Stability of solutions and the problem of Aizerman  for sixth-order differential equations

  • Boris S. Kalitine Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus
DOI: https://doi.org/10.33581/2520-6508-2020-2-49-58

Abstract

This article is devoted to the investigation of stability of equilibrium of ordinary differential equations using the method of semi-definite Lyapunov’s functions. Types of scalar nonlinear sixth-order differential equations for which regular constant auxiliary functions are used are emphasized. Sufficient conditions of global asymptotic stability and instability of the zero solution have been obtained and it has been established that the Aizerman problem has a positive solution concerning the roots of the corresponding linear differential equation. Studies highlight the advantages of using semi-definite functions compared to definitely positive Lyapunov’s functions.

Author Biography

Boris S. Kalitine, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

PhD (physics and mathematics), docent; professor at the department of analytical economics and econometrics, faculty of economy

References

  1. Ogurtsov AI. [On the stability in general of solutions of third-order and fourth-order nonlinear differential equations]. Izvestiya vysshikh uchebnykh zavedenii. Matematika. 1958;1:124–129. Russian.
  2. Ogurtsov AI. [On the stability of solutions of two nonlinear differential equations of the third and fourth order]. Prikladnaya matematika i mekhanika. 1959;23(1):179–181. Russian.
  3. Ogurtsov AI. [On the stability of solutions of certain third-order and fourth-order nonlinear differential equations]. Izvestiya vysshikh uchebnykh zavedenii. Matematika. 1959;3:200–209. Russian.
  4. Ogurtsov AI. [On the stability of solutions of certain nonlinear differential equations of the fifth and sixth orders]. Matematicheskie zapiski. 1962;3(2):78–93. Russian.
  5. Barbashin EA. Funktsii Lyapunova [Lyapunov’s functions]. Moscow: Nauka; 1970. 240 p. Russian.
  6. Lyapunov AM. Obshchaya zadacha ob ustoichivosti dvizheniya [The general problem of the stability of motion]. Moscow: Gostekhizdat; 1950. 472 p. Russian.
  7. Kalitine BS. Ustoichivost’ differentsial’nykh uravnenii (Metod znakopostoyannykh funktsii Lyapunova) [Stability of differential equations (Lyapunov’s method of semi-definite functions)]. Saarbrücken: LAP LAMBERT Academic Publishing; 2012. 223 p. Russian.
  8. Kalitine BS. Stability of Liénard equation. Izvestiya vysshikh uchebnykh zavedenii. Matematika. 2018;10:17–28. Russian.
  9. Kalitine BS. On the stability of third order differential equations. Journal of the Belarusian State University. Mathematics and Informatics. 2018;2:25–33. Russian.
  10. Kalitine BS. Stability of some differential equations of the fourth-order and fifth-order. Journal of the Belarusian State University. Mathematics and Informatics. 2019;1:18–27. Russian.
  11. Kalitine BS. [On the Aizerman problem for systems of two differential equations]. Matematicheskie zametki. 2019;105(2):240 –250. Russian.
  12. Kalitine BS. Ustoichivost’ dinamicheskikh sistem vtorogo poryadka [Stability of second-order dynamical systems]. Saarbrücken: LAP LAMBERT Academic Publishing; 2019. 138 p. Russian.
  13. Kalitine BS. Ustoichivost’ dinamicheskikh sistem (Metod znakopostoyannykh funktsii Lyapunova) [Stability of dynamical systems (Lyapunov’s method of semi-definite functions)]. Saarbrücken: LAP LAMBERT Academic Publishing; 2013. 259 p. Russian.
  14. Aizerman MA. [On one problem concerning stability «in large» dynamical systems]. Uspekhi matematicheskikh nauk. 1949;4(4):187–188. Russian.
Published
2020-07-30
Keywords: scalar differential equation, stability, semi-definite Lyapunov's function, equilibrium
How to Cite
Kalitine, B. S. (2020). Stability of solutions and the problem of Aizerman  for sixth-order differential equations. Journal of the Belarusian State University. Mathematics and Informatics, 2, 49-58. https://doi.org/10.33581/2520-6508-2020-2-49-58
Issue
No 2 (2020): Journal of the Belarusian State University. Mathematics and Informatics
Section
Differential Equations and Optimal Control

Copyright (c) 2020 Journal of the Belarusian State University. Mathematics and Informatics

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

The authors who are published in this journal agree to the following:

  1. The authors retain copyright on the work and provide the journal with the right of first publication of the work on condition of license Creative Commons Attribution-NonCommercial. 4.0 International (CC BY-NC 4.0).
  2. The authors retain the right to enter into certain contractual agreements relating to the non-exclusive distribution of the published version of the work (e.g. post it on the institutional repository, publication in the book), with the reference to its original publication in this journal.
  3. The authors have the right to post their work on the Internet (e.g. on the institutional store or personal website) prior to and during the review process, conducted by the journal, as this may lead to a productive discussion and a large number of references to this work. (See The Effect of Open Access.)