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.
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Phung, K.d. (2001). Observability of the Schrödinger Equation. In: Colombini, F., Zuily, C. (eds) Carleman Estimates and Applications to Uniqueness and Control Theory. Progress in Nonlinear Differential Equations and Their Applications, vol 46. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0203-5_12
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DOI: https://doi.org/10.1007/978-1-4612-0203-5_12
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