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Stabilization of Timoshenko Beam by Means of Pointwise Controls

Published online by Cambridge University Press:  15 September 2003

Gen-Qi Xu  and
Siu Pang Yung
Gen-Qi Xu
Affiliation:
Department of Mathematics of Shanxi University, TaiYuan 030006, P.R. China.; gqxu@mail.sxu.edu.cn.
Siu Pang Yung
Affiliation:
Department of Mathematics of the University of Hong Kong, Hong Kong, P.R. China; spyung@hku.hk.
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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.

Keywords

Timoshenko beampointwise feedback controlgeneralized eigenfunction systemRiesz basis.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 579 - 600
Copyright
© EDP Sciences, SMAI, 2003

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References

Chen, G., Delfour, C.M., Krall, A.M. and Payre, G., Modeling, stabilization and control of serially connected beam. SIAM J. Control Optim. 25 (1987) 526-546. CrossRef
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.
Chen, G., Krantz, S.G., Russell, D.L., Wayne, C.E. and West, H.H., Analysis, design and behavior of dissipative joints for coupled beams. SIAM J. Appl. Math. 49 (1989) 1665-1693. CrossRef
Conrad, F., Stabilization of beams by pointwise feedback control. SIAM J. Control Optim. 28 (1990) 423-437. CrossRef
J.E. Lagnese, G. Leugering and E. Schmidt, Modeling, analysis and control of dynamic Elastic Multi-link structures. Birkhauser, Basel (1994).
Rebarber, R., Exponential stability of coupled beam with dissipative joints: A frequency domain approach. SIAM J. Control Optim. 33 (1995) 1-28. CrossRef
Ammari, K. and Tucsnak, M., Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force. SIAM J. Control Optim. 39 (2000) 1160-1181. CrossRef
Kim, J.U. and Renardy, Y., Boundary control of the Timoshenko beam. SIAM. J. Control Optim. 25 (1987) 1417-1429. CrossRef
Ito, K. and Kunimatsu, N., Semigroup model and stability of the structurally damped Timoshenko beam with boundary inputs. Int. J. Control 54 (1991) 367-391. CrossRef
Morgül, Ö., Boundary control of a Timoshenko beam attached to a rigid body: Planar motion. Int. J. Control 54 (1991) 763-791. CrossRef
Shi, D.H. and Feng, D.X., Feedback stabilization of a Timoshenko beam with an end mass. Int. J. Control 69 (1998) 285-300. CrossRef
Feng, D.X., Shi, D.H. and Zhang, W.T., Boundary feedback stabilization of Timoshenko beam with boundary dissipation. Sci. China Ser. A 41 (1998) 483-490. CrossRef
Conrad, F. and Morgül, Ö., On the stabilization of a flexible beam with a tip mass. SIAM J. Control Optim. 36 (1998) 1962-1986. CrossRef
Guo, B.Z. and The Riesz, R.Y. Yu 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. CrossRef
Rao, B.P., Optimal energy decay rate in a damped Rayleigh beam, edited by S. Cox and I. Lasiecka. Contemp. Math. 209 (1997) 221-229.
G.Q. Xu, Boundary feedback control of elastic beams, Ph.D. Thesis. Institute of Mathematics and System Science, Chinese Academy of Sciences (2000).
A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York, Appl. Math. Sci. 44 (1983).
R.M. Young An introduction to nonharmonic Fourier series. Academic Press, Inc. New York (1980).