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. AksoyG. 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. YagubovF. 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). 

show all references

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. AksoyG. 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. YagubovF. 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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