Optimal bilinear control of nonlinear Schr\"{o}dinger equations with singular potentials

TL;DR

This paper develops an optimal control framework for nonlinear Schrödinger equations with singular potentials, establishing well-posedness, existence of optimal controls, and deriving the first-order optimality conditions, thus extending previous results.

Contribution

It introduces a generalized approach to bilinear control of nonlinear Schrödinger equations with singular potentials, including rigorous derivation of optimality systems.

Findings

Proved well-posedness of the control problem

Established existence of optimal controls

Derived first-order optimality conditions

Abstract

In this paper, we consider an optimal bilinear control problem for the nonlinear Schr\"{o}dinger equations with singular potentials. We show well-posedness of the problem and existence of an optimal control. In addition, the first order optimality system is rigorously derived. Our results generalize the ones in \cite{Sp} in several aspects.