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, (exp 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 = (exp 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.