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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 724190, 10 pages
http://dx.doi.org/10.1155/2013/724190
Research Article

Optimal Control Problems for Nonlinear Variational Evolution Inequalities

Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea

Received 12 November 2012; Revised 4 January 2013; Accepted 6 January 2013

Academic Editor: Ryan Loxton

Copyright © 2013 Eun-Young Ju and Jin-Mun Jeong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We deal with optimal control problems governed by semilinear parabolic type equations and in particular described by variational inequalities. We will also characterize the optimal controls by giving necessary conditions for optimality by proving the Gâteaux differentiability of solution mapping on control variables.

1. Introduction

In this paper, we deal with optimal control problems governed by the following variational inequality in a Hilbert space : Here, is a continuous linear operator from into which is assumed to satisfy Gårding's inequality, where is dense subspace in . Let be a lower semicontinuous, proper convex function. Let be a Hilbert space of control variables, and let be a bounded linear operator from into . Let be a closed convex subset of , which is called the admissible set. Let be a given quadratic cost function (see (61) or (103)). Then we will find an element which attains minimum of over subject to (1).

Recently, initial and boundary value problems for permanent magnet technologies have been introduced via variational inequalities in [1, 2] and nonlinear variational inequalities of semilinear parabolic type in [3, 4]. The papers treating the variational inequalities with nonlinear perturbations are not many. First of all, we deal with the existence and a variation of constant formula for solutions of the nonlinear functional differential equation (1) governed by the variational inequality in Hilbert spaces in Section 2.

Based on the regularity results for solution of (1), we intend to establish the optimal control problem for the cost problems in Section 3. For the optimal control problem of systems governed by variational inequalities, see [1, 5]. We refer to [6, 7] to see the applications of nonlinear variational inequalities. Necessary conditions for state constraint optimal control problems governed by semilinear elliptic problems have been obtained by Bonnans and Tiba [8] using methods of convex analysis (see also [9]).

Let stand for solution of (1) associated with the control . When the nonlinear mapping is Lipschitz continuous from into , we will obtain the regularity for solutions of (1) and the norm estimate of a solution of the above nonlinear equation on desired solution space. Consequently, in view of the monotonicity of , we show that the mapping is continuous in order to establish the necessary conditions of optimality of optimal controls for various observation cases.

In Section 4, we will characterize the optimal controls by giving necessary conditions for optimality. For this, it is necessary to write down the necessary optimal condition due to the theory of Lions [9]. The most important objective of such a treatment is to derive necessary optimality conditions that are able to give complete information on the optimal control.

Since the optimal control problems governed by nonlinear equations are nonsmooth and nonconvex, the standard methods of deriving necessary conditions of optimality are inapplicable here. So we approximate the given problem by a family of smooth optimization problems and afterwards tend to consider the limit in the corresponding optimal control problems. An attractive feature of this approach is that it allows the treatment of optimal control problems governed by a large class of nonlinear systems with general cost criteria.

2. Regularity for Solutions

If is identified with its dual space we may write densely and the corresponding injections are continuous. The norm on , , and will be denoted by , , and , respectively. The duality pairing between the element of and the element of is denoted by , which is the ordinary inner product in if .

For we denote by the value of at . The norm of as element of is given by Therefore, we assume that has a stronger topology than and, for brevity, we may regard that

Let be a bounded sesquilinear form defined in and satisfying Gårding's inequality where and is a real number. Let be the operator associated with this sesquilinear form: Then is a bounded linear operator from to by the Lax-Milgram Theorem. The realization of in which is the restriction of to is also denoted by . From the following inequalities where is the graph norm of , it follows that there exists a constant such that Thus we have the following sequence where each space is dense in the next one with continuous injection.

Lemma 1. With the notations (9) and (10), we have where denotes the real interpolation space between and (Section  1.3.3 of [10]).

It is also well known that generates an analytic semigroup in both and . For the sake of simplicity we assume that and hence the closed half plane is contained in the resolvent set of .

If is a Banach space, is the collection of all strongly measurable square integrable functions from into and is the set of all absolutely continuous functions on such that their derivative belongs to . will denote the set of all continuously functions from into with the supremum norm. If and are two Banach spaces, is the collection of all bounded linear operators from into , and is simply written as . Here, we note that by using interpolation theory we have

First of all, consider the following linear system:

By virtue of Theorem 3.3 of [11] (or Theorem 3.1 of [12, 13]), we have the following result on the corresponding linear equation of (13).

Lemma 2. Suppose that the assumptions for the principal operator stated above are satisfied. Then the following properties hold.(1) For (see Lemma 1) and , , there exists a unique solution of (13) belonging to and satisfying where is a constant depending on . (2) Let and . Then there exists a unique solution of (13) belonging to and satisfying where is a constant depending on .

Let be a nonlinear single valued mapping from into . (F) We assume that for every .

Let be another Hilbert space of control variables and take as stated in the Introduction. Choose a bounded subset of and call it a control set. Let us define an admissible control as

Noting that the subdifferential operator is defined by the problem (1) is represented by the following nonlinear functional differential problem on :

Referring to Theorem  3.1 of [3], we establish the following results on the solvability of (1).

Proposition 3. (1) Let the assumption (F) be satisfied. Assume that , , and where is the closure in of the set . Then, (1) has a unique solution which satisfies where is the minimum element of and there exists a constant depending on such that where is some positive constant and .
Furthermore, if then the solution belongs to and satisfies
(2) We assume the following. (A) is symmetric and there exists such that for every and any where .
Then for , , and (1) has a unique solution which satisfies

Remark 4. In terms of Lemma 1, the following inclusion is well known as seen in (9) and is an easy consequence of the definition of real interpolation spaces by the trace method (see [4, 13]).

The following Lemma is from Brézis [14; Lemma A.5].

Lemma 5. Let satisfying for all and be a constant. Let be a continuous function on satisfying the following inequality: Then,

For each , we can define the continuous solution mapping . Now, we can state the following theorem.

Theorem 6. (1) Let the assumption (F) be satisfied, , and . Then the solution of (1) belongs to and the mapping is Lipschtz continuous; that is, suppose that and be the solution of (1) with in place of for , where is a constant.
(2) Let the assumptions (A) and (F) be satisfied and let and . Then , and the mapping is continuous.

Proof. (1) Due to Proposition 3, we can infer that (1) possesses a unique solution with the data condition . Now, we will prove the inequality (33). For that purpose, we denote by . Then Multiplying on the above equation by , we have Put By integrating the above inequality over , we have Note that integrating the above inequality over , we have Thus, we get Combining this with (38) it holds that By Lemma 5, the following inequality implies that From (42) and (44) it follows that Putting The third term of the right hand side of (45) is estimated as The second term of the right hand side of (45) is estimated as Thus, from (47) and (48), we apply Gronwall's inequality to (15), and we arrive at where is a constant. Suppose in , and let and be the solutions (1) with and , respectively. Then, by virtue of (49), we see that in .
(2) It is easy to show that if and , then belongs to . Let , and be the solution of (1) with in place of for . Then in view of Lemma 2 and assumption (F), we have Since we get, noting that , Hence arguing as in (9) we get Combining (50) and (53) we obtain Suppose that and let and be the solutions (1) with and , respectively. Let be such that Then by virtue of (54) with replaced by we see that This implies that in . Hence the same argument shows that in Repeating this process we conclude that in .

3. Optimal Control Problems

In this section we study the optimal control problems for the quadratic cost function in the framework of Lions [9]. In what follows we assume that the embedding is compact.

Let be another Hilbert space of control variables, and be a bounded linear operator from into ; that is, which is called a controller. By virtue of Theorem 6, we can define uniquely the solution map of into . We will call the solution the state of the control system (1).

Let be a Hilbert space of observation variables. The observation of state is assumed to be given by where is an operator called the observer. The quadratic cost function associated with the control system (1) is given by where is a desire value of and is symmetric and positive; that is, for some . Let be a closed convex subset of , which is called the admissible set. An element which attains minimum of over is called an optimal control for the cost function (61).

Remark 7. The solution space of strong solutions of (1) is defined by endowed with the norm
Let be an open bounded and connected set of with smooth boundary. We consider the observation of distributive and terminal values (see [15, 16]).
(1) We take and and observe
(2) We take and and observe The above observations are meaningful in view of the regularity of (1) by Proposition 3.

Theorem 8. (1) Let the assumption (F) be satisfied. Assume that and . Let be the solution of (1) corresponding to . Then the mapping is compact from to .
(2) Let the assumptions (A) and (F) be satisfied. If and , then the mapping is compact from to .

Proof. (1) We define the solution mapping from to by In virtue of Lemma 2, we have Hence if is bounded in , then so is in . Since is compactly embedded in by assumption, the embedding is also compact in view of Theorem  2 of Aubin [17]. Hence, the mapping is compact from to .
(2) If is compactly embedded in by assumption, the embedding is compact. Hence, the proof of (2) is complete.

As indicated in the Introduction we need to show the existence of an optimal control and to give the characterizations of them. The existence of an optimal control for the cost function (61) can be stated by the following theorem.

Theorem 9. Let the assumptions (A) and (F) be satisfied and . Then there exists at least one optimal control for the control problem (1) associated with the cost function (61); that is, there exists such that

Proof. Since is nonempty, there is a sequence such that minimizing sequence for the problem (70) satisfies Obviously, is bounded. Hence by (62) there is a positive constant such that This shows that is bounded in . So we can extract a subsequence (denoted again by ) of and find a such that in . Let be the solution of the following equation corresponding to : By (15) and (17) we know that and are bounded in and , respectively. Therefore, by the extraction theorem of Rellich's, we can find a subsequence of , say again , and find such that However, by Theorem 8, we know that From (F) it follows that By the boundedness of we have Since are uniformly bounded from (73)–(77) it follows that and noting that is demiclosed, we have that Thus we have proved that satisfies a.e. on the following equation: Since is continuous and is lower semicontinuous, it holds that It is also clear from that Thus, But since by definition, we conclude is a desired optimal control.

4. Necessary Conditions for Optimality

In this section we will characterize the optimal controls by giving necessary conditions for optimality. For this it is necessary to write down the necessary optimal condition and to analyze (84) in view of the proper adjoint state system, where denote the Gâteaux derivative of at . Therefore, we have to prove that the solution mapping is Gâteaux differentiable at . Here we note that from Theorem 6 it follows immediately that The solution map of into is said to be Gâteaux differentiable at if for any there exists a such that The operator denotes the Gâteaux derivative of at and the function is called the Gâteaux derivative in the direction , which plays an important part in the nonlinear optimal control problems.

First, as is seen in Corollary 2.2 of Chapter II of [18], let us introduce the regularization of as follows.

Lemma 10. For every , define where . Then the function is Fréchet differentiable on and its Frećhet differential is Lipschitz continuous on with Lipschitz constant . In addition, where is the element of minimum norm in the set .

Now, we introduce the smoothing system corresponding to (1) as follows.

Lemma 11. Let the assumption (F) be satisfied. Then the solution map of into is Lipschtz continuous.
Moreover, let us assume the condition (A) in Proposition 3. Then the map of into is also Lipschtz continuous.

Proof. We set . From Theorem 6, it follows immediately that so the solution map of into is Lipschtz continuous. Moreover, since by the assumption (A) and (2) of Theorem 6, it holds and, by the relation (12), So we know that the map of into is also Lipschtz continuous.

Let the solution space of (1) of strong solutions is defined by as stated in Remark 7.

In order to obtain the optimality conditions, we require the following assumptions. (F1) The Gâteaux derivative in the second argument for is measurable in for and continuous in for a.e. , and there exist functions such that (F2) The map is Gâteaux differentiable, and the value is the Gâteaux derivative of at such that there exist functions such that

Theorem 12. Let the assumptions (A), (F1), and (F2) be satisfied. Let be an optimal control for the cost function in (61). Then the following inequality holds: where is a unique solution of the following equation:

Proof. We set . Let , . We set From (89), we have Then as an immediate consequence of Lemma 11 one obtains thus, in the sense of (F2), we have that satisfies (98) and the cost is Gâteaux differentiable at in the direction . The optimal condition (84) is rewritten as

With every control , we consider the following distributional cost function expressed by where the operator is bounded from to another Hilbert space and . Finally we are given that is a self adjoint and positive definite: Let stand for solution of (1) associated with the control . Let be a closed convex subset of .

Theorem 13. Let the assumptions in Theorem 12 be satisfied and let the operators and satisfy the conditions mentioned above. Then there exists an element such that Furthermore, the following inequality holds: holds, where is the canonical isomorphism onto and satisfies the following equation:

Proof. Let be a solution of (1) associated with the control 0. Then it holds that where The form is a continuous form in and from assumption of the positive definite of the operator , we have If is an optimal control, similarly for (97), (84) is equivalent to Now we formulate the adjoint system to describe the optimal condition:
Taking into account the regularity result of Proposition 3 and the observation conditions, we can assert that (112) admits a unique weak solution reversing the direction of time by referring to the well-posedness result of Dautray and Lions [19, pages 558–570].
We multiply both sides of (112) by of (98) and integrate it over . Then we have By the initial value condition of and the terminal value condition of , the left hand side of (113) yields