On the Solution of a Optimal Control Problem for a Hyperbolic System

In this study, the problem of determining the control function that is at the right hand side of a hyperbolic system from the final observation is investigated. Using the Fourier-Galerkin method, the weak solution of this hyperbolic system is obtained. The necessary conditions for the existence and uniqueness of the optimal solution are proved. We also find the approximate solutions of the test problems in numerical examples by a MAPLE program. Finally, the numerical results are presented in the form of tables.


Introduction
The problem of determining the control function that is at the right hand side of the hyperbolic system has been studied by different authors. Lions [3] examined the problems in detail when the control function is at the right hand side of the hyperbolic problem by using different cost function. Periago [4] has investigated the problem of optimizing the shape and position of the support of the internal exact control of minimal 0, norm for the 1-D wave equation. Yamamoto [5] has studied the inverse problem of determining from subject to the hyperbolic problem , ∆ , , ∈ Ω, 0 , 0 0, , 0 0, ∈ Ω , 0, ∈ Ω, 0 where ∈ 0, .
Benamou [6] has used the domain decomposition method to solve the optimal control problem in the hyperbolic system and has taken the set of admissible control as a convex subset of 0, Ω .
Kim and Pavol [7] have minimized the cost functional The necessary and sufficient conditions for an admissible pair * , * ∈ -ℚ -ℚ , ℚ 0, / 0, to be an optimal pair have given by authors.
Lopez at all. [8] have considered problem of controlling the function , related to the hyperbolic problem Privat et al. [9] have minimized the norm of the control for given initial data in the wave equation defined on 0, / with Homogeneous Dirichlet boundary condition when the control is in at the right hand side of the equation.
Subaşı and Saraç [10] have obtained a minimizer function for the optimal control problem of the initial velocity in a wave equation.
Saraç and Şener [11] have determined the transverse distributed load in Euler-Bernoulli beam problem from of admissible control. The set of admissible controls has been taken as a subspace of the space 56, 78.
Saraç [12] has obtained symbolic and numeric solutions by using the initial velocity as a control function in hyperbolic problem. Şener et al. [14] have explained applications of the Galerkin method to wave equation.
The problem of determining of unknown spatial load distributions in a vibrating Euler-Bernoulli beam from limited measured data has been solved in [16].
Let ? @A be closed, convex subset of 0, . In this study, we consider an optimal control problem for a wave equation with homogeneous Dirichlet boundary conditions, the control being the one from the functions that are at the right hand side of the equation. We determining the unknown function in the closed and convex subset ? @A ⊂ 0, from the target , ; , which correspond to final position using − norm. We are interested in generating Maple ® procedure easy to used for obtaining approximate optimal control. The useful approximate optimal control function is easily obtained in some numeric examples.
We consider the following final optimal control problem: Choose a control ∈ 0, and a corresponding such that the pair , minimizes the functional where E is given target function and I, J and are known functions.
With the choice of the functional in (1), we mentioned the observation of , ; in 0, for the control ∈ 0, . Our aim is to obtain suitable function * which approaches the solution of the problem (2) to desired target E ∈ 0, at the final time = . Another word, we want to find the function * ∈ ? @A such that Here F > 0 is a regularization parameter which ensures both the uniqueness of the solution and a balance between the norms ‖ , ; − E ‖ 9 : *,< and ‖ ‖ 9 : *,< . Detailed information as regards the regularization parameter can be found in [2]. The term ‖ ‖ 9 : is called penalization term; its role is to avoid using too large controls in the minimization of D .
In system (2), the term is considered to be an external force. External forces in this form of separation of variables are important in modelling vibrations. In [5] Yamamoto point out that the system (2) is regarded as approximation to a model for elastic waves from a point dislocation source.
This paper is organized as follows. In section 2, we state the definition for solution of the wave equation considered and give the necessary conditions for the existence and uniqueness of the optimal solution. In section 3, we give Frechet derivative of the cost functional and construct a minimizing sequence that converge to the optimal solution. In the last section, we obtain the approximate solutions on numeric examples.

Existence of Unique Optimal Solution
In this section, we give the solvability of the optimal control problem (1)-(2). First we state the generalized solution of the hyperbolic problem (2) in view of [1].
Definition 2.1. The generalized (weak) solution of the problem (2) will be defined as the function ∈ K * Ω , with , 0 = I , ∈ 0, which satisfies the following integral identity: for all L ∈ K * Ω with L , = 0. To have this solution the followings are needed; ∈ 0, , ∈ 0, , I ∈ K * 0, , J ∈ 0, Theorem 2.2. Suppose that the condition (4) holds, then the problem (2) has a unique generalized solution and the following estimate is valid for this solution; Proof of this theorem can easily be obtained by Galerkin method used in [1].
Since we have ‖∆ 1 ‖ 9 : *,< ≤ ! 5 ∆ 1 + 6 ∆ G 8( which implies the required estimate (7). We can write the cost functional (1) in the following way; Due to the linearity of the transform → 5 8 − 508, it can easily be seen that the functional / , is bilinear and symmetric. Further, we write the following; | / , | ≥ F‖ ‖ 9 : *,< (12) and this implies the coercivity of / , . Since for N =^6 _N , F`. Then / , L is continuous. The functional is linear. We can easily write that Proof of this theorem can easily be obtained by showing the weak lower semi-continuity of D same as in [3].

Frechet Differential of the Cost Functional and Minimizing Sequence
Let us introduce the Lagrangian So, we can state the following theorem in view of [2]. Theorem 3.1. The control * and the state * = * are optimal if there exists a multiplier i * ∈ ? @A such that i * and * satisfy the following optimality conditions: for ∀ ∈ ? @A . Now, we can apply standard steepest descent iteration. We write an iterative procedure to compute a sequence of controls _ p` convergent to the optimal one.
Select an initial control * . If p is known q ≥ 0 then pr is computed according to the following scheme.
for sufficiently small s p > 0. The term o s p is infinite decreasing term with high order respect to s p . Computations of the s p can be carried out by one of the methods shown in [13].
One of the following can be taken as a stopping criterion to the iteration process; ‖ pr − p ‖ < 0 , | D pr − D p | < 0 , ‖ D p ‖ < 0 a . Lemma 3.2. The cost functional (1) is strongly convex with the strong convexity constant F. From the following strongly convex functional definition for z ∈ 50,18: we can see that cost functional (1) is strongly convex the constant { = F. So, we can give the following theorem which states the convergence of the minimizer to optimal solution. Theorem 3.3. Let * be optimum solution of the problem (1)- (2). Then the minimizer given in (20) satisfies the following inequality; Proof of this theorem is obtained by taking z = in the definition of the above strongly convex functional.

Numerical Example
In this section we test the method in a numerical example. The used trigonometric basis functions are chosen such as; The function and its partial derivatives G , 1 belong to Ω . The function , is not a classical solution since GG ∉ Ω . Here the force function is discontinuous.
Rewrite the functional as

Conclusion
In this paper, we show that the external force in the wave equation be controlled by minimizing the distance between final situation distance and the desired target function. By using the adjoint approach in the mathematical analysis of the optimal control problem for wave equation, the gradient of the cost functional can be obtained. The minimizing sequence is constructed via this gradient.