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Nelin. Dinam., 2018, Volume 14, Number 3, Pages 387–407 (Mi nd621)  

On the Stability and Stabilization Problems of Volterra Integro-Differential Equations

A. S. Andreev, O. A. Peregudova

Ulyanovsk State University, ul. L’va Tolstogo 42, Ulyanovsk, 432017, Russia

Abstract: In this paper, the stability and stabilization problems for nonlinear Volterra integrodifferential equations with unlimited delay are considered. The development of the direct Lyapunov method in the study of the limiting properties of the solutions of these equations is carried out by using Lyapunov functionals with a semidefinite time derivative. The topological dynamics of these equations has been constructed revealing the limiting properties of their solutions. The assumption of the existence of a Lyapunov functional with a semidefinite time derivative gives a more complete solution to the positive limit set localization problem. On this basis new theorems on sufficient conditions for the asymptotic stability and instability of the zero solution of nonlinear Volterra integro-differential equations are proved. These theorems are applied to the problem of the equilibrium position stability of the hereditary mechanical systems as well as the regulation problem of the controlled mechanical systems using a proportional-integro-differential controller. As an example, the regulation problem of a mobile robot with three omnidirectional wheels and a displaced mass center is solved using the nonlinear integral controllers without velocity measurements.

Keywords: Volterra integro-differential equation, stability, Lyapunov functional, limiting equation, regulation problem

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 9.5994.2017/BP
Russian Foundation for Basic Research 18-01-00702
18-41-730022
This work was supported by the grant of the Ministry of Education and Science of Russia within the framework of the State task [9.5994.2017/BP] and the Russian Foundation for Basic Research [18-01-00702, 18-41-730022].


DOI: https://doi.org/10.20537/nd180309

Full text: PDF file (369 kB)
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MSC: 34K05, 34K20, 34K35, 37B25, 37B55, 45D05, 93D15, 93D20
Received: 14.05.2018
Accepted:13.09.2018
Language:

Citation: A. S. Andreev, O. A. Peregudova, “On the Stability and Stabilization Problems of Volterra Integro-Differential Equations”, Nelin. Dinam., 14:3 (2018), 387–407

Citation in format AMSBIB
\Bibitem{AndPer18}
\by A. S. Andreev, O. A. Peregudova
\paper On the Stability and Stabilization Problems of Volterra Integro-Differential Equations
\jour Nelin. Dinam.
\yr 2018
\vol 14
\issue 3
\pages 387--407
\mathnet{http://mi.mathnet.ru/nd621}
\crossref{https://doi.org/10.20537/nd180309}
\elib{http://elibrary.ru/item.asp?id=36296378}


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