论文标题
有限状态N-机构和均值野外控制问题
Finite state N-agent and mean field control problems
论文作者
论文摘要
我们在有限状态空间，连续时间和有限的时间范围内检查平均场控制问题。我们将平均场控制问题的价值函数表征为单纯形中汉密尔顿 - 雅各比 - 贝尔曼方程的唯一粘度解。在没有任何凸度假设的情况下，我们利用这种表征来证明，作为$ n $增长的融合，是集中式$ n $ n $的最佳控制问题的价值函数，以限制限制平均字段控制问题价值函数，并具有$ 1/\ sqrt {n} $的订单的收敛速率。然后，假设凸度，我们表明限制值函数是平稳的，并建立了混乱的传播，即$ n $ - 代理最佳轨迹与独特的限制最佳轨迹的收敛，并具有显式速率。
We examine mean field control problems on a finite state space, in continuous time and over a finite time horizon. We characterize the value function of the mean field control problem as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation in the simplex. In absence of any convexity assumption, we exploit this characterization to prove convergence, as $N$ grows, of the value functions of the centralized $N$-agent optimal control problem to the limit mean field control problem value function, with a convergence rate of order $1/\sqrt{N}$. Then, assuming convexity, we show that the limit value function is smooth and establish propagation of chaos, i.e. convergence of the $N$-agent optimal trajectory to the unique limiting optimal trajectory, with an explicit rate.
