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Stabilité de Lyapunov de systèmes couplés impliquant une équation de transport

Mohammed Safi 1
1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes
Abstract : This thesis in control theory aims at proposing a novel approach for the stability study of an infinite dimensional system where an ordinary differential equation is coupled to a transport equation through boundary terms. The idea is to exploit recent works on delay systems to quantify the stability of a system coupling a partial differential equations to ordinary differential equations. These works rely on Legendre’s polynomials and Bessel’s inequality to construct a novel approach of stability by the Lyapunov method and the use of linear matrix inequalities. Legendre’s polynomials allow construct a new structure of Lyapunov functional based partly on a polynomial approximation of the state of the transport equation (which is of infinite dimension). The manuscript is divided into several stages. After the presentation of a simple model coupling ordinary differential equations with one transport equation, the approximation of the infinite dimensional state using projection on Legendre polynomials is described. The Lyapunov method is then developed and it requires the production of stability conditions taking the shape of linear matrix inequalities. These conditions allow the production of numerical tests performed on academic examples. More difficult cases are discussed throughout the document, from a single equation of transport to several equations with different speeds, taking into account a term of coupling between them via a potential or the boundary of the domain. Finally, since such a coupling of a finite dimensional system with a transport equation can be an alternative description of a delay system, a study of the stability of the latter is developed using different models of the coupled system, in order to reduce the complexity of the stability conditions given in the form of matrix inequalities.
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Mohammed Safi. Stabilité de Lyapunov de systèmes couplés impliquant une équation de transport. Automatique. Institut supérieur de l'aéronautique et de l'espace, 2018. Français. ⟨tel-01975119⟩

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