We consider extended mean field control (extended MFC) problems, which seek optimal control of McKean-Vlasov dynamics whose coefficients involve mean field interactions both on the state and actions, and where objectives are optimized over open-loop strategies. We show that for a large class of linear-convex extended MFC problems, the unique optimal open-loop control admits the optimal 1/2-H\"{o}lder regularity in time. Based on the solution regularity, we prove that the value functions of such extended MFC problems can be approximated by those with piecewise constant controls and discrete-time state processes arising from Euler-Maruyama time stepping up to an order 1/2 error, which is optimal in our setting. We further show that any $\epsilon$-optimal controls of these discrete-time problems converge to the optimal control of the original problems.

中文翻译：

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扩展平均场控制的最优规律及其分段常数近似

我们考虑扩展平均场控制 (extended MFC) 问题，该问题寻求对 McKean-Vlasov 动力学的最优控制，其系数涉及状态和动作的平均场相互作用，并且在开环策略上优化目标。我们证明，对于一大类线性凸扩展 MFC 问题，独特的最优开环控制在时间上承认最优 1/2-H\"{o}lder 正则性。基于解正则性，我们证明了这种扩展 MFC 问题的价值函数可以通过分段常数控制和离散时间状态过程来近似，这些过程由欧拉 - 丸山时间步进到 1/2 阶误差，这在我们的设置中是最佳的。