Geometric conditions for local controllability

Abstract

Small time local controllability of a C 1 affine control system, with a generic drift vector field and n − 1 linearly independent input vector fields on an n-dimensional connected C 1 manifold M, is studied, showing that a necessary condition, namely the drift vector field belonging to the convex hull of the control vector fields at the given point, is also sufficient on an open dense subset H 1 of the set of points where it is verified. Local controllability at the points in the interior of H 1 is also considered when the input vector fields generate an involutive distribution, and a criterion based on the type of contract of H with the hyperplane spanned by the input vector fields is proved. An example shows a situation where the sufficient condition for small time local controllability of H. Sussmann [ SIAM J. Control Option. 25 (1987), 158–194] cannot be applied, but in which the method now presented distinguishes the possible cases.