Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hamilton--Jacobi equations for controlled gradient flows: the comparison principle

Giovanni Conforti, Richard Kraaij, Daniela Tonon

Abstract

Motivated by recent developments in the fields of large deviations for interacting particle system and mean field control, we establish a comparison principle for the Hamilton--Jacobi equation corresponding to linearly controlled gradient flows of an energy function $\cE$ defined on a metric space $(E,d)$. Our analysis is based on a systematic use of the regularizing properties of gradient flows in evolutional variational inequality (EVI) formulation, that we exploit for constructing rigorous upper and lower bounds for the formal Hamiltonian at hand and, in combination with the use of the Tataru's distance, for establishing the key estimates needed to bound the difference of the Hamiltonians in the proof of the comparison principle. Our abstract results apply to a large class of examples only partially covered by the existing theory, including gradient flows on Hilbert spaces and the Wasserstein space equipped with a displacement convex energy functional $\cE$ satisfying McCann's condition.

Hamilton--Jacobi equations for controlled gradient flows: the comparison principle

Abstract

Motivated by recent developments in the fields of large deviations for interacting particle system and mean field control, we establish a comparison principle for the Hamilton--Jacobi equation corresponding to linearly controlled gradient flows of an energy function defined on a metric space . Our analysis is based on a systematic use of the regularizing properties of gradient flows in evolutional variational inequality (EVI) formulation, that we exploit for constructing rigorous upper and lower bounds for the formal Hamiltonian at hand and, in combination with the use of the Tataru's distance, for establishing the key estimates needed to bound the difference of the Hamiltonians in the proof of the comparison principle. Our abstract results apply to a large class of examples only partially covered by the existing theory, including gradient flows on Hilbert spaces and the Wasserstein space equipped with a displacement convex energy functional satisfying McCann's condition.
Paper Structure (28 sections, 21 theorems, 159 equations)

This paper contains 28 sections, 21 theorems, 159 equations.

Table of Contents

  1. Data availability
  2. Introduction
  3. Hamilton--Jacobi equations in infinite dimensional spaces
  4. Master equation and Mean Field Games
  5. Contribution of this work
  6. Organization
  7. Frequently used notation
  8. The comparison principle
  9. EVI-gradient flows and statement of the main hypotheses
  10. A first attempt at defining upper and lower bounds
  11. The Tataru distance
  12. The comparison principle for a proper upper and lower bound
  13. Proof of Theorem \ref{['theorem:comparison_principle_tildeoperators']}
  14. Proof of proposition \ref{['prop:quadruplication_Ekeland']}
  15. Key estimates
  16. ...and 13 more sections

Key Result

Theorem 2.13

[The comparison Principle.] Let Assumptions assumption:distance_and_energy, assumption:gradientflow and assumption:regularized_geodesics be satisfied. Let $\lambda>0$ and $h^\dagger,h^\ddagger: E\to \mathbb{R}$ be bounded and uniformly continuous. Let $u : E\to \mathbb{R}$ be a viscosity subsolution

Theorems & Definitions (60)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.10
  • Definition 2.11
  • Definition 2.12
  • Theorem 2.13
  • ...and 50 more