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

Published online by Cambridge University Press:  10 February 2009

Camille Laurent
Camille Laurent*
Affiliation:
Université Paris-Sud, Bâtiment 425, 91405 Orsay, France. [email protected]
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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.

Keywords

Controllabilitystabilizationnonlinear Schrödinger equationBourgain spaces
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 2 , April 2010 , pp. 356 - 379
Copyright
© EDP Sciences, SMAI, 2009

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References

Bardos, C. and Masrour, T., 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).
Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part I. GAFA Geom. Funct. Anal. 3 (1993) 107–156. CrossRef
J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, in Colloquium publications 46, American Mathematical Society, Providence, RI (1999) 105.
Burq, N. and Zworski, M., Geometric control in the presence of a black box. J. Amer. Math. Soc. 17 (2004) 443–471. CrossRef
Burq, N., Gérard, P. and Tzvetkov, N., Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Amer. J. Math. 126 (2004) 569–605. CrossRef
B. Dehman and G. Lebeau, Analysis of the HUM Control Operator and Exact Controllability for Semilinear Waves in Uniform Time. Preprint.
Dehman, B., Lebeau, G. and Zuazua, E., Stabilization and control for the subcritical semilinear wave equation. Ann. Sci. École Norm. Sup. 36 (2003) 525–551. CrossRef
Dehman, B., Gérard, P. and Lebeau, G., Stabilization and control for the nonlinear Schrödinger equation on a compact surface. Math. Z. 254 (2006) 729–749. CrossRef
Gérard, P., Microlocal defect measures. Comm. Partial Diff. Equ. 16 (1991) 1762–1794. CrossRef
J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace, in Séminaire Bourbaki 37, exposé 796 (1994–1995) 163–187.
Isakov, V., Carleman type estimates in an anisotropic case and applications. J. Differ. Equ. 105 (1993) 217–238. CrossRef
Jaffard, S., 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).
Lebeau, G., 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.
Miller, L., Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal. 218 (2005) 425–444. CrossRef
L. Molinet, On ill-posedness for the one-dimensional periodic cubic Schrödinger equation. Math. Res. Lett. (to appear).
Phung, K.-D., Observability and control of Schrödinger equations. SIAM J. Control Optim. 40 (2001) 211–230. CrossRef
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 Mathematics 106. American Mathematical Society (2006).
G. Tenenbaum and M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation. Trans. Amer. Math. Soc. (to appear) iecn.u-nancy.fr.
Zuazua, E., Exact controllability for the semilinear wave equation. J. Math. Pures Appl. 69 (1990) 33–55.