Dynamic Programming and Its Application to Optimal Control by R. Boudarel, J. Delmas and P. Guichet (Eds.)

By R. Boudarel, J. Delmas and P. Guichet (Eds.)

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Extra resources for Dynamic Programming and Its Application to Optimal Control

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Nonetheless, Bellman [2] was able to establish the contraction property for the particular case : Ilf(x, u>II < a llxll , v(0, u ) = 0 , Vu; a < 1, vx, u ; r(x, u ) > 0 . It should be noted that the first condition requires that the system tend to zero regardless ofthe control applied, so that only the origin is reachable at infinity. From the point of view of applications, this condition is quite restrictive. 1 Initialization in Return Space The use of Picard’s method, when the iteration converges, will be the more efficient as the initial function cpo(x) approaches R(x).

1) is applied over these smaller regions b, , which leads to a better approximation since a smaller step can be used in constructing the table. If the regions b, are still too large for the number of points allowed in the table, the initial or final regions can be partitioned, and the problem worked separately for each subregion. This allows computing time to be traded for working memory. 2 The R e l a x a t i o n Method The relaxation method is particularly appropriate when a nominal initial state and its immediately adjacent states constitute the initial region.

4 27 SOLUTION OF THE OPTIMALITYEQUATION To each nm corresponds a positive return W m ( x )such that rn m gm+ 1(x) = 1 r(xn 3 0 g(xn)) = C0 r(xn > + R o [ f ( x m + 1, go(Xm+ 1))I g'(xn>> + r(xm+ g'(Xrn+ 1)) 9 13 + Ro[f(xrn+l,gl(xm+l))I. 12) that 4x3 g'(x)) + R o w >g'(x))l 5 4x3 gO(x)) + ROIf(x,g0(x))l so that 0 5 Brn+l 5 W,,, 5 - As m + 00, the policy 7cm 4 . 5 Wl < W o = Ro. 14) tends to the stationary policy g'(x), and thus 0 I R'(x) I Ro(x). Thus R'(x) is an improvement on Ro(x), and since R ' ( x ) is bounded, g'(x) is admissible.

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