Control and Stabilization of the Nonlinear Schroedinger Equation on Rectangles

TL;DR

This paper investigates the local controllability and stabilization of the nonlinear Schrödinger equation on rectangular domains, considering both internal and boundary controls with various boundary conditions, providing sharp controllability results.

Contribution

It offers new sharp local controllability and stabilization results for the nonlinear Schrödinger equation on rectangles, including boundary and internal control strategies.

Findings

Local controllability results are sharp regarding control localization and smoothness.

Exact controllability holds in H^{-1} for linear Schrödinger with Dirichlet control near a vertex.

Results apply to both periodic and Dirichlet/Neumann boundary conditions.

Abstract

This paper studies the local exact controllability and the local stabilization of the semilinear Schr\"odinger equation posed on a product of $n$ intervals ($n\ge 1$). Both internal and boundary controls are considered, and the results are given with periodic (resp. Dirichlet or Neumann) boundary conditions. In the case of internal control, we obtain local controllability results which are sharp as far as the localization of the control region and the smoothness of the state space are concerned. It is also proved that for the linear Schr\"odinger equation with Dirichlet control, the exact controllability holds in $H^{-1}(\Omega)$ whenever the control region contains a neighborhood of a vertex.