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On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well. (English) Zbl 1082.93002

Summary: 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.

MSC:

93B05 Controllability
35Q40 PDEs in connection with quantum mechanics
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[1] Beauchard, K., Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl. (9), 84, 7, 851-956 (2005) · Zbl 1124.93009
[2] K. Beauchard, J.-M. Coron, Controllability of a quantum particle in a moving potential well, J. Funct. Anal. (2005), in press; K. Beauchard, J.-M. Coron, Controllability of a quantum particle in a moving potential well, J. Funct. Anal. (2005), in press · Zbl 1188.93017
[3] Coron, J.-M., Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems, 5, 3, 295-312 (1992) · Zbl 0760.93067
[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., 1, 35-75 (1995/96), (electronic) · Zbl 0872.93040
[5] Coron, J.-M., On the controllability of 2-D incompressible perfect fluids, J. Math. Pures Appl. (9), 75, 2, 155-188 (1996) · Zbl 0848.76013
[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., 8, 513-554 (2002), A tribute to J.-L. Lions · Zbl 1071.76012
[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; 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), 6, 3, 367-398 (2004) · Zbl 1061.93054
[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., 4, 4, 429-448 (1996) · Zbl 0938.93030
[10] Fursikov, A. V.; Imanuvilov, O. Yu., Exact controllability of the Navier-Stokes and Boussinesq equations, Russian Math. Surveys, 54, 565-618 (1999) · Zbl 0970.35116
[11] Glass, O., Exact boundary controllability of 3-D Euler equation, ESAIM Control Optim. Calc. Var., 5, 1-44 (2000), (electronic) · Zbl 0940.93012
[12] Glass, O., On the controllability of the Vlasov-Poisson system, J. Differential Equations, 195, 2, 332-379 (2003) · Zbl 1109.93007
[13] Horsin, T., On the controllability of the Burgers equation, ESAIM Control Optim. Calc. Var., 3, 83-95 (1998), (electronic) · Zbl 0897.93034
[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; 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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