Interesting interpretations of the minimum principle exist for the
case of optimizing the time to reach the goal [424,903].
In this case,
in (15.26), and the cost term
can be ignored. For the remaining portion, let be defined
instead of using (15.25). In this case, the Hamiltonian
can be expressed as
which is an inner product between and the negative gradient
of . Using (15.40), the Hamiltonian should be
maximized instead of minimized (this is equivalent to Pontryagin's
original formulation ). An inner product of two
vectors increases as their directions become closer to parallel.
Optimizing (15.41) amounts to selecting so that
is as close as possible to the direction of steepest descent
of . This is nicely interpreted by considering how the
boundary of the reachable set
propagates through .
By definition, the points on
must correspond to
time-optimal trajectories. Furthermore,
interpreted as a propagating wavefront that is perpendicular to
. The minimum principle simply indicates
that should be chosen so that points into the propagating
boundary, as close to being orthogonal as possible .
Steven M LaValle