*Submitted Paper*

**Inserted:** 21 jun 2023

**Last Updated:** 21 jun 2023

**Year:** 2023

**Abstract:**

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.

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