HAL : hal-00450854, version 2

DOI : 10.1137/100784278

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SIAM Journal on Control and Optimization 48, 8 (2010) pp. 5357-5397

Versions disponibles :

v1 (27-01-2010)

v2 (28-08-2010)

Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications

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

(09/11/2010)

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 - Analyse, Probabilités, Modélisation - Orléans (MAPMO)
	Université d'Orléans – CNRS : UMR7349

Domaine

:

Mathématiques/Equations aux dérivées partielles Mathématiques/Analyse numérique

Mots Clés : Elliptic operator – Discrete and semi-discrete Carleman estimates – Spectral inequality – Control – Parabolic equations.

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Contributeur : Franck Boyer <>

Soumis le : Vendredi 27 Août 2010, 14:59:41

Dernière modification le : Dimanche 20 Mars 2011, 22:59:09