Abstract

This paper concerns a class of optimal control problems, where a central planner aims to control a multi-agent system in \({\mathbb {R}}^d\) in order to minimize a certain cost of Bolza type. At every time and for each agent, the set of admissible velocities, describing his/her underlying microscopic dynamics, depends both on his/her position, and on the configuration of all the other agents at the same time. So the problem is naturally stated in the space of probability measures on \({\mathbb {R}}^d\) equipped with the Wasserstein distance. The main result of the paper gives a new characterization of the value function as the unique viscosity solution of a first order partial differential equation. We introduce and discuss several equivalent formulations of the concept of viscosity solutions in the Wasserstein spaces suitable for obtaining a comparison principle of the Hamilton Jacobi Bellman equation associated with the above control problem.

中文翻译：

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Wasserstein空间中多主体系统的最优控制

摘要

本文涉及一类最优控制问题，其中中央计划者旨在控制\（{{mathbb {R}} ^ d \）中的多智能体系统，以最大程度地降低一定的Bolza类型成本。在每个时间，对于每个代理，描述他/她潜在的微观动力学的一组可允许的速度取决于他/她的位置以及同时所有其他代理的配置。因此，问题自然存在于\（{\ mathbb {R}} ^ d \）的概率测度空间中配备了Wasserstein距离。本文的主要结果给出了作为一阶偏微分方程唯一粘度解的值函数的新表征。我们介绍和讨论Wasserstein空间中粘度解的概念的几种等效公式，这些公式适合于获得与上述控制问题有关的Hamilton Jacobi Bellman方程的比较原理。