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Optimal control problems for hyperbolic equations with damping terms involving pLaplacian
Journal of Inequalities and Applications volume 2013, Article number: 92 (2013)
Abstract
In this paper we study optimal control problems for the hyperbolic equations with a damping term involving pLaplacian. 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.
1 Introduction
In this paper, we are concerned with optimal control problems for the hyperbolic equation with a damping term involving pLaplacian:
where Ω is a bounded domain in {\mathbb{R}}^{N} with sufficiently smooth boundary ∂ Ω, f is a forcing function, and {y}^{\mathrm{\prime}}=\frac{\partial y}{\partial t}, {y}^{\mathrm{\prime}\mathrm{\prime}}=\frac{{\partial}^{2}y}{\partial {t}^{2}}. 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, [15] showed the existence of global weak solutions for (1.1) and the asymptotic stability of the solution by using the Nakao lemma [9].
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 [16]. 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 [17], Hwang and Nakagiri [18]. Based on these methods, we intend to study optimal control problems for the hyperbolic hemivariational inequality (1.1) due to the theory of Lions [19] 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 [19] to the hyperbolic equation (1.1) involving pLaplacian with a damping term. Let H be a Hilbert space and let \mathcal{U} be another Hilbert space of control variables, and B be a bounded linear operator from \mathcal{U} into {L}^{2}(0,T;H), 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 u\in \mathcal{U} 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 v\to y(v), which is used to define the associated adjoint system.
2 Preliminaries
Throughout this paper we denote
For every q\in (1,\mathrm{\infty}), we denote {\parallel \cdot \parallel}_{q}={\parallel \cdot \parallel}_{{L}^{q}(\mathrm{\Omega})}. For brevity, we denote {\parallel \cdot \parallel}_{2} by \parallel \cdot \parallel. For a Banach space X, we denote by {\parallel \cdot \parallel}_{X} the norm of X.
Define A:V\to {V}^{\ast} by
where {V}^{\ast} denotes the dual space of V and \u3008\cdot ,\cdot \u3009 the dual pairing between V and {V}^{\ast}. Then the operator A is bounded, monotone, hemicontinuous (see, e.g., [20]) and
Now, we formulate the following assumption:

(H)
We assume that p is an even natural number satisfying
2\le p<\frac{(N1)p}{2(Np)}+1\phantom{\rule{1em}{0ex}}(2\le p<\mathrm{\infty}\text{if}p=N).
Definition 2.1 A function y(x,t) is a weak solution to problem (1.1) if for every T>0, y satisfies y\in {L}^{\mathrm{\infty}}(0,T;D(\mathrm{\u25b3})), {y}^{\mathrm{\prime}}\in {L}^{2}(0,T;D(\mathrm{\u25b3}))\cap {L}^{\mathrm{\infty}}(0,T;V), {y}^{\mathrm{\prime}\mathrm{\prime}}\in {L}^{2}(0,T;H). And for any z\in V and f\in {L}^{2}(0,T;H), the following relations hold:
The following theorem is from Jeong et al. [15].
Theorem 2.1 Let assumptions (H) be satisfied. Then, for {y}_{0}\in D(\mathrm{\u25b3}), {y}_{1}\in V and f\in {L}^{2}(0,T;H), problem (1.1) has a weak solution.
For any initial data ({y}_{0},{y}_{1})\in D(\mathrm{\u25b3})\times V, we define the solution space \mathcal{W}={L}^{\mathrm{\infty}}(0,T;V)\cap {W}^{1,2}(0,T;V)\cap {W}^{2,2}(0,T;H)\cap {W}^{1,\mathrm{\infty}}(0,T;V).
Here we remark that \mathcal{W} is continuously imbedded in C([0,T];V)\cap {C}^{1}([0,T];V), so that we assume that there exists
Theorem 2.2 Assume that {y}_{0}\in D(\mathrm{\u25b3}), {y}_{1}\in V and f\in {L}^{2}(0,T;H). The solution mapping p=({y}_{0},{y}_{1},f)\to y(p) of P\equiv D(\mathrm{\u25b3})\times V\times {L}^{2}(0,T;H) into \mathcal{W} is strongly continuous. Further, for each {p}_{1}=({y}_{0}^{1},{y}_{1}^{1},{f}_{1})\in P and {p}_{2}=({y}_{0}^{2},{y}_{1}^{2},{f}_{2})\in P, we have the inequality
Proof Due to [15], we can infer that a weak solution y\in \mathcal{W} under the data condition p=({y}_{0},{y}_{1},f)\in D(\mathrm{\u25b3})\times V\times {L}^{2}(0,T:H). Based on the above results, we prove the inequality (2.2). For that purpose, we denote {y}_{1}{y}_{2}\equiv y({p}_{1})y({p}_{2}) by ψ. Then we can observe from (1.1) that
where
Multiplying both sides of (2.3) by {\psi}^{\mathrm{\prime}}, we have
And we integrate (2.5) over [0,t] to have
The right member of (2.6) can be estimated as follows:
where {c}_{0}, {c}_{1} are constants. By using the boundedness of (2.1), we obtain (2.8) and (2.9). Finally, we replace the righthand 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 \mathcal{U} be a Hilbert space of control variables, and let B be a bounded linear operator from \mathcal{U} into {L}^{2}(0,T;H), which is denoted by
We consider the following nonlinear control system:
where {y}_{0}\in D(\mathrm{\u25b3}), {y}_{1}\in V, and v\in \mathcal{U} is a control. By virtue of Theorem 2.2 and (3.1), we can define uniquely the solution map v\to y(v) of \mathcal{U} into \mathcal{W}. We will call the solution y(v) 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 {z}_{d}\in M is a desired value of y(v) and R\in \mathcal{L}(\mathcal{U},\mathcal{U}) is symmetric and positive, i.e.,
for some d>0. Let {U}_{ad} be a closed convex subset of \mathcal{U}, which is called an admissible set. An element u\in {U}_{ad} which attains the minimum of J(v) over {U}_{ad} 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 J={inf}_{v\in {U}_{ad}}J(v). Since {U}_{ad} is nonempty, there is a sequence \{{v}_{n}\} in \mathcal{U} such that
Obviously, \{J({v}_{n})\} is bounded in {\mathbb{R}}^{+}. Then by (3.5) there exists a constant {K}_{0}>0 such that
This shows that \{{v}_{n}\} is bounded in \mathcal{U}. Since {U}_{ad} is closed and convex, we can choose a subsequence (denoted again by \{{v}_{n}\}) of \{{v}_{n}\} and find a u\in {U}_{ad} such that
as n\to \mathrm{\infty}. From now on, each state {y}_{n}=y({v}_{n})\in \mathcal{W} corresponding to {v}_{n} is the solution of
By (3.6) the term B{v}_{n} is estimated as
Hence it follows from the inequality (2.2), nothing y((0,0,0);t)\equiv 0, that
for some C>0. Combining (3.10) and (3.8), we deduce that
Therefore, by the extraction theorem of Rellich, we can find a subsequence of \{{y}_{n}\}, say again \{{y}_{n}\}, and find a y\in \mathcal{W} such that
By using the fact that D(\mathrm{\u25b3})\hookrightarrow V is compact and by virtue of (3.11), we can refer to the result of AubinLions Teman’s compact imbedding theorem (cf. Teman [21]) to verify that \{{y}_{n}\} is precompact in {L}^{2}(0,T;V). Hence there exists a subsequence \{{y}_{{n}_{k}}\}\subset \{{y}_{n}\} such that
Then we can choose a subsequence of \{{y}_{{n}_{k}}\}, denoted again by \{{y}_{{n}_{k}}\}, such that
Now we will show that
Let \varphi (\cdot )\in {L}^{2}(0,T;V) be given. Then we have
Since {\mathrm{\nabla}y(t)}^{p2}\mathrm{\nabla}\varphi \in {L}^{2}(0,T;H), it follows from (3.13) and the Hölder inequality that the last term of (3.16) converges to 0 as k\to \mathrm{\infty}. By the continuous imbedding \mathcal{W}\hookrightarrow C([0,T];V) and (3.12), we see that the set \{\mathrm{\nabla}{y}_{{n}_{k}}(t):t\in [0,T],k=1,2,\dots \} is bounded with a bound M>0. 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 k\to \mathrm{\infty}. This proves (3.15). We replace {y}_{n} by {y}_{{n}_{k}} and take k\to \mathrm{\infty} in (3.8). Since {y}_{{n}_{k}}\in \mathcal{W} are also the weak solutions, as in the standard argument in Dautray and Lions [22] and Lions and Magenes [23], we conclude that the limit y is a weak solution of
Also, since equation (3.18) has a unique strong solution y\in \mathcal{W} by Theorem 2.2, we conclude that y=y(v) in \mathcal{W} by the uniqueness of solutions, which implies y({v}_{n})\to y(u) weakly in \mathcal{W}. Since C is continuous on \mathcal{W} and {\parallel \cdot \parallel}_{M} is lower semicontinuous, it follows that
It is also clear from {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\parallel {R}^{\frac{1}{2}}{v}_{n}\parallel}_{\mathcal{U}}\ge {\parallel {R}^{\frac{1}{2}}u\parallel}_{\mathcal{U}} that {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{(R{v}_{n},{v}_{n})}_{\mathcal{U}}\ge {(Ru,u)}_{\mathcal{U}}. Hence
But since J(u)\ge J by definition, we conclude that J(u)={inf}_{v\in {U}_{ad}}J(v). 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 DJ(u) denotes the Gâteaux derivative of J(v) at v=u. That is, we have to prove that the mapping v\to y(v) of \mathcal{U}\to \mathcal{W} is Gâteaux differentiable at v=u. At first we can see the continuity of the mapping. The following lemma follows immediately from Theorem 2.2.
Lemma 4.1 Let w\in \mathcal{U} be arbitrarily fixed. Then
The solution map v\to y(v) of \mathcal{U} into \mathcal{W} is said to be Gâteaux differentiable at v=u if for any w\in \mathcal{U} there exists a Dy(u)\in \mathcal{L}(\mathcal{U},\mathcal{W}) such that
The operator Dy(u) denotes the Gâteaux derivative of y(u) at v=u and the function Dy(u)w\in \mathcal{W} is called the Gâteaux derivative in the direction w\in \mathcal{U}, which plays an important part in the nonlinear optimal control problem.
Theorem 4.1 The map v\to y(v) of \mathcal{U} into \mathcal{W} is Gâteaux differentiable at v=u and such a Gâteaux derivative of y(v) at v=u in the direction vu\in \mathcal{U}, say z=Dy(u)(vu), is a unique strong solution of the following problem:
Proof Let \lambda \in (1,1), \lambda \ne 0. We set w=vu and
Then {z}_{\lambda} satisfies
First we recall the simple equality
Multiplying the weak form of (4.3) by {z}_{\lambda}^{\mathrm{\prime}} and using the above equality, we have
We integrate (4.4) over [0,t] 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 \alpha >0. Combining (4.5) with (4.6)(4.8), we have
By applying Grownwall’s inequality to (4.9), we obtain
where {C}_{1} is a constant. The inequality (4.10) provides the boundedness of {z}_{\lambda} and {z}_{\lambda}^{\mathrm{\prime}} in appropriate spaces, and hence via (4.3) we can infer that there exists a z\in \mathcal{W} and a sequence \{{\lambda}_{k}\}\subset (1,1) 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 {z}_{\lambda}\to z=Dy(u)w weakly in \mathcal{W} as \lambda \to 0 in which z is a strong solution of (4.2). This convergency can be improved by showing the strong convergence of \{{z}_{\lambda}\} also in the topology of \mathcal{W}.
Subtracting (4.3) from (4.2) and denoting {z}_{\lambda}z by {\varphi}_{\lambda}, we see that
Here in (4.15), for \lambda \in (1,1), we set
From Lemma 4.1 and (4.14), we know that
To estimate {\varphi}_{\lambda}, we multiply both sides of (4.15) by {\varphi}_{\lambda}^{\mathrm{\prime}} and integrate it over [0,t] 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 righthand 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 {C}_{2} 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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Acknowledgements
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 (20120007560).
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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.
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Ju, EY., Jeong, JM. Optimal control problems for hyperbolic equations with damping terms involving pLaplacian. J Inequal Appl 2013, 92 (2013). https://doi.org/10.1186/1029242X201392
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DOI: https://doi.org/10.1186/1029242X201392
Keywords
 optimal control
 hyperbolic equation
 pLaplacian
 conditions for optimality
 Gâteaux differentiability