Abstract
This paper studies the exponential stability of nonlinear differential equations with constant decay rate under the assumption that the corresponding crisp equation (without delay, simply, nondelay equation) is exponentially stable. Different from most publications dealing with delay systems by applying Lyapunov-type methods, the perturbed system method is used in this paper. It shall be shown that the considered equations will remain exponentially stable provided the time lag is small enough. Moreover, we formulate and estimate the threshold of delay ensuring exponential stability when a constant decay rate appears explicitly in system model, which is better than the existing results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.-P. Richard, “Time-delay systems: An overview of some recent advances and open problems,” Automatica, vol. v39, pp. 1667–1694, 2003.
S.-I. Niculescu, Delay Effects on Stability: A Robust Control Approach, Springer-Verlag, Berlin, Germany, 2001.
S. D. Brierley, J. N. Chinasson, and S. H. Zak, “On stability independent of delay for linear systems,” IEEE Trans. Automat. Control, vol. AC-27, pp. 252–254, 1982.
T. Mori, “Criteria for asymptotic stability of lineardelay systems,” IEEE Trans. Automat. Control, vol. AC-30, pp. 185–161, 1985.
J.-H. Su, “Further results on the robust stability of linear systems with a single time delay,” Systems & Control Letters, vol. 23, pp. 375–379, 1994.
J.-H. Su, I.-K. Fong, and C.-L. Tseng, “Stability analysis of linear systems with linear delay,” IEEE Trans. Automat. Control, vol. AC-39, pp. 1341–1344, 1994.
C.-L. Tseng, I.-K. Fong, and J.-H Su, “Robust stability analysis for uncertain delay systems with output feedback controller,” Systems & Control Letters, vol. 23, pp. 271–278, 1994.
X. Mao, “Exponential stability of nonlinear differential delay equations,” Systems & Control Letters, vol. 28, pp. 159–165, 1996.
C. Li and Y. Qiu, Comments on ‘Exponential stability of nonlinear differential delay equations’,” Systems & Control Letters, vol. 57, pp. 834–835, 2008.
V. Singh, “Some remarks on global asymptotic stability of neural networks with constant time delay,” Chaos, Solitons and Fractals, vol. 32, pp. 1720–1724, 2007.
S. Arik, “Global asymptotic stability of a larger class of neural networks with constant time delays,” Phys. Lett., vol. 311, no. 6, pp. 504–511, 2003.
V. Singh, “A generalized LMI-based approach to the global asymptotic stability of delayed cellular neural networks,” IEEE Trans. Neural Network, vol. 15, no. 1, pp. 223–225, 2004.
A. Chen, J. Cao, and L. Huang, “An estimation of upperbound of delays for global asymptotic stability of delayed Hopfield neural networks,” IEEE Trans. Circuits Syst I, vol. 49, pp. 1028–1032, 2002.
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Editorial Board member Ju Hyun Park under the direction of Editor Young Il Lee. This work was supported by Fundamental Research Funds for the Central Universities of China (Project No. CDJZR10 18 55 01) and National Natural Science Foundation of China (Grant No.60972107, 60974020).
Hui Wang received her B.S. degree in Mathematics from Chongqing Normal University of China in 1999, an M.S. degree in Mathematics in 2003 and a Ph.D degree in Computer Application Tech-nology in 2007 both from Chongqing University, China. Her research interests include nonlinear systems, neural networks, stability theory, and bifurca-tion analysis.
Chuandong Li received his B.S. degree in Applied Mathematics from Sichuan University in 1992, an M.S. degree in Operational Research and control theory in 2001 and a Ph.D. degree in Computer Software and Theory in 2005 both from Chongqing University, China. His research interest covers nonlinear systems, neural networks, and chaos control.
Hongbing Xu received his B.S. degree in Automation from University of Electronic Science and Technology of China in 1988, an M.S. degree in automatic control from Southeast University in 1991 and a Ph.D degree in Circuits and Systems in 2000 from University of Electronic Science and Technology of China. His research interests include Signal processing technology, fault diagnosis of complex system and intelligent information processing and control.
Rights and permissions
About this article
Cite this article
Wang, H., Li, C. & Xu, H. Exponential stability of nonlinear delay equation with constant decay rate via perturbed system method. Int. J. Control Autom. Syst. 8, 1148–1152 (2010). https://doi.org/10.1007/s12555-010-0524-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12555-010-0524-x