Normal local controllability of order one

This paper gives a sufficient condition of normal local controllability at a point x0 for a family of analytical vector fields. The families considered are of the following $ where Ω is a bounded subset of Rκ, The condition is of the first order, that is it depends only on the values of the fields and their first derivatives at x0. It is proved that this condition is the most general condition of this type. In fact, if this condition is not satisfied, there are families of vector fields which at the first order coincide with the given family, and which are not normally locally controllable at x0