On the Control of Coefficient Function in a Hyperbolic Problem with Dirichlet Conditions

This paper presents theoretical results about control of the coefficient function in a hyperbolic problem with Dirichlet conditions. The existence and uniqueness of the optimal solution for optimal control problem are proved and adjoint problem is used to obtain gradient of the functional. However, a second adjoint problem is given to calculate the gradient on the spaceW1 2 (0, l). After calculating gradient of the cost functional and proving the Lipschitz continuity of the gradient, necessary condition for optimal solution is constructed.


Introduction
Hyperbolic boundary value problems have appeared as mathematical modelling of physical phenomena like small vibration of a string, in the fields of science and engineering. There has been much attention to studies related to optimal control problems involving hyperbolic problems [1]. There have been many studies about optimal control for hyperbolic systems which are considered [2][3][4].
Some of these important studies can be summarized as follows.

Statement of the Problem
In this study, we deal with the process of vibration in finite homogeneous string, occupying the interval (0, ). As the control function, we take the transverse elastic force which is in the coefficient of the vibration problem. Also, we propose the usage of a more regular space than the space of square integrable functions in the cost functional. In general, this process exposes some difficulties in the stage of acquiring the gradient. This study offers a second adjoint problem to overcome this case.
The initial status functions are in the following spaces: The aim of this study is to deal with the problem of * = inf under conditions (12)-(17).
International Journal of Differential Equations 3 Namely, we want to control the transverse elastic force on the space 1 2 (0, ) and the solution ( , ) corresponding to this control function must be close enough to ( ) in 2 (0, ). In order to get a stable solution, we choose the space 1 2 (0, ) which is more regular than 2 (0, ).
The inner product and norm in 1 2 (0, ) are defined, respectively, as The paper is organized as follows: in Section 3, we obtain the generalized solution for hyperbolic problem. In Section 4, we prove the existence and uniqueness of the optimal solution. In Section 5, we obtain the adjoint problem for the optimal control problem and find the gradient of the functional. The main contribution of this paper is executed in this section.
Because the controls are chosen in the space 1 2 (0, ), getting the gradient of the functional necessitates finding a second adjoint problem. In the last section, we demonstrate the Lipschitz continuity of the gradient and state the necessary condition for optimal solution.

Solvability of the Problem
In this section, we first give the definition of the generalized solution for hyperbolic problem.
The generalized solution of problem (14)-(15) is the function ∈ o 1,1 2 (Ω) satisfying the following integral equality: for ∀ ∈ o 1,1 2 (Ω), ( , ) = 0. It can be seen in [10] that solution in the sense of (20) exists, is unique, and satisfies the following inequality: Since 1 and 2 are given functions, it can be written as follows: By considering (23), we obtain that the solution of above difference initial-boundary problem holds the following inequality: Here 4 = ( 3 2 /3) 3 is independent from .

Existence and Uniqueness of the Optimal Solution
To demonstrate the existence and the uniqueness of optimal solution for problem (12)- (17), it is enough to show that conditions of the following theorem given by Goebel [11] hold. takes its minimum on the set for ∀ ∈ . If > 1 then minimum is unique.
Before showing that these conditions have been satisfied, we prove that the functional  Since ( ) ∈ 2 (0, ), if we consider inequalities (22) and (27), we conclude that this increment satisfies the following continuity inequality on the set : Here 5 is independent of . Thanks to this inequality, we can say that this functional is also lower semicontinuous and bounded from below on the set .
On the other hand, the set 1 2 (0, ) is a uniformly convex Banach space [12], the set is a closed, bounded, and convex subset of 1 2 (0, ), and = 2. Therefore the conditions of above theorem hold and optimal solution to the problem (18) is unique.

Adjoint Problem and Gradient of the Functional
In this section, we write the Lagrange functional used for finding adjoint problem, before we show the Frechet differentiability of the functional ( ) on the set . Lagrange functional to the problem is The difference problem (24)-(26) and the adjoint problem Inserting (40) in (39), we have By (27) and (38), the second and third integrals of the above equality give the following inequality: The statement (41) can be rewritten as  (46) Therefore, we have the following gradient: (47)

Lipschitz Continuity of the Gradient
In this section, we introduce a theorem about Lipschitz continuity of the gradient. By this means, we can express the necessary condition for optimal solution.
Here 8 is independent from .
Hence, it has been proven that the gradient ( ) is continuous on the set and it can be seen that it holds the Lipschitz condition with constant 8 > 0.
Proof. Increment of the functional ( ) by giving the increment of to the control ∈ is obtained: Taking the norm of (49) in the space 1 2 (0, ), we acquire the following inequality belonging to the functional ( ): ( ) There is a solution of problem (50) in 1 2 (0, ) and this solution satisfies the following inequality: The function in the right hand side of inequality (52) is the solution of the following problem: Here 9 is independent of . So, the function that takes place in the right hand side of (52) holds the same inequality given as follows: Hence inequality (52) has the following property: ( ) Here 10 is independent of . Considering inequality (58), the following is written: So the following inequality for the gradient ( ) is obtained: ( + ) − ( ) Once we take as 8 = 11 , then the proof is obtained.

6
International Journal of Differential Equations

The Necessary Condition for Optimal Solution
After showing Lipschitz continuity of the gradient, it can be said that the gradient ( ) is continuous on the set and it holds the Lipschitz constant 8 > 0. The fact that the functional ( ) is continuously differentiable on the set and the set is convex, in that case the following inequality is valid according to theorem in [13]: Therefore, the following inequality is written for optimal control problem: ⟨ + 2 ( * − ) , − * ⟩ 1 2 (0, ) ≥ 0, ∀ ∈ .