Necessary and sufficient conditions for reachability of discrete time affine systems on simplices

This paper studies reachability problems in discrete time affine systems on simplices-that is, how to steer the states in a simplex to a target polytope in finite time. Reachability problems in a discrete time setting have several distinct features from those in a continuous time setting and the solvability depends not only on the simplex itself but also on the target polytope. In this paper we first study the case where the union of the simplex and the target polytope is convex. Then we generalize the results by dropping the convex assumption. For theoretical completeness, the problem of making a polytope positive invariant is considered as well. Finally, a temperature control system is presented to illustrate our results.