  In this paper, we study the observability and controllability of wave
equations coupled by first or zero order terms on a compact manifold. We adopt
the approach in Dehman-Lebeau's paper \cite{DehmanLebeau09} to prove that: the
weak observability inequality holds for wave equations coupled by first order
terms on compact manifold without boundary if and only if a class of ordinary
differential equations related to the symbol of the first order terms along the
Hamiltonian flow are exactly controllable. We also compute the higher order
part of the observability constant and the observation time. By duality, we
obtain the controllability of the dual control system in a finite
co-dimensional space. This gives the full controllability under the assumption
of unique continuation of eigenfunctions. Moreover, these results can be
applied to the systems of wave equations coupled by zero order terms of cascade
structure after an appropriate change of unknowns and spaces. Finally, we
provide some concrete examples as applications where the unique continuation
property indeed holds.
