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 -Schrödinger-Poisson system, which couples a Schrödinger equation on a bounded domain of 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.
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@article{JEP_2017__4__287_0,
author = {Karine Beauchard and Camille Laurent},
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},
publisher = {\'Ecole polytechnique},
volume = {4},
year = {2017},
doi = {10.5802/jep.44},
zbl = {1375.35419},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.44/}
}
TY - JOUR AU - Karine Beauchard AU - Camille Laurent TI - Local exact controllability of the $2$D-Schrödinger-Poisson system JO - Journal de l’École polytechnique — Mathématiques PY - 2017 SP - 287 EP - 336 VL - 4 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.44/ DO - 10.5802/jep.44 LA - en ID - JEP_2017__4__287_0 ER -
%0 Journal Article %A Karine Beauchard %A Camille Laurent %T Local exact controllability of the $2$D-Schrödinger-Poisson system %J Journal de l’École polytechnique — Mathématiques %D 2017 %P 287-336 %V 4 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.44/ %R 10.5802/jep.44 %G en %F JEP_2017__4__287_0
Karine Beauchard; Camille Laurent. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.44/
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