In this work, we mainly focus on uncertain delayed neural network system with inertial term. Here, the existence, uniqueness, and exponential stability of inertial neural networks are derived without shifting the second order differential system into first order through substituting variables. Initially, we construct a proper Lyapunov–Krasovskii functional to investigate the stability of novel uncertain delayed inertial neural networks, which is different from the classical Lyapunov functional approach. By utilizing the Kirchhoff’s matrix tree theorem, Cauchy–Schwartz inequality, homeomorphism theorem, and some inequality techniques, the necessary and sufficient conditions are derived for the designed framework. Subsequently, to exhibit the strength of this outcome, we framed a quantitative example.

中文翻译：

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具有惯性项的不确定时滞神经网络的存在性，唯一性和指数稳定性：非约简案例

在这项工作中，我们主要关注具有惯性项的不确定时滞神经网络系统。在此，无需替换变量即可导出惯性神经网络的存在性，唯一性和指数稳定性，而无需将二阶微分系统转换为一阶微分系统。最初，我们构造一个适当的Lyapunov–Krasovskii函数来研究新颖的不确定延迟惯性神经网络的稳定性，这与经典的Lyapunov函数方法不同。通过利用基尔霍夫矩阵树定理，柯西-施瓦兹不等式，同胚定理和一些不等式技术，得出了设计框架的充要条件。随后，为了展示这一结果的优势，我们构筑了一个定量的例子。