By Dimitri P. Bertsekas
This study monograph is the authoritative and finished remedy of the mathematical foundations of stochastic optimum regulate of discrete-time structures, together with the remedy of the tricky measure-theoretic matters.
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Let u* be any point in the intersection. Then u* E U and f(u*) :os; }'n for all n, and it follows that f(u*) :os; f*. Hence, f attains its minimum over U at u*. D. 1 yields the following proposition. 4 Let the control space C be a· Hausdorff space and assume that for each XES, AER, and k = 0,1, ... ,N - 1, the set (16) is compact. Then and there exists a uniformly N-stage optimal policy. The compactness of the sets U k(X, },) of (16) may be verified in a number of important special cases. 1). Assume that 0 :os; «(x, u), b :os; g(x, u) < 00 for some bE R and all XES, U E U(x), and take J 0 == O.
The 32 2. MONOTONE MAPPINGS IN DYNAMIC PROGRAMMING MODELS probability distribution p(dwlx, u) on W written as Hix, u,J) = = ex I pi(X, u)max{O,g(x, u, w') i= 1 {w 1 , wZ , • • • }, then (15) can be + etJ[j(x, U, wi)J} 00 - I pi(X, u)max{O, - [g(x, U, w') i= 1 + etJ[j(x, U, wi)JJ}. : W -+ R* and :::z: W -+ R*, the equality E{Zl(W) + zz(w)} = E{:::l(W)} + E{:::z(w)} (16) need not always hold. 11). We always have, however, It is clear that the mapping H of (15) satisfies the monotonicity assumption.
It also provides examples of special cases which include wide classes of problems of practical interest. 1 Notation and Assumptions Our usage of mathematical notation is fairly standard. , R* = R u {- 00, co}, The sets ( - 00, 00] = R u {co} and [ - 00,(0) = R u {- co] will be written out explicitly. We will assume throughout that R is equipped with the usual topology generated by the open intervals (iX, f3), iX, f3 E R, and with the (Borel) a-algebra generated by this topology. Similarly R* is equipped with the topology generated by the open intervals («, f3), iX, f3 E R, together with the sets (y, 00], [ - 00, y), Y E R, and with the a-algebra generated by this topology.