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Theses

Theoretical and numerical study of the stability of some distributed systems with dynamic boundary control

Abstract : This thesis is devoted to the study of the stabilization of some distributed systems with dynamic boundary control. First, we consider the stabilization of the Rayleigh beam equation with only one dynamic boundary control moment or force. We show that the system is not uniformly (exponentially) stable. However, using a spectral method, we establish the optimal polynomial decay rate of the energy of the system. Next, we study the indirect stability of the wave equation with a fractional dynamic boundary control. We show that the decay rate of the energy depends on the nature of the geometry of the domain. Using a frequency approach and a spectral method, we show the non exponential stability of the system and we establish, different polynomial stability results. Finally, we consider the finite difference space discretization of the 1-d wave equation with dynamic boundary control. First, using a spectral approach, we show that the polynomial decay of the discretized energy is not uniform with respect to the mesh size, as the energy of the continuous system. Next, we introduce a viscosity term and we establish the uniform (with respect to the mesh size) polynomial energy decay of our discrete scheme.
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Mohamad Ali Sammoury. Theoretical and numerical study of the stability of some distributed systems with dynamic boundary control. General Mathematics [math.GM]. Université de Valenciennes et du Hainaut-Cambresis; Université Libanaise, 2016. English. ⟨NNT : 2016VALE0032⟩. ⟨tel-01523446⟩

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