#### Reverse-time system simulation

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

 (14.20)

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