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Strong stabilization of controlled vibrating systems

Published online by Cambridge University Press:  08 November 2010

Jean-François Couchouron
Jean-François Couchouron*
Affiliation:
Université Paul Verlaine de Metz, LMAM et INRIA Lorraine, Île du Saulcy, 57045 Metz, France. [email protected]
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Abstract

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

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

where A0 is a densely defined positive selfadjoint linear operator on areal Hilbert space H, with compact inverse and B0 is a linear map in diagonal form. It isproved here that the classical sufficient ad-condition of Jurdjevic-Quinn andBall-Slemrod with the feedback control u = ⟨yt, B0y⟩Himplies thestrong stabilization. This result is derived from a general compactnesstheorem for semigroup with compact resolvent and solves several open problems.

Keywords

Precompactnesscompactresolventalmost periodic functionsFourier seriesmild solutionintegral solutionControl TheoryStabilization
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 4 , October 2011 , pp. 1144 - 1157
Copyright
© EDP Sciences, SMAI, 2010

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