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On the dynamic behavior and stability of controlled connected Rayleighbeams under pointwise output feedback

Published online by Cambridge University Press:  18 January 2008

Bao-Zhu Guo ,
Jun-Min Wang  and
Cui-Lian Zhou
Bao-Zhu Guo
Affiliation:
Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100080, P.R. China; bzguo@iss.ac.cn School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa.
Jun-Min Wang
Affiliation:
Corresponding author: Department of Mathematics, Beijing Institute of Technology, Beijing 100081, P.R. China; wangjc@graduate.hku.hk
Cui-Lian Zhou
Affiliation:
Department of Mathematics, Beijing Institute of Technology, Beijing 100081, P.R. China.
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Abstract

We study the dynamic behavior and stability of two connected Rayleigh beams that are subject to, in addition to two sensors and two actuators applied at the joint point, one of the actuators also specially distributed along the beams. We show that with the distributed control employed, there is a set of generalized eigenfunctions of the closed-loop system, which forms a Riesz basis with parenthesis for the state space. Then both the spectrum-determined growth condition and exponential stability are concluded for the system. Moreover, we show that the exponential stability is independent of the location of the joint. The range of the feedback gains that guarantee the system to be exponentially stable is identified.

Keywords

Rayleigh beamcollocated controlspectral analysisexponential stability
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 3 , July 2008 , pp. 632 - 656
Copyright
© EDP Sciences, SMAI, 2008

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References

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
Ammari, K., Liu, Z. and Tucsnak, M., Decay rates for a beam with pointwise force and moment feedback. Math. Control Signals Systems 15 (2002) 229–255. CrossRef
S.A. Avdonin and S.A. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems. Cambridge University Press, Cambridge, UK (1995).
Avdonin, S.A. and Ivanov, S.A., Riesz bases of exponentials and divided differences. St. Petersburg Math. J. 13 (2002) 339–351.
Avdonin, S.A. and Moran, W., Simultaneous control problems for systems of elastic strings and beams. Syst. Control Lett. 44 (2001) 147–155. CrossRef
Castro, C. and Zuazua, E., A hybrid system consisting of two flexible beams connected by a point mass: spectral analysis and well-posedness in asymmetric spaces. ESAIM: Proc. 2 (1997) 17–53. CrossRef
C. Castro and E. Zuazua, Boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass. SIAM J. Control Optim. 36 (1998) 1576–1595.
Castro, C. and Zuazua, E., Exact boundary controllability of two Euler-Bernoulli beams connected by a point mass. Math. Comput. Modelling 32 (2000) 955–969. CrossRef
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. CrossRef
Chen, G., Krantz, S.G., Russell, D.L., Wayne, C.E., West, H.H. and Coleman, M.P., Analysis, designs, and behavior of dissipative joints for coupled beams. SIAM J. Appl. Math. 49 (1989) 1665–1693. CrossRef
Cox, S. and Zuazua, E., The rate at which energy decays in a damped string. Comm. Partial Diff. Eq. 19 (1994) 213–243. CrossRef
Cox, S. and Zuazua, E., The rate at which energy decays in a string damped at one end. Indiana Univ. Math. J. 44 (1995) 545–573. CrossRef
Curtain, R.F. and Weiss, G., Exponential stabilization of well-posed systems by colocated feedback. SIAM J. Control Optim. 45 (2006) 273–297. CrossRef
R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures, Mathématiques et Applications 50. Springer-Verlag, Berlin (2006).
Guo, B.Z. and Chan, K.Y., Riesz basis generation, eigenvalues distribution, and exponential stability for a Euler-Bernoulli beam with joint feedback control. Rev. Mat. Complut. 14 (2001) 205–229. CrossRef
Guo, B.Z. and Wang, J.M., Riesz basis generation of an abstract second-order partial differential equation system with general non-separated boundary conditions. Numer. Funct. Anal. Optim. 27 (2006) 291–328. CrossRef
B.Z. Guo and G.Q. Xu, Riesz basis and exact controllability of C 0-groups with one-dimensional input operators. Syst. Control Lett. 52 (2004) 221–232.
Guo, B.Z. and Expansion, G.Q. Xu of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition. J. Funct. Anal. 231 (2006) 245–268. CrossRef
Levin, B.Ya., On bases of exponential functions in L 2. Zapiski Math. Otd. Phys. Math. Facul. Khark. Univ. 27 (1961) 39–48 (in Russian).
Liu, K.S. and Liu, Z., Exponential decay of energy of vibrating strings with local viscoelasticity. Z. Angew. Math. Phys. 53 (2002) 265–280. CrossRef
Liu, K.S. and Rao, B., Exponential stability for the wave equations with local Kelvin-Voigt damping. Z. Angew. Math. Phys. 57 (2006) 419–432. CrossRef
Z.H. Luo, B.Z. Guo and Ö. Morgül, Stability and Stabilization of Linear Infinite Dimensional Systems with Applications. Springer-Verlag, London (1999).
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).
R. Rebarber, Exponential stability of coupled beams with dissipative joints: a frequency domain approach. SIAM J. Control Optim. 33 (1995) 1–28.
Renardy, M., On the linear stability of hyperbolic PDEs and viscoelastic flows. Z. Angew. Math. Phys. 45 (1994) 854–865. CrossRef
Shkalikov, A.A., Boundary problems for ordinary differential equations with parameter in the boundary conditions. J. Soviet Math. 33 (1986) 1311–1342. CrossRef
J.M. Wang and S.P. Yung, Stability of a nonuniform Rayleigh beam with internal dampings. Syst. Control Lett. 55 (2006) 863–870.
G. Weiss and R.F. Curtain, Exponential stabilization of a Rayleigh beam using colocated control. IEEE Trans. Automatic Control (to appear).
Xu, G.Q. and Guo, B.Z., Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation. SIAM J. Control Optim. 42 (2003) 966–984. CrossRef
Xu, G.Q. and Yung, S.P., Stabilization of Timoshenko beam by means of pointwise controls. ESAIM: COCV 9 (2003) 579–600. CrossRef
R.M. Young, An Introduction to Nonharmonic Fourier Series. Academic Press, Inc., London (1980).