Controllability of Schrödinger equations

Karine Beauchard[1]

  • [1] CMLA ENS Cachan 94230 Cachan

Séminaire Équations aux dérivées partielles (2005-2006)

  • Volume: 84, Issue: 7, page 1-18

Abstract

top
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].

How to cite

top

Beauchard, Karine. "Controllability of Schrödinger equations." Séminaire Équations aux dérivées partielles 84.7 (2005-2006): 1-18. <http://eudml.org/doc/11144>.

@article{Beauchard2005-2006,
abstract = {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].},
affiliation = {CMLA ENS Cachan 94230 Cachan},
author = {Beauchard, Karine},
journal = {Séminaire Équations aux dérivées partielles},
language = {eng},
number = {7},
pages = {1-18},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Controllability of Schrödinger equations},
url = {http://eudml.org/doc/11144},
volume = {84},
year = {2005-2006},
}

TY - JOUR
AU - Beauchard, Karine
TI - Controllability of Schrödinger equations
JO - Séminaire Équations aux dérivées partielles
PY - 2005-2006
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 84
IS - 7
SP - 1
EP - 18
AB - 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].
LA - eng
UR - http://eudml.org/doc/11144
ER -

References

top
  1. J.M. Ball, J.E. Marsden, M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control and Optim. 20 (1982) Zbl0485.93015MR661034
  2. K. Beauchard, Controllability of a quantum particle in a 1D infinite square potential well with variable length, ESAIM:COCV (accepted) Zbl1132.35446
  3. K. Beauchard, Local Controllability of a 1-D beam equation (submitted), SIAM J. of Contr. and Optim. (submitted) Zbl1124.93009
  4. K. Beauchard, Local Controllability of a 1-D Schrödinger equation, J. Math. Pures et Appl. 84 (2005), 851-956 Zbl1124.93009MR2144647
  5. K. Beauchard, J.-M. Coron, Controllability of a quantum particle in a moving potential well, accepted in J. Functional Analysis (2005) Zbl1188.93017MR2200740
  6. J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems 5 (1992), 295-312 Zbl0760.93067MR1164379
  7. J.-M. Coron, Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles bidimensionnels, C. R. Acad. Sci. Paris 317 (1993), 271-276 Zbl0781.76013
  8. J.-M. Coron, On the controllability of 2-D incompressible perfect fluids, J. Math. Pures Appl. 75 (1996), 155-188 Zbl0848.76013MR1380673
  9. J.-M. Coron, Local Controllability of a 1-D Tank Containing a Fluid Modeled by the shallow water equations, ESAIM: COCV 8 (2002), 513-554 Zbl1071.76012MR1932962
  10. J.-M. Coron, 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) Zbl1082.93002MR2193655
  11. J.-M. Coron, E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. Eur. Math. Soc. 6 (2004), 367-398 Zbl1061.93054MR2060480
  12. J.-M. Coron, A. Fursikov, Global exact controllability of the 2D Navier-Stokes equation on a manifold without boundary, Russian Journal of Mathematical Physics 4 (1996), 429-448 Zbl0938.93030MR1470445
  13. A. V. Fursikov, O. Yu. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations, Russian Math. Surveys 54 (1999), 565-618 Zbl0970.35116MR1728643
  14. O. Glass, Exact boundary controllability of 3-D Euler equation, ESAIM: COCV 5 (2000), 1-44 Zbl0940.93012MR1745685
  15. O. Glass, On the controllability of the Vlasov-Poisson system, Journal of Differential Equations 195 (2003), 332-379 Zbl1109.93007MR2016816
  16. A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire, J. Math. Pures et Appl. 68 (1989), 457-465 Zbl0685.93039
  17. Th. Horsin, On the controllability of the Burgers equation, ESAIM: COCV 3 (1998), 83-95 Zbl0897.93034MR1612027
  18. W. Krabs, On moment theory and controllability of one-dimensional vibrating systems and heating processes, (1992), Springer Verlag Zbl0955.93501MR1162111
  19. L. Hörmander, On the Nash-Moser Implicit Function Theorem, Annales Academiae Scientiarum Fennicae (1985), 255-259 Zbl0591.58003MR802486
  20. P. Rouchon, Control of a quantum particule in a moving potential well, 2nd IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Seville (2003) MR2082989
  21. G. Turinici, 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 74 of Lecture Notes in Chemistry (2000) Zbl1007.81019MR1857459

NotesEmbed ?

top

You must be logged in to post comments.