  We investigate the stability properties of strongly continuous semigroups
generated by operators of the form $A-BB^\ast$, where $A$ is a generator of a
contraction semigroup and $B$ is a possibly unbounded operator. Such systems
arise naturally in the study of hyperbolic partial differential equations with
damping on the boundary or inside the spatial domain. As our main results we
present general sufficient conditions for non-uniform stability of the
semigroup generated by $A-BB^\ast$ in terms of selected observability-type
conditions of the pair $(B^\ast,A)$. We apply the abstract results to obtain
rates of energy decay in one-dimensional and two-dimensional wave equations, a
damped fractional Klein--Gordon equation and a weakly damped beam equation.
