  We study the reach control problem for affine systems on simplices, and the
focus is on cases when it is known that the problem is not solvable by
continuous state feedback. We examine from a geometric viewpoint the structural
properties of the system which make continuous state feedbacks fail. This
structure is encoded by so-called reach control indices, which are defined and
developed in the paper. Based on these indices, we propose a subdivision
algorithm and associated piecewise affine feedback. The method is shown to
solve the reach control problem in all remaining cases, assuming it is solvable
by open-loop controls.
