Stabilization of Timoshenko beam by means of pointwise controls

Gen-Qi Xu; Siu Pang Yung

ESAIM: Control, Optimisation and Calculus of Variations (2003)

  • Volume: 9, page 579-600
  • ISSN: 1292-8119

Abstract

top
We intend to conduct a fairly complete study on Timoshenko beams with pointwise feedback controls and seek to obtain information about the eigenvalues, eigenfunctions, Riesz-Basis-Property, spectrum-determined-growth-condition, energy decay rate and various stabilities for the beams. One major difficulty of the present problem is the non-simplicity of the eigenvalues. In fact, we shall indicate in this paper situations where the multiplicity of the eigenvalues is at least two. We build all the above-mentioned results from an effective asymptotic analysis on both the eigenvalues and the eigenfunctions, and conclude with the Riesz-Basis-Property and the spectrum-determined-growth-condition. Finally, these results are used to examine the stability effects on the system by the location of the pointwise control relative to the length of the whole beam.

How to cite

top

Xu, Gen-Qi, and Yung, Siu Pang. "Stabilization of Timoshenko beam by means of pointwise controls." ESAIM: Control, Optimisation and Calculus of Variations 9 (2003): 579-600. <http://eudml.org/doc/245066>.

@article{Xu2003,
abstract = {We intend to conduct a fairly complete study on Timoshenko beams with pointwise feedback controls and seek to obtain information about the eigenvalues, eigenfunctions, Riesz-Basis-Property, spectrum-determined-growth-condition, energy decay rate and various stabilities for the beams. One major difficulty of the present problem is the non-simplicity of the eigenvalues. In fact, we shall indicate in this paper situations where the multiplicity of the eigenvalues is at least two. We build all the above-mentioned results from an effective asymptotic analysis on both the eigenvalues and the eigenfunctions, and conclude with the Riesz-Basis-Property and the spectrum-determined-growth-condition. Finally, these results are used to examine the stability effects on the system by the location of the pointwise control relative to the length of the whole beam.},
author = {Xu, Gen-Qi, Yung, Siu Pang},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Timoshenko beam; pointwise feedback control; generalized eigenfunction system; Riesz basis},
language = {eng},
pages = {579-600},
publisher = {EDP-Sciences},
title = {Stabilization of Timoshenko beam by means of pointwise controls},
url = {http://eudml.org/doc/245066},
volume = {9},
year = {2003},
}

TY - JOUR
AU - Xu, Gen-Qi
AU - Yung, Siu Pang
TI - Stabilization of Timoshenko beam by means of pointwise controls
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2003
PB - EDP-Sciences
VL - 9
SP - 579
EP - 600
AB - We intend to conduct a fairly complete study on Timoshenko beams with pointwise feedback controls and seek to obtain information about the eigenvalues, eigenfunctions, Riesz-Basis-Property, spectrum-determined-growth-condition, energy decay rate and various stabilities for the beams. One major difficulty of the present problem is the non-simplicity of the eigenvalues. In fact, we shall indicate in this paper situations where the multiplicity of the eigenvalues is at least two. We build all the above-mentioned results from an effective asymptotic analysis on both the eigenvalues and the eigenfunctions, and conclude with the Riesz-Basis-Property and the spectrum-determined-growth-condition. Finally, these results are used to examine the stability effects on the system by the location of the pointwise control relative to the length of the whole beam.
LA - eng
KW - Timoshenko beam; pointwise feedback control; generalized eigenfunction system; Riesz basis
UR - http://eudml.org/doc/245066
ER -

References

top
  1. [1] G. Chen, C.M. Delfour, A.M. Krall and G. Payre, Modeling, stabilization and control of serially connected beam. SIAM J. Control Optim. 25 (1987) 526-546. Zbl0621.93053MR885183
  2. [2] G. Chen, S.G. Krantz, D.W. Ma, C.E. Wayne and H.H. West, The Euler–Bernoulli beam equation with boundary energy dissipation, in Operator methods for optimal control problems, edited by Sung J. Lee. Marcel Dekker, New York (1988) 67-96. 
  3. [3] G. Chen, S.G. Krantz, D.L. Russell, C.E. Wayne and H.H. West, Analysis, design and behavior of dissipative joints for coupled beams. SIAM J. Appl. Math. 49 (1989) 1665-1693. Zbl0685.73046MR1025953
  4. [4] F. Conrad, Stabilization of beams by pointwise feedback control. SIAM J. Control Optim. 28 (1990) 423-437. Zbl0695.93089MR1040468
  5. [5] J.E. Lagnese, G. Leugering and E. Schmidt, Modeling, analysis and control of dynamic Elastic Multi-link structures. Birkhauser, Basel (1994). Zbl0810.73004MR1279380
  6. [6] R. Rebarber, Exponential stability of coupled beam with dissipative joints: A frequency domain approach. SIAM J. Control Optim. 33 (1995) 1-28. Zbl0819.93042MR1311658
  7. [7] K. Ammari and M. Tucsnak, Stabilization of Bernoulli–Euler beams by means of a pointwise feedback force. SIAM J. Control Optim. 39 (2000) 1160-1181. Zbl0983.35021
  8. [8] J.U. Kim and Y. Renardy, Boundary control of the Timoshenko beam. SIAM. J. Control Optim. 25 (1987) 1417-1429. Zbl0632.93057MR912448
  9. [9] K. Ito and N. Kunimatsu, Semigroup model and stability of the structurally damped Timoshenko beam with boundary inputs. Int. J. Control 54 (1991) 367-391. Zbl0735.35112MR1111166
  10. [10] Ö. Morgül, Boundary control of a Timoshenko beam attached to a rigid body: Planar motion. Int. J. Control 54 (1991) 763-791. Zbl0779.73034MR1125911
  11. [11] D.H. Shi and D.X. Feng, Feedback stabilization of a Timoshenko beam with an end mass. Int. J. Control 69 (1998) 285-300. Zbl0943.93051MR1684739
  12. [12] D.X. Feng, D.H. Shi and W.T. Zhang, Boundary feedback stabilization of Timoshenko beam with boundary dissipation. Sci. China Ser. A 41 (1998) 483-490. Zbl0903.73051MR1663171
  13. [13] F. Conrad and Ö. Morgül, On the stabilization of a flexible beam with a tip mass. SIAM J. Control Optim. 36 (1998) 1962-1986. Zbl0927.93031MR1638023
  14. [14] B.Z. Guo and R.Y. Yu, The Riesz basis property of discrete operators and application to a Euler–Bernoulli beam equation with boundary linear feedback control. IMA J. Math. Control Inform. 18 (2001) 241-251. Zbl1049.93017
  15. [15] B.P. Rao, Optimal energy decay rate in a damped Rayleigh beam, edited by S. Cox and I. Lasiecka. Contemp. Math. 209 (1997) 221-229. Zbl0910.35013MR1472297
  16. [16] G.Q. Xu, Boundary feedback control of elastic beams, Ph.D. Thesis. Institute of Mathematics and System Science, Chinese Academy of Sciences (2000). 
  17. [17] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York, Appl. Math. Sci. 44 (1983). Zbl0516.47023MR710486
  18. [18] R.M. Young An introduction to nonharmonic Fourier series. Academic Press, Inc. New York (1980). Zbl0493.42001MR591684

NotesEmbed ?

top

You must be logged in to post comments.