发表机构
Czech Technical University in Prague; CNRS; LAAS; Université de Toulouse（布拉格捷克理工大学; 法国国家科学研究中心；LAAS实验室；图卢兹大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了存在特定维度和次数的多项式向量场，其Lyapunov稳定性判定问题不可判定，从而证实了Arnold的猜想。
AI 中文摘要
我们证明存在整数$N$和奇数$D$，使得对于维度为$N$、次数为$D$的齐次多项式向量场$F$，其有理系数无法通过任何算法判定原点对于$\dot Y=F(Y)$是否Lyapunov稳定。这证明了V. I. Arnold的一个猜想，尽管所取的维度和次数较大且未优化。
英文摘要
We show that there are integers $N$ and odd $D$ such that no algorithm can decide, from the rational coefficients of a homogeneous polynomial vector field $F$ in dimension $N$ of degree $D$, whether the origin is Lyapunov stable for $\dot Y=F(Y)$. This proves a conjecture of V. I. Arnold. We provide a Lean formalization of the proof. The smallest dimension $N$ for which we were able to prove undecidability is $N = 5$. An analogous undecidability result holds for global asymptotic stability and global exponential stability for non-homogenous vector fields. For dimension $N=2$ and homogenous vector fields, we establish decidability for all commonly used stability notions, building heavily on existing results.
