This article discusses the fixed-time synchronization (FTS) of hypercomplex neural networks (HCNNs) with mixed time-varying delays. Unlike finite-time synchronization (FNTS) based on initial conditions, the settling time of FTS can be adjusted to meet the needs. The state vector, weight matrices, activation functions, and input vectors of HCNNs are all hypercomplex numbers. The techniques used in complex-valued neural networks (CVNNs) and quaternion-valued neural networks (QVNNs) cannot be used directly with HCNNs because they do not work with eight or more dimensions. To begin with, the decomposition method is used to split the HCNNs into \((n+1)\) real-valued neural networks (RVNNs) applying distributive law to handle non-commutativity and non-associativity. A nonlinear controller is constructed to synchronize the master-response systems of the HCNNs. Lyapunov-based method is used to prove the stability of an error system. The FTS of mixed time-varying delayed HCNNs is achieved using a suitable lemma, Lipschitz condition, appropriate Lyapunov functional construction, and designing suitable controllers. Two different algebraic criteria for settling time have been achieved by employing two distinct lemmas. It is demonstrated that the settling time derived from Lemma 1 produces a more precise result than that obtained from Lemma 2. Three numerical examples for CVNNs, QVNNs, and octonions-valued neural networks (OVNNs) are provided to demonstrate the efficacy and effectiveness of the proposed theoretical results.

摘要

本文讨论具有混合时变延迟的超复杂神经网络 (HCNN) 的固定时间同步 (FTS)。与基于初始条件的有限时间同步（FNTS）不同，FTS 的稳定时间可以根据需要进行调整。HCNN 的状态向量、权重矩阵、激活函数和输入向量都是超复数。复值神经网络 (CVNN) 和四元数值神经网络 (QVNN) 中使用的技术不能直接与 HCNN 一起使用，因为它们不适用于八个或更多维度。首先，使用分解方法将 HCNN 拆分为\((n+1)\)实值神经网络（RVNN），应用分配律来处理非交换性和非关联性。构建非线性控制器来同步 HCNN 的主响应系统。基于李亚普诺夫的方法用于证明误差系统的稳定性。混合时变延迟 HCNN 的 FTS 是通过使用合适的引理、Lipschitz 条件、合适的 Lyapunov 函数构造以及设计合适的控制器来实现的。通过使用两个不同的引理，已经实现了两种不同的稳定时间代数标准。结果表明，从引理 1 得出的稳定时间比从引理 2 得出的稳定时间产生更精确的结果。提供了 CVNN、QVNN 和八元值神经网络 (OVNN) 的三个数值示例，以证明该方法的有效性和有效性。提出的理论结果。