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Additional resources for Dynamic Programming and the Calculus of Variations (Mathematics in Science and Engineering, Volume 21)
The value of a path is taken to be equal to its terminal y-coordinate, hence we have a terminal control problem. The path of minimum value is sought. While this problem is admittedly trivial it will serve to illustrate some points. + + + 12. Solution of the Example We initiate the solution by defining the optimal value function by X(z, y) = . the value of the minimum-value admissible path starting at the point (2, y) and terminating at some vertex on the line x=6, where x and y are integers with x = 0, 1, .
28 11. THE CLASSICAL VARIATIONAL THEORY With regard to the problem of finding the curve of minimum length connecting two points, we can begin by defining a particular functional which associates numbers, which are lengths, with curves. In these terms, the problem of Section 2 is to find the admissible curve that has related to it, by the arc-length functional, the smallest numerical value. The calculus of variations can be defined as the study of certain problems involving functional optimization.
The appropriate boundary condition for this relation follows from the definition of the optimal value function and depends on the manner in which the problem was specified. If either the optimal value or the optimal policy function has been determined (by the algorithms of this chapter they are determined simultaneously), the specific multistage decision process problem that initiated the analysis (as well as all other problems in the family in which the particular problem was imbedded) has been solved.
Dynamic Programming and the Calculus of Variations (Mathematics in Science and Engineering, Volume 21) by Dreyfus