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Quelques problèmes de contrôle d'équations aux dérivées partielles : inégalités spectrales, systèmes couplés et limites singulières

Abstract : In this work, we are concerned with the controllability of different partial differential equations. The first part is devoted to the Lebeau-Robbiano method for the control of linear parabolic equations. We first extend this method to some non-selfadjoint elliptic operators, proving a spectral inequality together with the controllability of the associated parabolic equation. Then, we prove these two properties for a transmission model across an interface, where the transmission conditions involve a tangential diffusion. The proof relies on a Carleman estimate which is uniform with respect to the small parameter representing the thickness of the interface. In the second part, we analyse some linear systems of partial differential equations, coupled by zero order terms. After having studied the stabilization of two wave equations, only one of them being damped, we prove the controllability in large time of a similar system with one control force only, under optimal geometric conditions on the control and the coupling regions. Using microlocal analysis methods, we moreover obtain the controllability of cascade systems of wave equations, together with the exact minimal time needed to control. These results imply in particular the controllability of the associated parabolic systems, in some situations where the control and the coupling regions are disjoint. Finally, in a third part, we study the uniform controllability of viscous perturbations of scalar conservation laws in the vanishing viscosity limit. We prove the exact global controllability to constant states by means of controls that are uniformly bounded when the viscosity goes to zero.
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https://tel.archives-ouvertes.fr/tel-00607240
Contributor : Matthieu Léautaud <>
Submitted on : Friday, July 8, 2011 - 11:36:40 AM
Last modification on : Wednesday, December 9, 2020 - 3:12:28 PM
Long-term archiving on: : Monday, November 12, 2012 - 10:30:09 AM

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  • HAL Id : tel-00607240, version 1

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Matthieu Léautaud. Quelques problèmes de contrôle d'équations aux dérivées partielles : inégalités spectrales, systèmes couplés et limites singulières. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2011. Français. ⟨tel-00607240⟩

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