Damien Allonsius

• Thesis - Etude spectrale d’opérateurs de Sturm-Liouville et applications à la contrôlabilité de problèmes paraboliques discrets et continus
• Presentation - Thèse soutenue le 26 Septembre 2018

• Spectral analysis of discrete elliptic operators and applications in control theory, (Numerische Mathematik, 2018) (avec F. Boyer et M. Morancey). Also available on Hal
• Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries, (Mathematical Control and Related Fields, 2018) (avec F. Boyer). Also available on Hal
• Analysis of the null-controllability of degenerate parabolic systems of Grushin type via the moments method, (Mathematical Control and Related Fields, 2019) (avec F. Boyer et M. Morancey). Also available on Hal

• Soutenance Thèse - PhD defended on the 26th of Septembre 2018
• ISMP. Juillet 2018. Site de la conférence
• GTT Jussieu Groupe de Travail des Thésards. LJLL. Janvier 2018. Page du séminaire et Exposé.
• VII Partial differential equations, optimal design and numerics. Benasque, Espagne. Août 2017. Exposé .
• Workshop APACHE. Toulouse. Décembre 2016. Exposé .
• Porquerolles. Mai 2016. Exposé.
• PICOF Problèmes Inverses, Contrôle, Optimisation de Formes. Mini-symposium on Control of PDE. Autrans. Juin 2016. Site de la conférence et Exposé.
• Séminaire des doctorants de Marseille. Mai 2016. Page du séminaire et Exposé .
• CANUM 43e Congrès National d'Analyse Numérique. Obernai. Mai 2016. Site de la conférence et Exposé .

In this thesis, we study the null controllability of some continous and semi discretized parabolic systems.

We first consider cascade systems of parabolic equations of the form ∂ t − (∂ x γ∂ x + q). 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 φ(h) null controllability results, depending on the hypotheses on the mesh and on functions γ 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 φ(h) null controllability results on the discretized domain as well as null controllability results on the continous 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 param- eter 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.