Flux-limited and classical viscosity solutions for regional control problems

TL;DR

This paper compares classical and flux-limited viscosity solutions for regional control problems, showing how classical results can be interpreted within the flux-limited framework with simplified proofs.

Contribution

It bridges two approaches for regional control, providing simpler proofs and interpretations of classical results using flux-limited solutions.

Findings

Classical viscosity solutions can be interpreted as flux-limited solutions.

Simplified proofs of comparison principles for flux-limited solutions.

Identification of maximal and minimal solutions within the flux-limited framework.

Abstract

The aim of this paper is to compare two different approaches for regional control problems: the first one is the classical approach, using a standard notion of viscosity solutions, which is developed in a series of works by the three first authors. The second one is more recent and relies on ideas introduced by Monneau and the fourth author for problems set on networks in another series of works, in particular the notion of flux-limited solutions. After describing and even revisiting these two very different points of view in the simplest possible framework, we show how the results of the classical approach can be interpreted in terms of flux-limited solutions. In particular, we give much simpler proofs of three results: the comparison principle in the class of bounded flux-limited solutions of stationary multidimensional Hamilton-Jacobi equations and the identification of the maximal and minimal Ishii's solutions with flux-limited solutions which were already proved by Monneau and the fourth author, and the identification of the corresponding vanishing viscosity limit, already obtained by Vinh Duc Nguyen and the fourth author.