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Optimal control problems for hyperbolic equations with damping terms involving p-Laplacian
Journal of Inequalities and Applicationsvolume 2013, Article number: 92 (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.
In this paper, we are concerned with optimal control problems for the hyperbolic equation with a damping term involving p-Laplacian:
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.
Our goal in this paper is to extend the optimal control theory in the framework of Lions  to the hyperbolic equation (1.1) involving p-Laplacian with a damping term. Let H be a Hilbert space and let be another Hilbert space of control variables, and B be a bounded linear operator from into , which is called a controller. We formulate our optimal control problem as follows:
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.
Throughout this paper we denote
For every , we denote . For brevity, we denote by . For a Banach space X, we denote by the norm of X.
where denotes the dual space of V and the dual pairing between V and . Then the operator A is bounded, monotone, hemicontinuous (see, e.g., ) and
Now, we formulate the following assumption:
We assume that p is an even natural number satisfying
Definition 2.1 A function is a weak solution to problem (1.1) if for every , y satisfies , , . And for any and , the following relations hold:
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 .
Here we remark that is continuously imbedded in , so that we assume that there exists
Theorem 2.2 Assume that , and . The solution mapping of into is strongly continuous. Further, for each and , we have the inequality
Proof Due to , we can infer that a weak solution under the data condition . Based on the above results, we prove the inequality (2.2). For that purpose, we denote by ψ. Then we can observe from (1.1) that
Multiplying both sides of (2.3) by , we have
And we integrate (2.5) over [0,t] to have
The right member of (2.6) can be estimated as follows:
where , are constants. By using the boundedness of (2.1), we obtain (2.8) and (2.9). Finally, we replace the right-hand side of (2.6) by the right members of (2.7)-(2.9) and apply Gronwall’s inequality to the replaced inequality to obtain
where C is a constant. This completes the proof. □
3 Existence of an optimal control
Let be a Hilbert space of control variables, and let B be a bounded linear operator from into , which is denoted by
We consider the following nonlinear control system:
where , , and is a control. By virtue of Theorem 2.2 and (3.1), we can define uniquely the solution map of into . We will call the solution of (3.2) the state of the control system (3.2). The observation of the state is assumed to be given by
where C is an operator called the observer and M is a Hilbert space of observation variables. The quadratic cost function associated with the control system (3.2) is given by
where is a desired value of and is symmetric and positive, i.e.,
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).
Proof Set . Since is non-empty, there is a sequence in such that
Obviously, is bounded in . Then by (3.5) there exists a constant such that
This shows that is bounded in . Since is closed and convex, we can choose a subsequence (denoted again by ) of and find a such that
as . From now on, each state corresponding to is the solution of
By (3.6) the term is estimated as
Hence it follows from the inequality (2.2), nothing , that
for some . Combining (3.10) and (3.8), we deduce that
Therefore, by the extraction theorem of Rellich, we can find a subsequence of , say again , and find a such that
By using the fact that is compact and by virtue of (3.11), we can refer to the result of Aubin-Lions Teman’s compact imbedding theorem (cf. Teman ) to verify that is precompact in . Hence there exists a subsequence such that
Then we can choose a subsequence of , denoted again by , such that
Now we will show that
Let be given. Then we have
Since , it follows from (3.13) and the Hölder inequality that the last term of (3.16) converges to 0 as . By the continuous imbedding and (3.12), we see that the set is bounded with a bound . Then, by the Schwartz inequality, the middle term of (3.16) is estimated by
We use (3.14) and apply the Lebesgue dominated convergence theorem to have that the term of (3.17) converges to 0 as . This proves (3.15). We replace by and take in (3.8). Since are also the weak solutions, as in the standard argument in Dautray and Lions  and Lions and Magenes , we conclude that the limit y is a weak solution of
Also, since equation (3.18) has a unique strong solution by Theorem 2.2, we conclude that in by the uniqueness of solutions, which implies weakly in . Since C is continuous on and is lower semicontinuous, it follows that
It is also clear from that . Hence
But since by definition, we conclude that . This completes the proof. □
4 Necessary condition 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 optimality condition
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.
Lemma 4.1 Let be arbitrarily fixed. Then
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 problem.
Theorem 4.1 The map of into is Gâteaux differentiable at and such a Gâteaux derivative of at in the direction , say , is a unique strong solution of the following problem:
Proof Let , . We set and
First we recall the simple equality
Multiplying the weak form of (4.3) by and using the above equality, we have
We integrate (4.4) over and use the integration by parts to have
By virtue of (2.2) and Young’s inequality, we can estimate each of the integrands of (4.5) as follows:
where C is a positive constant depending only on the data and . Combining (4.5) with (4.6)-(4.8), we have
By applying Grownwall’s inequality to (4.9), we obtain
where is a constant. The inequality (4.10) provides the boundedness of and in appropriate spaces, and hence via (4.3) we can infer that there exists a and a sequence tending to 0 such that
From Lemma 4.1 and (4.12), we can easily show that
By Lemma 4.1,
so that by (4.12)
This implies that
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 .
Subtracting (4.3) from (4.2) and denoting by , we see that
Here in (4.15), for , we set
From Lemma 4.1 and (4.14), we know that
To estimate , we multiply both sides of (4.15) by and integrate it over and use integration by parts to have
The integral parts of the right member of (4.17) can be estimated as follows:
where C is a constant. We replace the right-hand side of (4.17) by the right members of (4.18)-(4.19) and we apply Gronwall’s inequality to the replaced inequality, then we arrive at
where is a constant. By virtue (4.16) and (4.20), we deduce that
Finally, by means of (4.15), (4.21) and (4.22), it follows that
This completes the proof. □
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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).
The authors declare that they have no competing interests.
EYJ carried out the main proof of this manuscript and participated in its design and coordination, JMJ drafted the manuscript and corrected the main theorems.