Comptes Rendus
no. 2
Partial Differential Equations/Optimal Control
On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well
[Sur la contrôlabilité en temps petit d'une particule quantique dans un puits de potentiel carré infini unidimensionnel mobile]

Présenté par : Haïm Brezis

Jean-Michel Coron1
1 Institut universitaire de France and département de mathématique, bâtiment 425, université Paris-Sud 11, 91405 Orsay, France
Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 103-108.

On considère une particule quantique chargée dans un puits de potentiel carré infini unidimensionnel se déplaçant le long d'une droite. On contrôle l'accélération du puits de potentiel. La contrôlabilité locale autour de l'état fondamental pour des temps grands de ce système de contrôle a été récemment démontrée. Nous montrons que l'on n'a pas contrôlabilité locale pour des temps petits, bien que l'équation de Schrödinger ait une vitesse de propagation infinie.

We consider a quantum charged particle in a one-dimensional infinite square potential well moving along a line. We control the acceleration of the potential well. The local controllability in large time of this nonlinear control system along the ground state trajectory has been proved recently. We prove that this local controllability does not hold in small time, even if the Schrödinger equation has an infinite speed of propagation.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.11.004
Jean-Michel Coron 1

1 Institut universitaire de France and département de mathématique, bâtiment 425, université Paris-Sud 11, 91405 Orsay, France 
@article{CRMATH_2006__342_2_103_0,
     author = {Jean-Michel Coron},
     title = {On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {103--108},
     publisher = {Elsevier},
     volume = {342},
     number = {2},
     year = {2006},
     doi = {10.1016/j.crma.2005.11.004},
     language = {en},
}
TY  - JOUR
AU  - Jean-Michel Coron
TI  - On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 103
EP  - 108
VL  - 342
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2005.11.004
LA  - en
ID  - CRMATH_2006__342_2_103_0
ER  - 
%0 Journal Article
%A Jean-Michel Coron
%T On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well
%J Comptes Rendus. Mathématique
%D 2006
%P 103-108
%V 342
%N 2
%I Elsevier
%R 10.1016/j.crma.2005.11.004
%G en
%F CRMATH_2006__342_2_103_0
Jean-Michel Coron. On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well. Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 103-108. doi : 10.1016/j.crma.2005.11.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.11.004/

[1] K. Beauchard Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl. (9), Volume 84 (2005) no. 7, pp. 851-956

[2] K. Beauchard, J.-M. Coron, Controllability of a quantum particle in a moving potential well, J. Funct. Anal. (2005), in press

[3] J.-M. Coron Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems, Volume 5 (1992) no. 3, pp. 295-312

[4] J.-M. Coron On the controllability of the 2-D incompressible Navier–Stokes equations with the Navier slip boundary conditions, ESAIM Control Optim. Calc. Var., Volume 1 (1995/96), pp. 35-75 (electronic)

[5] J.-M. Coron On the controllability of 2-D incompressible perfect fluids, J. Math. Pures Appl. (9), Volume 75 (1996) no. 2, pp. 155-188

[6] J.-M. Coron Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations, ESAIM Control Optim. Calc. Var., Volume 8 (2002), pp. 513-554 (A tribute to J.-L. Lions)

[7] J.-M. Coron, Local controllability of a 1-D tank containing a fluid, in: XVIII Congreso de Ecuaciones Diferenciales y Aplicaciones, VIII Congreso de Matemática Aplicada, Tarragona, 15–19 septiembre, 2003

[8] J.-M. Coron; E. Crépeau Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. Eur. Math. Soc. (JEMS), Volume 6 (2004) no. 3, pp. 367-398

[9] J.-M. Coron; A.V. Fursikov Global exact controllability of the 2D Navier–Stokes equations on a manifold without boundary, Russian J. Math. Phys., Volume 4 (1996) no. 4, pp. 429-448

[10] A.V. Fursikov; O.Yu. Imanuvilov Exact controllability of the Navier–Stokes and Boussinesq equations, Russian Math. Surveys, Volume 54 (1999), pp. 565-618

[11] O. Glass Exact boundary controllability of 3-D Euler equation, ESAIM Control Optim. Calc. Var., Volume 5 (2000), pp. 1-44 (electronic)

[12] O. Glass On the controllability of the Vlasov–Poisson system, J. Differential Equations, Volume 195 (2003) no. 2, pp. 332-379

[13] T. Horsin On the controllability of the Burgers equation, ESAIM Control Optim. Calc. Var., Volume 3 (1998), pp. 83-95 (electronic)

[14] P. Rouchon, Control of a quantum particule in a moving potential well, in: 2nd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Seville, 2003

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Semi-global weak stabilization of bilinear Schrödinger equations

Karine Beauchard; Vahagn Nersesyan

C. R. Math (2010)

Global controllability of nonviscous Burgers type equations

Marianne Chapouly

C. R. Math (2007)

A numerical study of the null boundary controllability of a convection diffusion equation

Ali Salem

C. R. Math (2009)