Strong stabilization of controlled vibrating systems

Université Paul Verlaine de Metz, LMAM et INRIA Lorraine, Île du Saulcy, 57045 Metz, France. couchour@univ-metz.fr

Received: 13 January 2010
Revised: 16 June 2010

Abstract

This paper deals with feedback stabilization of second order equations of the form

y_tt + A₀y + u (t) B₀y (t) = 0, t ∈ [0, +∞[,

where A₀ is a densely defined positive selfadjoint linear operator on a real Hilbert space H, with compact inverse and B₀ is a linear map in diagonal form. It is proved here that the classical sufficient ad-condition of Jurdjevic-Quinn and Ball-Slemrod with the feedback control u = ⟨y_t, B₀y⟩_H implies the strong stabilization. This result is derived from a general compactness theorem for semigroup with compact resolvent and solves several open problems.