Reachability by means of nice controls

We prove a stronger version of the Pontryagin Maximum Principle. Our version says that, if a point y can be reached from a point x by means of a control 𝒳 and corresponding trajectory γ of a control system x = f(x, u), and (χ,𝒰) is not a Pontryagin exiremal, then y belongs to the interior of the set. of points thai can be reached from x by means of piecewise constant controls. Under more restrictive conditions, the same conclusion also holds with "continuous", "smooth", or "polynomial" instead of "piecewise constant."