A Generalization of Chow’s Theorem and the Bang-Bang Theorem to Nonlinear Control Problems

The main results of this paper are two-fold. The first, Theorem 1, is a generalization of the work of Chow and others concerning the set of locally accessible points of a nonlinear control system. It is shown that under quite general conditions, this set lies on a surface in state space and has a nonemptyinterior in the relative topology of that surface. The second result, Theorem 3, generalizes the bang-bang theorem to nonlinear control systems using higher order control variations as developed by Kelley and others. As a corollary we obtain Halkin’s bang-bang theorem for a linear piecewise analytic control system.