Skip to main content
SpringerLink
Log in

On Necessary and Sufficient Conditions for the Asymptotic Stability of Impulsive Systems

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We prove that the sufficient conditions for the asymptotic stability of impulsive systems obtained by Gurgula and Perestyuk are also necessary conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations [in Russian], Vyshcha Shkola, Kiev (1987).

    Google Scholar 

  2. A. M. Samoilenko and M. N. Astaf'eva, “Averaging of multifrequency oscillating systems with pulse influence,” Dokl. Akad. Nauk Ukr., Ser. A, No. 8, 12-16 (1989).

    Google Scholar 

  3. A. M. Samoilenko and A. N. Stanzhitskii, “On fluctuations in the averaging scheme for differential equations with random pulse influence,” Ukr. Mat. Zh., 41, No. 5, 631-641 (1989).

    Google Scholar 

  4. A. M. Samoilenko, N. A. Perestyuk, and S. I. Trofimchuk, “Generalized solutions of impulsive systems and pulsation effect,” Ukr. Mat. Zh., 43, No. 5, 657-663 (1991).

    Google Scholar 

  5. A. Cabada and E. Liz, “Discontinuous impulsive differential equations with nonlinear boundary conditions,” Nonlin. Anal., 28, No. 9, 1491-1497 (1997).

    Google Scholar 

  6. A. M. Samoilenko and N. A. Perestyuk, “Stability of solutions of differential equations with pulse influence,” Differents. Uravn., 13, No. 11, 1981-1992 (1977).

    Google Scholar 

  7. A. M. Samoilenko and N. A. Perestyuk, “On stability of solutions of systems with pulse influence,” Differents. Uravn., 17, No. 11, 1995-2001 (1981).

    Google Scholar 

  8. S. I. Gurgula and N. A. Perestyuk, “On stability of equilibrium of impulsive systems,” Mat. Fiz., Issue 31, 9-14 (1982).

    Google Scholar 

  9. S. I. Gurgula and N. A. Perestyuk, “On the second Lyapunov method in systems with pulse influence,” Dokl. Akad. Nauk Ukr. SSR, Ser A, No. 10, 11-14 (1982).

    Google Scholar 

  10. D. D. Bainov and P. S. Simeonov, Systems with Pulse Influence: Stability, Theory, and Applications, Halsted Press, New York (1989).

    Google Scholar 

  11. M. L. Sverdan, “Second Lyapunov method for investigation of stability of differential equations with impulsive perturbations and Markov coefficients,” Ukr. Mat. Zh., 48, No. 6, 826-833 (1996).

    Google Scholar 

  12. N. A. Perestyuk and O. S. Chernikova, “Stability of solutions of impulsive systems,” Ukr. Mat. Zh., 49, No. 1, 98-111 (1997).

    Google Scholar 

  13. A. A. Martynyuk and I. P. Stavronlakis, “Stability analysis with respect to two measures of impulsive systems under structural perturbations,” Ukr. Mat. Zh., 51, No. 11, 1476-1484 (1999).

    Google Scholar 

  14. N. Rouche, P. Habets, and M. Laloy, Stability Theory by Liapunov's Direct Method [Russian translation], Mir, Moscow (1980).

    Google Scholar 

  15. I. G. Malkin, Theory of Stability of Motion [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  16. T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Springer, New York (1975).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk

    R. I. Gladilina & A. O. Ignat'ev

Authors
  1. R. I. Gladilina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. O. Ignat'ev
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gladilina, R.I., Ignat'ev, A.O. On Necessary and Sufficient Conditions for the Asymptotic Stability of Impulsive Systems. Ukrainian Mathematical Journal 55, 1254–1264 (2003). https://doi.org/10.1023/B:UKMA.0000010757.22157.1a

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:UKMA.0000010757.22157.1a

Keywords

Search

Navigation