Chattering in the Reach Control Problem

The Reach Control Problem (RCP) is a fundamental problem in hybrid control theory. The goal of the RCP is to find a feedback control that drives the state trajectories of an affine system to leave a polytope through a predetermined exit facet. In the current literature, the notion of leaving a polytope through a facet has an ambiguous definition. There are two different notions. In one, at the last time instance when the trajectory is inside the polytope, it must also be inside the exit facet. In the other, the trajectory is required to cross from the polytope into the outer open half-space bounded by the exit facet. In this paper, we provide a counterexample showing that these definitions are not equivalent for general continuous or smooth state feedback. On the other hand, we prove that analyticity of the feedback control is a sufficient condition for equivalence of these definitions. We generalize this result to several other classes of feedback control previously investigated in the RCP literature, most notably piecewise affine feedback. Additionally, we clarify or complete a number of previous results on the exit behaviour of trajectories in the RCP.