Convergence and turnpike properties of linear-quadratic mean field control problems with common noise

TL;DR

This paper studies the convergence and turnpike properties of linear-quadratic mean field control problems with common noise, providing a unified analysis of finite-horizon, mean field limit, and ergodic problems using Riccati equations.

Contribution

It introduces a comprehensive framework analyzing convergence and turnpike properties in mean field control with common noise, including new estimates for Riccati systems.

Findings

Established turnpike property for finite-horizon problems.

Derived quantitative convergence rates to mean field limit.

Reduced ergodic problem to algebraic Riccati equations.

Abstract

We investigate convergence and turnpike properties for linear-quadratic mean field control problems with common noise. Within a unified framework, we analyze a finite-horizon social optimization problem, its mean field control limit, and the corresponding ergodic mean field control problem. The finite-horizon problems are characterized by coupled Riccati differential equations, whereas the ergodic problem is addressed via a Bellman equation on the Wasserstein space, which reduces to a system of stabilizing algebraic Riccati equations. By deriving estimates for these Riccati systems, we establish a turnpike property for the finite-horizon mean field control problem and obtain quantitative convergence results from the social optimization problem to its mean field limit and the associated ergodic control problem.