On the necessity of the invariance conditions for reach control on polytopes

We study the Reach Control Problem (RCP) to make the solutions of an affine system defined on a polytopic state space reach and exit a prescribed facet of the polytope in finite time without first leaving the polytope. So-called invariance conditions are used to prevent solutions from leaving the polytope through facets which are not designated as the exit facet. These conditions are known to be necessary for solvability of the RCP on polytopes by continuous state feedback. We study whether the invariance conditions are also necessary for solvability of the RCP on polytopes by open-loop controls. We show by way of a counterexample that surprisingly the answer is negative. We identify a suitable class of polytopes for which the invariance conditions remain necessary conditions.