Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T19:14:13.790Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundary feedback stabilization of a three-layer sandwichbeam: Riesz basis approach

Published online by Cambridge University Press:  15 December 2005

Jun-Min Wang ,
Bao-Zhu Guo  and
Boumediène Chentouf
Jun-Min Wang
Affiliation:
Department of Mathematics, Beijing Institue of Technology, Beijing, 100081, P.R. China; [email protected] School of Computational and Applied Mathematics, University of the Witwatersrand, Private 3, Wits 2050, Johannesburg, South Africa.
Bao-Zhu Guo
Affiliation:
School of Computational and Applied Mathematics, University of the Witwatersrand, Private 3, Wits 2050, Johannesburg, South Africa. Institute of Systems Science, Academy of Mathematics and System Sciences, Academia Sinica, Beijing 100080, P.R. China; [email protected]
Boumediène Chentouf
Affiliation:
Department of Mathematics and Statistics, Sultan Qaboos University, PO Box 36 Al Khodh 123, Muscat, Sultanate of Oman.
Get access

Abstract

In this paper, we consider the boundary stabilization of asandwich beam which consists of two outer stiff layers and acompliant middle layer. Using Riesz basis approach, we show thatthere is a sequence of generalized eigenfunctions, which forms aRiesz basis in the state space. As a consequence, thespectrum-determined growth condition as well as the exponentialstability of the closed-loop system are concluded. Finally, thewell-posedness and regularity in the sense of Salamon-Weiss classas well as the exact controllability are also addressed.

Keywords

Riesz basissandwich beamexponential stabilityexact controllability.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 1 , January 2006 , pp. 12 - 34
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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).
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. American Academy Arts Sci. 58 (1923) 49–128. CrossRef
Curtain, R.F., The Salamon-Weiss class of well-posed infinite dimensional linear systems: a survey. IMA J. Math. Control Inform. 14 (1997) 207–223. CrossRef
Curtain, R.F., Linear operator inequalities for strongly stable weakly regular linear systems. Math. Control Signals Systems 14 (2001) 299–337. CrossRef
R.H. Fabiano and S.W. Hansen, Modeling and analysis of a three-layer damped sandwich beam. Discrete Contin. Dynam. Syst., Added Volume (2001) 143–155.
B.Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass. SIAM J. Control Optim. 39 (2001) 1736–1747.
Guo, B.Z. and Luo, Y.H., Controllability and stability of a second order hyperbolic system with collocated sensor/actuator. Syst. Control Lett. 46 (2002) 45–65. CrossRef
Hansen, S.W. and Spies, R., Structural damping in a laminated beams due to interfacial slip. J. Sound Vibration 204 (1997) 183–202. CrossRef
Hansen, S.W. and Lasiecla, I., Analyticity, hyperbolicity and uniform stability of semigroupsm arising in models of composite beams. Math. Models Methods Appl. Sci. 10 (2000) 555–580.
T. Kato, Perturbation theory of linear Operators. Springer, Berlin (1976).
V. Komornik, Exact Controllability and Stabilization: the Multiplier Method. John Wiley and Sons, Ltd., Chichester (1994).
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).
Russell, D.L. and Zhang, B.Y., Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain. SIAM J. Control Optim. 31 (1993) 659–676. CrossRef
Russell, D.L. and Zhang, B.Y., Exact controllability and stabilizability of the Korteweg-de Vries equation. Trans. Amer. Math. Soc. 348 (1996) 3643–3672. CrossRef
Tretter, C., Spectral problems for systems of differential equations $y^\prime+A_0y=\lambda A_1y$ with $\lambda-$ polynomial boundary conditions. Math. Nachr. 214 (2000) 129–172. 3.0.CO;2-X>CrossRef
Tretter, C., Boundary eigenvalue problems for differential equations $N\eta=\lambda P\eta$ with $\lambda-$ polynomial boundary conditions. J. Diff. Equ. 170 (2001) 408–471. CrossRef
Weiss, G., Transfer functions of regular linear systems I: Characterizations of regularity. Trans. Amer. Math. Soc. 342 (1994) 827–854.
R.M. Young, An Introduction to Nonharmonic Fourier Series. Academic Press, Inc., London (2001).