By Dimitri P. Bertsekas
ISBN-10: 0120932601
ISBN-13: 9780120932603
ISBN-10: 1886529035
ISBN-13: 9781886529038
This examine monograph is the authoritative and finished therapy of the mathematical foundations of stochastic optimum regulate of discrete-time structures, together with the remedy of the complex measure-theoretic concerns.
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Additional info for Stochastic optimal control: the discrete time case
Sample text
91 D l_)Vti \ ~v '" ~ t\ lItlHtV)(t) dt . 31) e Rem ark 2. Theorem 2 can be expressed in the following two ways: l-t + III 1 \ \ e 91 ~ l\tIHtV)lt)dt,. +t)dt e ~ t o + ~! ('l:) e and 91 satisfy the fol- and its first V derivatives. 33) can be derived from Theorem 6 with the help of obvious transformations and the properties of the Bernoulli polynomials. It is sometimes necessary to sum a function over integral pOints contained in some interval whose end points mayor may not be integers. The arbitrariness in the choice of e and at can be used.
T~) can be expanded in a Taylor's series converging to h~) in the interval Q+ h, t ~ -") "'r. ,,0.. + h,~, with ~ =I.... , 1-\,. Here, we have h, =- t, ~Q. 53) corresponds to the trapezoidal formula. 54) is a generalization of the concept of Riemann sums when e ~ [0, n . The other direction is the derivation of expansion formulas for functions. -Q. ) H J. 55) which with e = 0 and e = -I coincides with the Taylor's series. ), we can obtain a variety of expansions of functions in special series. 55) was the first to hand.
2). 9) for ~=~,2, ... and JA--O'~, ... ~ JA- .. ~ - '2. Let us separate the first term (with J .. p' (9) = (_tl e,,II-+1(9) • (f'~)! Theorem 4 has thus been proved. Rem ark 1. If the function f ('X,) and its first Vderivatives are continuous on the whole of the real axis, then there are no restrictions on the choice of e. This is all the more so when Hl) is an entire function. Rem ark 2. Theorem 4 is very curbersome and we will only use its limiting cases. 5)]. 1) vanishes. 5) when V,. ~ t "'-8 ttL) - \ e 8·1 e ~(~)d'X, +\ 1,9-i -t)t~' (i.
Stochastic optimal control: the discrete time case by Dimitri P. Bertsekas
by Daniel
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