论文标题
控制希尔伯特空间并应用于某些平均现场类型控制问题
Control on Hilbert Spaces and Application to Some Mean Field Type Control Problems
论文作者
论文摘要
我们提出了一种新的方法，用于研究Bellman方程的经典解决方案和平均场类型控制问题的主方程，并使用P.-L引入的“提升”想法的新形式。狮子。与其研究使用分析技术的汉密尔顿 - 雅各布（Hamilton-Jacobi/Fokker-Planck PDES的常用系统），不如研究在特殊构建的希尔伯特空间上的随机控制问题，这让人联想到最佳运输中瓦斯氏菌空间上的切线空间。在这个希尔伯特空间上，我们可以使用经典的控制理论技术，尽管它是无限的维度。我们构建的结果是，平均场类型控制问题似乎是一种特殊情况。因此，我们保留了实现程序的优势，同时消除了一些困难。我们的方法扩展了两位合着者的先前工作，该作品处理了希尔伯特空间可能是通用的确定性控制问题。
We propose a new approach to studying classical solutions of the Bellman equation and Master equation for mean field type control problems, using a novel form of the "lifting" idea introduced by P.-L. Lions. Rather than studying the usual system of Hamilton-Jacobi/Fokker-Planck PDEs using analytic techniques, we instead study a stochastic control problem on a specially constructed Hilbert space, which is reminiscent of a tangent space on the Wasserstein space in optimal transport. On this Hilbert space we can use classical control theory techniques, despite the fact that it is infinite dimensional. A consequence of our construction is that the mean field type control problem appears as a special case. Thus we preserve the advantages of the lifiting procedure, while removing some of the difficulties. Our approach extends previous work by two of the coauthors, which dealt with a deterministic control problem for which the Hilbert space could be generic.
