Convexity in Hamilton--Jacobi theory. II: Envelope representations (Q2706154)

The second part of the paper [see also the preceding review of Part I, Zbl 0998.49018] discusses upper and lower envelope representations for value functions associated with the problem of optimal control and calculus of variations that are fully convex in the sense of exhibiting convexity for both the state and the velocity. Convexity is used for 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 functions, extended Hopf-Lax formulas that operate beyond the case of state-independent Hamiltonians.NEWLINENEWLINENEWLINEThe derivation of the lower envelope representations leads to a new function called the dualizing kernel. This kernel propagates the Legendre-Fenchel envelope formula of convex analysis through the underlying dynamics. This kernel is characterized using the Hamilton-Jacobi equation. It furnishes the means for the determination of value functions and their subgradients using optimalization by the Hamilton-Jacobi equation for each choice of the initial or terminal cost data.