Local exact controllability of the 2D-Schrödinger-Poisson system
[Contrôlabilité locale exacte du système de Schrödinger-Poisson 2D]
Beauchard, Karine1 ; Laurent, Camille2
1 IRMAR, École Normale Supérieure de Rennes, UBL Avenue Robert Schumann, 35170 Bruz, France
2 Sorbonne Universités, CNRS UMR 7598 and UPMC Univ. Paris 06, Laboratoire Jacques-Louis Lions F-75005, Paris, France
Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 287-336.

Dans cet article, nous étudions la contrôlabilité exacte du système de Schrödinger-Poisson 2D, qui couple une équation de Schrödinger sur un ouvert borné 2D, avec une équation de Poisson pour le potentiel électrique. Le contrôle agit sur le système via une condition de Neumann sur le potentiel, localement distribuée sur le bord du domaine spatial. Nous démontrons plusieurs résultats, avec ou sans non-linéarité, avec différents types de conditions de bord sur la fonction d’onde, de type Dirichlet ou de type Neumann.

In this article, we investigate the exact controllability of the 2D-Schrödinger-Poisson system, which couples a Schrödinger equation on a bounded domain of ℝ 2 with a Poisson equation for the electrical potential. The control acts on the system through a Neumann boundary condition on the potential, locally distributed on the boundary of the space domain. We prove several results, with or without nonlinearity and with different boundary conditions on the wave function, of Dirichlet type or of Neumann type.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.44
Classification : 35Q40, 35Q41, 93C10, 93C20
Keywords: Control of partial differential equations, Schrödinger-Poisson system, bilinear control
Mot clés : Contrôle d’équations aux dérivées partielles, système de Schrödinger-Poisson, contrôle bilinéaire
Beauchard, Karine 1 ; Laurent, Camille 2

1 IRMAR, École Normale Supérieure de Rennes, UBL Avenue Robert Schumann, 35170 Bruz, France
2 Sorbonne Universités, CNRS UMR 7598 and UPMC Univ. Paris 06, Laboratoire Jacques-Louis Lions F-75005, Paris, France 
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     title = {Local exact controllability of the~$2${D-Schr\"odinger-Poisson} system},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {287--336},
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Beauchard, Karine; Laurent, Camille. Local exact controllability of the $2$D-Schrödinger-Poisson system. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 287-336. doi : 10.5802/jep.44. http://www.numdam.org/articles/10.5802/jep.44/

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