We study the reach control problem for affine systems on simplices, and the focus is on cases when 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

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... s shown to solve the reach control problem in all remaining cases, assuming it is solvable by open-loop controls.