Nilpotent and High-Order Approximations of Vector Field Systems

Abstract

This paper is concerned with the study of structurally stable properties associated with systems described by one, or several, real analytic vector fields via approximating systems which retain information pertinent to these properties. For example, let $X_0 , \cdots ,X_m $ be real analytic vector fields on an n-manifold M. An affine control system has the form (i) $\dot x = X_0 (x) + \sum\nolimits_{i - 1}^m {u_i } X_i (x)$; a second-order partial differential equation can be written as (ii) $\sum\nolimits_{i = 1}^m {X_i^2 } v - X_0 v = f$, while if $X_0 (p) = 0$ the asymptotic stability of the rest solution requires analysis of (iii) $\dot x = X_0 (x)$. If, in (ii), $m < n$ yet the vector fields in the Lie algebra generated by $X_1 , \cdots ,X_m $ , when evaluated at p, span the tangent space to M at p, the operator is hypoelliptic but an approximation of the vector fields needed to describe the singularity in a parametrix must retain more information than a linearization of the vector fields does. Similarly, the question of small time local controllability of (i) has been dealt with by constructing higher-order approximating vector fields which generate a nilpotent Lie algebra. The theory of these high-order approximations and their applications is concisely developed.