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This book presents the second part of a twovolume series devoted to a sys tematic exposition of some recent developments in the theory of discrete time Markov control processes (MCPs). As in the first part, hereafter re ferred to as 'Volume I' (see HernandezLerma and Lasserre [1]), interest is mainly confined to MCPs with Borel state and control spaces, and possibly unbounded costs. However, an important feature of the present volume is that it is essentially selfcontained and can be read independently of Volume I. The reason for this independence is that even though both volumes deal with similar classes of MCPs, the assumptions on the control models are usually different. For instance, Volume I deals only with nonnegative cost perstage functions, whereas in the present volume we allow cost functions to take positive or negative values, as needed in some applications. Thus, many results in Volume Ion, say, discounted or average cost problems are not applicable to the models considered here. On the other hand, we now consider control models that typically re quire more restrictive classes of controlconstraint sets and/or transition laws. This loss of generality is, of course, deliberate because it allows us to obtain more 'precise' results. For example, in a very general context, in 4.
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