# American Institute of Mathematical Sciences

June  2019, 12(3): 503-512. doi: 10.3934/dcdss.2019033

## On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation

 1 Iǧdır University, Faculty of Science and Art, Department of Mathematics, Iǧdır, Turkey 2 Kafkas University, Faculty of Science and Art, Department of Mathematics, Kars, Turkey 3 Atatürk University, Faculty of Science, Department of Mathematics, Erzurum, Turkey

* Corresponding author: gokce.kucuk@igdir.edu.tr

Received  March 2017 Revised  July 2017 Published  September 2018

In this paper, an optimal control problem for Schrödinger equation with complex coefficient which contains gradient is examined. A theorem is given that states the existence and uniqueness of the solution of the initial-boundary value problem for Schrödinger equation. Then for the solution of the optimal control problem, two different cases are investigated. Firstly, it is shown that the optimal control problem has a unique solution for $α >0$ on a dense subset $G$ on the space $H$ which contains the measurable square integrable functions on $\left(0,l\right)$ and secondly the optimal control problem has at least one solution for any $α ≥ 0$ on the space $H$.

Citation: Gökçe Dİlek Küçük, Gabil Yagub, Ercan Çelİk. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 503-512. doi: 10.3934/dcdss.2019033
##### References:
 [1] A. Abdon and D. Baleanu, Application of Fixed Point Theorem for Stability Analysis of a Nonlinear Schrodinger with Caputo-Liouville Derivative, Filomat, 31 (2017), 2243-2248.  doi: 10.2298/FIL1708243A. [2] N. Y. Aksoy, G. Yagubov and B. Yildiz, The finite difference approximations of the optimal control problem for nonlinear Schrodinger equation, International Journal of Mathematical Modeling and Numerical Optimization, 3 (2012), 158-183. [3] A. G. Butkovsky and Yu. I. Samoylenko, Control of Quantum-Mechanical Process and Systems, Kluwer Academic, Dordrecht, 1990. doi: 10.1007/978-94-009-1994-5. [4] M. Goebel, On Existence of optimal control, Math. Nachr., 53 (1979), 67-73.  doi: 10.1002/mana.19790930106. [5] D. N. Hao, Optimal control of quantum systems, Avtomatika i Telemechanika, 2 (1986), 14-21; (Russian)Automation and Remote Control, 47 (1986), 162-168. [6] K. Iosida, Functional Analysis, Mir, 1967, (in Russian). [7] A. D. Iskenderov and G. Ya. Yagubov, A variational method for solving inverse problem of determining the quantum mechanical potential, (Russian)Sov. Math. Doklady, 303 (1988), 1044-1048; Am. Math. Soc. , 38 (1989), 637-641. [8] A. D. Iskenderov and G. Ya. Yagubov, Optimal control of nonlinear quantum mechanical systems, Autom.Telemech., 12 (1989), 27-38. [9] A. D. Iskenderov, Definition of potantial in nonstationary Schrodinger equation, Mathematical Simulation and Optimal Control Problems, (2001), 6-36. [10] A. D. Iskenderov, On variational formulations of multidimensional inverse problems of mathematical physics, Dokl. Akad. Nauk SSSR, 274 (1984), 531-533. [11] A. D. Iskenderov and N. M. Makhmudov, Optimal control of a quantum mechanical system with the Lions quality criterion, Izv. Akad. Nauk Azerb. Ser. Fiz.-Tekh. Mat. Nauk, 16 (1995), 30-35. [12] O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, Nauka, Moscow, 1973 (in Russian). [13] J. L. Lions, Contrôle Des Systèmes Distribués Singuliers, Gauthier Villars, Paris, 1983. [14] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, New York, 1971. [15] N. M. Makhmudov, A difference method for solving an optimal control problem for the Schrödinger equation with the Lions quality criterion, Izv. Chelyabinsk. Nauchn. Tsentra, 3 (2009), 1-6 (in Russian). [16] N. M. Makhmudov, On an optimal control problem for the Schrödinger equation with a real-valued coefficient, Izv. Vyssh. Uchebn. Zaved. Mat., 11 (2010), 31-40 (in Russian).  doi: 10.3103/S1066369X10110034. [17] A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-posed Problems, Winston & Sons, Washington, 1977. [18] G. Yagubov, F. Toyoglu and M. Subasi, An optimal control problem for two-dimensional Schrodinger Equation, Applied Mathematics and Computation, 218 (2012), 6177-6187.  doi: 10.1016/j.amc.2011.12.028. [19] G. Y. Yagubov and M. A. Musayeva, On the identification problem for nonlinear Schrodinger equation, Differential Equation, 3 (1997), 1691-1698 (in Russian).

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##### References:
 [1] A. Abdon and D. Baleanu, Application of Fixed Point Theorem for Stability Analysis of a Nonlinear Schrodinger with Caputo-Liouville Derivative, Filomat, 31 (2017), 2243-2248.  doi: 10.2298/FIL1708243A. [2] N. Y. Aksoy, G. Yagubov and B. Yildiz, The finite difference approximations of the optimal control problem for nonlinear Schrodinger equation, International Journal of Mathematical Modeling and Numerical Optimization, 3 (2012), 158-183. [3] A. G. Butkovsky and Yu. I. Samoylenko, Control of Quantum-Mechanical Process and Systems, Kluwer Academic, Dordrecht, 1990. doi: 10.1007/978-94-009-1994-5. [4] M. Goebel, On Existence of optimal control, Math. Nachr., 53 (1979), 67-73.  doi: 10.1002/mana.19790930106. [5] D. N. Hao, Optimal control of quantum systems, Avtomatika i Telemechanika, 2 (1986), 14-21; (Russian)Automation and Remote Control, 47 (1986), 162-168. [6] K. Iosida, Functional Analysis, Mir, 1967, (in Russian). [7] A. D. Iskenderov and G. Ya. Yagubov, A variational method for solving inverse problem of determining the quantum mechanical potential, (Russian)Sov. Math. Doklady, 303 (1988), 1044-1048; Am. Math. Soc. , 38 (1989), 637-641. [8] A. D. Iskenderov and G. Ya. Yagubov, Optimal control of nonlinear quantum mechanical systems, Autom.Telemech., 12 (1989), 27-38. [9] A. D. Iskenderov, Definition of potantial in nonstationary Schrodinger equation, Mathematical Simulation and Optimal Control Problems, (2001), 6-36. [10] A. D. Iskenderov, On variational formulations of multidimensional inverse problems of mathematical physics, Dokl. Akad. Nauk SSSR, 274 (1984), 531-533. [11] A. D. Iskenderov and N. M. Makhmudov, Optimal control of a quantum mechanical system with the Lions quality criterion, Izv. Akad. Nauk Azerb. Ser. Fiz.-Tekh. Mat. Nauk, 16 (1995), 30-35. [12] O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, Nauka, Moscow, 1973 (in Russian). [13] J. L. Lions, Contrôle Des Systèmes Distribués Singuliers, Gauthier Villars, Paris, 1983. [14] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, New York, 1971. [15] N. M. Makhmudov, A difference method for solving an optimal control problem for the Schrödinger equation with the Lions quality criterion, Izv. Chelyabinsk. Nauchn. Tsentra, 3 (2009), 1-6 (in Russian). [16] N. M. Makhmudov, On an optimal control problem for the Schrödinger equation with a real-valued coefficient, Izv. Vyssh. Uchebn. Zaved. Mat., 11 (2010), 31-40 (in Russian).  doi: 10.3103/S1066369X10110034. [17] A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-posed Problems, Winston & Sons, Washington, 1977. [18] G. Yagubov, F. Toyoglu and M. Subasi, An optimal control problem for two-dimensional Schrodinger Equation, Applied Mathematics and Computation, 218 (2012), 6177-6187.  doi: 10.1016/j.amc.2011.12.028. [19] G. Y. Yagubov and M. A. Musayeva, On the identification problem for nonlinear Schrodinger equation, Differential Equation, 3 (1997), 1691-1698 (in Russian).
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