Triangulations, Subdivisions, and Covers for Control of Affine Hypersurface Systems on Polytopes

This paper studies the problem for an affine hypersurface system to reach a polytopic target set starting from inside a polytope in the state space. We present an exhaustive solution which begins with a characterization of states which can reach the target by open-loop control and concludes with a systematic 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 previous literature, the partition methods are guaranteed to yield a correct feedback synthesis, assuming the problem is solvable by open-loop control.