This chapter is complementary to Chapter 14 in that it provides tools and concepts that can be used to develop better local planning methods (LPMs). Most of the material was developed in the field of control theory, which focuses mainly on characterizing the behavior of particular classes of systems, and controlling them in the absence of obstacles. The two-point boundary value problem (BVP), which was a frequent nuisance in Chapter 14, can be better understood and solved for many systems by using the ideas of this chapter. Keep in mind that throughout this chapter there are no obstacles. Although planning for this case was trivial in Part II, the presence of differential constraints brings many challenges.

The style in this chapter is to provide a brief survey of concepts and techniques, with the hope of inspiring further study in other textbooks and research literature. Modern control theory is a vast and fascinating subject, of which only the surface can be scratched in one chapter. Section 15.1 introduces stability and controllability concepts, both of which characterize possible arrivals in a goal state. Stability characterizes how the integral curves of a vector field behave around a goal point, and controllability indicates whether an action trajectory exists that arrives at a specified goal.

Section 15.2 revisits dynamic programming one last time. Here it becomes a partial differential equation expressed in terms of the optimal cost-to-go function.