  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.
