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

Jun-Min Wang; Bao-Zhu Guo; Boumediène Chentouf

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 12, Issue: 1, page 12-34
  • ISSN: 1292-8119

Abstract

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In this paper, we consider the boundary stabilization of a sandwich beam which consists of two outer stiff layers and a compliant middle layer. Using Riesz basis approach, we show that there is a sequence of generalized eigenfunctions, which forms a Riesz basis in the state space. As a consequence, the spectrum-determined growth condition as well as the exponential stability of the closed-loop system are concluded. Finally, the well-posedness and regularity in the sense of Salamon-Weiss class as well as the exact controllability are also addressed.

How to cite

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Wang, Jun-Min, Guo, Bao-Zhu, and Chentouf, Boumediène. "Boundary feedback stabilization of a three-layer sandwich beam: Riesz basis approach." ESAIM: Control, Optimisation and Calculus of Variations 12.1 (2005): 12-34. <http://eudml.org/doc/90785>.

@article{Wang2005,
abstract = { In this paper, we consider the boundary stabilization of a sandwich beam which consists of two outer stiff layers and a compliant middle layer. Using Riesz basis approach, we show that there is a sequence of generalized eigenfunctions, which forms a Riesz basis in the state space. As a consequence, the spectrum-determined growth condition as well as the exponential stability of the closed-loop system are concluded. Finally, the well-posedness and regularity in the sense of Salamon-Weiss class as well as the exact controllability are also addressed. },
author = {Wang, Jun-Min, Guo, Bao-Zhu, Chentouf, Boumediène},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Riesz basis; sandwich beam; exponential stability; exact controllability.; exponential stability; exact controllability},
language = {eng},
month = {12},
number = {1},
pages = {12-34},
publisher = {EDP Sciences},
title = {Boundary feedback stabilization of a three-layer sandwich beam: Riesz basis approach},
url = {http://eudml.org/doc/90785},
volume = {12},
year = {2005},
}

TY - JOUR
AU - Wang, Jun-Min
AU - Guo, Bao-Zhu
AU - Chentouf, Boumediène
TI - Boundary feedback stabilization of a three-layer sandwich beam: Riesz basis approach
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2005/12//
PB - EDP Sciences
VL - 12
IS - 1
SP - 12
EP - 34
AB - In this paper, we consider the boundary stabilization of a sandwich beam which consists of two outer stiff layers and a compliant middle layer. Using Riesz basis approach, we show that there is a sequence of generalized eigenfunctions, which forms a Riesz basis in the state space. As a consequence, the spectrum-determined growth condition as well as the exponential stability of the closed-loop system are concluded. Finally, the well-posedness and regularity in the sense of Salamon-Weiss class as well as the exact controllability are also addressed.
LA - eng
KW - Riesz basis; sandwich beam; exponential stability; exact controllability.; exponential stability; exact controllability
UR - http://eudml.org/doc/90785
ER -

References

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  1. 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).  Zbl0866.93001
  2. G.D. Birkhoff and R.E. Langer, 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.  
  3. R.F. Curtain, The Salamon-Weiss class of well-posed infinite dimensional linear systems: a survey. IMA J. Math. Control Inform.14 (1997) 207–223.  Zbl0880.93021
  4. R.F. Curtain, Linear operator inequalities for strongly stable weakly regular linear systems. Math. Control Signals Systems14 (2001) 299–337.  Zbl1114.93029
  5. 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.  Zbl1301.74030
  6. B.Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass. SIAM J. ControlOptim. 39 (2001) 1736–1747.  Zbl1183.93111
  7. B.Z. Guo and Y.H. Luo, Controllability and stability of a second order hyperbolic system with collocated sensor/actuator. Syst. Control Lett.46 (2002) 45–65.  Zbl0994.93021
  8. S.W. Hansen and R. Spies, Structural damping in a laminated beams due to interfacial slip. J. Sound Vibration204 (1997) 183–202.  
  9. S.W. Hansen and I. Lasiecla, Analyticity, hyperbolicity and uniform stability of semigroupsm arising in models of composite beams. Math. Models Methods Appl. Sci.10 (2000) 555–580.  
  10. T. Kato, Perturbation theory of linear Operators. Springer, Berlin (1976).  Zbl0342.47009
  11. V. Komornik, Exact Controllability and Stabilization: the Multiplier Method. John Wiley and Sons, Ltd., Chichester (1994).  Zbl0937.93003
  12. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).  Zbl0516.47023
  13. D.L. Russell and B.Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain. SIAM J. Control Optim.31 (1993) 659–676.  Zbl0771.93073
  14. D.L. Russell and B.Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation. Trans. Amer. Math. Soc.348 (1996) 3643–3672.  Zbl0862.93035
  15. C. Tretter, Spectral problems for systems of differential equations y ' + A 0 y = λ A 1 y with λ - polynomial boundary conditions. Math. Nachr.214 (2000) 129–172.  Zbl0959.34067
  16. C. Tretter, Boundary eigenvalue problems for differential equations N η = λ P η with λ - polynomial boundary conditions. J. Diff. Equ.170 (2001) 408–471.  Zbl0984.34010
  17. G. Weiss, Transfer functions of regular linear systems I: Characterizations of regularity. Trans. Amer. Math. Soc.342 (1994) 827–854.  Zbl0798.93036
  18. R.M. Young, An Introduction to Nonharmonic Fourier Series. Academic Press, Inc., London (2001).  Zbl0981.42001

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