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Mean field first order optimality condition under low regularity of controls

Stefano Almi, Riccardo Durastanti, Francesco Solombrino

TL;DR

This work derives a first-order necessary condition for mean-field optimal control with low regularity by passing to the limit of a finite-particle Pontryagin maximum principle. The main technique combines a variational limit (Gamma-convergence) with measure-theoretic tools (Wasserstein differentiability, disintegration) to obtain a continuum MF problem without demanding spatial regularity on controls; the limit object is a pair $(\pmb{\nu},\mathbf{w})$ where $\pmb{\nu}$ evolves by a continuity equation in $(x,r)$ and $\mathbf{w}$ maximizes a mean-field Hamiltonian $\mathcal{H}$. The framework accommodates selective control, multi-population dynamics, and extensions to convex state spaces, including multi-label dynamics and entropy-regularized replicator models, illustrating broad applicability to kinetic-type mean-field systems. The results provide a robust MF-PMP that remains valid under low regularity and connects the finite-dimensional PMP with a measure-valued, velocity-field driven optimality structure, offering a principled basis for design of control laws in large-scale interacting systems.

Abstract

We show that mean field optimal controls satisfy a first order optimality condition (at a.e. time) without any a priori requirement on their spatial regularity. This principle is obtained by a careful limit procedure of the Pontryagin maximum principle for finite particle systems. In particular, our result applies to the case of mean field selective optimal control problems for multipopulation and replicator dynamics.

Mean field first order optimality condition under low regularity of controls

TL;DR

This work derives a first-order necessary condition for mean-field optimal control with low regularity by passing to the limit of a finite-particle Pontryagin maximum principle. The main technique combines a variational limit (Gamma-convergence) with measure-theoretic tools (Wasserstein differentiability, disintegration) to obtain a continuum MF problem without demanding spatial regularity on controls; the limit object is a pair where evolves by a continuity equation in and maximizes a mean-field Hamiltonian . The framework accommodates selective control, multi-population dynamics, and extensions to convex state spaces, including multi-label dynamics and entropy-regularized replicator models, illustrating broad applicability to kinetic-type mean-field systems. The results provide a robust MF-PMP that remains valid under low regularity and connects the finite-dimensional PMP with a measure-valued, velocity-field driven optimality structure, offering a principled basis for design of control laws in large-scale interacting systems.

Abstract

We show that mean field optimal controls satisfy a first order optimality condition (at a.e. time) without any a priori requirement on their spatial regularity. This principle is obtained by a careful limit procedure of the Pontryagin maximum principle for finite particle systems. In particular, our result applies to the case of mean field selective optimal control problems for multipopulation and replicator dynamics.

Paper Structure

This paper contains 8 sections, 10 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Assumptions and main results
  4. Comparison with the Pontryagin Maximum Principle for regular controls
  5. Proof of the results
  6. Generalization to optimal control problems with convex state space
  7. Control of multi-population systems
  8. Entropy regularized replicator dynamics

Key Result

Lemma 2.3

Let $F:\mathbb{R}^d \times \mathcal{P}_1(\mathbb{R}^d)\to \mathbb{R}^m$ be locally differentiable at $\psi^N$ in the sense of Definition Wmudiff, where $\psi^N$ is generated by $\bm{x}$ (i.e. $\psi^N=\frac{1}{N}\sum_{i=1}^N \delta_{x_i}$). Then $F_{|(\mathbb{R}^d)^N }\colon \mathbb{R}^d\times (\math Moreover, if, for every fixed $x\in \mathbb{R}^d$, the map $\mathcal{P}_c(\mathbb{R}^d) \times \mat

Theorems & Definitions (19)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: FLOS
  • Lemma 3.1
  • Proposition 3.2
  • Remark 3.3
  • Proposition 3.4
  • Theorem 3.5
  • ...and 9 more