The Structure of Small-Time Reachable Sets in Low Dimensions

This paper outlines a general method to determine the geometric structure of small-time reachable sets for a single-input control system with a bounded linear control. The authors’ analysis relies on free nilpotent systems as a guide, and hence their techniques only apply to nondegenerate situations. The paper illustrates the effectiveness of the method in low dimensions. Among other results is given a precise description of the small-time reachable set for a system x = f(x) + g(x)u, | u | 1 in dimension four, under the generic assumption that the constant controls u ≡ + 1 and u ≡ - 1 are not singular. As a corollary, a local synthesis is obtained in dimension three for the time-optimal control problem under the analogous generic condition.