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On the doubling of variables technique in first order Hamilton-Jacobi equations

Charles Bertucci, Giacomo Ceccherini Silberstein

TL;DR

This paper addresses the well-posedness issues for first-order Hamilton–Jacobi equations on finite-dimensional manifolds and on Wasserstein spaces by reinterpreting the doubling of variables through a geometry-driven penalization. It develops a penalization built from optimal-control/Lagrangian structure and extends it to the mean-field setting via a geometric transport distance, Fenchel duality, and a robust superdifferential calculus on P_p(M). The main contributions include a global doubling-variables framework that yields comparison principles for both convex and non-convex Hamiltonians in finite dimensions and a parallel mean-field theory using relaxed Hamiltonians, duality, and measure-space geometry, along with proof techniques that unify dynamic and static transport formulations. The work advances viscosity solutions theory on spaces of measures, with potential implications for mean-field control, mean-field games, and HJ equations on non-linear spaces, by exploiting intrinsic geometric structures rather than purely local Euclidean tools.

Abstract

In this paper, we revisit the technique of doubling variables in first order Hamilton-Jacobi equations, especially when the equations arise in optimal control. We show that by tuning the penalization between the two points, we can change drastically the proof, somehow shifting the regularity hypotheses into geometrical properties of the penalization. We present this idea in a finite dimensional setting and then exploit it on equations posed on Wasserstein spaces.

On the doubling of variables technique in first order Hamilton-Jacobi equations

TL;DR

This paper addresses the well-posedness issues for first-order Hamilton–Jacobi equations on finite-dimensional manifolds and on Wasserstein spaces by reinterpreting the doubling of variables through a geometry-driven penalization. It develops a penalization built from optimal-control/Lagrangian structure and extends it to the mean-field setting via a geometric transport distance, Fenchel duality, and a robust superdifferential calculus on P_p(M). The main contributions include a global doubling-variables framework that yields comparison principles for both convex and non-convex Hamiltonians in finite dimensions and a parallel mean-field theory using relaxed Hamiltonians, duality, and measure-space geometry, along with proof techniques that unify dynamic and static transport formulations. The work advances viscosity solutions theory on spaces of measures, with potential implications for mean-field control, mean-field games, and HJ equations on non-linear spaces, by exploiting intrinsic geometric structures rather than purely local Euclidean tools.

Abstract

In this paper, we revisit the technique of doubling variables in first order Hamilton-Jacobi equations, especially when the equations arise in optimal control. We show that by tuning the penalization between the two points, we can change drastically the proof, somehow shifting the regularity hypotheses into geometrical properties of the penalization. We present this idea in a finite dimensional setting and then exploit it on equations posed on Wasserstein spaces.

Paper Structure

This paper contains 25 sections, 17 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Doubling of variables and central question of the paper
  3. Structure of the paper
  4. Bibliographical comments
  5. Notation and assumptions
  6. Structure of the ambient space
  7. Assumptions on the Hamiltonian and the associated Lagrangian in the convex setting
  8. Action Functional
  9. Superdifferential calculus
  10. The case of equations in Riemannian manifold
  11. Viscosity solutions
  12. Result for non-convex Hamiltonians
  13. The case of a more geometric Hamiltonian
  14. Mean Field Setting
  15. Couplings and notation
  16. ...and 10 more sections

Key Result

Theorem 2.1

(Fathi2010) Suppose $L$ is a weak Lagrangian. Then for every $a,b \in \mathbb{R}$, $a<b$ and every $x,y \in M$, there exists an absolutely continuous curve $\gamma : [a,b] \to M$ which is a minimizer of $\mathcal{A}_{[a,b]}$ whith $\gamma(a)=x$ and $\gamma(b)=y$.

Theorems & Definitions (50)

  • Definition 2.1
  • Example 2.1
  • Example 2.2: Minima of the action functional and geodesics
  • Theorem 2.1
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Corollary 2.3
  • Definition 2.2
  • Example 2.3: Super-differential of a semiconcave function
  • ...and 40 more