Stabilisation de systèmes hyperboliques non-linéaires en dimension un d’espace

Abstract : This thesis is devoted to study the stabilization of nonlinear hyperbolic systems of partial differential equations. The main goal is to find boundary conditions ensuring the exponential stability of the system. In a first part, we study general systems that we aim at stabilizing in the $C^{1}$ norm by introducing a certain type of Lyapunov functions. Then we take a closer look at systems of two equations and we compare the results with the stabilization in the $H^{2}$ norm. In a second part we study a few physical equations: Burgers’ equation and the density-velocity systems, which include the Saint-Venant equations and the Euler isentropic equations. Using a local dissipative entropy, we show that these systems can be stabilized with very simple boundary controls which, remarkably, do not depend directly on the parameters of the system, provided some physical admissibility condition. Besides, we develop a way to stabilize shock steady-states in the case of Burgers’ and Saint-Venant equations. Finally, in a third part, we study proportional-integral (PI) controllers, which are very popular in practice but seldom understood mathematically for nonlinear infinite dimensional systems. For scalar systems we introduce an extraction method to find optimal conditions on the parameters of the controller ensuring the stability. Finally, we deal with the Saint-Venant equations with a single PI control.
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Amaury Hayat. Stabilisation de systèmes hyperboliques non-linéaires en dimension un d’espace. Equations aux dérivées partielles [math.AP]. Sorbonne Université UPMC, 2019. Français. ⟨tel-02274457⟩

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