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.
Mots clés : precompactness, compact resolvent, almost periodic functions, Fourier series, mild solution, integral solution, control theory, stabilization
@article{COCV_2011__17_4_1144_0,
author = {Couchouron, Jean-Fran\c{c}ois},
title = {Strong stabilization of controlled vibrating systems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1144--1157},
publisher = {EDP-Sciences},
volume = {17},
number = {4},
year = {2011},
doi = {10.1051/cocv/2010041},
mrnumber = {2859869},
zbl = {1254.93082},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2010041/}
}
TY - JOUR AU - Couchouron, Jean-François TI - Strong stabilization of controlled vibrating systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 1144 EP - 1157 VL - 17 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010041/ DO - 10.1051/cocv/2010041 LA - en ID - COCV_2011__17_4_1144_0 ER -
%0 Journal Article %A Couchouron, Jean-François %T Strong stabilization of controlled vibrating systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 1144-1157 %V 17 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2010041/ %R 10.1051/cocv/2010041 %G en %F COCV_2011__17_4_1144_0
Couchouron, Jean-François. Strong stabilization of controlled vibrating systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 1144-1157. doi : 10.1051/cocv/2010041. http://www.numdam.org/articles/10.1051/cocv/2010041/
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