A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems

This work is concerned with the boundary observability of an abstract system of two coupled second order evolution equations, the coupling operator being a compact perturbation of the uncoupled system. We assume that only one of the two components of the unknown is observed. This is indirect observability. We prove that by observing only one component, one can get back a full weakened energy of both components under a compatibility condition linking the operators of each equation and for small coupling. Using the Hilbert uniqueness method, we then establish an indirect exact controllability result. We apply this abstract result to several coupled systems of partial differential equations (wave-wave, coupled elastodynamic systems, Petrowsky-Petrowsky, and wave-Petrowsky systems).