Semi-global weak stabilization of bilinear Schrödinger equations

Karine Beauchard; Vahagn Nersesyan

doi:10.1016/j.crma.2010.09.002

Partial Differential Equations/Optimal Control

Semi-global weak stabilization of bilinear Schrödinger equations
[Stabilisation faible semi-globale d'équations de Schrödinger bilinéaires]

Présenté par : Gilles Lebeau

Karine Beauchard ¹ ; Vahagn Nersesyan ²

¹ CMLA, ENS Cachan, CNRS, UniverSud, 61, avenue du Président Wilson, 94230 Cachan, France
² Laboratoire de mathématiques de Versailles, bâtiment Fermat, 45, avenue des Etats-Unis, 78035 Versailles cedex, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1073-1078.

Résumé
Abstract

Nous considérons une équation de Schrödinger linéaire, sur un domaine borné, avec un contrôle bilinéaire, modélisant une particule quantique dans un champ électrique (la commande). Récemment, Nersesyan a proposé des lois de rétroaction explicites et démontré l'existence d'une suite de temps ${(t_{n})}_{n \in N}$ auxquels les valeurs de la solution du système bouclé convergent faiblement dans $H^{2}$ vers l'état fondamental. Ici, nous démontrons la convergence de toute la solution, quand $t \to + \infty$ . La preuve repose sur des fonctions de Lyapunov et une adaptation du principe d'invariance de LaSalle aux EDP.

We consider a linear Schrödinger equation, on a bounded domain, with bilinear control, representing a quantum particle in an electric field (the control). Recently, Nersesyan proposed explicit feedback laws and proved the existence of a sequence of times ${(t_{n})}_{n \in N}$ for which the values of the solution of the closed loop system converge weakly in $H^{2}$ to the ground state. Here, we prove the convergence of the whole solution, as $t \to + \infty$ . The proof relies on control Lyapunov functions and an adaptation of the LaSalle invariance principle to PDEs.

Reçu le : 2010-05-18
Accepté le : 2010-09-02
Publié le : 2010-09-28

DOI : 10.1016/j.crma.2010.09.002

Affiliations des auteurs :

Karine Beauchard ¹ ; Vahagn Nersesyan ²

¹ CMLA, ENS Cachan, CNRS, UniverSud, 61, avenue du Président Wilson, 94230 Cachan, France
² Laboratoire de mathématiques de Versailles, bâtiment Fermat, 45, avenue des Etats-Unis, 78035 Versailles cedex, France

@article{CRMATH_2010__348_19-20_1073_0,
     author = {Karine Beauchard and Vahagn Nersesyan},
     title = {Semi-global weak stabilization of bilinear {Schr\"odinger} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1073--1078},
     publisher = {Elsevier},
     volume = {348},
     number = {19-20},
     year = {2010},
     doi = {10.1016/j.crma.2010.09.002},
     language = {en},
}

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Karine Beauchard; Vahagn Nersesyan. Semi-global weak stabilization of bilinear Schrödinger equations. Comptes Rendus. Mathématique, Volume 348 (2010) no. 19-20, pp. 1073-1078. doi : 10.1016/j.crma.2010.09.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.09.002/

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