Optimal Control and Almost Analytic Feedback for Some Nonholonomic Systems

One possible approach to do path planning for nonholonomic systems is to use optimal control. It is intuitively clear that optimal control should give rise to solutions of the path planning problem that are in feedback form, and one can expect that the resulting feedback control laws will be \piecewise smooth" or \piecewise analytic" in some sense. Here we focus on the weaker property of analyticity on a large dense set (ALDS), and explain how such a property is related to the problem of the structure of the optimal trajectories and, in particular, to the existence of \proper subanalytic suucient families for optimality" that are made of a bounded number of pieces depending on nitely many parameters. The existence of such suucient families can be proved in a number of cases using techniques from diierential-geometric control theory. This is illustrated by brieey discussing the example of the \Reeds-Shepp car."