A Lyapunov approach to the stability of stochastic time delay systems

In this paper, we conduct a comprehensive stability analysis of time-delay systems influenced by various types of stochastic perturbations, including standard Brownian motion, randomly varying coefficients, and functions governed by stochastic processes. The primary objective is to assess mean-fourth stability and stochastic stability. To this end, we employ specifically designed Lyapunov functionals to derive sufficient conditions that ensure these stability criteria are met. We further extend the analysis to systems characterized by randomly changing coefficients, contributing new theoretical insights to the literature. The proposed results are substantiated with illustrative examples and a detailed exploration of the corresponding stability regions.