Strong stabilization of controlled vibrating systems

Jean-François Couchouron

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

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

Abstract

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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.

How to cite

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Couchouron, 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

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  10. [10] A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Z.41 (1936) 367–379. Zbl0014.21503MR1545625
  11. [11] V. Jurdjevic and J.P. Quinn, Controllability and stability. J. Differ. Equ.28 (1978) 381–389. Zbl0417.93012MR494275
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  14. [14] J. Simon, Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl.146 (1987) 65–96. Zbl0629.46031MR916688

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