Uniform-in-time convergence and turnpike properties of linear-quadratic mean field control problems with common noise

We investigate uniform-in-time convergence and turnpike properties for linear-quadratic mean field control problems with common noise. Within a unified framework, we analyze a finite-horizon social optimization problem, its mean field control limit, and the corresponding ergodic mean field control problem. The finite-horizon problems are characterized by coupled Riccati differential equations, whereas the ergodic problem is addressed via a Bellman equation on the Wasserstein space, which reduces to a system of stabilizing algebraic Riccati equations. By deriving estimates for these Riccati systems, we establish a turnpike property for the finite-horizon mean field control problem and obtain quantitative convergence results from the social optimization problem to its mean field limit and the associated ergodic control problem.