Comptes Rendus
Partial Differential Equations/Optimal Control
Semi-global weak stabilization of bilinear Schrödinger equations
[Stabilisation faible semi-globale d'équations de Schrödinger bilinéaires]
Comptes Rendus. Mathématique, Volume 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)nN 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)nN 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

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},
}
TY  - JOUR
AU  - Karine Beauchard
AU  - Vahagn Nersesyan
TI  - Semi-global weak stabilization of bilinear Schrödinger equations
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 1073
EP  - 1078
VL  - 348
IS  - 19-20
PB  - Elsevier
DO  - 10.1016/j.crma.2010.09.002
LA  - en
ID  - CRMATH_2010__348_19-20_1073_0
ER  - 
%0 Journal Article
%A Karine Beauchard
%A Vahagn Nersesyan
%T Semi-global weak stabilization of bilinear Schrödinger equations
%J Comptes Rendus. Mathématique
%D 2010
%P 1073-1078
%V 348
%N 19-20
%I Elsevier
%R 10.1016/j.crma.2010.09.002
%G en
%F CRMATH_2010__348_19-20_1073_0
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/

[1] J.M. Ball; M. Slemrod Feedback stabilization of distributed semilinear control systems, Appl. Math. Optim., Volume 5 (1979), pp. 169-179

[2] K. Beauchard; J.-M. Coron Controllability of a quantum particle in a moving potential well, J. Funct. Anal., Volume 232 (2006), pp. 328-389

[3] K. Beauchard; J.-M. Coron; M. Mirrahimi; P. Rouchon Implicit Lyapunov control of finite dimensional Schrödinger equations, Systems Control Lett., Volume 56 (2007) no. 5, pp. 388-395

[4] K. Beauchard; C. Laurent Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control, J. Math. Pures Appl. (2010) | DOI

[5] K. Beauchard; M. Mirrahimi Practical stabilization of a quantum particle in a one-dimensional infinite square potential well, SIAM J. Control Optim., Volume 48 (2009) no. 2, pp. 1179-1205

[6] T. Cazenave Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, AMS, 2003

[7] T. Chambrion; P. Mason; M. Sigalotti; M. Boscain Controllability of the discrete-spectrum Schrödinger equation driven by an external field, Ann. IHP Non Linear Anal., Volume 26 (2009) no. 1, pp. 329-349

[8] J.-M. Coron; B. d'Andréa-Novel Stabilization of a rotating body beam without damping, IEEE Trans. Automat. Control, Volume 43 (1998), pp. 608-618

[9] J.-M. Coron; B. d'Andréa-Novel; G. Bastin A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Trans. Automat. Control, Volume 52 (2007) no. 1, pp. 2-11

[10] J.-L. Lions Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969

[11] P. Mason; M. Sigalotti Generic controllability properties for the bilinear Schrödinger equation, Comm. Partial Differential Equations, Volume 35 (2010), pp. 685-706

[12] M. Mirrahimi Lyapunov control of a quantum particle in a decaying potential, Ann. IHP Non Linear Anal., Volume 2 (2009), pp. 1743-1765

[13] M. Mirrahimi; P. Rouchon; G. Turinici Lyapunov control of bilinear Schrödinger equations, Automatica, Volume 41 (2005), pp. 1987-1994

[14] V. Nersesyan Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications, Ann. IHP Non Linear Anal., Volume 27 (2010) no. 3, pp. 901-915

[15] V. Nersesyan Growth of Sobolev norms and controllability of the Schrödinger equation, Comm. Math. Phys., Volume 290 (2009) no. 1, pp. 371-387

[16] Y. Privat; M. Sigalotti The squares of the Laplacian–Dirichlet eigenfunctions are generically linearly independent, ESAIM: COCV, Volume 16 (July–September 2010) no. 3, pp. 794-805

Cité par Sources :

Commentaires - Politique