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

Coron, Jean-Michel1
1 Institut universitaire de France and département de mathématique, bâtiment 425, université Paris-Sud 11, 91405 Orsay, France
Comptes Rendus. Mathématique, Tome 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
Coron, Jean-Michel 1

1 Institut universitaire de France and département de mathématique, bâtiment 425, université Paris-Sud 11, 91405 Orsay, France 
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Coron, Jean-Michel. On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well. Comptes Rendus. Mathématique, Tome 342 (2006) no. 2, pp. 103-108. doi : 10.1016/j.crma.2005.11.004. http://www.numdam.org/articles/10.1016/j.crma.2005.11.004/

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

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

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

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

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

[13] Horsin, T. 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

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