The method of Lyapunov functionals and exponential stability of impulsive systems with time delay

Nonlinear Analysis, Theory, Methods and Applications

Volumn 66, Issue 7, 2007, Pages 1465-1484

  Liu, Xinzhi ^a   Wang, Qing ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

Author keywords

Exponential stability; Impulsive delay systems; Lyapunov functional

Indexed keywords

DELAY CONTROL SYSTEMS; FUNCTION EVALUATION; NONLINEAR CONTROL SYSTEMS; STABILITY CRITERIA;

EXPONENTIAL STABILITY; IMPULSIVE DELAY SYSTEMS; LYAPUNOV FUNCTIONAL;

LYAPUNOV METHODS;

EID: 33846079071     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2006.02.004     Document Type: Article

References (19)

1. Anokhin A., Berezansky L., and Braverman E. exponential stability of linear delay impulsive differential equations. J. Math. Anal. Appl. 193 (1995) 923-941
2. Ballinger G., and Liu X. Existence and uniqueness results for impulsive delay differential equations. Dyn. Contin. Discrete Impuls. Syst. 5 (1999) 579-591
3. Ballinger G., and Liu X. Practical stability of impulsive delay differential equations and applications to control problems. Optimization Methods and Applications (2001), Kluwer Acad. Publ., Dordrecht
4. Berezansky L., and Idels L. Exponential stability of some scalar impulsive delay differential equation. Commun. Appl. Math. Anal. 2 (1998) 301-309
5. Gopalsamy K., and Zhang B.G. On delay differential equations with impulses. J. Math. Anal. Appl. 139 (1989) 110-122
6. Hale J.K., and Lunel S.M.V. Introduction to Functional Differential Equations (1993), Springer-Verlag, New York
7. Kolmanovskii V.B., and Nosov V.R. Stability of Functional differential equations (1986), Academic Press Inc., London
8. Lakshmikantham V., Bainov D.D., and Simeonov P.S. Theory of Impulsive Differential Equations (1989), World Scientific, Singapore
9. Lakshmikantham V., and Liu X. Stability criteria for impulsive differential equations in terms of two measures. J. Math. Anal. Appl. 137 (1989) 591-604
10. Li X., and Weng P. Impulsive stabilization of two kinds of second-order linear delay differential equations. J. Math. Anal. Appl. 291 (2004) 270-281
11. Liu X. Impulsive stabilization of nonlinear systems. IMA J. Math. Control Inform. 10 (1993) 11-19
12. Liu X. Stability results for impulsive differential systems with applications to population growth models. Dyn. Stability Syst. 9 (1994) 163-174
13. Liu X., and Ballinger G. Existence and continuability of solutions for differential equations with delays and state-dependent impulses. Nonlinear Anal. 51 (2002) 633-647
14. Liu X., and Ballinger G. Boundedness for impulsive delay differential equations and applications to population growth models. Nonlinear Anal. 53 (2003) 1041-1062
15. Liu X., and Ballinger G. Uniform asymptotic stability of impulsive delay differential equations. Comput. Math. Appl. 41 (2001) 903-915
16. Liu X., Shen X., and Zhang Y. exponential stability of singularly perturbed systems with time delay. Appl. Anal. 82 (2003) 117-130
17. Wang Q., and Liu X. Exponential stability for impulsive delay differential equations by Razumikhin method. J. Math. Anal. Appl. 309 (2005) 462-473
18. Shen J., Luo Z., and Liu X. Impulsive stabilization of functional differential equations via Liapunov functionals. J. Math. Anal. Appl. 240 (1999) 1-5
19. Stamova I.M., and Stamov G.T. Lyapunov-Razumikhin method for impulsive functional equations and applications to the population dynamics. J. Comput. Appl. Math. 130 (2001) 163-171