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**Stability analysis of generalized impulsive functional differential equations.**
*(English)*
Zbl 1255.34077

Summary: The stability problem for a class of generalized impulsive functional differential equations in which the state variables on the impulses are related to the time delay is studied. By using Lyapunov functions and Razumikhin techniques, several global exponential stability and uniform stability criteria are derived, which can be applied to impulsive functional differential equations with any time delays. The results obtained improve and extend those in earlier publications. Moreover, our results show that delay differential equations can be exponentially stabilized by impulses in which the state variables are related to the time delay. Finally, two examples are given to illustrate the effectiveness and advantages of the results obtained.

### MSC:

34K20 | Stability theory of functional-differential equations |

34K45 | Functional-differential equations with impulses |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

### Keywords:

impulsive functional differential equations; Razumikhin technique; Lyapunov function; time delay; global exponential stability; uniform stability
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\textit{D. Lin} et al., Math. Comput. Modelling 55, No. 5--6, 1682--1690 (2012; Zbl 1255.34077)

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### References:

[1] | Gopalsamy, K.; Zhang, B., On delay differential equations with impulses, Journal of Mathematical Analysis and Applications, 139, 110-122 (1989) · Zbl 0687.34065 |

[2] | Liu, X.; Ballinger, G., Boundedness for impulsive delay differential equations and applications to population growth models, Nonlinear Analysis: Theory, Methods & Applications, 53, 1041-1062 (2003) · Zbl 1037.34061 |

[3] | Lakshmikantham, V.; Bainov, D.; Simeonov, P., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002 |

[4] | Bainov, D.; Simeonov, P., Systems with Impulse Effect: Stability Theory and Applications (1989), Ellis Horwood Limited: Ellis Horwood Limited Chichester · Zbl 0676.34035 |

[5] | Stamova, I.; Stamov, G., Lyapunov-Razumikhin method for impulsive functional equations and applications to the population dynamics, Journal of Computational and Applied Mathematics, 130, 163-171 (2001) · Zbl 1022.34070 |

[6] | Stamova, I., Vector Lyapunov functions for practical stability of nonlinear impulsive functional differential equations, Journal of Mathematical Analysis and Applications, 325, 612-623 (2007) · Zbl 1113.34058 |

[7] | Luo, Z.; Shen, J., New Razumikhin type theorems for impulsive functional differential equations, Applied Mathematics and Computation, 125, 375-386 (2002) · Zbl 1030.34078 |

[8] | Liu, X.; Ballinger, G., Uniform asymptotic stability of impulsive delay differential equations, Computers & Mathematics with Applications, 41, 903-915 (2001) · Zbl 0989.34061 |

[9] | Liu, X.; Wang, Q., The method of Lyapunov functionals and exponential stability of impulsive systems with time delay, Nonlinear Analysis: Theory, Methods & Applications, 66, 1465-1484 (2007) · Zbl 1123.34065 |

[10] | Wang, Q.; Liu, X., Exponential stability for impulsive delay differential equations by Razumikhin method, Journal of Mathematical Analysis and Applications, 309, 462-473 (2005) · Zbl 1084.34066 |

[11] | Wang, Q.; Liu, X., Impulsive stabilization of delay differential systems via the Lyapunov-Razumikhin method, Applied Mathematics Letters, 20, 839-845 (2007) · Zbl 1159.34347 |

[12] | Wu, Q.; Zhou, J.; Xiang, L., Global exponential stability of impulsive differential equations with any time delays, Applied Mathematics Letters, 23, 143-147 (2010) · Zbl 1210.34105 |

[13] | Zhang, Y.; Sun, J., Strict stability of impulsive functional differential equations, Journal of Mathematical Analysis and Applications, 301, 237-248 (2005) · Zbl 1068.34073 |

[14] | Zhang, Y.; Sun, J., Stability of impulsive functional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 68, 3665-3678 (2008) · Zbl 1152.34053 |

[15] | Li, X., Uniform asymptotic stability and global stability of impulsive infinite delay differential equations, Nonlinear Analysis: Theory, Methods & Applications, 70, 1975-1983 (2009) · Zbl 1175.34094 |

[16] | Hristova, S., Integral stability in terms of two measures for impulsive functional differential equations, Mathematical and Computer Modelling, 51, 100-108 (2010) · Zbl 1190.34091 |

[17] | Ignatyev, A.; Ignatyev, O.; Soliman, A., Asymptotic stability and instability of the solutions of systems with impulse action, Mathematical Notes, 80, 491-499 (2006) · Zbl 1125.34036 |

[18] | Ignatyev, A., On the stability of invariant sets of systems with impulse effect, Nonlinear Analysis, 69, 53-72 (2008) · Zbl 1145.34032 |

[19] | Nieto, J. J.; O’Regan, D., Variational approach to impulsive differential equations, Nonlinear Analysis. Real World Applications, 10, 680-690 (2009) · Zbl 1167.34318 |

[20] | Nieto, J. J.; Rodriguez-Lopez, R., Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations, Journal of Mathematical Analysis and Applications, 318, 593-610 (2006) · Zbl 1101.34051 |

[21] | Nieto, J. J.; Tisdell, C. C., On exact controllability of first-order impulsive differential equations, Advances in Difference Equations, 2010 (2010), 9 pages. Article ID 136504 · Zbl 1193.34125 |

[22] | Ballinger, G.; Liu, X., Existence and uniqueness results for impulsive delay differential equations, Dynamics of Continuous, Discrete and Impulsive Systems, 5, 579-591 (1999) · Zbl 0955.34068 |

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