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Optimal control of mean-field limit of multiagent systems with and without common noise

Giuseppe La Scala

TL;DR

The paper addresses optimal control for a two-species interacting system (herd and herders) under finite SDE-ODE constraints, and shows that in the mean-field limit with a large herd, the system converges to a $\text{McKean-Vlasov}$ SDE coupled with ODEs. It analyzes propagation of chaos, both in the absence and presence of common noise (conditional on the common noise), and derives the corresponding Fokker–Planck equations for the mean-field law. The authors prove well-posedness for both the discrete and mean-field problems and establish $\Gamma$-convergence of the discrete control problem to the mean-field control problem, ensuring that optimal policies for large finite systems approximate those of the mean-field model. The framework extends prior work by incorporating multiplicative idiosyncratic noise, common environmental noise, and control actions that affect both herd and herd dynamics, providing rigorous justification and tools for mean-field control in multi-agent systems with noise.

Abstract

We consider a generic, suitable class of optimal control problems under a constraint given by a finite-dimensional SDE-ODE system, describing a system of two interacting species of particles: the herd, described by SDEs, and the herders, described by ODEs with the addition of a control function. In particular, we firstly show that for a low number of herders and for the limit of large number of herd individuals, the SDE-ODE system can be approximated by an infinite-dimensional system given by a McKean-Vlasov single SDE coupled with ODEs. Then, thanks to this we show the $Γ-$convergence of the optimal control problem for the finite-dimensional system to a certain optimal control problem for the mean-field system. Differently from Ascione-Castorina-Solombrino [9] (SIAM J. Math. Anal., Vol. 55, No. 6, pp. 6965-6990 (2023)), we do not consider an additive noise for the herd, but a more general class, given by idiosyncratic noises (due to a single herd individual) together with common noise (due to how the environment affects the whole herd), and they are independent one from another. As well as this, we consider a more general class of control functions in the ODEs for herders, where the control is applied not only on the herd dynamics, but also on the herd one.

Optimal control of mean-field limit of multiagent systems with and without common noise

TL;DR

The paper addresses optimal control for a two-species interacting system (herd and herders) under finite SDE-ODE constraints, and shows that in the mean-field limit with a large herd, the system converges to a SDE coupled with ODEs. It analyzes propagation of chaos, both in the absence and presence of common noise (conditional on the common noise), and derives the corresponding Fokker–Planck equations for the mean-field law. The authors prove well-posedness for both the discrete and mean-field problems and establish -convergence of the discrete control problem to the mean-field control problem, ensuring that optimal policies for large finite systems approximate those of the mean-field model. The framework extends prior work by incorporating multiplicative idiosyncratic noise, common environmental noise, and control actions that affect both herd and herd dynamics, providing rigorous justification and tools for mean-field control in multi-agent systems with noise.

Abstract

We consider a generic, suitable class of optimal control problems under a constraint given by a finite-dimensional SDE-ODE system, describing a system of two interacting species of particles: the herd, described by SDEs, and the herders, described by ODEs with the addition of a control function. In particular, we firstly show that for a low number of herders and for the limit of large number of herd individuals, the SDE-ODE system can be approximated by an infinite-dimensional system given by a McKean-Vlasov single SDE coupled with ODEs. Then, thanks to this we show the convergence of the optimal control problem for the finite-dimensional system to a certain optimal control problem for the mean-field system. Differently from Ascione-Castorina-Solombrino [9] (SIAM J. Math. Anal., Vol. 55, No. 6, pp. 6965-6990 (2023)), we do not consider an additive noise for the herd, but a more general class, given by idiosyncratic noises (due to a single herd individual) together with common noise (due to how the environment affects the whole herd), and they are independent one from another. As well as this, we consider a more general class of control functions in the ODEs for herders, where the control is applied not only on the herd dynamics, but also on the herd one.

Paper Structure

This paper contains 9 sections, 18 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Random variables and stochastic calculus
  4. Wasserstein spaces
  5. Assumptions on interactions, noises and controls
  6. Well-posedness of discrete and mean-field equations
  7. The mean-field limit and the (conditional) propagation of chaos
  8. The Fokker-Plack equation for the mean-field problem
  9. Optimal control for discrete and mean-field problems

Key Result

Lemma 2.1

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $\mathcal{G},\mathcal{D}$ two independent sub-$\sigma-$algebras of $\mathcal{F}$. Let $X$ be a $\mathcal{D}-$measurable random variable taking values in the measurable space $(E,\mathcal{E})$, let $\Psi\colon\,E\times\Omega\longmapsto

Theorems & Definitions (43)

  • Lemma 2.1: Lemma 4.1, Baldi
  • Proposition 2.2
  • Proposition 2.3: Burkholder-Davis-Gundy Inequality
  • Proposition 2.4: $\hat{\text{I}}$to Formula
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • Remark 2.8
  • Definition 3.1
  • Remark 3.2
  • ...and 33 more