KUAZİ OPTİĞİN DURGUN OLMAYAN DENKLEMİ İÇİN BİR OPTİMAL KONTROL PROBLEMİ

Year 2014, Volume: 1 Issue: 1, 73 - 80, 31.12.2014

Gabil Yagub Nigar Yıldırım Aksoy , Eray Aksoy

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. Vorontsov M. A. and Shmalgauzen V. I., The Principles of Adaptive Optics, Izdatel’stvo Nauka, Moscow, 1985 (in Russian).
• 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. 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. 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. 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 .
• 7. Iskenderov A.D., Matematiksel fiziğin çok boyutlu denklemleri için ters problemlerin varyasyonel formülasyonu, Dokl. АN SSSR, 1984, c. 274, № 3, s.531-533.
• 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.
• 9. Mahmudov N.M., Reel katsayılı Schrödinger denklemi için bir optimal kontrol problemi, Izv. Vuzov., 2010, № 11, s. 31-40.
• 10. Ladyzhenskaya O.A. “ The Boundary value problems of mathematical physics” Springer Verlag, (1985).
• 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
• 12. Yoshida K., Functional analysis, Springer-Verlag, 1967.
• 13. Goebel M., On existence of optimal control, Math. Nachr.,1979, vol.93(1), pp. 67-73.
• 14.Vasilyev F.P., Methods of solving for extremal problems, Nauka, Moscow, 1981, p. 400.