#### Approximate value iteration

The continuous-space methods from Section 10.6 can be directly applied to produce an approximate solution by interpolating over to determine cost-to-go values. The initial cost-to-go value over the collection of samples is obtained by (12.6). Following (10.46), the dynamic programming recurrence is

 (12.10)

If is finite, the probability mass is distributed over a finite set of points, . This in turn implies that is also distributed over a finite subset of . This is somewhat unusual because is a continuous space, which ordinarily requires the specification of a probability density function. Since the set of future states is finite, this enables a sum to be used in (12.10) as opposed to an integral over a probability density function. This technically yields a probability density over , but this density must be expressed using Dirac functions.12.1 An approximation is still needed, however, because the points may not be exactly the sample points on which the cost-to-go function is represented.

Steven M LaValle 2012-04-20