Global controllability and stabilization for the nonlinear Schrödinger equation on an interval

Camille Laurent

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 2, page 356-379
  • ISSN: 1292-8119

Abstract

top
We prove global internal controllability in large time for the nonlinear Schrödinger equation on a bounded interval with periodic, Dirichlet or Neumann conditions. Our strategy combines stabilization and local controllability near 0. We use Bourgain spaces to prove this result on L2. We also get a regularity result about the control if the data are assumed smoother.

How to cite

top

Laurent, Camille. "Global controllability and stabilization for the nonlinear Schrödinger equation on an interval." ESAIM: Control, Optimisation and Calculus of Variations 16.2 (2010): 356-379. <http://eudml.org/doc/250771>.

@article{Laurent2010,
abstract = { We prove global internal controllability in large time for the nonlinear Schrödinger equation on a bounded interval with periodic, Dirichlet or Neumann conditions. Our strategy combines stabilization and local controllability near 0. We use Bourgain spaces to prove this result on L2. We also get a regularity result about the control if the data are assumed smoother. },
author = {Laurent, Camille},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Controllability; stabilization; nonlinear Schrödinger equation; Bourgain spaces; controllability},
language = {eng},
month = {4},
number = {2},
pages = {356-379},
publisher = {EDP Sciences},
title = {Global controllability and stabilization for the nonlinear Schrödinger equation on an interval},
url = {http://eudml.org/doc/250771},
volume = {16},
year = {2010},
}

TY - JOUR
AU - Laurent, Camille
TI - Global controllability and stabilization for the nonlinear Schrödinger equation on an interval
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/4//
PB - EDP Sciences
VL - 16
IS - 2
SP - 356
EP - 379
AB - We prove global internal controllability in large time for the nonlinear Schrödinger equation on a bounded interval with periodic, Dirichlet or Neumann conditions. Our strategy combines stabilization and local controllability near 0. We use Bourgain spaces to prove this result on L2. We also get a regularity result about the control if the data are assumed smoother.
LA - eng
KW - Controllability; stabilization; nonlinear Schrödinger equation; Bourgain spaces; controllability
UR - http://eudml.org/doc/250771
ER -

References

top
  1. C. Bardos and T. Masrour, Mesures de défaut : observation et contrôle de plaques. C. R. Acad. Sci. Paris Sér. I Math.323 (1996) 621–626.  
  2. J. Bergh and J. Löfstrom, Interpolation Spaces, An Introduction. Springer Verlag (1976).  
  3. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part I. GAFA Geom. Funct. Anal.3 (1993) 107–156.  
  4. J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, in Colloquium publications46, American Mathematical Society, Providence, RI (1999) 105.  
  5. N. Burq and M. Zworski, Geometric control in the presence of a black box. J. Amer. Math. Soc.17 (2004) 443–471.  
  6. N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Amer. J. Math.126 (2004) 569–605.  
  7. B. Dehman and G. Lebeau, Analysis of the HUM Control Operator and Exact Controllability for Semilinear Waves in Uniform Time. Preprint.  
  8. B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation. Ann. Sci. École Norm. Sup.36 (2003) 525–551.  
  9. B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface. Math. Z.254 (2006) 729–749.  
  10. P. Gérard, Microlocal defect measures. Comm. Partial Diff. Equ.16 (1991) 1762–1794.  
  11. J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace, in Séminaire Bourbaki37, exposé 796 (1994–1995) 163–187.  
  12. V. Isakov, Carleman type estimates in an anisotropic case and applications. J. Differ. Equ.105 (1993) 217–238.  
  13. S. Jaffard, Contrôle interne exact des vibrations d'une plaque rectangulaire. Portugal. Math.47 (1990) 423–429.  
  14. V. Komornik and P. Loreti, Fourier Series in Control Theory. Springer (2005).  
  15. G. Lebeau, Contrôle de l'équation de Schrödinger. J. Math. Pures Appl.71 (1992) 267–291.  
  16. E. Machtyngier, Exact controllability for the Schrödinger equation. SIAM J. Control Optim. 32 (1994) 24–34.  
  17. L. Miller, Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal.218 (2005) 425–444.  
  18. L. Molinet, On ill-posedness for the one-dimensional periodic cubic Schrödinger equation. Math. Res. Lett. (to appear).  
  19. K.-D. Phung, Observability and control of Schrödinger equations. SIAM J. Control Optim.40 (2001) 211–230.  
  20. L. Rosier and B.-Y. Zhang, Exact controllability and stabilization of the nonlinear Schrödinger equation on a bounded interval. SIAM J. Control Optim. (to appear).  
  21. T. Tao, Nonlinear Dispersive Equations, Local and global Analysis, CBMS Regional Conference Series in Mathematics106. American Mathematical Society (2006).  
  22. G. Tenenbaum and M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation. Trans. Amer. Math. Soc. (to appear) .  URIiecn.u-nancy.fr
  23. E. Zuazua, Exact controllability for the semilinear wave equation. J. Math. Pures Appl.69 (1990) 33–55.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.