  The energy of solutions of the scalar damped wave equation decays uniformly
exponentially fast when the geometric control condition is satisfied. A theorem
of Lebeau [leb93] gives an expression of this exponential decay rate in terms
of the average value of the damping terms along geodesics and of the spectrum
of the infinitesimal generator of the equation. The aim of this text is to
generalize this result in the setting of a vectorial damped wave equation on a
Riemannian manifold with no boundary. We obtain an expression analogous to
Lebeau's one but new phenomena like high frequency overdamping arise in
comparison to the scalar setting. We also prove a necessary and sufficient
condition for the strong stabilization of the vectorial wave equation.
