# Strong stabilization of controlled vibrating systems

ESAIM: Control, Optimisation and Calculus of Variations (2011)

- Volume: 17, Issue: 4, page 1144-1157
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topCouchouron, Jean-François. "Strong stabilization of controlled vibrating systems." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 1144-1157. <http://eudml.org/doc/272894>.

@article{Couchouron2011,

abstract = {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.},

author = {Couchouron, Jean-François},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {precompactness; compact resolvent; almost periodic functions; Fourier series; mild solution; integral solution; control theory; stabilization; bilinear systems in operator form; feedback stabilization; Jurdjevic-Quinn and Ball-Slemrod ad-condition},

language = {eng},

number = {4},

pages = {1144-1157},

publisher = {EDP-Sciences},

title = {Strong stabilization of controlled vibrating systems},

url = {http://eudml.org/doc/272894},

volume = {17},

year = {2011},

}

TY - JOUR

AU - Couchouron, Jean-François

TI - Strong stabilization of controlled vibrating systems

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2011

PB - EDP-Sciences

VL - 17

IS - 4

SP - 1144

EP - 1157

AB - 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.

LA - eng

KW - precompactness; compact resolvent; almost periodic functions; Fourier series; mild solution; integral solution; control theory; stabilization; bilinear systems in operator form; feedback stabilization; Jurdjevic-Quinn and Ball-Slemrod ad-condition

UR - http://eudml.org/doc/272894

ER -

## References

top- [1] J.M. Ball and M. Slemrod, Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim.5 (1979) 169–179. Zbl0405.93030MR533618
- [2] J.M. Ball and M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semilinear control systems. Commun. Pure Appl. Math.32 (1979) 555–587. Zbl0394.93041MR528632
- [3] J.-M. Coron and B. d'Andréa-Novel, Stabilization of a rotating body-beam without damping. IEEE Trans. Autom. Control.43 (1998) 608–618. Zbl0908.93055MR1618052
- [4] J.-F. Couchouron, Compactness theorems for abstract evolution problems. J. Evol. Equ.2 (2002) 151–175. Zbl1008.47057MR1914655
- [5] J.-F. Couchouron and M. Kamenski, An abstract topological point of view and a general averaging principle in the theory of differential inclusions. Nonlinear Anal.42 (2000) 1101–1129. Zbl0972.34049MR1780458
- [6] R. Courant and D. Hilbert, Methods of Mathematical Physics 1. Interscience, New York (1953). Zbl0053.02805MR65391
- [7] C.M. Dafermos and M. Slemrod, Asymptotic behaviour of nonlinear contraction semigroups. J. Funct. Anal.13 (1973) 97–106. Zbl0267.34062MR346611
- [8] A.M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics 377. Berlin-Heidelberg-New York, Springer-Verlag (1974). Zbl0325.34039MR460799
- [9] A. Haraux, Almost-periodic forcing for a wave equation with a nonlinear, local damping term. Proc. R. Soc. Edinb., Sect. A, Math. 94 (1983) 195–212. Zbl0589.35076MR709715
- [10] A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Z.41 (1936) 367–379. Zbl0014.21503MR1545625
- [11] V. Jurdjevic and J.P. Quinn, Controllability and stability. J. Differ. Equ.28 (1978) 381–389. Zbl0417.93012MR494275
- [12] A. Pazy, A class of semi-linear equations of evolution. Israël J. Math.20 (1975) 23–36. Zbl0305.47022MR374996
- [13] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag (1975). Zbl0516.47023MR710486
- [14] J. Simon, Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl.146 (1987) 65–96. Zbl0629.46031MR916688