具有凸状态空间的基于智能体模型的Pontryagin最大值原理
A Pontryagin Maximum Principle for agent-based models with convex state space
AI总结：
本文针对状态空间为Banach空间中可能无限维凸子集的基于智能体系统及其平均场对应，通过推广针状变分和线性化过程，推导出一阶最优性条件，即Pontryagin型最大值原理。
摘要：
我们推导了一类基于智能体的系统及其平均场对应的一阶最优性条件。我们分析的一个相关难点在于，状态方程是在Banach空间中可能为无限维的凸子集上表述的。这是多群体动力学中许多问题的典型特征，其中概率测度的凸集可以表示群体、影响程度或与每个智能体相关的策略。由于缺乏线性结构和局部紧致性，用于推导Pontryagin型条件的常用工具（针状变分和线性化过程）必须推广到当前设定。这是通过考虑适当的微分概念并仔细检查底层泛函结构来实现的。
英文摘要：
We derive a first order optimality condition for a class of agent-based systems, as well as for their mean-field counterpart. A relevant difficulty of our analysis is that the state equation is formulated on possibly infinite-dimensional convex subsets of Banach spaces. This is a typical feature of many problems in multi-population dynamics, where a convex set of probability measures may account for the population, the degree of influence or the strategy attached to each agent. Due to the lack of a linear structure and of local compactness, the usual tools of needle variations and linearisation procedures used to derive Pontryagin type conditions have to be generalised to the setting at hand. This is done by considering suitable notions of differentials and by a careful inspection of the underlying functional structures.
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