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 criterionfor the nonlinear Schrödinger equation with the specific gradient summand and the complex potential when theperformance criterion is the full-range integral. Concurrently, the first variation formula of the performancecriterion under consideration and via which the necessary condition is established in the form of a variationinequality 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-valuedpotential, when the potential depending on both spatial and time dependence and is sought in the class ofmeasurable 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 specificgradient term, complex potential and is relevant in its significance and also of scientific interest.