Advertisement

Strict Practical Stability of Impulsive Differential System in Terms of Two Measures

  • Pallvi MahajanEmail author
  • Sanjay Kumar Srivastava
  • Rakesh Dogra
Conference paper
  • 51 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 307)

Abstract

In this paper, an impulsive differential system is investigated to obtain sufficient conditions for strict practical stability. The investigations are carried out by perturbing Lyapunov function and by using the comparison principle. The stability properties are investigated in terms of two measures. Our results demonstrate that impulses do contribute to the system’s stability behaviour. An example is given to compliment on results.

Keywords

Impulsive differential system Strict practical stability Perturbed Lyapunov functions Two measures 

AMS Subject Classification:

34D20 

Notes

Acknowledgements

One of us (PM) would like to thank IKGPTU for providing the online library facility.

References

  1. 1.
    V. Lakshmikantham, S. Leela, A.A. Martynyuk, Practical Stability of Nonlinear Systems (World Scientific, 1990)Google Scholar
  2. 2.
    Y. Zhang, J. Sun, Practical stability of impulsive functional differential equations in terms of two measurements. Comput. Math. Appl. 48, 1549–1556 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    A.A. Soliman, On practical stability of perturbed differential systems. Appl. Math. Comput. 163, 1055–1060 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    S. Hristova, Generalization of practical stability for delay differential equations with respect to initial time difference. AIP Conf. Proc. 1570, 313–320 (2013)CrossRefGoogle Scholar
  5. 5.
    P. Singh, S.K. Srivastava, K. Kaur, Uniform practical stability in terms of two measures with effect of delay at the time of impulses. Nonlinear Eng. 5(2), 93–97 (2016)Google Scholar
  6. 6.
    V. Lakshmikantham, J.V. Devi, Strict stability criteria for impulsive differential systems. Nonlinear Anal., Theory, Methods Appl. 21(10), 785–794 (1993)MathSciNetCrossRefGoogle Scholar
  7. 7.
    R.P. Agarwal, S. Hristova, Strict stability in terms of two measures for impulsive differential equations with ‘supremum. Appl. Anal. 91, 1379–1392 (2012). JanuaryMathSciNetCrossRefGoogle Scholar
  8. 8.
    V. Lakshmikantham, Y. Zhang, Strict practical stability of delay differential equation. Appl. Math. Comput. 122, 341–351 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    S. Sun, W. Chen, Y. Zhao, Z. Han, Strict practical stability for discrete hybrid systems in terms of two measures. Appl. Mech. Mater. 643(2), 90–95 (2014)Google Scholar
  10. 10.
    S. Li, X. Song, A. Li, Strict practical stability of nonlinear impulsive systems by employing two lyapunov-like functions. Nonlinear Anal.: R. World Appl. 9(2), 2262–2269 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    V. Laksmhikantham, X. Liu, Stability criteria of impulsive differential equations in terms of two measures. J. Math. Anal. Appl. 137, 591–604 (1989)MathSciNetCrossRefGoogle Scholar
  12. 12.
    V. Lakshmikantham, X.Z. Liu, Perturbing families of lyapunov functions and stability in terms of two measures. J. Math. Anal. Appl. 140, 107–114 (1989)MathSciNetCrossRefGoogle Scholar
  13. 13.
    V. Lakshmikantham, S. Leela, On perturbing lyapunov functions. Math. Syst. Theory 10, 85–90 (1976)MathSciNetCrossRefGoogle Scholar
  14. 14.
    X. Liu, Stability analysis of impulsive system via perturbing families of lyapunov functions. Rocky Mt. J. Math. 23(2), 651–670 (1993)MathSciNetCrossRefGoogle Scholar
  15. 15.
    H. Zhao, E. Feng, Stability of impulsive system by perturbing lyapunov functions. Appl. Math. Lett. 20(2), 194–198 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    D. Stutson, A. Vatsala, Generalized practical stability results by perturbing lyapunov function. J. Appl. Math. Stoch. Anal. 9(1), 69–75 (1996)MathSciNetCrossRefGoogle Scholar
  17. 17.
    V. Lakshmikantham, D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations (World Scientific, 1989)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Pallvi Mahajan
    • 1
    • 2
    Email author
  • Sanjay Kumar Srivastava
    • 2
  • Rakesh Dogra
    • 2
  1. 1.Inder Kumar Gujral-Punjab Technical UniversityKapurthalaIndia
  2. 2.Beant College of Engineering and TechnologyGurdaspurIndia

Personalised recommendations