Oriented Measures With Continuous Densities and the Bang-bang Principle

We introduce the notion of an oriented measure. For such a measure mu, given nu in L(1)([a, b]), 0<1, there exist two sets E subset of[a,b] whose characteristic functions have less than n discontinuity points and such that integral upsilon d mu=mu(E). Given a solution x to the control problem L(x)=x((n)) + a(n-1)(t)x((n-1)) + ... + a(1)(t) x' + a(0)(t) x is an element of [phi(1), phi(2)] there exist two bang-bang solutions y, z having a contact of order n with x at a and b such that y less than or equal to x less than or equal to z. Reachable sets of bang-bang constrained solutions are convex; an application to the calculus of variations yields a density result. (C) 1994 Academic Press, Inc.