On the Stability and Synchronization of Distributed-Order Coupled Delayed Neural Networks: A Novel Halanay Inequality Technique

This paper investigates the stability and synchronization of distributed-order coupled delayed neural networks (DOCDNNs). First, an analytical solution to the distributed-order linear system is proved, thus resulting in an asymptotic stability criterion for distributed-order linear systems. This solution function is an extension of the Mittag–Leffler function. Then, a series of pivotal mathematical properties of the solution function are established, encompassing differential formula, monotonicity, weak additivity, and asymptotic property. It is further demonstrated that a novel distributed-order non-autonomous Halanay inequality can be derived from the unique properties of the solution function. Based on the proposed Halanay inequality technique, an asymptotic stability determination theorem for distributed-order nonlinear systems is derived. Moreover, the stability and synchronization of DOCNNs are analyzed using this theorem. The efficacy of the proposed method is substantiated by two numerical examples.