Convexity of reachable sets of nonlinear discrete-time systems

We present necessary and sufficient conditions for reachable sets of discrete-time systems x(k+1) = F (k, x(k)) to be convex. In particular, the set of states reachable at a given time from a sufficiently small ellipsoid of initial states is always convex if F is smooth enough, and we provide explicit bounds on the size of those ellipsoids. Our results imply that outer discrete approximations with approximation depth exceeding 1 can be readily computed up to arbitrary precision. A further potential application is outer polyhedral approximation of reachable sets, which becomes almost universally applicable if those sets are known to be convex.