On Bang-bang Constrained Solutions of A Control-system

Given phi(1), phi(2) is an element of L(1)([0, T]) and a function x is an element of W-2,W-1([0, T]) solving the control problem (P) x'' + a(1)(t)x' + a(0)(t)x is an element of [phi(1)(t), phi(2)(t)] a.e., x(0) = x(0), x(T) = x(1), x'(0) = v(0), x'(T) = v(1), there exists a bang-bang solution y to (P) satisfying y less than or equal to x; moreover there exists a finite union of intervals E such that y'' + a(1)y' + a(0)y = phi(1 chi E) + phi(2 chi[0, T]\E) . The reachable set of bang-bang constrained solutions is convex: an application to the calculus of variations.