HAL : hal-00366496, version 2

DOI : 10.1016/j.matpur.2009.11.003

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Journal de Mathématiques Pures et Appliqués 93, 3 (2010) 240-273

Versions disponibles :

v1 (08-03-2009)

v2 (05-11-2009)

Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations

Franck Boyer 1, Florence Hubert 1, Jérôme Le Rousseau 2

(03/2010)

We derive a semi-discrete two-dimensional elliptic global Carleman estimate, in which the usual large parameter is connected to the one-dimensional discretization step-size. The discretizations we address are some families of quasi-uniform meshes. As a consequence of the Carleman estimate, we derive a partial spectral inequality of the form of that proved by G.~Lebeau and L.~Robbiano, in the case of a discrete elliptic operator in one dimension. Here, this inequality concerns the lower part of the discrete spectrum. The range of eigenvalues/eigenfunctions we treat is however quasi-optimal and represents a constant portion of the discrete spectrum. For the associated parabolic problem, we then obtain a uniform null controllability result for this lower part of the spectrum. Moreover, with the control function that we construct, the $L^2$ norm of the final state converges to zero super-algebraically as the step-size of the discretization goes to zero. An observability-like estimate is then deduced.

1 :	Laboratoire d'Analyse, Topologie, Probabilités (LATP)
	CNRS : UMR6632 – Université de Provence - Aix-Marseille I – Université Paul Cézanne - Aix-Marseille III
2 :	Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO)
	Université d'Orléans – CNRS : UMR7349

Domaine

:

Mathématiques/Equations aux dérivées partielles

Mots Clés : Elliptic operator – discrete Carleman estimate – spectral inequality – parabolic equation – semi-discrete scheme – uniform controllability / observability

hal-00366496, version 2

http://hal.archives-ouvertes.fr/hal-00366496

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Contributeur : Franck Boyer <>

Soumis le : Mercredi 4 Novembre 2009, 22:15:05

Dernière modification le : Vendredi 27 Août 2010, 14:18:13