¹新疆大学数学与系统科学学院，新疆 乌鲁木齐

²新疆应用数学重点实验室，新疆 乌鲁木齐

收稿日期：2022年10月12日；录用日期：2022年11月11日；发布日期：2022年11月21日

摘要

本文解决了分数阶BAM模糊神经网络的全局Mittag-Leffler (M-L)镇定问题。首先回顾了与分数阶微积分相关的基础知识，并建立了网络模型。其次，基于一种新的压缩映射和二范数分析方法严格证明了模型平衡点的存在唯一性。最后，通过设计一种简洁有效的线性控制器导出了分数阶BAM模糊神经网络实现全局M-L镇定的充分性判据。

关键词

BAM神经网络，Mittag-Leffler镇定，分数阶，模糊逻辑

Global Mittag-Leffler Stabilization of BAM Fuzzy Neural Networks with Fractional-Order

Jie Li¹, Shenglong Chen¹, Hongli Li^1,2*

¹College of Mathematics and System Sciences, Xinjiang University, Urumqi Xinjiang

²Xinjiang Key Laboratory of Applied Mathematics, Urumqi Xinjiang

Received: Oct. 12^th, 2022; accepted: Nov. 11^th, 2022; published: Nov. 21^st, 2022

ABSTRACT

This paper deals with the issue of global Mittag-Leffler (M-L) stabilization for fractional-order BAM fuzzy neural networks (FBAMFNNs). Firstly, some necessary knowledge related to fractional calculus are reviewed, and the model of FBAMFNN is established. Next, the existence and uniqueness of equilibrium point is proved based on constructing a novel contraction mapping and 2-norm analysis method. Finally, the sufficient criterion is derived to realize global M-L stabilization of FBAMFNNs by designing a concise and effective linear controller.

Keywords:BAM Neural Networks, Mittag-Leffler Stabilization, Fractional-Order, Fuzzy Logic

Copyright © 2022 by author(s) and Hans Publishers Inc.

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

在过去的数十年中，神经网络因在信号处理、保密通信和模式识别等领域的广泛应用激发了许多国内外研究者的兴趣 [1] [2]。作为单层到双层模式匹配电路的推广，BAM神经网络最早由Kosko于1987年创立 [3]，其在自动控制与电力系统等方面具有重要应用前景 [4]，近年来BAM神经网络的动力学分析受到了学者们的广泛关注，并取得了许多有价值的成果 [5] [6]。

众所周知，不确定性或模糊性在现实中是不可避免的，模糊逻辑是解决上述问题的一种重要工具，通过考虑模糊因素，并将模糊AND与模糊OR运算融入经典的神经网络模型，学者们提出并陆续探究了各类模糊神经网络的动力学行为 [7] [8]。文献 [8] 采用非抖振量化控制研究了具有时滞和脉冲效应的不连续非恒等模糊BAM神经网络的有限时间同步。分数阶微积分不仅是整数阶微积分在阶数意义上的推广，与整数阶微积分相比，分数阶微积分在生物数学和工程控制等实际应用中具有更强的建模能力。无限记忆性与遗传性作为分数阶微积分的两大独特优势，将分数阶微积分与神经网络相结合有助于更精确地刻画神经网络的动力学行为。分数阶模糊神经网络作为一种重要的网络模型，得到了许多研究者的关注 [9] [10]。文献 [9] 基于非线性反馈控制考虑了分数阶模糊神经网络的完全同步与有限时间同步。文献 [10] 采用直接四元数法分析了分数阶四元值模糊BAM神经网络的有限时间镇定。

镇定性在控制工程与系统辨识等实际应用中至关重要，设计合适的控制策略是实现系统镇定的关键。迄今为止，反馈控制、间接控制、事件触发控制和混合控制等多种控制策略被陆续提出并用于神经网络的镇定性研究 [11] [12]，值得注意的是与其它控制策略相比，反馈控制更加简洁有效，然而目前很少有研究基于反馈控制策略分析分数阶BAM模糊神经网络的全局M-L镇定性问题，这激发了我们进一步研究的兴趣。

受上述分析的启发，本文将探究分数阶BAM模糊神经网络的全局M-L镇定问题。本文的创新点可归纳为以下三方面。首先，构建了分数阶BAM模糊神经网络模型。其次，基于压缩映射原理证明了分数阶BAM模糊神经网络平衡点的存在唯一性。最后，设计了一种简洁有效的线性反馈控制器，结合不等式分析技巧得到了分数阶BAM模糊神经网络实现全局M-L镇定的充分性判据。

本文结构安排如下：第二节给出了分数阶BAM模糊神经网络模型，回顾了分数阶微积分的相关定义与引理，为后文研究需要对激活函数做出来假设。第3节证明了分数阶BAM模糊神经网络平衡点的存在唯一性。第4节分析了分数阶BAM模糊神经网络的全局M-L镇定问题。第5节给出了总结与展望。

符号：R代表实数集， $R^{+}$ 表示正实值集， $N=\left\{1,2,\cdots ,n\right\},M=\left\{1,2,\cdots ,m\right\}$，对任意n维实值向量

$\tau ={\left({\tau }_1,{\tau }_2,\cdots ,{\tau }_n\right)}^{\text{T}}\in R^n$，$\tau$ 的2-范数定义为 $\Vert \tau \Vert =\sqrt{{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\left|{\tau }_{\iota }\right|}^2}}$。

2. 预备知识与模型描述

定义1. [13] 阶数为 $0<\nu <1$ 的函数 $\omega \left(t\right)$ 的Caputo分数阶导数定义为

${}_{t_0}{}^{c}D{}_t^{\nu }\omega \left(t\right)=\frac{1}{\Gamma \left(1-\nu \right)}{\displaystyle {\int }_{t_0}^t\frac{{\omega }^{\prime }\left(s\right)}{{\left(t-s\right)}^{\nu }}\text{d}s},$

其中 $\Gamma \left(\nu \right)={\displaystyle {\int }_{t_0}^{+\infty }{\text{e}}^{-t}}{t}^{\nu -1}\text{d}t$ 为Gamma函数。

考虑如下分数阶BAM模糊神经网络模型：

$\{\begin{array}{l}{}_{t_0}{}^{c}D{}_t^{\nu }{\alpha }_{\iota }\left(t\right)=-{\theta }_{\iota }{\alpha }_{\iota }\left(t\right)+{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\mu }_{\iota \kappa }}{\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)+\underset{\kappa =1}{\overset{m}{\vee }}{\varphi }_{\iota \kappa }{\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)+\underset{\kappa =1}{\overset{m}{\wedge }}{\psi }_{\iota \kappa }{\omega }_{\kappa }({\beta }_{\kappa }(t))+{\Theta }_{\iota }(t),\\ {}_{t_0}{}^{c}D{}_t^{\nu }{\beta }_{\kappa }\left(t\right)=-{\vartheta }_{\kappa }{\beta }_{\kappa }\left(t\right)+{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\zeta }_{\kappa \iota }}{\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)+\underset{\iota =1}{\overset{n}{\vee }}{\delta }_{\kappa \iota }{\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)+\underset{\iota =1}{\overset{n}{\wedge }}{\eta }_{\kappa \iota }{\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)+{\Lambda }_{\kappa }\left(t\right),\end{array}$ (1)

其中 $\iota \in N,\kappa \in M$，${\alpha }_{\iota }\left(t\right)$ 与 ${\beta }_{\kappa }\left(t\right)$ 分别代表第 $\iota$ 个和第 $\kappa$ 个神经元的状态。n与m依次表示第一层和第二层中神经元的数量。 ${\theta }_{\iota }$ 和 ${\vartheta }_{\kappa }$ 是第 $\iota$ 个和第 $\kappa$ 个神经元的衰减系数， ${\mu }_{\iota \kappa }$ 与 ${\zeta }_{\kappa \iota }$ 为连接权重， ${\varphi }_{\iota \kappa }$ 和 ${\delta }_{\kappa \iota }$ 表示模糊反馈最大模板的连接权重， ${\psi }_{\iota \kappa }$ 和 ${\eta }_{\kappa \iota }$ 代表模糊反馈最小模板的连接权重， ${\omega }_{\kappa }\left(t\right)$ 与 ${\varpi }_{\iota }\left(t\right)$ 表示第 $\kappa$ 个和第 $\iota$ 个神经元的激活函数。 $\wedge$ 和 $\vee$ 代表模糊OR与AND运算。 ${\Theta }_{\iota }\left(t\right)$ 与 ${\Lambda }_{\kappa }\left(t\right)$ 分别表示不同层中的外部输入。

为便于本文后续研究，对上述激活函数作出如下假设：

假设1. [8] 对任意 $\alpha ,\beta$，存在正常数 ${\lambda }_{\kappa }$ 与 ${\chi }_{\iota }$ 使得

$\begin{array}{l}\left|{\omega }_{\kappa }\left(\alpha \right)-{\omega }_{\kappa }\left(\beta \right)\right|\le {\lambda }_{\kappa }\left|\alpha -\beta \right|,\\ \left|{\varpi }_{\iota }\left(\alpha \right)-{\varpi }_{\iota }\left(\beta \right)\right|\le {\chi }_{\iota }\left|\alpha -\beta \right|.\end{array}$

引理1. [7] 若 ${\alpha }_{\iota },{\tilde{\alpha }}_{\iota }$ 与 ${\beta }_{\kappa },{\tilde{\beta }}_{\kappa }$ 分别为模型(1)的状态，则模型(1)中的激活函数满足下列不等式

$\begin{array}{l}\left|\underset{\kappa =1}{\overset{m}{\vee }}{\varphi }_{\iota \kappa }{\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-\underset{\kappa =1}{\overset{m}{\vee }}{\varphi }_{\iota \kappa }{\omega }_{\kappa }\left({\tilde{\beta }}_{\kappa }\left(t\right)\right)\right|\le {\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\left|{\varphi }_{\iota \kappa }\right|}\left|{\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\tilde{\beta }}_{\kappa }\left(t\right)\right)\right|,\\ \left|\underset{\kappa =1}{\overset{m}{\wedge }}{\psi }_{\iota \kappa }{\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-\underset{\kappa =1}{\overset{m}{\wedge }}{\psi }_{\iota \kappa }{\omega }_{\kappa }\left({\tilde{\beta }}_{\kappa }\left(t\right)\right)\right|\le {\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\left|{\psi }_{\iota \kappa }\right|}\left|{\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\tilde{\beta }}_{\kappa }\left(t\right)\right)\right|,\end{array}$

$\begin{array}{l}\left|\underset{\iota =1}{\overset{n}{\vee }}{\delta }_{\iota \kappa }{\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-\underset{\iota =1}{\overset{n}{\vee }}{\delta }_{\iota \kappa }{\varpi }_{\iota }\left({\tilde{\alpha }}_{\iota }\left(t\right)\right)\right|\le {\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\left|{\delta }_{\iota \kappa }\right|}\left|{\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\tilde{\alpha }}_{\iota }\left(t\right)\right)\right|,\\ \left|\underset{\iota =1}{\overset{n}{\wedge }}{\eta }_{\iota \kappa }{\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-\underset{\iota =1}{\overset{n}{\wedge }}{\eta }_{\iota \kappa }{\varpi }_{\iota }\left({\tilde{\alpha }}_{\iota }\left(t\right)\right)\right|\le {\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\left|{\eta }_{\iota \kappa }\right|}\left|{\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\tilde{\alpha }}_{\iota }\left(t\right)\right)\right|.\end{array}$

引理2. [14] 若 $\sigma =\left({\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_m\right)$ 与 $\pi =\left({\pi }_1,{\pi }_2,\cdots ,{\pi }_m\right)$ 是成比例的序列，则有

${\left|{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\sigma }_{\kappa }}{\pi }_{\kappa }\right|}^2\le \left({\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\left|{\sigma }_{\kappa }\right|}^2}\right)\left({\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\left|{\pi }_{\kappa }\right|}^2}\right).$

引理3. [15] 若 $V\left(t\right)$ 为定义在 $\left[t_0,a\right)$ 上的连续可微函数，则对任意的常数b与 $0<\nu <1$ 有

${}_{t_0}{}^{c}D{}_t^{\nu }{\left(V\left(t\right)-b\right)}^2\le 2\left(V\left(t\right)-b\right){}_{t_0}{}^{c}D{}_t^{\nu }V\left(t\right).$

引理4. [16] 如果 $V\left(t\right)$ 是定义在 $\left[t_0,+\infty \right)$ 上的非负连续函数，并且对任意初始时刻 $t_0$ 以及常数 $0<\nu <1,\Omega \in R$ 满足

${}_{t_0}{}^{c}D{}_t^{\nu }V\left(t\right)\le -\Omega V\left(t\right),$

那么有 $V\left(t\right)\le V\left(t_0\right)E_{\nu }\left(-\Omega {\left(t-t_0\right)}^{\nu }\right)$。

定义2. 如果存在 $\varsigma ={\left({\alpha }_1^{\ast },{\alpha }_2^{\ast },\cdots ,{\alpha }_n^{\ast },{\beta }_1^{\ast },{\beta }_2^{\ast },\cdots ,{\beta }_n^{\ast }\right)}^{\text{T}}$ 使得

$\{\begin{array}{l}0=-{\theta }_{\iota }{\alpha }_{\iota }^{\ast }\left(t\right)+{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\mu }_{\iota \kappa }}{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\left(t\right)\right)+\underset{\kappa =1}{\overset{m}{\vee }}{\varphi }_{\iota \kappa }{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\left(t\right)\right)+\underset{\kappa =1}{\overset{m}{\wedge }}{\psi }_{\iota \kappa }{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\left(t\right)\right)+{\Theta }_{\iota }\left(t\right),\\ 0=-{\vartheta }_{\kappa }{\beta }_{\kappa }^{\ast }\left(t\right)+{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\zeta }_{\kappa \iota }}{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\left(t\right)\right)+\underset{\iota =1}{\overset{n}{\vee }}{\delta }_{\kappa \iota }{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\left(t\right)\right)+\underset{\iota =1}{\overset{n}{\wedge }}{\eta }_{\kappa \iota }{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\left(t\right)\right)+{\Lambda }_{\kappa }\left(t\right),\end{array}$

那么 $\varsigma$ 是分数阶BAM模糊神经网络(1)的平衡点。

定义3. 如果存在正常数 $h,\Omega ,l$，使得对系统(1)的任意解 $\xi ={\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n,{\beta }_1,{\beta }_2,\cdots ,{\beta }_n\right)}^{\text{T}}$ 与 $t\ge t_0$ 有

$\Vert \xi -{\xi }^{\ast }\Vert \le {\left(h\Vert \xi \left(t_0\right)-{\xi }^{\ast }\Vert E_{\nu }\left(-\Omega {\left(t-t_0\right)}^{\nu }\right)\right)}^{\frac{1}{l}},$

那么称系统(1)在平衡点 ${\xi }^{\ast }={\left({\alpha }_1^{\ast },{\alpha }_2^{\ast },\cdots ,{\alpha }_n^{\ast },{\beta }_1^{\ast },{\beta }_2^{\ast },\cdots ,{\beta }_n^{\ast }\right)}^{\text{T}}$ 处是全局M-L镇定的，其中 $\xi \left(t_0\right)$ 为系统(1)的初始值。

3. 平衡点的存在唯一性

在本节中，通过构造新的压缩映射并结合二范数分析方法严格证明了分数阶BAM模糊神经网络平衡点的存在唯一性。

定理1. 在假设1下，如果满足下列条件

$0<\max \left\{\underset{1\le \iota \le n}{\max }\left\{{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\frac{{\lambda }_{\kappa }\left(\left|{\mu }_{\iota \kappa }\right|+\left|{\varphi }_{\iota \kappa }\right|+\left|{\psi }_{\iota \kappa }\right|\right)}{{\vartheta }_{\kappa }}}\right\},\underset{1\le \kappa \le m}{\max }\left\{{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\frac{{\chi }_{\iota }\left(\left|{\zeta }_{\kappa \iota }\right|+\left|{\delta }_{\kappa \iota }\right|+\left|{\eta }_{\kappa \iota }\right|\right)}{{\theta }_{\iota }}}\right\}\right\}<1,$ (2)

则分数阶BAM模糊神经网络(1)存在唯一的平衡点 $\varsigma ={\left({\alpha }_1^{\ast },{\alpha }_2^{\ast },\cdots ,{\alpha }_n^{\ast },{\beta }_1^{\ast },{\beta }_2^{\ast },\cdots ,{\beta }_n^{\ast }\right)}^{\text{T}}$。

证明：记 ${\rho }_{\iota }^{\ast }={\theta }_{\iota }{\alpha }_{\iota }^{\ast },{\gamma }_{\kappa }^{\ast }={\vartheta }_{\kappa }{\beta }_{\kappa }^{\ast }$，构造如下映射

$\begin{array}{l}{\Pi }_{\iota }\left(\rho ,\gamma \right)={\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\mu }_{\iota \kappa }}{\omega }_{\kappa }\left(\frac{{\gamma }_{\kappa }}{{\vartheta }_{\kappa }}\right)+\underset{\kappa =1}{\overset{m}{\vee }}{\varphi }_{\iota \kappa }{\omega }_{\kappa }\left(\frac{{\gamma }_{\kappa }}{{\vartheta }_{\kappa }}\right)+\underset{\kappa =1}{\overset{m}{\wedge }}{\psi }_{\iota \kappa }{\omega }_{\kappa }\left(\frac{{\gamma }_{\kappa }}{{\vartheta }_{\kappa }}\right)+{\Theta }_{\iota }\left(t\right),\\ {\Xi }_{\kappa }\left(\rho ,\gamma \right)={\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\zeta }_{\kappa \iota }}{\varpi }_{\iota }\left(\frac{{\rho }_{\iota }}{{\theta }_{\iota }}\right)+\underset{\iota =1}{\overset{n}{\vee }}{\delta }_{\kappa \iota }{\varpi }_{\iota }\left(\frac{{\rho }_{\iota }}{{\theta }_{\iota }}\right)+\underset{\iota =1}{\overset{n}{\wedge }}{\eta }_{\kappa \iota }{\varpi }_{\iota }\left(\frac{{\rho }_{\iota }}{{\theta }_{\iota }}\right)+{\Lambda }_{\kappa }\left(t\right),\end{array}$

其中 $\Pi \left(\rho ,\gamma \right)={\left({\Pi }_1\left(\rho ,\gamma \right),\cdots ,{\Pi }_n\left(\rho ,\gamma \right)\right)}^{\text{T}}$，$\Xi \left(\rho ,\gamma \right)={\left({\Xi }_1\left(\rho ,\gamma \right),\cdots ,{\Xi }_n\left(\rho ,\gamma \right)\right)}^{\text{T}}$，$\left(\rho ,\gamma \right)={\left(\left({\rho }_1,{\gamma }_1\right),\cdots ,\left({\rho }_m,{\gamma }_m\right)\right)}^{\text{T}}$ 并且 $m\ge n$，接下来证明 $\left(\Pi ,\Xi \right)$ 为一压缩映射。

对任意的 $\left(\rho ,\gamma \right)$ 与 $\left(\tilde{\rho },\tilde{\gamma }\right)$，基于假设1，引理1和2可得

$\begin{array}{l}\Vert \left(\Pi ,\Xi \right)\left(\rho ,\gamma \right)-\left(\Pi ,\Xi \right)\left(\tilde{\rho },\tilde{\gamma }\right)\Vert \\ =\Vert \Pi \left(\rho ,\gamma \right)-\Pi \left(\tilde{\rho },\tilde{\gamma }\right)\Vert +\Vert \Xi \left(\rho ,\gamma \right)-\Xi \left(\tilde{\rho },\tilde{\gamma }\right)\Vert \\ =({\displaystyle \underset{\iota =1}{\overset{n}{\sum }}|{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\mu }_{\iota \kappa }}{\omega }_{\kappa }\left(\frac{{\gamma }_{\kappa }}{{\vartheta }_{\kappa }}\right)}-{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\mu }_{\iota \kappa }}{\omega }_{\kappa }\left(\frac{{\tilde{\gamma }}_{\kappa }}{{\vartheta }_{\kappa }}\right)+\underset{\kappa =1}{\overset{m}{\vee }}{\varphi }_{\iota \kappa }{\omega }_{\kappa }\left(\frac{{\gamma }_{\kappa }}{{\vartheta }_{\kappa }}\right)\\ -\underset{\kappa =1}{\overset{m}{\vee }}{\varphi }_{\iota \kappa }{\omega }_{\kappa }\left(\frac{{\tilde{\gamma }}_{\kappa }}{{\vartheta }_{\kappa }}\right)+\underset{\kappa =1}{\overset{m}{\wedge }}{\psi }_{\iota \kappa }{\omega }_{\kappa }\left(\frac{{\gamma }_{\kappa }}{{\vartheta }_{\kappa }}\right)-{{\underset{\kappa =1}{\overset{m}{\wedge }}{\psi }_{\iota \kappa }{\omega }_{\kappa }\left(\frac{{\tilde{\gamma }}_{\kappa }}{{\vartheta }_{\kappa }}\right)|}^2)}^{\frac{1}{2}}\\ +({\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}|{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\zeta }_{\kappa \iota }}{\varpi }_{\iota }\left(\frac{{\rho }_{\iota }}{{\theta }_{\iota }}\right)-{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\zeta }_{\kappa \iota }}{\varpi }_{\iota }\left(\frac{{\tilde{\rho }}_{\iota }}{{\theta }_{\iota }}\right)+\underset{\iota =1}{\overset{n}{\vee }}{\delta }_{\kappa \iota }{\varpi }_{\iota }\left(\frac{{\rho }_{\iota }}{{\theta }_{\iota }}\right)}\end{array}$

$\begin{array}{l}-\underset{\iota =1}{\overset{n}{\vee }}{\delta }_{\kappa \iota }{\varpi }_{\iota }\left(\frac{{\tilde{\rho }}_{\iota }}{{\theta }_{\iota }}\right)+\underset{\iota =1}{\overset{n}{\wedge }}{\eta }_{\kappa \iota }{\varpi }_{\iota }\left(\frac{{\rho }_{\iota }}{{\theta }_{\iota }}\right)-{{\underset{\iota =1}{\overset{n}{\wedge }}{\eta }_{\kappa \iota }{\varpi }_{\iota }\left(\frac{{\tilde{\rho }}_{\iota }}{{\theta }_{\iota }}\right)|}^2)}^{\frac{1}{2}}\\ \le ({\displaystyle \underset{\iota =1}{\overset{n}{\sum }}|{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\left|{\mu }_{\iota \kappa }\right|}\left|{\omega }_{\kappa }\left(\frac{{\gamma }_{\kappa }}{{\vartheta }_{\kappa }}\right)-{\omega }_{\kappa }\left(\frac{{\tilde{\gamma }}_{\kappa }}{{\vartheta }_{\kappa }}\right)\right|}+\underset{\kappa =1}{\overset{m}{\vee }}\left|{\varphi }_{\iota \kappa }\right|\left|{\omega }_{\kappa }\left(\frac{{\gamma }_{\kappa }}{{\vartheta }_{\kappa }}\right)-{\omega }_{\kappa }\left(\frac{{\tilde{\gamma }}_{\kappa }}{{\vartheta }_{\kappa }}\right)\right|\\ +{\underset{\kappa =1}{\overset{m}{\wedge }}\left|{\psi }_{\iota \kappa }\right|{\left|{\omega }_{\kappa }\left(\frac{{\gamma }_{\kappa }}{{\vartheta }_{\kappa }}\right)-{\omega }_{\kappa }\left(\frac{{\tilde{\gamma }}_{\kappa }}{{\vartheta }_{\kappa }}\right)\right||}^2)}^{\frac{1}{2}}+({\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}|{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\left|{\zeta }_{\kappa \iota }\right|\left|{\varpi }_{\iota }\left(\frac{{\rho }_{\iota }}{{\theta }_{\iota }}\right)-{\varpi }_{\iota }\left(\frac{{\tilde{\rho }}_{\iota }}{{\theta }_{\iota }}\right)\right|}}\\ +\underset{\iota =1}{\overset{n}{\vee }}\left|{\delta }_{\kappa \iota }\right|\left|{\varpi }_{\iota }\left(\frac{{\rho }_{\iota }}{{\theta }_{\iota }}\right)-{\varpi }_{\iota }\left(\frac{{\tilde{\rho }}_{\iota }}{{\theta }_{\iota }}\right)\right|+{{\underset{\iota =1}{\overset{n}{\wedge }}\left|{\eta }_{\kappa \iota }\right|\left|{\varpi }_{\iota }\left(\frac{{\rho }_{\iota }}{{\theta }_{\iota }}\right)-{\varpi }_{\iota }\left(\frac{{\tilde{\rho }}_{\iota }}{{\theta }_{\iota }}\right)\right||}^2)}^{\frac{1}{2}}\\ \le {\left({\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\left({\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\frac{{\lambda }_{\kappa }\left(\left|{\mu }_{\iota \kappa }\right|+\left|{\varphi }_{\iota \kappa }\right|+\left|{\psi }_{\iota \kappa }\right|\right)}{{\vartheta }_{\kappa }}\left|{\gamma }_{\kappa }-{\tilde{\gamma }}_{\kappa }\right|}\right)}^2}\right)}^{\frac{1}{2}}\end{array}$

$\begin{array}{l}+{\left({{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\left({\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\frac{{\chi }_{\iota }\left(\left|{\zeta }_{\kappa \iota }\right|+\left|{\delta }_{\kappa \iota }\right|+\left|{\eta }_{\kappa \iota }\right|\right)}{{\theta }_{\iota }}\left|{\rho }_{\iota }-{\tilde{\rho }}_{\iota }\right|}\right)}}^2\right)}^{\frac{1}{2}}\\ \le {\left(\underset{1\le \iota \le n}{\max }{\left\{{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\frac{{\lambda }_{\kappa }\left(\left|{\mu }_{\iota \kappa }\right|+\left|{\varphi }_{\iota \kappa }\right|+\left|{\psi }_{\iota \kappa }\right|\right)}{{\vartheta }_{\kappa }}}\right\}}^2{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\left|{\gamma }_{\kappa }-{\tilde{\gamma }}_{\kappa }\right|}^2}\right)}^{\frac{1}{2}}\\ +{\left(\underset{1\le \kappa \le m}{\max }{\left\{{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\frac{{\chi }_{\iota }\left(\left|{\zeta }_{\kappa \iota }\right|+\left|{\delta }_{\kappa \iota }\right|+\left|{\eta }_{\kappa \iota }\right|\right)}{{\theta }_{\iota }}}\right\}}^2{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\left|{\rho }_{\iota }-{\tilde{\rho }}_{\iota }\right|}^2}\right)}^{\frac{1}{2}}\\ \le \underset{1\le \iota \le n}{\max }\left\{{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\frac{{\lambda }_{\kappa }\left(\left|{\mu }_{\iota \kappa }\right|+\left|{\varphi }_{\iota \kappa }\right|+\left|{\psi }_{\iota \kappa }\right|\right)}{{\vartheta }_{\kappa }}}\right\}\Vert \gamma -\tilde{\gamma }\Vert \\ +\underset{1\le \kappa \le m}{\max }\left\{{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\frac{{\chi }_{\iota }\left(\left|{\zeta }_{\kappa \iota }\right|+\left|{\delta }_{\kappa \iota }\right|+\left|{\eta }_{\kappa \iota }\right|\right)}{{\theta }_{\iota }}}\right\}\Vert \rho -\tilde{\rho }\Vert .\end{array}$

结合条件(2)与上式，我们有

${\Vert \left(\Pi ,\Xi \right)\left(\rho ,\gamma \right)-\left(\Pi ,\Xi \right)\left(\tilde{\rho },\tilde{\gamma }\right)\Vert }_2\le {\Vert \rho -\tilde{\rho }\Vert }_2+{\Vert \gamma -\tilde{\gamma }\Vert }_2={\Vert \left(\rho ,\gamma \right)-\left(\tilde{\rho },\tilde{\gamma }\right)\Vert }_2.$

故 $\left(\Pi ,\Xi \right)$ 是一个压缩映射，从而存在唯一的不动点 $\left({\rho }^{\ast },{\gamma }^{\ast }\right)$ 使得 $\left(\Pi ,\Xi \right)\left({\rho }^{\ast },{\gamma }^{\ast }\right)=\left({\rho }^{\ast },{\gamma }^{\ast }\right)$，即

$\begin{array}{l}{\rho }_{\iota }^{\ast }={\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\mu }_{\iota \kappa }}{\omega }_{\kappa }\left(\frac{{\gamma }_{\kappa }^{\ast }}{{\vartheta }_{\kappa }}\right)+\underset{\kappa =1}{\overset{m}{\vee }}{\varphi }_{\iota \kappa }{\omega }_{\kappa }\left(\frac{{\gamma }_{\kappa }^{\ast }}{{\vartheta }_{\kappa }}\right)+\underset{\kappa =1}{\overset{m}{\wedge }}{\psi }_{\iota \kappa }{\omega }_{\kappa }\left(\frac{{\gamma }_{\kappa }^{\ast }}{{\vartheta }_{\kappa }}\right)+{\Theta }_{\iota }\left(t\right),\\ {\gamma }_{\kappa }^{\ast }={\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\zeta }_{\kappa \iota }}{\varpi }_{\iota }\left(\frac{{\rho }_{\iota }^{\ast }}{{\theta }_{\iota }}\right)+\underset{\iota =1}{\overset{n}{\vee }}{\delta }_{\kappa \iota }{\varpi }_{\iota }\left(\frac{{\rho }_{\iota }^{\ast }}{{\theta }_{\iota }}\right)+\underset{\iota =1}{\overset{n}{\wedge }}{\eta }_{\kappa \iota }{\varpi }_{\iota }\left(\frac{{\rho }_{\iota }^{\ast }}{{\theta }_{\iota }}\right)+{\Lambda }_{\kappa }\left(t\right).\end{array}$ (3)

式等价于

$\{\begin{array}{l}0=-{\theta }_{\iota }{\alpha }_{\iota }^{\ast }\left(t\right)+{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\mu }_{\iota \kappa }}{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\left(t\right)\right)+\underset{\kappa =1}{\overset{m}{\vee }}{\varphi }_{\iota \kappa }{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\left(t\right)\right)+\underset{\kappa =1}{\overset{m}{\wedge }}{\psi }_{\iota \kappa }{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\left(t\right)\right)+{\Theta }_{\iota }\left(t\right),\\ 0=-{\vartheta }_{\kappa }{\beta }_{\kappa }^{\ast }\left(t\right)+{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\zeta }_{\kappa \iota }}{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\left(t\right)\right)+\underset{\iota =1}{\overset{n}{\vee }}{\delta }_{\kappa \iota }{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\left(t\right)\right)+\underset{\iota =1}{\overset{n}{\wedge }}{\eta }_{\kappa \iota }{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\left(t\right)\right)+{\Lambda }_{\kappa }\left(t\right).\end{array}$

由定义2可知，分数阶BAM模糊神经网络(1)有唯一的平衡点 $\varsigma ={\left({\alpha }_1^{\ast },{\alpha }_2^{\ast },\cdots ,{\alpha }_n^{\ast },{\beta }_1^{\ast },{\beta }_2^{\ast },\cdots ,{\beta }_n^{\ast }\right)}^{\text{T}}$。

注1. 通过将分数阶导数、模糊逻辑等因素考虑在内，分数阶BAM模糊神经网络模型比分数阶神经网络 [12] [15]、模糊神经网络 [7] [11]、BAM神经网络 [4] [5] 等更加一般化，并且实用性更广。

4. 全局M-L镇定性

本节设计了一种简洁有效的线性反馈控制器，基于分数阶理论与不等式分析技巧，我们得到了系统(1)实现全局M-L镇定的充分性判据。

接下来为将系统(1)的平衡点转换到原点，作变换 $r_{\iota }\left(t\right)={\alpha }_{\iota }\left(t\right)-{\alpha }_{\iota }^{\ast },{\hslash }_{\kappa }\left(t\right)={\hslash }_{\kappa }\left(t\right)-{\hslash }_{\kappa }^{\ast }$，从而系统(1)转换后的形式为

$\{\begin{array}{l}{}_{t_0}{}^{c}D{}_t^{\nu }{r}_{\iota }\left(t\right)=-{\theta }_{\iota }{r}_{\iota }\left(t\right)+{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\mu }_{\iota \kappa }}\left({\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right)+\underset{\kappa =1}{\overset{m}{\vee }}{\varphi }_{\iota \kappa }\left({\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right)\\ +\underset{\kappa =1}{\overset{m}{\wedge }}{\psi }_{\iota \kappa }\left({\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right),\\ {}_{t_0}{}^{c}D{}_t^{\nu }{\hslash }_{\kappa }\left(t\right)=-{\vartheta }_{\kappa }{\hslash }_{\kappa }(t)+{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\zeta }_{\kappa \iota }}\left({\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right)+\underset{\iota =1}{\overset{n}{\vee }}{\delta }_{\kappa \iota }\left({\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right)\\ +\underset{\iota =1}{\overset{n}{\wedge }}{\eta }_{\kappa \iota }\left({\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right),\end{array}$ (4)

(4)的受控形式为

$\{\begin{array}{l}{}_{t_0}{}^{c}D{}_t^{\nu }{r}_{\iota }\left(t\right)=-{\theta }_{\iota }{r}_{\iota }\left(t\right)+{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\mu }_{\iota \kappa }}\left({\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right)+\underset{\kappa =1}{\overset{m}{\vee }}{\varphi }_{\iota \kappa }\left({\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right)\\ +\underset{\kappa =1}{\overset{m}{\wedge }}{\psi }_{\iota \kappa }\left({\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right)+{\stackrel{⌢}{u}}_{\iota }\left(t\right),\\ {}_{t_0}{}^{c}D{}_t^{\nu }{\hslash }_{\kappa }\left(t\right)=-{\vartheta }_{\kappa }{\hslash }_{\kappa }\left(t\right)+{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\zeta }_{\kappa \iota }}\left({\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right)+\underset{\iota =1}{\overset{n}{\vee }}{\delta }_{\kappa \iota }\left({\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right)\\ +\underset{\iota =1}{\overset{n}{\wedge }}{\eta }_{\kappa \iota }\left({\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right)+{\stackrel{⌣}{u}}_{\iota }\left(t\right),\end{array}$ (5)

其中 ${\stackrel{⌢}{u}}_{\iota }\left(t\right)$ 与 ${\stackrel{⌣}{u}}_{\kappa }\left(t\right)$ 为如下所设计的线性反馈控制器

$\{\begin{array}{l}{\stackrel{⌢}{u}}_{\iota }\left(t\right)=-{\mathcal{l}}_{\iota }{r}_{\iota }\left(t\right),\\ {\stackrel{⌣}{u}}_{\iota }\left(t\right)=-{\wp }_{\kappa }{\hslash }_{\kappa }\left(t\right),\end{array}$ (6)

${\mathcal{l}}_{\iota }\left(t\right),{\wp }_{\kappa }\left(t\right)\in R^{+}.$

定理2. 基于假设1和控制器(6)，分数阶BAM模糊神经网络(1)在平衡点处是全局M-L镇定的。

证明：构造Lyapunov函数如下

$V\left(t\right)=\frac{1}{2}\left[{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{r}_{\iota }^2\left(t\right)}+{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\hslash }_{\kappa }^2\left(t\right)}\right].$

根据引理3，求 $V\left(t\right)$ 沿系统(5)在控制器(6)下的Caputo分数阶导数可得

$\begin{array}{c}{}_{t_0}{}^{c}D{}_t^{\nu }V\left(t\right)\le {\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{r}_{\iota }\left(t\right)}{}_{t_0}{}^{c}D{}_t^{\nu }{r}_{\iota }\left(t\right)+{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\hslash }_{\kappa }\left(t\right)}{}_{t_0}{}^{c}D{}_t^{\nu }{\hslash }_{\kappa }\left(t\right)\\ ={\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{r}_{\iota }\left(t\right)}[-{\theta }_{\iota }{r}_{\iota }\left(t\right)+{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\mu }_{\iota \kappa }}\left({\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right)\\ +\underset{\kappa =1}{\overset{m}{\vee }}{\varphi }_{\iota \kappa }\left({\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right)+\underset{\kappa =1}{\overset{m}{\wedge }}{\psi }_{\iota \kappa }\left({\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right)-{\mathcal{l}}_{\iota }{r}_{\iota }\left(t\right)]\end{array}$

$\begin{array}{c}+{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\hslash }_{\kappa }\left(t\right)}[-{\vartheta }_{\kappa }{\hslash }_{\kappa }\left(t\right)+{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\zeta }_{\kappa \iota }}\left({\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right)\\ +\underset{\iota =1}{\overset{n}{\vee }}{\delta }_{\kappa \iota }\left({\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right)+\underset{\iota =1}{\overset{n}{\wedge }}{\eta }_{\kappa \iota }\left({\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right)-{\wp }_{\kappa }{\hslash }_{\kappa }\left(t\right)]\\ \le -{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\left({\theta }_{\iota }+{\mathcal{l}}_{\iota }\right)r_{\iota }^2\left(t\right)}-{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\left({\vartheta }_{\kappa }+{\wp }_{\kappa }\right){\hslash }_{\kappa }^2\left(t\right)}+{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\left|{\mu }_{\iota \kappa }\right|}}\left|r_{\iota }\left(t\right)\right|\left|{\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right|\\ +{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\left|r_{\iota }\left(t\right)\right|}\left|\underset{\kappa =1}{\overset{m}{\vee }}{\varphi }_{\iota \kappa }\left({\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right)\right|+{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\left|r_{\iota }\left(t\right)\right|}\left|\underset{\kappa =1}{\overset{m}{\wedge }}{\psi }_{\iota \kappa }\left({\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right)\right|\\ +{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\left|{\zeta }_{\kappa \iota }\right|}\left|{\hslash }_{\kappa }\left(t\right)\right|\left|{\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right|+{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\left|{\hslash }_{\kappa }\left(t\right)\right|\left|\underset{\iota =1}{\overset{n}{\vee }}{\delta }_{\kappa \iota }\left({\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right)\right|}\\ +{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\left|{\hslash }_{\kappa }\left(t\right)\right|\left|\underset{\iota =1}{\overset{n}{\wedge }}{\eta }_{\kappa \iota }\left({\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right)\right|}.\end{array}$ (7)

由假设1可知

$\begin{array}{l}{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\left|{\mu }_{\iota \kappa }\right|}}\left|r_{\iota }\left(t\right)\right|\left|{\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right|\\ \le {\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\lambda }_{\kappa }\left|{\mu }_{\iota \kappa }\right|}}\left|r_{\iota }\left(t\right)\right|\left|{\hslash }_{\kappa }\left(t\right)\right|\\ \le \frac{1}{2}{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\lambda }_{\kappa }\left|{\mu }_{\iota \kappa }\right|\left(r_{\iota }^2\left(t\right)+{\hslash }_{\kappa }^2\left(t\right)\right),}}\end{array}$ (8)

$\begin{array}{l}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\left|{\zeta }_{\kappa \iota }\right|}}\left|{\hslash }_{\kappa }\left(t\right)\right|\left|{\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right|\\ \le {\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\chi }_{\iota }}}\left|{\zeta }_{\kappa \iota }\right|\left|{\hslash }_{\kappa }\left(t\right)\right|\left|r_{\iota }\left(t\right)\right|\\ \le \frac{1}{2}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\chi }_{\iota }\left|{\zeta }_{\kappa \iota }\right|\left({\hslash }_{\kappa }^2\left(t\right)+r_{\iota }^2\left(t\right)\right).}}\end{array}$ (9)

根据引理1和假设1有

$\begin{array}{l}{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\left|r_{\iota }\left(t\right)\right|}\left|\underset{\kappa =1}{\overset{m}{\vee }}{\varphi }_{\iota \kappa }\left({\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right)\right|\\ \le {\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\left|r_{\iota }\left(t\right)\right|}}\left|{\varphi }_{\iota \kappa }\right|\left|{\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right|\\ \le {\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\lambda }_{\kappa }\left|{\varphi }_{\iota \kappa }\right|\left|r_{\iota }\left(t\right)\right|}}\left|{\hslash }_{\kappa }\left(t\right)\right|\\ \le \frac{1}{2}{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\lambda }_{\kappa }\left|{\varphi }_{\iota \kappa }\right|}}\left(r_{\iota }^2\left(t\right)+{\hslash }_{\kappa }^2\left(t\right)\right),\end{array}$ (10)

同理依次可得

$\begin{array}{l}{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\left|r_{\iota }\left(t\right)\right|}\left|\underset{\kappa =1}{\overset{m}{\wedge }}{\psi }_{\iota \kappa }\left({\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right)\right|\\ \le {\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\left|r_{\iota }\left(t\right)\right|}}\left|{\psi }_{\iota \kappa }\right|\left|{\omega }_{\kappa }\left({\beta }_{\kappa }\left(t\right)\right)-{\omega }_{\kappa }\left({\beta }_{\kappa }^{\ast }\right)\right|\\ \le {\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\lambda }_{\kappa }\left|{\psi }_{\iota \kappa }\right|\left|r_{\iota }\left(t\right)\right|}}\left|{\hslash }_{\kappa }\left(t\right)\right|\\ \le \frac{1}{2}{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\lambda }_{\kappa }\left|{\psi }_{\iota \kappa }\right|}}\left(r_{\iota }^2\left(t\right)+{\hslash }_{\kappa }^2\left(t\right)\right),\end{array}$ (11)

$\begin{array}{l}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\left|{\hslash }_{\kappa }\left(t\right)\right|}\left|\underset{\iota =1}{\overset{n}{\vee }}{\delta }_{\kappa \iota }\left({\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right)\right|\\ \le {\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\left|{\hslash }_{\kappa }\left(t\right)\right|}}\left|{\delta }_{\kappa \iota }\right|\left|{\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right|\\ \le {\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\chi }_{\iota }\left|{\delta }_{\kappa \iota }\right|\left|r_{\iota }\left(t\right)\right|}}\left|{\hslash }_{\kappa }\left(t\right)\right|\\ \le \frac{1}{2}{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\chi }_{\iota }\left|{\delta }_{\kappa \iota }\right|}}\left(r_{\iota }^2\left(t\right)+{\hslash }_{\kappa }^2\left(t\right)\right),\end{array}$ (12)

$\begin{array}{l}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}\left|{\hslash }_{\kappa }\left(t\right)\right|}\left|\underset{\iota =1}{\overset{n}{\wedge }}{\eta }_{\kappa \iota }\left({\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right)\right|\\ \le {\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}\left|{\hslash }_{\kappa }\left(t\right)\right|}}\left|{\eta }_{\kappa \iota }\right|\left|{\varpi }_{\iota }\left({\alpha }_{\iota }\left(t\right)\right)-{\varpi }_{\iota }\left({\alpha }_{\iota }^{\ast }\right)\right|\\ \le {\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\chi }_{\iota }\left|{\eta }_{\kappa \iota }\right|\left|r_{\iota }\left(t\right)\right|}}\left|{\hslash }_{\kappa }\left(t\right)\right|\\ \le \frac{1}{2}{\displaystyle \underset{\iota =1}{\overset{n}{\sum }}{\displaystyle \underset{\kappa =1}{\overset{m}{\sum }}{\chi }_{\iota }\left|{\eta }_{\kappa \iota }\right|}}\left(r_{\iota }^2\left(t\right)+{\hslash }_{\kappa }^2\left(t\right)\right).\end{array}$ (13)