Lyapunov函数的构造及应用

The construction and application of Lyapunov function

微分方程解的稳定性研究中最为常用的是李雅普诺夫第二法(直接法),利用这种方法研究系统的解的稳定性,其关键就是构造李雅普诺夫函数,即V函数.本文目的在于分析、总结系统稳定性的李雅普诺夫第二法的相关理论,以及如何借助李雅普诺夫函数来判断系统的稳定性.在Lotka-Voltera模型和流行病模型(SIR模型和SI模型)中,通过构造李雅普诺夫函数(一个与ln有关的函数),并借助李雅普诺夫函数及导数的符号特征,直接判断系统模型在平衡状态下的稳定性.

The Lyapunov second method is the most common in the reaearch of the stability of differential equation solutions,by using this method,the key point is to construct Lyapunov Function,namely,V Function.This paper aims to analyze and summarize the relative theories about the Lyapunov Second Methods of systematic stability and how to determine the stability of system with Lyapunov Function.In the Lotka-Voltera Models and the Epidemiological Models including SIR and SI models,we determine directly the stability under equilibrium state through constructing the Lyapunov Function（a function related to ln）,as well as applying to Lyapunov Function and symbol characteristic of derivative.