Contrôlabilité de systèmes de réaction-diffusion non linéaires

Abstract : This thesis is devoted to the control of nonlinear partial differential equations. We are mostly interested in nonlinear parabolic reaction-diffusion systems in reaction kinetics. Our main goal is to prove local or global controllability results in small time or in large time.In a first part, we prove a local controllability result to nonnegative stationary states in small time, for a nonlinear reaction-diffusion system.In a second part, we solve a question concerning the global null-controllability in small time for a 2 × 2 nonlinear reaction-diffusion system with an odd coupling term.The third part focuses on the famous open problem due to Enrique Fernndez-Cara and Enrique Zuazua in 2000, concerning the global null-controllability of the weak semi-linear heat equation. We show that the equation is globally nonnegative controllable in small time and globally null-controllable in large time.The last part, which is a joint work with Karine Beauchard and Armand Koenig, enters the hyperbolic world. We study linear parabolic-transport systems with constant coeffcients. We identify their minimal time of control and the influence of their algebraic structure on the controllability properties.
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Kévin Le Balc'H. Contrôlabilité de systèmes de réaction-diffusion non linéaires. Mathématiques générales [math.GM]. École normale supérieure de Rennes, 2019. Français. ⟨NNT : 2019ENSR0016⟩. ⟨tel-02276541⟩

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