  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.
