,
, L, issue.0
0 (·) ? H 1 (0, ·) H 3 (0,L) + V 0 (·) ? V 1 (0, ·) H 3 (0,L) + |Z 0 | ,
, ·) H 2 (0,L) + |Z 0 | ? ?
, , vol.1, p.27
, is a C 2 function in u, t and x (and in particular C 1 ) and as
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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. Keywords: stabilization, partial differential equations, inhomogeneous, nonlinear, Boundary feedback control, Lyapunov functions, entropy, Saint-Venant equations, Burgers' equation. Stabilisation de systèmes hyperboliques non-linéaires en dimension un d'espace Résumé: Cette thèse est consacréeà l'étude de la stabilisation des systèmes d'équations aux dérivées partielles hyperboliques non-linéaires. L'objectif principal est de trouver des conditions de bords garantissant la stabilité exponentielle du système. Dans une première partie on s'intéresseà des systèmes généraux qu'on chercheà stabiliser en norme C 1 en introduisant un certain type de fonctions de Lyapunov, puis on regarde plus précisément les systèmes de deuxéquations pour lesquels on peut comparer nos résultats avec la stabilisation en norme H 2 . On s'intéresse ensuiteà quelqueséquations physiques: l'équation de Burgers et les systèmes densité-vélocité, dont font partie leséquations de Saint-Venant et leséquations d'Euler isentropiques. A l'aide d'une entropie locale dissipative, on montre qu'on peut stabiliser les systèmes densité-vélocité par des contrôles aux bords simples et,étonnement, ces contrôles ne dépendent pas explicitement des paramètres du système, pourvu qu'ils soient physiquement admissibles. Par ailleurs, on développe une méthode pour stabiliser lesétats-stationnaires avec un choc dans le cas de l'équation de Burgers et deséquations de Saint-Venant. Enfin, dans une troisième partie on s'intéresse aux contrôles proportionnels-intégraux (PI), très utilisés en pratique mais mal compris mathématiquement dans le cas des systèmes non-linéaires de dimension infinie, Classical Solutions for Quasilinear Hyperbolic Systems. PhD thesis. Stabilization of 1D nonlinear hyperbolic systems by boundary controls physical equations: Burgers' equation and the densityvelocity systems, which include the Saint-Venant equations and the Euler isentropic equations ,