Characterization of a topological obstruction to reach control by continuous state feedback

This paper studies a topological obstruction to solving the reach control problem (RCP) by continuous state feedback. Given a simplex and given an affine control system defined on the simplex, the RCP is to find a state feedback to drive closed-loop trajectories initiated in the simplex through an exit facet, without first exiting through other facets. We distill the problem as one of continuously extending a function that maps into a sphere from the boundary of a simplex to its interior. As such, we employ techniques from the extension problem of algebraic topology. Unlike previous work on the same problem, in this paper we remove unnecessary restrictions on the dimension of the simplex, the number of inputs of the system, and the particular geometry of the subset of the state space where the obstruction arises. Thus, the results of this paper represent the culmination of our efforts to characterize the topological obstruction. The conditions obtained in the paper are easily checkable and fully characterize the obstruction.