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Pré-Publication, Document De Travail Année : 2022

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 [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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hal-03592688 , version 1 (01-03-2022)

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  • HAL Id : hal-03592688 , version 1

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Alessandro Duca, Romain Joly, Dmitry Turaev. Control of the Schrödinger equation by slow deformations of the domain. 2022. ⟨hal-03592688⟩
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