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Theses

Stability and controlability of some locally coupled systems.

Abstract : This thesis is devoted to study the stabilization and exact controllability of some locally coupled systems. First, we studied the stabilization of a system of two wave equations coupled by velocities with only one localized damping and under appropriate geometric conditions. For the case involved waves propagating at the same speed, we established the exponential energy decay rate. However, the natural physical case also entails waves that do not propagate with equal speed, in such a case, we showed that our system is not uniformly stable and we established an optimal polynomial energy decay rate.Second, we investigated the exact controllability of locally coupled wave equations. The main tool is a result of A. Haraux by which the observability inequality is equivalent to the exponential stability of the system. More precisely, we provided a complete stability analysis of the system in two different Hilbert spaces and under appropriate geometric conditions. Then, using the HUM method, we proved that the system is exactly controllable. Later, we performed numerical experiments to valid our obtained theoretical results.Last, we analyzed the stability of a Bresse system with local Kelvin-Voight damping with fully Dirichlet or Dirichlet- Neumann-Neumann boundary conditions. Here we trait several cases.In the case of three local damping, according to their properties (smoothness), we established an exponential or a polynomial energy decay rate. However, when the waves are only subjected to one or two damping and under Dirichlet-Neumann-Neumann boundary conditions, we demonstrated that the Bresse system is not uniformly stable. In this case, we established a polynomial energy decay rate.In this thesis, the frequency domain approach and the multiplier technique were used.
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Chiraz Kassem. Stability and controlability of some locally coupled systems.. Analysis of PDEs [math.AP]. Université Grenoble Alpes; Université libanaise; École Doctorale des Sciences et de Technologie (Beyrouth), 2019. English. ⟨NNT : 2019GREAM072⟩. ⟨tel-02908913⟩

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