Some planning algorithms require integration in the reverse-time
direction. For some given and action trajectory that runs from
to 0, the *backward system simulator* computes a
state trajectory,
, which when
integrated from to 0 under the application of
yields . This may seem like an *inverse control problem*
[856] or a BVP as
shown in Figure 14.10; however, it is much simpler.
Determining the action trajectory for given initial and goal states is
more complicated; however, in reverse-time integration, the action
trajectory and final state are given, and the initial state does not
need to be fixed.

The reverse-time version of (14.14) is

which relies on the fact that is time-invariant. Thus, reverse-time integration is obtained by simply negating the state transition equation. The Euler and Runge-Kutta methods can then be applied in the usual way to .

Steven M LaValle 2020-08-14