On the endpoint maps of fully nonlinear control systems that use locally Lp controls

We show that the endpoint maps of certain time-dependent fully nonlinear control systems have high-order differentiability properties. We then derive a closed-form expression for the differentials of these maps. The systems we consider evolve on manifolds, and use locally L
^p
controls, where p ≥ 1. Control-affine systems, possibly time-dependent, constitute a special case.