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Modélisation asymptotique de plaques : contrôlabilité exacte frontière, piézoélectricité

Abstract : This dissertation deals with various aspects of plate modelling : boundary exact controllability of 2D structures, construction of models for piezoelectric plates, and analysis of singularities. The first chapter presents a result of boundary exact controllability for a 2D elastic plate. First, we solve an exact controllability problem for a plate with thickness h, by controlling only its interior and its lateral boundary. We choose interior controls that vanish as h tends to 0. In Chapters 2, 3, 4, we study the behavior of a piezoelectric plate when its thickness tends to 0. Particularly, in the dynamic case where the magnetic contribution is taken into account in the Maxwell equations. So, in one hand, we justify the thin plate models which assume that the electric potential is a second order polynomial in the thickness direction. On the other hand, we prove that in 2D models, the equilibrium equations depend on the electric potential only through the difference of potential between the horizontal faces. Moreover, we obtain the very contribution of the piezoelectric constants in the bending operator.
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Abdou Sène. Modélisation asymptotique de plaques : contrôlabilité exacte frontière, piézoélectricité. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 1999. Français. ⟨tel-00004849⟩

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