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arXiv · 2203.00486

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

Alessandro Duca·Romain Joly·Dmitry Turaev
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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.

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Alessandro Duca, Romain Joly, Dmitry Turaev. 2022-03-01. Control of the Schr\"odinger equation by slow deformations of the domain. https://arxiv.org/abs/2203.00486

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