解决高维控制问题中维度降解导致的精度漏洞，提升计算效率与准确性。
Threshold Strategy for Leaking Corner-Free Hamilton-Jacobi Reachability with Decomposed Computations
- 提出基于值函数的泄漏角问题定义与发生必要条件
- 设计局部更新方法，在不牺牲效率前提下修正错误值函数
- 适用于多种降维技术，尤其适合复杂系统分解场景
哈密顿-雅可比（HJ）可达性广泛用于计算满足特定控制目标的状态值函数。然而，由于维数灾难，其在高维问题中变得难以处理。维度降低方法对缓解这一挑战至关重要，但可能引发“泄漏角问题”，导致结果不准确。本文从值函数角度定义了“泄漏角问题”，并提出了其发生的必要条件。基于这些理论成果，我们提出一种新的局部更新方法，可在保持维度降低方法计算效率的同时，有效修正不准确的值函数。通过数值仿真验证了该方法的有效性。尽管我们在自包含子系统分解（SCSD）框架下验证了该方法，但其适用范围可扩展至其他引入“泄漏角”的维度降低技术。
原文摘要 · Abstract (English)
Hamilton-Jacobi (HJ) Reachability is widely used to compute value functions for states satisfying specific control objectives. However, it becomes intractable for high-dimensional problems due to the curse of dimensionality. Dimensionality reduction approaches are essential for mitigating this challenge, whereas they could introduce the ``leaking corner issue", leading to inaccuracies in the results. In this paper, we define the ``leaking corner issue" in terms of value functions, propose and prove a necessary condition for its occurrence. We then use these theoretical contributions to introduce a new local updating method that efficiently corrects inaccurate value functions while maintaining the computational efficiency of the dimensionality reduction approaches. We demonstrate the effectiveness of our method through numerical simulations. Although we validate our method with the self-contained subsystem decomposition (SCSD), our approach is applicable to other dimensionality reduction techniques that introduce the ``leaking corners".
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