Abstract
The paper deals with the problem of global asymptotic stability of the zero solution of a system of autonomous differential equations. A proposed method for studying this problem is based on the use of an auxiliary positive definite function whose derivative can be sign-alternating along solutions of the system.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. N. Barbashin and E. A. Krasovskii, “On stability of motion in the whole,” Dokl. Akad. Nauk SSSR 86(6), 453–456 (1952).
N. N. Krasovskii, Certain Problems in the Theory of Stability of Motion (Fizmatgiz, Moscow, 1959) [in Russian].
A. O. Ignat’ev, “Some generalizations of the Barbashin-Krasovskii theorems,” Matem. Fizika 34, 19–22 (1983).
A. O. Ignat’ev, “Stability of almost periodic systems with respect to some of the variables,” Differ. Uravn. 25(8), 1446–1448 (1989).
A. O. Ignatyev, “On the stability of equilibriumfor almost periodic systems,” Nonlinear Anal. Theory Methods Appl. 29(8), 957–962 (1997).
A. O. Ignatyev, “On the asymptotic stability in functional differential equations,” Proc. Amer. Math. Soc. 127(6), 1753–1760 (1999).
A. O. Ignatyev, “On the partial equiasymptotic stability in functional differential equations,” J. Math. Anal. Appl. 268(2), 615–628 (2002).
A. S. Andreev, “On asymptotic stability and instability of the zero solution of a non-autonomous functionaldifferential equation,” in Problems of Analytic Mechanics, Stability, and Motion Control (Nauka, Novosibirsk, 1991) [in Russian].
A. S. Andreev, Stability ofNon-Autonomous Functional-Differential Equations (Izd.UlGU, Ul’yanovsk, 2005) [in Russian].
S.V. Pavlikov, “Limit equations and Lyapunov functionals in the problem of stability in some of the variables,” Uch. Zap. Ul’yanovsk. Gos. Univ. Ser. Fundament. Probl. Matem. iMekh. 1(13), 63–74 (2003).
S. V. Pavlikov, Method of Lyapunov Functionals in Problems of Stability (Institut Upravleniya, Naberezhnye Chelny, 2006) [in Russian].
A. S. Andreev, “The method of Lyapunov functionals in the problem of stability of functional-differential equations,” Avtomat. i Telemekh., No. 9, 4–55 (2009) [Autom. Remote Control 70 (9), 1438–1486 (2009)].
O. A. Ignatyev and V. Madrekar, “Barbashin-Krasovskii theorem for stochastic differential equations,” Proc. Amer. Math. Soc. 138(11), 4123–4128 (2010).
E. L. Panasenko and E. A. Tonkov, “Generalization of E. A. Barbashin and N. N. Krasovskii theorems of stability to controlled dynamic systems,” in Trudy Inst. Math. Mech. Ural Branch Russian Academy of Sciences (2009), Vol. 15, No. 3, pp. 185–201 [in Russian].
G.-Q. Xu and S. P. Yung, “Lyapunov stability of abstract nonlinear dynamic system in Banach space,” IMA J. Math. Control Inform. 20(1), 105–127 (2003).
O. A. Ignat’ev, “On equiasymptotic stability with respect to part of the variables,” J. Appl. Math. Mech. 63(5), 821–824 (1999).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A. O. Ignat’ev, 2014, published in Matematicheskie Zametki, 2014, Vol. 96, No. 2, pp. 212–216.
Rights and permissions
About this article
Cite this article
Ignat’ev, A.O. Using an analog of the Lyapunov function with sign-alternating derivative in the study of global asymptotic stability of equilibria. Math Notes 96, 204–207 (2014). https://doi.org/10.1134/S0001434614070219
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434614070219