Exponential Stability of an Abstract Non-dissipative Linear System.

Kangsheng Liu , Zhuangyi Liu , Bopeng Rao (1)

1 IRMA - Institut de Recherche Mathématique Avancée

In this paper we consider an abstract linear system with perturbation of the form ${dyover dt}= Ay + epsilon By$ on a Hilbert space $H$, where $A$ is skew-adjoint, $B$ is bounded, and $epsilon$ is a positive parameter. Motivated by a work of Freitas and Zuazua (1996) on the one-dimensional wave equation with indefinite viscous damping, 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 non-co-located observation and control.

Mots clés: linear elastic system; stabilization; "linear elastic system; exponential stability; exact controllability; Hautus-type criterion; indefinite damping; stabilization"

Domaines: Optimisation et contrôle [math.OC]

Kangsheng Liu, Zhuangyi Liu, Bopeng Rao. Exponential Stability of an Abstract Non-dissipative Linear System.. 2000. ⟨hal-00129638⟩