Stable representation of convex Hamiltonians

Existence and uniqueness of solutions to a Hamilton–Jacobi equation vt+H(t,x,vx)=0,v(0,⋅)=φ(⋅) with HH convex with respect to the last variable can be proved by associating to HH either a Calculus of Variations or an optimal control problem. The data of the new problem should be so that its Hamiltonian coincides with HH and should also inherit appropriate continuity/local Lipschitz continuity properties of HH. In other words, HH can be represented by functions describing an optimization problem. In this paper we provide further developments of representation theorems from Rampazzo (2005). In particular, our results imply continuous dependence of representations on the mapping HH. We apply them to study existence of solutions to the Hamilton–Jacobi equation with HH possibly discontinuous in tt.