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 space After calculating gradient of the cost functional and proving the Lipschitz continuity of the gradient, necessary condition for optimal solution is constructed.
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 . There have been many studies about optimal control for hyperbolic systems which are considered [2–4].
Some of these important studies can be summarized as follows.
Hasanov  has considered problem of controlling the function for the following problem:with the conditionsusing the functionalsMajewski  has controlled the function for hyperbolic equation:using the functional Yeloğlu and Subaşı  have dealt with determination pair in the following problem:for the functionalKröner  has specified the function for nonlinear hyperbolic equation:using the functionalTagiyev  has studied the problem of controlling the coefficients for linear hyperbolic equation:using the functional
2. Statement of the Problem
In this study, we deal with the process of vibration in finite homogeneous string, occupying the interval . 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.
In the domain , we consider the functionalon the setsubject to the hyperbolic problemHere is the desired target function to which must be close enough. The function is an initial guess for optimal solution. is regularization parameter. are given positive numbers.
Namely, we want to control the transverse elastic force on the space and the solution corresponding to this control function must be close enough to in . In order to get a stable solution, we choose the space which is more regular than .
The inner product and norm in are defined, respectively, asThe 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 , 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.
3. Solvability of the Problem
In this section, we first give the definition of the generalized solution for hyperbolic problem.
It can be seen in  that solution in the sense of (20) exists, is unique, and satisfies the following inequality:where and orSince and are given functions, it can be written as follows:Now, we give an increment to the control function such as Then the difference function is the solution of the following difference initial-boundary problem:By considering (23), we obtain that the solution of above difference initial-boundary problem holds the following inequality:Here is independent from .
4. Existence and Uniqueness of the Optimal Solution
Theorem 1. Let be a uniformly convex Banach space and the set be a closed, bounded, and convex subset of . If and are given numbers and the functional is lower semicontinuous and bounded from below on the set , then there is a dense set of that the functionaltakes its minimum on the set for . If then minimum is unique.
Before showing that these conditions have been satisfied, we prove that the functional is continuous. For this, we write the following increment of the functional:Since , if we consider inequalities (22) and (27), we conclude that this increment satisfies the following continuity inequality on the set :Here 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 is a uniformly convex Banach space , the set is a closed, bounded, and convex subset of , and .
Therefore the conditions of above theorem hold and optimal solution to the problem (18) is unique.
5. 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 isThe first variation of this functional according to the function is obtained such asBy means of stationary condition , the following adjoint boundary value problem is found:For , the function which satisfies the following equalityis the solution of adjoint boundary value problem (34)-(36).
This solution satisfies the following inequality:Now, we can pass the calculation of the gradient. In order to do this, we must evaluate the increment of the functional . The increment can be written such asThe difference problem (24)-(26) and the adjoint problem (34)-(36) give together the equality ofInserting (40) in (39), we haveBy (27) and (38), the second and third integrals of the above equality give the following inequality: The statement (41) can be rewritten asorIn order to pass the inner product in , we rearrange (44) such asHere function is the solution of the second adjoint problem:Therefore, we have the following gradient:
6. 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.
Theorem 2. Gradient satisfies the following Lipschitz inequality:Here 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 .
Proof. Increment of the functional by giving the increment of to the control is obtained:where the function is the solution of the increment problem:Taking the norm of (49) in the space , we acquire the following inequality belonging to the functional :There is a solution of problem (50) in and this solution satisfies the following inequality:The function in the right hand side of inequality (52) is the solution of the following problem:and this function holds the following inequality:Here 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:If inequalities (27), (38), (55), and (56) about functions , , , and are written in (57), then the following assessment is obtained:Here is independent of .
Considering inequality (58), the following is written: So the following inequality for the gradient is obtained:Once we take as , then the proof is obtained.
7. 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 . 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 :Therefore, the following inequality is written for optimal control problem:
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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