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 state and the velocity. Such convexity is used in dualizing the upper envelope representations to get the lower ones, which have advantages not previously perceived in such generality and in some situations can be regarded as furnishing, at least for value functions, extended Hopf--Lax formulas that operate beyond the case of state-independent Hamiltonians. The derivation of the lower envelope representations centers on a new function called the dualizing kernel, which propagates the Legendre--Fenchel envelope formula of convex analysis through the underlying dynamics. This kernel 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 through optimization without having to deal with a separate, and typically much less favorable, Hamilton--Jacobi equation for each choice of the initial or terminal cost data.