Accessible sets for analytic control systems in R2

AbstractLet S be a control system on R2 given by ẋ(x) = ∑ni = 1 uiXi(x), where X={Xi}ni=1 is a fixed set of analytic vector fields on R2, with control constraints ui ⩾ 0, i = 1, 2, …, n, and ∑ui = 1, and suppose S has the accessibility property at every point of R2. Let R(x0) be the accessible set from a point x0, and A(x0) be the set of points accessible from x0 by bang-bang controls. Then R(x0) = A(x0) and R(x0) belongs to a class of sets which is a small extension of the class of semianalytic sets, and which, like the latter, has the property that the sets in it admit stratifications into locally finite unions of analytic manifolds