Exponential stability of nondissipative linear system in Banach space and application to unbounded bilinear systems

We consider linear control systems of the form dz(t)dt=Az(t)−ρBz(t), where ρ is a real parameter, A is the infinitesimal generator of a linear C0-semigroup of contractions S0(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.