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Статья
2019

The Aizerman Problem for Scalar Differential Equations


B. S. KalitineB. S. Kalitine
Русская математика
https://doi.org/10.3103/S1066369X19090044
Abstract / Full Text

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.

Author information
  • Belarusian State University, 4 Nezavisimosti Ave., Minsk, 220030, Republic of BelarusB. S. Kalitine
References
  1. Aizerman, M.A. “On a Problem Concerning Stability “in the Large” of Dynamical Systems”, Usp. Matem. Nauk 4 (4), 187–188 (1949).
  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).
  3. Erugin, N.P. “Some General Questions in the Theory of Stability of Motion”, Prikl. Matem. i Mekhan. 15 (2), 227–236 (1951).
  4. Malkin, I.G. Theory of Motion Stability (Nauka, Moscow, 1966) [in Russian].
  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).
  6. Pliss, V.A. Some Problems in the Theory of the Stability of Motion (Izd. LGU, Leningrad, 1958) [in Russian].
  7. Leonov, G.A. “On the Aizerman Problem”, Avtomat. i Telemekhan. 7, 37–49 (2009).
  8. Hahn, W. Stability of Motion (Springer-Verlag, New York, 1967).
  9. Barbashin, E.A. Lyapunov Functions (Nauka, Moscow, 1970) [in Russian].
  10. Kalitine, B.S. Stability of Differential Equations (Method of Fixed Sign Lyapunov Functions) (LAP Lambert Academic Publishing, Saarbrücken, 2012).
  11. Rouche, N., Habets, P., Laloy, M. Stability Theory by Liapunov’s Direct Method (Springer-Verlag, New York, Heidelberg, Berlin, 1977; Mir, Moscow, 1980).
  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).
  13. Kalitine, B.S. Stability of Nonautonomous Differential Equations (BGU, Minsk, 2013) [in Russian].
  14. Demidovich, B.P. Lectures on Mathematical Stability Theory (Nauka, Moscow, 1967) [in Russian].
  15. Liénard, A. “Étude des oscillations autoentretienes”, Rev. Gen. Elec. 23, 901–902, 946–954 (1928).
  16. Kalitine, B.S. “Stability of Liénard Equation”, Izv. Vyssh. Uchebn. Zaved., Matem. 10, 17–28 (2018).