Stability estimates in nonlinear differential equations of a special kind

Authors

DOI:

https://doi.org/10.17721/1812-5409.2022/1.9

Keywords:

mathematical model, stability, Lyapunov's second method, convergence, quadratic form

Abstract

Quite a lot of works have been devoted to problems of stability theory and, in particular, to the use of the second Lyapunov method for this. The main ones are the following [1-7]. The main attention in these works is paid to obtaining stability conditions. At the same time, when solving practical problems, it is important to obtain quantitative characteristics of the convergence of solutions to an equilibrium position. In this paper, we consider nonlinear scalar differential equations with nonlinearity of a special form (weakly nonlinear equations). Differential equations of this type are encountered in the study of processes in neurodynamics [8,9]. In this paper, we obtain stability conditions for a stationary solution of scalar equations of this type. And also the characteristics of the convergence of the process are calculated. It is shown that the solution of stability problems is closely related to optimization problems [10-12].

Pages of the article in the issue: 67 - 71

Language of the article: Ukrainian

References

BARBASHIN EA (1967) Introduction to the theory of sustainability. - M., Nauka, 224р.

BARBASHIN EA (1970) Lyapunov functions. M., Nauka, 240 p.

VALEEV KG (1981) Construction of Lyapunov functions. - Kiev, Scientific Opinion, 412 p.

DEMIDOVICH BP (1967) Lectures on the mathematical theory of stability. - M., Nauka, 472 p.

LYAPUNOV AM (1950) The general problem of stability of movement. - M.-L., Gos. issue. Tech.-Theory Lit., 1950. - 471 p.

MALKIN IG (1966) Theory of stability of motion. M., Nauka, 532 p.

PERESTYUK MO, CHERNIKOVA OS (2012) Sustainability theory. - Ukrainian Orthodox Church “Kyiv University”, 103 p.

ARKHANGELSKY VI (1999) Neural networks in automation systems Rumshin. - Kiev: "Technology", 364 p. (VA 590706)

HAIKIN S. (2006) Neural networks: full course, 2nd edition M: Williams Publishing House, 1104 p.

SHORE NZ, STETSENKO SI (1989) Quadratic extremal problems and undifferentiable optimization , K., Naukova Dumka, 208 p.

WERNER KRATZ (1995) Quadratic Functionals in Variational Analysis and Control Theory, Berlin: AcademieVerlagGmbh, 293 pp.

BELLMAN R. (1976) Introduction to the theory of matrices. M., Nauka, 352 p.

Downloads

Published

2022-04-26

How to Cite

Khusainov, D. Y., & Shakotko, T. I. (2022). Stability estimates in nonlinear differential equations of a special kind. Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, (1), 67–71. https://doi.org/10.17721/1812-5409.2022/1.9

Issue

Section

Computer Science and Informatics