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 problem with 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.
We study the minimax optimal control problems for the following viscoelastic equation with long memory:
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 , and Messaoudi  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 . For the bilinear control problems, we refer to Bradley and Lenhart [5–7] in which they studied the bilinear control problems on linear hyperbolic state equations. Recently, Belmiloudi  employed the bilinear minimax control framework to study robust control in a linear parabolic state equation. In , we extended the results in  to a quasilinear beam equation by proving the Fréchet differentiability of the nonlinear solution map.
The minimax control methods have been employed by many mathematicians for various control problems (see Arada and Raymond , Lasiecka and Triggiani , and Li and Yong ). 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 in equation (1) by , where is a control variable that belongs to the admissible control set and is a disturbance (or noise) that belongs to the admissible disturbance set . We also introduce the following cost function to be minimized within and maximized within :where is the solution of equation (1) with , is a Hilbert space of observation variables, is a continuous observation operator, is a desired value, and the positive constants and 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). This leads to the problem of proving the existence of an admissible control and disturbance (or noise) such that is a saddle point of the functional of (2), that is,
As a main tool to prove the existence of an optimal pair satisfying (3), we use the minimax theorem in infinite dimensions given in Barbu and Precupanu . 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 [5–7] 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  (cf. ) to overcome some difficulties. This is another novelty of the paper.
2.1. Notations and Assumptions
Let be an open, connected, and bounded set of with the smooth boundary . We set and for . We consider the following viscoelastic equation with long memory with the Dirichlet boundary condition:where and are given initial values, is a given forcing function, is a memory kernel function, and is a bilinear forcing control function applied to the system together with displacement .
We denote , where We endow the space with the usual scalar product and norm:where means the inner product on . Let us denote the topological dual space of by and the dual pairing between and by . Then, operator defined byis a nonnegative self-adjoint operator with domain . Also, it is obvious that each natural topological imbeddingis continuous and compact.
According to Adams , we know the following embeddings:
Throughout this paper, the following assumptions will be effective: For the exponent in equation (4), we assume that For the above , we define the nonlinear function by . Then, it is easily verified that and . For given , we can deduce from (8) and (11) that the nonlinear operator , defined by , is well defined. For the memory kernel function 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:
2.2. Well-Posedness and Continuity of the Solution Map
The Hilbert space of the weak solutions of equation (13) is defined byequipped with the normwhere and denote the first- and second-order time derivatives of in the sense of distribution.
Definition 1. A function is said to be a weak solution of equation (13) if and satisfieswhere is the space of distributions in .
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 (, pp. 212–214). In addition, by referring to the results in  dealing with the Volterra-type semilinear evolution equations, one can verify the well-posedness of equation (13).
Let be a Banach space. Set
Lemma 1. Assume thatand is a corresponding weak solution of equation (13). Then, we can verify that
Proof. By regarding in equation (13) as in  and noting (18), we can use the result of  to obtainThen, as shown in  (cf. Lions and Magenes , pp. 275–278), we can obtain (19) from (20). This completes the proof.
The following lemma is used frequently and importantly in this paper.
Proof. By (19) and the uniform boundedness theorem, and for each . Thus, we know that all terms in (21) are meaningful. From (18) and (20), we know that . Thus, by applying the result in Proposition 2.1 of  to equation (13), we obtainSincewe can combine (22) and (23) to obtain (21).
This completes the proof.
With the help of (21) or (22), we can address the following theorem.
Throughout this paper, we use as a generic constant and omit the integral variables in any definite integrals without confusion.
Theorem 1. For given condition (18), let be the corresponding weak solution of equation (13). Then, we can assure that . Moreover, the solution mapping of into is locally Lipschitz continuous. Indeed, for each and , the following is satisfied:where is a constant depending on the data.
Proof. Referring to Dautray and Lions (, pp. 578–581), we can make use of (22) to show that is continuous, which implies the following:Thus, every weak solution of equation (13) with data condition (18) exists in . From this result, we show inequality (24): we denote by . Then, from equation (13), we can know that satisfies the following in the weak sense:whereBy using , (8), (11), and the Hölder inequality, we note thatTo obtain the estimation for in equation (13), we apply energy equality (22) to equation (26). Then, the energy equality for can be given byBy (28), we can verify thatBy estimating other terms in the right-hand side of (29) as in , we can getHere, we note thatThus, plugging (32) to (31) and applying Gronwall’s lemma, we have the following estimation:which immediately impliesSince is an isomorphism, by conducting similar estimations in equation (26), we can obtain from (34) thatHence, by (34) and (35), we can prove (24).
This completes the proof.
Remark 2. From Theorem 1, for fixed , we can define the continuous solution map from the term to satisfying equation (13). Indeed, for each , we haveThus, from now on, to emphasize the fact that the only varying variable of the weak solution of equation (37) is the bilinear multiplier , we use the notation where satisfiesFurthermore, 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 ). Let , and be Banach spaces such that each embedding is continuous and the embedding is compact. Then, a bounded set of is relatively compact in .
Proposition 1. For fixed , the solution map from to of equation (37) is weakly continuous.
Proof. Let , and let be a bounded sequence such thatFrom now on, each state is a solution ofFrom Theorem 1, we havewhich implies that remains in a bounded set of . Therefore, we can find a subsequence of , say again , and find such thatSince the embedding is compact, we can apply Lemma 3 to (42) and (43) with to verify thatThus, we can find a subsequence of , if necessary, still denoted by , such thatFor any given , using notation (27), we can deduceTherefore, from (45) and (46), we can readily find a subsequence of , if necessary, still denoted by itself, such thatFrom (38) and (45), we may extract a subsequence of , denoted again by , such thatReplacing by in equation (39), if necessary, and letting , we can conclude by the standard arguments as in Dautray and Lions (, pp.561–565) that the limit is a weak solution ofMoreover, by the uniqueness of the weak solutions, we can conclude that in , which implies that weakly in .
This completes the proof.
3. 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 and are lower and upper bounds of the control variables, respectively. Let the following be the set of the admissible disturbance or noises:where and are lower and upper bounds of the disturbance (noise) variables, respectively. For simplicity, let be a product space defined by .
Using Theorem 1, for fixed , we can uniquely define the solution map , which maps from via to the weak solution , where satisfies
The weak solution is called to be the state of the control system.
The quadratic cost function associated with control system (52) is given bywhere is a Hilbert space of observation variables, is a continuous linear operator, that is to say, observer, is a desired value, and the positive constants and 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 and a disturbance (or noise) such that is a saddle point of the functional of (53), that is, In this paper, we call such a pair in (54) as the optimal pair for the minimax optimal control problem with cost (53). To show the existence of the above saddle point (optimal pair), we need to show the solution map is differentiable in some sense. Then, with the assumptions on the weight constants and the exponent , we utilize the arguments in Barbu and Precupanu  to show the existence of the optimal pairs.(ii)Characterize (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.
3.1. 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. The solution map of into is said to be Fréchet differentiable on if for any , there exists such that, for any ,The operator is called the Fréchet derivative of at , which we denote by . is called the Fréchet derivative of at in the direction of .
Theorem 2. The solution map is Fréchet differentiable on , and the Fréchet derivative of at in the direction , that is to say , is given by the solution of
Proof. LetThen, by (11) and Theorem 1, we can verifyHence, from (58), we see thatSince and , we know that . By (59), we can use the results in  to verify that equation (56) admits a unique weak solution . And by Theorem 1, we can know that the weak solution of equation (56) satisfiesHence, from (60), the mapping is linear and bounded.
We set . Then, using notation (27) and notingwe know that satisfiesin the weak sense, whereArguing as in the proof of Theorem 1, we can know that the weak solution of equation (62) exists, and the following is fulfilled:By (60), we can deduceBy Theorem 1, we can see thatThus, if , then by (66), we can extract a subsequence, still denoted by , such thatAnd from (11), we know thatThus, by the Lebesgue dominated convergence theorem through (67) and (68), we can obtainHence, from (64) to (69), we can obtainwhich immediately implies that as .
This completes the proof.
As before, for simplicity, we denote and by and , respectively. From Theorem 2, we know that the map of to is Fréchet differentiable at , and the Fréchet derivative of at in the direction , say , is a unique weak solution of the following problem:The following results will be used in showing the existence of optimal pairs.
Proposition 2. Assume that the exponent . For given , the Fréchet derivative is locally Lipschitz continuous on . Indeed, it is satisfied thatwhere is a constant depending on the data.
Proof. Let be the weak solutions of equation (56) corresponding to . If we set , then we can know that satisfiesin the weak sense, whereAs before, arguing as in the proof of Theorem 1, we can deduce thatIn estimating , the case of will be clarified as we will see later. Thus, we restrict to the case of to estimate . By (11), Theorem 1, and (60), we can obtain