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Additional resources for Dynamic Programming and the Calculus of Variations (Mathematics in Science and Engineering, Volume 21)
We might mention that, in a sense, the dynamic programming approach is conceptualIy more general than the calculus of variations, including, for example, the above maximum-value functional. We shall not! however, pursue this topic further in this book. 10. The Nature of Necessary Conditions The classical variational theory begins by deducing conditions that the minimizing curve must satisfy. These are called necessary conditions. While the minimizing curve must satisfy a necessary condition, other nonminimizing curves may also meet the condition.
DISCRETE DYNAMIC PROQRAMMINQ condition is S(4, y) = 0. 2) Note by comparison with Eqs. 2) that allowing the terminal condition to be any vertex of an admissible set rather than a specified vertex affects only the boundary condition and not the recurrence relation. 2) for the example of Fig. 8, are shown by the numbers and arrows, respectively, in Fig. 9. The optimal path from point A to line B has sum 7 and consists of two arcs going diagonally up, then one going down, and finally one going up.
THE CLASSICAL VARIATIONAL THEORY length connecting two given curves. In Fig. 2, the curve C is the curve of minimum length connecting curves A and B. A variational problem is not termed “simple” if there are supplementary side conditions that must be satisfied by admissible curves (see, for example, Section 36, this chapter). Such problems still lie, however, in the domain of the calculus of variations. 9. The Maximum-Value Functional While the definite integral is an excellent and common example of a functional, it is by no means the only one.
Dynamic Programming and the Calculus of Variations (Mathematics in Science and Engineering, Volume 21) by Dreyfus