Numerical Solution of an Optimal Control Problem Governed by Two Dimensional Schrodinger Equation
Applied and Computational Mathematics
Volume 4, Issue 2, April 2015, Pages: 30-38
Received: Feb. 11, 2015;
Accepted: Feb. 26, 2015;
Published: Mar. 4, 2015
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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; Department of Mathematics, Faculty of Art and Science, Kafkas University, Kars, Turkey
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.
Numerical Solution of an Optimal Control Problem Governed by Two Dimensional Schrodinger Equation, Applied and Computational Mathematics.
Vol. 4, No. 2,
2015, pp. 30-38.
H. Yetişkin, M. Subaşı, “On the optimal control problem for Schrödinger equation with complex potential,” Applied Mathematics and Computation, 216, 1896-1902, 2010.
B. Yıldız, O. Kılıçoğlu, G. Yagubov, “Optimal control problem for non stationary Schrödinger equation,” Numerical Methods for Partial Differential Equations, 25, 1195-1203, 2009.
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.
B. Yıldız, M. Subaşı, “On the optimal control problem for linear Schrödinger equation. Applied Mathematics and Computation,” 121, 373-381, 2001.
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.
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.
G. D. Smith,“Numerical Solution of Partial Differential Equations,” Oxford University Press, 1985.
J. V. Thomas, “Numerical Partial Differential Equations,” Springer- Verlag, 1995.
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.
F. P. Vasilyev, “Numerical Methods for Extremal Problems,” Nauka, Moskow, 1981.