Convexity of reachable sets of nonlinear ordinary differential equations

We present a necessary and sufficient condition for the reachable set, i.e., the set of states reachable from a ball of initial states at some time, of an ordinary differential equation to be convex. In particular, convexity is guaranteed if the ball of initial states is sufficiently small, and we provide an upper bound on the radius of that ball, which can be directly obtained from the right hand side of the differential equation. In finite dimensions, our results cover the case of ellipsoids of initial states. A potential application of our results is inner and outer polyhedral approximation of reachable sets, which becomes extremely simple and almost universally applicable if these sets are known to be convex. We demonstrate by means of an example that the balls of initial states for which the latter property follows from our results are large enough to be used in actual computations.