The turnpike property for mean-field optimal control problems

We study the turnpike phenomenon for optimal control problems with mean field dynamics that are obtained as the limit $N\to \mathrm{\infty }$ of systems governed by a large number $N$ of ordinary differential equations. We show that the optimal control problems with with large time horizons give rise to a turnpike structure of the optimal state and the optimal control. For the proof, we use the fact that the turnpike structure for the problems on the level of ordinary differential equations is preserved under the corresponding mean-field limit.