On the problem of stability of a motion of essentially nonlinear systems
DOI:
https://doi.org/10.15407/dopovidi2021.02.003Keywords:
estimation of the Lyapunov function, essentially nonlinear system, motion, resistance to large disturbancesAbstract
This article discusses essentially nonlinear systems. Following the approach of applying the pseudolinear inequalities
developed in a number of works, new estimates for the variation of Lyapunov functions along solutions
of the considered systems of equations are obtained. Based on these estimates, we obtain sufficient conditions
for the equiboundedness of solutions of second-order systems and sufficient conditions for the stability of an
essentially nonlinear system under large initial perturbations. Conditions for the stability of affine systems are
also obtained.
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References
Lyapunov, А.M. (1950). The general problem of the stability of motion. Moscow, Leningrad: Gostehteoretizdat
(in Russian).
Lakshmikantham, V., Leela S. & Martynyuk, A.A. (2015). Stability analysis of nonlinear systems. 2 ed. Basel:
Birkhäuser.
Corduneanu, C., Li, Y. & Mahdavi, M. (2016). Functional differential equations: Advances and applications.
New York: Wiley.
Martynyuk, A.A. (2007). Stability of motion. The role of multicomponent Liapunov’s functions. Cambridge:
Cambridge Scientific Publishers.
Pachpatte, B.G. (1998). Inequalities for differential and integral equations. San Diego: Academic Press.
N’Doye, I. (2011). Généralisation du lemme de Gronwall–Bellman pour la stabilisation des systèmes
fractionnaires. (These PhD). Université Henri Poincaré — Nancy I; Université Hassan II Aïn Chock de
Casablanca. Nancy, Français.
Director, S.W. & Rohrer, R.A. (1972). Introduction to systems theory. New York: Mc Gram — Hill Book
Com.
Yoshizawa, T. (1966). Stability theory by Liapunov’s second method. Tokyo: Mathematical Society of Japan.
Lakshmikantham, V., Leela, S. & Martynyuk, A.A. (1990). Practical stability of nonlinear systems. Singapore:
World Scientific.
Martynyuk, A.A. & Chernienko, V.A. (2020). Sufficient stability conditions for polynomial systems. Int.
Appl. Mech., 56, No. 1, pp. 13-21. https://doi.org/10.1007/s10778-020-00992-1
Martynyuk, A.A., Khusainov, D.Ya. & Chernienko, V.A. (2018). Constructive estimation of the Lyapunov
function for quadratic nonlinear systems. Int. Appl. Mech., 54, No. 3, pp. 346-357. https://doi.org/10.1007/
s10778-018-0886-y
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