Global exact controllability of 1d Schrödinger equations with a polarizability term

Morgan Morancey; Vahagn Nersesyan

doi:10.1016/j.crma.2014.03.013

Partial differential equations/Optimal control

Global exact controllability of 1d Schrödinger equations with a polarizability term
[Contrôle exact global d'équations de Schrödinger 1d avec un terme de polarisabilité]

Présenté par : Gilles Lebeau

Morgan Morancey ^1,² ; Vahagn Nersesyan ³

¹ CMLS UMR 7640, École polytechnique, 91128 Palaiseau, France
² CMLA ENS Cachan, 61, avenue du Président-Wilson, 94235 Cachan, France
³ Laboratoire de mathématiques, UMR CNRS 8100, Université de Versailles–Saint-Quentin-en-Yvelines, 78035 Versailles, France

Comptes Rendus. Mathématique, Volume 352 (2014) no. 5, pp. 425-429.

Résumé
Abstract

On considère une particule quantique dans un intervalle 1d, soumise à un potentiel. L'évolution de cette particule est contrôlée par un champ électrique extérieur. En prenant en compte dans le modèle le terme dit de polarisabilité (quadratique par rapport au contrôle), on prouve la contrôlabilité exacte globale dans un espace approprié pour des potentiels et des moments dipolaires arbitraires. Ce terme est intéressant à la fois d'un point de vue mathématique et physique. La preuve utilise des outils issus du cadre bilinéaire et un argument de perturbation.

We consider a quantum particle in a 1d interval submitted to a potential. The evolution of this particle is controlled using an external electric field. Taking into account the so-called polarizability term in the model (quadratic with respect to the control), we prove global exact controllability in a suitable space for arbitrary potential and arbitrary dipole moment. This term is relevant both from the mathematical and physical points of view. The proof uses tools from the bilinear setting and a perturbation argument.

Reçu le : 2013-09-26
Accepté le : 2014-03-17
Publié le : 2014-03-31

DOI : 10.1016/j.crma.2014.03.013

Affiliations des auteurs :

Morgan Morancey ^{1, 2} ; Vahagn Nersesyan ³

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     author = {Morgan Morancey and Vahagn Nersesyan},
     title = {Global exact controllability of 1d {Schr\"odinger} equations with a polarizability term},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {425--429},
     publisher = {Elsevier},
     volume = {352},
     number = {5},
     year = {2014},
     doi = {10.1016/j.crma.2014.03.013},
     language = {en},
}

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Morgan Morancey; Vahagn Nersesyan. Global exact controllability of 1d Schrödinger equations with a polarizability term. Comptes Rendus. Mathématique, Volume 352 (2014) no. 5, pp. 425-429. doi : 10.1016/j.crma.2014.03.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.03.013/

Bibliographie
Cité par

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