Abstract
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.
Keywords:
stationary equation of quasi optic; boundary value problem; optimal control problem; variational problem1 Introduction
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 quasioptics equation is a form of the Schrödinger equation with complex potential. Such problems were investigated in [15]. Optimal control problem for nonstationary Schrödinger equation of quasi optics was investigated for the first time in [6].
2 Formulation of the problem
We are interested in finding the problem of the minimum of the functional
in the set
under the condition
where
respectively,
The problem of finding a function
Generalized solution of this problem is a function
for
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).
Theorem 1Suppose that a functionfsatisfies the condition (5). So, for each
is valid for
Proof Proof can be done by processes similar to those given in [7]. □
Theorem 2Let us accept that the conditions of Theorem 1 hold and
Proof Firstly, let us show that
is continuous on the set V. Let us take an arbitrary ∈V, and let
Because the problem (10)(12) and the problem (2)(4) are the same type problems, we can write the following estimate the same as (8):
If we use estimate (13) then we can write the following estimate:
Now, let us evaluate the increment of the functional
Using the CauchyBunyakowski inequality and estimates (8) and (14), we write the inequality as
where
3.1 Fréchet diffrentiability of the functional
In this section, we prove the Fréchet differentiability of a given functional. For this purpose, we consider the following adjoint boundary value problem:
Here, the function
As seen, the problem (17)(19) is an initial boundary value problem. This can easily
be obtained by a transform
where
If we write the complex conjugate of this boundary value problem, we obtain the following boundary value problem:
where
This problem is a type of (2)(4) boundary value problem. As the righthand side is
equal to zero, and initial function
If we use the problem (24)(26) as a type of the conjugate problem (17)(19), we
obtain the initial bounded value problem (17)(19) has a unique solution belonging
to the space
Here, the number
Here, the number
Theorem 3Let us accept that the conditions of Theorem 2 hold and
Proof Let us evaluate the increment of the functional
The last formula can be written as follows:
where
Applying the CauchyBunyakowski inequality, we obtain:
If we use estimates (13) and (28) in this inequality, we obtain
Here,
By using equality (33), the increment of the functional can be written as
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.
Theorem 4Accept that the conditions of Theorem 3 hold and
Here, the functions
Proof Now, we prove that the gradient
for
In order to show (36), using the formula
Here,
This bounded value problem is a type of a conjugate problem. For this solution, the following estimate is valid:
Here, the number
Using (13) and (28), we write
Here, the number
and then
If we use estimate (8), we can write the following inequality:
Using this inequality and estimates (13), (28), and (45), we obtain
Here, the number of
If we use inequalities (48) and (49), we see that the correlations limit (36) and (37) is valid.
Now, we prove (38). To prove this using the formula
Here,
If this inequality is used in (49), we easily can write
Similarly, if we use (39), we obtain
We can see that (38) and (39) are valid by inequalities (51) and (52). That is,
for
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
YK carried out the optimal control problem studies, participated in the sequence alignment and drafted the manuscript. EÇ conceived of the study and, participated in its design and coordination. All authors read and approved the final manuscript.
References

İskenderov, AD, Yagubov, GY: Optimal control of nonlinear quantummechanical systems. Autom. Remote Control. 50, 1631–1641 (1989)

İskenderov, AD, Yagubov, GY: A variational method for solving the inverse problem of determining the quantumnmechanical potential. Sov. Math. Doklady (Engl. Trans.) Am. Math. Soc.. 38, 637–641 (1989)

Yagubov, GY, Musayeva, MA: About the problem of identification for nonlinear Schrödinger equation. J. Differ. Equ.. 33(12), 1691–1698 (1997)

Yagubov, GY: Optimal control by coefficient of quasilinear Schrödinger equation. Abstract of these doctors sciences, Kiev, p. 25 (1994)

Yetişkin, H, Subaşı, M: On the optimal control problem for Schrödinger equation with complex potential. Appl. Math. Comput.. 216, 1896–1902 (2010). Publisher Full Text

Yıldız, B, Yagubov, G: On the optimal control problem. J. Comput. Appl. Math.. 88, 275–287 (1997). Publisher Full Text

Ladyzhenskaya, OA, Solonnikov, VA, Uralsteva, NN: Linear and QuasiLinear Equations of Parabolic Type, Am. Math. Soc., Providence (1968)

Goebel, M: On existence of optimal control. Math. Nachr.. 93, 67–73 (1979). Publisher Full Text