Internal control of the Schr\"odinger equation

TL;DR

This paper reviews known results on the internal controllability of linear and nonlinear Schrödinger equations, including proofs, geometric conditions, and open questions, highlighting recent advances and challenges.

Contribution

It provides a comprehensive overview of controllability results, proofs, and open problems for Schrödinger equations on various manifolds and in nonlinear settings.

Findings

Controllability in 1D using propagation results

Extension of controllability to compact manifolds with Geometric Control Condition

Discussion of open problems when the Geometric Control Condition is not met

Abstract

In this paper, we intend to present some already known results about the internal controllability of the linear and nonlinear Schr\"odinger equation. After presenting the basic properties of the equation, we give a self contained proof of the controllability in dimension 1 using some propagation results. We then discuss how to obtain some similar results on a compact manifold where the zone of control satisfies the Geometric Control Condition. We also discuss some known results and open questions when this condition is not satisfied. Then, we present the links between the controllability and some resolvent estimates. Finally, we discuss the new difficulties when we consider the Nonlinear Schr\"odinger equation.