Control of the Schrödinger equation by slow deformations of the domain

Abstract

The aim of this work is to study the controllability of the Schrödinger equation $i{\partial }_{t}u(t)=-\Delta u(t)$ on $\Omega (t)$ with Dirichlet boundary conditions, where $\Omega (t)\subset {\mathbb{R}}^N$ is a time-varying domain. We prove the global approximate controllability of the equation in $L^2(\Omega )$, via an adiabatic deformation $\Omega (t)\subset {\mathbb{R}}^N$ ($t\in [0,T]$) such that $\Omega (0)=\Omega (T)=\Omega$. This control is strongly based on the Hamiltonian structure of the equation provided by Duca and Joly [Ann. Henri Poincaré 22 (2021), 2029–2063], which enables the use of adiabatic motions.We also discuss several explicit interesting controls that we perform in the specific framework of rectangular domains.