Convexity in Hamilton-Jacobi theory - II: Envelope representations

Upper and lower envelope representations are developed for value functions associated with problems of optimal control and the calculus of variations that are fully convex, in the sense of exhibiting convexity in both the sta te and the velocity. Such convexity is used in dualizing the upper envelope representations to get the lower ones, which have advantages not previousl y perceived in such generality and in some situations can be regarded as fu rnishing, at least for value functions, extended Hopf-Lax formulas that ope rate beyond the case of state-independent Hamiltonians. The derivation of the lower envelope representations centers on a new funct ion called the dualizing kernel, which propagates the Legendre-Fenchel enve lope formula of convex analysis through the underlying dynamics. This kerne l is shown to be characterized by a kind of double Hamilton-Jacobi equation and, despite overall nonsmoothness, to be smooth with respect to time and concave-convex in the primal and dual states. It furnishes a means whereby, in principle, value functions and their subgradients can be determined thr ough optimization without having to deal with a separate, and typically muc h less favorable, Hamilton-Jacobi equation for each choice of the initial o r terminal cost data.