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Weak epigraphical solutions to Hamilton-Jacobi-Bellman equations on infinite horizon

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.