Global exponential stability of impulsive functional differential equations via Razumikhin technique.

*(English)*Zbl 1203.34119This paper deals with some new Razumikhin-type results on global exponential stability of impulsive differential equations with any finite delays, which allow to develop an effective impulsive mechanism to a certain class of functional differential systems even if it may be unstable. Finally, two numerical examples are given to verify the effectiveness of the results above.

Reviewer: Hong Zhang (Umeå)

##### MSC:

34K20 | Stability theory of functional-differential equations |

34K25 | Asymptotic theory of functional-differential equations |

34K45 | Functional-differential equations with impulses |

##### Keywords:

impulsive functional differential equation; global exponential stability; Razumikhin technique
PDF
BibTeX
XML
Cite

\textit{S. Peng} and \textit{L. Yang}, Abstr. Appl. Anal. 2010, Article ID 987372, 11 p. (2010; Zbl 1203.34119)

**OpenURL**

##### References:

[1] | J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993. · Zbl 0938.34536 |

[2] | V. B. Kolmanovskiĭ and V. R. Nosov, Stability of Functional Differential Equations, vol. 180 of Mathematics in Science and Engineering, Academic Press, London, UK, 1986. · Zbl 0593.34070 |

[3] | A. M. Samoĭlenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, Singapore, 1995. · Zbl 0939.16010 |

[4] | X. Liu and G. Ballinger, “Uniform asymptotic stability of impulsive delay differential equations,” Computers & Mathematics with Applications, vol. 41, no. 7-8, pp. 903-915, 2001. · Zbl 0989.34061 |

[5] | X. Liu, “Stability of impulsive control systems with time delay,” Mathematical and Computer Modelling, vol. 39, no. 4-5, pp. 511-519, 2004. · Zbl 1081.93021 |

[6] | Y. Zhang and J. Sun, “Strict stability of impulsive functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 301, no. 1, pp. 237-248, 2005. · Zbl 1068.34073 |

[7] | K. Liu and X. Fu, “Stability of functional differential equations with impulses,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 830-841, 2007. · Zbl 1118.34079 |

[8] | F. Chen and X. Wen, “Asymptotic stability for impulsive functional differential equation,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 1149-1160, 2007. · Zbl 1130.34057 |

[9] | R. Liang and J. Shen, “Uniform stability theorems for delay differential equations with impulses,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 62-74, 2007. · Zbl 1116.34062 |

[10] | Z. Chen and X. Fu, “New Razumikhin-type theorems on the stability for impulsive functional differential systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 9, pp. 2040-2052, 2007. · Zbl 1120.34056 |

[11] | Y. Zhang and J. Sun, “Stability of impulsive functional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 12, pp. 3665-3678, 2008. · Zbl 1152.34053 |

[12] | 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. |

[13] | K. Liu and G. Yang, “The improvement of Razumikhin type theorems for impulsive functional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3104-3109, 2009. · Zbl 1157.34332 |

[14] | Q. Wang and X. Liu, “Exponential stability for impulsive delay differential equations by Razumikhin method,” Journal of Mathematical Analysis and Applications, vol. 309, no. 2, pp. 462-473, 2005. · Zbl 1084.34066 |

[15] | Q. Wang and X. Liu, “Impulsive stabilization of delay differential systems via the Lyapunov-Razumikhin method,” Applied Mathematics Letters, vol. 20, no. 8, pp. 839-845, 2007. · Zbl 1159.34347 |

[16] | 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. · Zbl 1123.34065 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.