Skip to main content
SpringerLink
Log in

Development of the Lyapunov function method in the stability problem for functional-differential equations

  • Ordinary Differential Equations
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

In the present paper, we consider the stability problem for delay functional-differential equations with finite delay. We suggest a development of the Lyapunov function method involving the use of scalar comparison equations and limit functions and equations. We prove a localization theorem for the positive limit set of a bounded solution and a theorem on the asymptotic stability of the zero solution. We present examples of sufficient conditions for the asymptotic stability of solutions of systems of the first, second, and arbitrary orders.

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. Razumikhin, B.S., Prikl. Mat. Mekh., 1956, vol. 20, no. 4, pp. 500–512.

    Google Scholar 

  2. Krasovskii, N.N., Prikl. Mat. Mekh., 1956, vol. 20, no. 4, pp. 513–518.

    MathSciNet  Google Scholar 

  3. Kato, J., Funkcial. Ekvac., 1973, vol. 16, pp. 225–239.

    MATH  MathSciNet  Google Scholar 

  4. Lakshmikantham, V., Leela, S., and Martynyuk, A.A., Ustoichivost’ dvizheniya: metod sravneniya (Stability of Motion: The Comparison Method), Kiev: Naukova Dumka, 1991.

    Google Scholar 

  5. Lakshmikantham, V. and Martynyuk, A.A., Prikl. Mekh., 1993, vol. 29, no. 2, pp. 3–16.

    MathSciNet  Google Scholar 

  6. Andreev, A.S., Dokl. Akad. Nauk, 1997, vol. 356, no. 7, pp. 151–153.

    MathSciNet  Google Scholar 

  7. Andreev, A.S. and Khusanov, D.Kh., Differ. Uravn., 1998, vol. 34, no. 7, pp. 435–440.

    Google Scholar 

  8. Andreev, A.S., Ustoichivost’ neavtonomnykh funktsional’no-differentsial’nykh uravnenii (Stability of Nonautonomous Functional-Differential Equations), Ul’yanovsk, Ul’yanovsk State University, 2005.

    Google Scholar 

  9. Sell, G.R., Trans. Amer. Math. Soc., 1967, vol. 22, pp. 241–283.

    Article  MathSciNet  Google Scholar 

  10. Razumikhin, B.S., Ustoichivost’ ereditarnykh sistem (Stability of Hereditary Systems), Moscow: Nauka, 1988.

    Google Scholar 

  11. Lozinskii, S.M., Izv. Vyssh. Uchebn. Zaved. Mat., 1958, no. 5, pp. 52–90.

  12. Xu, Bu Gong and Liu, Yong Qing, IEEE Trans. Automat. Control, 1994, vol. 39, no. 4, pp. 839–841.

    Article  MATH  MathSciNet  Google Scholar 

  13. Kolmanovskii, V.B., Nonlinear Differ. Equat., 1995, vol. 2, pp. 880–892.

    MathSciNet  Google Scholar 

  14. Kolmanovskii, V.B., Prikl. Mat. Mekh., 1996, vol. 60, no. 2, pp. 210–222.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ul’yanovsk State University, Ul’yanovsk, Russia

    O. A. Peregudova

Authors
  1. O. A. Peregudova
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © O.A. Peregudova, 2008, published in Differentsial’nye Uravneniya, 2008, Vol. 44, No. 12, pp. 1638–1647.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Peregudova, O.A. Development of the Lyapunov function method in the stability problem for functional-differential equations. Diff Equat 44, 1701–1710 (2008). https://doi.org/10.1134/S0012266108120069

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266108120069

Keywords

Search

Navigation