Optimal control problem for stationary quasi-optic equations

Koçak Yusuf, Çelik, Çelik Ercan

Description

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.

Sujets

• Boundary value problem

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 18

Extrait

Koçak and ÇelikBoundary Value Problems2012,2012:151 http://www.boundaryvalueproblems.com/content/2012/1/151

R E S E A R C H

Optimal control problem quasi-optic equations 1* 2 Yusuf Koçak and Ercan Çelik

* Correspondence: [e-mail] 1 DepartmentofMathematics,A ˘ grı ˙ Ibrahim Çeçen University Faculty of Science and Art, Ag˘ rı, Turkey Full list of author information is available at the end of the article

for stationary

Open Access

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 problem

1 Introduction Optimal control theory for the quantum mechanic systems described with the Schrö- dinger 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 [–]. Optimal control problem for nonsta- tionary Schrödinger equation of quasi optics was investigated for the first time in [].

2 Formulation of the problem We are interested in finding the problem of the minimum of the functional

    J α (v) =ψ(∙,L) –y+αv–ω H L(,l) 

in the set

  V≡v= (v  ,v  ,ϕ  ,ϕ  ),v m ∈L  (,l),v  (z)≥, ∀ z∈(,L),  v,l),ϕ m L  (,l) ≤b m ,ϕ m ∈L  ( m L  (,l) ≤d m ,m= , 

under the condition

 ∂ψ ∂ ψ i+a  +v  (z)ψ+iv  (z)ψ=f(x,z),  ∂z∂x ψ(x, ) =ϕ(x) =ϕ  (x) +iϕ  (x),x∈(,l),

ψ(,z) =ψ(l,z) = ,

z∈(,L),

(x,z)∈,

()

()

()

()

©2012 Koçak and Çelik; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.