American Institute of Mathematical Sciences

Advanced
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

Gökçe Dİlek Küçük 1,, , Gabil Yagub 2, and  Ercan Çelİk 3,

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 $H$.

Keywords: Optimal control problem, Lions functional, Schrödinger equation.
Mathematics Subject Classification: Primary: 49N05, 34A12.
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 & 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

M. Goebel, On Existence of optimal control, Math. Nachr., 53 (1979), 67-73.  doi: 10.1002/mana.19790930106.  Google Scholar

[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. Google Scholar

[6]

K. Iosida, Functional Analysis, Mir, 1967, (in Russian).  Google Scholar

[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.  Google Scholar

[8]

A. D. Iskenderov and G. Ya. Yagubov, Optimal control of nonlinear quantum mechanical systems, Autom.Telemech., 12 (1989), 27-38.   Google Scholar

[9]

A. D. Iskenderov, Definition of potantial in nonstationary Schrodinger equation, Mathematical Simulation and Optimal Control Problems, (2001), 6-36.   Google Scholar

[10]

A. D. Iskenderov, On variational formulations of multidimensional inverse problems of mathematical physics, Dokl. Akad. Nauk SSSR, 274 (1984), 531-533.   Google Scholar

[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.   Google Scholar

[12]

O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, Nauka, Moscow, 1973 (in Russian).  Google Scholar

[13]

J. L. Lions, Contrôle Des Systèmes Distribués Singuliers, Gauthier Villars, Paris, 1983.  Google Scholar

[14]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, New York, 1971.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[17]

A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-posed Problems, Winston & Sons, Washington, 1977.  Google Scholar

[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.  Google Scholar

[19]

G. Y. Yagubov and M. A. Musayeva, On the identification problem for nonlinear Schrodinger equation, Differential Equation, 3 (1997), 1691-1698 (in Russian).   Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

M. Goebel, On Existence of optimal control, Math. Nachr., 53 (1979), 67-73.  doi: 10.1002/mana.19790930106.  Google Scholar

[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. Google Scholar

[6]

K. Iosida, Functional Analysis, Mir, 1967, (in Russian).  Google Scholar

[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.  Google Scholar

[8]

A. D. Iskenderov and G. Ya. Yagubov, Optimal control of nonlinear quantum mechanical systems, Autom.Telemech., 12 (1989), 27-38.   Google Scholar

[9]

A. D. Iskenderov, Definition of potantial in nonstationary Schrodinger equation, Mathematical Simulation and Optimal Control Problems, (2001), 6-36.   Google Scholar

[10]

A. D. Iskenderov, On variational formulations of multidimensional inverse problems of mathematical physics, Dokl. Akad. Nauk SSSR, 274 (1984), 531-533.   Google Scholar

[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.   Google Scholar

[12]

O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, Nauka, Moscow, 1973 (in Russian).  Google Scholar

[13]

J. L. Lions, Contrôle Des Systèmes Distribués Singuliers, Gauthier Villars, Paris, 1983.  Google Scholar

[14]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, New York, 1971.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[17]

A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-posed Problems, Winston & Sons, Washington, 1977.  Google Scholar

[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.  Google Scholar

[19]

G. Y. Yagubov and M. A. Musayeva, On the identification problem for nonlinear Schrodinger equation, Differential Equation, 3 (1997), 1691-1698 (in Russian).   Google Scholar

[1]

Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022
[2]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303
[3]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032
[4]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052
[5]

Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125
[6]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[7]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376
[8]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[9]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
[10]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284
[11]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264
[12]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019
[13]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[14]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298
[15]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[16]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121
[17]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294
[18]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020298
[19]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316
[20]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (150)
  • HTML views (171)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences