Lack of convexity for tangent cones of needle variations

The article provides a carefully constructed sequence of simple systems which show that even for very benign systems, the usual conditions that needle variations do not collide (or that they are moveable by sufficiently large amounts) are essential for guaranteeing convexity of the tangent objects. This, in turn, is essential for practical applicability to decide optimality. These examples also further raise deep questions about the structural stability of nonlinear controllability properties: they demonstrate that the controllability (or the lack thereof) of nilpotent approximating systems need not reflect the controllability (or the lack thereof) of the original systems. These suggest limitations to extending the usual arguments using nilpotent approximations have played a critical role in obtaining many classical controllability and optimality results.