Before entering the continuous realm, the concepts will first be described for discrete planning, which is often easier to understand. Recall from Section 2.3 that if , , and the stages are discrete, then optimal planning can be performed by using value iteration or Dijkstra's algorithm on the search graph. The stationary, optimal cost-to-go function can be used as a navigation function that encodes the optimal feedback plan. This was suggested in Section 8.2.2, and an example was shown in Figure 8.3.
Suppose that has been computed under Formulation 8.1 (or Formulation 2.3). Let the state transition equation be denoted as
Let denote the optimal action that is applied in the of (15.6). Imagine that is hypothesized as the optimal action but needs to be tested in (15.6) to make sure. If it is truly optimal, then
Steven M LaValle 2012-04-20