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Russian Mathematics

, Volume 63, Issue 10, pp 13–17 | Cite as

Application of Quadratic Lyapunov Functions to Investigation of Stability of Systems with Delay

  • O. G. AntonovskayaEmail author
Article
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Abstract

When studying systems with delay, a particular attention is paid to such important property as their stability. In investigation of stability of systems of ODEs by the direct Lyapunov method, quadratic Lyapunov functions are widely used. For a system with delay, we suggest to use as Lyapunov functions some quadratic Lyapunov functions constructed for a system of ODEs and satisfying some restrictions on the first derivative. For such systems we find conditions which make possible to use such quadratic functions as Lyapunov functions.

Key words

system with delay stability direct Lyapunov method quadratic Lyapunov function 

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References

  1. 1.
    Lyapunov, A.M. The General Problem of the Stability of Motion (Izd. Tehn.-Teor. Lit, Moscow-Leningrad, 1950) fin Russian].zbMATHGoogle Scholar
  2. 2.
    Barbashin, E.A. Lyapunov Functions (Nauka, Moscow, 1970) fin Russian].Google Scholar
  3. 3.
    Kim, A.V. Direct Lyapunov’s Method for Systems with Delays (Ural University, Ekaterinburg, 1992) fin Russian].Google Scholar
  4. 4.
    Razumikhin, B.S. Stability of Ereditar Systems (Nauka, Moscow, 1988) fin Russian].Google Scholar
  5. 5.
    Krasovskii, N.N. “On asymptotical stability of systems with delay”, Prikl. Mat. Mech.20(4), 513–518 (1956) fin Russian].Google Scholar
  6. 6.
    Krasovskii, N.N. Some Problems of the Theory of Stability of Motions (Gostehizdat, Moscow, 1959) [in Russian].Google Scholar
  7. 7.
    Gorbunov, A.V., Kamenetskii, V.A. “Attraction Domains of Delay Systems: Construction by the Lyapunov Function Method”, Automation and Remote Control66(10), 1569–1579 (2005).MathSciNetCrossRefGoogle Scholar
  8. 8.
    Rasumikhin, B.S. “Application of method of Liapunov to problems of stability of delay systems”, Avtomat. i Telemekh.21(6), 740–748 (1960) fin Russian].Google Scholar
  9. 9.
    Antonovskaya, O.G. “On the Construction of a Quadratic Lyapunov Function with Given Properties”, Diff. Equations 49(9), 1187–1191 (2013).MathSciNetCrossRefGoogle Scholar
  10. 10.
    Antonovskaya, O.G. “Determination of the Coefficients of a Quadratic Lyapunov Function with Given Properties”, Diff. Equations52(3), 265–271 (2016).MathSciNetCrossRefGoogle Scholar
  11. 11.
    Antonovskaya, O.G., Goryunov, V.I. “On a method of evaluation of attraction domain for fixed point of nonlinear point mapping of arbitrary dimension”, Russian Math. (Iz. VUZ)60(12), 9–14 (2016).MathSciNetCrossRefGoogle Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Nizhny Novgorod State UniversityNizhny NovgorodRussia

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