Needle variations that cannot be summed

This article analyzes sets of higher order tangent vectors to reachable sets of analytic control systems (affine in the control). Both small-time local output controllability and small-time local controllability about a nonstationary reference trajectory are considered. In a series of purposefully constructed examples it is shown that the cones generated by needle variations may fail to be convex. The examples demonstrate that the usual technical condition that needle variations must be movable is essential to guarantee desirable convexity properties. Moreover, new doubts are cast on the structural stability of controllability properties, as apparently higher order perturbations can reverse the (lack of) controllability of lower order nilpotent approximating systems, thereby providing new insights about the ultimate question of whether controllability is finitely determined.