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Observability of the Schrödinger Equation

  • Kim dang Phung
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 46)

Abstract

The goal here is to present two approaches concerning results on observability and control of the Schrödinger equation in a bounded domain. Our results are obtained from different works on the control of the heat equation or of the wave equation. From the theory of exact and approximate controllability, introduced by J.L. Lions [10], we know that observation is equivalent to approximate controllability and stable observation is equivalent to exact controllability.

Our first result is based on a gaussian transform which traduces any estimate of stable observability of the heat equation to an estimate of unstable observability for the Schrödinger equation(see 1 below).This work is similar to those done by L.Robbiano [14] for hyperbolic problems on the domain where the geometrical control condition of C. Bardos, G. Lebeau and J. Rauch on the exact controllability of the wave equation [1] is not satisfied.

Our second result is about exact control for the Schrödinger equation see 2) and is inspired by a transform introduced by L. Boutet de Monvel [2] for the study of the propagation of singularities of an analogous solution of the Schrödinger equation. Our strategy is to construct an exact control for the Schrödinger equation from an exact controllability result for the wave equation.

Keywords

Initial Data Wave Equation Heat Equation Boundary Control Evolution Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30(5) (1992), 1024–1065.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    L. Boutet de Monvel, Propagation des singularités des solutions d’équations analogues à l’équation de Schrödinger, Lecture Notes in Mathematics, 459, 1975.Google Scholar
  3. [3]
    N. Burq, Contrôle de l’équation des plaques en présence d’obstacles strictement convexes, Mémoires S.M.F., 55, nouvelle série, 1993.Google Scholar
  4. [4]
    N. Burq and P. Gérard, condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Acad. Sci. Paris, t. 325, série 1, 1997,749–752.CrossRefGoogle Scholar
  5. [5]
    C. Fabre, Résultats de contrôlabilité exacte interne pour l’équation de Schrödinger et leurs limites asymptotiques: application à certaines équations de plaques vibrantes, Asymptotic Analysis, 5 (1992), 343 - 379.MathSciNetzbMATHGoogle Scholar
  6. [6]
    E. Fernandez-Cara and E. Zuazua, Conférence à Cortona, 1999.Google Scholar
  7. [7]
    A.V. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, no. 34.Google Scholar
  8. [8]
    M.A. Horn and W. Littman, Boundary Control of a Schrödinger Equation with Nonconstant Principal Part, Lecture Notes in Pure and Applied Mathematics, no. 174.Google Scholar
  9. [9]
    G. Lebeau, Contrôle de l’équation de Schrödinger, J. Math. Pures et Applic., 71 (1992), 267–291.MathSciNetzbMATHGoogle Scholar
  10. [10]
    J.-L. Lions, Contrôlabilité exacte, stabilisation et perturbation des systèmes distribués, 1, Coll. RMA, Masson, Paris, 1998.Google Scholar
  11. [11]
    G. Lebeau and L. Robbiano, Contrôle exacte de l’équation de la chaleur, Comm. Part. Diff. Eq., 20 (1995), 335–356.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilisation of Schrödinger equations with Dirichlet control, Differential and Integral Equations, 5 (1992), 521–535.MathSciNetzbMATHGoogle Scholar
  13. [13]
    E. Machtyngier, Exact controllability for the Schrödinger equation, SIAM J. Control Optm., 32(1) (1994), 24–34.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    L. Robbiano, Fonction de coût et contrôle des solutions des équations hyperboliques, Asymptotic Analysis, 10 (1995), 95–115.MathSciNetzbMATHGoogle Scholar
  15. [15]
    D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems, J. Math. Pures et Applic., 75 (1996), 367–408.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Kim dang Phung
    • 1
  1. 1.ChatillonFrance

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