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/221938>.

@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},
month = {11},
number = {4},
pages = {1144-1157},
publisher = {EDP Sciences},
title = {Strong stabilization of controlled vibrating systems},
url = {http://eudml.org/doc/221938},
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
DA - 2011/11//
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/221938
ER -

References

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  8. A.M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics377. Berlin-Heidelberg-New York, Springer-Verlag (1974).  Zbl0325.34039
  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.  Zbl0589.35076
  10. A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Z.41 (1936) 367–379.  Zbl0014.21503
  11. V. Jurdjevic and J.P. Quinn, Controllability and stability. J. Differ. Equ.28 (1978) 381–389.  Zbl0417.93012
  12. A. Pazy, A class of semi-linear equations of evolution. Israël J. Math.20 (1975) 23–36.  Zbl0305.47022
  13. A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag (1975).  Zbl0516.47023
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