We consider linear control systems of the form d z ( t ) d t = A z ( t ) − ρ B z ( t ) , where ρ is a real parameter, A is the infinitesimal generator of a linear C 0 -semigroup of contractions S 0 ( t ) on a Banach space X and the linear perturbation B is a relatively bounded operator with respect to A . We provide necessary and sufficient conditions for exponential stability of the above system. Our stability results are then applied to establish uniform exponential stabilization of linear and bilinear systems with unbounded control operators. Applications to hyperbolic like equations are considered.