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In this study we investigate an optimal control problem for the nonstationary Schrödinger equation; the questions needed to correctly identify the optimal control problem were answered, and the existence and uniqueness of the solution and the necessary and sufficient conditions for the solution were investigated. We consider the initial situation as control for the controlled system.

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Using variational methods, the solution of the inverse problem of finding the refractive index of a nonlinear medium in a multidimensional Schrbdinger equation is studied. The correctness of the statement of the problem under consideration is investigated, and a necessary condition that must be satisfied by the solution of this problem is found. Bibliography: 8 titles. Finding the refractive index of a nonlinear medium is one of the fundamental problems of nonlinear optics [1]. Inverse problems of finding the refractive index of a nonlinear medium in variational statements were studied in [2, 3] for the one-dimensional Schr6dinger equation. Let D be a bounded domain of the Euclidean space E3; F the boundary of the domain D, which is assumed to be sufficiently smooth; T > 0 a given number; 0 < t < T; ~t = D • (0, t); ~ = ~T, S = F x (0, T); x = (Xl,X2, x3) an arbitrary point of the domain D; C~ T], B) the Banach space consisting of all welldefined and continuous functions on [0, iF] with values in the Banach space B; Lp(D) the Lebesgue space of functions, summable up to the power p > 1; -~-E , , where the spaces L~ ([0, T], B) and Wk(D), w~'m(~), p >_ 1, k, m ~ O, are as defined in [4, 5]; the symbol means that the property is valid for almost all values of a variable. Consider a system whose state is described by the nonlinear Schrbdinger equation .0r 3 02 r l-~ +no C Ox~. v(x,t)r162162 = f(x,t), (1) j=l where i = x/-L-f, a0, al > 0 are given numbers, and f E W21'1 (~t) is a given function. Let Eq. (1) be subject to the following initial and boundary conditions: 0r s = O, (2) r I =o = x e D, where ~(x) is a given function from W2(D), for which 0~ r Ou = O, and u is the direction of the outward normal of the boundary F. Our goal is to find the coefficient v(x, t) on the basis of additional information r Is = Y(~, t) about the solution of the Schr6dinger equation (1). The function v(x, t) is sought on the set { Ov(x,t) Ov(x,t) E o } V-v:v=v(x,t), veW#'~(fi), Iv(z,t)l<_bo, at <_bl, ~ <--52, V(x,t) efi , where bo, bt, b2 > 0 are given numbers. The variational statement of the problem at hand lies in the minimization of the functional J(v) II% = = -llL=(s) on the set V under conditions (1), (2), where y E W~/2'1/4(S) is a given function.From this point on, we assume that the above conditions on the problem are fulfilled.(3)

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.