黎曼密度驱动最优控制：弯曲流形上二阶多智能体系统的切空间LQR
Riemannian Density-Driven Optimal Control: Tangent-Space LQR for Second-Order Multi-Agent Systems on Curved Manifolds
- Texas Tech University（德州理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出黎曼密度驱动最优控制（R-D2OC），通过切空间LQR和并行传输，将D2OC扩展至弯曲流形上的二阶多智能体系统，并给出近似界与下降保证。
AI中文摘要：
密度驱动最优控制（D2OC）为将多智能体系统引导至指定空间分布提供了有效框架。然而，现有的D2OC公式主要针对欧几里得域开发，并未直接考虑流形的内在几何结构。本文将D2OC扩展至在黎曼流形上演化的二阶多智能体系统。所提出的黎曼D2OC（R-D2OC）通过对数映射在每个智能体的切空间中构建局部分布目标，并将其加权中心用作有限时域LQR的参考。所得控制通过内在二阶动力学和并行传输在流形上以滚动时域方案执行。我们建立了局部曲率相关界，以量化切空间简化引入的近似，并刻画由此产生的目标偏差。此外，我们推导了一个条件性离散下降结果，表明当局部速度对齐条件满足时，闭环目标会减小。在三维椭球流形上的数值模拟展示了分布级控制，并实证支持所提出的近似和下降结果。
英文摘要：
Density-Driven Optimal Control (D2OC) provides an effective framework for steering multi-agent systems toward prescribed spatial distributions. However, existing D2OC formulations are primarily developed for Euclidean domains and do not directly account for intrinsic manifold geometry. This paper extends D2OC to second-order multi-agent systems evolving on Riemannian manifolds. The proposed Riemannian D2OC (R-D2OC) constructs a local distribution objective in the tangent space of each agent through logarithmic maps and uses its weighted center as the reference for a finite-horizon LQR. The resulting control is executed on the manifold through intrinsic second-order dynamics and parallel transport within a receding-horizon scheme. We establish local curvature-dependent bounds that quantify the approximation introduced by the tangent-space reduction and characterize the resulting target bias. Furthermore, we derive a conditional discrete-descent result showing that the closed-loop objective decreases when a local velocity-alignment condition is satisfied. Numerical simulations on a 3D ellipsoidal manifold demonstrate distribution-level control and empirically support the proposed approximation and descent results.
