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

By Franck Boyer, Florence Hubert, Jérôme Le Rousseau. 2010

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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.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/65256bcae3ddac8b868827be4402f49867858d86

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In Computational Theory and Mathematics

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Citations in the corpus

The Wave Equation: Control and Numerics (2012, methodology): “The convergence properties of controls for discrete heat eq uations has also been developed lately in [70, 61, 30, 8, 9, 10].”