Exponential stability of an abstract nondissipative linear system (Q2719157)

In this very important publication an abstract linear system with perturbation of the form: \({dy\over dt}= Ay+\varepsilon By\) on a Hilbert space \(H\) is considered. Here the operator \(A_\varepsilon= A+\varepsilon B\) generates a \(C_0\) semigroup \(S_\varepsilon(t)\) on \(H\). The authors (mainly) study the exponential stability of the system.NEWLINENEWLINENEWLINEMain result: The sufficient condition for exponential stability when \(B\) is not a dissipative operator is proposed. Moreover, a Hautus-type criterion for exact controllability of the system \((A,G)\) \((G\) is a bounded linear operator from another Hilbert space to \(H)\) is presented. Finally the result about stability is then applied to establish the exponential stability of several elastic systems with indefinite viscous damping, as well as the exponential stabilization of elastic systems with noncolocated observation and control.