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

TL;DR

This paper establishes the uniqueness of weak epigraphical solutions to Hamilton-Jacobi-Bellman equations on infinite horizon, extending solution concepts and analyzing conditions under which these solutions are viscosity solutions.

Contribution

It introduces a new notion of locally absolutely continuous tubes for set-valued maps and extends the theory of weak solutions for HJB equations with less restrictive conditions.

Findings

Weak epigraphical solutions are viscosity solutions under Lipschitz continuity.

Extension of solution concepts to set-valued maps with bounded variation.

Controllability assumptions ensure uniqueness of solutions.

Abstract

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.