ASYMPTOTIC EXPANSIONS OF SINGULARLY PERTURBED SYSTEMS INVOLVING RAPIDLY FLUCTUATING MARKOV-CHAINS

A class of singularly perturbed time-varying systems with a small para meter epsilon > 0 is considered in this paper. The importance of the s tudy stems from the fact that many problems arise in various applicati ons involve a rapidly fluctuating Markov chain. To investigate the lim it behavior of such systems, it is necessary to consider the correspon ding singular-perturbation problems. Existing results in singular pert urbation of ordinary differential equations cannot be applied since th e 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 expan sion in this paper. Thanks to the specific structure and the propertie s of Markovian generators, it is established that the solution of the system can be approximated ''as close as possible'' by a series expans ion in terms of the small parameter epsilon > 0.