Stability of Impulsive Systems with Time-Delay

  • Xinzhi Liu
  • Kexue Zhang
Part of the IFSR International Series in Systems Science and Systems Engineering book series (IFSR, volume 33)


This chapter introduces the fundamental theory of impulsive functional differential equations and studies exponential stability of general nonlinear time-delay systems with delayed impulsive effects, including discrete delays and distributed delays. Stability results are constructed by using the method of Lyapunov functionals and Razumikhin technique, respectively. Some results will be used in Chapters  5 and  6.


  1. 1.
    C.K. Ahn, P. Shi, L. Wu, Receding horizon stabilization and disturbance attenuation for neural networks with time-varying delay. IEEE Trans. Cybern. 45(12), 2680–2692 (2015)CrossRefGoogle Scholar
  2. 6.
    G. Ballinger, Qualitative theory of impulsive delay differential equations. PhD Thesis, University of Waterloo (2000)Google Scholar
  3. 7.
    G. Ballinger, X. Liu, Existence and uniqueness results for impulsive delay differential equations. Dynam. Contin. Discrete Impuls. Syst. Ser. A: Math. Anal. 5(1–4), 579–591 (1999)MathSciNetzbMATHGoogle Scholar
  4. 21.
    W.H. Chen, W.X. Zeng, Exponential stability of nonlinear time delay systems with delayed impulse effects. Automatica 47(5), 1075–1083 (2011)MathSciNetCrossRefGoogle Scholar
  5. 22.
    P. Cheng, F. Deng, F. Yao, Exponential stability analysis of impulsive functional differential equations with delayed impulses. Commun. Nonlinear Sci. Numer. Simul. 19(6), 2104–2114 (2014)MathSciNetCrossRefGoogle Scholar
  6. 35.
    L. Gao, Y. Wu, H. Shen, Exponential stability of nonlinear impulsive and switched time-delay systems with delayed impulse effects. Circuits Syst. Signal Process. 33(7), 2107–2129 (2014)MathSciNetCrossRefGoogle Scholar
  7. 56.
    V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations (World Scientific Publishing, Singapore, 1989)CrossRefGoogle Scholar
  8. 63.
    X. Li, S. Song, Stabilization of delay systems: delay-dependent impulsive control. IEEE Trans. Autom. Control 62(1), 406–411 (2017)MathSciNetCrossRefGoogle Scholar
  9. 78.
    X. Liu, Q. Wang, The method of Lyapunov functionals and exponential stability of impulsive systems with time delay. Nonlinear Anal. 66(7), 1465–1484 (2007)MathSciNetCrossRefGoogle Scholar
  10. 145.
    T. Yang, Impulsive Control Theory (Springer, New York, 2001)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Xinzhi Liu
    • 1
  • Kexue Zhang
    • 1
  1. 1.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations