  In this article, we study energy decay of the damped wave equation on compact
Riemannian manifolds where the damping coefficient is anisotropic and modeled
by a pseudodifferential operator of order zero. We prove that the energy of
solutions decays at an exponential rate if and only if the damping coefficient
satisfies an anisotropic analogue of the classical geometric control condition,
along with a unique continuation hypothesis. Furthermore, we compute an
explicit formula for the optimal decay rate in terms of the spectral abscissa
and the long-time averages of the principal symbol of the damping over
geodesics, in analogy to the work of Lebeau for the isotropic case. We also
construct genuinely anisotropic dampings which satisfy our hypotheses on the
flat torus.
