International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 1999 / Article

Open Access

Volume 12 |Article ID 490250 | https://doi.org/10.1155/S1048953399000015

Lambros Katafygiotis, Yevgeny Tsarkov, "Averaging and stability of quasilinear functional differential equations with Markov parameters", International Journal of Stochastic Analysis, vol. 12, Article ID 490250, 15 pages, 1999. https://doi.org/10.1155/S1048953399000015

Averaging and stability of quasilinear functional differential equations with Markov parameters

Received01 Mar 1997
Revised01 Jan 1998

Abstract

An asymptotic method for stability analysis of quasilinear functional differential equations, with small perturbations dependent on phase coordinates and an ergodic Markov process, is presented. The proposed method is based on an averaging procedure with respect to: 1) time along critical solutions of the linear equation; and 2) the invariant measure of the Markov process. For asymptotic analysis of the initial random equation with delay, it is proved that one can approximate its solutions (which are stochastic processes) by corresponding solutions of a specially constructed averaged, deterministic ordinary differential equation. Moreover, it is proved that exponential stability of the resulting deterministic equation is sufficient for exponential p-stability of the initial random system for all positive numbers p, and for sufficiently small perturbation terms.

Copyright © 1999 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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