Advertisement

Russian Mathematics

, Volume 63, Issue 9, pp 31–42 | Cite as

The Aizerman Problem for Scalar Differential Equations

  • B. S. KalitineEmail author
Article
  • 4 Downloads

Abstract

We study the stability of the equilibrium point of a scalar differential equation of an n-th order. We obtain a positive solution to the Aizerman problem for special-type equations. We prove that one can replace the parameter in the real part of the root of the characteristic equation with an arbitrary continuous function, which depends on all phase variables and preserves the global asymptotic stability property.

Key words

scalar differential equation equilibrium stability Lyapunov function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aizerman, M.A. “On a Problem Concerning Stability “in the Large” of Dynamical Systems”, Usp. Matem. Nauk 4 (4), 187–188 (1949).MathSciNetzbMATHGoogle Scholar
  2. 2.
    Erugin, N.P. “On Certain Questions of Stability of Motion and the Qualitative Theory of Differential Equations in the Large”, Prikl. Matem. i Mekhan. 14 (5), 459–512 (1950).MathSciNetGoogle Scholar
  3. 3.
    Erugin, N.P. “Some General Questions in the Theory of Stability of Motion”, Prikl. Matem. i Mekhan. 15 (2), 227–236 (1951).Google Scholar
  4. 4.
    Malkin, I.G. Theory of Motion Stability (Nauka, Moscow, 1966) [in Russian].zbMATHGoogle Scholar
  5. 5.
    Krasovskii, N.N. “Theorems on Stability of Motions Determined by a System of Two Equations”, Prikl. Matem. i Mekhan. 16 (3), 546–554 (1952).MathSciNetGoogle Scholar
  6. 6.
    Pliss, V.A. Some Problems in the Theory of the Stability of Motion (Izd. LGU, Leningrad, 1958) [in Russian].Google Scholar
  7. 7.
    Leonov, G.A. “On the Aizerman Problem”, Avtomat. i Telemekhan. 7, 37–49 (2009).zbMATHGoogle Scholar
  8. 8.
    Hahn, W. Stability of Motion (Springer-Verlag, New York, 1967).CrossRefGoogle Scholar
  9. 9.
    Barbashin, E.A. Lyapunov Functions (Nauka, Moscow, 1970) [in Russian].Google Scholar
  10. 10.
    Kalitine, B.S. Stability of Differential Equations (Method of Fixed Sign Lyapunov Functions) (LAP Lambert Academic Publishing, Saarbrücken, 2012).Google Scholar
  11. 11.
    Rouche, N., Habets, P., Laloy, M. Stability Theory by Liapunov’s Direct Method (Springer-Verlag, New York, Heidelberg, Berlin, 1977; Mir, Moscow, 1980).CrossRefGoogle Scholar
  12. 12.
    Bulgakov, N.G., Kalitine, B.S. “Generalization of Theorems of Ljapunov’s Second Method. I. Theory”, Izv. Akad. Nauk BSSR. Ser. Fiz.-Matem. Nauk 3, 32–36 (1978).MathSciNetGoogle Scholar
  13. 13.
    Kalitine, B.S. Stability of Nonautonomous Differential Equations (BGU, Minsk, 2013) [in Russian].Google Scholar
  14. 14.
    Demidovich, B.P. Lectures on Mathematical Stability Theory (Nauka, Moscow, 1967) [in Russian].zbMATHGoogle Scholar
  15. 15.
    Liénard, A. “Étude des oscillations autoentretienes”, Rev. Gen. Elec. 23, 901–902, 946–954 (1928).Google Scholar
  16. 16.
    Kalitine, B.S. “Stability of Liénard Equation”, Izv. Vyssh. Uchebn. Zaved., Matem. 10, 17–28 (2018).zbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Belarusian State UniversityMinskRepublic of Belarus

Personalised recommendations