Averaging and stability of quasilinear functional differential equations with Markov parameters. (1999)

• Record Type: Journal Article
• Authors: Katafygiotis, Lambros; Tsarkov, Yevgeny
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 12:Number 1(1999)
• Page Start: 1
• Page End: 15
• Publication Date: 1999
• Subjects: functional differential equations -- small random perturbations -- stochastic stability -- averaging procedures; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S1048953399000015
• Languages: English
• ISSNs: 1048-9533

Abstract

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.