Dynamic programming

The numerical dynamic programming approach of Section 14.5 can be applied to provide optimal steering for virtual any system. To apply it here, the obstacle region ${X_{free}}$ is empty. The main drawback, however, is that the computational cost is usually too high, particularly if the dimension of $X$ is high. On the other hand, it applies in a very general setting, and Lie group symmetries can be used to apply precomputed trajectories from any initial state. This is certainly a viable approach with systems for which the state space is $SE(2)$ or $SO(3)$.