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By Dimitri P. Bertsekas

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Extra resources for Dynamic Programming and Stochastic Control

Example text

In this text we are primarily interested only in a subclass of such decision problems. These problems involve a dynamic system. Such systems have an input-output description and furthermore in such systems inputs are selected sequentially after observing past outputs. This allows the possibility of feedback. Let us first give an abstract description of the type of problems with which we shall be dealing. Let us consider a system characterized by three sets U,W, and Y and a function S : U x W + Y.

Find n E I-I that maximizes F ( 4 = E { u ( u , w, Y)) n' where u and y are expressed in terms of n and w by means of u = n[S(u,w)] and y = S(u, w). We shall be mostly dealing with problems of this second type. Of course, on the basis of the formulation given one can say that problems of decision under uncertainty can be reduced to problems of decision under certainty-the problem of maximizing over l7 the numerical function F(n). However, it is important to realize that due to the feedback possibility the set n is a set offunctions (of the system output).

2 47 THE DYNAMIC PROGRAMMING ALGORITHM sets is a countable set). Now the system equation (3), the probability distributions Pk( lxkrpk(xk)),the initial state xo, and the control law {polp i , . . , p N - l }define a probability distribution on the countable set x 9, x ... x and the expectation in (4) is defined with respect to this latter distribution. In fact, it is easy to see that in this formulation the states {xo,xl, x 2 , . . , xN} form a finite Markov sequence [P2] that can be described completely by the corresponding conditional probabilities P(xk + I xk), which depend on the control law { p o , pi, .