Consider a (discrete) *potential function*, defined as
. The
potential function can be used to define a feedback plan through the
use of a *local operator*, which is a function that selects the
action that reduces the potential as much as possible. First,
consider the case of a feasible planning problem. The potential
function, , defines a feedback plan by selecting through the
*local operator*,

which means that is chosen to reduce as much as possible. The local operator yields a kind of

In the case of optimal planning, the local operator is defined as

which looks similar to the dynamic programming condition, (2.19). It becomes identical to (2.19) if is interpreted as the optimal cost-to-go. A simplification of (8.5) can be made if the planning problem is

When is a potential function useful? Many useless potential functions
can be defined that fail to reach the goal, or cause states to cycle
indefinitely, and so on. The most desirable potential function is one
that for any initial state causes arrival in , if it is
reachable. This requires only a few simple properties. A potential
function that satisfies these will be called a *navigation
function*.^{8.2}

Suppose that the cost functional is isotropic. Let
,
which is the state reached after applying the action
that was selected by (8.4). A potential function,
, is called a *(feasible) navigation function* if

- for all .
- if and only if no point in is reachable from .
- For every reachable state, , the local operator produces a state for which .

An *optimal navigation function* is defined as the optimal
cost-to-go, . This means that in addition to the three
properties above, the navigation function must also satisfy the
principle of optimality:

which is just (2.18) with replaced by . See Section 15.2.1 for more on this connection.

At any state, an action is applied that reduces the potential value.
This corresponds to selecting the action using (8.4).
The process may be repeated from any state until is reached.
This example clearly illustrates how a navigation function can be used
as an alternative definition of a feedback plan.

Steven M LaValle 2012-04-20