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MSC2010 45K05

On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations

A. S. Andreev1, O. A. Peregudova2

AnnotationIn this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.
Keywordsnonlinear systems of integro-differential equations, Lyapounov functional, stability, topological dynamics, limiting equation

1Aleksandr S. Andreev, Head of the Department of Information Security and Control Theory, Ulyanovsk State University, (42 Leo Tolstoi Str., Ulyanovsk 432017, Russia), Dr. Sci. (Physics and Mathematics), ORCID: https://orcid.org/0000-0002-9408-0392, asa5208@mail.ru

2Olga A. Peregudova, Professor, Department of Information Security and Control Theory, Ulyanovsk State University, (42 Leo Tolstoi Str., Ulyanovsk 432017, Russia), Dr. Sci. (Physics and Mathematics), ORCID: https://orcid.org/0000-0003-2701-9054, peregudovaoa@gmail.com

Citation: A. S. Andreev, O. A. Peregudova, "[On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,20:3 (2018) 260–272 (In Russian)

DOI 10.15507/2079-6900.20.201803.260-272