Francis Clarke 
Necessary Conditions in Dynamic Optimization 

This monograph derives necessary conditions of optimality for a general control problem formulated in terms of a differential inclusion. These conditions constitute a new state of the art, subsuming, unifying, and substantially extending the results in the literature. The Euler, Weierstrass and transversality conditions are expressed in their sharpest known forms. No assumptions of boundedness or convexity are made, no constraint qualifications imposed, and only weak pseudo-Lipschitz behavior is postulated on the underlying multifunction. The conditions also incorporate a ‘stratified’ feature of a novel nature, in which both the hypotheses and the conclusion are formulated relative to a given radius function. When specialized to the calculus of variations, the results yield necessary conditions and regularity theorems that go significantly beyond the previous standard. They also apply to parametrized control systems, giving rise to new and stronger maximum principles of Pontryagin type. The final chapter is devoted to a different issue, that of the Hamiltonian necessary condition. It is obtained here, for the first time, in the case of nonconvex values and in the absence of any constraint qualification; this has been a longstanding open question in the subject. Apart from the final chapter, the treatment is self-contained, and calls upon only standard results in functional and nonsmooth analysis.