Contrôle d'équations de Schrödinger et d'équations paraboliques dégénérées singulières

Abstract : This memoir presents the achievements of my thesis on the control of partial differential equations. The first part mainly deals with two aspects of bilinear Schrödinger equations : negative controllability results in small time with small controls and simultaneous controllability. We propose a general setting for the existence of a positive minimal time for local exact controllability to hold with small controls. The negative result, based on the coercivity of a quadratic form associated to a second order power series expansion, is extended to simultaneous controllability. Using J.-M.~Coron's return method, we prove simultaneous local exact controllability for two or three equations, up to a global phase and/or up to a global delay. The reference trajectory is designed using partial control results. Using a perturbation argument, this idea is extended to prove simultaneous global exact controllability of an arbitrary number of equations without restrictions on the potential. In the second part, we add, in the model, a polarizability term which is quadratic with respect to the control. Taking into account this physically meaningfull term, usually neglected, is interesting when the dipole moment is not sufficient to control the particle. In any space dimension, we obtain approximate controllability towards the ground state with explicit controls. The previous perturbation argument together with tools from bilinear control lead to global exact controllability of the 1D Schrödinger equation with a polarizability term. The last part deals with a Grushin-type operator on 2D rectangle. This operator is both degenerate and singular on a line that separates the domain in two components. A necessary and sufficient condition on the coefficient of the singular potential for unique continuation to hold.
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Submitted on : Wednesday, December 4, 2013 - 11:27:37 AM
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Morgan Morancey. Contrôle d'équations de Schrödinger et d'équations paraboliques dégénérées singulières. Optimisation et contrôle [math.OC]. Ecole Polytechnique X, 2013. Français. ⟨NNT : 2013EPXX0125⟩. ⟨pastel-00913679⟩

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