Year 2016,
Volume: 2 Issue: 2, 67 - 71, 30.09.2016
Abstract
In this paper, difference method is applied to the optimal control problem arising in non-linear optics. Firstly, the difference scheme is established for the problem. Then stability of the difference scheme is given and the error analysis for this scheme is evaluated. Finally, the covergence according to the functional of the difference approximation is proved.
References
- Ladyzenskaja, O.A., Solonnikov, V.A. and Ural’ceva, N.N. (1968) Linear and Quasilinear Equations of Parabolic Type, English trans., Amer. Math. Soc., Providence, RI. [2] Potapov, M.N. and Razgulin, A.V. (1990) ‘The difference methods for optimal control problems of the stationary light beam with self-interaction’, Comput. Math. and Math. Phys., Vol. 30, No. 8, pp.1157–1169, in Russian. [3] Vasilyev, F.P. (1981) Methods of Solving for Extremal Problems, in Russian, Nauka, Moscow . [4] Vorontsov, M.A. and Shmalgauzen, V.I. (1985) The Principles of Adaptive Optics, Izdatel’stvo Nauka, Moscow, in Russian. [5] Yagubov, G.Y. (1994) Optimal Control by Coefficient of the Quasilinear Schrödinger Equation, Thesis Doctora Science, Kiev State University. [6] Yagubov, G.Y. and Musayeva, M.A. (1994) ‘The finite difference method for solution of variational formulation of an inverse problem for nonlinear Schrödinger equation’, Izv. AN. Azerb.- Ser. Physics Tech. Math. Science, Vol. 15, Nos. 5–6, pp.58–61. [7] Yagubov, G.Y. and Musayeva, M.A. (1997) ‘On the identification problem for nonlinear Schrödinger equation’, Differentsial’nye Uravneniya, Vol. 3, No. 12, pp.1691–1698, in Russian. [8] Yıldırım, N., Yagubov, G.Y. and Yıldız B. ‘The finite difference approximations of the optimal control problem for non-linear Schrödinger equation’ Int. J. Mathematical Modelling and Numerical Optimisation, Vol. 3, No. 3, 2012 [9] Fatma Toyoğlu and Gabil Yagub, (2015) ‘ Numerical solution of an Optimal Control Problem Governed by Two Dimensional Schrödinger Equation, Applied and Computational Mathematics, 4(2): 30-38 [10] İbrahimov N. S. Solubilitiy of initial-boundary value problems for linear stationary equation of quasi optic. Journal of Qafqaz University, Vol. 1, No.29, 2010 [11] Yusuf Koçak, Ercan Çelik, Nigar Yildrim Aksoy, (2015) On a Stability Theorem of the Optimal Control Problem For Quasi Optic Equation, Journal of Progressive Research in Mathematics, 5(2), 487-492 [12] Y. Koçak, E. Çelik, Optimal control problem for stationary quasioptic equations, Boundary Value Problems 2012, 2012:151 [13] Y. Koçak, E. Çelik, N. Y. Aksoy, A Note on Optimal Control Problem Governed by Schrödinger Equation, Open Physics. Volume 13, Issue 1, ISSN (Online) 2391-5471 [14] Y. Koçak, M.A. Dokuyucu, E. Çelik, Well-Posedness of Optimal Control Problem for the Schrödinger Equations with Complex Potential, International Journal of Mathematics and Computation, 2015, 26 (4), 11-16 [15] Anil V. Rao , A Survey of Numerıcal Methods for Optimal Control, (Preprint) AAS 09334
Year 2016,
Volume: 2 Issue: 2, 67 - 71, 30.09.2016
Abstract
References
- Ladyzenskaja, O.A., Solonnikov, V.A. and Ural’ceva, N.N. (1968) Linear and Quasilinear Equations of Parabolic Type, English trans., Amer. Math. Soc., Providence, RI. [2] Potapov, M.N. and Razgulin, A.V. (1990) ‘The difference methods for optimal control problems of the stationary light beam with self-interaction’, Comput. Math. and Math. Phys., Vol. 30, No. 8, pp.1157–1169, in Russian. [3] Vasilyev, F.P. (1981) Methods of Solving for Extremal Problems, in Russian, Nauka, Moscow . [4] Vorontsov, M.A. and Shmalgauzen, V.I. (1985) The Principles of Adaptive Optics, Izdatel’stvo Nauka, Moscow, in Russian. [5] Yagubov, G.Y. (1994) Optimal Control by Coefficient of the Quasilinear Schrödinger Equation, Thesis Doctora Science, Kiev State University. [6] Yagubov, G.Y. and Musayeva, M.A. (1994) ‘The finite difference method for solution of variational formulation of an inverse problem for nonlinear Schrödinger equation’, Izv. AN. Azerb.- Ser. Physics Tech. Math. Science, Vol. 15, Nos. 5–6, pp.58–61. [7] Yagubov, G.Y. and Musayeva, M.A. (1997) ‘On the identification problem for nonlinear Schrödinger equation’, Differentsial’nye Uravneniya, Vol. 3, No. 12, pp.1691–1698, in Russian. [8] Yıldırım, N., Yagubov, G.Y. and Yıldız B. ‘The finite difference approximations of the optimal control problem for non-linear Schrödinger equation’ Int. J. Mathematical Modelling and Numerical Optimisation, Vol. 3, No. 3, 2012 [9] Fatma Toyoğlu and Gabil Yagub, (2015) ‘ Numerical solution of an Optimal Control Problem Governed by Two Dimensional Schrödinger Equation, Applied and Computational Mathematics, 4(2): 30-38 [10] İbrahimov N. S. Solubilitiy of initial-boundary value problems for linear stationary equation of quasi optic. Journal of Qafqaz University, Vol. 1, No.29, 2010 [11] Yusuf Koçak, Ercan Çelik, Nigar Yildrim Aksoy, (2015) On a Stability Theorem of the Optimal Control Problem For Quasi Optic Equation, Journal of Progressive Research in Mathematics, 5(2), 487-492 [12] Y. Koçak, E. Çelik, Optimal control problem for stationary quasioptic equations, Boundary Value Problems 2012, 2012:151 [13] Y. Koçak, E. Çelik, N. Y. Aksoy, A Note on Optimal Control Problem Governed by Schrödinger Equation, Open Physics. Volume 13, Issue 1, ISSN (Online) 2391-5471 [14] Y. Koçak, M.A. Dokuyucu, E. Çelik, Well-Posedness of Optimal Control Problem for the Schrödinger Equations with Complex Potential, International Journal of Mathematics and Computation, 2015, 26 (4), 11-16 [15] Anil V. Rao , A Survey of Numerıcal Methods for Optimal Control, (Preprint) AAS 09334
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Details
| Authors | |
|---|---|
| Publication Date | September 30, 2016 |
| IZ | https://izlik.org/JA23NE83GW |
| Published in Issue | Year 2016 Volume: 2 Issue: 2 |