Faithful representations for convex Hamilton-Jacobi equations

When a Hamiltonian H = H(t, x, p) is convex in the adjoint variable p, the corresponding Hamilton - Jacobi equation (0.1) u(t) + H(t, x, u(x)) = 0 is known to be the Bellman equation of a suitable optimal control problem. Of course, the latter is not unique, so it is interesting to select a good optimal control problem among those representing ( 0.1). We call such a representation faithful if (i) it involves a dynamics which is locally Lipschitz continuous in the state variable - so that a unique trajectory corresponds to any given control and initial point - and (ii) the Lagrangian displays the same regularity as H in the x variable. The main result of the present paper establishes the existence of faithful representations for a large class of Hamiltonians, including those for which the standard comparison theorems ( of viscosity solution theory) are valid. Moreover, our investigation includes t-measurable Hamiltonians as well. If a faithful control-theoretical representation does exist ( and ( 0.1) enjoys uniqueness properties), one can infer sharp regularity results for the solution of ( 0.1) just by studying the regularity of the value function of the associated optimal control problem. A further application consists of a simple interpretation of the front propagation phenomenon in terms of optimal trajectories of the underlying minimum problem.