Global controllability and stabilization for the nonlinear Schrödinger equation on an interval
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 2, pp. 356-379.

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.

DOI: 10.1051/cocv/2009001
Classification: 93B05,  93D15,  35Q55,  35A21
Keywords: controllability, stabilization, nonlinear Schrödinger equation, Bourgain spaces
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author = {Laurent, Camille},
title = {Global controllability and stabilization for the nonlinear {Schr\"odinger} equation on an interval},
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url = {http://archive.numdam.org/articles/10.1051/cocv/2009001/}
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Laurent, Camille. Global controllability and stabilization for the nonlinear Schrödinger equation on an interval. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 2, pp. 356-379. doi : 10.1051/cocv/2009001. http://archive.numdam.org/articles/10.1051/cocv/2009001/

[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. | Zbl

[2] J. Bergh and J. Löfstrom, Interpolation Spaces, An Introduction. Springer Verlag (1976). | Zbl

[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. | Zbl

[4] J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, in Colloquium publications 46, American Mathematical Society, Providence, RI (1999) 105. | Zbl

[5] N. Burq and M. Zworski, Geometric control in the presence of a black box. J. Amer. Math. Soc. 17 (2004) 443-471. | Zbl

[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. | Zbl

[7] B. Dehman and G. Lebeau, Analysis of the HUM Control Operator and Exact Controllability for Semilinear Waves in Uniform Time. Preprint. | Zbl

[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. | Numdam | Zbl

[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. | Zbl

[10] P. Gérard, Microlocal defect measures. Comm. Partial Diff. Equ. 16 (1991) 1762-1794. | Zbl

[11] 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. | Numdam | Zbl

[12] V. Isakov, Carleman type estimates in an anisotropic case and applications. J. Differ. Equ. 105 (1993) 217-238. | Zbl

[13] S. Jaffard, Contrôle interne exact des vibrations d'une plaque rectangulaire. Portugal. Math. 47 (1990) 423-429. | Zbl

[14] V. Komornik and P. Loreti, Fourier Series in Control Theory. Springer (2005). | Zbl

[15] G. Lebeau, Contrôle de l'équation de Schrödinger. J. Math. Pures Appl. 71 (1992) 267-291. | Zbl

[16] E. Machtyngier, Exact controllability for the Schrödinger equation. SIAM J. Control Optim. 32 (1994) 24-34. | Zbl

[17] L. Miller, Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal. 218 (2005) 425-444. | Zbl

[18] L. Molinet, On ill-posedness for the one-dimensional periodic cubic Schrödinger equation. Math. Res. Lett. (to appear). | Zbl

[19] K.-D. Phung, Observability and control of Schrödinger equations. SIAM J. Control Optim. 40 (2001) 211-230. | Zbl

[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). | Zbl

[21] T. Tao, Nonlinear Dispersive Equations, Local and global Analysis, CBMS Regional Conference Series in Mathematics 106. American Mathematical Society (2006). | Zbl

[22] G. Tenenbaum and M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation. Trans. Amer. Math. Soc. (to appear) iecn.u-nancy.fr. | Zbl

[23] E. Zuazua, Exact controllability for the semilinear wave equation. J. Math. Pures Appl. 69 (1990) 33-55. | Zbl

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