扩展与统一哈密顿-雅可比可达性分析的基础任务
Extending and Unifying the Fundamental Tasks of Hamilton-Jacobi Reachability Analysis
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中文总结 AI 辅助
本研究提出广义可达-回避（GRA）任务，扩展并统一了哈密顿-雅可比可达性（HJR）的基础任务，可求解复合任务的值函数，其结果在HJR框架中具有理论与实用价值。
中文摘要 AI 辅助
在本研究中，我们提出了广义可达-回避（GRA）任务，该任务既扩展又统一了哈密顿-雅可比可达性（HJR）的经典任务。我们证明，GRA不仅是该类基础任务中的通用原语，还严格扩展了可通过HJR求解的基础任务。此外，GRA公式允许通过将复合任务的值函数分解为GRA任务的值函数，来计算某些复合任务的值函数，包括来自时间时序逻辑的复合任务。我们还从偏微分方程（PDE）视角证明，GRA也是值得考虑的自然原语，因为它可用于表示HJR对应的哈密顿-雅可比偏微分方程（HJ-PDE）的所有足够正则的解。总体而言，本研究的结果表明，该任务在日益重要的HJR框架中具有理论和实用价值。
英文摘要
In this work, we introduce the generalized reach-avoid (GRA) task, which both extends and unifies the canonical tasks of Hamilton-Jacobi Reachability (HJR). We show that the GRA not only serves as a common primitive in this class of fundamental tasks, but also strictly extends the fundamental tasks that can be solved with HJR. Moreover, the GRA formulation enables one to compute the value functions of certain composite tasks, including ones from timed temporal logic, by decomposing the value function of the composite task into value functions of GRA tasks. We additionally show that the GRA is also a natural primitive to consider from a PDE perspective, as it can be used to represent all sufficiently regular solutions of the HJ-PDE that is canonical to HJR. Collectively, the results in this work show the theoretical and practical utility of this task within the increasingly important framework of HJR.
