提出概率密度最优控制的普适理论框架，解决多智能体大规模协同问题。
Maximum Principle of Optimal Probability Density Control
- 基于测度空间建立概率分布控制的极大值原理
- 推导出定义在概率分布空间上的哈密顿-雅可比-贝尔曼方程
- 用神经网络实现高维场景下可扩展的数值求解
我们构建了一个适用于标准测度空间上最优概率密度控制的一般理论框架，旨在应对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建立了极大值原理（MP），并推导了定义在概率分布空间上的哈密顿-雅可比-贝尔曼方程。两项成果均以简洁形式呈现，并经严格的数学分析验证，支持高效数值处理。基于所提极大值原理，我们设计了一种可扩展的数值算法，利用深度神经网络处理高维情形。通过涉及领域障碍与智能体间相互作用的多个多智能体控制实例，验证了该方法的有效性。
原文摘要 · Abstract (English)
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.
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