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internal control and stabilization of some 1-D hyperbolic systems

Abstract : In this thesis we study controllability and stabilization questions for some hyperbolic systems in one space dimension, with an internal control. The first question we study is the indirect internal controllability of a system of two coupled semilinear wave equations, the control being a function of time and space. Using the so-called fictitious control method, we give sufficient conditions for such a system to be locally controllable around 0, and a natural condition linking the minimal control time to the support of the control. Then, we study a particular case where the aforementioned sufficient conditions are not satisfied, applying the return method. The second question in this thesis is the design of explicit scalar feedbacks to stabilize controllable systems. The method we use draws from the backstepping method for PDEs elaborated by Miroslav Krstic, and its most recent developments: thus, the controllability of the system under consideration plays a crucial role. The method yields explicit stationary feedbacks which stabilize the linear periodic transport equation exponentially, and even in finite time. Finally, we implement this method on a more complex system, the so-called water tank system. We prove that the linearized systems around constant acceleration equilibria are controllable if the acceleration is not too strong. Our method then yields explicit feedbacks which, although they remain explicit, are no longer stationary and require the addition of an integrator in the feedback loop.
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Submitted on : Sunday, February 2, 2020 - 6:43:17 PM
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  • HAL Id : tel-02464011, version 1

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Christophe Zhang. internal control and stabilization of some 1-D hyperbolic systems. Optimization and Control [math.OC]. Sorbonne Université, 2019. English. ⟨tel-02464011⟩

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