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By Peter S. Maybeck

This quantity builds upon the principles set in Volumes 1 and a pair of. bankruptcy thirteen introduces the fundamental options of stochastic keep watch over and dynamic programming because the primary technique of synthesizing optimum stochastic regulate legislation.

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Although the replacement of % by %‘* in the right hand side of (13-58) accomplishes nothing substantive here, this equation provides insight into the structure of the general dynamic programming step. ;) to show that the same form of %* equation as (13-58) can be written down for time t,_l as well, and induction will then lead to its validity for any time t , . From the first step backward, we obtained (13-58), where u affects both L and J’. 5 29 OPTIMAL STOCHASTIC CONTROL using that particular law.

Process, because we could just define &[X(tjL ti] = +[x(tj),u(x(ti),t i ) ,t i ] (1 3-48) We can now write the transition probability density for x ( t i + L as ) ('; =(-t(<-+[P,U(P, ti),t;])T[Gd(P,t i ) Q d ( t i ) G d T ( P , f i ) ] - 1 ( 5 - q 5 [ P , U ( P , f i ) r t ; ] ) i (13-49b) From this expression, we can see that the only effect of the control is to be able to move the conditional mean, while the conditional covariance is unchanged. Now assume that (13-43)and (13-47)are replaced by the more general solution to It6 stochastic differential equations (13-32)and (1 3-l),respectively, with u ( t ) defined by u(t) u[x(t,),ti] for all t E [ t i ,t i + ,) (1 3-50) In practice, online numerical integration would yield approximate solutions.

From the first step backward, we obtained (13-58), where u affects both L and J’. 5 29 OPTIMAL STOCHASTIC CONTROL using that particular law. Now step back to time tN- 1 . By definition, %‘CX(lN - tN- 1 1 1 1 9 = E{L[X(tN- >. I)? tN - I 119 u(x(tN- tN- 1 1 + L[x(tN), u { x ( t N ) , t N ) . , t N - 1 ] so that we can completely specify t N - 1 ] can be written as l ) . Then % [ x ( t N - J;(c, tNI q, t N %[X(tN- 1)- t N - 1 1 =“[x(tN- l h U { X ( r N - l), r N - 1 1 3 t N - 1 1 +m: J L[<,u{c, t N } , t N ] f x ( c ?

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