Journal of Applied Mathematics and Stochastic Analysis
Volume 12 (1999), Issue 1, Pages 1-15

Averaging and stability of quasilinear functional differential equations with Markov parameters

Lambros Katafygiotis1 and Yevgeny Tsarkov1

1Hong Kong University of Science and Technology, Department of Civil and Structural Engineering, Clear Water Bay, Kowloon, Hong Kong
2Riga Technical University, Institute of Information Technology, 9, Ausekla Iela, Riga LV-1010, Latvia

Received 1 March 1997; Revised 1 January 1998

Copyright © 1999 Lambros Katafygiotis and Yevgeny Tsarkov. 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.


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.