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.