Asymptotic Expansions of Singularly Perturbed Systems Involving Rapidly Fluctuating Markov Chains

A class of singularly perturbed time-varying systems with a small parameter $\varepsilon > 0$ is considered in this paper. The importance of the study stems from the fact that many problems arise in various applications involve a rapidly fluctuating Markov chain. To investigate the limit behavior of such systems, it is necessary to consider the corresponding singular-perturbation problems. Existing results in singular perturbation of ordinary differential equations cannot be applied since the coefficient matrix of the equation is a generator of a finite-state Markov chain, and as a result it is singular. Asymptotic properties of the aforementioned systems are developed via matched asymptotic expansion in this paper. Thanks to the specific structure and the properties of Markovian generators, it is established that the solution of the system can be approximated “as close as possible” by a series expansion in terms of the small parameter $\varepsilon > 0$.