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Journal of Convex Analysis 31 (2024), No. 2, 619--670
Copyright Heldermann Verlag 2024

Equivalence between Strict Viscosity Solution and Viscosity Solution in the Wasserstein Space and Regular Extension of the Hamiltonian in L2P

Chloé Jimenez
Univ Brest, CNRS UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique, Brest, France


\newcommand{\p}{\mathbb P} 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 $L^2_\p\times L^2_\p$. 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 $L^2_\p\times L^2_\p$. 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, viscosity solutions, Hamilton-Jacobi equations, multi-agent optimal control.

MSC: 49L25.

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