A remark on reachable sets of linear systems

Any compact convex set of R^dcan occur as a reachable set of some -dimensional linear differential system with compact control variable constraint set. However, with the constraint sets most frequently considered in the literature, the reachable sets must be zonoids, a severe restriction on possible shapes when . On the other hand, given a zonoid, there is a differential system with scalar bang-bang control that has this zonoid as a reachable set.