Turnpike Property of Mean-Field Linear-Quadratic Optimal Control Problems in Infinite-Horizon with Regime Switching

TL;DR

This paper proves the strong turnpike property for infinite-horizon mean-field linear-quadratic control problems with regime switching, using Riccati and backward differential equations to analyze asymptotic behavior.

Contribution

It establishes the strong turnpike property for mean-field LQ control problems with regime switching over infinite horizons, a novel extension in stochastic control theory.

Findings

Strong turnpike property is proven for the problem.

Convergence of Riccati and backward equations is demonstrated.

Optimality of the limit pair is verified in specific cases.

Abstract

This paper considers an optimal control problem for a linear mean-field stochastic differential equation having regime switching with quadratic functional in the large time horizons. Our main contribution lies in establishing the strong turnpike property for the optimal pairs when the time horizon tends to infinity. To work with the mean-field terms, we apply the orthogonal decomposition method to derive a closed-loop representation of the optimal control problem in a finite time horizon. To analyze the asymptotic behavior of the optimal controls, we examine the convergence of the solutions of Riccati equations and backward differential equations as the time horizon tends to infinity. The strong turnpike property can be obtained based on these convergence results. Finally, we verify the optimality of the limit optimal pair in two cases: integrable case and local-integrable case.