Oriented Measures with Continuous Densities and the Bang-Bang Principle

We introduce the notion of an oriented measure. For such a measure μ, given ν in L1([a, b]), 0 < ν < 1, there exist two sets E ⊂ [a,b] whose characteristic functions have less than n discontinuity points and such that ∫ ν dμ = μ(E). Given a solution x to the control problem L(x) = x(n) + an − 1 (t) x(n − 1) + … + a1(t) x′ + a0(t) x ∈ [φ1, φ2] there exist two bang-bang solutions y, z having a contact of order n with x at a and b such that y ≤ x ≤ z. Reachable sets of bang-bang constrained solutions are convex; an application to the calculus of variations yields a density result.