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Romeo and A. Sangiovanni-Vincentelli,A theoretical framework for simulated annealing, Algorithmica, vol. 6, pp. 302-345, 1991. 23. N. Wojciech, Tails events of simulated annealing Markov chains, J. Appl. , vol. 32, pp. 867-876, 1995. 24. A. S. Poznyak and K. Najim, Learning Automata and Stochastic Optimization, Springer-Verlag, Berlin, 1997. 25. M. L. Tsetlin, Automaton Theory and Modeling of Biological Systems, Academic Press, New York, 1973. 26. R. M. Jr. Wheeler and K. S. Narendra, Decentralized learning in finite Markov chains, IEEE Trans.
L ) is a transition matrix describing the transition probabilities from a group of nonessential states X ( 0 ) to X(Z). 18) changes its properties from time to time. It can correspond, for example, to ergodic homogeneous finite Markov chain, then to a chain with two ergodic subclasses, to a chain with five ergodic subclasses and so on. The next lemma proved by V. Sragovitch [l21 (see also [13]) clarifies the notion of a communicating homogeneous controlled chain and states the conditions when a given chain is a communicating chain.
K ) I j=1 These initial probabilities satisfy the following relation But for the stationary initial distribution, we have from which, we conclude that K - y p * ( i ) = 1. i=l The theorem is proved. In the next section we shall be concerned with the so-called controlled finite Markov chains which represent the basic model investigated in this book. 5 Controlled finite Markov chains We start by discussing the properties of controlled finite Markov chains. This discussion will be followed by the consideration and classification of control strategies (orpolicies).
