Optimal Control of a Nonlinear Stochastic Schrödinger Equation

This paper deals with a nonlinear controlled Schrödinger equation which is perturbed by an additive Gaussian noise in the form of a Q-Wiener process. The nonlinearity is especially characterized by Lipschitz-continuity and growth-boundedness. Moreover, suitable initial and Neumann boundary conditions are introduced. Referring to the concept of a variational solution throughout this work, the existence and uniqueness of such a solution of the Schrödinger problem will be investigated. The main goal is to minimize a given objective functional by an optimal control. However, the objective functional depends on the solution of the controlled Schrödinger problem and the control itself. Therefore, the adjoint equation (of the stochastic Schrödinger equation) with appropriate final and Neumann boundary conditions is required. Based on a generalized solution of this adjoint problem and some important estimates concerning the difference of two solutions of the Schrödinger problem corresponding to two controls, a gradient formula in the sense of Gâteaux is calculated. Thus, a necessary optimality condition can be stated.