Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks*

Zhong-Jie Han; Gen-Qi Xu

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 2, page 552-574
  • ISSN: 1292-8119

Abstract

top

In this paper, the stability of a Timoshenko beam with time delays in the boundary input is studied. The system is fixed at the left end, and at the other end there are feedback controllers, in which time delays exist. We prove that this closed loop system is well-posed. By the complete spectral analysis, we show that there is a sequence of eigenvectors and generalized eigenvectors of the system operator that forms a Riesz basis for the state Hilbert space. Hence the system satisfies the spectrum determined growth condition. Then we conclude the exponential stability of the system under certain conditions. Finally, we give some simulations to support our results.


How to cite

top

Han, Zhong-Jie, and Xu, Gen-Qi. "Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks*." ESAIM: Control, Optimisation and Calculus of Variations 17.2 (2011): 552-574. <http://eudml.org/doc/276334>.

@article{Han2011,
abstract = {
In this paper, the stability of a Timoshenko beam with time delays in the boundary input is studied. The system is fixed at the left end, and at the other end there are feedback controllers, in which time delays exist. We prove that this closed loop system is well-posed. By the complete spectral analysis, we show that there is a sequence of eigenvectors and generalized eigenvectors of the system operator that forms a Riesz basis for the state Hilbert space. Hence the system satisfies the spectrum determined growth condition. Then we conclude the exponential stability of the system under certain conditions. Finally, we give some simulations to support our results.
},
author = {Han, Zhong-Jie, Xu, Gen-Qi},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Timoshenko beam; exponential stability; time delay; Riesz basis; feedback control},
language = {eng},
month = {5},
number = {2},
pages = {552-574},
publisher = {EDP Sciences},
title = {Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks*},
url = {http://eudml.org/doc/276334},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Han, Zhong-Jie
AU - Xu, Gen-Qi
TI - Exponential stability of Timoshenko beam system with delay terms in boundary feedbacks*
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/5//
PB - EDP Sciences
VL - 17
IS - 2
SP - 552
EP - 574
AB - 
In this paper, the stability of a Timoshenko beam with time delays in the boundary input is studied. The system is fixed at the left end, and at the other end there are feedback controllers, in which time delays exist. We prove that this closed loop system is well-posed. By the complete spectral analysis, we show that there is a sequence of eigenvectors and generalized eigenvectors of the system operator that forms a Riesz basis for the state Hilbert space. Hence the system satisfies the spectrum determined growth condition. Then we conclude the exponential stability of the system under certain conditions. Finally, we give some simulations to support our results.

LA - eng
KW - Timoshenko beam; exponential stability; time delay; Riesz basis; feedback control
UR - http://eudml.org/doc/276334
ER -

References

top
  1. J.W. Brown and R.V. Churchill, Complex variables and applications. Seventh Edition, China Machine Press, Beijing (2004).  
  2. R. Datko, Two examples of ill-posedness with respect to small time delays in stabilized elastic systems. IEEE Trans. Automat. Contr.38 (1993) 163–166.  
  3. J.U. Kim and Y. Renardy, Boundary control of the Timoshenko beam. SIAM J. Control Optim.25 (1987) 1417–1429.  
  4. W.H. Kwon, G.W. Lee and S.W. Kim, Performance improvement, using time delays in multi-variable controller design. Int. J. Control52 (1990) 1455–1473.  
  5. J.S. Liang, Y.Q. Chen and B.Z. Guo, A new boundary control method for beam equation with delayed boundary measurement using modified smith predictors, in Proceedings of the 42nd IEEE Conference on Decision and Control, Hawaii (2003) 809–814.  
  6. Yu.I. Lyubich and V.Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces. Studia Math.88 (1988) 34–37.  
  7. R. Mennicken and M. Möller, Non-self-adjoint boundary eigenvalue problem, North-Holland Mathematics Studies192. North-Holland, Amsterdam (2003).  
  8. W. Michiels and S.I. Niculescu, Stability and stabilization of time-delay systems: An Eigenvalue-based approach. Society for Industrial and Applied Mathematics, Philadelphia (2007).  
  9. O. Mörgul, On the stabilization and stability robustness against small delays of some damped wave equation. IEEE Trans. Automat. Contr.40 (1995) 1626–1630.  
  10. S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim.45 (2006) 1561–1585.  
  11. S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay. Differential and Integral Equations21 (2008) 935–958.  
  12. S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks. NHM2 (2007) 425–479.  
  13. A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, Berlin (1983).  
  14. K. Sriram and M.S. Gopinathan, A two variable delay model for the circadian rhythm of Neurospora crassa. J. Theor. Biol.231 (2004) 23–38.  
  15. J. Srividhya and M.S. Gopinathan, A simple time delay model for eukaryotic cell cycle. J. Theor. Biol.241 (2006) 617–627.  
  16. H. Suh and Z. Bien, Use of time-delay actions in the controller design. IEEE Trans. Automat. Contr.25 (1980) 600–603.  
  17. S. Timoshenko, Vibration Problems in Engineering. Van Norstrand, New York (1955).  
  18. Q.P. Vu, J.M. Wang, G.Q. Xu and S.P. Yung, Spectral analysis and system of fundamental solutions for Timoshenko beams. Appl. Math. Lett.18 (2005) 127–134.  
  19. G.Q. Xu and D.X. Feng, The Riesz basis property of a Timoshenko beam with boundary feedback and application. IMA J. Appl. Math.67 (2002) 357–370.  
  20. G.Q. Xu and B.Z. Guo, Riesz basis property of evolution equations in Hilbert space and application to a coupled string equation. SIAM J. Control Optim.42 (2003) 966–984.  
  21. G.Q. Xu and J.G. Jia, The group and Riesz basis properties of string systems with time delay and exact controllability with boundary control. IMA J. Math. Control Inf.23 (2006) 85–96.  
  22. G.Q. Xu and S.P. Yung, The expansion of semigroup and criterion of Riesz basis J. Differ. Equ.210 (2005) 1–24.  
  23. G.Q. Xu, Z.J. Han and S.P. Yung, Riesz basis property of serially connected Timoshenko beams. Int. J. Control80 (2007) 470–485.  
  24. G.Q. Xu, S.P. Yung and L.K. Li, Stabilization of wave systems with input delay in the boundary control. ESAIM: COCV12 (2006) 770–785.  
  25. R.M. Young, An introduction to nonharmonic Fourier series. Academic Press, London (1980) 80–84.  

NotesEmbed ?

top

You must be logged in to post comments.