分数阶中立型四元数值神经网络的同步 Synchronization of Fractional Order Neutral Quaternion Numerical Neural Networks

doi:10.12677/AAM.2023.129378

Advances in Applied Mathematics
Vol. 12 No. 09 ( 2023 ), Article ID: 71887 , 9 pages
10.12677/AAM.2023.129378

分数阶中立型四元数值神经网络的同步

王丽^*，赵晴晴

●How to Cite this Article

云南财经大学统计与数学学院，云南昆明

收稿日期：2023年8月4日；录用日期：2023年9月1日；发布日期：2023年9月6日

摘要

本文研究了具有分数阶中立型四元数值神经网络的同步问题。首先构造了李雅普诺夫函数，其次根据利普希茨条件和不等式原理也称为李雅普诺夫直接法导出了同步的条件。最后给出了一个数值例子，验证了其满足定理所需的条件。

关键词

分数阶，四元数值，中立型神经网络

Synchronization of Fractional Order Neutral Quaternion Numerical Neural Networks

Li Wang^*, Qingqing Zhao

School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming Yunnan

Received: Aug. 4^th, 2023; accepted: Sep. 1^st, 2023; published: Sep. 6^th, 2023

ABSTRACT

This article investigates the synchronization problem of a fractional order neutral type quaternion numerical neural network. Firstly, the Lyapunov function was constructed, and secondly, the synchronization conditions were derived based on the Lipschitz condition and inequality principle, also known as the Lyapunov direct method. Finally, a numerical example was provided to verify the conditions required for the theorem to be satisfied.

Keywords:Fractional Order, Quaternion Numerical, Neutral Neural Networks

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 引言

众所周知，分数阶微积分具有记忆和遗传特性，而神经网络是模仿生物神经元信号之间的相互传递，将二者结合起来的动力学性质被广泛研究，如文献 [1] 利用非光滑分析、微分包含理论和分数阶微积分的性质，研究了一类具有时滞的分数阶神经网络的全局耗散性；文献 [2] 根据压缩映射原理讨论了一类分数阶非自治神经网络系统的渐近周期解的存在唯一性；文献 [3] 考虑一类具有时变时滞的脉冲Caputo分数阶细胞神经网络，应用分数Lyapunov方法和Mittag-Leffler函数，给出了神经网络平衡点全局渐近稳定的充分条件。文献 [4] 利用符号函数、Banach不动点定理和两类激活函数，研究了具有线性脉冲和固定时滞的分数阶复值神经网络的一致稳定性。

然而神经网络中的时间延迟是不可避免的，经发现一些系统的动力学特性依赖于状态导数的延迟，这被称为中立型系统。研究者们将中立型系统和神经网络相结合，并涌现出了丰富的研究成果，如文献 [5] 利用矩阵不等式的方法和随机分析方法，讨论了具有马尔可夫切换参数的中立型神经网络的同步问题。文献 [6] 利用指数二分法理论和李雅普诺夫法，讨论了中立型双向联想记忆神经网络概周期解的存在性和稳定性。文献 [7] 根据线性矩阵不等式给出了中立型神经网络全局鲁棒耗散的充分条件，进一步研究了具有时滞的中立型双向联想记忆神经网络的概周期解。在文献 [8] 研究了一类具有常时滞的中立型神经网络的稳定性问题，通过构造适当的李雅普诺夫泛函，给出了一类中立型神经系统平衡点全局稳定的新的充分条件。文献 [9] 利用一类新的李雅普诺夫–克拉索夫斯基泛函，和线性矩阵不等式，研究了一类中立型神经网络的渐近稳定性。

到目前为止，涉及多维数据的实际应用中，实值神经网络或复值神经网络的神经元不能很好地处理这些数据。而四元数拥有一个实部和三个虚部，它可以处理多维数据，因此在许多领域受到了广泛的关注，比如文献 [10] ，利用图像压缩问题，四元数版本的反向传播算法在三维空间和颜色空间中实现了正确的几何变换，并且指出四元数神经网络在收敛速度方面也优于实值神经网络。文献 [11] 为了弥补被忽略的颜色通道之间结构的相关信息，利用四元数的特性提出了基于协同表示的分类和基于稀疏表示的分类。近年来，结合四元数和神经网络的优势，建立了四元数值神经网络，其性能相对较好。如文献 [12] 报道了一个延迟分数阶四元数值神经网络的稳定性和分岔的新结果，进一步证明了分岔振荡的振幅随着时间延迟的增大而增大，随着阶数的递增，分岔现象产生得更早。文献 [13] 基于非光滑分析、李雅普诺夫直接方法和比较理论，研究了具有时滞和不确定参数的分数阶忆阻四元数值神经网络的同步和稳定性问题。

最近，由于神经网络在安全通信、信息科学和生物等领域的广泛应用，其同步问题成为一个有趣的话题，因此关于神经网络同步问题的研究涌现出许多，如文献 [14] 通过应用分数微分包含理论、不等式分析技术和所提出的收敛性，用线性矩阵不等式的形式讨论了具有不连续激活和时滞的分数阶神经网络的全局Mittag-Leffler同步和有限时间内的同步问题。文献 [15] 利用复变函数理论，考虑了复值动态网络的同步问题。文献 [16] 在不将分数阶复值耦合神经网络分为两个实值系统的情况下，设计了两种可行的自适应协议：1) 基于全局信息的分数阶自适应策略；2) 基于连通支配集理论，提出了一种基于局部信息的分数阶自适应律，研究了具有时变耦合强度的分数阶复值神经网络的同步问题。

因此，本文将模拟人脑神经系统的神经网络，与具有记忆和遗传特性的分数阶微积分，与依赖于状态导数延迟的中立型系统和可以处理多维数据的四元数相结合，利用李亚普诺夫直接方法讨论分数阶中立型四元数神经网络的同步，为神经网络提供更丰富的信息处理能力和更高的性能。

基于上述，本文的主要目的是利用李亚普诺夫直接方法讨论分数阶中立型四元数神经网络的同步。主要优点如下所示：1) 由于四元数域中不存在符号函数，因此系统中采用了一种不借助符号函数的新控制器，并且此控制器是不同于大多数线性控制器的非线性控制器。2) 文章在处理四元数时，未将四元数神经网络分解成一个实部和三个虚部进行研究的。3) 本文中神经网络模型的系数是函数，是区别于大多数神经网络模型中的常系数。4) 基于以上三点的分数阶中立型四元数值神经网络的同步问题未被研究过。

2. 预备知识和符号说明

为了更方便地阅读这篇文章，对一些符号进行了说明，若无特别说明，其在本文中是通用的。R表示实数，Q表示四元数。四元数x的形式如下 $x = x^{R} + i x^{I} + j x^{J} + k x^{K} \in Q$ ，其中 $x^{R}, x^{I}, x^{J}, x^{K} \in R$ ， $i, j, k$ 是虚数单位，并遵循汉密尔顿规则： $i^{2} = j^{2} = k^{2} = - 1$ ， $i j = k = - j i$ ， $j k = i = - k j$ ， $k i = j = - i k$ ， $\bar{x} = x^{R} - i x^{I} - j x^{J} - k x^{K}$ 是x的共轭， $| x | = \sqrt{x \bar{x}} = \sqrt{\bar{x} x} = \sqrt{{(x^{R})}^{2} + {(x^{I})}^{2} + {(x^{J})}^{2} + {(x^{K})}^{2}}$ 表示x的模， $‖ a_{i} (t) ‖$ 表示 $a_{i} (t)$ 的范数，四元数的范数表示为： $‖ a_{i} (t) ‖ = {[a_{i}^{R} (t)]}^{2} + {[a_{i}^{I} (t)]}^{2} + {[a_{i}^{J} (t)]}^{2} + {[a_{i}^{K} (t)]}^{2}$ 。

在本节中，介绍Caputo分数阶导数的一些必要知识，分数阶中立型四元数神经网络模型和引理。

定义1 [17] [18] 函数 $x (t)$ 的 $α$ 阶Caputo分数阶微分定义如下：

$D^{α} x (t) = \frac{1}{Γ (1 - α)} \int_{0}^{t} {(t - τ)}^{- α} h^{'} (τ) d τ, t > 0$ (1)

定义2 [17] [18] 函数 $x (t)$ 的 $α$ 阶分数积分定义如下：

$D^{- α} x (t) = \frac{1}{Γ (α)} \int_{0}^{t} \frac{x (τ)}{{(t - τ)}^{1 - α}} d τ,$ (2)

当 $t > 0$ ， $Γ (α) = \int_{0}^{\infty} ω^{α - 1} e^{- ω} d ω$ ，其中 $α > 0$ 。

考虑的分数阶中立型四元数神经网络的模型

$D^{α} [x_{i} (t) + h x_{i} (t - τ)] = - a_{i} (t) x_{i} (t) + \sum_{j = 1}^{n} b_{i j} (t) f_{j} (x_{j} (t)) + I_{i} (t), i = 1, 2, \dots, n,$ (3)

其中 $0 < α < 1$ ， $x_{i} (t) \in Q$ 是第i个神经元在时间t的状态， $\forall t > 0$ ， $a_{i} (t) \in Q$ 是第i个神经元的自反馈权重， $b_{i j} (t)$ 是神经元互连值，h是神经元的中立型权重， $τ > 0$ 是时滞， $f_{j} (x_{j} (t)) : Q \to Q$ 表示激活函数， $\forall j = 1, 2, \dots, n$ ， $I_{i} (t) \in Q$ 是第i个神经元的外部输入，并且系统(3)的初始条件为 $x_{i} (0) = ξ_{i} \in Q$ 。

假设1 激活函数 $f_{i} (\cdot)$ 满足如下利普希茨条件：

$| f_{i} (p) - f_{i} (\tilde{p}) | \leq l_{i} | p - \tilde{p} |$ (4)

其中 $l_{i} > 0, \forall p, \tilde{p} \in Q, i = 1, 2, \dots, n$ 。

假设2 存在函数 $e_{i} (t)$ 有如下条件成立：

$\bar{[e_{i} (t) + h e_{i} (t - τ)]} [e_{i} (t) + h e_{i} (t - τ)] > 0$ 。

为了研究分数阶中立型四元数神经网络同步，将模型(3)视为驱动系统，相应的响应系统模型由下式给出

$D^{α} [{\tilde{x}}_{i} (t) + h {\tilde{x}}_{i} (t - τ)] = - a_{i} (t) {\tilde{x}}_{i} (t) + \sum_{j = 1}^{n} b_{i j} (t) f_{j} ({\tilde{x}}_{j} (t)) + I_{i} (t) + u_{i} (t), i = 1, 2, \dots, n,$ (5)

其中 ${\tilde{x}}_{i} (t)$ 表示系统(5)的状态向量， $u_{i} (t)$ 是外部控制器，系统(5)的初始条件是 ${\tilde{x}}_{i} (0) = {\tilde{ξ}}_{i} \in Q$ 。

接下来，同步误差表示如下：

$\begin{matrix} D^{α} [e_{i} (t) + h e_{i} (t - τ)] = D^{α} [{\tilde{x}}_{i} (t) + h {\tilde{x}}_{i} (t - τ) - x_{i} (t) - h x_{i} (t - τ)] \\ = - a_{i} (t) [{\tilde{x}}_{i} (t) - x_{i} (t)] + \sum_{j = 1}^{n} b_{i j} (t) [{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))] + u_{i} (t) \end{matrix}$ (6)

令 $e_{i} (t) = {\tilde{x}}_{i} (t) - x_{i} (t)$ ， $f_{j} (e_{i} (t)) = f_{j} ({\tilde{x}}_{i} (t)) - f_{j} (x_{i} (t))$ ，误差系统(6)的初始条件为 $e_{i} (0) = {\tilde{ξ}}_{i} - ξ_{i}$ 。

引理1 [19] 设 $x (t) \in Q$ 是一个可微函数，接下来有

$D^{α} \bar{x (t)} x (t) \leq D^{α} \bar{x (t)} x (t) + \bar{x (t)} D^{α} x (t), t \geq 0$ (7)

其中 $1 > α > 0$ 。

引理2 [20] 设 $ζ, η \in Q$ ，则有：

$\bar{ζ} η + \bar{η} ζ \leq \bar{ζ} ζ + \bar{η} η$ (8)

成立。

3. 主要结果

在本节中，基于一些控制方案，建立了分数阶中立型四元数神经网络的一些同步准则。

为了实现系统(3)和(5)的同步，控制器 $u_{i} (t) (i = 1, 2, \dots, n)$ 设置如下：

$u_{i} (t) = {\begin{cases} q_{i} (t) e_{i} (t) + \sum_{j = 1}^{n} d_{i j} (t) e_{i} (t - τ), e_{i} (t) \neq 0 \\ 0, e_{i} (t) = 0 \end{cases}$ (9)

其中 $q_{i} (t), d_{i j} (t) \in Q$ 。

注记1 与文献 [21] [22] 中的线性控制器不同，本文设置了一种非线性控制器，是一种新的控制器，为神经网络提供更丰富的信息处理能力和更高的性能。

定理1 根据假设1，存在常量 $δ_{i} > 0$ ，使得

${\begin{cases} X_{1} > X_{2} \\ X_{1} = \min \sum_{i = 1}^{n} \sum_{j = 1}^{n} δ_{i} {h a_{i} (t) + 2 a_{i}^{R} (t) - 2 q_{i}^{R} (t) - 1 - 2 l_{i}^{2} b_{i j}^{2} (t) - \bar{d_{i j} (t)} - h q_{i} (t)} > 0 \\ X_{2} = \max \sum_{i = 1}^{n} \sum_{j = 1}^{n} δ_{i} {2 d_{i}^{R} (t) + h^{2} + \bar{h q_{i} (t)} + d_{i j} (t) - \bar{h a_{i} (t)}} > 0 \end{cases}$ (10)

则系统(3)和(5)是同步的。

证选择如下V函数

$V (t) = \sum_{i = 1}^{n} δ_{i} \bar{[e_{i} (t) + h e_{i} (t - τ)]} [e_{i} (t) + h e_{i} (t - τ)]$ (11)

根据引理1的微分不等式可得(11)式结果如下：

$\begin{matrix} D^{α} V (t) \leq \sum_{i = 1}^{n} δ_{i} {D^{α} \bar{[e_{i} (t) + h e_{i} (t - τ)]} [e_{i} (t) + h e_{i} (t - τ)] \\ + \bar{[e_{i} (t) + h e_{i} (t - τ)]} D^{α} [e_{i} (t) + h e_{i} (t - τ)]} \\ = \sum_{i = 1}^{n} δ_{i} {\bar{- a_{i} (t)} \bar{[{\tilde{x}}_{i} (t) - x_{i} (t)]} + \sum_{j = 1}^{n} \bar{b_{i j} (t)} \bar{[{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))]} + \bar{u_{i} (t)}} [e_{i} (t) - h e_{i} (t - τ)] \\ + \sum_{i = 1}^{n} δ_{i} \bar{[e_{i} (t) + h e_{i} (t - τ)]} {- a_{i} (t) [{\tilde{x}}_{i} (t) - x_{i} (t)] + \sum_{j = 1}^{n} b_{i j} (t) [{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))] + u_{i} (t)} \end{matrix}$ (12)

将(9)式的控制器代入(12)式中得到结果如下：

$\begin{matrix} D^{α} V (t) \leq \sum_{i = 1}^{n} δ_{i} {\bar{- a_{i} (t)} \bar{[{\tilde{x}}_{i} (t) - x_{i} (t)]} + \sum_{j = 1}^{n} \bar{b_{i j} (t)} \bar{[{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))]} \\ + \bar{q_{i} (t) e_{i} (t)} + \sum_{j = 1}^{n} \bar{d_{i j} (t)} \bar{e_{i} (t - τ)}} [e_{i} (t) + h e_{i} (t - τ)] \\ + \sum_{i = 1}^{n} δ_{i} [e_{i} (t) + h e_{i} (t - τ)] {- a_{i} (t) [{\tilde{x}}_{i} (t) - x_{i} (t)] \begin{matrix} \end{matrix} \\ + \sum_{j = 1}^{n} b_{i j} (t) [{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))] + q_{i} (t) e_{i} (t) + \sum_{j = 1}^{n} d_{i j} (t) e_{i} (t - τ)} \end{matrix}$

$\begin{matrix} = \sum_{i = 1}^{n} δ_{i} {\bar{- a_{i} (t)} \bar{e_{i} (t)} + \sum_{j = 1}^{n} \bar{b_{i j} (t)} \bar{[{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))]} \\ + \bar{q_{i} (t) e_{i} (t)} + \sum_{j = 1}^{n} \bar{d_{i j} (t)} \bar{e_{i} (t - τ)}} [e_{i} (t) + h e_{i} (t - τ)] \\ + \sum_{i = 1}^{n} δ_{i} \bar{[e_{i} (t) + h e_{i} (t - τ)]} {- a_{i} (t) e_{i} (t) + \sum_{j = 1}^{n} b_{i j} (t) [{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))] \\ + q_{i} (t) e_{i} (t) + \sum_{j = 1}^{n} d_{i j} (t) e_{i} (t - τ)} \end{matrix}$ (13)

根据假设1利普希茨条件和引理2不等式原理，(13)式中部分结果可得如下

$\begin{array}{l} \sum_{i = 1}^{n} \sum_{j = 1}^{n} {\bar{b_{i j} (t)} \bar{[{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))]} e_{i} (t) + \bar{e_{i} (t)} b_{i j} (t) [{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))]} \\ \leq \sum_{i = 1}^{n} e_{i} (t) \bar{e_{i} (t)} + \sum_{i = 1}^{n} \sum_{j = 1}^{n} \bar{b_{i j} (t)} b_{i j} (t) \bar{[{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))]} [{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))] \\ \leq \sum_{i = 1}^{n} e_{i} (t) \bar{e_{i} (t)} + \sum_{i = 1}^{n} \sum_{j = 1}^{n} l_{i}^{2} \bar{b_{i j} (t)} b_{i j} (t) \bar{e_{i} (t)} e_{i} (t) \end{array}$ (14)

同理

$\begin{array}{l} \sum_{i = 1}^{n} \sum_{j = 1}^{n} {\bar{b_{i j} (t)} \bar{[{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))]} h e_{i} (t - τ) \bar{h e_{i} (t - τ)} b_{i j} (t) [{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))]} \\ \leq \sum_{i = 1}^{n} \sum_{j = 1}^{n} h e_{i} (t - τ) \bar{h e_{i} (t - τ)} + \sum_{i = 1}^{n} \sum_{j = 1}^{n} \bar{b_{i j} (t)} b_{i j} (t) \bar{[{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))]} [{\tilde{f}}_{j} (x_{j} (t)) - f_{j} (x_{j} (t))] \\ \leq \sum_{j = 1}^{n} h^{2} \bar{e_{i} (t - τ)} e_{i} (t - τ) + \sum_{i = 1}^{n} \sum_{j = 1}^{n} l_{i}^{2} \bar{b_{i j} (t)} b_{i j} (t) \bar{e_{i} (t)} e_{i} (t) \end{array}$ (15)

将(14)式和(15)式代入(13)式中，可得结果如下：

$\begin{matrix} D^{α} V (t) \leq \sum_{i = 1}^{n} δ_{i} {\bar{- a_{i} (t)} \bar{e_{i} (t)} [e_{i} (t) + h e_{i} (t - τ)] + \bar{[e_{i} (t) + h e_{i} (t - τ)]} [- a_{i} (t) e_{i} (t)] \begin{matrix} \end{matrix} \\ + [\bar{q_{i} (t) e_{i} (t)} + \sum_{j = 1}^{n} \bar{d_{i j} (t)} \bar{e_{i} (t - τ)}] [e_{i} (t) + h e_{i} (t - τ)] \\ + \bar{[e_{i} (t) + h e_{i} (t - τ)]} [q_{i} (t) e_{i} (t) + \sum_{j = 1}^{n} d_{i j} (t) e_{i} (t - τ)] \\ + \sum_{i = 1}^{n} e_{i} (t) \bar{e_{i} (t)} + \sum_{i = 1}^{n} \sum_{j = 1}^{n} l_{i}^{2} \bar{b_{i j} (t)} b_{i j} (t) \bar{e_{i} (t)} e_{i} (t) \\ + \sum_{j = 1}^{n} h^{2} \bar{e_{i} (t - τ)} e_{i} (t - τ) + \sum_{i = 1}^{n} \sum_{j = 1}^{n} l_{i}^{2} \bar{b_{i j} (t)} b_{i j} (t) \bar{e_{i} (t)} e_{i} (t)} \end{matrix}$

$\begin{matrix} = \sum_{i = 1}^{n} δ_{i} (\bar{- a_{i} (t)} - a_{i} (t) + \bar{q_{i} (t)} + q_{i} (t) + 1 + 2 \sum_{i = 1}^{n} \sum_{j = 1}^{n} l_{i}^{2} \bar{b_{i j} (t)} b_{i j} (t)) \bar{e_{i} (t)} e_{i} (t) \\ + \sum_{i = 1}^{n} δ_{i} (\sum_{j = 1}^{n} d_{i j} (t) + \sum_{j = 1}^{n} \bar{d_{i j} (t)} + \sum_{j = 1}^{n} h^{2}) \bar{e_{i} (t - τ)} e_{i} (t - τ) \\ + \sum_{i = 1}^{n} δ_{i} (h \times \bar{(- a_{i} (t))} + h \bar{q_{i} (t)} + \sum_{j = 1}^{n} d_{i j} (t)) \bar{e_{i} (t)} e_{i} (t - τ) \\ + \sum_{i = 1}^{n} δ_{i} (- h a_{i} (t) + \sum_{j = 1}^{n} \bar{d_{i j} (t)} + h q_{i} (t)) \bar{e_{i} (t - τ)} e_{i} (t) \end{matrix}$ (16)

由假设1和引理2，(16)式可得

$\begin{array}{l} \sum_{i = 1}^{n} δ_{i} (h \times \bar{(- a_{i} (t))} + h \bar{q_{i} (t)} + \sum_{j = 1}^{n} d_{i j} (t)) \bar{e_{i} (t)} e_{i} (t - τ) \\ + \sum_{i = 1}^{n} δ_{i} (- h a_{i} (t) + \sum_{j = 1}^{n} \bar{d_{i j} (t)} + h q_{i} (t)) \bar{e_{i} (t - τ)} e_{i} (t) \\ \leq \sum_{i = 1}^{n} δ_{i} (- h a_{i} (t) + \sum_{j = 1}^{n} \bar{d_{i j} (t)} + h q_{i} (t)) \bar{e_{i} (t)} e_{i} (t) \\ + \sum_{i = 1}^{n} δ_{i} (h \times \bar{(- a_{i} (t))} + h \bar{q_{i} (t)} + \sum_{j = 1}^{n} d_{i j} (t)) \bar{e_{i} (t - τ)} e_{i} (t - τ) \end{array}$ (17)

根据(17)式可知，(16)式可得如下结果：

$\begin{matrix} D^{α} V (t) \leq \sum_{i = 1}^{n} δ_{i} (\bar{- a_{i} (t)} - a_{i} (t) + \bar{q_{i} (t)} + q_{i} (t) + 1 + 2 \sum_{i = 1}^{n} \sum_{j = 1}^{n} l_{i}^{2} \bar{b_{i j} (t)} b_{i j} (t)) \bar{e_{i} (t)} e_{i} (t) \\ + \sum_{i = 1}^{n} δ_{i} (\sum_{j = 1}^{n} d_{i j} (t) + \sum_{j = 1}^{n} \bar{d_{i j} (t)} + \sum_{j = 1}^{n} h^{2}) \bar{e_{i} (t - τ)} e_{i} (t - τ) \\ + \sum_{i = 1}^{n} δ_{i} (- h a_{i} (t) + \sum_{j = 1}^{n} \bar{d_{i j} (t)} + h q_{i} (t)) \bar{e_{i} (t)} e_{i} (t) \\ + \sum_{i = 1}^{n} δ_{i} (h \times \bar{(- a_{i} (t))} + h \bar{q_{i} (t)} + \sum_{j = 1}^{n} d_{i j} (t)) \bar{e_{i} (t - τ)} e_{i} (t - τ) \end{matrix}$

$\begin{matrix} = \sum_{i = 1}^{n} δ_{i} (\bar{- a_{i} (t)} - a_{i} (t) + \bar{q_{i} (t)} + q_{i} (t) + 1 + 2 \sum_{i = 1}^{n} \sum_{j = n}^{n} l_{i}^{2} \bar{b_{i j} (t)} b_{i j} (t) - h a_{i} (t) \\ + \sum_{j = 1}^{n} \bar{d_{i j} (t)} + h q_{i} (t)) \bar{e_{i} (t)} e_{i} (t) + \sum_{i = 1}^{n} δ_{i} (\sum_{j = 1}^{n} d_{i j} (t) + \sum_{j = 1}^{n} \bar{d_{i j} (t)} + \sum_{j = 1}^{n} h^{2} \\ + h \times [\bar{- a_{i} (t)}] + h \bar{q_{i} (t)} + \sum_{j = 1}^{n} d_{i j} (t)) \bar{e_{i} (t - τ)} e_{i} (t - τ) \\ = \sum_{i = 1}^{n} \sum_{j = 1}^{n} δ_{i} (- 2 a_{i}^{R} (t) + 2 q_{i}^{R} (t) + 1 + 2 l_{i}^{2} b_{i j}^{2} (t) - h a_{i} (t) + \bar{d_{i j} (t)} + h q_{i} (t)) \bar{e_{i} (t)} e_{i} (t) \\ + \sum_{i = 1}^{n} \sum_{j = 1}^{n} δ_{i} (2 d_{i}^{R} (t) + h^{2} + h \times [\bar{- a_{i} (t)}] + h \bar{q_{i} (t)} + d_{i j} (t)) \bar{e_{i} (t - τ)} e_{i} (t - τ) \end{matrix}$

当 $\bar{e_{i} (t - τ)} e_{i} (t - τ) > \bar{e_{i} (t)} e_{i} (t)$ 时，有

$D^{α} V (t) \leq - (X_{1} - X_{2}) \bar{e_{i} (t)} e_{i} (t)$ ，

当 $\bar{e_{i} (t - τ)} e_{i} (t - τ) < \bar{e_{i} (t)} e_{i} (t)$ 时，有

$D^{α} V (t) \leq - (X_{1} - X_{2}) \bar{e_{i} (t - τ)} e_{i} (t - τ)$ 。

因此，根据定理1可知， $D^{α} V (t) \leq 0$ ，所以系统(3)和系统(5)是同步的。

注记2 根据文献 [21] [22] ，我们可以看出本文和这些文献在模型的设计上有些不同之处，文献中的模型系数都是常数，而本文中模型的系数是函数，使得神经网络的模型更加通用，这可以看作是本文的一个主要新颖之处。

注记3 根据文献 [23] [24] ，我们可以看出，这些研究者们是将四元数神经网络分解成一个实部和三个虚部进行研究的，本文在处理四元数时，未将四元数神经网络分解成一个实部和三个虚部进行研究的。利用四元数理论实现了分数阶中立型四元数神经网络的同步，并导出了一些简单的准则。该方法更加实用有效，减少了计算的繁琐性。

4. 数值例子

在本节中，给出一个数值例子对主要内容来进行说明。

考虑以下分数阶中立型四元数神经网络，它被视为驱动系统：

$D^{α} [x_{i} (t) + h x_{i} (t - τ)] = - a_{i} (t) x_{i} (t) + \sum_{j = 1}^{n} b_{i j} (t) f_{j} (x_{j} (t)) + I_{i} ( t )$

$i = 1, 2,$ (18)

当 $α = 0.95$ ， $x_{i} (t) = x_{1}^{R} (t) + i x_{2}^{I} (t) + j x_{3}^{J} (t) + k x_{4}^{K} (t)$ ，有 $x_{i}^{R} (t), x_{i}^{I} (t), x_{i}^{J} (t), x_{i}^{K} (t) \in R$ ， $f_{j} (x_{j} (t)) = \tanh (x_{i}^{R} (t)) + i \tanh (x_{i}^{I} (t)) + j \tanh (x_{i}^{J} (t)) + k \tanh (x_{i}^{K} (t))$ ， $I_{1} = I_{2} = 0$ ， $τ = 1$ ， $l_{1} = l_{2} = 1$ ， $h = 1$ 。

$A = {(a_{i} (t))}_{2 \times 2} = (\begin{matrix} 2 + 2 i + 2 j + 2 k & - 0.1 - 0.1 i - 0.1 j - 0.1 k \\ - 2 - 3 i - 2 j - 3 k & 1 + i + j + k \end{matrix}) (\begin{matrix} t & t \\ t & t \end{matrix})$ ，

$B = {(b_{i j} (t))}_{2 \times 2} = (\begin{matrix} - 1 - 1.5 i - 1 j - 1.5 k & - 0.1 - 0.1 i - 0.1 j - 0.1 k \\ - 0.3 - 0.2 i - 0.3 j - 0.2 k & 2 + 2.4 + 2.4 j + 2.4 k \end{matrix}) (\begin{matrix} t & t \\ t & t \end{matrix})$ ，

模型(18)的初始值为 $x_{1} = 1 + 2 i + j + 2 k$ ， $x_{2} = - 2 + 2 i + j + 3 k$ 。

相应系统如下：

$D^{α} [{\tilde{x}}_{i} (t) + h {\tilde{x}}_{i} (t - τ)] = - a_{i} (t) {\tilde{x}}_{i} (t) + \sum_{j = 1}^{n} b_{i j} (t) f_{j} ({\tilde{x}}_{j} (t)) + I_{i} (t) + u_{i} (t),$ (19)

其中 ${\tilde{x}}_{i} (t) = {\tilde{x}}_{1}^{R} (t) + i {\tilde{x}}_{2}^{I} (t) + j {\tilde{x}}_{3}^{J} (t) + k {\tilde{x}}_{4}^{K} (t)$ ， ${\tilde{x}}_{i}^{R} (t), {\tilde{x}}_{i}^{I} (t), {\tilde{x}}_{i}^{J} (t), {\tilde{x}}_{i}^{K} (t) \in R$ ， $u_{i} (t)$ 是控制器，模型(19)的参数与模型(18)的参数相似。模型(19)的初始值为 ${\tilde{x}}_{1} = 0.2 + 0.4 i + j + k$ ， ${\tilde{x}}_{2} = - 0.1 + 0.2 i + 0.1 j - 0.1 k$ 。

$Q = {(q_{i} (t))}_{2 \times 2} = (\begin{matrix} - 2 - 3 i - 2 j - 3 k & 3 + 2 i + 3 j + 2 k \\ 2 + j + j + k & 2 + 3 i + 2 j + 3 k \end{matrix}) (\begin{matrix} t & t \\ t & t \end{matrix})$ ，

$D = {(d_{i j} (t))}_{2 \times 2} = (\begin{matrix} - 3 - 2 i - j - k & - 1 - 2 i - 3 j - 4 k \\ - 1 - 2 i - 3 j - 4 k & 3 + 2 + j + k \end{matrix}) (\begin{matrix} t & t \\ t & t \end{matrix})$ ，

经计算可知，以上数值例子满足定理1的所需条件。

5. 总结

本文主要研究了基于李雅普诺夫直接法的分数阶中立型四元数神经网络的同步。首先在不借助于符号函数的情况下尝试使用非线性的控制器，其次构造了V函数，并给出了一些同步的充分准则，最后给出的数值例子与定理条件相融合。与现有文献中的解方法不同，该模型被视为一个整体。此外，除了所提出的控制器具有新颖性，本文中模型的系数也是不常见的函数形式，这使得模型具有一般性。在未来的工作中，继续将四元数不分解为一个实部和三个虚部来处理，用李亚普诺夫直接法和Mittag-Leffler函数相结合来研究分数阶中立型四元数值神经网络的同步，为神经网络提供更丰富的信息处理能力和更高的性能。

基金项目

云南财经大学研究生创新基金项目(编号2023YUFEYC071)资助。

文章引用

王丽,赵晴晴. 分数阶中立型四元数值神经网络的同步
Synchronization of Fractional Order Neutral Quaternion Numerical Neural Networks[J]. 应用数学进展, 2023, 12(09): 3842-3850. https://doi.org/10.12677/AAM.2023.129378

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NOTES

^*通讯作者。

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