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

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

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

## Access Full Article

top## Abstract

top## How to cite

topLaurent, 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- 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.
- J. Bergh and J. Löfstrom, Interpolation Spaces, An Introduction. Springer Verlag (1976).
- 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.
- J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, in Colloquium publications46, American Mathematical Society, Providence, RI (1999) 105.
- N. Burq and M. Zworski, Geometric control in the presence of a black box. J. Amer. Math. Soc.17 (2004) 443–471.
- N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Amer. J. Math.126 (2004) 569–605.
- B. Dehman and G. Lebeau, Analysis of the HUM Control Operator and Exact Controllability for Semilinear Waves in Uniform Time. Preprint.
- 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.
- 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.
- P. Gérard, Microlocal defect measures. Comm. Partial Diff. Equ.16 (1991) 1762–1794.
- 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.
- V. Isakov, Carleman type estimates in an anisotropic case and applications. J. Differ. Equ.105 (1993) 217–238.
- S. Jaffard, Contrôle interne exact des vibrations d'une plaque rectangulaire. Portugal. Math.47 (1990) 423–429.
- V. Komornik and P. Loreti, Fourier Series in Control Theory. Springer (2005).
- G. Lebeau, Contrôle de l'équation de Schrödinger. J. Math. Pures Appl.71 (1992) 267–291.
- E. Machtyngier, Exact controllability for the Schrödinger equation. SIAM J. Control Optim. 32 (1994) 24–34.
- L. Miller, Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal.218 (2005) 425–444.
- L. Molinet, On ill-posedness for the one-dimensional periodic cubic Schrödinger equation. Math. Res. Lett. (to appear).
- K.-D. Phung, Observability and control of Schrödinger equations. SIAM J. Control Optim.40 (2001) 211–230.
- 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).
- T. Tao, Nonlinear Dispersive Equations, Local and global Analysis, CBMS Regional Conference Series in Mathematics106. American Mathematical Society (2006).
- G. Tenenbaum and M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation. Trans. Amer. Math. Soc. (to appear) . URIiecn.u-nancy.fr
- E. Zuazua, Exact controllability for the semilinear wave equation. J. Math. Pures Appl.69 (1990) 33–55.

## NotesEmbed ?

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