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

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.