KUAZİ OPTİĞİN DURGUN OLMAYAN DENKLEMİ İÇİN BİR OPTİMAL KONTROL PROBLEMİ

Abstract

In this paper, an optimal control problem for nonstationary quasi-optics equation that show the scattering of the light beam inhomogeneous mediums is considered. In this problem, controls are a refraction and absortion indicators of the scattered medium of the light beam. As a cost fuctional is used the Lions functional that is based on Dirichlet- Neumann operator. For the considered optimal control problem, the existence and uniqueness theorems are obtained. Frechet-differentiability of the cost functional is shown and its gradient is obtained. Finally, a necessary optimality condition in variational inequality form for the optimal control problem is given.

Keywords

• Kuazi optiğin durgun olmayan denklemi,kompleks değerli kontrol,optimal kontrol problemi

References

1. 1. Vorontsov M. A. and Shmalgauzen V. I., The Principles of Adaptive Optics, Izdatel’stvo Nauka, Moscow, 1985 (in Russian).
2. 2. Yagubov G.Y., Ibrahimov N.S. The optimal control problem for a time-dependent equation of quasi-optics I, Sat: "Problems of Math. and wholesale. Management ", Baku, 2001, pp. 49-57
3. 3. Ibrahimov N.S., On the existence of solutions for the identification problem for the unsteady boundary observation equation quasioptics, Bulletin of the Lankaran State. Univ. Ser. Science, 2010, Lankaran, p. 27-44.
4. 4. Ibrahimov N.S., The task of identifying non-stationary equation for the quasi-optics, Tauride Journal of Computer Science and Mathematics, 2010, № 2, p. 45-55.
5. 5. Ibrahimov N.S., On a problem of identification on the final observation for linear time-dependent equation quasioptics, Journal of Computational and Applied Mathematics, Kiev. Zap them. Shevchenko, 2010, № 4, p. 26-37. 6. Lions J.-L., Optimal control of systems governed by partial differential equations, Springer-Verlag, 1972. p 416 .
6. 7. Iskenderov A.D., Matematiksel fiziğin çok boyutlu denklemleri için ters problemlerin varyasyonel formülasyonu, Dokl. АN SSSR, 1984, c. 274, № 3, s.531-533.
7. 8. Iskenderov A.D . Mahmudov N.M., Kuantum Mekanik sistemlerin Lions fonksiyoneli kriterli optimal kontrolü, - Izv.AN. Azerb. SSR, Serie fiz. tekn. mat. Bilimleri, 1995,c. XVI, №5-6, s. 30-35.
8. 9. Mahmudov N.M., Reel katsayılı Schrödinger denklemi için bir optimal kontrol problemi, Izv. Vuzov., 2010, № 11, s. 31-40.

9. 10. Ladyzhenskaya O.A. “ The Boundary value problems of mathematical physics” Springer Verlag, (1985).
10. 11. Iskenderov A.D., Ibrahimov N.S., The initial-boundary value problems for the unsteady equation quasioptics, Bulletin of the Lankaran State. Univ. Ser. Science, 2009, Lankaran, p. 47-66
11. 12. Yoshida K., Functional analysis, Springer-Verlag, 1967.
12. 13. Goebel M., On existence of optimal control, Math. Nachr.,1979, vol.93(1), pp. 67-73.
13. 14.Vasilyev F.P., Methods of solving for extremal problems, Nauka, Moscow, 1981, p. 400.