Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-27T19:14:49.214Z Has data issue: false hasContentIssue false
  • English
  • Français

A comparison principle and stability for large-scale impulsive delay differential systems

Published online by Cambridge University Press:  17 February 2009

Xinzhi Liu ,
Xuemin Shen  and
Yi Zhang
Xinzhi Liu
Affiliation:
Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada; e-mail: xzliu@math.uwaterloo.ca.
Xuemin Shen
Affiliation:
Department of Electric and Computer Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.
Yi Zhang
Affiliation:
Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada; e-mail: xzliu@math.uwaterloo.ca. China University of Petroleum, Beijing 102249, China.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper studies the stability of large-scale impulsive delay differential systems and impulsive neutral systems. By developing some impulsive delay differential inequalities and a comparison principle, sufficient conditions are derived for the stability of both linear and nonlinear large-scale impulsive delay differential systems and impulsive neutral systems. Examples are given to illustrate the main results.

Type
Research Article
Information
The ANZIAM Journal , Volume 47 , Issue 2 , October 2005 , pp. 203 - 235
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Anokhin, A., Berezansky, L. and Braverman, E., “Exponential stability of linear delay impulsive differential equations”. J. Math. Anal. Appl. 193 (1995) 923–941.CrossRefGoogle Scholar
[2]Ballinger, G. and Liu, X., “On boundedness of solutions of impulsive systems”, Nonlinear Stud. 4 (1997) 121–131.Google Scholar
[3]Ballinger, G. and Liu, X., “Existence and uniqueness results for impulsive delay differential equations”, Dynam. Contin. Discrete Impuls. Systems 5 (1999) 579–591.Google Scholar
[4]Berezansky, L. and Idels, L., “Exponential stability of some scalar impulsive delay differential equation”, Commun. Appl. Anal. 2 (1998) 301–309.Google Scholar
[5]Deo, S. G. and Pandit, S. G., Differential systems involving impulses, Lecture Notes 954 (Springer, Berlin, 1982).Google Scholar
[6]Guan, Z., “Decentralized stabilization for impulsive large-scale systems with delays”, Dynam. Contin. Discrete Impuls. Systems 6 (1999) 367–379.Google Scholar
[7]Guo, D. and Liu, X., “First order impulsive integro-differential equations on unbounded domain in a Banach space”, Dynam. Contin. Discrete Impuls. Systems 2 (1996) 381–394.Google Scholar
[8]Krishna, S. V. and Anokhin, A. V., “Delay differential systems with discontinuous initial data and existence and uniqueness theorems for systems with impulse and delay”, J. Appl. Math. Stochastic Anal. 7 (1994) 49–67.CrossRefGoogle Scholar
[9]Lakshmikantham, V., Bainov, D. D. and Simeonov, P. S., Theory of impulsive differential equations (World Scientific, Teaneck, NJ, 1989).CrossRefGoogle Scholar
[10]Lakshmikantham, V. and Leela, S., Differential and integral inequalities (Academic Press, New York, 1969).Google Scholar
[11]Lakshmikantham, V. and Liu, X., “Stability criteria for impulsive differential equations in term of two measures”, J. Math. Anal. Appl. 137 (1989) 591–604.CrossRefGoogle Scholar
[12]Liu, X., “Stability results for impulsive differential systems with applications to population growth models”, Dynam. Stability Systems 9 (1994) 163–174.CrossRefGoogle Scholar
[13]Liu, X. and Liao, X., “Comparison method and robust stability of large-scale dynamic systems”, Dynam. Contin. Discrete Impuls. Systems Ser. A Math. Anal. 11 (2004) 413–430.Google Scholar
[14]Liu, X. and Shen, J., “Asymptotic behavior of solutions of impulsive neutral differential equations”, Appl. Math. Lett. 12 (1999) 51–58.CrossRefGoogle Scholar
[15]Mil'man, V. D. and Myshkis, A. D., “On the stability of motion in the presence of impulses”, Siberian Math. J. 1 (1960) 233–237.Google Scholar
[16]Pavlids, T., “Stability of systems described by differential equations containing impulses”, IEEE Trans. Automatic Control 12 (1967) 43–45.CrossRefGoogle Scholar
[17]Shen, J., Luo, Z. and Liu, X., “Impulsive stabilization of functional differential equations via Liapunov functionals”, J. Math. Anal. Appl. 240 (1999) 1–15.CrossRefGoogle Scholar
[18]Shen, J. and Yan, J., “Razumikhim type stability theorems for impulsive functional differential equations”, Nonlinear Anal. 33 (1998) 519–531.CrossRefGoogle Scholar
[19]Wen, L. and Weng, P., “Weakly exponentially asymptotic stability of functional differential equation with impulses”, Dynam. Contin. Discrete Impuls. Systems 6 (1999) 251–269.Google Scholar
[20]Yang, T., Impulsive control theory, Lecture Notes in Control and Information Sciences 272 (Springer, Berlin, 2000).Google Scholar