On smoothness of the boundary of a reachable set under integral control constraints

The paper considers reachable sets at a given time of linear control systems with integral constraints on control in the form of a ball in the space $L_p$ for $p>1$. The reachable sets are convex compact sets in the finite-dimensional Euclidean space $\mathbb R^n$. For $p=2$, it is known that the reachable set, under the controllability condition, is an ellipsoid in $\mathbb R^n$ whose boundary is a compact smooth manifold diffeomorphic to a sphere. In this paper we obtain sufficient conditions under which the boundary of the attainability set turns out to be a smooth manifold of dimension $n-1$ for all $1<p\leq 2$.