Distinguished vector fields over smooth manifolds with applications to ensemble control

Motivated by the controllability problem of steering an ensemble of driftless bilinear control systems, we introduce the notion of a distinguished set of vector fields over a smooth manifold. Roughly speaking, a distinguished set of vector fields is such that it spans the tangent space of the manifold at every point, and moreover, is invariant under Lie brackets (up to scaling by real numbers). By providing an example about an ensemble of driftless bilinear control systems, we demonstrate that ensemble controllability follows when the underlying control vector fields form a distinguished set. With the example at hand, we then propose the following question: Given a smooth manifold, does there exist a distinguished set of vector fields? One of the contributions of the paper is to provide a partial solution to the question by exhibiting a few classes of smooth manifolds that admit distinguished sets of vector fields. More specifically, we show that all semi-simple Lie groups admit distinguished sets of vector fields. Furthermore, homogeneous spaces whose Lie transformation groups are semisimple admit distinguished sets of vector fields. A few examples are also given along the presentation of the paper.