Observability inequality for the finite-difference semi-discretization of the 1-d coupled wave equations

In this paper we consider a system of two coupled conservative wave equations and we prove an observability estimate. We treat the question of uniform observability for the finite difference semi-discretization. As for a single wave equation (see Infante and Zuazua (Math. Model. Num. Ann. 33, 407–438, 1999)), we prove that there exists a lack of numerical observability, i.e., the constant on the observability inequality blows-up as the mesh-size h tends to zero. However, there exists a uniform observability in a subspace of solutions generated by the low frequencies and we prove it. The method of proof combines discrete multiplier techniques and Fourier series developments. We see also an Ingham type approach.