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Exponential decay rate of the energy of a Timoshenko beam with locally distributed feedback

Published online by Cambridge University Press:  17 February 2009

Dong-Hua Shi  and
De-Xing Feng
Dong-Hua Shi
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China; e-mail: dhshi@math.tsinghua.edu.cn.
De-Xing Feng
Affiliation:
Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences Beijing 100080, China; e-mail: dxfeng@iss03.iss.ac.cn.
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Abstract

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The problem of the energy exponential decay rate of a Timoshenko beam with locally distributed controls is investigated. Consider the case in which the beam is nonuniform and the two wave speeds are different. Then, using Huang's theorem and Birkhoff's asymptotic expansion method, it is shown that, under some locally distributed controls, the energy exponential decay rate is identical to the supremum of the real part of the spectrum of the closed loop system. Furthermore, explicit asymptotic locations of eigenfrequencies are derived.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 2 , October 2002 , pp. 205 - 220
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Banks, H. T., Smith, R. C. and Wang, Yun, “The modeling of piezoceramic patch interactions with shells, plates, and beams”, Quart. Appl. Math. 53 (1995) 353–381.CrossRefGoogle Scholar
[2]Birkhoff, G. D. and Langer, R. E., “The boundary problems and developments associated with a system of ordinary linear differential equations of the first order”, Proc. Amer. Acad. Arts Sci. 2 (1923) 51–128.CrossRefGoogle Scholar
[3]Chen, G., Krantz, S. G., Ma, D. W., Wayne, C. E. and West, H. H., “The Euler-Bernoulli beam equation with boundary energy dissipation”, in Operator Methods for Optimal Control Problems (ed. Lee, S. J.), Lecture Notes in Pure and Appl. Math. Series, (Marcel Dekker, New York, 1987) 67–96.Google Scholar
[4]Chen, G., Delfour, M. C., Krall, A. M. and Payre, G., “Modeling, stabilization and control of serially connected beams”, SIAM J. Control Optim. 25 (1987) 526–546.CrossRefGoogle Scholar
[5]Chen, G., Fulling, S. A., Narcowich, F. J. and Sun, S., “Exponential decay of energy of evolution equations with locally distributed damping”, SIAM J. Appl. Math. 51 (1991) 266–301.CrossRefGoogle Scholar
[6]Desch, W., Hannsgen, K. B., Renardy, Y. and Wheeler, R. L., “Boundary stabilization of an Euler-Bernoulli beam with viscoelastic damping”, in Proc. IEEE Conf. on Decision and Control, (Los Angeles, CA, 1987) 1792–1795.Google Scholar
[7]Ho, L. F., “Exact controllability of the one-dimensional wave equation with locally distributed control”, SIAMJ. Control Optim. 28 (1990) 733–748.CrossRefGoogle Scholar
[8]Huang, F. L., “Characteristic conditions for exponential stability of linear dynamical system in Hilbert spaces”, Ann. Diff. Eqs. 1 (1985) 43–53.Google Scholar
[9]Kim, J. U. and Renardy, Y., “Boundary control of the Timoshenko beam”, SIAM J. Control Optim. 25 (1987)1417–1429.CrossRefGoogle Scholar
[10]Lagnese, J., “Control of wave processes with distributed controls supported on a subregion”, SIAM J. Control Optim. 21 (1983) 68–85.CrossRefGoogle Scholar
[11]Liu, K. S., “Locally distributed control and damping for the conservative systems”, SIAMJ. Control optim. 35 (1997) 1574–1590.CrossRefGoogle Scholar
[12]Liu, K. S., Huang, F. L. and Chen, G., “Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage”, SIAM J. Appl. Math. 49 (1989) 1694–1707.CrossRefGoogle Scholar
[13]Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer-Verlag, New York, 1983).CrossRefGoogle Scholar
[14]Russell, D. L., “Controllability and stabilizability theory for linear partial differential equations, recent progress and open questions”, SIAM Rev. 20 (1978) 639–740.CrossRefGoogle Scholar
[15]Russell, D. L., “Mathematical models for the elastic beam and their control-theoretic implications”, in Semigroups, Theory and Applications(eds. Brezis, H., Crandall, H. G. and Kapell, F.), Volume 2, (Longman, Essex, 1986) 177–217.Google Scholar
[16]Stark, H. M., An Introduction to Number Theory (The MIT Press, Cambridge, Massachusetts, 1979).Google Scholar
[17]Timoshenko, S., Vibration Problems in Engineering (Van Nostand, New York, 1955).Google Scholar
[18]Wang, H. K. and Chen, G., “Asymptotic locations of eigenfrequencies of Euler-Bernoulli beam with nonhomogeneous structural and viscous damping coefficients”, SIAMJ. Control optim. 29 (1991) 347–367.CrossRefGoogle Scholar