Stability analysis for delayed high-order type of ... - Semantic Scholar
Neurocomputing 165 (2015) 312–329
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Stability analysis for delayed high-order type of Hopfield neural networks with impulses Adnène Arbi a, Chaouki Aouiti a, Farouk Chérif b, Abderrahmen Touati a, Adel M. Alimi c,n a
University of Carthage, Department of Mathematics, Faculty of Sciences of Bizerta, BP W, Jarzouna 7021, Bizerta, Tunisia University of Sousse, Department of Computer science, ISSATS, Laboratory of Math Physics; Specials Functions and Applications LR11ES35, Ecole Supérieure des Sciences et de Technologie, Sousse 4002, Tunisia c University of Sfax, ENIS, REGIM-Lab. (Research Groups in Intelligent Machines), Bp 1173, Sfax 3038, Tunisia b
art ic l e i nf o
a b s t r a c t
Article history: Received 22 January 2014 Received in revised form 5 March 2015 Accepted 7 March 2015 Communicated by Ligang Wu Available online 9 April 2015
This paper can be regarded as the continuation of the work of the authors contained in papers (2015). At the same time, it represents the extension of the papers Lou and Cui (2007, [24]), Sannay (2007, [34]) and Acka et al. (2004, [1]). This work discusses a generalized model of high-order Hopfield-type neural networks with time-varying delays. By utilizing Lyapunov functional method and the linear inequality approach, some new stability criteria for such system are derived. The results are related to the size of delays and impulses. The exponential convergence rate of the equilibrium point is also estimated. Finally, we analyze and interpret four numerical examples proving the efficiency of our theoretical results and showing that impulse can be used to stabilize and exponentially stabilize some high-order Hopfield-type neural networks. & 2015 Elsevier B.V. All rights reserved.
Keywords: High-order Hopfield neural networks Lyapunov functional Time varying delay Impulse Global exponential stability Uniform asymptotic stability Global asymptotic stability Uniform stability
1. Introduction Hopfield neural networks have been extensively studied and developed in recent years (see [3,10,32]), and there has been considerable attention in the literature on Hopfield neural networks with time delays [2,4,5,22,27,43,44]. Continuous-time and discrete-time Hopfield-type neural networks (HNN) have been applied to model identification, optimization, etc. (see [8,11,18,20, 25,26,28,33,35–37,39,40]). On the other hand, there are many impulsive phenomena in biological systems, economics systems, control systems, telecommunication systems and engineering applications, etc., which can be well described by impulsive systems (see [12,14,23]). However, impulsive control has been used for stabilization of chaos and synchronization for secure communications (see [19,21]), impulsive neural networks have been considered in [1,4,5,15,30,34], and the stability, existence of the equilibrium of such networks have been investigated. In [34], an exponential stability of a unique equilibrium state for a n
Corresponding author. E-mail addresses: [email protected] (A. Arbi), [email protected] (C. Aouiti), [email protected] (F. Chérif), [email protected] (A. Touati), [email protected] (A.M. Alimi). http://dx.doi.org/10.1016/j.neucom.2015.03.021 0925-2312/& 2015 Elsevier B.V. All rights reserved.
Hopfield-type neural network with impulses is demonstrated. An exponential stability of Hopfield neural networks with impulses and periodic coefficients is presented in [6]. The authors in [38], proved an exponential stability of the two models: impulsive discrete systems with time delay and impulsive discrete-time stochastic neural networks. It is well known that the artificial recurrent neural networks are subject to instantaneous perturbations and experience change of the state abruptly, that is, do exhibit impulsive effects. Such systems are described by impulsive differential systems which have been used in modeling many practical problems of engineering. Because the fact that high-order neural networks have a stronger approximation property, faster convergence rate, greater storage capacity, higher fault tolerance than lower order neural networks, we consider impulsive highorder Hopfield type neural networks with delays in the present paper. Hence, it is very important to investigate the issue of the stability of high-order HNN with impulses. In the literature, there are few results on the stability of delayedhigh-order HNN with impulses. Higher-order Hopfield neural networks have been investigated recently in [41,42,46]. In [42], Xu et al. obtained some sufficient conditions for ensuring global asymptotic stability of impulsive high-order HNN with time-varying delays by using Lyapunov functional method. In [41], Xu et al. investigate the
A. Arbi et al. / Neurocomputing 165 (2015) 312–329
high-order Hopfield type neural networks with bounded uncertainties, some sufficient conditions are derived to ensure the existence of a globally parametrically asymptotically stable equilibrium point by Lyapunov method. In [31], Rakkiyappan et al. consider existence, uniqueness, and the global asymptotic stability for a class of high-order Hopfield neural networks with mixed delays and impulses. The purpose of this paper is to present some new criteria concerning various types of stability for a class of high-order delayed HNN with impulses by utilizing Lyapunov functional method and the linear inequality approach. The conditions on impulses are different from that presented in [34]. The effects of impulses and delays on the solutions are stressed here. As a special case, several new criteria on global exponential stability and uniform stability for the corresponding high-order HNN without impulses (see [24]) are obtained. To illustrate the validity of these results, three examples are given to illustrate the effectiveness of the outcomes obtained. Those criteria generalize and improve some of the recent results reported in the literature [3,5,9,24,34]. This paper is organized as follows. In Section 2.1, an impulsive high-order Hopfield type neural network model with delays is described. Based on the Lyapunov stability theory and analytic technique, three theorems and six corollaries established the global exponential stability, the uniform asymptotic stability and the global asymptotic stability for high-order HNNs are given in Section 3. The effectiveness, the feasibility and the applicability of the developed results are shown by four numerical examples, in Sections 4. Finally, we draw conclusion and future works. It should be mentioned that the main theoretical results include Theorems 3.1, 3.5 and 3.8.
2. Preliminaries Let R denote the set of real numbers, R þ the set of nonnegative numbers, Z þ the set of positive integers and Rn the n-dimensional real space equipped with the Euclidean norm J J . Consider the following high-order delayed HNN model with impulses 8 n n X X > > > aij f j ðxj ðtÞÞ þ bij g j ðxj ðt τðtÞÞÞ > x_ i ðtÞ ¼ ci xi ðtÞ þ > > > j¼1 j¼1 > < n X n X ð1Þ > þ T ijk g k ðxk ðt τðtÞÞÞg j ðxj ðt τðtÞÞÞ þ I i if t a t k ; t Z t 0 ; > > > > j ¼ 1k ¼ 1 > > > : Δxi nt ¼ t ¼ xi ðt k Þ xi ðt Þ; i ¼ 1; …; n; n; k A Z þ ; k k where n Z 2 corresponds to the number of units in a neural network; the impulse time tk satisfies 0 r t 0 o t 1 o⋯ o t k o ⋯, limk⟶ þ 1 t k ¼ þ 1; the xi(t) corresponds to the membrane potential of the unit i at time t; the ci is a positive constant; the activation functions fj, gj denote, respectively, the measures of response or activation to its incoming potentials of the unit j at time t and t τðtÞ; the Tijk are the second-order synaptic weights of the neural networks (see Fig. 2); the constant aij is the synaptic connection weights of the unit j on the unit i at time t; the constant bij denotes the synaptic connection weights of the unit j on the unit i at time t τðtÞ (see Fig. 1); the Ii is the input unit i; the τðtÞ is the transmission delay such that 0 oτðtÞ r τ and τ_ ðtÞ r ρo 1, t Z t 0 , where τ and ρ are constants. The initial conditions associated with system (1) are of the form xðsÞ ¼ ϕðsÞ;
s A ½t 0 τ; t 0 ;
ð2Þ n
n
n
where x(s), ϕðsÞ A PCð½ τ; 0; R Þ, PCð½ τ; 0; R Þ ¼ ψ : ½ τ; 0⟶R is continuous everywhere except at a finite number of points tk, at which ψðt kþ Þ and ψðt k Þ exist and ψðt kþ Þ ¼ ψðt k Þ . For ψ A PC
313
ð½ τ; 0; Rn Þ, the norm of ψ is defined by Jψ Jτ ¼
sup τrθr0
J ψðθÞ J :
For any t 0 Z 0, let PC δ ðt 0 Þ ¼ fψ A PCð½ τ; 0; Rn Þ : J ψ J τ o δg: Notations 2.1. In the following, the notations:
XT and X 1 mean the transpose of and the inverse of a square matrix X. We will use the notation X 4 0 (or X o0, X Z 0, X r 0) to denote that the matrix X is a symmetric and positive definite (negative definite, positive semidefinite, negative semidefinite) matrix. Id denote the n-dimensional identity matrix. Let λmax ðXÞ, λmin ðXÞ, respectively, denote the largest and smallest eigenvalue of matrix X. cmax ¼ maxi A Λ ci , cmin ¼ mini A Λ ci , iA Λ ¼ f1; 2; …; ng. f ðxðtÞÞ ¼ ½f 1 ðx1 ðtÞÞ; …; f n ðxn ðtÞÞ and gðxðt τðtÞÞÞ ¼ ½g 1 ðx1 ðt τðtÞÞÞ; …; g n ðxn ðt τðtÞÞÞ.
Assume that x ¼ ðx 1 ; x 2 ; …; x n Þ is an equilibrium point of system (1) (see Definition 2.1). Impulsive operator is viewed as perturbation of the equilibrium point x of such system without impulsive effects. We assume that ðiÞ
Δxi nt ¼ t k ¼ xi ðt k Þ xi ðt k Þ ¼ dk ðxi ðt k Þ x i Þ; ðiÞ
dk A R;
i ¼ 1; 2; …; n; k ¼ 1; 2; …
This model is the special case when T ijk ¼ 0, which is the Hopfield neural networks with time delays. In this paper, the following assumptions on the neuron activation functions and amplification function are made. Assumption 2.1. Assume that there exists positive constant χ i 40 such that j g i ðxi Þj r χ i ;
i ¼ 1; …; n:
Assumption 2.2. There exist positive constants M, N Z 0, such that T
f ðyÞf ðyÞ r MyT y;
g T ðyÞgðyÞ rNyT y; 8 y A Rn :
Assumption 2.3. zf i ðzÞ Z0, zg i ðzÞ Z 0, 8 z A R. Some definitions and lemmas of stability for system (1) at its equilibrium point are introduced as follows: Definitions 2.1 (Chen et al. [9]). The constant x ¼ ðx 1 ; …; x n Þ A Rn is said to be an equilibrium point of system (1), if ci x i þ
n X j¼1
aij f j ðx j Þ þ
n X
bij g j ðx j Þ þ
j¼1
n X n X
T ijk g k ðx k Þg j ðx j Þ þI i ¼ 0;
i ¼ 1; …; n:
j¼1k¼1
Now, we need the following basic lemmas used in our work. Lemma 2.1 (Berman and Plemmons [7]). Let X A Rnn , then λmin ðXÞaT a r aT Xa rλmax ðXÞaT a for any a A Rn if X is a symmetric matrix. Lemma 2.2 (Lou and Cui [24]). For any a, bA Rn , the inequality 7 2aT b r aT Xa þ b X 1 b T
holds, where X is any n n matrix with X 4 0.
314
A. Arbi et al. / Neurocomputing 165 (2015) 312–329
a11
b11
X1(t-1)
X1(t-1)
a12
a 13
a21
X2(t-1)
b12
b13
a31
b21
a 33
X3(t-1)
a32
X2(t-1)
b 31
X3(t-1)
b 32
a23
b33
b23
a22
b22
Fig. 1. In (a), the topological synaptic connection weights between the three neurons at time t. In (b), the topological synaptic connection weights between the neurons at time t τðtÞ, with τðtÞ 1.
T111
T211
X1(t-1)
X1(t-1)
T 121
T131
T112
X2(t-1)
T 221
T 113
T231
T212
X3(t-1)
T 123
T133
X2(t-1)
T 213
T 223
T132
X3(t-1)
T233
T232
T122
T222 T311
X1(t-1)
T 321
T331
T312
X2(t-1)
T 313
T 323
X3(t-1)
T333
T332 T322 Fig. 2. In (a), the topological second-order synaptic weights between the three neurons at time t τðtÞ for the dynamics x_ 1 ðtÞ, with τðtÞ 1. In (b), the topological secondorder synaptic weights between the three neurons at time t τðtÞ for the dynamics x_ 2 ðtÞ, with τðtÞ 1. In (c), the topological second-order synaptic weights between the three neurons at time t τðtÞ for the dynamics x_ 3 ðtÞ, with τðtÞ 1.
A. Arbi et al. / Neurocomputing 165 (2015) 312–329
Since x is an equilibrium point of system (1). While using the transformation yi ¼ xi x i , and denote F j ðyj ðtÞÞ ¼ f j ðyj ðtÞ þ x j Þ f j ðx j Þ;
Definitions 2.2 (Zhang and Wang [45]). Letting V : R þ Rn -R þ , for any ðt; xÞ A ½t k 1 ; t k Þ Rn , the upper right-hand Dini derivative of V ðt; xÞ along the solution of (4) is defined by D þ Vðt; xÞ ¼ lim sup
Gj ðyj ðtÞÞ ¼ g j ðyj ðtÞ þ x j Þ g j ðx j Þ;
h-0
n X
aij F j ðyj ðtÞÞ þ
j¼1
þ
n X n X
n X
T ijk ðg j ðxj ðt τðtÞÞ g j ðx j ÞÞÞg k ðxk ðt τðtÞÞÞ
¼ ci yi ðtÞ þ
n X
aij F j ðyj ðtÞÞ þ
j¼1
þ
n X n X
Assume that yðtÞ ¼ yðt 0 ; φÞðtÞ be the solution of (4) through ðt 0 ; φÞ, then the zero solution of (4) is said to be
j¼1
þ ðg k ðxk ðt τðtÞÞÞ g k ðx k ÞÞg j ðx j Þ
o
n X
Definitions 2.3 (Zhang and Wang [45]). Globally exponentially stable, if there exist constants μ 40, M Z 1 such that for any initial value φ,
bij Gj ðyj ðt τðtÞÞÞ
J yðt 0 ; φÞðtÞ J r M J φ J τ e μðt t 0 Þ :
j¼1
ðT ijk g k ðxk ðt τðtÞÞÞ þ T ikj g k ðx k ÞÞg j ðyj ðt τðtÞÞÞ;
j¼1k¼1
so, y_ i ðtÞ ¼ ci yi ðtÞ þ
n X
aij F j ðyj ðtÞÞ þ
j¼1
þ
n X n X
n X
bij g j ðyj ðt τðtÞÞÞ
j¼1
ðT ijk þ T ikj Þξk Gj ðyj ðt τðtÞÞÞ;
ð3Þ
j¼1k¼1
where ξk ¼
T ijk T ikj g ðx ðt τðtÞÞÞþ g ðx Þ: T ijk þ T ikj k k T ijk þ T ikj k k
The high order Hopfield neural networks (3) can be rewritten in the following matrix-vector form: 8 _ ¼ CyðtÞ þ AFðyðtÞÞ þ BGðyðt τðtÞÞÞ yðtÞ > > > > < þ Γ T T n Gðyðt τðtÞÞÞ if t a t k ; t Z t 0 ; ð4Þ yðt Þ ¼ Dk yðt k Þ; k A Z þ ; > > > k > : yðt 0 þ θÞ ¼ φðθÞ; θ A ½ τ; 0; where φðθÞ ¼ xðt 0 þ θÞ x;
yðtÞ ¼ ½y1 ðtÞ; …; yn ðtÞT ;
FðyðtÞÞ ¼ ½F 1 ðy1 ðtÞÞ; …; F n ðyn ðtÞÞT ; Gðyðt τðtÞÞÞ ¼ ½G1 ðy1 ðt τðtÞÞÞ; …; Gn ðyn ðt τðtÞÞÞT ; C ¼ diag½c1 ; …; cn ; B ¼ ½bij nn ;
Γ ¼ diag½ξ; …; ξ;
in which
T i ¼ ½T ijk nn ;
ξ ¼ ½ξ1 ; …; ξn T ;
3. Qualitative analysis of impulsive high-order delayed HNNs In this section, we shall obtain some sufficient conditions for global exponential stability, uniform asymptotic stability, global asymptotic stability and uniform stability of the high-order Hopfield-type neural networks with time-varying delays of system (1). We start with establishing a theorem which proves the global exponential stability of the equilibrium point of system (1)
eτϵ þ eτϵ λmax C 1 BQ 1 BT C 1 þ 2 J χ J 2 o 2; cmin
T n ¼ ½T 1 þ T T1 ; …; T n þ T Tn T ; ð1Þ
ð2Þ
ðnÞ
Dk ¼ ½1 þ dk ; 1 þ dk ; …; 1 þ dk T ;
ξk A ½g k ðxk ðt τðtÞÞÞ; g k ðx k Þ;
Definitions 2.4 (Fu et al. [12]). (P1) Stable, if for any ϵ 4 0 and t 0 Z 0, there exists some δðϵ; t 0 Þ 4 0 such that φ A PC δ ðt 0 Þ implies J yðt 0 ; φÞðtÞ J o ϵ, t Z t 0 . (P2) Uniformly stable, if the δ in (P1) is independent of t0. (P3) Uniformly attractive, if there exists some δ 40 such that for any ϵ 4 0, there exists some T ¼ Tðϵ; δÞ 4 0 such that t 0 Z 0 and φ A PC δ ðt 0 Þ implies J yðt 0 ; φÞðtÞ J o ϵ, t Z t 0 þ T. (P4) Uniformly asymptotically stable, if (P2) and (P3) hold. (P5) Globally asymptotically stable, if (P1) holds and for any given initial value y0 ¼ φ, J yðt 0 ; φÞðtÞ J ⟶0 as t⟶ þ1.
Theorem 3.1. Assume that there exist ϵ 40, δ A ½0; ϵÞ and n n matrix Q 4 0 such that (i) 8 9 ( ) n n > j¼1 j¼1 > > > > 3 X 3 < X T ijk g k ðxk ðt τðtÞÞÞg j ðxj ðt τðtÞÞÞ if t a t k ; t Z t 0 ; þ > > j ¼ 1k ¼ 1 > > > > > with impulse > > > > : Δx j t ¼ t ¼ x ðt Þ x ðt Þ ¼ dðiÞ ðx ðt Þ x Þ; k A Z þ ; i ¼ 1; 2; 3;
In this case, we easily observe that τ ¼ 0.5, M ¼0.36, N ¼0.64, J χ J 2 ¼ 64. For Theorem 3.1, choosing Q¼ Id, then from
2 1 1 i ðiÞ ∏ 1 þdk ¼ ∏ 1 þ 2 o 1; i ¼ 1; 2; 3; k k¼1 k¼1
i
k
i
k
i
k
k
i
k
i
ð5Þ
0:08
0:01
2
we may choose, ϵ ¼ 0:3, δ ¼ 0, γ ¼ ∏1 k ¼ 1 ð1 þ3=k Þ o1. On the other hand, we can compute 2 3 0:2059 0:0832 0:0535 6 0:0832 nT n 0:2895 0:2735 7 T T ¼4 5 and λnmax ¼ 0:0542; 0:0535 0:2735 0:4220
318
A. Arbi et al. / Neurocomputing 165 (2015) 312–329
Fig. 4. The solution trajectory of system (5) with impulses for t A ½0; 10.
Fig. 5. The solution trajectory of system (5) with impulses for t A ½0; 100.
Fig. 6. Phase trajectory of system (5) for t A ½0; 2500.
implies that eτϵ λnmax 2 o
ϵ cmin
max
8 3 > > aij f j ðxj ðtÞÞ þ bij g j ðxj ðt τðtÞÞÞ x_i ðtÞ ¼ ci xi ðtÞ þ > > > > j¼1 j¼1 > > > > 2 X 2 < X T ijk g k ðxk ðt τðtÞÞÞg j ðxj ðt τðtÞÞÞ þ I i if t a t k ; t Z t 0 ; þ > > j ¼ 1k ¼ 1 > > > > > with impulse > > > > : Δx j t ¼ t ¼ x ðt Þ x ðt Þ ¼ dðiÞ ðx ðt Þ x Þ; k A Z þ ; i ¼ 1; 2; i
k
i
i
k
k
k
i
k
i
ð6Þ
B ¼ ½bij 22 ¼
0:09
0:25
0:21
0:45
T 2 ¼ ½T 2jk 22 ¼
τ ¼ 1;
f 1 ðx1 Þ ¼ g 1 ðx1 Þ ¼ tanhð0:53x1 Þ; C ¼ diag½c1 ; c2 ¼
1:9 0
0:29
0:10
0:23
0:14
;
I¼
0:05
0:14
0:06
0:05
;
1:5 : 2
k ¼ 1m P ln maxfηk ; 1g δðt m t 0 Þ ¼ 0:125m 0:128m ¼ 0:03m o γ;
8mAZþ :
On the other hand, we can compute maxn λ
¼ 0:0885;
T
Tn Tn ¼
0:4617
0:1150
0:1150
0:2037
and ϵ cmin
max
8 > > aij f j ðxj ðtÞÞ þ bij g j ðxj ðt τðtÞÞ > x_i ðtÞ ¼ ci xi ðtÞ þ > > > j¼1 j¼1 > > > > 3 X 3 < X T ijk g k ðxk ðt τðtÞÞÞg j ðxj ðt τðtÞÞÞ þ I i if t a t k ; t Z t 0 ; þ > > j ¼ 1k ¼ 1 > > > > > with impulse > > > > : Δx j t ¼ t ¼ x ðt Þ x ðt Þ ¼ dðiÞ ðx ðt Þ x Þ; k A Z þ ; i ¼ 1; 2; 3; i
k
i
k
i
k
k
i
k
ð1Þ
ð2Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ 1=5k 1, t k ¼ k þ 2, k A Z þ . 3 2 0:2 0:1 0 0 7 6 0:2 6 0 5; A ¼ ½aij 33 ¼ 4 0:2 0:2 0:2 0 0 6
and dk ¼ dk ¼ 2 6 6 C ¼ ½cij 33 ¼ 4 0
2
2:2
2
3
6 7 I ¼ 4 1:8 5; 1
0:3
6 T 1 ¼ ½T 1jk 33 ¼ 4 0:1 0:2 2 0:2 6 T 2 ¼ ½T 2jk 33 ¼ 4 0:1 0:2 ð7Þ
2
0:2
pffiffiffi pffiffiffi where τðtÞ ¼ sin ð 2tÞ þ cos ð 2tÞ, the activation functions are the following:
6 T 3 ¼ ½T 3jk 33 ¼ 4 0:1 0:2
f 1 ðx1 Þ ¼ tanhð0:5x1 Þ;
f 2 ðx2 Þ ¼ tanhð0:48x2 Þ;
f 3 ðx3 Þ ¼ tanhð0:6x3 Þ;
Thus, we have
g 1 ðx1 Þ ¼ tanhð0:3x1 Þ;
g 2 ðx2 Þ ¼ tanhð0:8x2 Þ;
g 3 ðx3 Þ ¼ tanhð0:73x3 Þ
τ ¼ 2;
0:1
6 B ¼ ½bij 33 ¼ 4 0:1 0:3 2
i
0:5
M ¼ 0:36;
0:1 0:3 0:2 0:2 0:2 0:1 0:1
0:2
0:2 0:1 3
7 0:2 5; 0:2 0:2
0:2 7 5; 0:2 0:2
3
0:2
0:2 7 5:
0:1
0:2
N ¼ 0:64;
3
J χ J 2 ¼ 0:64:
0:3
3
0:5 7 5; 0:2
0:3
3
0:2 7 5; 0:2
A. Arbi et al. / Neurocomputing 165 (2015) 312–329
321
Fig. 11. The solution trajectory of system (7) without impulses for t A ½0; 30 and t A ½0; 5.
2
T
nT
1:0900 6 T ¼ 4 0:0000 0:0300 n
0:0000 1:0300 0:4400
3 0:0300 0:4400 7 5;
λmax ðI þ T
nT
n
T Þ ¼ 2:4864:
1:0600
By choosing ϵn ¼ 0:2 and σ ¼ 0:17, we obtain 8 9 n <X aij = ϵn þ max þ Mλmax ðAC 1 Þ cmin i ¼ 1;…;n:j ¼ 1 ci ; 1 1 T þ λmax C 1 BQ 1 BT C 1 þ Nλmax Q þ T n T n þ 2 J χ J 2 σ σcmin ¼
0:2 0:2 þ þ 0:64 0:067 þ5:8823 0:0147 6 6 þ :64 2:4864 þ
0:64 o 2; 0:17 36
and ∏t 0 o tk r t maxfξk cmax ; 1g 1 þ ϵn ðt t 0 Þ2
⟶0
while t⟶ þ 1:
Thus, while applying Theorem 3.5, the equilibrium point of system (7) is uniformly asymptotically stable and globally asymptotically stable. This fact is verified by the numerical simulations in Figs. 11–13. On the other hand, by using the Mathematica software, we noticed that
2 þ1 1 2 ðiÞ ∏ max ci 1 þ ds ¼ 6 ∏ 1 þ 2 o8:4; 5s s ¼ 1 i ¼ 1;2;3 s¼1 þ1
so, þ1 2 1 ðiÞ ∏ max ci 1 þ ds λmax C 1 BBT C 1 þ Mλmax ðAC 1 Þ i ¼ 1;2;3 s¼1 þ1 2 1 ðiÞ Jχ J2 þ ∏ max ci 1 þ ds c2min s ¼ 1 i ¼ 1;2;3 8 9 n > > aij f j ðxj ðtÞÞ þ bij g j ðxj ðt τðtÞÞÞ > x_i ðtÞ ¼ ci xi ðtÞ þ > > > j¼1 j¼1 > < 2 X 2 X ð8Þ > T ijk g k ðxk ðt τðtÞÞÞg j ðxj ðt τðtÞÞÞ if t a t k ; þ > > > > j ¼ 1k ¼ 1 > > > : k A Z ; t Z t ; i ¼ 1; 2; þ 0
322
A. Arbi et al. / Neurocomputing 165 (2015) 312–329
Fig. 12. The solution trajectory of system (7) with impulses for t A ½0; 10 and t A ½0; 100.
Fig. 13. Phase trajectory of system (7) for t A ½0; 2500.
with
0 ; 1
A ¼ ½aij 22 ¼
1:6
0:1
0:2
2:4
;
C ¼ diag½c1 ; c2 ¼ B ¼ ½bij 22 ¼
1 0
7
6
5:3
3:2
;
0:05
0 0:02
T 1 ¼ ½T 1jk 22 ¼ T2 ¼ ½T 2jk 22 ¼
f 1 ðx1 Þ ¼ tanhðx1 Þ;
0
; 0:01 0:01 ; 0:01
0:01
f 2 ðx2 Þ ¼ tanhðx2 Þ;
g 1 ðx1 Þ ¼ tanhðx1 Þ;
A. Arbi et al. / Neurocomputing 165 (2015) 312–329
323
Fig. 14. The solution of neural network model (8) for t A ½0; 150.
Fig. 15. Phase trajectory of neural network model (8) without impulse for t A ½0; 150.
g 2 ðx2 Þ ¼ tanhðx2 Þ; τ ¼ 1;
τ_ ðtÞ ¼
τðtÞ ¼
expðtÞ ð1 þ expðtÞÞ2
expðtÞ ; 8t ARþ : 1 þ expðtÞ
o 1:
Now, using the impulsive operator ðiÞ
Δxi ðt k Þ ¼ dk ðxi ðt k Þ x i Þ; ð1Þ dk
i ¼ 1; 2;
ð2Þ ¼ dk
¼ 1:9 A ½ 2; 0, x 1 ¼ 0:3166 and x 2 ¼ 0:3337, where for exponentially stabilizing the system (8). It is clear that the hypothesis of Corollary 3.3 is verified 8 9 ( ) n n
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Stability analysis for delayed high-order type of Hopfield neural networks with impulses Adnène Arbi a, Chaouki Aouiti a, Farouk Chérif b, Abderrahmen Touati a, Adel M. Alimi c,n a
University of Carthage, Department of Mathematics, Faculty of Sciences of Bizerta, BP W, Jarzouna 7021, Bizerta, Tunisia University of Sousse, Department of Computer science, ISSATS, Laboratory of Math Physics; Specials Functions and Applications LR11ES35, Ecole Supérieure des Sciences et de Technologie, Sousse 4002, Tunisia c University of Sfax, ENIS, REGIM-Lab. (Research Groups in Intelligent Machines), Bp 1173, Sfax 3038, Tunisia b
art ic l e i nf o
a b s t r a c t
Article history: Received 22 January 2014 Received in revised form 5 March 2015 Accepted 7 March 2015 Communicated by Ligang Wu Available online 9 April 2015
This paper can be regarded as the continuation of the work of the authors contained in papers (2015). At the same time, it represents the extension of the papers Lou and Cui (2007, [24]), Sannay (2007, [34]) and Acka et al. (2004, [1]). This work discusses a generalized model of high-order Hopfield-type neural networks with time-varying delays. By utilizing Lyapunov functional method and the linear inequality approach, some new stability criteria for such system are derived. The results are related to the size of delays and impulses. The exponential convergence rate of the equilibrium point is also estimated. Finally, we analyze and interpret four numerical examples proving the efficiency of our theoretical results and showing that impulse can be used to stabilize and exponentially stabilize some high-order Hopfield-type neural networks. & 2015 Elsevier B.V. All rights reserved.
Keywords: High-order Hopfield neural networks Lyapunov functional Time varying delay Impulse Global exponential stability Uniform asymptotic stability Global asymptotic stability Uniform stability
1. Introduction Hopfield neural networks have been extensively studied and developed in recent years (see [3,10,32]), and there has been considerable attention in the literature on Hopfield neural networks with time delays [2,4,5,22,27,43,44]. Continuous-time and discrete-time Hopfield-type neural networks (HNN) have been applied to model identification, optimization, etc. (see [8,11,18,20, 25,26,28,33,35–37,39,40]). On the other hand, there are many impulsive phenomena in biological systems, economics systems, control systems, telecommunication systems and engineering applications, etc., which can be well described by impulsive systems (see [12,14,23]). However, impulsive control has been used for stabilization of chaos and synchronization for secure communications (see [19,21]), impulsive neural networks have been considered in [1,4,5,15,30,34], and the stability, existence of the equilibrium of such networks have been investigated. In [34], an exponential stability of a unique equilibrium state for a n
Corresponding author. E-mail addresses: [email protected] (A. Arbi), [email protected] (C. Aouiti), [email protected] (F. Chérif), [email protected] (A. Touati), [email protected] (A.M. Alimi). http://dx.doi.org/10.1016/j.neucom.2015.03.021 0925-2312/& 2015 Elsevier B.V. All rights reserved.
Hopfield-type neural network with impulses is demonstrated. An exponential stability of Hopfield neural networks with impulses and periodic coefficients is presented in [6]. The authors in [38], proved an exponential stability of the two models: impulsive discrete systems with time delay and impulsive discrete-time stochastic neural networks. It is well known that the artificial recurrent neural networks are subject to instantaneous perturbations and experience change of the state abruptly, that is, do exhibit impulsive effects. Such systems are described by impulsive differential systems which have been used in modeling many practical problems of engineering. Because the fact that high-order neural networks have a stronger approximation property, faster convergence rate, greater storage capacity, higher fault tolerance than lower order neural networks, we consider impulsive highorder Hopfield type neural networks with delays in the present paper. Hence, it is very important to investigate the issue of the stability of high-order HNN with impulses. In the literature, there are few results on the stability of delayedhigh-order HNN with impulses. Higher-order Hopfield neural networks have been investigated recently in [41,42,46]. In [42], Xu et al. obtained some sufficient conditions for ensuring global asymptotic stability of impulsive high-order HNN with time-varying delays by using Lyapunov functional method. In [41], Xu et al. investigate the
A. Arbi et al. / Neurocomputing 165 (2015) 312–329
high-order Hopfield type neural networks with bounded uncertainties, some sufficient conditions are derived to ensure the existence of a globally parametrically asymptotically stable equilibrium point by Lyapunov method. In [31], Rakkiyappan et al. consider existence, uniqueness, and the global asymptotic stability for a class of high-order Hopfield neural networks with mixed delays and impulses. The purpose of this paper is to present some new criteria concerning various types of stability for a class of high-order delayed HNN with impulses by utilizing Lyapunov functional method and the linear inequality approach. The conditions on impulses are different from that presented in [34]. The effects of impulses and delays on the solutions are stressed here. As a special case, several new criteria on global exponential stability and uniform stability for the corresponding high-order HNN without impulses (see [24]) are obtained. To illustrate the validity of these results, three examples are given to illustrate the effectiveness of the outcomes obtained. Those criteria generalize and improve some of the recent results reported in the literature [3,5,9,24,34]. This paper is organized as follows. In Section 2.1, an impulsive high-order Hopfield type neural network model with delays is described. Based on the Lyapunov stability theory and analytic technique, three theorems and six corollaries established the global exponential stability, the uniform asymptotic stability and the global asymptotic stability for high-order HNNs are given in Section 3. The effectiveness, the feasibility and the applicability of the developed results are shown by four numerical examples, in Sections 4. Finally, we draw conclusion and future works. It should be mentioned that the main theoretical results include Theorems 3.1, 3.5 and 3.8.
2. Preliminaries Let R denote the set of real numbers, R þ the set of nonnegative numbers, Z þ the set of positive integers and Rn the n-dimensional real space equipped with the Euclidean norm J J . Consider the following high-order delayed HNN model with impulses 8 n n X X > > > aij f j ðxj ðtÞÞ þ bij g j ðxj ðt τðtÞÞÞ > x_ i ðtÞ ¼ ci xi ðtÞ þ > > > j¼1 j¼1 > < n X n X ð1Þ > þ T ijk g k ðxk ðt τðtÞÞÞg j ðxj ðt τðtÞÞÞ þ I i if t a t k ; t Z t 0 ; > > > > j ¼ 1k ¼ 1 > > > : Δxi nt ¼ t ¼ xi ðt k Þ xi ðt Þ; i ¼ 1; …; n; n; k A Z þ ; k k where n Z 2 corresponds to the number of units in a neural network; the impulse time tk satisfies 0 r t 0 o t 1 o⋯ o t k o ⋯, limk⟶ þ 1 t k ¼ þ 1; the xi(t) corresponds to the membrane potential of the unit i at time t; the ci is a positive constant; the activation functions fj, gj denote, respectively, the measures of response or activation to its incoming potentials of the unit j at time t and t τðtÞ; the Tijk are the second-order synaptic weights of the neural networks (see Fig. 2); the constant aij is the synaptic connection weights of the unit j on the unit i at time t; the constant bij denotes the synaptic connection weights of the unit j on the unit i at time t τðtÞ (see Fig. 1); the Ii is the input unit i; the τðtÞ is the transmission delay such that 0 oτðtÞ r τ and τ_ ðtÞ r ρo 1, t Z t 0 , where τ and ρ are constants. The initial conditions associated with system (1) are of the form xðsÞ ¼ ϕðsÞ;
s A ½t 0 τ; t 0 ;
ð2Þ n
n
n
where x(s), ϕðsÞ A PCð½ τ; 0; R Þ, PCð½ τ; 0; R Þ ¼ ψ : ½ τ; 0⟶R is continuous everywhere except at a finite number of points tk, at which ψðt kþ Þ and ψðt k Þ exist and ψðt kþ Þ ¼ ψðt k Þ . For ψ A PC
313
ð½ τ; 0; Rn Þ, the norm of ψ is defined by Jψ Jτ ¼
sup τrθr0
J ψðθÞ J :
For any t 0 Z 0, let PC δ ðt 0 Þ ¼ fψ A PCð½ τ; 0; Rn Þ : J ψ J τ o δg: Notations 2.1. In the following, the notations:
XT and X 1 mean the transpose of and the inverse of a square matrix X. We will use the notation X 4 0 (or X o0, X Z 0, X r 0) to denote that the matrix X is a symmetric and positive definite (negative definite, positive semidefinite, negative semidefinite) matrix. Id denote the n-dimensional identity matrix. Let λmax ðXÞ, λmin ðXÞ, respectively, denote the largest and smallest eigenvalue of matrix X. cmax ¼ maxi A Λ ci , cmin ¼ mini A Λ ci , iA Λ ¼ f1; 2; …; ng. f ðxðtÞÞ ¼ ½f 1 ðx1 ðtÞÞ; …; f n ðxn ðtÞÞ and gðxðt τðtÞÞÞ ¼ ½g 1 ðx1 ðt τðtÞÞÞ; …; g n ðxn ðt τðtÞÞÞ.
Assume that x ¼ ðx 1 ; x 2 ; …; x n Þ is an equilibrium point of system (1) (see Definition 2.1). Impulsive operator is viewed as perturbation of the equilibrium point x of such system without impulsive effects. We assume that ðiÞ
Δxi nt ¼ t k ¼ xi ðt k Þ xi ðt k Þ ¼ dk ðxi ðt k Þ x i Þ; ðiÞ
dk A R;
i ¼ 1; 2; …; n; k ¼ 1; 2; …
This model is the special case when T ijk ¼ 0, which is the Hopfield neural networks with time delays. In this paper, the following assumptions on the neuron activation functions and amplification function are made. Assumption 2.1. Assume that there exists positive constant χ i 40 such that j g i ðxi Þj r χ i ;
i ¼ 1; …; n:
Assumption 2.2. There exist positive constants M, N Z 0, such that T
f ðyÞf ðyÞ r MyT y;
g T ðyÞgðyÞ rNyT y; 8 y A Rn :
Assumption 2.3. zf i ðzÞ Z0, zg i ðzÞ Z 0, 8 z A R. Some definitions and lemmas of stability for system (1) at its equilibrium point are introduced as follows: Definitions 2.1 (Chen et al. [9]). The constant x ¼ ðx 1 ; …; x n Þ A Rn is said to be an equilibrium point of system (1), if ci x i þ
n X j¼1
aij f j ðx j Þ þ
n X
bij g j ðx j Þ þ
j¼1
n X n X
T ijk g k ðx k Þg j ðx j Þ þI i ¼ 0;
i ¼ 1; …; n:
j¼1k¼1
Now, we need the following basic lemmas used in our work. Lemma 2.1 (Berman and Plemmons [7]). Let X A Rnn , then λmin ðXÞaT a r aT Xa rλmax ðXÞaT a for any a A Rn if X is a symmetric matrix. Lemma 2.2 (Lou and Cui [24]). For any a, bA Rn , the inequality 7 2aT b r aT Xa þ b X 1 b T
holds, where X is any n n matrix with X 4 0.
314
A. Arbi et al. / Neurocomputing 165 (2015) 312–329
a11
b11
X1(t-1)
X1(t-1)
a12
a 13
a21
X2(t-1)
b12
b13
a31
b21
a 33
X3(t-1)
a32
X2(t-1)
b 31
X3(t-1)
b 32
a23
b33
b23
a22
b22
Fig. 1. In (a), the topological synaptic connection weights between the three neurons at time t. In (b), the topological synaptic connection weights between the neurons at time t τðtÞ, with τðtÞ 1.
T111
T211
X1(t-1)
X1(t-1)
T 121
T131
T112
X2(t-1)
T 221
T 113
T231
T212
X3(t-1)
T 123
T133
X2(t-1)
T 213
T 223
T132
X3(t-1)
T233
T232
T122
T222 T311
X1(t-1)
T 321
T331
T312
X2(t-1)
T 313
T 323
X3(t-1)
T333
T332 T322 Fig. 2. In (a), the topological second-order synaptic weights between the three neurons at time t τðtÞ for the dynamics x_ 1 ðtÞ, with τðtÞ 1. In (b), the topological secondorder synaptic weights between the three neurons at time t τðtÞ for the dynamics x_ 2 ðtÞ, with τðtÞ 1. In (c), the topological second-order synaptic weights between the three neurons at time t τðtÞ for the dynamics x_ 3 ðtÞ, with τðtÞ 1.
A. Arbi et al. / Neurocomputing 165 (2015) 312–329
Since x is an equilibrium point of system (1). While using the transformation yi ¼ xi x i , and denote F j ðyj ðtÞÞ ¼ f j ðyj ðtÞ þ x j Þ f j ðx j Þ;
Definitions 2.2 (Zhang and Wang [45]). Letting V : R þ Rn -R þ , for any ðt; xÞ A ½t k 1 ; t k Þ Rn , the upper right-hand Dini derivative of V ðt; xÞ along the solution of (4) is defined by D þ Vðt; xÞ ¼ lim sup
Gj ðyj ðtÞÞ ¼ g j ðyj ðtÞ þ x j Þ g j ðx j Þ;
h-0
n X
aij F j ðyj ðtÞÞ þ
j¼1
þ
n X n X
n X
T ijk ðg j ðxj ðt τðtÞÞ g j ðx j ÞÞÞg k ðxk ðt τðtÞÞÞ
¼ ci yi ðtÞ þ
n X
aij F j ðyj ðtÞÞ þ
j¼1
þ
n X n X
Assume that yðtÞ ¼ yðt 0 ; φÞðtÞ be the solution of (4) through ðt 0 ; φÞ, then the zero solution of (4) is said to be
j¼1
þ ðg k ðxk ðt τðtÞÞÞ g k ðx k ÞÞg j ðx j Þ
o
n X
Definitions 2.3 (Zhang and Wang [45]). Globally exponentially stable, if there exist constants μ 40, M Z 1 such that for any initial value φ,
bij Gj ðyj ðt τðtÞÞÞ
J yðt 0 ; φÞðtÞ J r M J φ J τ e μðt t 0 Þ :
j¼1
ðT ijk g k ðxk ðt τðtÞÞÞ þ T ikj g k ðx k ÞÞg j ðyj ðt τðtÞÞÞ;
j¼1k¼1
so, y_ i ðtÞ ¼ ci yi ðtÞ þ
n X
aij F j ðyj ðtÞÞ þ
j¼1
þ
n X n X
n X
bij g j ðyj ðt τðtÞÞÞ
j¼1
ðT ijk þ T ikj Þξk Gj ðyj ðt τðtÞÞÞ;
ð3Þ
j¼1k¼1
where ξk ¼
T ijk T ikj g ðx ðt τðtÞÞÞþ g ðx Þ: T ijk þ T ikj k k T ijk þ T ikj k k
The high order Hopfield neural networks (3) can be rewritten in the following matrix-vector form: 8 _ ¼ CyðtÞ þ AFðyðtÞÞ þ BGðyðt τðtÞÞÞ yðtÞ > > > > < þ Γ T T n Gðyðt τðtÞÞÞ if t a t k ; t Z t 0 ; ð4Þ yðt Þ ¼ Dk yðt k Þ; k A Z þ ; > > > k > : yðt 0 þ θÞ ¼ φðθÞ; θ A ½ τ; 0; where φðθÞ ¼ xðt 0 þ θÞ x;
yðtÞ ¼ ½y1 ðtÞ; …; yn ðtÞT ;
FðyðtÞÞ ¼ ½F 1 ðy1 ðtÞÞ; …; F n ðyn ðtÞÞT ; Gðyðt τðtÞÞÞ ¼ ½G1 ðy1 ðt τðtÞÞÞ; …; Gn ðyn ðt τðtÞÞÞT ; C ¼ diag½c1 ; …; cn ; B ¼ ½bij nn ;
Γ ¼ diag½ξ; …; ξ;
in which
T i ¼ ½T ijk nn ;
ξ ¼ ½ξ1 ; …; ξn T ;
3. Qualitative analysis of impulsive high-order delayed HNNs In this section, we shall obtain some sufficient conditions for global exponential stability, uniform asymptotic stability, global asymptotic stability and uniform stability of the high-order Hopfield-type neural networks with time-varying delays of system (1). We start with establishing a theorem which proves the global exponential stability of the equilibrium point of system (1)
eτϵ þ eτϵ λmax C 1 BQ 1 BT C 1 þ 2 J χ J 2 o 2; cmin
T n ¼ ½T 1 þ T T1 ; …; T n þ T Tn T ; ð1Þ
ð2Þ
ðnÞ
Dk ¼ ½1 þ dk ; 1 þ dk ; …; 1 þ dk T ;
ξk A ½g k ðxk ðt τðtÞÞÞ; g k ðx k Þ;
Definitions 2.4 (Fu et al. [12]). (P1) Stable, if for any ϵ 4 0 and t 0 Z 0, there exists some δðϵ; t 0 Þ 4 0 such that φ A PC δ ðt 0 Þ implies J yðt 0 ; φÞðtÞ J o ϵ, t Z t 0 . (P2) Uniformly stable, if the δ in (P1) is independent of t0. (P3) Uniformly attractive, if there exists some δ 40 such that for any ϵ 4 0, there exists some T ¼ Tðϵ; δÞ 4 0 such that t 0 Z 0 and φ A PC δ ðt 0 Þ implies J yðt 0 ; φÞðtÞ J o ϵ, t Z t 0 þ T. (P4) Uniformly asymptotically stable, if (P2) and (P3) hold. (P5) Globally asymptotically stable, if (P1) holds and for any given initial value y0 ¼ φ, J yðt 0 ; φÞðtÞ J ⟶0 as t⟶ þ1.
Theorem 3.1. Assume that there exist ϵ 40, δ A ½0; ϵÞ and n n matrix Q 4 0 such that (i) 8 9 ( ) n n > j¼1 j¼1 > > > > 3 X 3 < X T ijk g k ðxk ðt τðtÞÞÞg j ðxj ðt τðtÞÞÞ if t a t k ; t Z t 0 ; þ > > j ¼ 1k ¼ 1 > > > > > with impulse > > > > : Δx j t ¼ t ¼ x ðt Þ x ðt Þ ¼ dðiÞ ðx ðt Þ x Þ; k A Z þ ; i ¼ 1; 2; 3;
In this case, we easily observe that τ ¼ 0.5, M ¼0.36, N ¼0.64, J χ J 2 ¼ 64. For Theorem 3.1, choosing Q¼ Id, then from
2 1 1 i ðiÞ ∏ 1 þdk ¼ ∏ 1 þ 2 o 1; i ¼ 1; 2; 3; k k¼1 k¼1
i
k
i
k
i
k
k
i
k
i
ð5Þ
0:08
0:01
2
we may choose, ϵ ¼ 0:3, δ ¼ 0, γ ¼ ∏1 k ¼ 1 ð1 þ3=k Þ o1. On the other hand, we can compute 2 3 0:2059 0:0832 0:0535 6 0:0832 nT n 0:2895 0:2735 7 T T ¼4 5 and λnmax ¼ 0:0542; 0:0535 0:2735 0:4220
318
A. Arbi et al. / Neurocomputing 165 (2015) 312–329
Fig. 4. The solution trajectory of system (5) with impulses for t A ½0; 10.
Fig. 5. The solution trajectory of system (5) with impulses for t A ½0; 100.
Fig. 6. Phase trajectory of system (5) for t A ½0; 2500.
implies that eτϵ λnmax 2 o
ϵ cmin
max
8 3 > > aij f j ðxj ðtÞÞ þ bij g j ðxj ðt τðtÞÞÞ x_i ðtÞ ¼ ci xi ðtÞ þ > > > > j¼1 j¼1 > > > > 2 X 2 < X T ijk g k ðxk ðt τðtÞÞÞg j ðxj ðt τðtÞÞÞ þ I i if t a t k ; t Z t 0 ; þ > > j ¼ 1k ¼ 1 > > > > > with impulse > > > > : Δx j t ¼ t ¼ x ðt Þ x ðt Þ ¼ dðiÞ ðx ðt Þ x Þ; k A Z þ ; i ¼ 1; 2; i
k
i
i
k
k
k
i
k
i
ð6Þ
B ¼ ½bij 22 ¼
0:09
0:25
0:21
0:45
T 2 ¼ ½T 2jk 22 ¼
τ ¼ 1;
f 1 ðx1 Þ ¼ g 1 ðx1 Þ ¼ tanhð0:53x1 Þ; C ¼ diag½c1 ; c2 ¼
1:9 0
0:29
0:10
0:23
0:14
;
I¼
0:05
0:14
0:06
0:05
;
1:5 : 2
k ¼ 1m P ln maxfηk ; 1g δðt m t 0 Þ ¼ 0:125m 0:128m ¼ 0:03m o γ;
8mAZþ :
On the other hand, we can compute maxn λ
¼ 0:0885;
T
Tn Tn ¼
0:4617
0:1150
0:1150
0:2037
and ϵ cmin
max
8 > > aij f j ðxj ðtÞÞ þ bij g j ðxj ðt τðtÞÞ > x_i ðtÞ ¼ ci xi ðtÞ þ > > > j¼1 j¼1 > > > > 3 X 3 < X T ijk g k ðxk ðt τðtÞÞÞg j ðxj ðt τðtÞÞÞ þ I i if t a t k ; t Z t 0 ; þ > > j ¼ 1k ¼ 1 > > > > > with impulse > > > > : Δx j t ¼ t ¼ x ðt Þ x ðt Þ ¼ dðiÞ ðx ðt Þ x Þ; k A Z þ ; i ¼ 1; 2; 3; i
k
i
k
i
k
k
i
k
ð1Þ
ð2Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 þ 1=5k 1, t k ¼ k þ 2, k A Z þ . 3 2 0:2 0:1 0 0 7 6 0:2 6 0 5; A ¼ ½aij 33 ¼ 4 0:2 0:2 0:2 0 0 6
and dk ¼ dk ¼ 2 6 6 C ¼ ½cij 33 ¼ 4 0
2
2:2
2
3
6 7 I ¼ 4 1:8 5; 1
0:3
6 T 1 ¼ ½T 1jk 33 ¼ 4 0:1 0:2 2 0:2 6 T 2 ¼ ½T 2jk 33 ¼ 4 0:1 0:2 ð7Þ
2
0:2
pffiffiffi pffiffiffi where τðtÞ ¼ sin ð 2tÞ þ cos ð 2tÞ, the activation functions are the following:
6 T 3 ¼ ½T 3jk 33 ¼ 4 0:1 0:2
f 1 ðx1 Þ ¼ tanhð0:5x1 Þ;
f 2 ðx2 Þ ¼ tanhð0:48x2 Þ;
f 3 ðx3 Þ ¼ tanhð0:6x3 Þ;
Thus, we have
g 1 ðx1 Þ ¼ tanhð0:3x1 Þ;
g 2 ðx2 Þ ¼ tanhð0:8x2 Þ;
g 3 ðx3 Þ ¼ tanhð0:73x3 Þ
τ ¼ 2;
0:1
6 B ¼ ½bij 33 ¼ 4 0:1 0:3 2
i
0:5
M ¼ 0:36;
0:1 0:3 0:2 0:2 0:2 0:1 0:1
0:2
0:2 0:1 3
7 0:2 5; 0:2 0:2
0:2 7 5; 0:2 0:2
3
0:2
0:2 7 5:
0:1
0:2
N ¼ 0:64;
3
J χ J 2 ¼ 0:64:
0:3
3
0:5 7 5; 0:2
0:3
3
0:2 7 5; 0:2
A. Arbi et al. / Neurocomputing 165 (2015) 312–329
321
Fig. 11. The solution trajectory of system (7) without impulses for t A ½0; 30 and t A ½0; 5.
2
T
nT
1:0900 6 T ¼ 4 0:0000 0:0300 n
0:0000 1:0300 0:4400
3 0:0300 0:4400 7 5;
λmax ðI þ T
nT
n
T Þ ¼ 2:4864:
1:0600
By choosing ϵn ¼ 0:2 and σ ¼ 0:17, we obtain 8 9 n <X aij = ϵn þ max þ Mλmax ðAC 1 Þ cmin i ¼ 1;…;n:j ¼ 1 ci ; 1 1 T þ λmax C 1 BQ 1 BT C 1 þ Nλmax Q þ T n T n þ 2 J χ J 2 σ σcmin ¼
0:2 0:2 þ þ 0:64 0:067 þ5:8823 0:0147 6 6 þ :64 2:4864 þ
0:64 o 2; 0:17 36
and ∏t 0 o tk r t maxfξk cmax ; 1g 1 þ ϵn ðt t 0 Þ2
⟶0
while t⟶ þ 1:
Thus, while applying Theorem 3.5, the equilibrium point of system (7) is uniformly asymptotically stable and globally asymptotically stable. This fact is verified by the numerical simulations in Figs. 11–13. On the other hand, by using the Mathematica software, we noticed that
2 þ1 1 2 ðiÞ ∏ max ci 1 þ ds ¼ 6 ∏ 1 þ 2 o8:4; 5s s ¼ 1 i ¼ 1;2;3 s¼1 þ1
so, þ1 2 1 ðiÞ ∏ max ci 1 þ ds λmax C 1 BBT C 1 þ Mλmax ðAC 1 Þ i ¼ 1;2;3 s¼1 þ1 2 1 ðiÞ Jχ J2 þ ∏ max ci 1 þ ds c2min s ¼ 1 i ¼ 1;2;3 8 9 n > > aij f j ðxj ðtÞÞ þ bij g j ðxj ðt τðtÞÞÞ > x_i ðtÞ ¼ ci xi ðtÞ þ > > > j¼1 j¼1 > < 2 X 2 X ð8Þ > T ijk g k ðxk ðt τðtÞÞÞg j ðxj ðt τðtÞÞÞ if t a t k ; þ > > > > j ¼ 1k ¼ 1 > > > : k A Z ; t Z t ; i ¼ 1; 2; þ 0
322
A. Arbi et al. / Neurocomputing 165 (2015) 312–329
Fig. 12. The solution trajectory of system (7) with impulses for t A ½0; 10 and t A ½0; 100.
Fig. 13. Phase trajectory of system (7) for t A ½0; 2500.
with
0 ; 1
A ¼ ½aij 22 ¼
1:6
0:1
0:2
2:4
;
C ¼ diag½c1 ; c2 ¼ B ¼ ½bij 22 ¼
1 0
7
6
5:3
3:2
;
0:05
0 0:02
T 1 ¼ ½T 1jk 22 ¼ T2 ¼ ½T 2jk 22 ¼
f 1 ðx1 Þ ¼ tanhðx1 Þ;
0
; 0:01 0:01 ; 0:01
0:01
f 2 ðx2 Þ ¼ tanhðx2 Þ;
g 1 ðx1 Þ ¼ tanhðx1 Þ;
A. Arbi et al. / Neurocomputing 165 (2015) 312–329
323
Fig. 14. The solution of neural network model (8) for t A ½0; 150.
Fig. 15. Phase trajectory of neural network model (8) without impulse for t A ½0; 150.
g 2 ðx2 Þ ¼ tanhðx2 Þ; τ ¼ 1;
τ_ ðtÞ ¼
τðtÞ ¼
expðtÞ ð1 þ expðtÞÞ2
expðtÞ ; 8t ARþ : 1 þ expðtÞ
o 1:
Now, using the impulsive operator ðiÞ
Δxi ðt k Þ ¼ dk ðxi ðt k Þ x i Þ; ð1Þ dk
i ¼ 1; 2;
ð2Þ ¼ dk
¼ 1:9 A ½ 2; 0, x 1 ¼ 0:3166 and x 2 ¼ 0:3337, where for exponentially stabilizing the system (8). It is clear that the hypothesis of Corollary 3.3 is verified 8 9 ( ) n n
