Ingham type approach for uniform observability inequality of the semi-discrete coupled wave equations

Abstract

This article concerns an observability inequality for a system of coupled wave equations for the continuous models as well as for the space semi-discrete finite difference approximations. For finite difference and standard finite elements methods on uniform numerical meshes it is known that a numerical pathology produces a blow-up of the constant on the observability inequality as the mesh-size tends to zero. We identify this numerical anomaly for coupled wave equations and we prove that there exists a uniform observability inequality in a subspace of solutions generated by low frequencies. We use the Ingham type approach for getting a uniform boundary observability.