Novel LMI-based condition on global asymptotic stability for BAM ...
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Author's personal copy Neurocomputing 136 (2014) 213–223
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Novel LMI-based condition on global asymptotic stability for BAM neural networks with reaction–diffusion terms and distributed delays$ Zhiyong Quan a, Lihong Huang a,b, Shenghua Yu c,n, Zhengqiu Zhang a a
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, PR China Department of Information Technology, Hunan Women's University, Changsha, Hunan 410004, PR China c School of Economics and Trade, Hunan University, Changsha, Hunan 410079, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 27 June 2013 Received in revised form 4 January 2014 Accepted 6 January 2014 Communicated by S. Arik Available online 12 February 2014
In this paper, under the assumption that the activation functions only satisfy global Lipschitz conditions, a novel LMI-based sufficient condition for global asymptotic stability of equilibrium point of a class of BAM neural networks with reaction–diffusion terms and distributed delays is obtained by using degree theory, LMI method, inequalities technique and constructing Lyapunov functionals. In our results, the assumptions for boundedness and monotonicity in existing papers on the activation functions are removed. & 2014 Elsevier B.V. All rights reserved.
Keywords: BAM neural network Reaction–diffusion term Global asymptotic stability LMI method Degree theory Lyapunov functional
1. Introduction Bidirectional associative memory (BAM) neural networks model, known as an extension of the unidirectional auto-associator of Hopfield [1–3], was first introduced by Kosko [4,5]. This class of networks possess good application prospects in some fields such as pattern recognition, signal and image process, and artificial intelligence. Such applications heavily depend on the dynamical behavior of the neural networks. Thus, the study of the dynamical behavior is a very necessary step for practical design of neural networks. So far, many stability results such as global exponential stability and global asymptotic stability have been obtained for BAM neural networks with discrete time delays [6–11]. Because neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths, there will be a
☆ Project supported by the Funds of National Natural Science Foundation of China (Nos. 11371127, 71171077), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China (No. 20091341) and the Fund of National Social Science (No. 12BTJ014). n Corresponding author. Tel.: +86 073188821296; fax: +86 073188684825. E-mail addresses: [email protected] (Z. Quan), [email protected] (L. Huang), [email protected] (S. Yu), [email protected] (Z. Zhang).
http://dx.doi.org/10.1016/j.neucom.2014.01.011 0925-2312 & 2014 Elsevier B.V. All rights reserved.
distribution of propagation delays. It is desired to model them by introducing continuously distributed delays [1–4]. There are generally two kinds of continuously distributed delays in the neural networks system, i.e., finitely distributed delays and infinitely distributed delays. The following neural networks with finitely distributed delays: Z t n x0i ðtÞ ¼ ai ðxi ðtÞÞ þ ∑ wij g j ðxj ðsÞÞ ds ð1:1Þ j¼1
t τðtÞ
and their variants have been studied in [5–9] based on linear matrix inequality method and other methods. Similarly, the following neural networks with infinitely distributed delays: Z t n x0i ðtÞ ¼ ai ðxi ðtÞÞ þ ∑ wij K ij ðt sÞg j ðxj ðsÞÞ ds ð1:2Þ j¼1
1
and their variants have been studied in [2–4,10–20]. The global stability results for system (1.2) and its variants are expressed in different forms, such as M matrix form and algebraic inequality forms. If we consider dynamic behavior of neural networks which only depends on time, the model of BAM neural networks is an ordinary equation or a functional differential equation (see [21,22]). In the strict sense, however, diffusion effect cannot be avoided in the neural network models when electrons are moving in an asymmetrical electromagnetic field. So we must consider that the space is
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varying with the time. Thus it is desired to model neural networks by introducing continuously distributed delays and reaction–diffusion terms. So, the study of stability for neural networks with distributed time delays and reaction–diffusion terms should be a focused topic of theoretical as well as practical importance. So far, some authors have discussed the stability of some one-dimensional neural networks with reaction–diffusion terms and distributed delays, for example, see [12,16,23–31] by different ways, such as M matrix method, LMI method and algebraic method. Since two-dimensional neural networks with reaction–diffusion terms and distributed delays consider the interaction between two neural networks system, then two-dimensional neural networks with reaction–diffusion terms and distributed delays will be a greater neural network system and will have more colorful functions in pattern recognition, parallel computing, associative memory, and combinatorial optimization. Hence, the studies of stability behavior of two-dimensional neural networks with reaction–diffusion terms and distributed delays are of greater interest than the studies of stability of one-dimensional neural networks with reaction–diffusion terms and distributed delays. Motivated by the above idea, in [23], the authors considered the following two-dimensional BAM neural networks with distributed delays and reaction–diffusion terms: 8 l m > ∂u ðt; xÞ ∂ ∂ui > > i ¼ ∑ ai ui þ ∑ wji g j ðvj Þ D > ik > ∂t ∂x ∂x > k j¼1 k¼1 k > > > Z t > n > > n > > kji ðt sÞg j ðvj ðs; xÞÞ ds þ I i ; þ ∑ wji > < 1 j¼1 ð1:3Þ l n > ∂vj ðt; xÞ ∂ > n ∂vj > > ¼ ∑ Dik bj vj þ ∑ hij f i ðui Þ > > ∂xk ∂t > i¼1 k ¼ 1∂xk > > Z > t > n > n n > > kij ðt sÞf i ðui ðs; xÞÞ ds þ J j ; þ ∑ hij > : 1 i¼1 for i; j ¼ 1; 2; …; n; t 4 0, where x ¼ ðx1 ; x2 ; …; xl ÞT A Ω Rl ; Ω is a bounded compact set with smooth boundary ∂Ω and μðΩÞ 4 0 in space Rl ; u ¼ ðu1 ; u2 ; …; un ÞT A Rn ; v ¼ ðv1 ; v2 ; …; vn ÞT A Rn ; ui ðt; xÞ and vj ðt; xÞ are the state of the ith neurons and the jth neurons at time t and in space x, respectively; I i and J j denote the external inputs on the ith neurons and the jth neurons, respectively; ai 4 0; n bj 4 0; wji ; wnji ; hij ; hij are constants, ai and bj denote the rate with which the ith neurons and the jth neurons will reset their potential to the resting state in isolation when disconnected from n the networks and external inputs respectively; wji ; wnji ; hij ; hij denote the connection weights. Smooth functions Dik ¼ Dik ðt; x; uÞ Z 0 and Dnjk ¼ Dnjk ðt; x; vÞ Z 0 correspond to the transmission diffusion operators along the ith neurons and the jth neurons, respectively. The boundary conditions and initial conditions of system (1.3) are given by 8 > ∂ui ∂ui ∂ui ∂ui T > > ¼ ; ; …; ¼ 0; i ¼ 1; 2; …; n; > < ∂n ∂x1 ∂x2 ∂xl T > ∂vj ∂vj ∂vj ∂vj > > ¼ ; ; …; ¼ 0; j ¼ 1; 2; …; n > : ∂n ∂x1 ∂x2 ∂xl
terms: 8 l n ∂ui ðt; xÞ ∂ ∂ui > > ¼ ∑ ai ui þ ∑ wji f j ðvj Þ D > ik > > ∂t ∂xk > j¼1 k ¼ 1∂xk > > > n > > n > > þ ∑ wji g j ðvj ðt sji ðtÞ; xÞÞ þ I i ; > < j¼1 l n > ∂v ðt; xÞ ∂vj ∂ > j > > ¼ ∑ Dnik bj vj þ ∑ hij f i ðui Þ > > ∂x ∂x ∂t > k k i ¼ 1 k¼1 > > > n > > n > > þ ∑ hij g i ðui ðt τij ðtÞ; xÞÞ þ J j : : i¼1
In [40], by using the method of variation parameter and inequality technique, the sufficient condition in inequality form for the global exponential stability of equilibrium point of system (1.4) is obtained under the assumptions that the activation functions satisfy bounded condition and global Lipschitz condition. In [32], the authors are concerned with the following interval general two-dimensional BAM neural networks with reaction– diffusion terms and multiple time-varying delays: 8 l n > > > ∂ui ðt; xÞ ¼ ∑ ∂ Dik ∂ui ai ui þ ∑ T ji f j ½uj ðt; xÞ; vj ðt τj ðtÞ; xÞ þ Ii ; > > > ∂x ∂t ∂x k j¼1 k¼1 k > < l n ∂vj ðt; xÞ ∂vj ∂ > bj vj þ ∑ Sij g i ½ui ðt δi ðtÞ; xÞ; vi ðt; xÞ þ J j ; ¼ ∑ C jk > > ∂t ∂x ∂x > k k i¼1 k¼1 > > > : i ¼ 1; 2; …; n; j ¼ 1; …; n:
ð1:5Þ The global exponential stability of equilibrium point of system (1.5) is established by using degree theory and analysis technique. The stability result is expressed in algebraic inequality form. So far, some LMI-based global stability results have been obtained for one-dimensional neural networks with reaction– diffusion terms and time delays under the assumption that the activation functions satisfy boundedness or monotonicity conditions. We only find a paper [31] which established LMI-based condition for global exponential stability of equilibrium point of a class of neural networks with reaction–diffusion terms and time delays under the assumption that the activation functions only satisfy global Lipschitz conditions. To the best of the authors' knowledge, no LMI-based stability results have been published for two-dimensional neural networks with reaction–diffusion terms and time delays. Motivated by the above discussion, in this paper, our purpose is to establish a novel LMI-based sufficient condition on global asymptotic stability for system (1.3) under the assumption that the activation functions only satisfy global Lipschitz conditions. The paper is organized as follows. In Section 2, some lemmas are given. In Section 3, a novel LMI-based sufficient condition is derived for the existence of an equilibrium point of system (1.3). In Section 4, the novel LMI-based sufficient condition on global asymptotic stability of equilibrium point of system (1.3) is obtained. In Section 5, an example is given to show the effectiveness of our results.
2. Preliminaries
and ( ui ðs; xÞ ¼ ϕui ðs; xÞ;
s A ð 1; 0;
i ¼ 1; 2; …; n;
vj ðs; xÞ ¼ ϕvj ðs; xÞ;
s A ð 1; 0;
j ¼ 1; 2; …; n:
In [23], by using degree theory, M-matrix theory and constructing proper Lyapunov functionals, the authors established global exponential stability of equilibrium point of system (1.3). Their global stability results are expressed in algebraic inequality form. In [40], the authors considered the following two-dimensional BAM neural networks with time-varying delays and reaction–diffusion
ð1:4Þ
First we introduce some notations as follows: Fð0; 0Þ ¼ ðF 1 ð0; 0Þ; F 2 ð0; 0Þ; …; F n ð0; 0Þ; F n þ 1 ð0; 0Þ; …; F 2n ð0; 0ÞÞ; r 1 ¼ 2 min max fpi ai g; min fqj bj g ; 1rirn
1rjrn
r 2 ¼ 2 max max fpi jF i ð0; 0Þjg; max fqj jF j þ n ð0; 0Þjg ; 1rirn
r¼n
max
1 r i r n;1 r j r n
fr 2ij w2ij g;
1rjrn
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rn ¼ n l¼n
1 r i r n;1 r j r n
2 2
max
1 r i r n;1 r j r n
n
l ¼n
a11 a 21 ¼ ⋮ a n1 y1
fðr nij Þ2 ðwnij Þ2 g;
max
max
flji hji g;
1 r i r n;1 r j r n
n
n
fðlji Þ2 ðhji Þ2 g;
A7 ¼ α12 α23 α13 α22 ; B4 ¼ β22 β33 β223 þ jβ12 β33 β13 β23 j;
a22 ⋮
⋯ ⋱
a2n ⋮
an2
⋯
ann
y2
⋯
yn
0 0 ðy1 z1 þ ⋯ þ yn zn Þ 0
3. Existence of an equilibrium point
B5 ¼ β11 β33 β213 þ jβ12 β33 β23 β13 jþ jβ11 β23 β13 β12 j; B6 ¼ β11 β22 β212 þ jβ11 β23 β13 β12 j; B7 ¼ β12 β23 β13 β22 : Now, we introduce several lemmas which will be used in Sections 3 and 4. Lemma 1 (Guo et al. [33]). Let Hðλ; xÞ : ½0; 1 Ω-Rn be a continuous homotopic mapping. If Hðλ; xÞ ¼ y has no solutions in ∂Ω for λ A ½0; 1 and y A Rn \Hðλ; ∂ΩÞ, where ∂Ω denotes the boundary of Ω, then the topological degree degðHðλ; xÞ; Ω; yÞ of Hðλ; xÞ is a constant which is independent of λ. In this case, degðHð0; xÞ; Ω; yÞ ¼ degðHð1; xÞ; Ω; yÞ. Lemma 2 (Guo et al. [33]). Let HðxÞ : Ω-Rn be a continuous mapping. If degðHðxÞ; Ω; yÞ a 0, then there exists at least one solution of HðxÞ ¼ y in Ω. Lemma 3 (Guo et al. [33]). Let Ω Rn be a nonempty, bounded, open set and f : Rn -Rn be an Ω-admissible map, i.e., f ðxÞ a 0 for all x A ∂Ω. Then degðf ; ΩÞ ¼ ∑x A f 1 ð0Þ \ Ω signdet Df ðxÞ, where det Df ðxÞ denotes Jacobi determinant of f(x) at point x, signdet Df ðxÞ denotes the symbol of Jacobi determinant of f(x) at point x. !2 n
a1n
Since ∑ni¼ 1 ∑nj¼ 1 aij xi xj is a positive definite quadratic form, then A is a positive definite matrix. Hence jAjzT Az o0. Namely, f ðy1 ; y2 ; …; yn Þ is a negative definite quadratic form. □
A6 ¼ α11 α22 α212 þ jα11 α23 α13 α12 j;
n
n
i¼1
i¼1
r ∑ jak j2 ∑ jbk j2 :
∑ jak bk j
⋯
¼ jAjyT z ¼ jAjzT Az:
A5 ¼ α11 α33 α213 þ jα12 α33 α23 α13 j þ jα11 α23 α13 α12 j;
i¼1
a12
¼ jAjðy1 z1 þ ⋯ þ yn zn Þ
A4 ¼ α22 α33 α223 þ jα12 α33 α13 α23 j;
Lemma 4.
215
Proof. This is a well-known inequality and its proof is omitted.
□
Lemma 5. If ∑ni¼ 1 ∑nj¼ 1 aij xi xj ðaij ¼ aji Þ is a positive definite quadratic form, then a11 a12 ⋯ a1n y1 a 21 a22 ⋯ a2n y2 ⋮ ⋱ ⋮ f ðy1 ; y2 ; …; yn Þ ¼ ⋮ a n1 an2 ⋯ ann yn y1 y2 ⋯ yn 0
Theorem 3.1. We assume that the following conditions hold. ðH 1 Þ There exist positive constants βj ; αi ði; j ¼ 1; 2; …; nÞ such that for 8 u; v A R, jf i ðuÞ f i ðvÞj rαi ju vj; jg j ðuÞ g i ðvÞj r βj ju vj: n
ðH 2 Þ The delay kernels kij ; kji : ½0; 1Þ-½0; 1Þ ði; j ¼ 1; 2; …; nÞ are real-valued non-negative continuous functions which satisfy the following conditions: Z 1 Z 1 n kij ðsÞ ds ¼ 1; kji ðsÞ ds ¼ 1: 0
0
ðH 3 Þ There exist positive diagonal n order matrices P; Q , Y i ði ¼ 1; 2; 3; 4Þ, K; N, positive definite3-order matrices M ¼ ðαij Þ; Nn ¼ ðβij Þ ði; j ¼ 1; 2; 3Þ with α22 α33 4 α223 ; α11 α33 4α213 ; β22 β33 4 β223 ; β11 β33 4 β213 and n-order matrices P 2 ¼ diagðp11 ; p22 ; …; pnn Þ; M 2 ¼ diagðmn11 ; mn22 ; …; mnnn Þ; P 3 ¼ ðr ij Þnn ; P 4 ¼ ðr nij Þnn ; M 3 n ¼ ðlji Þnn ; M 4 ¼ ðlji Þnn such that 0 1 0 0 T 11 T 12 T 13 B n T 22 0 0 0 C B C B C n C o 0; n n T 0 0 Q1 ¼ B 33 B C B n n n T 44 T 45 C @ A n n n n T 55 0 1 0 t 11 t 12 t 13 0 B n t 0 0 0 C 22 B C B C n C o0; n n t 0 0 Q2 ¼ B 33 B C B n n n t 44 t 45 C @ A n n n n t 55 where T 11 ¼ PAþ FY 1 F þ FKF þ A6 I;
T 12 ¼ PW þA7 I;
is a negative definite quadratic form. Proof. Let y¼Az, that is 0 1 0 10 1 y1 a11 a12 ⋯ a1n z1 B C B B z2 C B y2 C B a21 a22 ⋯ a2n C CB C B C¼B CB C: B⋱C @ ⋮ ⋮ ⋱ ⋮ A@ ⋱ A @ A yn an1 an2 ⋯ ann zn Then a11 a 21 f ðy1 ; y2 ; …; yn Þ ¼ ⋮ a n1 y1
a12
⋯
a1n
a22
⋯
a2n
⋮
⋱
⋮
an2
⋯
ann
y2
⋯
yn
a11 z1 þ a12 z2 þ⋯ þ a1n zn a21 z1 þ a22 z2 þ⋯ þ a2n zn an1 z1 þ an2 z2 þ⋯ þ ann zn 0
T 13 ¼ PW n ;
T 22 ¼ Y 2 þ n2 A4 I;
T 44 ¼ PAþ FY 3 F þ P 2 A þ P 22 A2 ; T 33 ¼ N þ n2 A5 I;
T 55 ¼ 2nrI þ 2nr n I þnr n P 2 A Y 4 ;
t 11 ¼ QB þ EY 2 E þ ENE þ B6 I; t 13 ¼ QH n ;
T 45 ¼ P 2 AL1 ; L1 ¼ ðr ij wij Þnn ;
t 12 ¼ QH þ B7 I;
t 22 ¼ Y 1 þ n2 B4 I;
t 44 ¼ QB þ EY 4 E þ M 2 B þM 22 B2 ;
t 45 ¼ M 2 BL2 ; L2 ¼ ðlji hji Þnn ;
t 33 ¼ K þn2 B5 I;
n
n
t 55 ¼ 2nlI þ 2nl I þ nl M 2 B Y 3 ;
p ¼ diagðp1 ; p2 ; …; pn Þ;
q ¼ diagðq1 ; q2 ; …; qn Þ;
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Z. Quan et al. / Neurocomputing 136 (2014) 213–223
F ¼ diagðα1 ; α2 ; …; αn Þ;
E ¼ diagðβ1 ; β2 ; …; βn Þ;
Z 2ð1 þλÞ
n
where l; r; l ; r n are defined in Preliminaries, * denotes the corresponding part in a matrix. Then system (1.3) has at least one equilibrium point.
!
n
∑
j¼1
qj bj v2j
n
2 ∑ pi jui jjF i ð0; 0Þj i¼1
n
j¼1
Proof. Note that if ðunT ; v Þ is an equilibrium point of system (1.3) with un ¼ ðun1 ; un2 ; …; unn ÞT , vn ¼ ðvn1 ; vn2 ; …; vnn ÞT , then ðunT ; vnT ÞT satisfies for i; j ¼ 1; 2; …; n,
2λuT PðW þ W n ÞgðvÞ 2λvT Q ðH þH n Þf ðuÞ:
ð3:1Þ
From ðH 1 Þ, noting that Y i 4 0 ði ¼ 1; 2Þ; K 4 0; N 4 0 are diagonal, we have
ai uni ∑ ðwij þ wnij Þg j ðvnj Þ I i ¼ 0; j¼1 n
∑
i¼1
pi ai u2i þ
2 ∑ qj jvj jjF n þ j ð0; 0Þj
nT T
n
n
n
bj vnj ∑ ðhji þ hji Þf i ðuni Þ J j ¼ 0:
ðf ðuÞÞT Y 1 f ðuÞ ¼ ½f ðuÞ f ð0ÞT Y 1 ½f ðuÞ f ð0Þ r uT FY 1 Fu;
ð3:2Þ
i¼1
Define the following map associated with system (1.3): ! ! ! gðvÞ In Au W þWn 0 Fðu; vÞ ¼ n ; f ðuÞ J Bv 0 H þ Hn
ðgðvÞÞT Y 2 gðvÞ ¼ ½gðvÞ gð0ÞT Y 2 ½gðvÞ gð0Þ r vT EY 2 Ev;
ð3:3Þ
ðf ðuÞÞT Kf ðuÞ r uT FKFu
ð3:4Þ
where
ðgðvÞÞT Ng ðvÞ r vT ENEv: n
W ¼ ðwij Þnn ;
n
W ¼ ðwij Þnn ;
u ¼ ðu1 ; u2 ; …; un ÞT ;
n
n
H ¼ ðhji Þnn ;
and
H ¼ ðhji Þnn ;
v ¼ ðv1 ; v2 ; …; vn ÞT ;
For i; j ¼ 1; 2; …; n; the following inequalities hold: n n n ∑ pii ai ui þ ∑ r ij wij ½g j ðvj Þ g j ð0Þ þ ∑ r nij wnij : i¼1
I n ¼ ðI 1 ; I 2 ; …; I n ÞT ;
Bv ¼ ðb1 v1 ; b2 v2 ; …; bn vn ÞT ;
f ðuÞ ¼ ðf 1 ðu1 Þ; f 2 ðu2 Þ; …; f n ðun ÞÞT ;
and
gðvÞ ¼ ðg 1 ðv1 Þ; g 2 ðv2 Þ; …; g n ðvn ÞÞT :
Obviously, the equilibrium point of system (1.3) is the solution of equation F(u, v)¼ 0. Rewrite F(u, v) as follows: ! ! gðvÞ Au 0 W þWn Fðu; vÞ ¼ þFð0; 0Þ; f ðuÞ Bv 0 H þ Hn where gðvÞ ¼ gðvÞ gð0Þ;
f ðuÞ ¼ f ðuÞ f ð0Þ:
We define a bounded open Ω and a mapping Gðλ; u; vÞ as follows: ( ) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ωo ¼
j¼1
n
n
i¼1
j¼1
n
(
pffiffiffiffiffiffiffiffiffiffi where R 4 2nr 2 =r 1 and Au W þWn Gðλ; u; vÞ ¼ ð1 þ λÞ λ Bv 0
j¼1
¼ 2ð1 þ λÞ
∑
i¼1
n
∑
j¼1
n
n
)2 Z 0:
n
n
j¼1
n
n
j¼1
j¼1
)
þ p2ii a2i u2i þ 2 ∑ r ij wij ðg j ðvj Þ g j ð0ÞÞ ∑ r nij wnij ðg j ðvj Þ g j ð0ÞÞ !
0
gðvÞ f ðuÞ
H þH n
! þ λFð0; 0Þ;
Z0 and n
∑
ð3:6Þ
8"
∂ui ðt; xÞ ∂ ∂u > > ¼ ∑ Dik i ai ui þ ∑ wji g j ðvj Þ > > ∂t ∂xk > j¼1 k ¼ 1∂xk > > > Z t > m > > > > kji ðt sÞg j ðvj ðs; xÞÞ ds þ I i ; þ ∑ wnji > < 1 j¼1 ð5:1Þ l n > ∂vj ðt; xÞ ∂ > n ∂vj > > ¼ ∑ D v þ ∑ h f ðu Þ b j j ij i i > ik > ∂xk ∂t > i¼1 k ¼ 1∂xk > > Z t > > n > n n > > kij ðt sÞf i ðui ðs; xÞÞ ds þ J j ; þ ∑ hij > : 1 i¼1 where
From (4.27), it follows that
i; j; k ¼ 1; 2; Dik ¼ Dik ðt; x; uÞ Z0; Z 1 Z 1 n kji ðsÞ ds ¼ kij ðsÞ ds ¼ 1;
V 0 ðtÞ r 0:
I 1 ¼ 1;
Thus, from the Lyapunov stability theory, it follows that the equilibrium point of system (1.3) is globally asymptotically stable. The completes the proof. □ Remark 1. In our result on global stability, the two inequalities of parameters in stability result in [23,32] are replaced with Q n1 o 0 and
0
Dnjk ¼ Dnjk ðt; x; vÞ Z 0;
0
I 2 ¼ 3;
J 1 ¼ 5; n
wji ¼ wnji ¼ hij ¼ hij ¼ 1; f i ðyÞ ¼ jyj þ 1;
J 2 ¼ 7; ai ¼ bj ¼ 2;
g j ðxÞ ¼ jxj þ 1:
In Theorem 3.1, αi ¼ βj ¼ 1, it is easy to verify that the two inequalities of parameters in [23,32] are not satisfied. Therefore, the result on global stability in [23,32] cannot ensure the global
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reaction–diffusion terms and distributed delays by using degree theory and LMI method. In the results obtained, the assumptions for boundedness and monotonicity on the activation functions in existing papers are removed.
u1 u2 v1 v2
8 6 4
References
2 0 −2 −4 −6
0
2
4
6
8
10
12
14
t Fig. 1. Transient response of state variables in Example 1.
stability of equilibrium point of system (5.1). Since the activation functions in [40] are bounded, while the activation functions in system (5.1) are only global Lipschitz continuous, then the result in [40] cannot ensure the global stability of equilibrium point of system (5.1). When reaction–diffusion terms in system (5.1) become zero, system (5.1) reduces to a BAM neural network with delays. Since the activation functions in system (5.1) are not bounded and monotonic, and are only globally Lipschitz continuous, then the result on global stability in [36–39,41] cannot ensure the global stability of equilibrium point of system (5.1). However, letting α11 ¼ α12 ¼ α13 ¼ α21 ¼ α31 ¼ 1, α22 ¼ 2, α23 ¼ α32 ¼ 1, α33 ¼ 6, β11 ¼ β12 ¼ β13 ¼ β21 ¼ β31 ¼ 1, β22 ¼ 2, β23 ¼ β32 ¼ 1, β33 ¼ 6, then all the conditions in Theorem 3.1 in our paper can be satisfied. By using the Matlab LMI Control toolbox, it can be verified that the LMI in Theorem 3.1 is feasible and 166:6811 0 116:5557 0 P¼ ; Q¼ ; 0 166:6811 0 116:5557 109:2066 0 146:2646 0 ; Y2 ¼ ; Y1 ¼ 0 109:2066 0 146:2646 538:3772 0 90:9774 0 ; Y4 ¼ ; Y3 ¼ 0 538:3772 0 90:9774 129:2227 0 2:34 0 N¼ ; K¼ ; 0 129:2227 0 2:34 2 0 3 0 ; M2 ¼ ; P2 ¼ 0 1 0 4 1:2 2:5 2:4 4 ; M3 ¼ ; P3 ¼ 2:0 1:6 1 1:8 6 4 2 4 ; M4 ¼ : P4 ¼ 3 2 3 1 The conditions ðH 1 Þ; ðH 2 Þ in Theorem 3.1 are obviously satisfied. Then all conditions in Theorem 3.1 are satisfied. By Theorem 3.1 we can conclude that system (5.1) has a unique equilibrium point, which is globally asymptotically stable. The global asympotic stability of system (5.1) in Example 1 is shown in Fig. 1. 6. Conclusions In this paper, under the assumption that the activation functions only satisfy global Lipschitz conditions, a novel LMI-based sufficient condition is obtained for global asymptotic stability of equilibrium point of two-dimensional BAM neural networks with
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Zhiyong Quan was born in 1974 and is a teacher of applied mathematics of Hunan University in China. His field of study is neural networks theory and applications. So far, he has been an author of more than 2 papers.
223
Lihong Huang was born in 1963 and is a professor of applied mathematics of Hunan University in China. His field of study is neural networks theory and applications. So far, he has been an author of more than 100 papers.
Shenghua Yu was born in 1966 and is a professor of applied mathematics of Hunan University in China. His field of study is economic mathematics, neural networks theory and applications. So far, he has been an author of more than 30 papers.
Zhengqiu Zhang was born in 1963 and is a professor of applied mathematics of Hunan University in China. His field of study is neural networks theory and applications. So far, he has been an author of more than 50 papers.
Author's personal copy Neurocomputing 136 (2014) 213–223
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Novel LMI-based condition on global asymptotic stability for BAM neural networks with reaction–diffusion terms and distributed delays$ Zhiyong Quan a, Lihong Huang a,b, Shenghua Yu c,n, Zhengqiu Zhang a a
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, PR China Department of Information Technology, Hunan Women's University, Changsha, Hunan 410004, PR China c School of Economics and Trade, Hunan University, Changsha, Hunan 410079, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 27 June 2013 Received in revised form 4 January 2014 Accepted 6 January 2014 Communicated by S. Arik Available online 12 February 2014
In this paper, under the assumption that the activation functions only satisfy global Lipschitz conditions, a novel LMI-based sufficient condition for global asymptotic stability of equilibrium point of a class of BAM neural networks with reaction–diffusion terms and distributed delays is obtained by using degree theory, LMI method, inequalities technique and constructing Lyapunov functionals. In our results, the assumptions for boundedness and monotonicity in existing papers on the activation functions are removed. & 2014 Elsevier B.V. All rights reserved.
Keywords: BAM neural network Reaction–diffusion term Global asymptotic stability LMI method Degree theory Lyapunov functional
1. Introduction Bidirectional associative memory (BAM) neural networks model, known as an extension of the unidirectional auto-associator of Hopfield [1–3], was first introduced by Kosko [4,5]. This class of networks possess good application prospects in some fields such as pattern recognition, signal and image process, and artificial intelligence. Such applications heavily depend on the dynamical behavior of the neural networks. Thus, the study of the dynamical behavior is a very necessary step for practical design of neural networks. So far, many stability results such as global exponential stability and global asymptotic stability have been obtained for BAM neural networks with discrete time delays [6–11]. Because neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths, there will be a
☆ Project supported by the Funds of National Natural Science Foundation of China (Nos. 11371127, 71171077), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China (No. 20091341) and the Fund of National Social Science (No. 12BTJ014). n Corresponding author. Tel.: +86 073188821296; fax: +86 073188684825. E-mail addresses: [email protected] (Z. Quan), [email protected] (L. Huang), [email protected] (S. Yu), [email protected] (Z. Zhang).
http://dx.doi.org/10.1016/j.neucom.2014.01.011 0925-2312 & 2014 Elsevier B.V. All rights reserved.
distribution of propagation delays. It is desired to model them by introducing continuously distributed delays [1–4]. There are generally two kinds of continuously distributed delays in the neural networks system, i.e., finitely distributed delays and infinitely distributed delays. The following neural networks with finitely distributed delays: Z t n x0i ðtÞ ¼ ai ðxi ðtÞÞ þ ∑ wij g j ðxj ðsÞÞ ds ð1:1Þ j¼1
t τðtÞ
and their variants have been studied in [5–9] based on linear matrix inequality method and other methods. Similarly, the following neural networks with infinitely distributed delays: Z t n x0i ðtÞ ¼ ai ðxi ðtÞÞ þ ∑ wij K ij ðt sÞg j ðxj ðsÞÞ ds ð1:2Þ j¼1
1
and their variants have been studied in [2–4,10–20]. The global stability results for system (1.2) and its variants are expressed in different forms, such as M matrix form and algebraic inequality forms. If we consider dynamic behavior of neural networks which only depends on time, the model of BAM neural networks is an ordinary equation or a functional differential equation (see [21,22]). In the strict sense, however, diffusion effect cannot be avoided in the neural network models when electrons are moving in an asymmetrical electromagnetic field. So we must consider that the space is
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varying with the time. Thus it is desired to model neural networks by introducing continuously distributed delays and reaction–diffusion terms. So, the study of stability for neural networks with distributed time delays and reaction–diffusion terms should be a focused topic of theoretical as well as practical importance. So far, some authors have discussed the stability of some one-dimensional neural networks with reaction–diffusion terms and distributed delays, for example, see [12,16,23–31] by different ways, such as M matrix method, LMI method and algebraic method. Since two-dimensional neural networks with reaction–diffusion terms and distributed delays consider the interaction between two neural networks system, then two-dimensional neural networks with reaction–diffusion terms and distributed delays will be a greater neural network system and will have more colorful functions in pattern recognition, parallel computing, associative memory, and combinatorial optimization. Hence, the studies of stability behavior of two-dimensional neural networks with reaction–diffusion terms and distributed delays are of greater interest than the studies of stability of one-dimensional neural networks with reaction–diffusion terms and distributed delays. Motivated by the above idea, in [23], the authors considered the following two-dimensional BAM neural networks with distributed delays and reaction–diffusion terms: 8 l m > ∂u ðt; xÞ ∂ ∂ui > > i ¼ ∑ ai ui þ ∑ wji g j ðvj Þ D > ik > ∂t ∂x ∂x > k j¼1 k¼1 k > > > Z t > n > > n > > kji ðt sÞg j ðvj ðs; xÞÞ ds þ I i ; þ ∑ wji > < 1 j¼1 ð1:3Þ l n > ∂vj ðt; xÞ ∂ > n ∂vj > > ¼ ∑ Dik bj vj þ ∑ hij f i ðui Þ > > ∂xk ∂t > i¼1 k ¼ 1∂xk > > Z > t > n > n n > > kij ðt sÞf i ðui ðs; xÞÞ ds þ J j ; þ ∑ hij > : 1 i¼1 for i; j ¼ 1; 2; …; n; t 4 0, where x ¼ ðx1 ; x2 ; …; xl ÞT A Ω Rl ; Ω is a bounded compact set with smooth boundary ∂Ω and μðΩÞ 4 0 in space Rl ; u ¼ ðu1 ; u2 ; …; un ÞT A Rn ; v ¼ ðv1 ; v2 ; …; vn ÞT A Rn ; ui ðt; xÞ and vj ðt; xÞ are the state of the ith neurons and the jth neurons at time t and in space x, respectively; I i and J j denote the external inputs on the ith neurons and the jth neurons, respectively; ai 4 0; n bj 4 0; wji ; wnji ; hij ; hij are constants, ai and bj denote the rate with which the ith neurons and the jth neurons will reset their potential to the resting state in isolation when disconnected from n the networks and external inputs respectively; wji ; wnji ; hij ; hij denote the connection weights. Smooth functions Dik ¼ Dik ðt; x; uÞ Z 0 and Dnjk ¼ Dnjk ðt; x; vÞ Z 0 correspond to the transmission diffusion operators along the ith neurons and the jth neurons, respectively. The boundary conditions and initial conditions of system (1.3) are given by 8 > ∂ui ∂ui ∂ui ∂ui T > > ¼ ; ; …; ¼ 0; i ¼ 1; 2; …; n; > < ∂n ∂x1 ∂x2 ∂xl T > ∂vj ∂vj ∂vj ∂vj > > ¼ ; ; …; ¼ 0; j ¼ 1; 2; …; n > : ∂n ∂x1 ∂x2 ∂xl
terms: 8 l n ∂ui ðt; xÞ ∂ ∂ui > > ¼ ∑ ai ui þ ∑ wji f j ðvj Þ D > ik > > ∂t ∂xk > j¼1 k ¼ 1∂xk > > > n > > n > > þ ∑ wji g j ðvj ðt sji ðtÞ; xÞÞ þ I i ; > < j¼1 l n > ∂v ðt; xÞ ∂vj ∂ > j > > ¼ ∑ Dnik bj vj þ ∑ hij f i ðui Þ > > ∂x ∂x ∂t > k k i ¼ 1 k¼1 > > > n > > n > > þ ∑ hij g i ðui ðt τij ðtÞ; xÞÞ þ J j : : i¼1
In [40], by using the method of variation parameter and inequality technique, the sufficient condition in inequality form for the global exponential stability of equilibrium point of system (1.4) is obtained under the assumptions that the activation functions satisfy bounded condition and global Lipschitz condition. In [32], the authors are concerned with the following interval general two-dimensional BAM neural networks with reaction– diffusion terms and multiple time-varying delays: 8 l n > > > ∂ui ðt; xÞ ¼ ∑ ∂ Dik ∂ui ai ui þ ∑ T ji f j ½uj ðt; xÞ; vj ðt τj ðtÞ; xÞ þ Ii ; > > > ∂x ∂t ∂x k j¼1 k¼1 k > < l n ∂vj ðt; xÞ ∂vj ∂ > bj vj þ ∑ Sij g i ½ui ðt δi ðtÞ; xÞ; vi ðt; xÞ þ J j ; ¼ ∑ C jk > > ∂t ∂x ∂x > k k i¼1 k¼1 > > > : i ¼ 1; 2; …; n; j ¼ 1; …; n:
ð1:5Þ The global exponential stability of equilibrium point of system (1.5) is established by using degree theory and analysis technique. The stability result is expressed in algebraic inequality form. So far, some LMI-based global stability results have been obtained for one-dimensional neural networks with reaction– diffusion terms and time delays under the assumption that the activation functions satisfy boundedness or monotonicity conditions. We only find a paper [31] which established LMI-based condition for global exponential stability of equilibrium point of a class of neural networks with reaction–diffusion terms and time delays under the assumption that the activation functions only satisfy global Lipschitz conditions. To the best of the authors' knowledge, no LMI-based stability results have been published for two-dimensional neural networks with reaction–diffusion terms and time delays. Motivated by the above discussion, in this paper, our purpose is to establish a novel LMI-based sufficient condition on global asymptotic stability for system (1.3) under the assumption that the activation functions only satisfy global Lipschitz conditions. The paper is organized as follows. In Section 2, some lemmas are given. In Section 3, a novel LMI-based sufficient condition is derived for the existence of an equilibrium point of system (1.3). In Section 4, the novel LMI-based sufficient condition on global asymptotic stability of equilibrium point of system (1.3) is obtained. In Section 5, an example is given to show the effectiveness of our results.
2. Preliminaries
and ( ui ðs; xÞ ¼ ϕui ðs; xÞ;
s A ð 1; 0;
i ¼ 1; 2; …; n;
vj ðs; xÞ ¼ ϕvj ðs; xÞ;
s A ð 1; 0;
j ¼ 1; 2; …; n:
In [23], by using degree theory, M-matrix theory and constructing proper Lyapunov functionals, the authors established global exponential stability of equilibrium point of system (1.3). Their global stability results are expressed in algebraic inequality form. In [40], the authors considered the following two-dimensional BAM neural networks with time-varying delays and reaction–diffusion
ð1:4Þ
First we introduce some notations as follows: Fð0; 0Þ ¼ ðF 1 ð0; 0Þ; F 2 ð0; 0Þ; …; F n ð0; 0Þ; F n þ 1 ð0; 0Þ; …; F 2n ð0; 0ÞÞ; r 1 ¼ 2 min max fpi ai g; min fqj bj g ; 1rirn
1rjrn
r 2 ¼ 2 max max fpi jF i ð0; 0Þjg; max fqj jF j þ n ð0; 0Þjg ; 1rirn
r¼n
max
1 r i r n;1 r j r n
fr 2ij w2ij g;
1rjrn
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rn ¼ n l¼n
1 r i r n;1 r j r n
2 2
max
1 r i r n;1 r j r n
n
l ¼n
a11 a 21 ¼ ⋮ a n1 y1
fðr nij Þ2 ðwnij Þ2 g;
max
max
flji hji g;
1 r i r n;1 r j r n
n
n
fðlji Þ2 ðhji Þ2 g;
A7 ¼ α12 α23 α13 α22 ; B4 ¼ β22 β33 β223 þ jβ12 β33 β13 β23 j;
a22 ⋮
⋯ ⋱
a2n ⋮
an2
⋯
ann
y2
⋯
yn
0 0 ðy1 z1 þ ⋯ þ yn zn Þ 0
3. Existence of an equilibrium point
B5 ¼ β11 β33 β213 þ jβ12 β33 β23 β13 jþ jβ11 β23 β13 β12 j; B6 ¼ β11 β22 β212 þ jβ11 β23 β13 β12 j; B7 ¼ β12 β23 β13 β22 : Now, we introduce several lemmas which will be used in Sections 3 and 4. Lemma 1 (Guo et al. [33]). Let Hðλ; xÞ : ½0; 1 Ω-Rn be a continuous homotopic mapping. If Hðλ; xÞ ¼ y has no solutions in ∂Ω for λ A ½0; 1 and y A Rn \Hðλ; ∂ΩÞ, where ∂Ω denotes the boundary of Ω, then the topological degree degðHðλ; xÞ; Ω; yÞ of Hðλ; xÞ is a constant which is independent of λ. In this case, degðHð0; xÞ; Ω; yÞ ¼ degðHð1; xÞ; Ω; yÞ. Lemma 2 (Guo et al. [33]). Let HðxÞ : Ω-Rn be a continuous mapping. If degðHðxÞ; Ω; yÞ a 0, then there exists at least one solution of HðxÞ ¼ y in Ω. Lemma 3 (Guo et al. [33]). Let Ω Rn be a nonempty, bounded, open set and f : Rn -Rn be an Ω-admissible map, i.e., f ðxÞ a 0 for all x A ∂Ω. Then degðf ; ΩÞ ¼ ∑x A f 1 ð0Þ \ Ω signdet Df ðxÞ, where det Df ðxÞ denotes Jacobi determinant of f(x) at point x, signdet Df ðxÞ denotes the symbol of Jacobi determinant of f(x) at point x. !2 n
a1n
Since ∑ni¼ 1 ∑nj¼ 1 aij xi xj is a positive definite quadratic form, then A is a positive definite matrix. Hence jAjzT Az o0. Namely, f ðy1 ; y2 ; …; yn Þ is a negative definite quadratic form. □
A6 ¼ α11 α22 α212 þ jα11 α23 α13 α12 j;
n
n
i¼1
i¼1
r ∑ jak j2 ∑ jbk j2 :
∑ jak bk j
⋯
¼ jAjyT z ¼ jAjzT Az:
A5 ¼ α11 α33 α213 þ jα12 α33 α23 α13 j þ jα11 α23 α13 α12 j;
i¼1
a12
¼ jAjðy1 z1 þ ⋯ þ yn zn Þ
A4 ¼ α22 α33 α223 þ jα12 α33 α13 α23 j;
Lemma 4.
215
Proof. This is a well-known inequality and its proof is omitted.
□
Lemma 5. If ∑ni¼ 1 ∑nj¼ 1 aij xi xj ðaij ¼ aji Þ is a positive definite quadratic form, then a11 a12 ⋯ a1n y1 a 21 a22 ⋯ a2n y2 ⋮ ⋱ ⋮ f ðy1 ; y2 ; …; yn Þ ¼ ⋮ a n1 an2 ⋯ ann yn y1 y2 ⋯ yn 0
Theorem 3.1. We assume that the following conditions hold. ðH 1 Þ There exist positive constants βj ; αi ði; j ¼ 1; 2; …; nÞ such that for 8 u; v A R, jf i ðuÞ f i ðvÞj rαi ju vj; jg j ðuÞ g i ðvÞj r βj ju vj: n
ðH 2 Þ The delay kernels kij ; kji : ½0; 1Þ-½0; 1Þ ði; j ¼ 1; 2; …; nÞ are real-valued non-negative continuous functions which satisfy the following conditions: Z 1 Z 1 n kij ðsÞ ds ¼ 1; kji ðsÞ ds ¼ 1: 0
0
ðH 3 Þ There exist positive diagonal n order matrices P; Q , Y i ði ¼ 1; 2; 3; 4Þ, K; N, positive definite3-order matrices M ¼ ðαij Þ; Nn ¼ ðβij Þ ði; j ¼ 1; 2; 3Þ with α22 α33 4 α223 ; α11 α33 4α213 ; β22 β33 4 β223 ; β11 β33 4 β213 and n-order matrices P 2 ¼ diagðp11 ; p22 ; …; pnn Þ; M 2 ¼ diagðmn11 ; mn22 ; …; mnnn Þ; P 3 ¼ ðr ij Þnn ; P 4 ¼ ðr nij Þnn ; M 3 n ¼ ðlji Þnn ; M 4 ¼ ðlji Þnn such that 0 1 0 0 T 11 T 12 T 13 B n T 22 0 0 0 C B C B C n C o 0; n n T 0 0 Q1 ¼ B 33 B C B n n n T 44 T 45 C @ A n n n n T 55 0 1 0 t 11 t 12 t 13 0 B n t 0 0 0 C 22 B C B C n C o0; n n t 0 0 Q2 ¼ B 33 B C B n n n t 44 t 45 C @ A n n n n t 55 where T 11 ¼ PAþ FY 1 F þ FKF þ A6 I;
T 12 ¼ PW þA7 I;
is a negative definite quadratic form. Proof. Let y¼Az, that is 0 1 0 10 1 y1 a11 a12 ⋯ a1n z1 B C B B z2 C B y2 C B a21 a22 ⋯ a2n C CB C B C¼B CB C: B⋱C @ ⋮ ⋮ ⋱ ⋮ A@ ⋱ A @ A yn an1 an2 ⋯ ann zn Then a11 a 21 f ðy1 ; y2 ; …; yn Þ ¼ ⋮ a n1 y1
a12
⋯
a1n
a22
⋯
a2n
⋮
⋱
⋮
an2
⋯
ann
y2
⋯
yn
a11 z1 þ a12 z2 þ⋯ þ a1n zn a21 z1 þ a22 z2 þ⋯ þ a2n zn an1 z1 þ an2 z2 þ⋯ þ ann zn 0
T 13 ¼ PW n ;
T 22 ¼ Y 2 þ n2 A4 I;
T 44 ¼ PAþ FY 3 F þ P 2 A þ P 22 A2 ; T 33 ¼ N þ n2 A5 I;
T 55 ¼ 2nrI þ 2nr n I þnr n P 2 A Y 4 ;
t 11 ¼ QB þ EY 2 E þ ENE þ B6 I; t 13 ¼ QH n ;
T 45 ¼ P 2 AL1 ; L1 ¼ ðr ij wij Þnn ;
t 12 ¼ QH þ B7 I;
t 22 ¼ Y 1 þ n2 B4 I;
t 44 ¼ QB þ EY 4 E þ M 2 B þM 22 B2 ;
t 45 ¼ M 2 BL2 ; L2 ¼ ðlji hji Þnn ;
t 33 ¼ K þn2 B5 I;
n
n
t 55 ¼ 2nlI þ 2nl I þ nl M 2 B Y 3 ;
p ¼ diagðp1 ; p2 ; …; pn Þ;
q ¼ diagðq1 ; q2 ; …; qn Þ;
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F ¼ diagðα1 ; α2 ; …; αn Þ;
E ¼ diagðβ1 ; β2 ; …; βn Þ;
Z 2ð1 þλÞ
n
where l; r; l ; r n are defined in Preliminaries, * denotes the corresponding part in a matrix. Then system (1.3) has at least one equilibrium point.
!
n
∑
j¼1
qj bj v2j
n
2 ∑ pi jui jjF i ð0; 0Þj i¼1
n
j¼1
Proof. Note that if ðunT ; v Þ is an equilibrium point of system (1.3) with un ¼ ðun1 ; un2 ; …; unn ÞT , vn ¼ ðvn1 ; vn2 ; …; vnn ÞT , then ðunT ; vnT ÞT satisfies for i; j ¼ 1; 2; …; n,
2λuT PðW þ W n ÞgðvÞ 2λvT Q ðH þH n Þf ðuÞ:
ð3:1Þ
From ðH 1 Þ, noting that Y i 4 0 ði ¼ 1; 2Þ; K 4 0; N 4 0 are diagonal, we have
ai uni ∑ ðwij þ wnij Þg j ðvnj Þ I i ¼ 0; j¼1 n
∑
i¼1
pi ai u2i þ
2 ∑ qj jvj jjF n þ j ð0; 0Þj
nT T
n
n
n
bj vnj ∑ ðhji þ hji Þf i ðuni Þ J j ¼ 0:
ðf ðuÞÞT Y 1 f ðuÞ ¼ ½f ðuÞ f ð0ÞT Y 1 ½f ðuÞ f ð0Þ r uT FY 1 Fu;
ð3:2Þ
i¼1
Define the following map associated with system (1.3): ! ! ! gðvÞ In Au W þWn 0 Fðu; vÞ ¼ n ; f ðuÞ J Bv 0 H þ Hn
ðgðvÞÞT Y 2 gðvÞ ¼ ½gðvÞ gð0ÞT Y 2 ½gðvÞ gð0Þ r vT EY 2 Ev;
ð3:3Þ
ðf ðuÞÞT Kf ðuÞ r uT FKFu
ð3:4Þ
where
ðgðvÞÞT Ng ðvÞ r vT ENEv: n
W ¼ ðwij Þnn ;
n
W ¼ ðwij Þnn ;
u ¼ ðu1 ; u2 ; …; un ÞT ;
n
n
H ¼ ðhji Þnn ;
and
H ¼ ðhji Þnn ;
v ¼ ðv1 ; v2 ; …; vn ÞT ;
For i; j ¼ 1; 2; …; n; the following inequalities hold: n n n ∑ pii ai ui þ ∑ r ij wij ½g j ðvj Þ g j ð0Þ þ ∑ r nij wnij : i¼1
I n ¼ ðI 1 ; I 2 ; …; I n ÞT ;
Bv ¼ ðb1 v1 ; b2 v2 ; …; bn vn ÞT ;
f ðuÞ ¼ ðf 1 ðu1 Þ; f 2 ðu2 Þ; …; f n ðun ÞÞT ;
and
gðvÞ ¼ ðg 1 ðv1 Þ; g 2 ðv2 Þ; …; g n ðvn ÞÞT :
Obviously, the equilibrium point of system (1.3) is the solution of equation F(u, v)¼ 0. Rewrite F(u, v) as follows: ! ! gðvÞ Au 0 W þWn Fðu; vÞ ¼ þFð0; 0Þ; f ðuÞ Bv 0 H þ Hn where gðvÞ ¼ gðvÞ gð0Þ;
f ðuÞ ¼ f ðuÞ f ð0Þ:
We define a bounded open Ω and a mapping Gðλ; u; vÞ as follows: ( ) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ωo ¼
j¼1
n
n
i¼1
j¼1
n
(
pffiffiffiffiffiffiffiffiffiffi where R 4 2nr 2 =r 1 and Au W þWn Gðλ; u; vÞ ¼ ð1 þ λÞ λ Bv 0
j¼1
¼ 2ð1 þ λÞ
∑
i¼1
n
∑
j¼1
n
n
)2 Z 0:
n
n
j¼1
n
n
j¼1
j¼1
)
þ p2ii a2i u2i þ 2 ∑ r ij wij ðg j ðvj Þ g j ð0ÞÞ ∑ r nij wnij ðg j ðvj Þ g j ð0ÞÞ !
0
gðvÞ f ðuÞ
H þH n
! þ λFð0; 0Þ;
Z0 and n
∑
ð3:6Þ
8"
∂ui ðt; xÞ ∂ ∂u > > ¼ ∑ Dik i ai ui þ ∑ wji g j ðvj Þ > > ∂t ∂xk > j¼1 k ¼ 1∂xk > > > Z t > m > > > > kji ðt sÞg j ðvj ðs; xÞÞ ds þ I i ; þ ∑ wnji > < 1 j¼1 ð5:1Þ l n > ∂vj ðt; xÞ ∂ > n ∂vj > > ¼ ∑ D v þ ∑ h f ðu Þ b j j ij i i > ik > ∂xk ∂t > i¼1 k ¼ 1∂xk > > Z t > > n > n n > > kij ðt sÞf i ðui ðs; xÞÞ ds þ J j ; þ ∑ hij > : 1 i¼1 where
From (4.27), it follows that
i; j; k ¼ 1; 2; Dik ¼ Dik ðt; x; uÞ Z0; Z 1 Z 1 n kji ðsÞ ds ¼ kij ðsÞ ds ¼ 1;
V 0 ðtÞ r 0:
I 1 ¼ 1;
Thus, from the Lyapunov stability theory, it follows that the equilibrium point of system (1.3) is globally asymptotically stable. The completes the proof. □ Remark 1. In our result on global stability, the two inequalities of parameters in stability result in [23,32] are replaced with Q n1 o 0 and
0
Dnjk ¼ Dnjk ðt; x; vÞ Z 0;
0
I 2 ¼ 3;
J 1 ¼ 5; n
wji ¼ wnji ¼ hij ¼ hij ¼ 1; f i ðyÞ ¼ jyj þ 1;
J 2 ¼ 7; ai ¼ bj ¼ 2;
g j ðxÞ ¼ jxj þ 1:
In Theorem 3.1, αi ¼ βj ¼ 1, it is easy to verify that the two inequalities of parameters in [23,32] are not satisfied. Therefore, the result on global stability in [23,32] cannot ensure the global
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reaction–diffusion terms and distributed delays by using degree theory and LMI method. In the results obtained, the assumptions for boundedness and monotonicity on the activation functions in existing papers are removed.
u1 u2 v1 v2
8 6 4
References
2 0 −2 −4 −6
0
2
4
6
8
10
12
14
t Fig. 1. Transient response of state variables in Example 1.
stability of equilibrium point of system (5.1). Since the activation functions in [40] are bounded, while the activation functions in system (5.1) are only global Lipschitz continuous, then the result in [40] cannot ensure the global stability of equilibrium point of system (5.1). When reaction–diffusion terms in system (5.1) become zero, system (5.1) reduces to a BAM neural network with delays. Since the activation functions in system (5.1) are not bounded and monotonic, and are only globally Lipschitz continuous, then the result on global stability in [36–39,41] cannot ensure the global stability of equilibrium point of system (5.1). However, letting α11 ¼ α12 ¼ α13 ¼ α21 ¼ α31 ¼ 1, α22 ¼ 2, α23 ¼ α32 ¼ 1, α33 ¼ 6, β11 ¼ β12 ¼ β13 ¼ β21 ¼ β31 ¼ 1, β22 ¼ 2, β23 ¼ β32 ¼ 1, β33 ¼ 6, then all the conditions in Theorem 3.1 in our paper can be satisfied. By using the Matlab LMI Control toolbox, it can be verified that the LMI in Theorem 3.1 is feasible and 166:6811 0 116:5557 0 P¼ ; Q¼ ; 0 166:6811 0 116:5557 109:2066 0 146:2646 0 ; Y2 ¼ ; Y1 ¼ 0 109:2066 0 146:2646 538:3772 0 90:9774 0 ; Y4 ¼ ; Y3 ¼ 0 538:3772 0 90:9774 129:2227 0 2:34 0 N¼ ; K¼ ; 0 129:2227 0 2:34 2 0 3 0 ; M2 ¼ ; P2 ¼ 0 1 0 4 1:2 2:5 2:4 4 ; M3 ¼ ; P3 ¼ 2:0 1:6 1 1:8 6 4 2 4 ; M4 ¼ : P4 ¼ 3 2 3 1 The conditions ðH 1 Þ; ðH 2 Þ in Theorem 3.1 are obviously satisfied. Then all conditions in Theorem 3.1 are satisfied. By Theorem 3.1 we can conclude that system (5.1) has a unique equilibrium point, which is globally asymptotically stable. The global asympotic stability of system (5.1) in Example 1 is shown in Fig. 1. 6. Conclusions In this paper, under the assumption that the activation functions only satisfy global Lipschitz conditions, a novel LMI-based sufficient condition is obtained for global asymptotic stability of equilibrium point of two-dimensional BAM neural networks with
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Zhiyong Quan was born in 1974 and is a teacher of applied mathematics of Hunan University in China. His field of study is neural networks theory and applications. So far, he has been an author of more than 2 papers.
223
Lihong Huang was born in 1963 and is a professor of applied mathematics of Hunan University in China. His field of study is neural networks theory and applications. So far, he has been an author of more than 100 papers.
Shenghua Yu was born in 1966 and is a professor of applied mathematics of Hunan University in China. His field of study is economic mathematics, neural networks theory and applications. So far, he has been an author of more than 30 papers.
Zhengqiu Zhang was born in 1963 and is a professor of applied mathematics of Hunan University in China. His field of study is neural networks theory and applications. So far, he has been an author of more than 50 papers.
