Exponential Stability of an Abstract Nondissipative Linear System

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