Differentiability of the functional and the necessary condition to solve optimal control problem with integral criterion qualities over the entire domain for a nonlinear Schrödinger equation with a special gradient term
Abstract:
This work is devoted to the study of the optimal control problem for the three-dimensional nonlinear Schrödinger equation with a special gradient term and with a complex potential. The controls are the real and imaginary parts of the complex potential and are selected from the class of measurable bounded functions depending on the time variable, and the performance criterion is an integral over throughout the region is of considerable scientific interest. We note that the existence and uniqueness of a solution to an optimal control problem with an integral performance criterion over the entire domain for a nonlinear Schrödinger equation with a special gradient term was previously studied by the author. And in this article, the differentiability of the previously considered problem is studied. A theorem on the uniqueness of the adjoint problem to the stated problem is given, which is proved by the Galerkin method. A theorem on the differentiability of the solution of the optimal control problem under consideration is given and proved. Along with these, a formula is found for the first variation of the quality criterion under consideration and with the help of which a necessary condition is established in the form of a variational inequality.
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