AI 中文总结
该研究针对局部由渐近稳定齐次向量场主导的系统，构造锚定保正神经网络族，证明其可半全局通用逼近最大李雅普诺夫函数，还给出候选函数认证条件并通过数值示例验证架构有效性。
AI 中文摘要
最大李雅普诺夫函数编码了渐近稳定平衡点的整个吸引域，但由于其在平衡点处的裕度消失，难以在神经逼近下保持严格递减。对于局部由渐近稳定齐次向量场主导的系统，我们构造了一个连续可微的最大目标函数和一个锚定的保正神经网络族。我们证明了半全局通用逼近：严格神经李雅普诺夫函数及其一阶导数可以在穷尽吸引域的嵌套不变子水平集上逼近该目标函数。我们还提供了可直接验证的条件，用于正式认证候选神经李雅普诺夫函数，并通过数值示例说明了所提出神经架构的有效性。
英文摘要
Maximal Lyapunov functions encode the entire domain of attraction of an asymptotically stable equilibrium, but preserving strict decrease under neural approximation is difficult because its margin vanishes at the equilibrium. For systems locally dominated by an asymptotically stable homogeneous vector field, we construct a continuously differentiable maximal target and an anchored, positivity-preserving neural family. We prove semiglobal universal approximation: strict neural Lyapunov functions and their first derivatives can approximate the target on nested invariant sublevel sets that exhaust the domain of attraction. We also provide directly verifiable conditions under which a candidate neural Lyapunov function can be formally certified, and illustrate the effectiveness of the proposed neural architecture through numerical examples.
