BIRS Workshop Lecture Videos
Rough controls for SchrÃ¶dinger equation on the torus Burq, Nicolas
In this talk I will present some results on the exact controllability of SchrÃ¶dinger equation on the torus. In a general setting, these questions are well understood for wave equations with continuous localisation functions, while for SchrÃ¶dinger, we only have partial results. For rough localisation functions, I will first present some partial results for waves. Then I will show how one can take benefit from the particular simplicity of the geodesic flow on the torus to get (for continuous localisation functions) strong results (works by Haraux, Jaffard, Burq-Zworski, Anantharaman-Macia). Finally, for general localisation functions (typically characteristic functions of measurable sets) I will show how one can go further, by taking benefit from dispersive properties (on the 2 dimensional torus), to show that in this setting the SchrÃ¶dinger equation is exactly controllable by any $L^2$ (non trivial) localisation function (and in particular by the characteristic function of any set (with positive measure).
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