Contributions pour les systèmes non-linéaires de dimension infinie : commande et optimisation
Contributions pour les systèmes non-linéaires de dimension infinie : commande et optimisation
Résumé
This manuscript aims at presenting my research results since my PhD defense which took place in 2017. The main topic of this manuscript is the analysis of nonlinear infinite-dimensional systems (modeling for instance partial differential equations) from a control and an optimization point of view. Infinite-dimensional systems model numerous physical phenomena in electronics, mechanics, etc. Their study is hence crucial and requires the development of control and numerical tools specific to these models. This manuscript proposes some contributions allowing us to analyze the well-posedness of these equations, their asymptotic stability, and some numerical schemes which solve them We will propose general results of existence, uniqueness, asymptotic stability and regulation which concern large classes of infinite dimensional systems modeled thanks to the semigroup theory. Some contributions will deal with more specific PDEs (namely, the wave equation and the Korteweg-de Vries equation for which particular techniques are necessary for their study. Finally, the numerical schemes proposed in this manuscript are based on the Moment-SOS hierarchy (a polynomial optimization method) and on the Christoffel Darboux kernel.
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