Normality of the maximum principle for non convex constrained Bolza problems∗

We consider a Bolza optimal control problem with state constraints. It is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle. In general, the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls. In the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate, cf. [2, 19, 20]. Our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle. In particular we allow the initial condition to be fixed and the state constraints to be non smooth. To prove normality we use J.Yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined, in turn, by linearization of the original set of constraints along an extremal trajectory.