We develop a general theoretical framework for optimal probability density control on standard measure spaces, aimed at addressing large-scale multi-agent control problems. In particular, we establish a maximum principle (MP) for control problems posed on infinite-dimensional spaces of probability distributions and control vector fields. We further derive the Hamilton--Jacobi--Bellman equation for the associated value functional defined on the space of probability distributions. Both results are presented in a concise form and supported by rigorous mathematical analysis, enabling efficient numerical treatment of these problems. Building on the proposed MP, we introduce a scalable numerical algorithm that leverages deep neural networks to handle high-dimensional settings. The effectiveness of the approach is demonstrated through several multi-agent control examples involving domain obstacles and inter-agent interactions.

翻译：我们针对标准测度空间上的最优概率密度控制问题，建立了一个通用的理论框架，旨在解决大规模多智能体控制问题。特别地，我们在概率分布与控制向量场构成的无限维空间上，为控制问题建立了最大值原理。我们进一步推导了定义在概率分布空间上的值函数所满足的Hamilton--Jacobi--Bellman方程。这两项结果均以简洁形式呈现，并辅以严格的数学分析支撑，从而能够对这些问题进行高效的数值处理。基于所提出的最大值原理，我们引入了一种可扩展的数值算法，该算法利用深度神经网络处理高维场景。通过多个涉及区域障碍与智能体间相互作用的多智能体控制实例，验证了该方法的有效性。