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

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-00366496
Contributor : Franck Boyer <>
Submitted on : Wednesday, November 4, 2009 - 10:15:05 PM
Last modification on : Friday, April 12, 2019 - 4:46:03 PM
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### Citation

Franck Boyer, Florence Hubert, Jérôme Le Rousseau. Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations. Journal de Mathématiques Pures et Appliquées, Elsevier, 2010, 93 (3), pp.240-273. ⟨10.1016/j.matpur.2009.11.003⟩. ⟨hal-00366496v2⟩

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