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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 641736, 9 pages
http://dx.doi.org/10.1155/2014/641736
Research Article

Optimal Control of Pseudoparabolic Variational Inequalities Involving State Constraint

1School of Mathematics and Physics, University of South China, Hunan 421001, China
2School 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 [13]. 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. [612]). However, these problems studied in [712] do not involve state constraints and maximal monotone graph. On the other hand, optimal control problems governed by some parabolic variational inequalities (cf. [4, 1319]) 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 [79, 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 [24]. 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 [24], 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).

References

  1. X. J. Li and J. M. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Birkhäauser, Boston, Mass, USA, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  2. G. S. Wang, “Optimal control of parabolic differential equations with two point boundary state constraints,” SIAM Journal on Control and Optimization, vol. 38, no. 5, pp. 1639–1654, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. G. S. Wang, “Optimal control of parabolic variational inequality involving state constraint,” Nonlinear Analysis: Theory, Methods & Applications, vol. 42, no. 5, pp. 789–801, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Academic Press, Boston, Mass, USA, 1993. View at MathSciNet
  5. M. Tsutsumi and T. Matahashi, “On some nonlinear pseudoparabolic equations,” Journal of Differential Equations, vol. 32, no. 1, pp. 65–75, 1979. View at Publisher · View at Google Scholar · View at MathSciNet
  6. W. Liu, Elementary Feedback Stabilization of the Linear Reaction-convection-diffusion Equation and the Wave Equation, vol. 66 of Mathmatiques et Applications, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  7. S. I. Lyashko, Generalized Control in Linear Systems, Naukova Dumka, Kiev, Ukraine, 1998 (Russian).
  8. G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP International Science, 2003. View at MathSciNet
  9. V. E. Fedorov and M. V. Plekhanova, “Optimal control of Sobolev type linear equations,” Differential Equations, vol. 40, no. 11, pp. 1627–1637, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  10. L. W. White, “Control problems governed by a pseudo-parabolic partial differential equation,” Transactions of the American Mathematical Society, vol. 250, pp. 235–246, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. L. W. White, “Convergence properties of optimal controls of pseudoparabolic problems,” Applied Mathematics and Optimization, vol. 7, no. 2, pp. 141–147, 1981. View at Publisher · View at Google Scholar · View at MathSciNet
  12. I. Bock and J. Lovíšek, “On pseudoparabolic optimal control problems,” Kybernetika, vol. 29, no. 3, pp. 222–230, 1993. View at Zentralblatt MATH · View at MathSciNet
  13. K. Kunisch and D. Wachsmuth, “Path-following for optimal control of stationary variational inequalities,” Computational Optimization and Applications, vol. 51, no. 3, pp. 1345–1373, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. J. C. de Los Reyes, “Optimal control of a class of variational inequalities of the second kind,” SIAM Journal on Control and Optimization, vol. 49, no. 4, pp. 1629–1658, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. K. Ito and K. Kunisch, “Optimal control of parabolic variational inequalities,” Journal de Mathématiques Pures et Appliquées, vol. 93, no. 4, pp. 329–360, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. K. Ito and K. Kunisch, “An augmented Lagrangian technique for variational inequalities,” Applied Mathematics and Optimization, vol. 21, no. 3, pp. 223–241, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. K. Ito and K. Kunisch, “Optimal control of elliptic variational inequalities,” Applied Mathematics and Optimization, vol. 41, no. 3, pp. 343–364, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. X. M. Guo and S. X. Zhou, “Optimal control of parabolic variational inequalities with state constraint,” Applied Mathematics and Mechanics (English Edition), vol. 24, no. 7, pp. 756–762, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. D. R. Adams and S. Lenhart, “Optimal control of the obstacle for a parabolic variational inequality,” Journal of Mathematical Analysis and Applications, vol. 268, no. 2, pp. 602–614, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. E. Casas, “Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations,” SIAM Journal on Control and Optimization, vol. 35, no. 4, pp. 1297–1327, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. Z.-X. He, “State constrained control problems governed by variational inequalities,” SIAM Journal on Control and Optimization, vol. 25, no. 5, pp. 1119–1144, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. H. Brézis, “Quelques Propriétés des Opérateurs Monotones et des Semi-groupes Non Linéaires,” in Nonlinear operators and the calculus of variations, vol. 543 of Lecture Notes in Mathematics, pp. 56–82, Springer, Berlin, Germany, 1976. View at Zentralblatt MATH · View at MathSciNet
  23. T. B. Benjamin, J. L. Bona, and J. J. Mahony, “Model equations for long waves in nonlinear dispersive systems,” Philosophical Transactions of the Royal Society of London A, vol. 272, no. 1220, pp. 47–78, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet