Abstract :
When a Hamiltonian H = H(t,x,p) is convex in the adjoint variable p, the corresponding Hamilton-Jacobi equation (0.1) ut+ H(t,x,ux)= 0 is known to be the Bellman equation of a suitable optimal control problem. Of course, the latter is notunique, 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.
