A new condition for global optimality in optimal control theory

We consider how a well-known sufficient condition for global optimality, expressed in terms of a continuously differentiable function ?? satisfying a certain partial differential inequality, may be modified to broaden its applicability. We present a new condition in which ?? is permitted to be Lipschitz continuous. The new condition always applies under a certain ´calmness´ hypothesis, a local version of which has previously arisen in the literature as the weakest available hypothesis assuring non-triviality of the Pontryagin Maximum Principle multipliers.