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Observation et contrôle de quelques systèmes conservatifs

Abstract : In this work, we focus on the internal controllability and its cost for some linear partial differential equations. In the first part, we introduce and describe two methods to provide precise estimates of the cost of control (and by duality, of the observability constant) for general one dimensional wave equations with potential. The first one is based on a propagation argument along the characteristics relying on the symmetrical roles of the time and space variables. The second one uses a spectral decomposition of the solution of the wave equation and ingham's inequalities. This relates the estimation of the observability constant to the study of an optimal problem involving dirichlet eigenfunctions of laplacian with potential. We provide some qualitative properties of the minimizers, and also precise bounds on the minimum. In the second part, we are concerned with the controllability of some systems of equations by a reduced number of controls (i.e. the number of controls is less that the number of equations). In particular, in the case of coupled systems of schrödinger equations, we exactly characterize the initial conditions that can be controlled and we give a necessary and sufficient condition of kalman type for the controllability of coupled systems of wave equations. The proof relies on the fictitious control method coupled with the proof of an algebraic solvabilityproperty for some related underdetermined system, as well as on some regularity results.
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Contributor : Thibault Liard <>
Submitted on : Monday, November 7, 2016 - 3:34:04 PM
Last modification on : Thursday, March 26, 2020 - 9:14:17 PM
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  • HAL Id : tel-01393503, version 1


Thibault Liard. Observation et contrôle de quelques systèmes conservatifs. Variables complexes [math.CV]. Université Pierre et Marie Curie - Paris VI, 2016. Français. ⟨NNT : 2016PA066364⟩. ⟨tel-01393503v1⟩



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