AI 中文总结
该研究提出循环李雅普诺夫函数（RLF）概念，放松李雅普诺夫函数的不变性要求，开发GPU算法仅用轨迹数据即可验证系统稳定性，揭示了性能与成本间的固有权衡。
AI 中文摘要
李雅普诺夫直接法是稳定性与控制领域的基石，但该方法依赖于寻找李雅普诺夫函数，而这项任务需要创造力或大量计算。一个关键难点在于，该函数的每个子水平集必须是前向不变的，这将其几何结构与系统轨迹耦合在一起。我们通过用循环性替代不变性来放松这一要求：若从某集合出发的每条轨迹都能在有限时间内返回该集合，则称该集合是循环的。这引出了循环李雅普诺夫函数（Recurrent Lyapunov Function，RLF）的概念，其仅要求子水平集具备循环性即可。我们证明，在温和条件下，循环李雅普诺夫函数可保证稳定性，并引入更强的概念以实现渐近稳定性与指数稳定性。我们还给出了基于范数的逆定理：在相应的稳定性条件下，任意范数都是其实用版本的循环李雅普诺夫函数。随后，我们开发了基于GPU的算法，该算法仅通过轨迹数据即可验证（实用）稳定性，无需预先构造李雅普诺夫函数。验证至ε邻域内的稳定性仅需O(log(1/ε))次轨迹评估，且常数会随着经验证的衰减率接近真实衰减率而增长，这揭示了固有的性能-成本权衡关系。
英文摘要
Lyapunov's direct method is a cornerstone of stability and control, but it hinges on finding a Lyapunov function, a task demanding ingenuity or computation. A key difficulty is that every sub-level set of the function must be forward invariant, coupling its geometry to the system's trajectories. We relax this by replacing invariance with recurrence: a set is recurrent if every trajectory starting in it returns within a finite time. This yields the notion of a Recurrent Lyapunov Function (RLF), whose sub-level sets need only be recurrent. We show that, under mild conditions, RLFs guarantee stability, and we introduce stronger notions yielding asymptotic and exponential stability. We also give norm-based converse theorems: under the corresponding stability conditions, any norm is an RLF for their practical versions. We then develop GPU-based algorithms that certify (practical) stability from trajectory data alone, without a Lyapunov function. Certifying stability up to an $\varepsilon$-neighborhood needs only $O(\log(1/\varepsilon))$ trajectory evaluations, with constants growing as the certified decay rate nears the true one, exposing an intrinsic performance-cost trade-off.
Comments16 pages
