  In the present paper, it is provided a representation result for the weak
solutions of a class of evolutionary Hamilton-Jacobi-Bellman equations on
infinite horizon, with Hamiltonians measurable in time and fiber convex. Such
Hamiltonians are associated with a - faithful - representation, namely
involving two functions measurable in time and locally Lipschitz in the state
and control. Our results concern the recovering of a representation of convex
Hamiltonians under a relaxed assumption on the Fenchel transform of the
Hamiltonian with respect to the fiber. We apply them to investigate the
uniqueness of weak solutions, vanishing at infinity, of a class of
time-dependent Hamilton-Jacobi-Bellman equations. Assuming a viability
condition on the domain of the aforementioned Fenchel transform, these weak
solutions are regarded as an appropriate value function of an infinite horizon
control problem under state constraints.
