Characterizing Equilibria in Reach Control Under Affine Feedback

The Reach Control Problem (RCP) deals with driving the states of an affine system on a simplex to leave the simplex through a pre-determined facet. A necessary condition for the solvability of the RCP by a given feedback is that there are no closed-loop equilibria in the simplex. As a stepping stone to fully characterizing when equilibria can be removed from the simplex using feedback, this paper studies the geometric structure of open-loop equilibria. Using a triangulation in which the set of potential equilibria intersects the interior of the simplex, we prove that the equilibrium set contains at most one point, in both the single-input and multi-input case. We additionally improve on the currently available results on reach controllability to characterize when the closed-loop equilibria can be pushed off the simplex using affine feedback.