Title: Distributed Lyapunov Functions for Nonlinear Networks Show Chinese title

Title: 分布式李雅普诺夫函数用于非线性网络

Abstract: Nonlinear networks are often multistable, exhibiting coexisting stable states with competing regions of attraction (ROAs). As a result, ROAs can have complex "tentacle-like" morphologies that are challenging to characterize analytically or computationally. In addition, the high dimensionality of the state space prohibits the automated construction of Lyapunov functions using state-of-the-art optimization methods, such as sum-of-squares (SOS) programming. In this letter, we propose a distributed approach for the construction of Lyapunov functions based solely on local information. To this end, we establish an augmented comparison lemma that characterizes the existence conditions of partial Lyapunov functions, while also accounting for residual effects caused by the associated dimensionality reduction. These theoretical results allow us to formulate an SOS optimization that iteratively constructs such partial functions, whose aggregation forms a composite Lyapunov function. The resulting composite function provides accurate convex approximations of both the volumes and shapes of the ROAs. We validate our method on networks of van der Pol and Ising oscillators, demonstrating its effectiveness in characterizing high-dimensional systems with non-convex ROAs. Show Chinese abstract

Abstract: 非线性网络通常具有多稳定性，表现出共存的稳定状态以及竞争的吸引区域（ROAs）。 因此，ROAs可能具有复杂的“触须状”形态，难以通过分析或计算方法进行表征。 此外，状态空间的高维性使得无法使用最先进的优化方法（如平方和（SOS）编程）自动构造李雅普诺夫函数。 在本文中，我们提出了一种基于局部信息的李雅普诺夫函数构造的分布式方法。 为此，我们建立了一个扩展的比较引理，该引理描述了部分李雅普诺夫函数的存在条件，同时考虑了由相关降维引起的残余效应。 这些理论结果使我们能够制定一个SOS优化问题，该问题迭代地构造此类部分函数，其聚合形成一个复合李雅普诺夫函数。 所得的复合函数能够准确地近似ROAs的体积和形状。 我们在范德波尔和伊辛振荡器网络上验证了我们的方法，证明了其在表征具有非凸ROAs的高维系统方面的有效性。