Flux-limited and classical viscosity solutions for regional control problems^★

¹ Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964), Université François Rabelais, Parc de Grandmont, 37200 Tours, France
² CNRS & Département de mathématiques et applications (UMR CNRS 8553), École Normale Supérieure (Paris), 45 rue d’Ulm, 75230 Paris Cedex 5, France

^** Corresponding author: emmanuel.chasseigne@lmpt.univ-tours.fr

Received: 27 April 2017
Accepted: 23 November 2017

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.