INDIRECT BOUNDARY OBSERVABILITY OF SEMI-DISCRETE COUPLED WAVE EQUATIONS

This work concerns the indirect observability properties for the finite-difference space semi-discretization of the 1-d coupled wave equations with homogeneous Dirichlet boundary conditions. We assume that only one of the two components of the unknown is observed. As for a single wave equation, as well as for the direct (complete) observability of the coupled wave equations, we prove the lack of the numerical observability. However, we show that a uniform observability holds in the subspace of solutions in which the initial conditions of the observed component is generated by the low frequencies. Our main proofs use a two-level energy method at the discrete level and a Fourier decomposition of the solutions.