Abstract and Applied Analysis

Practical Stability of Impulsive Discrete Systems with Time Delays

Liangji Sun, Chengyan Liu, and Xiaodi Li

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Abstract

The purpose of this paper is to investigate the practical stability problem for impulsive discrete systems with time delays. By using Lyapunov functions and the Razumikhin-type technique, some criteria which guarantee the practical stability and uniformly asymptotically practical stability of the addressed systems are provided. Finally, two examples are presented to illustrate the criteria.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 954121, 10 pages.

Dates
First available in Project Euclid: 3 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412364372

Digital Object Identifier
doi:10.1155/2014/954121

Mathematical Reviews number (MathSciNet)
MR3186989

Zentralblatt MATH identifier
07023392

Citation

Sun, Liangji; Liu, Chengyan; Li, Xiaodi. Practical Stability of Impulsive Discrete Systems with Time Delays. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 954121, 10 pages. doi:10.1155/2014/954121. https://projecteuclid.org/euclid.aaa/1412364372


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References

  • Z. Zhang and X. Liu, “Robust stability of uncertain discrete impulsive switching systems,” Computers & Mathematics with Applications, vol. 58, no. 2, pp. 380–389, 2009.
  • Q. Song and J. Cao, “Dynamical behaviors of discrete-time fuzzy cellular neural networks with variable delays and impulses,” Journal of the Franklin Institute, vol. 345, no. 1, pp. 39–59, 2008.
  • R. Nigmatulin and M. Kipnis, “Stability of the discrete population model with two delays,” in Proceedings of the International Conference on Physics and Control, vol. 1, pp. 313–315, IEEE, 2003.
  • M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001.
  • Y. Zhang, J. Sun, and G. Feng, “Impulsive control of discrete systems with time delay,” IEEE Transactions on Automatic Control, vol. 54, no. 4, pp. 871–875, 2009.
  • B. Liu and H. J. Marquez, “Razumikhin-type stability theorems for discrete delay systems,” Automatica, vol. 43, no. 7, pp. 1219–1225, 2007.
  • X. Liu and Z. Zhang, “Uniform asymptotic stability of impulsive discrete systems with time delay,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 15, pp. 4941–4950, 2011.
  • Y. Zhang, “Exponential stability of impulsive discrete systems with time delays,” Applied Mathematics Letters, vol. 25, no. 12, pp. 2290–2297, 2012.
  • V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Practical Stability of Nonlinear Systems, World Scientific, Singapore, 1990.
  • C. H. Kou and S. N. Zhang, “Practical stability for finite delay differential systems in terms of two measures,” Acta Mathematicae Applicatae Sinica, vol. 25, no. 3, pp. 476–483, 2002.
  • D. D. Bainov and I. M. Stamova, “On the practical stability of the solutions of impulsive systems of differential-difference equations with variable impulsive perturbations,” Journal of Mathematical Analysis and Applications, vol. 200, no. 2, pp. 272–288, 1996.
  • Y. Zhang and J. Sun, “Practical stability of impulsive functional differential equations in terms of two measurements,” Computers & Mathematics with Applications, vol. 48, no. 10-11, pp. 1549–1556, 2004.
  • Y. Zhang and J. Sun, “Eventual practical stability of impulsive differential equations with time delay in terms of two measurements,” Journal of Computational and Applied Mathematics, vol. 176, no. 1, pp. 223–229, 2005.
  • R. Villafuerte, S. Mondié, and A. Poznyak, “Practical stability of time-delay systems: LMI's approach,” European Journal of Control, vol. 17, no. 2, pp. 127–138, 2011.
  • I. M. Stamova, “Vector Lyapunov functions for practical stability of nonlinear impulsive functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 612–623, 2007.
  • D. D. Baĭnov and P. S. Simeonov, Systems with Impulse Effect: Stability, Theory and Applications, Halsted Press, New York, NY, USA, 1989.
  • V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Singapore, 1989.
  • T. Yang, Impulsive Systems and Control: Theory and Applications, Nova Science Publishers, Huntington, NY, USA, 2001.
  • X. Fu, B. Yan, and Y. Liu, Introduction of Impulsive Differential Systems, Science Press, Beijing, China, 2005.
  • V. Lakshmikantham, V. M. Matrosov, and S. Sivasundaram, Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems, vol. 63 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1991.
  • J. H. Shen, “Razumikhin techniques in impulsive functional-differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 36, no. 1, pp. 119–130, 1999.
  • Z. Luo and J. Shen, “New Razumikhin type theorems for impulsive functional differential equations,” Applied Mathematics and Computation, vol. 125, no. 2-3, pp. 375–386, 2002.
  • X. Li, “Further analysis on uniform stability of impulsive infinite delay differential equations,” Applied Mathematics Letters, vol. 25, no. 2, pp. 133–137, 2012.
  • X. Fu and X. Li, “Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems,” Journal of Computational and Applied Mathematics, vol. 224, no. 1, pp. 1–10, 2009.
  • X. Li, “New results on global exponential stabilization of impulsive functional differential equations with infinite delays or finite delays,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 4194–4201, 2010.
  • Z. Luo and J. Shen, “Stability of impulsive functional differential equations via the Liapunov functional,” Applied Mathematics Letters, vol. 22, no. 2, pp. 163–169, 2009.
  • X. Liu and Q. Wang, “The method of Lyapunov functionals and exponential stability of impulsive systems with time delay,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 7, pp. 1465–1484, 2007.
  • X. Li and M. Bohner, “An impulsive delay differential inequality and applications,” Computers & Mathematics with Applications, vol. 64, no. 6, pp. 1875–1881, 2012.
  • X. Li, “Uniform asymptotic stability and global stabiliy of impulsive infinite delay differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 5, pp. 1975–1983, 2009.
  • X. Li, H. Akca, and X. Fu, “Uniform stability of impulsive infinite delay differential equations with applications to systems with integral impulsive conditions,” Applied Mathematics and Computation, vol. 219, no. 14, pp. 7329–7337, 2013. \endinput