{"title":"Small-time local controllability of the bilinear Schrödinger equation with a nonlinear competition","authors":"Mégane Bournissou","doi":"10.1051/cocv/2023077","DOIUrl":null,"url":null,"abstract":"We consider the local controllability near the ground state of a 1D Schrödinger equation with bilinear control. Specifically, we investigate whether nonlinear terms can restore local controllability when the linearized system is not controllable. In such settings, it is known that the quadratic terms induce drifts in the dynamics which prevent small-time local controllability when the controls are small in very regular spaces. In this paper, using oscillating controls, we prove that the cubic terms can entail the small-time local controllability of the system, despite the presence of such a quadratic drift. This result, which is new for PDEs, is reminiscent of Sussmann's $\\S(\\theta)$ sufficient condition of controllability for ODEs. Our proof however relies on a different general strategy involving a new concept of tangent vector, better suited to the infinite-dimensional setting.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Esaim-Control Optimisation and Calculus of Variations","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1051/cocv/2023077","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: 0
Abstract
We consider the local controllability near the ground state of a 1D Schrödinger equation with bilinear control. Specifically, we investigate whether nonlinear terms can restore local controllability when the linearized system is not controllable. In such settings, it is known that the quadratic terms induce drifts in the dynamics which prevent small-time local controllability when the controls are small in very regular spaces. In this paper, using oscillating controls, we prove that the cubic terms can entail the small-time local controllability of the system, despite the presence of such a quadratic drift. This result, which is new for PDEs, is reminiscent of Sussmann's $\S(\theta)$ sufficient condition of controllability for ODEs. Our proof however relies on a different general strategy involving a new concept of tangent vector, better suited to the infinite-dimensional setting.
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.
Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines.
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in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory;
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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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