AI 中文总结
针对大规模机器人群体的平均场控制问题，建立基于优化的框架，将其提升到占用测度空间并采用弗兰克 - 沃尔夫算法求解，避免状态空间离散化，具有O(1/k)收敛速度，可扩展到三维环境，展现出实用性和可扩展性。
AI 中文摘要
大规模机器人群体推动了平均场控制（MFC）的应用。基于经典偏微分方程（PDE）的公式提供了一个有原则的框架，但在高维中计算上具有挑战性，而机器学习以近似和保证为代价实现了可扩展性。在这项工作中，我们建立了一个基于优化的框架，将MFC问题提升到占用测度空间，得到一个作为测度优化的凸松弛问题。使用测度空间中的弗兰克 - 沃尔夫（FW）算法解决所得问题，每次迭代简化为一个易于处理的最优控制问题。该方法保留了FW的O(1/k)收敛速度，避免了状态空间离散化，并自然地纳入了相互作用和安全约束。数值实验表明在二维中与解析和基于PDE的基线一致，且该方法可扩展到具有多个障碍物的三维环境，在标准工作站上几分钟内就能解决有十个障碍物的完整3D实例，凸显了所提框架的实用性和可扩展性。
英文摘要
Large-scale robotic swarms motivate the use of mean-field control (MFC). Classical partial differential equation (PDE)-based formulations provide a principled framework but can become computationally challenging in higher dimensions, whereas machine learning achieves scalability at the cost of approximation and guarantees. In this work, we establish an optimization-based framework that lifts the MFC problem into the space of occupation measures, resulting in a convex relaxation formulated as an optimization over measures. The resulting problem is solved using a Frank-Wolfe (FW) algorithm in the measure space, with each iteration reduced to a tractable optimal control problem. This approach retains the O(1/k) convergence rate of FW, avoids discretization of the state space, and naturally incorporates interaction and safety constraints. Numerical experiments demonstrate agreement with analytic and PDE-based baselines in two dimensions and show that the method scales to three-dimensional environments with multiple obstacles, where standard grid-based PDE solvers become impractical. A full 3D instance with ten obstacles is solved in minutes on a standard workstation, underscoring the practicality and scalability of the proposed framework.
Comments7 pages, 5 figures. Accepted to the 2026 American Control Conference (ACC)
