Lie Algebras of Vector Fields and Local Approximation of Attainable Sets

Consider an analytic, n-dimensional control system described by {dx / dt} = X (x) + u (t) Y (x), x (0) = p and let A (t, p) denote the set of states attainable at time t by use of all admissible control functions u, which we take as measurable functions with values |u (t) | 1. Our goal is to derive second order conditions to determine if the reference solution, (tX) (p), corresponding to u (t) 0, lies on the boundary or interior of A (t, p) for small t > 0. If t₁ > 0 and p¹ = (t₁ X) (p) the approach is to use the Campbell–Baker–Hausdorff formula to obtain second order tangent vectors to A (t₁, p) at p¹. These involve elements of the Lie algebra generated by X and Y having an arbitrary number of X factors and two Y factors. With certain hypotheses, for n = 2, 3 relatively complete results are obtained.