no. 19-20
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

Beauchard, Karine1 ; Nersesyan, Vahagn2
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, Tome 348 (2010) no. 19-20, pp. 1073-1078.

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 (tn)n∈N auxquels les valeurs de la solution du système bouclé convergent faiblement dans H2 vers l'état fondamental. Ici, nous démontrons la convergence de toute la solution, quand t→+∞. 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 (tn)n∈N for which the values of the solution of the closed loop system converge weakly in H2 to the ground state. Here, we prove the convergence of the whole solution, as t→+∞. The proof relies on control Lyapunov functions and an adaptation of the LaSalle invariance principle to PDEs.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.09.002
Beauchard, Karine 1 ; Nersesyan, Vahagn 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 
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     title = {Semi-global weak stabilization of bilinear {Schr\"odinger} equations},
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     pages = {1073--1078},
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Beauchard, Karine; Nersesyan, Vahagn. Semi-global weak stabilization of bilinear Schrödinger equations. Comptes Rendus. Mathématique, Tome 348 (2010) no. 19-20, pp. 1073-1078. doi : 10.1016/j.crma.2010.09.002. http://www.numdam.org/articles/10.1016/j.crma.2010.09.002/

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