揭示轨迹优化中80%特征值集中在[-2,4]区间，提升求解效率。
Hessians in Birkhoff-Theoretic Trajectory Optimization
- 基于Birkhoff理论推导轨迹优化的海森矩阵结构
- 80%特征值位于[-2,4]窄区间，具高度聚集性
- 为百万点轨迹优化提供理论基础，适合控制与优化研究者
本文推导了与Birkhoff理论轨迹优化相关的一系列海森矩阵。根据本文证明的定理，所有Birkhoff离散化的最优控制问题中，约80%的特征值均落入[-2, 4]的狭窄区间。同时初步分析了计算复杂度，并就求解百万点轨迹优化这一重大挑战进行了进一步讨论。
原文摘要 · Abstract (English)
This paper derives various Hessians associated with Birkhoff-theoretic methods for trajectory optimization. According to a theorem proved in this paper, approximately 80% of the eigenvalues are contained in the narrow interval [-2, 4] for all Birkhoff-discretized optimal control problems. A preliminary analysis of computational complexity is also presented with further discussions on the grand challenge of solving a million point trajectory optimization problem.
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