On an abstract linear elastic system with indefinite damping

Free access article

Issue		ESAIM: PROC Volume 8, 2000 Contrôle des systèmes gouvernés par des équations aux dérivées partielles


Page(s)		107 - 117
DOI		10.1051/proc:2000008

ESAIM: Proc., 2000, Vol. 8, pp. 107-117
DOI: 10.1051/proc:2000008

On an abstract linear elastic system with indefinite damping

Kangsheng Liu¹, Zhuangyi Liu² and Bopeng Rao³

¹ Department of Applied Mathematics, Zhejiang University, Hangzhou, 310027, China. Supported partially by the National Key Project of China and by the NSFC grants 69874034
² Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812, USA
³ Institut de Recherche Mathématique Avancée, Université de Loius Pasteur de Strasbourg, 7 Rue René-Descartes, 67084 Strasbourg Cedex, France

Abstract
In this paper we consider an abstract linear system with perturbation of the form (dy/dt) = Ay + εBy on a Hilbert space H, where A is skew-adjoint, B is bounded, and ε is a positive parameter. Motivated by a result of Freitas and Zuazua on the one-dimensional wave equation with indefinite viscous damping [JDE, 1996], we obtain sufficient conditions for the exponential stability for the above system when B is not a dissipative operator. Our result is then applied to many elastic systems with indefinite viscous damping.

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