Optimal control problems for hyperbolic equations with damping terms involving p-Laplacian
© Ju and Jeong; licensee Springer 2013
Received: 10 October 2012
Accepted: 19 February 2013
Published: 6 March 2013
In this paper we study optimal control problems for the hyperbolic equations with a damping term involving p-Laplacian. We prove the existence of an optimal control and the Gâteaux differentiability of a solution mapping on control variables. And then we characterize the optimal controls by giving necessary conditions for optimality.
AMS Subject Classification:49K20, 93C20.
Keywordsoptimal control hyperbolic equation p-Laplacian conditions for optimality Gâteaux differentiability
where Ω is a bounded domain in with sufficiently smooth boundary ∂ Ω, f is a forcing function, and , . The background of these variational problems are in physics, especially in solid mechanics. The precise hypotheses on the above system will be given in the next section.
Recently, much research has been devoted to the study of hemivariational inequalities [1–8]. The research works have mainly considered the existence of weak solutions for differential inclusions of various forms [2, 3, 5, 6, 9]. In particular, the case where the nonlinear wave equation includes a nonlinear damping term in a bounded domain was proved by many authors for both existence and nonexistence of global solutions [10–14]. Especially,  showed the existence of global weak solutions for (1.1) and the asymptotic stability of the solution by using the Nakao lemma .
On the other hand, it is interesting to mention that optimal control problems for the equation of Kirchhoff type with a damping term have been studied by Hwang and Nakagiri . We can find some articles about the studies on some kinds of semilinear partial differential equations and on quasilinear partial differential equations; see Ha and Nakagiri , Hwang and Nakagiri . Based on these methods, we intend to study optimal control problems for the hyperbolic hemivariational inequality (1.1) due to the theory of Lions  in which the optimal control problems are surveyed on many types of linear partial differential equations.
The plan of this paper is as follows. In Section 2, the main results besides notations and assumptions are stated. In Section 3, we show the existence of an optimal control which minimizes the quadratic cost function. In Section 4, we characterize the optimal controls by giving necessary conditions for optimality. For this we prove the Gâteaux differentiability of the nonlinear mapping , which is used to define the associated adjoint system.
For every , we denote . For brevity, we denote by . For a Banach space X, we denote by the norm of X.
- (H)We assume that p is an even natural number satisfying
The following theorem is from Jeong et al. .
Theorem 2.1 Let assumptions (H) be satisfied. Then, for , and , problem (1.1) has a weak solution.
For any initial data , we define the solution space .
where C is a constant. This completes the proof. □
3 Existence of an optimal control
for some . Let be a closed convex subset of , which is called an admissible set. An element which attains the minimum of over is called an optimal control for the cost function (3.4).
As indicated in Introduction, we need to show the existence of an optimal control and to give the characterizations of it. The existence of an optimal control u for the cost function (3.4) can be stated by the following theorem.
Theorem 3.1 Assume that the hypotheses of Theorem 2.2 are satisfied. Then there exists at least one optimal control u for the control problem (3.2) with (3.4).
But since by definition, we conclude that . This completes the proof. □
4 Necessary condition for optimality
and to analyze (4.1) in view of the proper adjoint state system, where denotes the Gâteaux derivative of at . That is, we have to prove that the mapping of is Gâteaux differentiable at . At first we can see the continuity of the mapping. The following lemma follows immediately from Theorem 2.2.
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 problem.
Hence we can see from (4.11)-(4.13) that weakly in as in which z is a strong solution of (4.2). This convergency can be improved by showing the strong convergence of also in the topology of .
This completes the proof. □
This research was supported by the Basic Science Research Program through the National research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0007560).
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