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Theses

Stabilité de quelques problèmes d'évolution

Abstract : In this PhD thesis we study the stabilization of some evolution equations by feedback laws. First we consider the stabilization of the wave equation on 1-d networks with nodal feedbacks. In chapter 1, assuming that the weight of the feedback without delay is smaller than the one with delay, we give spectral conditions to obtain the strong, exponential or polynomial stability, by studying an observability inequality for the conservative system. In chapter 2 we transfer known observability results for another conservative system into a weighted observability estimate for the dissipative one without delay. Thanks to an interpolation inequality, we obtain explicit decay rates which depend on the geometric and topological properties of the network. Then we develop, in chapter 3, an abstract theory for second order evolution equation with delay, which generalizes the results of chapter 1. We study the case where the delay depends on time for the heat and wave equations in chapter 4. Using some assumptions about the delay and an appropriate Lyapunov functional, we prove that the energy is exponentially decreasing and we give explicitely its decay rate. Finally, we show, in chapter 4, that a filtering technique allows to obtain a quasi-exponential decay of a finite difference space discretization of the wave equation by pointwise interior stabilization.
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https://hal-uphf.archives-ouvertes.fr/tel-03012114
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Julie Valein. Stabilité de quelques problèmes d'évolution. Mathématiques [math]. Université de Valenciennes et du Hainaut-Cambrésis, 2008. Français. ⟨NNT : 2008VALE0032⟩. ⟨tel-03012114⟩

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