Accessible sets for analytic control systems in R2

Abstract

Let S be a control system on R 2 given by x ̇ (x) = ∑ n i = 1 u i X i (x) , where X ={ X i } n i =1 is a fixed set of analytic vector fields on R 2 , with control constraints u i ⩾ 0, i = 1, 2, …, n , and ∑ u i = 1, and suppose S has the accessibility property at every point of R 2 . Let R ( x 0 ) be the accessible set from a point x 0 , and A ( x 0 ) be the set of points accessible from x 0 by bang-bang controls. Then R ( x 0 ) = A ( x 0 ) and R ( x 0 ) 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.