Boundary controllability for finite-differences semi-discretizations of a clamped beam equation

This article deals with the boundary observability properties of a space finite-differences semi-discretization of the clamped beam equation. We make a detailed spectral analysis of the system and, by combining numerical estimates with asymptotic expansions, we localize all the eigenvalues of the corresponding discrete operator depending on the mesh size $h$. Then, an Ingham's type inequality and a discrete multiplier method allow us to deduce that the uniform (with respect to $h$) observability property holds if and only if the eigenfrequencies are filtered out in the range $O(1/hˆ4)$