  In this paper we show a uniqueness result for weak epigraphical solutions of
Hamilton-Jacobi-Bellman (HJB) equations on infinite horizon for a class of
lower semicontinuous functions vanishing at infinity. Weak epigraphical
solutions of HJB equations, with time-measurable data and fiber-convex, turn
out to be viscosity solutions - in the classical sense - whenever they are
locally Lipschitz continuous. Here we extend the notion of locally absolutely
continuous tubes to set-valued maps with continuous epigraph of locally bounded
variations. This new notion fits with the lack of uniform lower bound of the
Fenchel transform of the Hamiltonian with respect to the fiber. Controllability
assumptions are considered.
