Representation of Weak Solutions of Convex Hamilton-Jacobi-Bellman Equations on Infinite Horizon

TL;DR

This paper provides a new representation for weak solutions of infinite horizon Hamilton-Jacobi-Bellman equations with convex Hamiltonians, aiding in understanding their uniqueness and connection to control problems.

Contribution

It introduces a relaxed representation theorem for convex Hamiltonians, extending previous results and linking weak solutions to infinite horizon control problems with state constraints.

Findings

Established a representation for weak solutions under relaxed assumptions.

Proved uniqueness of solutions vanishing at infinity.

Connected solutions to infinite horizon control problems with constraints.

Abstract

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.