Quadratic Cost Minimax Optimal Control Problems for a Semilinear Viscoelastic Equation with Long Memory

In this paper, we study the quadratic cost minimax optimal control problems for a semilinear viscoelastic equation with long memory. A global well-posedness theorem regarding the solutions to its Cauchy problem is given. We formulate the minimax control problemwith bilinear control inputs and corresponding disturbances. Under some assumptions, we prove the existence of optimal pairs and give necessary optimality conditions for optimal pairs in some observation cases.


Introduction
We study the minimax optimal control problems for the following viscoelastic equation with long memory: k(t − s)Δy(s)ds +|y| c y � uy + f. (1) As is widely known, equation (1) is considered a typical model of viscoelastic waves with a long-memory damping. By analyzing the above equation with various boundary conditions or additional linear or nonlinear terms, one can study the eventual properties of the viscoelastic dynamics, such as exponential decay or blowup. We can find many research articles in this direction, to name just a few, refer to Cavalcanti et al. [1,2], Xiao and Liang [3], and Messaoudi [4] and references therein. However, the research on other applications including optimal control or identification problems for the above state equation is few.
In this paper, we study the quadratic cost bilinear minimax control problems for equation (1) with the control or disturbance function u. For the bilinear control problems, we refer to Bradley and Lenhart [5][6][7] in which they studied the bilinear control problems on linear hyperbolic state equations. Recently, Belmiloudi [8] employed the bilinear minimax control framework to study robust control in a linear parabolic state equation. In [9], we extended the results in [8] to a quasilinear beam equation by proving the Fréchet differentiability of the nonlinear solution map. e minimax control methods have been employed by many mathematicians for various control problems (see Arada and Raymond [10], Lasiecka and Triggiani [11], and Li and Yong [12]). As explained in [8,9], in this paper, the minimax control framework is employed to consider the effects of disturbance (or noise) in control inputs such that a cost function achieves its optimum (minimum) even in the case of the worst disturbances of the system. For the purpose, we replace the bilinear multiplier u in equation (1) by c + η, where c is a control variable that belongs to the admissible control set U ad and η is a disturbance (or noise) that belongs to the admissible disturbance set V ad . We also introduce the following cost function to be minimized within U ad and maximized within V ad : where y(c + η) is the solution of equation (1) with u � c + η, M is a Hilbert space of observation variables, C is a continuous observation operator, Y d ∈ M is a desired value, and the positive constants α and β are the relative weights of the second and third terms on the right-hand side of (2). In this paper, our goal is to find and characterize the optimal controls of the cost function (2) even in the worst disturbances through the control input in equation (1). is leads to the problem of proving the existence of an admissible control c * ∈ U ad and disturbance (or noise) η * ∈ V ad such that (c * , η * ) is a saddle point of the functional J(c, η) of (2), that is, en, we derive optimality conditions for such (c * , η * ) in (3) corresponding to some observation cases. As in [9], we use the term optimal pair to indicate such a saddle point (c * , η * ) in (3).
As a main tool to prove the existence of an optimal pair (c * , η * ) satisfying (3), we use the minimax theorem in infinite dimensions given in Barbu and Precupanu [13]. For this, we prove the Fréchet differentiability of the nonlinear solution map and its local Lipschitz continuity.
Next, we derive the necessary optimal conditions for some observation cases that should be satisfied by the optimal pairs in these observation cases. To derive these conditions, we refer to the previous results in [5][6][7] dealing with linear problems. Especially, we deduce the necessary optimality condition for the velocity distribution observation case which is physically meaningful. We propose an appropriate adjoint system for the velocity observation case. To author's knowledge, this is a newly developed adjoint system to deal with the second-order Volterra-type state equation. Finally, when deducing the optimality condition for the velocity observation case, we use the regularization method proposed by Lions [14] (cf. [15]) to overcome some difficulties. is is another novelty of the paper.

Notations and Assumptions.
Let Ω be an open, connected, and bounded set of R n (n ≤ 3) with the smooth boundary Γ. We set Q � (0, T) × Ω and Σ � (0, T) × Γ for T > 0. We consider the following viscoelastic equation with long memory with the Dirichlet boundary condition: where y 0 (x) and y 1 (x) are given initial values,f is a given forcing function, k is a memory kernel function, and u is a bilinear forcing control function applied to the system together with displacement y. We denote ‖ · ‖ p � ‖ · ‖ L p (Ω) , H � L 2 (Ω), and V � H 1 0 (Ω), where p ≥ 1. We endow the space V with the usual scalar product and norm: where (·, ·) 2 means the inner product on H. Let us denote the topological dual space of V by V ′ (� H − 1 (Ω)) and the dual pairing between V and V ′ by 〈·, ·〉 V,V′ . en, operator A defined by is a nonnegative self-adjoint operator with domain is continuous and compact. According to Adams [16], we know the following embeddings: roughout this paper, the following assumptions will be effective: For the above c, we define the nonlinear function g: R ⟶ R by g(s) � |s| c s. en, it is easily verified that g ∈ C 1 (R) and g ′ (s) � (c + 1)|s| c . For given ϕ ∈ V, we can deduce from (8) and (11) that the nonlinear operator g(·): For the memory kernel function k(·) in equation (4), we assume that Now, equation (4) can be rewritten as the abstract initial value problem described by the following second-order semilinear Volterra integrodifferential system: equipped with the norm where ϕ ′ and ϕ ″ denote the first-and second-order time derivatives of ϕ in the sense of distribution.

Definition 1.
A function y is said to be a weak solution of equation (13) if y ∈ W(0, T) and y satisfies where D ′ (0, T) is the space of distributions in (0, T).

Remark 1.
For the existence and uniqueness of weak or strong solutions of equation (13) without the integral term, one can refer to the results of an abstract semilinear wave equation given in Teman ( [17], pp. 212-214). In addition, by referring to the results in [18] dealing with the Volterra-type semilinear evolution equations, one can verify the wellposedness of equation (13). Let X be a Banach space. Set Lemma 1. Assume that and y is a corresponding weak solution of equation (13). en, we can verify that Proof. By regarding − g(y) + uy + f in equation (13) as f(t, y, y ′ ) in [18] and noting (18), we can use the result of [18] to obtain en, as shown in [18] (cf. Lions and Magenes [19], pp. 275-278), we can obtain (19) from (20). is completes the proof. e following lemma is used frequently and importantly in this paper. □ Lemma 2. Let y be a weak solution of equation (13) with condition (18). en, for each t ∈ [0, T], we have the following energy equality: Proof. By (19) and the uniform boundedness theorem, y(t) ∈ V and y ′ (t) ∈ H for each t ∈ [0, T]. us, we know that all terms in (21) are meaningful. From (18) and (20), we know that − g(y) + uy + f ∈ L 2 (0, T; H). us, by applying the result in Proposition 2.1 of [18] to equation (13), we obtain we can combine (22) and (23) to obtain (21). is completes the proof. With the help of (21) or (22), we can address the following theorem.
roughout this paper, we use C as a generic constant and omit the integral variables in any definite integrals without confusion. □ Theorem 1. For given condition (18), let y be the corresponding weak solution of equation (13). en, we can assure ∈ P, the following is satisfied: where C > 0 is a constant depending on the data.
en, from equation (13), we can know that ϕ satisfies the following in the weak sense: By using y i ∈ S(0, T)(i � 1, 2), (8), (11), and the Hölder inequality, we note that To obtain the estimation for ϕ in equation (13), we apply energy equality (22) to equation (26). en, the energy equality for ϕ can be given by By estimating other terms in the right-hand side of (29) as in [18], we can get Here, we note that us, plugging (32) to (31) and applying Gronwall's lemma, we have the following estimation: which immediately implies Since A ∈ L(V, V ′ ) is an isomorphism, by conducting similar estimations in equation (26), we can obtain from (34) that Hence, by (34) and (35), we can prove (24). is completes the proof.
we can define the continuous solution map from the term u ∈ L 2 (0, T) to y(p) ∈ S(0, T) satisfying equation (13). Indeed, for each p i � (y 0 , y 1 , f, u i ) ∈ P(i � 1, 2), we have us, from now on, to emphasize the fact that the only varying variable of the weak solution of equation (37) is the bilinear multiplier u, we use the notation y(u) where y(u) satisfies Furthermore, we present the following weakly continuous results to study the existence of the optimal pair in the subsequent section. For this, we need the following lemma.
Lemma 3 (see Simon [21]). Let X, Y, and Z be Banach spaces such that each embedding X ⊂ Y ⊂ Z is continuous and the embedding X ⊂ Y is compact. en, a bounded set of Proof. Let u ∈ L 2 (0, T), and let u n ⊂ L 2 (0, T) be a bounded sequence such that u n ⟶ u weakly in L 2 (0, T), as n ⟶ ∞.
From now on, each state y n � y(u n ) is a solution of Journal of Mathematics y n ″ + Ay n + t 0 k(t − s)Ay n ds + g y n � u n y n + f, in(0, T), From eorem 1, we have which implies that y n remains in a bounded set of . erefore, we can find a subsequence of y n , say again y n , and find y ∈ W(0, T) Since the embedding V↪H is compact, we can apply Lemma 3 to (42) and (43) us, we can find a subsequence of y n , if necessary, still denoted by y n , such that For any given ϕ ∈ L 2 (0, T; V), using notation (27), we can deduce g y n − g(y), ϕ 2 � (c + 1) B y n , y y n − y , ϕ 2 ≤ (c + 1) y n +|y| c y n − y , |ϕ| 2 erefore, from (45) and (46), we can readily find a subsequence of y n , if necessary, still denoted by itself, such that g y n ⟶ g(y) strongly in L 2 0, T; V ′ , as n ⟶ ∞.

(47)
From (38) and (45), we may extract a subsequence of u n , denoted again by u n , such that u n y n ⟶ uy weakly in L 2 (0, T; H).
Replacing y n by y n k in equation (39), if necessary, and letting k ⟶ ∞, we can conclude by the standard arguments as in Dautray and Lions ([20], pp.561-565) that the limit y is a weak solution of Moreover, by the uniqueness of the weak solutions, we can conclude that y � y(u) in W(0, T), which implies that y(u n ) ⟶ y(u) weakly in W(0, T).
is completes the proof.

Quadratic Cost Minimax Control Problems
In this section, we study the quadratic cost minimax optimal control problems for equation (52). Let the following be the set of the admissible controls: where c l and c u are lower and upper bounds of the control variables, respectively. Let the following be the set of the admissible disturbance or noises: where η l and η u are lower and upper bounds of the disturbance (noise) variables, respectively. For simplicity, let F ad be a product space defined by F ad � U ad × V ad . Using eorem 1, for fixed (y 0 , y 1 , f) ∈ V × H × L 2 (0, T; H), we can uniquely define the solution map F ad ⟶ S(0, T), which maps from (c, η) ∈ F ad via q � c + η to the weak solution y(q) ∈ S(0, T), where y(q) satisfies e weak solution y(q) is called to be the state of the control system. e quadratic cost function associated with control system (52) is given by where M is a Hilbert space of observation variables, C ∈ L(W(0, T), M) is a continuous linear operator, that is to say, observer, Y d ∈ M is a desired value, and the positive constants α and β are the relative weights of the second and third terms on the right-hand side of (53). As indicated in the introduction, the minimax optimal control problem can be summarized as follows: (i) Find an admissible control c * ∈ U ad and a disturbance (or noise) η * ∈ V ad such that (c * , η * ) is a saddle point of the functional J(c, η) of (53), that is, In this paper, we call such a pair (c * , η * ) in (54) as the optimal pair for the minimax optimal control problem with cost (53). To show the existence of the above saddle point (c * , η * ) (optimal pair), we need to show the solution map is differentiable in some sense. en, with the assumptions on the weight constants α, β and the exponent c, we utilize the arguments in Barbu and Precupanu [13] to show the existence of the optimal pairs.
(ii) Characterize (c * , η * ) (optimality condition): in characterizing these optimal pairs, we introduce an appropriate adjoint system corresponding to the observed case and deduce the necessary optimal conditions through the variational inequality.

Differentiability of the Nonlinear Solution Map.
In this section, we address the Fréchet differentiability of the nonlinear solution map, which is desirable for many applications. For our study, we define the Fréchet differentiability of the nonlinear solution map as follows.

Definition 2.
e solution map u ⟶ y(u) of L 2 (0, T) into S(0, T) is said to be Fréchet differentiable on L 2 (0, T) if for any u ∈ L 2 (0, T), there exists T(u) ∈ L(L 2 (0, T), S(0, T)) such that, for any w ∈ L 2 (0, T), e operator T(u) is called the Fréchet derivative of y at u, which we denote by Dy(u). T(u)w � Dy(u)w ∈ S(0, T) is called the Fréchet derivative of y at u in the direction of w ∈ L 2 (0, T).

Theorem 2.
e solution map u ⟶ y(u) is Fréchet differentiable on L 2 (0, T), and the Fréchet derivative of y(u) at u in the direction w ∈ L 2 (0, T), that is to say z � Dy(u)w, is given by the solution of en, by (11) and eorem 1, we can verify Hence, from (58), we see that Since w ∈ L 2 (0, T) and y(u) ∈ S(0, T), we know that wy(u) ∈ L 2 (0, T; H). By (59), we can use the results in [18] to verify that equation (56) admits a unique weak solution z ∈ S(0, T). And by eorem 1, we can know that the weak solution z(� z(w)) of equation (56) Hence, from (60), the mapping w ∈ L 2 (0, T)↦z (w) ∈ S(0, T) is linear and bounded.
Proof. Let z i � Dy(u i )w(i � 1, 2) be the weak solutions of equation (56) corresponding to u i (i � 1, 2). If we set ϕ � z 1 − z 2 , then we can know that ϕ satisfies in the weak sense, where dθ y u 1 − y u 2 z 2 , As before, arguing as in the proof of eorem 1, we can deduce that In estimating I 1 , the case of c � 1 will be clarified as we will see later. us, we restrict to the case of c > 1 to estimate I 1 . By (11), eorem 1, and (60), we can obtain Journal of Mathematics From (75) to (77), we can get the following: is completes the proof. □ 3.2. Existence of the Optimal Pairs. By the minimax theorem in infinite dimensions in Barbu and Precupanu [13], we study the existence of the optimal pairs in the underlying control system with quadratic cost function (53).
In order to prove this theorem, we mainly illustrate the following as in [8] (cf. [9]): (i) For sufficiently large α and β in (53), the maps c ⟶ J(c, η) and η ⟶ J(c, η) are convex and concave for all η ∈ V ad and for all c ∈ U ad , respectively (ii) e maps c ⟶ J(c, η) and η ⟶ J(c, η) are lower and upper semicontinuous for all η ∈ V ad and for all c ∈ U ad , respectively for any fixed η ∈ V ad . (79) means where D c y(c i + η)(c 1 − c 2 )(i � 1, 2) are weak solutions of equation (71), in which q * z + (h + l)y(p) is � 1, 2), respectively. We can deduce that (80) is equivalent to By eorem 1, (60), and Proposition 2, we can have the following estimations: is satisfied for any fixed c ∈ U ad and β > β l (P, F ad , Y d , C, M).
is also indicates the concavity of the map η ⟶ J(c, η).
(ii) Since c n is bounded in L 2 (0, T), we can extract a subsequence c n k ⊂ c n such that en, since C is a continuous linear operator on W(0, T), by Proposition 1, we obtain ∀η ∈ V ad . Since the norm is weakly lower semicontinuous, we can see from (85) and (86) that the map c ⟶ J(c, η) is lower semicontinuous for all η ∈ V ad . By similar arguments, we can prove that the map η ⟶ J(c, η) is upper semicontinuous for all c ∈ U ad .
Next, we prove the existence of an optimal pair (c * , η * ) ∈ F ad : let c n ⊂ U ad be a minimizing sequence of J(c, η) where η ∈ V ad is fixed. us, en, by (i) and (ii), we know that Similarly, we also know that there exists η * ∈ V ad such that J 0 η * � sup η∈V ad J 0 (η). (90) From (89) and (90), we can conclude that (c * , η * ) ∈ F ad is an optimal pair for cost (53).
is completes the proof.

Necessary Conditions of Optimal Pairs.
In this section, we study the necessary optimality conditions that have to be satisfied by each optimal pair of the minimax optimal control problem with cost (53) in which the following two types of observations are considered: and observe C 1 y(q) � (y(q), y(q; T)) ∈ L 2 (0, T; H) × H (2) We take M 2 � L 2 (0, T; H) and C 2 ∈ L(W(0, T), M 2 ) and observe C 2 y(q) � y ′ (q) ∈ L 2 (0, T; H) Since y(q) ∈ W(0, T) ⊂ C([0, T]; H), the above observations are meaningful.

Case of Distributive and Terminal Values' Observations
We consider the cost functional expressed by where y d ∈ L 2 (0, T; H) and y T d ∈ H are desired values. Let (c * , η * ) ∈ [L 2 (0, T)] 2 (q * � c * + η * ) be an optimal pair subject to equation (52) and quadratic cost (91). Now, we are about to formulate the following adjoint equation corresponding to cost (91) and equation (52) in which (c, η) with q � c + η is replaced by (c * , η * ) with q * � c * + η * : Proof. From the observation conditions y(q * ) − y d ∈ L 2 (0, T; H) and y(q * ; T) − y T d ∈ H and (59), we can refer to the results in [18] (cf. Dautray and Lions [20], pp.655-659) to ensure that equation (92), after reversing the direction of time t ⟶ T − t, admits a unique weak solution p ∈ S(0, T). is completes the proof.
We now discuss the first-order optimality conditions for minimax optimal control problem (54) for quadratic cost function (91).

Journal of Mathematics
Theorem 4. If α and β in cost (91) are large enough and the exponent c ≥ 1, then an optimal control c * ∈ U ad and a disturbance η * ∈ V ad , namely, an optimal pair (c * , η * ) ∈ F ad satisfying (54), can be given by where p is the weak solution of equation (92).
Proof. By the assumptions of eorem 4, we can verify through eorem 3 that there exists an optimal pair in (54) with cost (91). Let (c * , η * ) ∈ F ad be an optimal pair in (54) with cost (91) and y(q * ) be the corresponding weak solution of equation (52).

Case of Velocity's Distributive Value Observation
C 2 ∈ L(W(0, T), M 2 ). In this case, we consider the following cost functional: where y d ∈ L 2 (0, T; H) is the desired value. Let (c * , η * ) ∈ F ad be an optimal pair subject to equation (52) and quadratic cost (103). In this case, we introduce the following adjoint equation corresponding to cost (103) and equation (52) in which q � c + η is replaced by q * � c * + η * : p ″ + Ap +