Contrôle d'équations aux dérivées partielles non linéaires dispersives

Abstract : In this thesis, we study the controllability and the stabilization of some dispersive partial differential equations. We are first interested in the internal control. Thanks to some microlocal analysis methods and the use of Bourgain spaces, we prove stabilization and control in large time for the nonlinear Schrödinger equation on an interval and then on some manifolds of dimension 3. In the case of an interval, we work at the L^2 regularity, which allows to deal with both focusing and defocusing nonlinearity. We also obtain additional results about the regularity of the control. Moreover, we prove the controllability near trajectories, from which we deduce a second proof of global controllability. We then apply these methods to the Korteweg-de Vries equation on a periodic domain. For this equation, we also provide a time dependent damping term which enables an arbitrary exponential decay rate. We also study the Klein Gordon equation with a critical nonlinearity on some compact manifolds. Under some assumptions slightly stronger than the geometric control condition, we prove the stabilization and controllability in large time for high frequency data. The proof requires the statement of a prole decomposition on manifolds for which some geometric effects have to be analysed. In a last part, we study the bilinear control. Thanks to a regularizing effect, we establish the local controllability of the Schrödinger equation on an interval with a proof simpler than in the available litterature, allowing to reach the optimal spaces and in an arbitrary time. The method is robust enough to be extended to other situations: radial data on a ball, non linear Schrödinger equation and non linear wave equation on an interval.
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https://tel.archives-ouvertes.fr/tel-00536082
Contributor : Camille Laurent <>
Submitted on : Monday, November 15, 2010 - 12:32:25 PM
Last modification on : Thursday, October 10, 2019 - 1:08:53 AM
Long-term archiving on : Wednesday, February 16, 2011 - 2:51:16 AM

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Camille Laurent. Contrôle d'équations aux dérivées partielles non linéaires dispersives. Mathématiques [math]. Université Paris Sud - Paris XI, 2010. Français. ⟨tel-00536082⟩

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