Necessary conditions in optimal control without differentiability

Abstract

In traditional proofs of the maximum principle, continuous differentiability, or at least Lipschitz continuity, of the dynamic constraint with respect to the state variable is invoked, along arbitrary control functions. Recently Sussmann, following up ideas proposed by Lojasiewicz, has demonstrated the validity of the maximum principle in circumstances when the dynamic constraint is Lipschitz continuous merely along the optimal control function. We provide a simple derivation of the maximum principle, under this milder hypothesis, for problems with unilateral state constraints and where the right endpoint constraint takes the form of a family of functional inequalities. Ekeland's theorem (1976) is used to construct a sequence of perturbed 'Lipschitz' optimal control problems. The maximum principle (for nonLipschitz data off the optimal control) is then proved by applying the standard maximum principle to each of the perturbed problems and passing to the limit.