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

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.