An optimal control problem for the Schrödinger equation with a real-valued factor

We study an optimal control problem for the Schrödinger equation with a real-valued factor in its nonlinear part where the control function is square summable and the quality criterion is Lions' functional. First, we examine the correctness of the statement of the reduced problem and, second, we do that of the optimal control problem. We also study the differentiability of Lions' functional and obtain a necessary optimality condition in the form of a variational inequality.