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 LP2×LP2L^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 LP2×LP2L^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.

Author information

Contact details are reproduced from the original publication and may be historical.

Chloé Jimenez

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

chloe.jimenez@univ-brest.fr

Keywords

  • Optimal transport
  • viscosity solutions
  • Hamilton-Jacobi equations
  • multi-agent optimal control

Mathematics Subject Classification

49L25

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Suggested citation

C. Jimenez. “Equivalence between Strict Viscosity Solution and Viscosity Solution in the Wasserstein Space and Regular Extension of the Hamiltonian in L^2_ℙ.” Journal of Convex Analysis 31 (2024), No. 2, 619–670. https://doi.org/10.68381/jca31031

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Published by Heldermann Verlag, 2024. Rights now held by Banach Press.