Download Discrete-Time Markov Control Processes: Basic Optimality by Onesimo Hernandez-Lerma, Jean B. Lasserre PDF

By Onesimo Hernandez-Lerma, Jean B. Lasserre

This booklet provides the 1st a part of a deliberate two-volume sequence dedicated to a scientific exposition of a few fresh advancements within the concept of discrete-time Markov keep an eye on methods (MCPs). curiosity is especially constrained to MCPs with Borel nation and regulate (or motion) areas, and doubtless unbounded charges and noncompact keep an eye on constraint units. MCPs are a category of stochastic keep an eye on difficulties, sometimes called Markov determination strategies, managed Markov methods, or stochastic dynamic seasoned­ grams; occasionally, quite whilst the country area is a countable set, also they are referred to as Markov selection (or managed Markov) chains. whatever the identify used, MCPs look in lots of fields, for instance, engineering, economics, operations examine, statistics, renewable and nonrenewable re­ resource administration, (control of) epidemics, and so on. despite the fact that, lots of the lit­ erature (say, at the least 90%) is targeted on MCPs for which (a) the nation area is a countable set, and/or (b) the costs-per-stage are bounded, and/or (c) the keep watch over constraint units are compact. yet interestingly sufficient, the main generic keep an eye on version in engineering and economics--namely the LQ (Linear system/Quadratic price) model-satisfies none of those stipulations. furthermore, whilst facing "partially observable" structures) a customary procedure is to rework them into an identical "completely observable" sys­ tems in a bigger country area (in truth, an area of likelihood measures), that's uncountable no matter if the unique country strategy is finite-valued.

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D) If an a-discount optimal policy exists, then there exists one that is deterministic stationary. 3 requires several lemmas. The first is a general result [from Hermindez-Lerma and Munoz de Ozak (1992)J on the interchange of limits and minima, which is interesting in itself. 4 Lemma. Let U and Un (n = 1,2, ... c. functions, bounded below, and inf-compact on lK. If Un i u, then lim min un(x, a) n->oo A(x) = min u(x, a) \fx A(x) E X. Proof. Define, for x E X, l ( x) := lim min Un (X , a) , and U* ( x) := min U(x, a) .

13) Jt(x) :::; c(x, I) Jt+l(y)Q(dYlx, I), which implies Jt(x) :::; min [c(x, a) A(x) as 1f + J Jt+1(y)Q(dylx, a)] f was arbitrary. 1) is obvious. 1(a)-(b)] that a policy is optimal if and only if for any t = 0, ... , N and any initial state Xo = x E;Ct (1f,Xt) = E;Jt(xt). The "only if" (or necessity) statement is usually referred to as Bellman's Principle of Optimality. 2). 1 hold. Without these conditions, the theorem may fail, even in elementary cases. For example, consider the one-stage problem in which X = {O}, A = {1, 2, ...

2) with CN, the terminal cost function, a given measurable function on X. 3) IT such that = J*(x) '

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