Optimal control problem for stationary quasi-optic equations
© Koçak and Çelik; licensee Springer 2012
Received: 1 October 2012
Accepted: 30 November 2012
Published: 28 December 2012
In this paper, an optimal control problem was taken up for a stationary equation of quasi optic. For the stationary equation of quasi optic, at first judgment relating to the existence and uniqueness of a boundary value problem was given. By using this judgment, the existence and uniqueness of the optimal control problem solutions were proved. Then we state a necessary condition to an optimal solution. We proved differentiability of a functional and obtained a formula for its gradient. By using this formula, the necessary condition for solvability of the problem is stated as the variational principle.
Keywordsstationary equation of quasi optic boundary value problem optimal control problem variational problem
Optimal control theory for the quantum mechanic systems described with the Schrödinger equation is one of the important areas of modern optimal control theory. Actually, a stationary quasi-optics equation is a form of the Schrödinger equation with complex potential. Such problems were investigated in [1–5]. Optimal control problem for nonstationary Schrödinger equation of quasi optics was investigated for the first time in .
2 Formulation of the problem
respectively, and . is a Hilbert space that consists of all functions in , which are measurable and square-integrable. is the well-known Lebesgue space consisting of all functions in Ω, which are measurable and square-integrable.
The problem of finding a function under the condition (2)-(4) for each , which is a boundary value problem, is a function for Eq. (2).
3 Existence and uniqueness of a solution of the optimal control problem
In this section, we prove the optimal control problem using the Galerkin method and the existence and uniqueness of a solution of the problem (1)-(4).
is valid for . Here, the number is independent of z.
Proof Proof can be done by processes similar to those given in . □
Theorem 2 Let us accept that the conditions of Theorem 1 hold and is a given function. Then there is such a set G dense in that the optimal control problem (1)-(4) has a unique solution and .
is constant that does not depend on Δv.
where is a constant that does not depend on Δv. This inequality shows that the functional is continuous on the set V. On the other hand, for ; therefore, is bounded on V. The set V is closed, bounded on a Hilbert space H. According to Theorem (Goebel) in , there is such a set G dense in H that optimalcontrol problem (1)-(4) has a unique solution for and . Theorem 2 is proven. □
3.1 Fréchet diffrentiability of the functional
Here, the number is constant.
Considering this equality (34), and by using the definition of Fréchet differentiable, we can easily obtain the validity of the rule. Theorem 3 is proved. □
3.2 A necessary condition for an optimal solution
In this section, we prove the continuity of a gradient and state a necessary condition to an optimal solution in the variational inequality form using the gradient.
Here, the functions , are solutions of the problems (2)-(4) and a conjugate problem corresponding to , respectively.
Here, the number is constant.
If we use inequalities (48) and (49), we see that the correlations limit (36) and (37) is valid.
for . Here, using (29), it is seen that the statement of Theorem 4 is valid. □
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