On a reachability problem for affine hypersurface systems on polytopes

This paper studies the problem for an affine hypersurface system (with n−1n−1 inputs) to reach a polytopic target set starting from inside a polytope in the state space. We present a solution which begins with a characterization of solvability by open-loop control and concludes with a procedure to synthesize a feedback control. Our emphasis is on methods of subdivision, triangulation, and covers which explicitly account for the capabilities of the control system. In contrast with the previous literature, the partition methods are guaranteed to yield a correct feedback synthesis, assuming the problem is solvable by open-loop control.