Download Stochastic Optimal Control: The Discrete Time Case by Dimitri P. Bertsekas PDF

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.

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