# Stabilization of Timoshenko Beam by Means of Pointwise Controls

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

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topXu, Gen-Qi, and Yung, Siu Pang. "Stabilization of Timoshenko Beam by Means of Pointwise Controls." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 579-600. <http://eudml.org/doc/90712>.

@article{Xu2010,

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.; generalized eigenfunction system; Riesz basis},

language = {eng},

month = {3},

pages = {579-600},

publisher = {EDP Sciences},

title = {Stabilization of Timoshenko Beam by Means of Pointwise Controls},

url = {http://eudml.org/doc/90712},

volume = {9},

year = {2010},

}

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

DA - 2010/3//

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.; generalized eigenfunction system; Riesz basis

UR - http://eudml.org/doc/90712

ER -

## References

top- 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.
- 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.
- 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.
- F. Conrad, Stabilization of beams by pointwise feedback control. SIAM J. Control Optim.28 (1990) 423-437.
- J.E. Lagnese, G. Leugering and E. Schmidt, Modeling, analysis and control of dynamic Elastic Multi-link structures. Birkhauser, Basel (1994).
- R. Rebarber, Exponential stability of coupled beam with dissipative joints: A frequency domain approach. SIAM J. Control Optim.33 (1995) 1-28.
- 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.
- J.U. Kim and Y. Renardy, Boundary control of the Timoshenko beam. SIAM. J. Control Optim.25 (1987) 1417-1429.
- K. Ito and N. Kunimatsu, Semigroup model and stability of the structurally damped Timoshenko beam with boundary inputs. Int. J. Control54 (1991) 367-391.
- Ö. Morgül, Boundary control of a Timoshenko beam attached to a rigid body: Planar motion. Int. J. Control54 (1991) 763-791.
- D.H. Shi and D.X. Feng, Feedback stabilization of a Timoshenko beam with an end mass. Int. J. Control69 (1998) 285-300.
- D.X. Feng, D.H. Shi and W.T. Zhang, Boundary feedback stabilization of Timoshenko beam with boundary dissipation. Sci. China Ser. A41 (1998) 483-490.
- F. Conrad and Ö. Morgül, On the stabilization of a flexible beam with a tip mass. SIAM J. Control Optim.36 (1998) 1962-1986.
- 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.
- 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.
- 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).

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