Well-posedness of a Hamilton-Jacobi-Bellman equation in the strong coupling regime

TL;DR

This paper establishes a comparison principle for viscosity solutions of Hamilton-Jacobi-Bellman equations in a strong coupling regime, accommodating discontinuous and unbounded cost functions, with applications in large deviations and homogenisation.

Contribution

It introduces a novel approach to handle discontinuous, unbounded cost functions in Hamilton-Jacobi-Bellman equations within a strong coupling framework, expanding the scope of solvable problems.

Findings

Proved comparison principle for viscosity solutions in strong coupling regime.

Established existence of viscosity solutions under new conditions.

Applied results to an example from biochemistry.

Abstract

We prove comparison principle for viscosity solutions of a Hamilton-Jacobi-Bellman equation in a strong coupling regime considering a stationary and a time-dependent version of the equation. We consider a Hamiltonian that has a representation as the supremum of a difference of two functions: an internal Hamiltonian depending on a control variable and a function interpreted as a cost of applying the controls. Our major innovation lies in the use of a cost function that can be discontinuous, unbounded and depending on momenta, enabling us to address previously unexplored scenarios such as cases arising from the theory of large deviations and homogenisation. For completeness, we also state the existence of viscosity solutions and we verify the assumptions for an example arising from biochemistry.