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  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
Steven M LaValle 2020-08-14