel measurable in (t , x, u); (f0 2) f0 (t , ·, ·) is lower semicontinuous on H × V for a.e. t ∈ J (i.e., for all x ∈ H , u ∈ V , {xn } ⊂ H , {un } ⊂ V such that xn → x in H and un → u in V , we have lim inf f0 (t , xn , un ) ≥ f0 (t , x, u)) and there exists a constant M1 > 0 such that f0 (t , x, u) ≥ −M1 ,

( t , x , u) ∈ J × H × V .

For any (t , x) ∈ J × H, we set the set

ε(t , x) = {(z 0 , z 1 , z 2 ) ∈ R × H × V |z 0 ≥ f0 (t , x, z 2 ), z 1 ∈ ∂ F (t , x), z 2 ∈ U (t , x)}. In order to obtain the existence result of optimal state–control pairs for Problem (ϕ), we assume that:

(Hε ) : for a.e. t ∈ J, the map ε(t , ·) : X → P (R × H × V ) has the Cesari property, i.e.,  coε(t , Oδ (x)) = ε(t , x), ∀x ∈ H . δ>0

Theorem 4.1. If conditions (HT ), (F1)–(F3), (U1), (U2), (f0 1), (f0 2), (Hε ) are satisfied, then Problem (ϕ) admits at least one optimal state–control pair. Proof. Without considering the situation inf{ϕ(x, u)|(x, u) ∈ H [0, T ]} = +∞, we assume that inf{ϕ(x, u)|(x, u) ∈ H [0, T ]} = m < +∞. By (f0 2), we have ϕ(x, u) ≥ m ≥ −M1 T > −∞. Then there exists a sequence {(xn , un )}n≥1 ⊂ H [0, T ] such that

ϕ(xn , un ) → m. From the proof of Theorem 3.8, without loss of generality, we obtain that xn (·) → x(·)

in C (J , H ),

and un (·) ⇀ u(·) in L2 (J , V ),

f n (·) ⇀ f (·) in L2 (J , H ),

where x(t ) = T (t )x0 +

t



T (t − s)(Bu(s) + f (s))ds,

t ∈ J.

0

By Mazur Theorem again, let ail , bil ≥ 0 and

φl (·) =





ail ui+l (·) → u(·) in L2 (J , V ),

i≥1

ail =

ψl (·) =

i ≥1



i≥1

 i ≥1

Let

ψl (·) =



bkl f0 (·, xk+l (·), uk+l (·)),

k≥1

and f0 (t ) = liml→+∞ ψl (t ) ≥ −M1 ,

a.e. t ∈ J .

For any δ > 0 and l large enough, from (f0 2) we have

(ψl (t ), ψl (t ), φl (t )) ∈ ε(t , Oδ (x(t ))),

a.e. t ∈ J .

(f0 (t ), f (t ), u(t )) ∈ coε(t , Oδ (x(t ))),

a.e. t ∈ J .

Then

From (Hε ), we have

(f0 (t ), f (t ), u(t )) ∈ ε(t , x(t )),

a.e. t ∈ J ,

bil = 1 such that bil f i+l (·) → f (·)

in L2 (J , H ).

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i.e.,

 f0 (t ) ≥ f0 (t , x(t ), u(t )), t ∈ J , f (t ) ∈ ∂ F (t , x(t )), t ∈ J ,  u(t ) ∈ U (t , x(t )). Therefore,

(x, u) ∈ H [0, T ]. By Fatou’s Lemma, we obtain T



f0 (t )dt =

T



0

liml→+∞ ψl (t )dt ≤ liml→+∞ 0

T



ψl (t )dt 0

T

 = liml→+∞ 0

= liml→+∞



 k≥1

= liml→+∞

T



f0 (t , xk+l (t ), uk+l (t ))dt

qkl 0

k≥1



qkl f0 (t , xk+l (t ), uk+l (t ))dt

T



f0 (t , xk+l (t ), uk+l (t ))dt

qkl liml→+∞ 0

k≥1

= m. Therefore, m ≤ ϕ(x, u) =

T



f0 (t , x(t ), u(t ))dt ≤ m, 0

i.e., T



f0 (t , x(t ), u(t ))dt = m = 0

inf

(x,u)∈H [0,T ]

ϕ(x, u).

Thus, (x, u) is an optimal state–control pair. The proof is complete.



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