  We consider damped wave (resp. Schr{\"o}dinger and plate) equations driven by
a hypoelliptic "sum of squares" operator L on a compact manifold and a damping
function b(x). We assume the Chow-Rashevski-H{\"o}rmander condition at rank k
(at most k Lie brackets needed to span the tangent space) together with
analyticity of M and the coefficients of L. We prove decay of the energy at
rate $log(t)^{-1/k}$ (resp. $log(t)^{-2/k}$ ) for data in the domain of the
generator of the associated group. We show that this decay is optimal on a
family of Grushin-type operators. This result follows from a perturbative
argument (of independent interest) showing, in a general abstract setting, that
quantitative approximate observability/controllability results for wave-type
equations imply a priori decay rates for associated damped wave,
Schr{\"o}dinger and plate equations. The adapted quantitative approximate
observability/controllability theorem for hypoelliptic waves is obtained by the
authors in [LL19, LL17].
