Controllability of Schrödinger equations
Séminaire Équations aux dérivées partielles (Polytechnique) (2005-2006), Talk no. 9, 18 p.

One considers a quantum particle in a 1D moving infinite square potential well. It is a nonlinear control system in which the state is the wave function of the particle and the control is the acceleration of the potential well. One proves the local controllability around any eigenstate, and the steady state controllability (controllability between eigenstates) of this control system. In particular, the wave function can be moved from one eigenstate to another one, exactly and in finite time, by moving the potential well in a suitable way.

The proof uses moment theory, a Nash-Moser theorem, Coron’s return method and expansions to the second order.

This article summarizes two works : [4] and a joint work with Jean-Michel Coron [5].

@article{SEDP_2005-2006____A9_0,
     author = {Beauchard, Karine},
     title = {Controllability of Schr\"odinger equations},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2005-2006},
     note = {talk:9},
     mrnumber = {2276075},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_2005-2006____A9_0}
}
Beauchard, Karine. Controllability of Schrödinger equations. Séminaire Équations aux dérivées partielles (Polytechnique) (2005-2006), Talk no. 9, 18 p. http://www.numdam.org/item/SEDP_2005-2006____A9_0/

[1] Ball, J.M.; Marsden, J.E.; Slemrod, M. Controllability for distributed bilinear systems, SIAM J. Control and Optim., Tome 20 (1982) | MR 661034 | Zbl 0485.93015

[2] Beauchard, K. Controllability of a quantum particle in a 1D infinite square potential well with variable length, ESAIM:COCV (accepted)

[3] Beauchard, K. Local Controllability of a 1-D beam equation (submitted), SIAM J. of Contr. and Optim. (submitted)

[4] Beauchard, K. Local Controllability of a 1-D Schrödinger equation, J. Math. Pures et Appl., Tome 84 (2005), pp. 851-956 | MR 2144647 | Zbl 02228731

[5] Beauchard, K.; Coron, J.-M. Controllability of a quantum particle in a moving potential well, accepted in J. Functional Analysis (2005) | MR 2200740 | Zbl 05017416

[6] Coron, J.-M. Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems, Tome 5 (1992), pp. 295-312 | MR 1164379 | Zbl 0760.93067

[7] Coron, J.-M. Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles bidimensionnels, C. R. Acad. Sci. Paris, Tome 317 (1993), pp. 271-276 | Zbl 0781.76013

[8] Coron, J.-M. On the controllability of 2-D incompressible perfect fluids, J. Math. Pures Appl., Tome 75 (1996), pp. 155-188 | MR 1380673 | Zbl 0848.76013

[9] Coron, J.-M. Local Controllability of a 1-D Tank Containing a Fluid Modeled by the shallow water equations, ESAIM: COCV, Tome 8 (2002), pp. 513-554 | Numdam | MR 1932962 | Zbl 1071.76012

[10] Coron, J.-M. On the small-time local controllability of a quantum particule in a moving one-dimensional infinite square potential well, C.R.A.S. (to appear) (2005) | MR 2193655 | Zbl 1082.93002

[11] Coron, J.-M.; Crépeau, E. Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. Eur. Math. Soc., Tome 6 (2004), pp. 367-398 | MR 2060480 | Zbl 1061.93054

[12] Coron, J.-M.; Fursikov, A. Global exact controllability of the 2D Navier-Stokes equation on a manifold without boundary, Russian Journal of Mathematical Physics, Tome 4 (1996), pp. 429-448 | MR 1470445 | Zbl 0938.93030

[13] Fursikov, A. V.; Imanuvilov, O. Yu. Exact controllability of the Navier-Stokes and Boussinesq equations, Russian Math. Surveys, Tome 54 (1999), pp. 565-618 | MR 1728643 | Zbl 0970.35116

[14] Glass, O. Exact boundary controllability of 3-D Euler equation, ESAIM: COCV, Tome 5 (2000), pp. 1-44 | Numdam | MR 1745685 | Zbl 0940.93012

[15] Glass, O. On the controllability of the Vlasov-Poisson system, Journal of Differential Equations, Tome 195 (2003), pp. 332-379 | MR 2016816 | Zbl 1109.93007

[16] Haraux, A. Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire, J. Math. Pures et Appl., Tome 68 (1989), pp. 457-465 | Zbl 0685.93039

[17] Horsin, Th. On the controllability of the Burgers equation, ESAIM: COCV, Tome 3 (1998), pp. 83-95 | Numdam | MR 1612027 | Zbl 0897.93034

[18] Krabs, W. On moment theory and controllability of one-dimensional vibrating systems and heating processes, Springer Verlag (1992) | MR 1162111 | Zbl 0955.93501

[19] L. Hörmander On the Nash-Moser Implicit Function Theorem, Annales Academiae Scientiarum Fennicae (1985), pp. 255-259 | MR 802486 | Zbl 0591.58003

[20] Rouchon, P. Control of a quantum particule in a moving potential well, 2nd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Seville (2003) | MR 2082989

[21] Turinici, G. On the controllability of bilinear quantum systems, In C. Le Bris and M. Defranceschi, editors, Mathematical Models and Methods for Ab Initio Quantum Chemistry, Tome volume 74 of Lecture Notes in Chemistry (2000) | MR 1857459 | Zbl 1007.81019