Sufficient conditions for controllability

The problem is to find sufficient conditions for the system
\dot{x}(t)= f(x((t))+ \sum_{i = 1}^{m} u_{i}(t)g_{i}(x(t)), x(0)=x_{0} \in M
to be controllable. Here
M
is a connected
{\cal C}^{\infty}
n
-dimensional manifold,
f
,
g_{1}, \cdots ,g_{m}
are complete
{\cal C}^{\infty}
vector fields on
M
, and
u_{1}, \cdots , u_{m}
are real-valued controls. If
m = n - 1, M,f, g_{1}, \cdots ,g_{n-1}
are real-analytic,
M
is simply connected, and
g_{1}, \cdots ,g_{n-1}
are linearly independent on
M
, then necessary and sufficient conditions are known. For the case of our
{\cal C}^{\infty}
system with general
m
, we assume that the space spanned by the Lie algebra
L_{A}
generated by
f, g_{1}, \cdots ,g_{m}
and successive Lie brackets has constant dimension
p
on
M
and the algebra
L_{A}^{\prime}
generated by
g_{1}, \cdots ,g_{m}
and successive Lie brackets has constant dimension
p^{\prime} \leq p
on
M
. If
p^{\prime}=p
, Chow's Theorem implies controllability for a
p
-dimensional submanifold of
M
containing
x_{0}
. If
p^{\prime}< p
, sufficient conditions are found involving the computation of certain Lie brackets at points where the vector field
f
is tangent to the integral manifolds of
L_{A}^{\prime}
. Here we assume that every integral manifold of
L_{A}^{\prime}
contains such a point.