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Etude spectrale d’opérateurs de Sturm-Liouville et applications à la contrôlabilité de problèmes paraboliques discrets et continus

Abstract : In this thesis, we study the null controllability of some continuous and semi discretized parabolic systems. We first consider cascade systems of parabolic equations of the form $\partial_t - \left(\partial_x \gamma \partial_x + q \right)$. The space variable belongs to a real and bounded interval and this system is semi-discretized in space by a finite differences scheme. Applying the so called moments method, we prove null controllability and $\phi (h)$ null controllability results, depending on the hypotheses on the mesh and on functions $\gamma$ and $q$. Then, we extend this results when the space variable belongs to a cylindrical domain which control zone is in a section at the border of the cylinder. This cylindrical domain is decomposed into a product of two spaces. On the first, of dimension 1, we apply the results described previously. On the second, we use the Lebeau-Robbiano's procedure. In this framework, we prove $\phi (h)$ null controllability results on the discretized domain as well as null controllability results on the continuous problem. In another section, we investigate the computation of minimal time of null controllability of Grushin's equation defined on a rectangular domain which control region is a vertical strip. This problem of control amounts to study a countably infinite family, indexed by the Fourier parameter $n$, of null control problems of parabolic equations, tackled, once again, with the moments method. The latter requires precise estimates on the spectrum of Sturm-Liouville operators. We prove lower bounds on quantities depending on the eigenfunctions of these operators and we study the gap property of their eigenvalues. To tackle control problems addressed in this manuscript, it is crucial that our estimates are uniform with respect to the discretization parameter or the parameter $n$. Spectral theory of these operators is therefore the keystone of this thesis. Our results are illustrated and complemented by numerical simulations, based on the HUM approach.
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Contributor : Damien Allonsius <>
Submitted on : Wednesday, October 3, 2018 - 3:08:45 PM
Last modification on : Thursday, January 23, 2020 - 6:22:13 PM
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Damien Allonsius. Etude spectrale d’opérateurs de Sturm-Liouville et applications à la contrôlabilité de problèmes paraboliques discrets et continus. Equations aux dérivées partielles [math.AP]. Aix Marseille Université, 2018. Français. ⟨tel-01887023⟩



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