Observability of the Schrödinger Equation
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 , 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  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  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  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.
KeywordsInitial Data Wave Equation Heat Equation Boundary Control Evolution Problem
Unable to display preview. Download preview PDF.
- 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
- N. Burq, Contrôle de l’équation des plaques en présence d’obstacles strictement convexes, Mémoires S.M.F., 55, nouvelle série, 1993.Google Scholar
- E. Fernandez-Cara and E. Zuazua, Conférence à Cortona, 1999.Google Scholar
- A.V. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, no. 34.Google Scholar
- 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
- J.-L. Lions, Contrôlabilité exacte, stabilisation et perturbation des systèmes distribués, 1, Coll. RMA, Masson, Paris, 1998.Google Scholar