Control of the Schr\"odinger equation by slow deformations of the domain

TL;DR

This paper demonstrates that the Schr"odinger equation on a domain can be approximately controlled by slowly deforming the domain shape over time, leveraging the Hamiltonian structure and adiabatic processes.

Contribution

It proves global approximate controllability of the Schr"odinger equation via slow domain deformations, a novel approach using adiabatic motions and Hamiltonian structure.

Findings

Achieved approximate controllability in $L^2( abla)$

Utilized adiabatic deformations of the domain

Implemented explicit controls in rectangular domains

Abstract

The aim of this work is to study the controllability of the Schr\"odinger equation \begin{equation}\label{eq_abstract} i\partial_t u(t)=-\Delta u(t)~~~~~\text{ on }\Omega(t) \tag{$\ast$} \end{equation} with Dirichlet boundary conditions, where $\Omega(t)\subset\mathbb{R}^N$ is a time-varying domain. We prove the global approximate controllability of \eqref{eq_abstract} in $L^2(\Omega)$, via an adiabatic deformation $\Omega(t)\subset\mathbb{R}$ ($t\in[0,T]$) such that $\Omega(0)=\Omega(T)=\Omega$. This control is strongly based on the Hamiltonian structure of \eqref{eq_abstract} provided by [18], which enables the use of adiabatic motions. We also discuss several explicit interesting controls that we perform in the specific framework of rectangular domains.