Étude de quelques problèmes de contrôlabilité exacte, de contrôle optimal et de stabilisation pour des domaines minces à frontières ondulées

Abstract : This thesis is devoted to study of some problems of exact controllability, optimal control and stabilization in a three-dimensional domain with low thickness and non-flat boundary. We study also the asymptotic behaviour corresponding to each problem when the thickness tends to zero. In a first part, we show, using the multipliers adapted to the domain and acting with a Dirichlet control on the lateral part and a Neumann control on top-bottom surfaces, exact controllability for the wave equation. Then we consider, in the second part, a problem pf optimal control for a state equation given by an elliptic operator of second order and cost function to be minimized. Using suitable functions test, we show that the optimal control converges to the optimal control of the limiting problem when the thickness tends to zero. Lastly, in the third and last part, we consider the wave equation with an interior and boundary damping. Using multipliers adapted to the domain, we weaken the conditions on the localization of the dissipation and we show that energy decrease exponentially. For each one of these problems, we give a two-dimensional description of the problem limits corresponding. It is seen how the two-dimensional limiting operator depends on the undulations and how boundary controls on top-bottom surfaces become interior control
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Nabil Laanaia. Étude de quelques problèmes de contrôlabilité exacte, de contrôle optimal et de stabilisation pour des domaines minces à frontières ondulées. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 2001. Français. ⟨NNT : 2001METZ016S⟩. ⟨tel-01775484⟩

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