By G. George Yin, Qing Zhang
Prologue and Preliminaries: advent and evaluate- Mathematical preliminaries.- Markovian models.- Two-Time-Scale Markov Chains: Asymptotic Expansions of recommendations for ahead Equations.- career Measures: Asymptotic houses and Ramification.- Asymptotic Expansions of recommendations for Backward Equations.- Applications:MDPs, Near-optimal Controls, Numerical tools, and LQG with Switching: Markov choice Problems.- Stochastic regulate of Dynamical Systems.- Numerical equipment for regulate and Optimization.- Hybrid LQG Problems.- References.- Index
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For the related applications in control theory and optimization, we refer the reader to Bensoussan , Kokotovic , Kokotovic, Bensoussan, and Blankenship , Kokotovic and Khalil , Kokotovic, Khalil, and O’Reilly , Pervozvanskii and Gaitsgori , Phillips and Kokotovic , and the large reference citations contained therein. The idea of two-time-scale expansion has also found emerging applications in communication theory (see Tse, Gallager, and Tsitsiklis  among others).
Starting with the case that the Markov chain is weakly irreducible, by use of the asymptotic expansions developed in Chapter 4, it is shown that a sequence of unscaled occupation measures converges to that of an “integrated” quasi-stationary distribution, and a scaled sequence of the occupation measures veriﬁes a mixing condition and is tight in an appropriate function space. We establish the weak convergence of the sequence and derive the explicit representation of the covariance of the Gaussian process.
Introduction and Overview applications. 1 might be considered as a small quantity from a practical point of view. The mathematical results to be presented can serve as a guide for various approximations and for estimating error bounds; the asymptotic results of the underlying systems (as ε → 0) provide insights into the structure of the system and heuristics in applications. A thorough understanding of the intrinsic behavior of the systems will be instructive and beneﬁcial for in-depth studies of applications in hierarchical production planning, Markov decision processes, random evolution, and control and optimization of stochastic dynamic systems involving singularly perturbed Markov chains.