在Hopfield时滞神经网络的研究中，人们通常是利用构造李亚普诺夫函数来分析系统的稳定性。本文利用一类Gronwall积分不等式研究了Hopfield神经网络的稳定性问题，我们得出Hopfield神经网络及其时滞系统全局指数稳定性的新判据，并通过实例仿真验证了结果的有效性和可行性。 When people study Hopfield time-delay neural network, Lyapunov function is usually used to an-alyze the stability of the system. But, in this paper, we study the stability of Hopfield neural network by using Gronwall integral inequalities, and obtain the new criterion of global exponential stability of Hopfield neural network and its delay system. Finally, we demonstrate the validity of the results by a numerical example.
Gronwall积分不等式，Hopfield神经网络，时滞，指数稳定性, Gronwall Integral Inequalities
Hopfield Neural Networks
Time Delays
Exponential Stability
基于Gronwall积分不等式的Hopfield神经网络稳定性的新判据<sup> </sup>
x ˙ ( t ) = − A x ( t ) + B g ( x ( t ) ) + C g ( x ( t − τ ) ) + I (1)
其中 x ( t ) = ( x 1 ( t ) , x 2 ( t ) , ⋯ , x n ( t ) ) T 表示神经元的状态变量， g ( x ( t ) ) = ( g 1 ( x 1 ( t ) ) , g 2 ( x 2 ( t ) ) , ⋯ , g n ( x n ( t ) ) ) 表示激活函数， g ( x ( t − τ ) ) 为时滞项激活函数， A = d i a g { a 1 , a 2 , ⋯ , a n } ( a i > 0 , i = 1 , 2 , ⋯ , n )， B = ( b i j ) n × n ， C = ( c i j ) n × n ，分别表示对应神经元的连接权重系数矩阵，其中，ai表示第i个神经元的自反馈强度， b i j 表示第j个神经元的输出 g j ( x j ( t ) ) 对于第i个神经元的输入的反馈连接强度，如果第j个神经元的输出使第i个神经元激活(或抑制)，则 b i j > 0 (或 b i j < 0 )； c i j 类似。τ表示传输时滞，为某一正常数。
这里我们定义n × n矩阵 M = ( a i j ) n × n 和n维向量 x = ( x 1 , x 2 , ⋯ , x n ) T 的范数为：
‖ M ‖ = λ max ( A T A ) ，其中， λ max ( A T A ) 表示 A T A 的最大特征值， ‖ x ‖ = x 1 2 + x 2 2 + ⋯ + x n 2 。
我们假设激活函数f满足李普希兹条件： ‖ f ( y ) − f ( x ) ‖ ≤ L ‖ y − x ‖ 。
由于系统(2)的激活函数f满足李普希兹条件，所以存在唯一平衡点，这在很多文献已经做了证明 [
15
] - [
20
] ，这里不再重复。令系统(2)的平衡点为x*，设 y ( t ) = x ( t ) − x * 。那么系统(2)可变为
y ˙ ( t ) = − A y ( t ) + B f ( y ( t ) ) + C f ( y ( t − τ ) ) , (2)
其中， f ( y ) = g ( y + x * ) − g ( x * ) ，其它各个符号的含义与系统(1)相同。
u ( t ) ≤ K + ∫ α t u ( s ) g ( s ) d s , α ≤ t ≤ β
则有
u ( t ) ≤ K ⋅ exp [ ∫ α t g ( s ) d s ] , α ≤ t ≤ β .
2. 稳定性分析2.1. 无时滞的情形分析
考虑系统
y ˙ ( t ) = − A y ( t ) + B f ( y ( t ) ) (3)
系统(3)的线性系统为
y ˙ ( t ) = − A y ( t ) (4)
基解矩阵为
exp [ − A t ] = [ e − a 1 t 0 e − a 2 t ⋱ 0 e − a n t ] ,
设初值 t = 0 时，对应初值为 η = ( η 1 , η 2 , ⋯ , η n ) ，那么系统(4)的解可以表示为 Y ( t ) = ( η 1 e − a 1 t , η 2 e − a 2 t , ⋯ , η n e − a n t ) ，取范数 ‖ Y ( t ) ‖ = ( | η 1 | e − a 1 t ) 2 + ( | η 2 | e − a 2 t ) 2 + ⋯ + ( | η n | e − a n t ) 2 ，做以下记号：
I) M = n ⋅ max { | η 1 | , | η 2 | , ⋯ , | η n | } ，显然，M > 0。
II) σ = min { a 1 , a 2 , ⋯ , a n } 。
可得 ‖ Y ( t ) ‖ ≤ M e − σ t 。
定理1：激活函数 f ( y ( t ) ) 满足利普希茨条件下，当 L ‖ B ‖ − σ < 0 时，系统(3)的零解是全局指数稳定的。
证明：设Y(t)为方程(4)的解，由常数变易法得方程(3)的解可以表示为
y ( t ) = Y ( t ) + ∫ 0 t e − A ( t − s ) B f ( y ( s ) ) d s ，实际上，这里 Y ( t ) = e − A t η ，两边取范数
‖ y ( t ) ‖ = ‖ Y ( t ) ‖ + ∫ 0 t ‖ e − A ( t − s ) B f ( y ( s ) ) ‖ d s ≤ M e − σ t + ∫ 0 t ‖ e − A ( t − s ) ‖ ‖ B ‖ ‖ f ( y ( s ) ) ‖ d s ≤ M e − σ t + L ‖ B ‖ ∫ 0 t e − σ ( t − s ) ‖ y ( s ) ‖ d s
两边同时乘以 e σ t ，可得
e σ t ‖ y ( t ) ‖ ≤ M + L ‖ B ‖ ∫ 0 t e σ s ‖ y ( s ) ‖ d s
根据引理1 (Gronwall不等式)得
e σ t ‖ y ( t ) ‖ ≤ M e L ‖ B ‖ ∫ 0 t d s = M e L ‖ B ‖ t
两边同时除以 e σ t ，得 ‖ y ( t ) ‖ ≤ M e ( L ‖ B ‖ − σ ) t ，由已知条件 L ‖ B ‖ − σ < 0 ，可得系统(3)的零解是全局指数稳定的。定理得证。
2.2. 具有时滞的情况分析
设
D ( y ) = [ f 1 ( y 1 ) / y 1 0 f 2 ( y 2 ) / y 2 ⋱ 0 f n ( y n ) / y n ] (5)
系统(3)可以表示为
y ˙ ( t ) = ( − A + B D ( y ) ) y ( t ) (6)
设初值 t = 0 时，对应初值为 η = ( η 1 , η 2 , ⋯ , η n ) ，系统(6)的解为向量函数 φ ( t ) ，设 σ − L ‖ B ‖ = σ 1 (>0)那么由定理1可得
‖ φ ( t ) ‖ ≤ ‖ η ‖ e − σ 1 t (7)
定理2：激活函数 f ( · ) 满足利普希茨条件下，当 e σ 1 τ ‖ C ‖ L − σ 1 < 0 时，系统(2)是全局指数稳定的。
证明：由(5)式，系统(2)改写为
y ˙ ( t ) = ( − A + B D ( y ) ) y ( t ) + C f ( y ( t − τ ) ) (8)
当 t = 0 时，对应初值为 η = ( η 1 , η 2 , ⋯ , η n ) ，方程(8)的解可以表示为
y ( t ) = φ ( t ) + ∫ 0 t e ( L ‖ B ‖ E − A ) ( t − s ) C f ( y ( s − τ ) ) d s
两边取范数，有
‖ y ( t ) ‖ ≤ ‖ φ ( t ) ‖ + ∫ 0 t ‖ e ( L ‖ B ‖ E − A ) ( t − s ) C f ( y ( s − τ ) ) ‖ d s ≤ ‖ φ ( t ) ‖ + ∫ 0 t ‖ e ( L ‖ B ‖ E − A ) ( t − s ) ‖ ‖ C ‖ ‖ f ( y ( s − τ ) ) ‖ d s ≤ ‖ η ‖ e − σ 1 t + ∫ − τ t − τ e ( L ‖ B ‖ − σ ) ( t − τ − s ) ‖ C ‖ L ‖ y ( s ) ‖ d s = ‖ η ‖ e − σ 1 t + e σ 1 t ‖ C ‖ L ∫ − τ t − τ e − σ 1 ( t − s ) ‖ y ( s ) ‖ d s
两边同时乘以 e σ 1 t ，可得
e σ 1 t ‖ y ( t ) ‖ = ‖ η ‖ + e σ 1 t ‖ C ‖ L ∫ − τ t − τ e σ 1 t e − σ 1 ( t − s ) ‖ y ( s ) ‖ d s = ‖ η ‖ + e σ 1 t ‖ C ‖ L ∫ − τ t − τ e σ 1 s ‖ y ( s ) ‖ d s
根据引理1 (Gronwall不等式)得
e σ 1 t ‖ y ( t ) ‖ ≤ ‖ η ‖ e e σ 1 τ ‖ C ‖ L ∫ − τ t − τ d s = ‖ η ‖ e e σ 1 τ ‖ C ‖ L t
所以 ‖ y ( t ) ‖ ≤ ‖ η ‖ e e σ 1 τ ‖ C ‖ L ∫ − τ t − τ d s e − σ 1 t = ‖ η ‖ e ( e σ 1 τ ‖ C ‖ L − σ 1 ) t ，由于 e σ 1 τ ‖ C ‖ L − σ 1 < 0 ，所以系统(8)即时滞系统(2)是全局指数稳定的。
3. 数值仿真
这一节，我们用一个实际例子来验证定理结果的有效性。
例1：考虑如下无时滞二维神经网络模型：
( x ˙ ( t ) y ˙ ( t ) ) = − A ( x ( t ) y ( t ) ) + B ( f 1 ( x ( t ) ) f 2 ( y ( t ) ) ) (9)
其中，激活函数 f i ( u ) = e u − 1 e u + 1 ， i = 1 , 2 ，显然满足李普希兹条件，并且 L = 0. 5 ，
显然，如果我们取 A = [ 1 0 0 1 ] ， B = [ 0.9 0.8 0.4 − 0.5 ] ，那么， σ = 1 ， ‖ B ‖ = 1.3 ， L ‖ B ‖ − σ = − 0.35 ，当 t = 0 时，初值取 ( x ( 0 ) , y ( 0 ) ) = ( 10 , 10 ) ，状态轨线图如图1所示，系统(9)的零解是指数稳定的。
图1. 系统(9)的状态轨线图( L ‖ B ‖ − σ < 0 )
如果我们取 A = [ 2 0 0 2 ] ， B = [ 10 − 20 25 − 10 ] ，那么 σ = 2 ， ‖ B ‖ = 35 ， L ‖ B ‖ − σ = 15.5 ，当 t = 0 时，初值取 ( x ( 0 ) , y ( 0 ) ) = ( 10 , 10 ) ，状态轨线图如图2所示，此时系统(9)是稳定的，但不是指数稳定的。
例2：考虑如下无时滞二维神经网络模型：
( x ˙ ( t ) y ˙ ( t ) ) = − A ( x ( t ) y ( t ) ) + B ( f 1 ( x ( t ) ) f 2 ( y ( t ) ) ) + C ( f 1 ( x ( t − τ ) ) f 2 ( y ( t − τ ) ) ) (10)
其中，激活函数 f i ( u ) = e u − 1 e u + 1 ， i = 1 , 2 ，显然满足李普希兹条件，并且 L = 0. 5 ，
如果我们取 A = [ 1 0 0 1 ] ， B = [ 0.3 0.8 0.2 − 0.5 ] ， C = [ 0.1 − 0.3 0.2 0.5 ] ，时滞 τ = 0. 5 ，那么， σ = 1 ， ‖ B ‖ = 0.9565 ， ‖ C ‖ = 0.5966 ， σ 1 = σ − L ‖ B ‖ = 0.5217 ， e σ 1 τ ‖ C ‖ L − σ 1 = − 0.1344 < 0 ，根据定理2，系统(10)的零解是指数稳定的，当 t = 0 时，初值取 ( x ( 0 ) , y ( 0 ) ) = ( − 10 , 10 ) ，系统状态轨线图如图3所示。
图2. 系统(9)的状态轨线图( L ‖ B ‖ − σ > 0 )
图3. 系统(10)的状态轨线图( e σ 1 τ ‖ C ‖ L − σ 1 < 0 )
图4. 系统(10)的状态轨线图( e σ 1 τ ‖ C ‖ L − σ 1 > 0 )
如果我们取 A = [ 1 0 0 1 ] ， B = [ 0.3 0.8 0.2 − 0.5 ] ， C = [ 1 − 5 4 − 4 ] ，时滞 τ = 0.5 ，那么， σ = 1 ， ‖ B ‖ = 0.9565 ， ‖ C ‖ = 7.2929 ， σ 1 = σ − L ‖ B ‖ = 0.5217 ， e σ 1 τ ‖ C ‖ L − σ 1 = 4.2115 > 0 ，当 t = 0 时，初值取 ( x ( 0 ) , y ( 0 ) ) = ( − 15 , 15 ) ，系统的状态轨线图如图4，显然此时系统(10)是稳定的，但不是指数稳定的。
基金项目
广西自然科学基金项目(2016GXNSFAA380125, 2016GXNSFAA380090)。
文章引用
黄星寿,罗日才,王五生. 基于Gronwall积分不等式的Hopfield神经网络稳定性的新判据A New Criterion for Stability of Hopfield Neural Network Based on Gronwall Integral Inequality[J]. 应用数学进展, 2018, 07(12): 1658-1666. https://doi.org/10.12677/AAM.2018.712194
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