Nilpotent bases for distributions and control systems

Let V(M) be the Lie algebra (infinite dimensional) of real analytic vector fields on the n-dimensional manifold M. Necessary conditions that a real analytic k-dimensional distibution on M have a local basis which generates a nilpotent subalgebra of V(M) are derived. Two methods for sufficient conditions are given, the first depending on the existence of a solution to a system of partial differential equations, the second using Darboux's theorem to give a computable test for an (n - 1)-dimensional distribution. A nonlinear control system in which the control variables appear linearly can be transformed into an orbit equivalent system whose describing vector fields generate a nilpotent algebra if the distribution generated by the original describing vector fields admits a nilpotent basis. When this is the case, local analysis of the control system is greatly simplified.