Turnpike Properties for Mean-Field Linear-Quadratic Optimal Control Problems

TL;DR

This paper investigates the long-term behavior of optimal controls in mean-field linear-quadratic stochastic systems, establishing exponential and mean-square turnpike properties and highlighting differences from deterministic cases.

Contribution

It introduces conditions under which the exponential and mean-square turnpike properties hold for mean-field LQ problems, emphasizing the formulation of static optimization and correction equations.

Findings

Proves exponential and mean-square turnpike properties for the problem

Identifies key differences between stochastic and deterministic cases

Provides a framework for analyzing long-term optimal control in stochastic mean-field systems

Abstract

This paper is concerned with an optimal control problem for a mean-field linear stochastic differential equation with a quadratic functional in the infinite time horizon. Under suitable conditions, including the stabilizability, the (strong) exponential, integral, and mean-square turnpike properties for the optimal pair are established. The keys are to correctly formulate the corresponding static optimization problem and find the equations determining the correction processes. These have revealed the main feature of the stochastic problems which are significantly different from the deterministic version of the theory.