Coupled linear Schr\"odinger equations: Control and stabilization results

TL;DR

This paper establishes controllability and exponential stabilization for a coupled system of linear Schrödinger equations in one dimension, using new Carleman estimates and boundary feedback control techniques.

Contribution

It introduces a novel Carleman estimate for coupled Schrödinger equations and demonstrates boundary controllability and exponential stabilization in a bounded domain.

Findings

Null boundary controllability achieved.

Exponential decay of solutions proven.

New Carleman estimate developed.

Abstract

This article presents some controllability and stabilization results for a system of two coupled linear Schr\"odinger equations in the one-dimensional case where the state components are interacting through the Kirchhoff boundary conditions. Considering the system in a bounded domain, the null boundary controllability result is shown. The result is achieved thanks to a new Carleman estimate, which ensures a boundary observation. Additionally, this boundary observation together with some trace estimates, helps us to use the Gramian approach, with a suitable choice of feedback law, to prove that the system under consideration decays exponentially to zero at least as fast as the function $e^{-2\omega t}$ for some $\omega>0$.