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Hamilton--Jacobi equations for Wasserstein controlled gradient flows: existence of viscosity solutions

Giovanni Conforti, Richard C. Kraaij, Luca Tamanini, Daniela Tonon

Abstract

This work is the third part of a program initiated in arXiv:2111.13258, arXiv:2302.06571 aiming at the development of an intrinsic geometric well-posedness theory for Hamilton-Jacobi equations related to controlled gradient flow problems in metric spaces. In this paper, we finish our analysis in the context of Wasserstein gradient flows with underlying energy functional satisfying McCann's condition. More prescisely, we establish that the value function for a linearly controlled gradient flow problem whose running cost is quadratic in the control variable and just continuous in the state variable yields a viscosity solution to the Hamilton-Jacobi equation in terms of two operators introduced in our former works, acting as rigorous upper and lower bounds for the formal Hamiltonian at hand. The definition of these operators is directly inspired by the Evolutional Variational Inequality formulation of gradient flows (EVI): one of the main innovations of this work is to introduce a controlled version of EVI, which turns out to be crucial in establishing regularity properties, energy and metric bounds along optimzing sequences in the controlled gradient flow problem that defines the candidate solution.

Hamilton--Jacobi equations for Wasserstein controlled gradient flows: existence of viscosity solutions

Abstract

This work is the third part of a program initiated in arXiv:2111.13258, arXiv:2302.06571 aiming at the development of an intrinsic geometric well-posedness theory for Hamilton-Jacobi equations related to controlled gradient flow problems in metric spaces. In this paper, we finish our analysis in the context of Wasserstein gradient flows with underlying energy functional satisfying McCann's condition. More prescisely, we establish that the value function for a linearly controlled gradient flow problem whose running cost is quadratic in the control variable and just continuous in the state variable yields a viscosity solution to the Hamilton-Jacobi equation in terms of two operators introduced in our former works, acting as rigorous upper and lower bounds for the formal Hamiltonian at hand. The definition of these operators is directly inspired by the Evolutional Variational Inequality formulation of gradient flows (EVI): one of the main innovations of this work is to introduce a controlled version of EVI, which turns out to be crucial in establishing regularity properties, energy and metric bounds along optimzing sequences in the controlled gradient flow problem that defines the candidate solution.
Paper Structure (17 sections, 14 theorems, 170 equations)

This paper contains 17 sections, 14 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. McKean Vlasov control and controlled gradient flows
  3. Literature review
  4. Our contribution
  5. Organization
  6. Setting, assumptions and main results
  7. Setting and assumptions
  8. The candidate solution
  9. The rigorous upper and lower Hamiltonian
  10. Main results
  11. Preliminary results
  12. The dynamic programming principle
  13. A controlled EVI
  14. Continuity of the transport map
  15. The subsolution property
  16. ...and 2 more sections

Key Result

Theorem 2.7

Under Assumption ass: OT energy functional, let $h \in C_b(\mathcal{P}_p(\mathbb{R}^d))$ for some $p<2$. Let $\Phi^*$ and $\Phi_*$ be respectively the upper and the lower semicontinuous relaxation, w.r.t. the $W_p$-topology, of the value function $\Phi$ defined in def:value_fun. Then $\Phi^*$ is a v

Theorems & Definitions (32)

  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Proposition 3.1: Dynamic Programming Principle
  • proof
  • Corollary 3.2
  • ...and 22 more