{"title":"Control of the Schrödinger equation by slow deformations of the domain","authors":"Alessandro Duca, R. Joly, D. Turaev","doi":"10.4171/aihpc/86","DOIUrl":null,"url":null,"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.","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":"2 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/aihpc/86","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 4
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.
期刊介绍:
The Nonlinear Analysis section of the Annales de l''Institut Henri Poincaré is an international journal created in 1983 which publishes original and high quality research articles. It concentrates on all domains concerned with nonlinear analysis, specially applicable to PDE, mechanics, physics, economy, without overlooking the numerical aspects.