Optimal Control Problem and Integral Quality Criteria with a Special Gradient Term

The paper deals with the optimal control problem with the full-range integral performance criterion for the nonlinear Schrödinger equation with the specific gradient summand and the complex potential when the performance criterion is the full-range integral. Concurrently, the first variation formula of the performance criterion under consideration and via which the necessary condition is established in the form of a variation inequality is found. This article investigates the differentiability of the optimal control problem for a threedimensional nonlinear no stationary Schrödinger equation with a special gradient term and a real-valued potential, when the potential depending on both spatial and time dependence and is sought in the class of measurable bounded functions, and the coefficient in the nonlinear part of the equation is a complex number. Therefore, this work differs from previously studied problems nonlinear Schrödinger equation with a specific gradient term, complex potential and is relevant in its significance and also of scientific interest.