Title: Mean-field limit for stochastic control problems under state constraint Show Chinese title

Title: 随机控制问题在状态约束下的平均场极限

Abstract: We study the convergence problem of mean-field control theory in the presence of state constraints and non-degenerate idiosyncratic noise. Our main result is the convergence of the value functions associated to stochastic control problems for many interacting particles subject to symmetric, almost-sure constraints toward the value function of a control problem of mean-field type, set on the space of probability measures. The key step of the proof is to show that admissible controls for the limit problem can be turned into admissible controls for the $N$-particle problem up to a correction which vanishes as the number of particles increases. The rest of the proof relies on compactness methods. We also provide optimality conditions for the mean-field problem and discuss the regularity of the optimal controls. Finally we present some applications and connections with large deviations for weakly interacting particle systems. Show Chinese abstract

Abstract: 我们研究在存在状态约束和非退化个性噪声的情况下平均场控制理论的收敛问题。 我们的主要结果是，对于受对称、几乎必然约束的多个相互作用粒子的随机控制问题，其价值函数收敛到一个平均场类型控制问题的价值函数，该问题定义在概率测度的空间上。 证明的关键步骤是证明可以将极限问题的可行控制转换为$N$粒子问题的可行控制，修正项随着粒子数的增加而消失。 证明的其余部分依赖于紧致性方法。 我们还提供了平均场问题的最优条件，并讨论了最优控制的正则性。 最后，我们展示了一些应用，并讨论了与弱相互作用粒子系统的大偏差之间的联系。