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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 641736, 9 pages

http://dx.doi.org/10.1155/2014/641736

## Optimal Control of Pseudoparabolic Variational Inequalities Involving State Constraint

^{1}School of Mathematics and Physics, University of South China, Hunan 421001, China^{2}School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received 5 March 2013; Accepted 27 May 2013; Published 20 January 2014

Academic Editor: Victor Kovtunenko

Copyright © 2014 Youjun Xu and Shu Zhou. 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 establish the necessary condition of optimality for optimal control problem governed by some pseudoparabolic differential equations involving monotone graphs. Some approximating control process and examples are given.

#### 1. Introduction

We will study the following optimal control problem governed by nonlinear pseudoparabolic variational inequalities of the following form: with the state constraint The pay-off function is given by where , is a bounded domain with smooth boundary.

For the problem (1)–(3), we have the following assumptions.(H1) is a selfadjoint operator in with such that for every , Throughout in the sequel, we will denote by and the norm and the scalar product of , respectively. The norm of the control set will be denoted by and the scalar product , respectively. denote the domain of operator , respectively.(H2) is a real Hilbert space such that is dense in and algebraically and topologically, where is the dual of . Further, the injection of into is compact. is a linear continuous and symmetric operator from to satisfying the coercivity condition where and .(H3) is a maximal monotone graph in with . Let be the lower semicontinuous convex function defined by , where is such that . Moreover, where for all , . For every , there exists a constant such that where if and if . denotes the generalized Clarke subdifferential of the function .(H4) is a linear continuous operator from a real Hilbert space to .(H5) Let be a Banach space with the dual strictly convex. is a closed convex subset with finite codimensionality [1–3]. is of class .(H6) The functional is convex and lower semicontinuous (l. s. c), such that where , for all .(H7) is measurable in , and for every , there exists independent of such that and

*Remark 1. *Note that, by , system (1) is equivalent to

As we know, by Barbu [4] (see Chapter 4) and Theorem 1.1 of [5], we have the following.

Lemma 2. *Let hold. Then, for any , , (1) admits a unique solution satisfying
*

Now we formulate the optimal control problems as follows.

Let is the solution of (10) with (2)*}*.

We will find

Recently, some optimal control problems governed by pseudoparabolic equations have already been discussed. Linear optimal control problems for pseudoparabolic equations were considered by many authors (cf. [6–12]). However, these problems studied in [7–12] do not involve state constraints and maximal monotone graph. On the other hand, optimal control problems governed by some parabolic variational inequalities (cf. [4, 13–19]) have already been discussed. Li and Yong [1] studied the maximal principle for optimal control governed by some nonlinear parabolic equations with two point boundary (time variable) state constraints. In Cases’ work [20], the state constraint was considered, but the state equation did not involve monotone graph. He [21] studied the optimal control problems involving some special maximal monotone graph (Lipschitz continuous) with state constraint. Wang [2, 3] also discussed the optimal control problem governed by the state equation involving some maximal monotone graph.

The present work in this paper considers the optimal control problem governed by the pseudoparabolic equations which is different from what they discussed in [7–9, 12], with the state constraints which is similar to those in [3, 4, 21].

The plan of this paper is as follows. Section 2 gives an approximating control process. In Section 3, we state and prove the necessary conditions on optimality for the problem . In Section 4, some examples are given.

#### 2. The Approximating Control Process

Let be optimal for the problem . Then with

From a perturbation theorem for m-accretive operators ([22], Lemma 5) and , , we easily know that is m-accretive in .

Now consider the following approximating equation: where and . By Lemma 2, for any , , (14) has a unique solution in .

Besides, we have the following result on (14).

Lemma 3. *For given, let weakly in , and the solutions of (14) corresponding to and , respectively. Then, there exists some subsequence of , still denoted by itself, such that strongly in .*

*Proof. *Multiplying (14) by and using the self-adjointness of , we see the following:

Then yield
where . Integrating the above inequality from to and using Gronwall’s inequality, we see the following:

Note that from , has a bounded inverse operator on and

Together (17) and (18), we have the following:

Since for every , taking into account (17) and (19), we have the following:

Multiplying (14) by , we see

Then we get the following:

Applying Gronwall’s inequality to the above inequality and noting that is bounded, we have the following:

From and (18), we see

Then in view of (14), (24) gives
thus we see
which implies

Here, is the norm in . For every

By some calculation, we see

Hence and are Cauchy sequences in . Note that ; then there exists a function such that as

This completes the proof.

Next, we define the approximation of and of as follows. For the details, we refer to [2–4]. Let

Here, is a mollifier in , . is the projection of on , which is the finite dimensional space generated by , where is an orthonormal basis in . is the operator defined by .

We define :

Now we define the penalty by where is the solution of (14). denotes the distance of to .

The approximating optimal control problems are as follows:

From Lemma 3, we easily show the following existence of the optimal solutions for (see [2, 3]).

Theorem 4. * has at least one optimal solution.*

The following results are useful in discussing the approximating control problems.

Lemma 5. *Let weakly in as . Then there exists a subsequence , still denoted itself
**
as , where is the solutions of (14) corresponding to and is the solutions of (10) corresponding to .*

*Proof. *Rewrite (14) as follows:

Multiplying (35) by , we see

Then, yield
Integrating the above inequality from to and using Gronwall’s inequality, we have the following:
together with (18) implies

Since for every , taking into account (36) (39), we see

Multiplying (35) by , we see

Then we get the following:
from which it follows that

From , , and (18), we see

Then in view of (14) and (24) give

Thus, we see
which implies

For every ,

Using the identities for every , , we see

Because of (43) and (44), we obtain the following:
where is a constant independent of and . Then Gronwall’s inequality yields

Hence, and are Cauchy sequences in . Note that ; then there exists a function such that as , ,

Thus, we deduce that as ,

Note that

Indeed, we see
for all . From (43) and (46), is uniformly bounded and equicontinuous in . Hence the Ascoli-Arzela theorem gives that as , for every strongly in . In virtue of (46) and (48), weak closedness of , and , it is shown that

Therefore, and . By , we denote the space of all -valued strongly absolutely continuous functions on . We easily get that and there exists a function such that as ,
and . Thus, letting in (35), we see

Lemma 6. *Let , ; then strongly in as , where is the solutions of (14) corresponding to and is the solutions of (1) corresponding to with the initial condition . Furthermore,
*

*Proof. *By the same argument in the proof of Lemma 5, we have the following:

We have for all and ,

Multiplying (62) by , we have

Using the identities for every , and so forth, we get the following:

Thus, we see
then

Because of (61), letting in (66), we get (60).

Lemma 7. *Let be optimal for the problem and be the solution of (14) corresponding to . For , then
*

*Proof. *For any , we have the following:

By Lemma 5, we know strongly in . So we have the following:

So

Similarly, by (60) and , we obtain the following:

Then, we get the following:

On the other hand, since is bounded in , there exists such that, on some subsequence , still denoted by itself, as ,
and so, by Lemma 5,

By (66), one can check easily that

Thus, as . Since is closed and convex, . Since the function is weakly lower semicontinuous on , we see
Together with (72), we obtain

Therefore,

Hence, , . This completes the proof.

#### 3. Necessary Condition on Optimality

Let the generalized gradient of . Let which is the dual of with .

Firstly, we consider the following Cauchy problem: where , , , and is a -mollifier on .

Lemma 8. *Problem (79) has a unique absolutely continuous function with , such that
*

*Proof. *From and , it is seen that is demicontinuous monotone operator that satisfies
where and . It follows by Theorem of [4] that (79) has a unique solution with . Multiplying (79) by and using the self-adjointness of and integrating over , we see
Because of , and . And so by Gronwall’s lemma we obtain the following:

Combining the above equalities, we see

Since for every , taking into account the above equalities, we have the following:

Thus, we obtain (80).

Multiplying (79) by and integrate on , where is a smooth monotonically increasing approximation of the sign function such that . For instance
where for for , and is a -mollifier. Then ; therefore,

Then, letting tend to the sign function, we get (81).

We state the main results of the necessary conditions on optimality as follows.

Theorem 9. *Suppose that hold. Let be an optimal pair of problem . Then, there exists function , a measure , satisfying
*

*Proof. *Since is optimal for problem , we see

Here . Thus,

By some calculation, we have the following:
where is the following solution to the linear equation

Hence, we also have the following:
where
and . Since is convex and closed, we see

So, we see
It follows from Lemma 7 that strongly in . By the same arguments as those in [2–4], there exists and such that, on some subsequence , still denoted itself
where is the space of all -valued functions with bounded variation on . On the other hand, by (80), we see

Note that is compact, for every , there is such that

This yields

Moreover, by (81) we infer that there is such that, on some generalized subsequence ,

Since is continuously differentiable from to ,

Now letting in (79), it follows that

It follows from (93), (94), and (79) that

By Lemma 7, strongly in , it follows

Thus,

Since , we get for all . Now we claim that . Indeed, if , we have that is bounded in . By (H3), has finite codimentionality, so dose . Thus, it follows that weakly in and

Finally, if , it follows from (105) that . So in the case that , we must have . Together with (104), (105), and (109), we completes the proof.

#### 4. Some Examples

In this section, we present two examples.

*Example 1. *Consider the initial value controlled system
where 1 is a function on and is a multivalued function on .

If , rewrite (110) in the form
(111) was introduced by Benjamin et al. [23] as an approximate equation of the propagation of one-dimensional waves of small amplitude in water. If , satisfies . Since is a Lipschitz continuous and monotone increasing function, integration by parts yields

Thus, is m-accretive in . We easily proof the following result.

Theorem 10. *Suppose that hold. Let be an optimal pair of problem . Then there exists function , a measure and with satisfying
*

*Example 2. *Consider the initial boundary value controlled system
where is a bounded domain with smooth boundary. , satisfies with , . Since is a monotone function,

Then, is m-accretive in . We easily obtain similar necessary condition of optimality of problem .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The work is supported by the NNSFC (Grants nos. 10671211, 11126170), Hunan Provincial Natural Science Foundation of China (Grant no. 11JJ4006), and Doctor Priming Fund Project of University of South China (Grant no. 5-2011-XQD-008).

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