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。 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 unique-ness 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.
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。
BAM神经网络,Mittag-Leffler镇定,分数阶,模糊逻辑
Jie Li1, Shenglong Chen1, Hongli Li1,2*
1College of Mathematics and System Sciences, Xinjiang University, Urumqi Xinjiang
2Xinjiang Key Laboratory of Applied Mathematics, Urumqi Xinjiang
Received: Oct. 12th, 2022; accepted: Nov. 11th, 2022; published: Nov. 21st, 2022
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).
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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 [
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 [
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 [
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 = { 1 , 2 , ⋯ , n } , M = { 1 , 2 , ⋯ , m } ,对任意n维实值向量
τ = ( τ 1 , τ 2 , ⋯ , τ n ) T ∈ R n , τ 的2-范数定义为 ‖ τ ‖ = ∑ ι = 1 n | τ ι | 2 。
定义1. [
D t 0 c t ν ω ( t ) = 1 Γ ( 1 − ν ) ∫ t 0 t ω ′ ( s ) ( t − s ) ν d s ,
其中 Γ ( ν ) = ∫ t 0 + ∞ e − t t ν − 1 d t 为Gamma函数。
考虑如下分数阶BAM模糊神经网络模型:
{ D t 0 c t ν α ι ( t ) = − θ ι α ι ( t ) + ∑ κ = 1 m μ ι κ ω κ ( β κ ( t ) ) + ∨ κ = 1 m ϕ ι κ ω κ ( β κ ( t ) ) + ∧ κ = 1 m ψ ι κ ω κ ( β κ ( t ) ) + Θ ι ( t ) , D t 0 c t ν β κ ( t ) = − ϑ κ β κ ( t ) + ∑ ι = 1 n ζ κ ι ϖ ι ( α ι ( t ) ) + ∨ ι = 1 n δ κ ι ϖ ι ( α ι ( t ) ) + ∧ ι = 1 n η κ ι ϖ ι ( α ι ( t ) ) + Λ κ ( t ) , (1)
其中 ι ∈ N , κ ∈ M , α ι ( t ) 与 β κ ( t ) 分别代表第 ι 个和第 κ 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。 θ ι 和 ϑ κ 是第 ι 个和第 κ 个神经元的衰减系数, μ ι κ 与 ζ κ ι 为连接权重, ϕ ι κ 和 δ κ ι 表示模糊反馈最大模板的连接权重, ψ ι κ 和 η κ ι 代表模糊反馈最小模板的连接权重, ω κ ( t ) 与 ϖ ι ( t ) 表示第 κ 个和第 ι 个神经元的激活函数。 ∧ 和 ∨ 代表模糊OR与AND运算。 Θ ι ( t ) 与 Λ κ ( t ) 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. [
| ω κ ( α ) − ω κ ( β ) | ≤ λ κ | α − β | , | ϖ ι ( α ) − ϖ ι ( β ) | ≤ χ ι | α − β | .
引理1. [
| ∨ κ = 1 m ϕ ι κ ω κ ( β κ ( t ) ) − ∨ κ = 1 m ϕ ι κ ω κ ( β ˜ κ ( t ) ) | ≤ ∑ κ = 1 m | ϕ ι κ | | ω κ ( β κ ( t ) ) − ω κ ( β ˜ κ ( t ) ) | , | ∧ κ = 1 m ψ ι κ ω κ ( β κ ( t ) ) − ∧ κ = 1 m ψ ι κ ω κ ( β ˜ κ ( t ) ) | ≤ ∑ κ = 1 m | ψ ι κ | | ω κ ( β κ ( t ) ) − ω κ ( β ˜ κ ( t ) ) | ,
| ∨ ι = 1 n δ ι κ ϖ ι ( α ι ( t ) ) − ∨ ι = 1 n δ ι κ ϖ ι ( α ˜ ι ( t ) ) | ≤ ∑ ι = 1 n | δ ι κ | | ϖ ι ( α ι ( t ) ) − ϖ ι ( α ˜ ι ( t ) ) | , | ∧ ι = 1 n η ι κ ϖ ι ( α ι ( t ) ) − ∧ ι = 1 n η ι κ ϖ ι ( α ˜ ι ( t ) ) | ≤ ∑ ι = 1 n | η ι κ | | ϖ ι ( α ι ( t ) ) − ϖ ι ( α ˜ ι ( t ) ) | .
引理2. [
| ∑ κ = 1 m σ κ π κ | 2 ≤ ( ∑ κ = 1 m | σ κ | 2 ) ( ∑ κ = 1 m | π κ | 2 ) .
引理3. [
D t 0 c t ν ( V ( t ) − b ) 2 ≤ 2 ( V ( t ) − b ) D t 0 c t ν V ( t ) .
引理4. [
D t 0 c t ν V ( t ) ≤ − Ω V ( t ) ,
那么有 V ( t ) ≤ V ( t 0 ) E ν ( − Ω ( t − t 0 ) ν ) 。
定义2. 如果存在 ς = ( α 1 ∗ , α 2 ∗ , ⋯ , α n ∗ , β 1 ∗ , β 2 ∗ , ⋯ , β n ∗ ) T 使得
{ 0 = − θ ι α ι ∗ ( t ) + ∑ κ = 1 m μ ι κ ω κ ( β κ ∗ ( t ) ) + ∨ κ = 1 m ϕ ι κ ω κ ( β κ ∗ ( t ) ) + ∧ κ = 1 m ψ ι κ ω κ ( β κ ∗ ( t ) ) + Θ ι ( t ) , 0 = − ϑ κ β κ ∗ ( t ) + ∑ ι = 1 n ζ κ ι ϖ ι ( α ι ∗ ( t ) ) + ∨ ι = 1 n δ κ ι ϖ ι ( α ι ∗ ( t ) ) + ∧ ι = 1 n η κ ι ϖ ι ( α ι ∗ ( t ) ) + Λ κ ( t ) ,
那么 ς 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 , Ω , l ,使得对系统(1)的任意解 ξ = ( α 1 , α 2 , ⋯ , α n , β 1 , β 2 , ⋯ , β n ) T 与 t ≥ t 0 有
‖ ξ − ξ ∗ ‖ ≤ ( h ‖ ξ ( t 0 ) − ξ ∗ ‖ E ν ( − Ω ( t − t 0 ) ν ) ) 1 l ,
那么称系统(1)在平衡点 ξ ∗ = ( α 1 ∗ , α 2 ∗ , ⋯ , α n ∗ , β 1 ∗ , β 2 ∗ , ⋯ , β n ∗ ) T 处是全局M-L镇定的,其中 ξ ( t 0 ) 为系统(1)的初始值。
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 { max 1 ≤ ι ≤ n { ∑ κ = 1 m λ κ ( | μ ι κ | + | ϕ ι κ | + | ψ ι κ | ) ϑ κ } , max 1 ≤ κ ≤ m { ∑ ι = 1 n χ ι ( | ζ κ ι | + | δ κ ι | + | η κ ι | ) θ ι } } < 1 , (2)
则分数阶BAM模糊神经网络(1)存在唯一的平衡点 ς = ( α 1 ∗ , α 2 ∗ , ⋯ , α n ∗ , β 1 ∗ , β 2 ∗ , ⋯ , β n ∗ ) T 。
证明:记 ρ ι ∗ = θ ι α ι ∗ , γ κ ∗ = ϑ κ β κ ∗ ,构造如下映射
Π ι ( ρ , γ ) = ∑ κ = 1 m μ ι κ ω κ ( γ κ ϑ κ ) + ∨ κ = 1 m ϕ ι κ ω κ ( γ κ ϑ κ ) + ∧ κ = 1 m ψ ι κ ω κ ( γ κ ϑ κ ) + Θ ι ( t ) , Ξ κ ( ρ , γ ) = ∑ ι = 1 n ζ κ ι ϖ ι ( ρ ι θ ι ) + ∨ ι = 1 n δ κ ι ϖ ι ( ρ ι θ ι ) + ∧ ι = 1 n η κ ι ϖ ι ( ρ ι θ ι ) + Λ κ ( t ) ,
其中 Π ( ρ , γ ) = ( Π 1 ( ρ , γ ) , ⋯ , Π n ( ρ , γ ) ) T , Ξ ( ρ , γ ) = ( Ξ 1 ( ρ , γ ) , ⋯ , Ξ n ( ρ , γ ) ) T , ( ρ , γ ) = ( ( ρ 1 , γ 1 ) , ⋯ , ( ρ m , γ m ) ) T 并且 m ≥ n ,接下来证明 ( Π , Ξ ) 为一压缩映射。
对任意的 ( ρ , γ ) 与 ( ρ ˜ , γ ˜ ) ,基于假设1,引理1和2可得
‖ ( Π , Ξ ) ( ρ , γ ) − ( Π , Ξ ) ( ρ ˜ , γ ˜ ) ‖ = ‖ Π ( ρ , γ ) − Π ( ρ ˜ , γ ˜ ) ‖ + ‖ Ξ ( ρ , γ ) − Ξ ( ρ ˜ , γ ˜ ) ‖ = ( ∑ ι = 1 n | ∑ κ = 1 m μ ι κ ω κ ( γ κ ϑ κ ) − ∑ κ = 1 m μ ι κ ω κ ( γ ˜ κ ϑ κ ) + ∨ κ = 1 m ϕ ι κ ω κ ( γ κ ϑ κ ) − ∨ κ = 1 m ϕ ι κ ω κ ( γ ˜ κ ϑ κ ) + ∧ κ = 1 m ψ ι κ ω κ ( γ κ ϑ κ ) − ∧ κ = 1 m ψ ι κ ω κ ( γ ˜ κ ϑ κ ) | 2 ) 1 2 + ( ∑ κ = 1 m | ∑ ι = 1 n ζ κ ι ϖ ι ( ρ ι θ ι ) − ∑ ι = 1 n ζ κ ι ϖ ι ( ρ ˜ ι θ ι ) + ∨ ι = 1 n δ κ ι ϖ ι ( ρ ι θ ι )
− ∨ ι = 1 n δ κ ι ϖ ι ( ρ ˜ ι θ ι ) + ∧ ι = 1 n η κ ι ϖ ι ( ρ ι θ ι ) − ∧ ι = 1 n η κ ι ϖ ι ( ρ ˜ ι θ ι ) | 2 ) 1 2 ≤ ( ∑ ι = 1 n | ∑ κ = 1 m | μ ι κ | | ω κ ( γ κ ϑ κ ) − ω κ ( γ ˜ κ ϑ κ ) | + ∨ κ = 1 m | ϕ ι κ | | ω κ ( γ κ ϑ κ ) − ω κ ( γ ˜ κ ϑ κ ) | + ∧ κ = 1 m | ψ ι κ | | ω κ ( γ κ ϑ κ ) − ω κ ( γ ˜ κ ϑ κ ) | | 2 ) 1 2 + ( ∑ κ = 1 m | ∑ ι = 1 n | ζ κ ι | | ϖ ι ( ρ ι θ ι ) − ϖ ι ( ρ ˜ ι θ ι ) | + ∨ ι = 1 n | δ κ ι | | ϖ ι ( ρ ι θ ι ) − ϖ ι ( ρ ˜ ι θ ι ) | + ∧ ι = 1 n | η κ ι | | ϖ ι ( ρ ι θ ι ) − ϖ ι ( ρ ˜ ι θ ι ) | | 2 ) 1 2 ≤ ( ∑ ι = 1 n ( ∑ κ = 1 m λ κ ( | μ ι κ | + | ϕ ι κ | + | ψ ι κ | ) ϑ κ | γ κ − γ ˜ κ | ) 2 ) 1 2
+ ( ∑ κ = 1 m ( ∑ ι = 1 n χ ι ( | ζ κ ι | + | δ κ ι | + | η κ ι | ) θ ι | ρ ι − ρ ˜ ι | ) 2 ) 1 2 ≤ ( max 1 ≤ ι ≤ n { ∑ κ = 1 m λ κ ( | μ ι κ | + | ϕ ι κ | + | ψ ι κ | ) ϑ κ } 2 ∑ κ = 1 m | γ κ − γ ˜ κ | 2 ) 1 2 + ( max 1 ≤ κ ≤ m { ∑ ι = 1 n χ ι ( | ζ κ ι | + | δ κ ι | + | η κ ι | ) θ ι } 2 ∑ ι = 1 n | ρ ι − ρ ˜ ι | 2 ) 1 2 ≤ max 1 ≤ ι ≤ n { ∑ κ = 1 m λ κ ( | μ ι κ | + | ϕ ι κ | + | ψ ι κ | ) ϑ κ } ‖ γ − γ ˜ ‖ + max 1 ≤ κ ≤ m { ∑ ι = 1 n χ ι ( | ζ κ ι | + | δ κ ι | + | η κ ι | ) θ ι } ‖ ρ − ρ ˜ ‖ .
结合条件(2)与上式,我们有
‖ ( Π , Ξ ) ( ρ , γ ) − ( Π , Ξ ) ( ρ ˜ , γ ˜ ) ‖ 2 ≤ ‖ ρ − ρ ˜ ‖ 2 + ‖ γ − γ ˜ ‖ 2 = ‖ ( ρ , γ ) − ( ρ ˜ , γ ˜ ) ‖ 2 .
故 ( Π , Ξ ) 是一个压缩映射,从而存在唯一的不动点 ( ρ ∗ , γ ∗ ) 使得 ( Π , Ξ ) ( ρ ∗ , γ ∗ ) = ( ρ ∗ , γ ∗ ) ,即
ρ ι ∗ = ∑ κ = 1 m μ ι κ ω κ ( γ κ ∗ ϑ κ ) + ∨ κ = 1 m ϕ ι κ ω κ ( γ κ ∗ ϑ κ ) + ∧ κ = 1 m ψ ι κ ω κ ( γ κ ∗ ϑ κ ) + Θ ι ( t ) , γ κ ∗ = ∑ ι = 1 n ζ κ ι ϖ ι ( ρ ι ∗ θ ι ) + ∨ ι = 1 n δ κ ι ϖ ι ( ρ ι ∗ θ ι ) + ∧ ι = 1 n η κ ι ϖ ι ( ρ ι ∗ θ ι ) + Λ κ ( t ) . (3)
式等价于
{ 0 = − θ ι α ι ∗ ( t ) + ∑ κ = 1 m μ ι κ ω κ ( β κ ∗ ( t ) ) + ∨ κ = 1 m ϕ ι κ ω κ ( β κ ∗ ( t ) ) + ∧ κ = 1 m ψ ι κ ω κ ( β κ ∗ ( t ) ) + Θ ι ( t ) , 0 = − ϑ κ β κ ∗ ( t ) + ∑ ι = 1 n ζ κ ι ϖ ι ( α ι ∗ ( t ) ) + ∨ ι = 1 n δ κ ι ϖ ι ( α ι ∗ ( t ) ) + ∧ ι = 1 n η κ ι ϖ ι ( α ι ∗ ( t ) ) + Λ κ ( t ) .
由定义2可知,分数阶BAM模糊神经网络(1)有唯一的平衡点 ς = ( α 1 ∗ , α 2 ∗ , ⋯ , α n ∗ , β 1 ∗ , β 2 ∗ , ⋯ , β n ∗ ) 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 [
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 ι ( t ) = α ι ( t ) − α ι ∗ , ℏ κ ( t ) = ℏ κ ( t ) − ℏ κ ∗ ,从而系统(1)转换后的形式为
{ D t 0 c t ν r ι ( t ) = − θ ι r ι ( t ) + ∑ κ = 1 m μ ι κ ( ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) ) + ∨ κ = 1 m ϕ ι κ ( ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) ) + ∧ κ = 1 m ψ ι κ ( ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) ) , D t 0 c t ν ℏ κ ( t ) = − ϑ κ ℏ κ ( t ) + ∑ ι = 1 n ζ κ ι ( ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) ) + ∨ ι = 1 n δ κ ι ( ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) ) + ∧ ι = 1 n η κ ι ( ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) ) , (4)
(4)的受控形式为
{ D t 0 c t ν r ι ( t ) = − θ ι r ι ( t ) + ∑ κ = 1 m μ ι κ ( ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) ) + ∨ κ = 1 m ϕ ι κ ( ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) ) + ∧ κ = 1 m ψ ι κ ( ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) ) + u ⌢ ι ( t ) , D t 0 c t ν ℏ κ ( t ) = − ϑ κ ℏ κ ( t ) + ∑ ι = 1 n ζ κ ι ( ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) ) + ∨ ι = 1 n δ κ ι ( ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) ) + ∧ ι = 1 n η κ ι ( ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) ) + u ⌣ ι ( t ) , (5)
其中 u ⌢ ι ( t ) 与 u ⌣ κ ( t ) 为如下所设计的线性反馈控制器
{ u ⌢ ι ( t ) = − l ι r ι ( t ) , u ⌣ ι ( t ) = − ℘ κ ℏ κ ( t ) , (6)
l ι ( t ) , ℘ κ ( t ) ∈ 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 ( t ) = 1 2 [ ∑ ι = 1 n r ι 2 ( t ) + ∑ κ = 1 m ℏ κ 2 ( t ) ] .
根据引理3,求 V ( t ) 沿系统(5)在控制器(6)下的Caputo分数阶导数可得
D t 0 c t ν V ( t ) ≤ ∑ ι = 1 n r ι ( t ) D t 0 c t ν r ι ( t ) + ∑ κ = 1 m ℏ κ ( t ) D t 0 c t ν ℏ κ ( t ) = ∑ ι = 1 n r ι ( t ) [ − θ ι r ι ( t ) + ∑ κ = 1 m μ ι κ ( ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) ) + ∨ κ = 1 m ϕ ι κ ( ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) ) + ∧ κ = 1 m ψ ι κ ( ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) ) − l ι r ι ( t ) ]
+ ∑ κ = 1 m ℏ κ ( t ) [ − ϑ κ ℏ κ ( t ) + ∑ ι = 1 n ζ κ ι ( ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) ) + ∨ ι = 1 n δ κ ι ( ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) ) + ∧ ι = 1 n η κ ι ( ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) ) − ℘ κ ℏ κ ( t ) ] ≤ − ∑ ι = 1 n ( θ ι + l ι ) r ι 2 ( t ) − ∑ κ = 1 m ( ϑ κ + ℘ κ ) ℏ κ 2 ( t ) + ∑ ι = 1 n ∑ κ = 1 m | μ ι κ | | r ι ( t ) | | ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) | + ∑ ι = 1 n | r ι ( t ) | | ∨ κ = 1 m ϕ ι κ ( ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) ) | + ∑ ι = 1 n | r ι ( t ) | | ∧ κ = 1 m ψ ι κ ( ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) ) | + ∑ κ = 1 m | ζ κ ι | | ℏ κ ( t ) | | ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) | + ∑ κ = 1 m | ℏ κ ( t ) | | ∨ ι = 1 n δ κ ι ( ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) ) | + ∑ κ = 1 m | ℏ κ ( t ) | | ∧ ι = 1 n η κ ι ( ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) ) | . (7)
由假设1可知
∑ ι = 1 n ∑ κ = 1 m | μ ι κ | | r ι ( t ) | | ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) | ≤ ∑ ι = 1 n ∑ κ = 1 m λ κ | μ ι κ | | r ι ( t ) | | ℏ κ ( t ) | ≤ 1 2 ∑ ι = 1 n ∑ κ = 1 m λ κ | μ ι κ | ( r ι 2 ( t ) + ℏ κ 2 ( t ) ) , (8)
∑ κ = 1 m ∑ ι = 1 n | ζ κ ι | | ℏ κ ( t ) | | ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) | ≤ ∑ κ = 1 m ∑ ι = 1 n χ ι | ζ κ ι | | ℏ κ ( t ) | | r ι ( t ) | ≤ 1 2 ∑ κ = 1 m ∑ ι = 1 n χ ι | ζ κ ι | ( ℏ κ 2 ( t ) + r ι 2 ( t ) ) . (9)
根据引理1和假设1有
∑ ι = 1 n | r ι ( t ) | | ∨ κ = 1 m ϕ ι κ ( ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) ) | ≤ ∑ ι = 1 n ∑ κ = 1 m | r ι ( t ) | | ϕ ι κ | | ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) | ≤ ∑ ι = 1 n ∑ κ = 1 m λ κ | ϕ ι κ | | r ι ( t ) | | ℏ κ ( t ) | ≤ 1 2 ∑ ι = 1 n ∑ κ = 1 m λ κ | ϕ ι κ | ( r ι 2 ( t ) + ℏ κ 2 ( t ) ) , (10)
同理依次可得
∑ ι = 1 n | r ι ( t ) | | ∧ κ = 1 m ψ ι κ ( ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) ) | ≤ ∑ ι = 1 n ∑ κ = 1 m | r ι ( t ) | | ψ ι κ | | ω κ ( β κ ( t ) ) − ω κ ( β κ ∗ ) | ≤ ∑ ι = 1 n ∑ κ = 1 m λ κ | ψ ι κ | | r ι ( t ) | | ℏ κ ( t ) | ≤ 1 2 ∑ ι = 1 n ∑ κ = 1 m λ κ | ψ ι κ | ( r ι 2 ( t ) + ℏ κ 2 ( t ) ) , (11)
∑ κ = 1 m | ℏ κ ( t ) | | ∨ ι = 1 n δ κ ι ( ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) ) | ≤ ∑ κ = 1 m ∑ ι = 1 n | ℏ κ ( t ) | | δ κ ι | | ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) | ≤ ∑ ι = 1 n ∑ κ = 1 m χ ι | δ κ ι | | r ι ( t ) | | ℏ κ ( t ) | ≤ 1 2 ∑ ι = 1 n ∑ κ = 1 m χ ι | δ κ ι | ( r ι 2 ( t ) + ℏ κ 2 ( t ) ) , (12)
∑ κ = 1 m | ℏ κ ( t ) | | ∧ ι = 1 n η κ ι ( ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) ) | ≤ ∑ κ = 1 m ∑ ι = 1 n | ℏ κ ( t ) | | η κ ι | | ϖ ι ( α ι ( t ) ) − ϖ ι ( α ι ∗ ) | ≤ ∑ ι = 1 n ∑ κ = 1 m χ ι | η κ ι | | r ι ( t ) | | ℏ κ ( t ) | ≤ 1 2 ∑ ι = 1 n ∑ κ = 1 m χ ι | η κ ι | ( r ι 2 ( t ) + ℏ κ 2 ( t ) ) . (13)
将(8)~(13)代入(7),我们有
D t 0 c t ν V ( t ) ≤ − 1 2 ∑ ι = 1 n [ 2 θ ι + 2 l ι + χ ι ( | ζ κ ι | + | δ κ ι | + | η κ ι | ) − ∑ κ = 1 m λ κ ( | μ ι κ | + | ϕ ι κ | + | ψ ι κ | ) ] r ι 2 ( t ) − 1 2 ∑ κ = 1 m [ 2 ϑ κ + 2 ℘ κ + χ ι ( | ζ κ ι | + | δ κ ι | + | η κ ι | ) − λ κ ∑ ι = 1 n ( | μ ι κ | + | ϕ ι κ | + | ψ ι κ | ) ] ℏ κ 2 ( t ) , (14)
令 Ω 1 = min 1 ≤ ι ≤ n { 2 θ ι + 2 l ι + χ ι ( | ζ κ ι | + | δ κ ι | + | η κ ι | ) − ∑ κ = 1 m λ κ ( | μ ι κ | + | ϕ ι κ | + | ψ ι κ | ) } , Ω 2 = min 1 ≤ ι ≤ n { 2 ϑ κ + 2 ℘ κ + χ ι ( | ζ κ ι | + | δ κ ι | + | η κ ι | ) − λ κ ∑ ι = 1 n ( | μ ι κ | + | ϕ ι κ | + | ψ ι κ | ) } , Ω = min { Ω 1 , Ω 2 } ,则由(14)可得
D t 0 c t ν V ( t ) ≤ − Ω × 1 2 ( ∑ ι = 1 n r ι 2 ( t ) + ∑ κ = 1 m ℏ κ 2 ( t ) ) = − Ω V ( t ) . (15)
对(15)使用引理4有
V ( t ) ≤ V ( t 0 ) E ν ( − Ω ( t − t 0 ) ν ) , (16)
即
‖ ξ − ξ ∗ ‖ 2 ≤ ‖ ξ ( t 0 ) − ξ ∗ ‖ 2 E ν ( − Ω ( t − t 0 ) ν ) ,
其中 ξ = ( α 1 , α 2 , ⋯ , α n , β 1 , β 2 , ⋯ , β m ) T , ξ ∗ = ( α 1 ∗ , α 2 ∗ , ⋯ , α n ∗ , β 1 ∗ , β 2 ∗ , ⋯ , β m ∗ ) T ,由定义3可知分数阶BAM模糊神经网络(1)在平衡点处是全局M-L镇定的。
注2. 当 ν = 1 时,分数阶BAM模糊神经网络将退化为整数阶BAM模糊神经网络模型,此时定理1和2的结论仍成立。
注3. 由于全局M-L镇定意味着全局渐近镇定,因此分数阶BAM模糊神经网络(1)在平衡点处也是全局渐近镇定的。
本文研究了分数阶BAM模糊神经网络的全局M-L镇定,首先通过构造新的压缩映射并结合不等式技巧与2-范数分析方法严格证明了该模型平衡点的存在唯一性。此外,设计了一种简洁有效的线性反馈控制器,基于分数阶理论得到了分数阶BAM模糊神经网络实现全局M-L镇定的充分性准则。考虑到放大器有限的切换速度以及现实中不可避免的外部扰动,分析具有时滞与外部扰动的神经网络的动力学行为具有重要的应用前景,如何分析具有上述因素的分数阶BAM神经网络的动力学有待未来进一步探究。
国家级大学生创新创业训练计划(202110755094)。
李 洁,陈胜龙,李洪利. 分数阶BAM模糊神经网络的全局Mittag-Leffler镇定Global Mittag-Leffler Stabilization of BAM Fuzzy Neural Networks with Fractional-Order[J]. 理论数学, 2022, 12(11): 1925-1933. https://doi.org/10.12677/PM.2022.1211207