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://archive.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://archive.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://archive.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://archive.numdam.org/articles/10.1051/cocv/2010041/
[1] and , Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. 5 (1979) 169-179. | MR | Zbl
[2] and , Nonharmonic Fourier series and the stabilization of distributed semilinear control systems. Commun. Pure Appl. Math. 32 (1979) 555-587. | MR | Zbl
[3] and , Stabilization of a rotating body-beam without damping. IEEE Trans. Autom. Control. 43 (1998) 608-618. | MR | Zbl
[4] , Compactness theorems for abstract evolution problems. J. Evol. Equ. 2 (2002) 151-175. | MR | Zbl
[5] and , An abstract topological point of view and a general averaging principle in the theory of differential inclusions. Nonlinear Anal. 42 (2000) 1101-1129. | MR | Zbl
[6] and , Methods of Mathematical Physics 1. Interscience, New York (1953). | MR | Zbl
[7] and , Asymptotic behaviour of nonlinear contraction semigroups. J. Funct. Anal. 13 (1973) 97-106. | MR | Zbl
[8] , Almost Periodic Differential Equations, Lecture Notes in Mathematics 377. Berlin-Heidelberg-New York, Springer-Verlag (1974). | MR | Zbl
[9] , Almost-periodic forcing for a wave equation with a nonlinear, local damping term. Proc. R. Soc. Edinb., Sect. A, Math. 94 (1983) 195-212. | MR | Zbl
[10] , Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41 (1936) 367-379. | MR | Zbl
[11] and , Controllability and stability. J. Differ. Equ. 28 (1978) 381-389. | MR | Zbl
[12] , A class of semi-linear equations of evolution. Israël J. Math. 20 (1975) 23-36. | MR | Zbl
[13] , Semigroups of linear operators and applications to partial differential equations. Springer-Verlag (1975). | MR | Zbl
[14] , Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl. 146 (1987) 65-96. | MR | Zbl
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