Control of the Schrödinger equation by slow deformations of the domain - Archive ouverte HAL
Article Dans Une Revue Annales de l'Institut Henri Poincaré C, Analyse non linéaire Année : 2023

Control of the Schrödinger equation by slow deformations of the domain

Résumé

The aim of this work is to study the controllability of the Schrödinger 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 $\Omega(t)\subset\mathbb{R}^N$ ($t\in[0,T]$) such that $\Omega(0)=\Omega(T)=\Omega$. This control is strongly based on the Hamiltonian structure of \eqref{eq_abstract} provided by \cite{SEmoving}, 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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Dates et versions

hal-04734979 , version 1 (14-10-2024)

Identifiants

Citer

Alessandro Duca, Romain Joly, Dmitry Turaev. Control of the Schrödinger equation by slow deformations of the domain. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2023, 41 (3), pp.511-553. ⟨10.4171/AIHPC/86⟩. ⟨hal-04734979⟩
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