We consider linear damped wave (resp., Schrödinger and plate) equations driven by a hypoelliptic “sum of squares” operator $\mathscr{L}$ on a compact manifold $\mathcal{M}$ and a damping function $b(x)$. We assume the Chow--Rashevski--Hörmander condition at rank $k$ (at most $k$ Lie brackets are needed to span the tangent space) together with analyticity of $\mathcal{M}$ and the coefficients of $\mathscr{L}$. We prove that the energy decays at rate $\log(t)^{-\frac{1}{k}}$ (resp., $\log(t)^{-\frac{2}{k}}$) for data in the domain of the generator of the associated group. We show that this decay is optimal on a family of Baouendi--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, Schrödinger, and plate equations. The adapted quantitative approximate observability/controllability theorem for hypoelliptic waves is obtained by the authors in [J. Eur. Math. Soc. (JEMS), 21 (2019), pp. 957--1069] and [Mem. Amer. Math. Soc., to appear]. Read More: https://epubs.siam.org/doi/10.1137/20M1354969