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Controllability of a quantum particle in a moving potential well. (English) Zbl 1188.93017
Summary: We consider a nonrelativistic charged particle in a 1D moving potential well. This quantum system is subject to a control, which is the acceleration of the well. It is represented by a wave function solution of a Schrödinger equation, the position of the well together with its velocity. We prove the following controllability result for this bilinear control system: given \(\psi_{0}\) close enough to an eigenstate and \(\psi_{f}\) close enough to another eigenstate, the wave function can be moved exactly from \(\psi_{0}\) to \(\psi_{f}\) in finite time. Moreover, we can control the position and the velocity of the well. Our proof uses moment theory, a Nash–Moser implicit function theorem, the return method and expansion to the second order.

MSC:
93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
35Q55 NLS equations (nonlinear Schrödinger equations)
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[1] K. Beauchard, Local controllability of a 1-D Schrödinger equation, Prépubl. Univ. Paris Sud, (accepted in Journal de Mathématiques Pures et Appliquées), 2004.
[2] Coron, J.-M., Global asymptotic stabilization for controllable systems without drift, Math. control signals systems, 5, 295-312, (1992) · Zbl 0760.93067
[3] Coron, J.-M., Contrôlabilité exacte frontière de l’équation d’euler des fluides parfaits incompressibles bidimensionnels, C.R. acad. sci. Paris, 317, 271-276, (1993) · Zbl 0781.76013
[4] Coron, J.-M., On the controllability of 2-D incompressible perfect fluids, J. math. pures appl., 75, 155-188, (1996) · Zbl 0848.76013
[5] Coron, J.-M., On the controllability of the 2-D incompressible navier – stokes equations with the Navier slip boundary conditions, Esaim: cocv, 1, 35-75, (1996) · Zbl 0872.93040
[6] Coron, J.-M., Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations, Esaim: cocv, 8, 513-554, (2002) · Zbl 1071.76012
[7] Coron, J.-M.; Crépeau, E., Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. European math. soc., 6, 367-398, (2004) · Zbl 1061.93054
[8] Coron, J.-M.; Fursikov, A., Global exact controllability of the 2-D navier – stokes equations on a manifold without boundary, Russian journal of mathematical physics, 4, 429-448, (1996) · Zbl 0938.93030
[9] A.V. Fursikov, O.Yu. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations, Russian Math. Surveys 54 (3) (1999) 565-618; Uspekhi Mat. Nauk 54 (3) (1999) 565-618 (in Russian). · Zbl 0970.35116
[10] Glass, O., Exact boundary controllability of 3-D Euler equation, Esaim: cocv, 5, 1-44, (2000) · Zbl 0940.93012
[11] Glass, O., On the controllability of the vlasov – poisson system, Journal of differential equations, 195, 332-379, (2003) · Zbl 1109.93007
[12] Gromov, G., Partial differential relations, (1986), Springer Berlin, New York, London · Zbl 0651.53001
[13] Hörmander, L., On the nash – moser implicit function theorem, Ann. acad. sci. fennicae, 255-259, (1985) · Zbl 0591.58003
[14] Horsin, Th., On the controllability of the Burgers equation, Esaim: cocv, 3, 83-95, (1998) · Zbl 0897.93034
[15] Kato, T., Perturbation theory for linear operators, (1966), Springer Berlin, New York · Zbl 0148.12601
[16] P. Rouchon, Control of a quantum particule in a moving potential well, Second IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Seville, 2003.
[17] E. Zuazua, Remarks on the controllability of the Schrödinger equation, CRM Proceedings and Lecture Notes, vol. 33, 2003.
[18] G. Turinici, On the controllability of bilinear quantum systems, in: M. Defranceschi, C. Le Bris (Eds.), Mathematical models and methods for ab initio quantum chemistry. Lecture Notes in Chem. Vol. 74, Springer, Berlin, 2000, pp. 75-92.
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