Stabilization of Timoshenko beam by means of pointwise controls

Xu, Gen-Qi; Yung, Siu Pang

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 579-600.

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.

[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. | MR | Zbl

[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] 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. | MR | Zbl

[4] F. Conrad, Stabilization of beams by pointwise feedback control. SIAM J. Control Optim. 28 (1990) 423-437. | MR | Zbl

[5] J.E. Lagnese, G. Leugering and E. Schmidt, Modeling, analysis and control of dynamic Elastic Multi-link structures. Birkhauser, Basel (1994). | MR | Zbl

[6] R. Rebarber, Exponential stability of coupled beam with dissipative joints: A frequency domain approach. SIAM J. Control Optim. 33 (1995) 1-28. | MR | Zbl

[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. | Zbl

[8] J.U. Kim and Y. Renardy, Boundary control of the Timoshenko beam. SIAM. J. Control Optim. 25 (1987) 1417-1429. | MR | Zbl

[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. | MR | Zbl

[10] Ö. Morgül, Boundary control of a Timoshenko beam attached to a rigid body: Planar motion. Int. J. Control 54 (1991) 763-791. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[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] A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York, Appl. Math. Sci. 44 (1983). | MR | Zbl

[18] R.M. Young An introduction to nonharmonic Fourier series. Academic Press, Inc. New York (1980). | MR | Zbl