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Habilitation à diriger des recherches

Problèmes inverses et commande robuste de quelques équations aux dérivées partielles

Lucie Baudouin 1
1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes
Abstract : This manuscript gathers several works on partial differential equations models, focusing on inverse problems on the one hand and on robust control with respect to disturbances on the other hand. The first part of this document is introduced by a glimpse of the bibliography concerning coefficient inverse problems in the main evolution equations. The first chapter is structured around the proof of stability in the determination of a potential in a basic wave equation posed in a bounded domain. It allows me to describe some more exotic cases where the wave equation is either with transmission conditions on an interface, or posed on a 1-d network, or even semi-discretized in space. The main tool is an appropriate Carleman inequality. The method that allows to deduce the stability of the inverse problem is then used to prove the convergence of a new scheme for the numerical reconstruction of the potential.We also prove convergence results for discretized inverse problems. The second chapter contains the study of the stability of coefficient inverse problems for some parabolic (Kuramoto-Sivashinisky) and dispersive (Schrödinger and Korteweg-de-Vries) equations. The second part of the manuscript begins with a presentation of the H1 robust control of systems described by partial differential equations and a theorem giving, as in the finite dimensional setting, the equivalence between robust control and the resolution of Riccati equations. Our aim was to formulate, on specific applicative settings, theoretical control results guaranteeing a certain robustness of the stability of the system ensuring the disturbance rejection. The third chapter focuses on the study of an adaptive optics system (the dynamics of the deformable mirror being modeled by a plate equation) that we manage to put in this framework. The fourth chapter presents the study of a coupled fluid/structure system, modeled using partial differential equations (plate, Laplace and Bernoulli equations) and then reduced to a finite-dimensional system, studied using usual tools of control (pole placement, observers, reduced order controllers) prior to actual plant testing.
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Habilitation à diriger des recherches
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Submitted on : Tuesday, September 23, 2014 - 2:58:40 PM
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Lucie Baudouin. Problèmes inverses et commande robuste de quelques équations aux dérivées partielles. Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2014. ⟨tel-01067485⟩

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