Skip to main content
SpringerLink
Log in

Global Asymptotic Stability with Respect to Part of the Variables for Solutions of Systems of Ordinary Differential Equations

  • ORDINARY DIFFERENTIAL EQUATIONS
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We propose a new approach to studying the global asymptotic stability of an equilibrium position of a system of ordinary differential equations with respect to part of the variables. New functions, different from Lyapunov ones, are used to study state trajectories of systems of ordinary differential equations and prove the global stability theorem. An application of the theorem to proving the global stability for some systems of ordinary differential equations studied by Krasovskii and Pliss is considered.

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.

Fig. 1.
Fig. 2.

Similar content being viewed by others

Notes

  1. Stability with respect to part of the variables is sometimes also called partial stability.

REFERENCES

  1. Poincaré, H., O krivykh, opredelyaemykh differentsial’nymi uravneniyami (Mémoire sur les courbes définies par les équations différentielles (I–IV)), Moscow–Leningrad: OGIZ GITTL, 1947.

    Google Scholar 

  2. Lyapunov, A.M., Obshchaya zadacha ob ustoichivosti dvizheniya (General Problem of the Stability of Motion), Moscow–Leningrad: GITTL, 1951.

    Google Scholar 

  3. Malkin, I.G., On the stability of motion in Lyapunov sense, Mat. Sb., 1938, vol. 3, no. 1, pp. 47–100.

    Google Scholar 

  4. Rumyantsev, V.V., On the stability of motion with respect to part of the variables, Vestn. Mosk. Gos. Univ. Ser. Mat. Mekh. Fiz. Astronom. Khim., 1957, no. 4, pp. 9–16.

  5. Rumyantsev, V.V. and Oziraner, A.S., Ustoichivost’ i stabilizatsiya dvizheniya po otnosheniyu k chasti peremennykh (Stability and Stabilization of Motion with Respect to Part of the Variables), Moscow: Nauka, 1987.

    MATH  Google Scholar 

  6. Vorotnikov, V.I., Partial Stability and Control, Boston: Birkhäuser, 1998.

    MATH  Google Scholar 

  7. Vorotnikov, V.I. and Rumyantsev, V.V., Osnovy teorii chastichnoi ustoichivosti i upravleniya, vvedenie v teoriyu ustoichivosti i upravleniya (Basics of the Theory of Partial Stability and Control. Introduction to the Theory of Stability and Control), Nizhnii Tagil: Nizhnii Tagil Fil. Ural. Fed. Univ, 2014.

    Google Scholar 

  8. Matrosov, V.M., Development of the method of Lyapunov functions in the stability theory, in Tr. II Vsesoyuz. s’ezda po teoreticheskoi i prikladnoi mekhanike. T. 1 (Proc. II All-Union Congr. Theor. Appl. Mech. Vol. 1), Moscow: Nauka, 1965, pp. 112–125.

    Google Scholar 

  9. Matrosov, V.M., Metod vektornykh funktsii Lyapunova: analiz dinamicheskikh svoistv nelineinykh sistem (Method of Lyapunov Vector Functions: Analysis of Dynamic Properties of Nonlinear Systems), Moscow: Fizmatlit, 2001.

    Google Scholar 

  10. Filimonov, M., On a new method for studying stability in the whole by a part of the variables of solutions of systems of ordinary differential equations, AIP Conf. Proc., 2016, vol. 1789, p. 040023.

    Article  Google Scholar 

  11. Hartman, Ph., Ordinary Differential Equations, New York–London–Sydney: John Wiley & Sons, 1964. Translated under the title: Obyknovennye differentsial’nye uravneniya, Moscow: Mir, 1970.

    MATH  Google Scholar 

  12. Filimonov, M.Y., On the justification of the applicability of the Fourier method to the solution of nonlinear partial differential equations, Russ. J. Numer. Anal. Math. Model., 1996, vol. 11, no. 1, pp. 27–39.

    Article  MathSciNet  Google Scholar 

  13. Filimonov, M.Yu., On applying Lyapunov functions when justifying the Fourier method for nonlinear partial differential equations, Vychislit. Tekhnol., 1993, vol. 2, no. 5, pp. 214–216.

    Google Scholar 

  14. Nemytskii, V.V. and Stepanov, V.V., Kachestvennaya teoriya differentsial’nykh uravnenii (Qualitative Theory of Differential Equations), Moscow–Leningrad: GITTL, 1949.

    Google Scholar 

  15. Stepanov, V.V., Kurs differentsial’nykh uravnenii (A Course in Differential Equations), Moscow–Leningrad: GIFML, 1945.

    Google Scholar 

  16. Barbashin, E.P., Vvedenie v teoriyu ustoichivosti (Introduction to Stability Theory), Moscow: Nauka, 1967.

    Google Scholar 

  17. Corduneanu, C., Sur la stability partielle, Rev. Roum. Math. Pure Appl., 1964, vol. 7, no. 2, pp. 247–252.

    MATH  Google Scholar 

  18. Erugin, N.P., On some issues of the stability of motion and qualitative theory of differential equations in the large, Prikl. Mat. Mekh., 1950, vol. 14, no. 5, pp. 459–512.

    Google Scholar 

  19. Lyusternik, L.A. and Sobolev, V.I., Elementy funktsional’nogo analiza (Elements of Functional Analysis), Moscow: Nauka, 1965.

    Google Scholar 

  20. Vorotnikov, V.I., Ustoichivost’ dinamicheskikh sistem po otnosheniyu k chasti peremennykh (Stability of Dynamical Systems with Respect to Part of the Variables), Moscow: Nauka, 1991.

    MATH  Google Scholar 

  21. Krasovskii, N.N., Nekotorye zadachi teorii ustoichivosti dvizheniya (Some Problems in the Theory of Stability of Motion), Moscow: Fizmatgiz, 1959.

    Google Scholar 

  22. Vorotnikov, V.I. and Prokop’ev, V.P., On the stability with respect to part of the variables for linear systems, Prikl. Mat. Mekh., 1978, vol. 42, no. 2, pp. 268–271.

    MathSciNet  Google Scholar 

  23. Pliss, V.A., Nekotorye problemy teorii ustoichivosti v tselom (Some Problems in the Theory of Global Stability), Leningrad: Izd. Leningr. Univ., 1958.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, 620108, Russia

    M. Yu. Filimonov

  2. Ural Federal University, Yekaterinburg, 620002, Russia

    M. Yu. Filimonov

Authors
  1. M. Yu. Filimonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Yu. Filimonov.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Filimonov, M.Y. Global Asymptotic Stability with Respect to Part of the Variables for Solutions of Systems of Ordinary Differential Equations. Diff Equat 56, 710–720 (2020). https://doi.org/10.1134/S001226612006004X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S001226612006004X

Search

Navigation