Necessary conditions in optimal control: a new approach

We present necessary conditions of optimality for a general control problem formulated in terms of a differential inclusion. These conditions unify and substantially extend previous results in the literature, and also incorporate a `stratified' feature of a novel nature. When specialized to the calculus of variations, the results yield necessary conditions and regularity theorems that go significantly beyond the previous standard. They also give rise to new and stronger maximum principles of Pontryagin type. The principal focus here is the development of structural criteria on the problem which guarantee a priori that the necessary conditions apply to any local minimizer.