Periodic Exponential Turnpike Phenomenon in Mean-Field Stochastic Linear-Quadratic Optimal Control

TL;DR

This paper proves the exponential turnpike property for mean-field stochastic LQ control problems with periodic coefficients, showing long-term optimal control behavior converges to a steady-state solution.

Contribution

It introduces stability, stabilizability, and detectability concepts for stochastic systems and analyzes Riccati equations to establish the periodic exponential turnpike phenomenon.

Findings

Established exponential turnpike property for periodic mean-field stochastic LQ problems

Analyzed Riccati equations under stabilizability and detectability conditions

Showed the optimal control converges to a periodic steady-state

Abstract

The paper establishes the exponential turnpike property for a class of mean-field stochastic linear-quadratic (LQ) optimal control problems with periodic coefficients. It first introduces the concepts of stability, stabilizability, and detectability for stochastic linear systems. Then, the long-term behavior of the associated Riccati equations is analyzed under stabilizability and detectability conditions.Subsequently, a periodic mean-field stochastic LQ problem is formulated and solved. Finally, a linear transformation of the periodic extension of its optimal pair is shown to be the turnpike limit of the initial optimal control problem.