Controllability of Schrödinger equations
Beauchard, Karine1
1 CMLA ENS Cachan 94230 Cachan
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé 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].

Beauchard, Karine 1

1 CMLA ENS Cachan 94230 Cachan 
@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) dit aussi "S\'eminaire Goulaouic-Schwartz"},
     note = {talk:9},
     pages = {1--18},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2005-2006},
     mrnumber = {2276075},
     language = {en},
     url = {http://archive.numdam.org/item/SEDP_2005-2006____A9_0/}
}
TY  - JOUR
AU  - Beauchard, Karine
TI  - Controllability of Schrödinger equations
JO  - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
N1  - talk:9
PY  - 2005-2006
SP  - 1
EP  - 18
PB  - Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://archive.numdam.org/item/SEDP_2005-2006____A9_0/
LA  - en
ID  - SEDP_2005-2006____A9_0
ER  - 
%0 Journal Article
%A Beauchard, Karine
%T Controllability of Schrödinger equations
%J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
%Z talk:9
%D 2005-2006
%P 1-18
%I Centre de mathématiques Laurent Schwartz, École polytechnique
%U http://archive.numdam.org/item/SEDP_2005-2006____A9_0/
%G en
%F SEDP_2005-2006____A9_0
Beauchard, Karine. Controllability of Schrödinger equations. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 9, 18 p. http://archive.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., Volume 20 (1982) | MR | Zbl

[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., Volume 84 (2005), pp. 851-956 | MR | Zbl

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

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

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

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

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

[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 | Zbl

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

[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, Volume 4 (1996), pp. 429-448 | MR | Zbl

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

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

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

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

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

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

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

[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

[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, Volume volume 74 of Lecture Notes in Chemistry (2000) | MR | Zbl