Lie Algebras of Vector Fields and Local Approximation of Attainable Sets

Abstract

Consider an analytic, n-dimensional control system described by ${{dx} / {dt}} = X(x) + u(t)Y(x)$, $x(0) = p$ and let $\mathcal{A}(t,p)$ 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 $|u(t)| \leqq 1$. Our goal is to derive second order conditions to determine if the reference solution, $(\exp tX)(p)$, corresponding to $u(t) \equiv 0$, lies on the boundary or interior of $\mathcal{A}(t,p)$ for small $t > 0$. If $t_1 > 0$ and $p^1 = (\exp t_1 X)(p)$ the approach is to use the Campbell–Baker–Hausdorff formula to obtain second order tangent vectors to $\mathcal{A}(t_1 ,p)$ at $p^1 $. 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.