The Maximum Principle under Minimal Hypotheses

We consider the optimal control system \[ x(t) = f(t,x(t),u(t)), u(t) ∈ U(t) {a.e.}\] with given initial and terminal constraints and a cost functional. We derive necessary conditions for optimality in a form similar to Pontryagin’s maximum principle under hypotheses which are in a certain sense minimal in order that the problem be meaningful. In particular we do not assume $f(t,s,u)$ continuous in u or differentiable in s, nor do we require $U(t)$ or $f(t,s,U(t))$ to be bounded or closed. These necessary conditions, which are expressed in terms of certain “generalized Jacobians,” reduce to the usual ones when classical hypotheses are imposed.