Ergodicity and turnpike properties of linear-quadratic mean field control problems

TL;DR

This paper investigates the long-term behavior of linear-quadratic mean field control problems, establishing convergence to ergodic solutions and demonstrating the turnpike property where optimal solutions stay close to equilibrium over time.

Contribution

It introduces an ergodic control framework for mean field problems and proves the turnpike property, linking finite horizon solutions to their long-term ergodic limits.

Findings

Finite horizon solutions converge to ergodic solutions.

Optimal pairs exhibit exponential proximity to ergodic equilibrium.

Turnpike property holds for mean field control systems.

Abstract

We study the asymptotic behavior of solutions to linear-quadratic mean field stochastic optimal control problems. By formulating an ergodic control framework, we characterize the convergence between the finite time horizon control problem and its ergodic counterpart. Leveraging these convergence results, we establish the turnpike property for the optimal pairs, demonstrating that solutions to the finite time horizon control problem remain exponentially close to the ergodic equilibrium except near the temporal boundaries. This result reveals the intrinsic connection between long-term dynamics and their asymptotic behavior in mean field control systems.