Sharp Observability Estimates for a System of Two Coupled Nonconservative Hyperbolic Equations

We consider a system of two coupled nonconservative wave equations. For this system, we prove several observability estimates. Those observability estimates are sharp in the sense that they lead by duality to the controllability (exact or approximate) of the coupled system with a single control acting through one of the equations only while keeping the same controllability time as for a single equation. Existing results in the literature either require two controls, or in the case of a single control, they have a controllability time that blows up as the coupling parameter goes to zero. Our proofs rely on: (i) Carleman estimates, (ii) energy estimates and (iii) localizing arguments. The results obtained complement and improve, in some sense, earlier results while at the same time providing new uniqueness results.