n-Dimensional controllability with(n-1)controls

Let
M
be a connected real-analytic
n
-dimensional manifold,
f, g_{1}, ... ,g_{n-1}
be complete real-analytic vector fields on
M
which are linearly independent at some point of
M
, and
u_{1}, ... , u_{n-1}
be real-valued controls. Consider the controllability of the system
\dot{x}(t)=f(x(t)) + \sum\min{i=1}\max{n-1} u_{i}(t)g_{i}(x(t)), x(0)=x_{0} \in M
. Necessary and sufficient conditions are given so that this system is controllable on any simply connected domain
D
contained in
M
on which
g_{1},... ,g_{n-1}
are linearly independent. These conditions depend on the computation of Lie brackets at those points where
f, g_{1}, ... ,g_{n-1}
are linearly dependent.