This paper considers a wide class of smooth continuous dynamic nonlinear systems (control objects) with a measurable vector of state. The problem is to find a special function (Lyapunov function), which in the framework of the second Lyapunov method guarantees asymptotic stability for the above described class of nonlinear systems. It is well known that the search for a Lyapunov function is the "cornerstone" of mathematical stability theory. Methods for selecting or finding the Lyapunov function to analyze the stability of closed linear stationary systems, as well as for nonlinear objects with explicit linear dynamic and nonlinear static parts, have been well studied (see works by Lurie, Yakubovich, Popov, and many others). However, universal approaches to the search for the Lyapunov function for a more general class of nonlinear systems have not yet been identified. There is a large variety of methods for finding the Lyapunov function for nonlinear systems, but they all operate within the constraints imposed on the structure of the control object. In this paper we propose another approach, which allows to give specialists in the field of automatic control theory a new tool/mechanism of Lyapunov function search for stability analysis of smooth continuous dynamic nonlinear systems with measurable state vector. The essence of proposed approach consists in representation of some function through sum of nonlinear terms, which are elements of object's state vector, multiplied by unknown coefficients, raised to positive degrees. Then the unknown coefficients are selected using genetic algorithm, which should provide the function with all necessary conditions for Lyapunov function (in the framework of the second Lyapunov method).

翻译：本文研究了一类具有可测状态向量的光滑连续动态非线性系统（控制对象）。问题在于寻找一种特殊函数（李雅普诺夫函数），在第二李雅普诺夫方法框架下，该函数能保证上述非线性系统的渐近稳定性。众所周知，李雅普诺夫函数的搜索是数学稳定性理论的"基石"。对于封闭线性定常系统以及具有显式线性动态部分与非线性静态部分的非线性对象，已有成熟的李雅普诺夫函数选择或搜索方法（参见Lurie、Yakubovich、Popov等学者的工作）。然而，针对更一般非线性系统的李雅普诺夫函数通用搜索方法尚未明确。现有大量非线性系统李雅普诺夫函数搜索方法均受限于控制对象的结构约束。本文提出一种新方法，为自动控制理论领域的研究人员提供一种全新的李雅普诺夫函数搜索工具/机制，用于分析具有可测状态向量的光滑连续动态非线性系统的稳定性。该方法的核心思想是将某函数表示为对象状态向量元素（乘以未知系数并赋予正指数）的非线性项之和。随后通过遗传算法选择未知系数，确保该函数满足李雅普诺夫函数的所有必要条件（基于第二李雅普诺夫方法框架）。