Nonuniform stability of damped contraction semigroups

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