扩展平均场控制的最优正则性及其分段常数逼近
Optimal regularity of extended mean field controls and their piecewise constant approximation
摘要：
我们考虑McKean-Vlasov动力学的控制，其系数在状态和控制中具有平均场相互作用。证明了对于一类线性凸平均场控制问题，唯一的最优开环控制在时间上具有最优1/2-Hölder正则性.因此，我们证明了值函数可以近似为一个分段常数控制和离散时间状态过程所产生的Euler-Maruyama时间步进，高达1/2阶误差，和最优控制可以近似为1/4阶误差。这些结果是新颖的，即使在没有平均场相互作用的情况下。
英文摘要：
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-Hölder 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.
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