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Etude de la géométrie optimale des zones de contrôle dans des problèmes de stabilisation

Abstract : In this Ph.D thesis, we deal with the optimization of the uniform exponential decay rate of the wave equation on a one- or two-dimensional domain [omeg]. The energy decrease is due to a constant damping k on a subset [omega]. The decay rate is given by the spectral abscissa [mu] of the operator associated to the problem, and in the two-dimensional case by a geometrical quantity g, first introduced by Bardos, Lebeau and Rauch. We establish that the spectral abscissa is differentiable with respect to k at the origin, and we study this derivative J in order to approximate m by the product of k and J. In the first part, we address the theoretical properties of the functionals J and g. We characterize the optimal geometries in the case of an interval or a square for some particular values of the area constraint. In the case of a square, we obtain an algorithm for the exact calculus of the geometrical quantity in the case where w is the union of square based on a new theorem of limits inversion. The second part of the thesis is dedicated to the numerical optimization of the quantities J and g by different types of genetic algorithms. The obtained results are not intuitive.
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Submitted on : Thursday, March 29, 2018 - 10:43:36 AM
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  • HAL Id : tel-01746730, version 1


Pascal Hebrard. Etude de la géométrie optimale des zones de contrôle dans des problèmes de stabilisation. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 2002. Français. ⟨NNT : 2002NAN10188⟩. ⟨tel-01746730v1⟩



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