Nilpotent Approximations of Control Systems and Distributions

Abstract

A constructive method is given to approximate the vector fields in a nonlinear control system, which is linear in the controls, by a system of similar form and on the same state space, with the describing vector fields of the approximating system generating a Lie algebra which has a certain, relevant, sub-algebra nilpotent. Given a k-dimensional distribution on $R^n $, say locally defined near zero, the method leads to an approximating k-distribution which has nilpotent basis, agrees with the original distribution at zero, and each derived distribution of the approximating distribution and original distribution also agree at zero.