We consider the control of McKean-Vlasov dynamics whose coefficients have mean field interactions in the state and control. We show that for a class of linear-convex mean field control problems, the unique optimal open-loop control admits the optimal 1/2-H\"{o}lder regularity in time. Consequently, we prove that the value function can be approximated by one with piecewise constant controls and discrete-time state processes arising from Euler-Maruyama time stepping, up to an order 1/2 error, and the optimal control can be approximated up to an order 1/4 error. These results are novel even for the case without mean field interaction.

翻译：我们考虑控制McKan-Vlasov动态,其系数意味着国家和控制中的实地互动。我们显示,对于某类线性电流意味着实地控制问题,独特的最佳开放环控制在时间上接受最佳的1/2-H\"{o}lder 常规性。因此,我们证明,价值函数可以被一个具有零星常量控制和由Euler-Maruyama时间加速产生的离散时间状态进程的人所近似,最高可达到第1/2号命令的错误,而最佳控制可以近似于第1/4号命令的错误。这些结果甚至对本案来说都是新奇的,没有明显的实地互动。