Moment exponential stability of random delay systems with two-time-scale Markovian switching

Facing the pressing needs of many applications in network and control systems, this paper introduces a class of nonlinear systems with random time delays and derives conditions on moment exponential stability of the underlying systems. The system model is versatile and can accommodate a wide variety of situations. The stability analysis to date in the literature is mostly delay independent. To highlight the role of random delay for stability, this paper focuses on delay-dependent stability. Dependence of stability on random time delays introduces technical difficulties beyond the existing literature. We model the random time delays by a continuous-time Markov chain involving two-time scales defined by a small parameter ε. leading to a two-time scale framework. The random delays change their values with a fast varying mode and a slowly evolving effect. Under broad conditions, the stability of the system is studied using a limit system in the sense of weak convergence of probability measures. Using the limit system as a bridge, this paper establishes the Razumikhin-type criteria on the moment exponential stability. These criteria show that the mean of the random time delay with respect to the stationary distribution of the fast changing part of the Markov chain plays an important role in the moment exponential stability, which presents a novel feature of our work. In particular, we show that the overall system may be stabilized by the Markov switching even when some of the underlying subsystems are unstable, which shows that the Markov chain may serve as a stabilization factor. Explicit conditions for moment exponential stability are derived when the system is linear. Examples are given to illustrate our results