Lack of convexity for tangent cones of needle variations

Cones of tangent vectors (generated by families of control variations) are the principal analytical tool for obtaining necessary and sufficient conditions for optimality and nonlinear controllability. Unlike the classical Pontryagin cone, it is known that in general high-order variational cones may fail to be convex. For control variations that converge in L to the reference control, this was shown in 1985 by Bressan. This article announces a counter example for high order tangent vectors generated by families of needle variations. In particular, we show that for arbitrarily small positive times the the reachable sets exhibit inward corners. The counter examples are polynomial cascade systems that are affine in the control. We also point out some repercussions for nilpotent approximating systems as what appear to be higher order perturbations may both yield and destroy controllability, raising further doubts about whether controllability can be decided upon evaluating a finite number of derivatives.