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Solutions globales, limite de relaxation, contrôlabilité et observabilité exactes, frontières pour des systèmes hyperboliques quasi-linéaires

Abstract : This thesis is essentially composed of two parts. In the ¯rst part, I study the Euler- Maxwell system. Using the classical method of energy integral, I prove the existence and uniqueness of global solutions to the system with small initial data. After that, I study the relaxation limit. I prove that, as the relaxation time tends to zero, the Euler-Maxwell system converges to the drift-diffusion models. In the second part, I study the exact boundary controllability and observability of quasilinear hyperbolic systems in a tree-like network. In this part, based on the theory of the semi-global C1 solution of the mixed initial-boundary value problem for first order quasilinear hyperbolic systems, I deal with the controllability and observability with a constructive method. Taking the unsteady flows in a tree-like network of open canals as a physical model, I consider the exact boundary controllability and observability in subcritical and super- critical situations, respectively. By the comparison of these two cases, I find some duality of the controllability and observability. Meanwhile, using the similar way, I get the exact boundary controllability of quasilinear wave equations on a tree-like planar network of stings.
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Qilong Gu. Solutions globales, limite de relaxation, contrôlabilité et observabilité exactes, frontières pour des systèmes hyperboliques quasi-linéaires. Analyse numérique [math.NA]. Université Blaise Pascal - Clermont-Ferrand II, 2009. Français. ⟨NNT : 2009CLF21933⟩. ⟨tel-00725524⟩

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