Volume 17, Number 4, October-December 2011
|Page(s)||1144 - 1157|
|Published online||08 November 2010|
Strong stabilization of controlled vibrating systems
Université Paul Verlaine de Metz, LMAM et INRIA Lorraine, Île du Saulcy,
57045 Metz, France. firstname.lastname@example.org
Revised: 16 June 2010
This paper deals with feedback stabilization of second order equations of the form
ytt + 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 = ⟨yt, 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.
Mathematics Subject Classification: 37L05 / 43A60 / 47D06 / 47H20 / 93D15
Key words: Precompactness / compact resolvent / almost periodic functions / Fourier series / mild solution / integral solution / Control Theory / Stabilization
© EDP Sciences, SMAI, 2010
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