Numerical Solution of an Optimal Control Problem Governed by Two Dimensional Schrodinger Equation

Fatma Toyoglu; Gabil Yagubov

doi:doi:10.11648/j.acm.20150402.11

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Numerical Solution of an Optimal Control Problem Governed by Two Dimensional Schrodinger Equation

Fatma Toyoglu, Gabil Yagubov

Published in Applied and Computational Mathematics (Volume 4, Issue 2)

Received: 11 February 2015 Accepted: 26 February 2015 Published: 4 March 2015

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Abstract

In this study, the finite difference method is applied to an optimal control problem controlled by two functions which are in the coefficients of two-dimensional Schrodinger equation. Convergence of the finite difference approximation according to the functional is proved. We have used the implicit method for solving the two-dimensional Schrodinger equation. Although the implicit scheme obtained from solution of the system of the linear equations is generally numerically stable and convergent without time-step condition, the solution of considered equation is numerically stable with time-step condition, due to the gradient term.

Published in	Applied and Computational Mathematics (Volume 4, Issue 2)
DOI	10.11648/j.acm.20150402.11
Page(s)	30-38
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Optimal Control, Schrodinger Operator, Finite Difference Methods, Stability, Convergence of Numerical Methods

References

[1]	H. Yetişkin, M. Subaşı, “On the optimal control problem for Schrödinger equation with complex potential,” Applied Mathematics and Computation, 216, 1896-1902, 2010.
[2]	B. Yıldız, O. Kılıçoğlu, G. Yagubov, “Optimal control problem for non stationary Schrödinger equation,” Numerical Methods for Partial Differential Equations, 25, 1195-1203, 2009.
[3]	K. Beauchard, C. Laurent, “Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control,” Journal de Mathematiques Pures et Appliquees, 94 (5), 520-554, 2010.
[4]	B. Yıldız, M. Subaşı, “On the optimal control problem for linear Schrödinger equation. Applied Mathematics and Computation,” 121, 373-381, 2001.
[5]	L. Baudouin, O. Kavian, J. P. Puel, “Regularity for a Schrödinger equation with singuler potentials and application to bilinear optimal control,” Journal Differential Equations, 216, 188-222, 2005.
[6]	G. Ya. Yagubov, M. A. Musayeva, “Finite-difference method solution of variation formulation of an inverse problem for nonlinear Schrodinger equation,” Izv. AN Azerb.-Ser. Physictex. matem. nauk, vol.16, No 1-2,46-51, 1995.
[7]	G. D. Smith,“Numerical Solution of Partial Differential Equations,” Oxford University Press, 1985.
[8]	J. V. Thomas, “Numerical Partial Differential Equations,” Springer- Verlag, 1995.
[9]	O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, “Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs,” American Mathematical Society, Rhode Island, 1968.
[10]	F. P. Vasilyev, “Numerical Methods for Extremal Problems,” Nauka, Moskow, 1981.

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Fatma Toyoglu, Gabil Yagubov. (2015). Numerical Solution of an Optimal Control Problem Governed by Two Dimensional Schrodinger Equation. Applied and Computational Mathematics, 4(2), 30-38. https://doi.org/10.11648/j.acm.20150402.11

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ACS Style

Fatma Toyoglu; Gabil Yagubov. Numerical Solution of an Optimal Control Problem Governed by Two Dimensional Schrodinger Equation. Appl. Comput. Math. 2015, 4(2), 30-38. doi: 10.11648/j.acm.20150402.11

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AMA Style

Fatma Toyoglu, Gabil Yagubov. Numerical Solution of an Optimal Control Problem Governed by Two Dimensional Schrodinger Equation. Appl Comput Math. 2015;4(2):30-38. doi: 10.11648/j.acm.20150402.11

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@article{10.11648/j.acm.20150402.11,
  author = {Fatma Toyoglu and Gabil Yagubov},
  title = {Numerical Solution of an Optimal Control Problem Governed by Two Dimensional Schrodinger Equation},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {2},
  pages = {30-38},
  doi = {10.11648/j.acm.20150402.11},
  url = {https://doi.org/10.11648/j.acm.20150402.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150402.11},
  abstract = {In this study, the finite difference method is applied to an optimal control problem controlled by two functions which are in the coefficients of two-dimensional Schrodinger equation. Convergence of the finite difference approximation according to the functional is proved. We have used the implicit method for solving the two-dimensional Schrodinger equation. Although the implicit scheme obtained from solution of the system of the linear equations is generally numerically stable and convergent without time-step condition, the solution of considered equation is numerically stable with time-step condition, due to the gradient term.},
 year = {2015}
}

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TY  - JOUR
T1  - Numerical Solution of an Optimal Control Problem Governed by Two Dimensional Schrodinger Equation
AU  - Fatma Toyoglu
AU  - Gabil Yagubov
Y1  - 2015/03/04
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150402.11
DO  - 10.11648/j.acm.20150402.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 30
EP  - 38
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150402.11
AB  - In this study, the finite difference method is applied to an optimal control problem controlled by two functions which are in the coefficients of two-dimensional Schrodinger equation. Convergence of the finite difference approximation according to the functional is proved. We have used the implicit method for solving the two-dimensional Schrodinger equation. Although the implicit scheme obtained from solution of the system of the linear equations is generally numerically stable and convergent without time-step condition, the solution of considered equation is numerically stable with time-step condition, due to the gradient term.
VL  - 4
IS  - 2
ER  -

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Author Information

Fatma Toyoglu

Department of Mathematics, Faculty of Art and Science, Erzincan University, Erzincan, Turkey
Gabil Yagubov

Department of Mathematics, Faculty of Art and Science, Erzincan University, Erzincan, Turkey

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APA Style

Fatma Toyoglu, Gabil Yagubov. (2015). Numerical Solution of an Optimal Control Problem Governed by Two Dimensional Schrodinger Equation. Applied and Computational Mathematics, 4(2), 30-38. https://doi.org/10.11648/j.acm.20150402.11

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ACS Style

Fatma Toyoglu; Gabil Yagubov. Numerical Solution of an Optimal Control Problem Governed by Two Dimensional Schrodinger Equation. Appl. Comput. Math. 2015, 4(2), 30-38. doi: 10.11648/j.acm.20150402.11

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AMA Style

Fatma Toyoglu, Gabil Yagubov. Numerical Solution of an Optimal Control Problem Governed by Two Dimensional Schrodinger Equation. Appl Comput Math. 2015;4(2):30-38. doi: 10.11648/j.acm.20150402.11

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@article{10.11648/j.acm.20150402.11,
  author = {Fatma Toyoglu and Gabil Yagubov},
  title = {Numerical Solution of an Optimal Control Problem Governed by Two Dimensional Schrodinger Equation},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {2},
  pages = {30-38},
  doi = {10.11648/j.acm.20150402.11},
  url = {https://doi.org/10.11648/j.acm.20150402.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150402.11},
  abstract = {In this study, the finite difference method is applied to an optimal control problem controlled by two functions which are in the coefficients of two-dimensional Schrodinger equation. Convergence of the finite difference approximation according to the functional is proved. We have used the implicit method for solving the two-dimensional Schrodinger equation. Although the implicit scheme obtained from solution of the system of the linear equations is generally numerically stable and convergent without time-step condition, the solution of considered equation is numerically stable with time-step condition, due to the gradient term.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Numerical Solution of an Optimal Control Problem Governed by Two Dimensional Schrodinger Equation
AU  - Fatma Toyoglu
AU  - Gabil Yagubov
Y1  - 2015/03/04
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150402.11
DO  - 10.11648/j.acm.20150402.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 30
EP  - 38
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150402.11
AB  - In this study, the finite difference method is applied to an optimal control problem controlled by two functions which are in the coefficients of two-dimensional Schrodinger equation. Convergence of the finite difference approximation according to the functional is proved. We have used the implicit method for solving the two-dimensional Schrodinger equation. Although the implicit scheme obtained from solution of the system of the linear equations is generally numerically stable and convergent without time-step condition, the solution of considered equation is numerically stable with time-step condition, due to the gradient term.
VL  - 4
IS  - 2
ER  -

Copy | Download