Convexity of reachable sets of nonlinear ordinary differential equations

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 is presented. In particular, convexity is guaranteed if the ball of initial states is sufficiently small, an upper bound on the radius of that ball being obtained directly from the right hand side of the differential equation. In finite dimensions, the results cover the case of ellipsoids of initial states. A potential application of the 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. An example demonstrates that the balls of initial states for which the latter property follows from the results are large enough to be used in actual computations.