Strong convexity of reachable sets of linear systems

The reachable set on some time interval of a linear control system $x'\in Ax\,{+}\,U$, $x(0)=0$, is considered. A number of cases is examined when the reachable set is the intersection of some balls of fixed radius $R$ (that is, a strongly convex set of radius $R$). In some cases the radius $R$ is estimated from above. It turns out that strong convexity is fairly typical for this class of reachable sets in a certain sense.
Among possible applications of this result are the possibility of constructing outer polyhedral approximation of reachable sets with better accuracy in the Hausdorff metric than in the general case, and applications to linear differential games and some optimization problems.
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