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Stability results for some hyperbolic systems with direct or indirect local dampings

Abstract : In this thesis, we study the indirect stability of some coupled systems with different kinds of local discontinuous dampings. We also study the stability and the instability results of the Kirchhoff plate equation with delay terms on the boundary or dynamical boundary controls. First, we investigate the stabilization of locally coupled wave equations with non-smooth localized viscoelastic damping of Kelvin-Voigt type and localized time delay. Using a general criteria of Arendt-Batty, we show the strong stability of our system in the absence of the compactness of the resolvent. However, by combining the frequency domain approach with the multiplier method, we prove a polynomial energy decay rate. Second, we investigate the stabilization of locally coupled wave equations with local viscoelastic damping of past history type acting only on one equation via non-smooth coefficients. We prove the strong stability of our system. Next, we establish the exponential stability of the solution if the two waves have the same speed of propagation. In the case of different propagation speeds, we prove that the energy of our system decays polynomially. Moreover, we show the lack of exponential stability if the speeds of wave propagation are different with a global damping and a global coupling. Third, we investigate the stabilization of a linear Bresse system with one discontinuous local internal viscoelastic damping of Kelvin-Voigt type acting on the axial force, under fully Dirichlet boundary conditions. We prove the strong and polynomial stabilities of our system. Finally, we consider two models of the Kirchhoff plate equation, the first one with delay terms on the dynamical boundary controls, and the second one where delay terms on the boundary control are added. For the first system, we prove its well-posedness, strong stability, non-exponential stability, and polynomial stability under a multiplier geometric control condition. For the second one, we prove its well-posedness, strong stability, and exponential stability under the same multiplier geometric control condition. Finally, we give some instability examples of the second system for some choices of delays.
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Submitted on : Friday, April 29, 2022 - 5:27:16 PM
Last modification on : Tuesday, August 23, 2022 - 5:36:41 PM
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Haidar Badawi. Stability results for some hyperbolic systems with direct or indirect local dampings. Optimization and Control [math.OC]. Université Polytechnique Hauts-de-France; Institut national des sciences appliquées Hauts-de-France, 2022. English. ⟨NNT : 2022UPHF0005⟩. ⟨tel-03648293⟩

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