Exponential stability of an abstract nondissipative linear system

In this paper we consider an abstract linear system with perturbation of th e form dy/dt = Ay + epsilon By on a Hilbert space H, where A is skew-adjoint, B is bounded, and is a posit ive parameter. Motivated by a work of Freitas and Zuazua on the one-dimensi onal wave equation with indefinite viscous damping [P. Freitas and E. Zuazu a, J. Differential Equations, 132 ( 1996), pp. 338-352], we obtain a suffic ient condition for exponential stability of the above system when B is not a dissipative operator. We also obtain a Hautus-type criterion for exact co ntrollability of system (A,G), where G is a bounded linear operator from an other Hilbert space to H. Our result about the stability is then applied to establish the exponential stability of several elastic systems with indefi nite viscous damping, as well as the exponential stabilization of the elast ic systems with noncolocated observation and control.