Turnpike Property of Mean-Field Linear-Quadratic Optimal Control Problems in Infinite-Horizon with Regime Switching

Abstract: This paper considers an optimal control problem for a linear mean-field stochastic differential equation having regime switching with quadratic functional in the large time horizons. Our main contribution lies in establishing the strong turnpike property for the optimal pairs when the time horizon tends to infinity. To work with the mean-field terms, we apply the orthogonal decomposition method to derive a closed-loop representation of the optimal control problem in a finite time horizon. To analyze the asymptotic behavior of the optimal controls, we examine the convergence of the solutions of Riccati equations and backward differential equations as the time horizon tends to infinity. The strong turnpike property can be obtained based on these convergence results. Finally, we verify the optimality of the limit optimal pair in two cases: integrable case and local-integrable case.