Calculus of Variations and Geometric Measure Theory

C. Jimenez

Equivalence between strict viscosity solution and viscosity solution in the space of Wasserstein and regular extension of the Hamiltonian in $L^2_{I\! P}$

created by jimenez on 21 Jun 2023


Submitted Paper

Inserted: 21 jun 2023
Last Updated: 21 jun 2023

Year: 2023


This article aims to build bridges between several notions of viscosity solution of first order dynamic Hamilton-Jacobi equations. The first main result states that, under assumptions, the definitions of Gangbo-Nguyen-Tudorascu and Marigonda-Quincampoix are equivalent. Secondly, to make the link with Lions' definition of solution, we build a regular extension of the Hamiltonian in $L^2_{I\! P}\times L^2_{I\! P}$. This extension allows to give an existence result of viscosity solution in the sense of Gangbo-Nguyen-Tudorascu, as a corollary of the existence result in $L^2_{I\! P}\times L^2_{I\! P}$. We also give a comparison principle for rearrangement invariant solutions of the extended equation. Finally we illustrate the interest of the extended equation by an example in Multi-Agent Control.

Keywords: Optimal Transport, Hamilton-Jacobi equations, Multi-Agent Optimal Control.