For linear time-invariant time delay systems, the so-called Lyapunov–Krasovskii functionals of complete type (Kharitonov and Zhabko, 2003) are known to be effective in the stability analysis and a number of applications. More precisely, there exist the necessary and sufficient asymptotic stability and instability conditions expressed in terms of these functionals. The case excluded from consideration in the theory (since the functionals either do not exist or are not uniquely defined) is violation of the Lyapunov condition, i.e. the case of systems with the eigenvalues placed symmetrically with respect to the origin of the complex plane. In this paper, an analogue of this theory for a class of nonlinear time delay systems with homogeneous right-hand sides of degree greater than one and a constant delay is developed. An explicit expression for the Lyapunov–Krasovskii functionals as well as necessary and sufficient conditions for the asymptotic stability and instability of the trivial solution based on these functionals are given. An important assumption, which constitutes an analogue of the Lyapunov condition for linear systems, is existence of the Lyapunov function for a delay free system, obtained from the original one setting the delay equal to zero. Furthermore, this Lyapunov function is a key element in the construction of the functional. The functionals are applied to estimating the region of attraction.

对于线性时不变时滞系统，已知完整类型的所谓Lyapunov–Krasovskii功能（Kharitonov和Zhabko，2003年）在稳定性分析和许多应用中都是有效的。更准确地说，存在以这些功能表示的必要和充分的渐近稳定性和不稳定性条件。理论上没有考虑的情况（因为功能不存在或未唯一定义）违反了Lyapunov条件，即特征值相对于复杂平面的原点对称放置的系统的情况。在本文中，针对一类非线性均质系统，其右手边的均方度大于1且具有恒定的时滞，对该理论进行了模拟。给出了Lyapunov–Krasovskii泛函的显式表达式，并给出了基于这些泛函的平凡解的渐近稳定性和不稳定性的充要条件。构成线性系统Lyapunov条件的一个重要假设，是存在一个无延迟系统的Lyapunov函数，该函数是从将延迟设置为零的原始函数获得的。此外，该李雅普诺夫功能是功能构建中的关键要素。该功能用于估计吸引力区域。它构成了线性系统的Lyapunov条件的一个类似物，它是无延迟系统的Lyapunov函数的存在，它是从将延迟设置为零的原始值获得的。此外，该李雅普诺夫功能是功能构建中的关键要素。该功能用于估计吸引力区域。它构成了线性系统的Lyapunov条件的类似物，是无延迟系统的Lyapunov函数的存在，该函数是从将延迟设置为零的原始值获得的。此外，该李雅普诺夫功能是功能构建中的关键要素。该功能用于估计吸引力区域。