By R. Boudarel, J. Delmas and P. Guichet (Eds.)
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Extra info for Dynamic Programming and Its Application to Optimal Control
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The solution proceeds in two steps. 1 Calculation of the Optimal Control Functions Starting with R(x, N ) = 0 [or with R(x, N - k)], the R ( x , n) are calculated successively for decreasing n. This can be done by supposing that R(x, n + 1) is known either literally or numerically. 3). If a particular value x = a is considered, then R(a, n ) = opt [r(a, u, n ) U€R" + R ( f ( a , u, n ) , n + 111. The expression in brackets depends only on the variable u, and calculation of R(a, n) reduces to a search for the optimum of a function of u , which is a problem in nonlinear programming (see Boudarel et al.
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). Unfortunately, it is difficult to determine the form of R ( x ) . One method is to choose cpo(x) = 0.
The optimal recurrence equation need be calculated only one inore time to obtain the additional control functions. This is not generally the case with variational methods, which calculate the optimal control law directly relative to a particular horizon. The solution obtained in that case is of no use in solving a problem only slightly different. Remark 4. I n dynamic programming methods of solution, it is possible to calculate the optimal return without explicitly calculating the control law. From the practical point of view, this is a considerable advantage.