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Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications

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

(27/01/2010)

In arbitrary dimension, we consider the semi-discrete elliptic operator $- \d_t^2 + \Am$, where $\Am$ is a finite difference approximation of the operator $-\nabla_x (\Gamma(x) \nabla_x)$. For this operator we derive a global Carleman estimate, in which the usual large parameter is connected to the discretization step-size. We address discretizations on some families of smoothly varying meshes. We present consequences of this estimate such as a partial spectral inequality of the form of that proven by G.~Lebeau and L.~Robbiano for $A^m$ and a null controllability result for the parabolic operator $\partial_t + A^m$, for the lower part of the spectrum of $A^m$. With the control function that we construct (whose norm is uniformly bounded) we prove that the $L^2$-norm of the final state converges to zero exponentially, as the step-size of the discretization goes to zero. A relaxed observability 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 et Applications, Physique Mathématique d'Orléans (MAPMO)
	CNRS : UMR6628 – Université d'Orléans

Domaine

:

Mathématiques/Equations aux dérivées partielles Mathématiques/Analyse numérique

Mots Clés : Elliptic operator -- discrete and semi-discrete Carleman estimates -- spectral inequality -- control -- parabolic equations.

hal-00450854, version 1

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Contributeur : Franck Boyer <>

Soumis le : Mercredi 27 Janvier 2010, 15:02:52

Dernière modification le : Mercredi 27 Janvier 2010, 15:04:24