{"title":"Controllability of the Schrödinger equation on unbounded domains without geometric control condition","authors":"Matthias Taufer","doi":"10.1051/cocv/2023037","DOIUrl":null,"url":null,"abstract":"We prove controllability of the Schr\\\"odinger equation in $\\mathbb{R}^d$ in any time $T > 0$ with internal control supported on nonempty, periodic, open sets.\n This demonstrates in particular that controllability of the Schr\\\"odinger equation in full space holds for a strictly larger class of control supports than for the wave equation and suggests that the control theory of Schr\\\"odinger equation in full space might be closer to the diffusive nature of the heat equation than to the ballistic nature of the wave equation.\n Our results are based on a combination of Floquet-Bloch theory with Ingham-type estimates on lacunary Fourier series.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"4 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2022-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Esaim-Control Optimisation and Calculus of Variations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1051/cocv/2023037","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 6
Abstract
We prove controllability of the Schr\"odinger equation in $\mathbb{R}^d$ in any time $T > 0$ with internal control supported on nonempty, periodic, open sets.
This demonstrates in particular that controllability of the Schr\"odinger equation in full space holds for a strictly larger class of control supports than for the wave equation and suggests that the control theory of Schr\"odinger equation in full space might be closer to the diffusive nature of the heat equation than to the ballistic nature of the wave equation.
Our results are based on a combination of Floquet-Bloch theory with Ingham-type estimates on lacunary Fourier series.
期刊介绍：
ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations.
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in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.
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