Ergodicity and Turnpike Properties of Linear-Quadratic Mean Field Control Problems

We study the asymptotic behavior of linear-quadratic mean field stochastic optimal control problems, in which the expectation of the control enters both the drift and diffusion coefficients, and the cost functional incorporates general quadratic terms involving the expectations of both the state and the control. By formulating an ergodic control framework, we characterize the convergence between the finite-horizon control problem and its ergodic counterpart. Leveraging these convergence results, we establish the turnpike property for the optimal pairs, demonstrating that solutions to the finite-horizon control problem remain exponentially close to the ergodic optimal pair except near the temporal boundaries. This result reveals the intrinsic connection between long-term dynamics and their asymptotic behavior in mean field control systems.