SIAM J. CONTROL AND OPTIMIZATION Vol. 18, No. 6, November 1980 1980 Society for Industrial and Applied Mathematics 0363-0129/80/1806-0001 $01.00/0 A CHARACTERIZATION OF THE REACHABLE SET FOR NONLINEAR CONTROL SYSTEMS* RICHARD VINTER" Abstract. The question of whether a set is reachable by a nonlinear control system is answered in terms of the properties of a convex optimization problem. The set is reachable or not according to whether the value of the optimization problem is zero or infinity. Our findings strengthen earlier sufficient conditions for a point not to be reachable, given in terms of Lyapunov-like functions, in that we assure that the functions exist. Our approach is to embed admissible trajectories in a space of measures, and to apply recently obtained results on the properties of measures arising in this way. 1. Introduction. In this paper we provide a characterization of the set of points reachable from a point (x0, to) along solutions of the differential equation with control (1.1) Yc(t) f(x(t), t, u(t)). Our main result is that, under very mild conditions, if no point in a closed set F is reachable from (x0, to), then there exists a continuously differentiable function b (.,.) which satisfies the partial differential inequality t(x, t)+&x(x, t)f(x, t, u)<-O, (c, 04/Ot etc.), and which takes positive values on F and nonpositive values on points which are reachable from (x0, to). A function with these properties may always be obtained from a maximizing sequence for a convex optimization problem. Our methods are grounded in an idea originally due to L. C. Young, that "trajectories" can be embedded in a space of linear functionals [17]. We introduce a notion of "weak" reachability which turns out to be equivalent to reachability as conventionally defined; weak reachability concerns existence of a linear functional satisfying a certain "convex" constraint. We can exploit this convexity, and obtain reachability criteria using the methods of convex analysis. Our characterization is somewhat in the spirit of a theorem of Carath6odory [3]" a Pfaffian is integrable at a point if every neighborhood of the point contains an inaccessible point, The inequality (1.2) replaces the Pfaffian identified with the system of partial differential equations expressing the dynamical constraint (1.1) on the trajectories, and our function b(., .) replaces the "complete integral" (which is nonzero on locally inaccessible points and zero on locally accessible ones). Of course, our results hold under much milder conditions than would be needed to make this parallel precise [7]. There are points of contact also with the Control Theory literature. Concerning nonlinear systems, sufficient conditions for either reachability or nonreachability are known, which involve functions similar to our b(., .) function (see for example [6], [14]). But we show that such b(., )’s always exist, an apparently new result. It is well known that, for some linear systems, a necessary and sufficient condition that a convex set in the output space be reachable at some fixed time can be given in terms of the value * Received by the editors June 15, 1979, and in final revised form March 17, 1980. " Imperial College of Science and Technology, London SW7, England. (This research was carried out in part while the author was visiting the Laboratory of Information and Decision Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139.) am indebted to R. W. Brockett for this observation. 599