关于具有不可分离哈密顿量的平均场控制系统的策略迭代算法的研究
A Study on the Policy Iteration Method for Time-Dependent Mean Field Controls Systems with Non-Separable Hamiltonians
DOI: 10.12677/AAM.2023.125239, PDF,   
作者: 唐毅雄：中国地质大学(武汉)数学与物理学院，湖北 武汉
关键词: 平均场控制；微分博弈；最优控制；数值计算方法；Mean Field Type Control； Differential Games； Optimal Control； Numerical Methods
摘要: 平均场控制研究的是在大量智能体参与的系统中，由一个中心控制者计算系统的帕累托最优，且所有个体先验地使用完全相同的反馈控制。本文研究的对象是一类来自于平均场控制问题的倒向–正向偏微分方程组有限差分数值计算方法，使用的算法基于最优控制问题中的策略迭代方法。本文讨论了较短时间区域内算法的收敛性和收敛速度。本文通过一些例子验证了算法的有效性。本文通过数值试验讨论了平均场博弈纳什均衡和平均场控制帕累托最优之间的总体效益比较。
Abstract: Mean field type control studies the Pareto optimum computed by a central planner in a system with large number of agents, with each agent using a priori the same feedback control. This paper stud-ies the finite difference numerical approximation of a type of Mean field type control problem, using algorithms based on the policy iteration method from optimal control problems. Under the as-sumption of a short time horizon, convergence of the algorithm and the rate of convergence are discussed. Some numerical examples are supplemented to show the efficacy of the algorithms. The comparisons between Nash equilibrium and Pareto optimum, in terms of aggregate utility, are il-lustrated by numerical examples.
文章引用：唐毅雄. 关于具有不可分离哈密顿量的平均场控制系统的策略迭代算法的研究[J]. 应用数学进展, 2023, 12(5): 2364-2375. https://doi.org/10.12677/AAM.2023.125239

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