asymptotic stability of cellular neural networks with multiple proportional delays
TRANSCRIPT
Applied Mathematics and Computation 229 (2014) 457–466
Contents lists available at ScienceDirect
Applied Mathematics and Computation
journal homepage: www.elsevier .com/ locate/amc
Asymptotic stability of cellular neural networks with multipleproportional delays q
0096-3003/$ - see front matter � 2014 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.12.061
q This work is supported by the National Science Foundation of China (Nos. 61374009, 61272514, 61003287, 61170272, 61121061, 61161Tianjin Municipal Education Commission (No. 20100813), NCET (No. NCET-13-0681), the Specialized Research Fund for the Doctoral Program oEducation (No. 20100005120002), the Fok Ying Tong Education Foundation (No. 131067) and the Fundamental Research Funds for the Central Un(No. BUPT2012RC0221).⇑ Corresponding author.
E-mail addresses: [email protected], [email protected] (L. Zhou).
Liqun Zhou a,⇑, Xiubo Chen b,c, Yixian Yang b,c
a School of Mathematics Science, Tianjin Normal University, Tianjin 300387, Chinab Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing100876, Chinac State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China
a r t i c l e i n f o a b s t r a c t
Keywords:Cellular neural networksProportional delayGlobal asymptotic stabilityUniform asymptotic stabilityLinear matrix inequality
Proportional delay is a kind of unbounded time-varying delay, which is different fromunbounded distributed delays. In this paper, asymptotic stability of the equilibrium pointof cellular neural networks with multiple proportional delays is presented. Sufficient con-ditions for delay-dependent global asymptotic stability and delay-independent uniformasymptotic stability of the system are obtained by employing matrix theory and construct-ing Lyapunov functional. Two examples are given to illustrate the effectiveness of theobtained results and less conservative than previously existing results.
� 2014 Elsevier Inc. All rights reserved.
1. Introduction
Cellular neural networks (CNNs) [1] have been widely investigated owing to their widespread applications in image pro-cessing, pattern recognition, optimization and associative memories. These practical application of CNNs depends on theexistence and stability of the equilibrium point of the CNNs. Moreover, time delay is inevitable due to the finite switchingspeed of information processing and the inherent communication time of neurons, and its existence may cause the instabil-ity of the system. Therefore, a number of stability criteria of CNNs with delays have been proposed [2–20]. At present, sta-bility results of CNNs with delays studied in [4,6,11,13,16,18] are mainly based on such approaches as M-matrix, algebraicinequalities, and so on. As pointed out in [14], the characteristic of those results, which means to take absolute value oper-ation on the interconnection matrix, leads to the ignorance of neuron’s inhibitory and excitatory effects on neural networks.In recent years, linear matrix inequality (LMI) technique has been used to deal with the stability problem for neural networks[2,3,5,7–10,12,14,15,17,19,20]. The feature of LMI-based results is that it can consider the neuron’s inhibitory and excitatoryeffects on neural networks. However, few stability results have been obtained for CNNs with proportional delays on basis ofLMI, which is important, as did for neural networks model studied in [32,33]. The exponential stability of CNNs with multi-pantograph delays was studied by nonlinear measure in [32]. In [33], by employing matrix theory and Lyapunov functional,global exponential stability of a class of CNNs with multi-proportional delays was investigated and delay-dependent suffi-cient conditions were obtained. The proportional delay system as an important mathematical model often rises in somefields such as physics, biology systems and control theory and it has attracted many scholars’ interest [25–31].
140320),f Higher
iversities
458 L. Zhou et al. / Applied Mathematics and Computation 229 (2014) 457–466
As a neural network usually has a spatial nature for the presence of an amount of parallel pathways of a variety of axonsizes and lengths, it is desired to model by introducing continuously proportional delay over a certain duration of time.The proportional delay function is that sðtÞ ¼ ð1� qÞt ! þ1 as q – 1; t ! þ1 (0 < q 6 1), so it is time-varying, unbounded,and monotonically increasing. Therefore, the network’s running time may be controlled according to the network alloweddelays. At the present time, besides the distributed delays, the delay function sðtÞ is usually required to be boundedin the stability discussion of neural networks with delays, such as, these results in [9,12,19–24], are required that thedelay function sðtÞ satisfies 0 6 sðtÞ 6 s and other conditions. It is seldom that the delay function sðtÞ is unbounded, i.e.sðtÞ ! þ1ðt ! þ1Þ. Compared with the distributed delay [6–8,10,13], whose delay kernel functions satisfy some condi-tions such that the distributed delay is easier to be handled [8,10,13] in the use inequality, but the proportional delay isnot easy to be controlled. Thus, it is important to study stability of neural networks with proportional delays in theoryand practice.
Motivated by the discussion above, the asymptotic stability of CNNs with multiple proportional delays is further dis-cussed in this paper, inspired by Liao et al. [4], Zhang and Wang [14] and Zhang and Zhou [32]. This paper is organized asfollows. Model description and preliminaries are given in Section 2. Two delay-dependent and delay-independent sufficientconditions are given in Section 3 to ascertain the asymptotic stability of the CNNs with multiple proportional delays, whichare easy to be verified. Numerical examples and their simulation are presented in Section 4 to illustrate the effectiveness andless conservative of obtained results. Finally, conclusions are given in Section 5.
2. Model description and preliminaries
Consider the following CNNs with multiple proportional delays
_xiðtÞ ¼ �dixiðtÞ þXn
j¼1
aijfjðxjðtÞÞ þXn
j¼1
bijfjðxjðqijtÞÞ þ Ii;
xiðsÞ ¼ xi0; s 2 ½q;1�; i ¼ 1;2 . . . ;n
8><>: ð2:1Þ
for t P 1, where n is the number of neurons; di > 0 is a constant; xiðtÞ is the state variable; aij and bij are constants whichdenote the strengths of connectivity between the cells j and i at time t and qijt, respectively; fið�Þ denotes a nonlinear acti-vation function; Ii denotes the constant external inputs; qij; i; j ¼ 1;2; . . . ;n are proportional delay factors and satisfy0 < qij 6 1, and qijt ¼ t � ð1� qijÞt, in which sijðtÞ ¼ ð1� qijÞt is the transmission delay function, and ð1� qijÞt ! þ1 asqij – 1; t ! þ1; q ¼min16i;j6nfqijg; xi0 denotes constant initial value of xiðsÞ at s 2 ½q;1�. Assume that fjð�Þ; j ¼ 1;2; . . . ;nare bounded, monotonically nondecreasing and satisfying
0 6fiðuÞ � fiðvÞ
u� v 6 li; fið0Þ ¼ 0; u – v; f iðuÞ – 0; u – 0; u; v 2 R: ð2:2Þ
Remark 2.1. In (2.1), if qij ¼ 1; i; j ¼ 1;2; . . . ;n, then system (2.1) is a class of standard cellular neural networks model.
Throughout this paper, the notation A > 0 means that the matrix A is symmetric positive definite. The notation AT and A�1
denote the transpose and the inverse of a square matrix A, respectively. Rn denotes the n-dimension Euclidean Space,Rþ ¼ ½0;þ1Þ; kxk denotes the Euclidean norm in Rn; x 2 Rn. Cð½�s;0�;RÞ represents a set of all continuous functions from½�s;0� to R.
It is known that there always exists an equilibrium point for system (2.1). Let x� ¼ ðx�1; x�2; . . . ; x�nÞT be an equilibrium point
of (2.1). By shifting x� to the origin, then (2.1) are converted into the following form
_ziðtÞ ¼ �diziðtÞ þXn
j¼1
aijgjðzjðtÞÞ þXn
j¼1
bijgjðzjðqijtÞÞ;
ziðsÞ ¼ zi0; s 2 ½q;1�; i ¼ 1;2; . . . ;n;
8><>: ð2:3Þ
where zið�Þ ¼ xið�Þ � x�i , and gjðzjð�ÞÞ ¼ fjðzjð�Þ þ x�j Þ � fjðx�j Þ.Note that since each function fið�Þ satisfies the condition (2.2), hence, each gi satisfies
g2i ðzið�ÞÞ 6 lizið�Þgiðzið�ÞÞ; gið0Þ ¼ 0; i ¼ 1;2; . . . ;n: ð2:4Þ
Let yiðtÞ ¼ ziðetÞ. It is easy to prove that system (2.3) is equivalent to the following CNNs with constant delays and variablecoefficients (see [33])
_yiðtÞ ¼ et �diyiðtÞ þXn
j¼1
aijgjðyjðtÞÞ þXn
j¼1
bijgjðyjðt � sijÞÞ( )
; t P 0;
yiðsÞ ¼ uiðsÞ; s 2 ½�s;0�; i ¼ 1;2; . . . ; n;
8><>: ð2:5Þ
where sij ¼ � log qij � 0; s ¼max16i;j6nfsijg and uiðsÞ ¼ zi0 2 Cð½�s;0�;RÞ.
L. Zhou et al. / Applied Mathematics and Computation 229 (2014) 457–466 459
From (2.4), one obtains
g2i ðyið�ÞÞ 6 liyið�Þgiðyið�ÞÞ; gið0Þ ¼ 0; i ¼ 1;2; . . . ;n: ð2:6Þ
It is easy to verify that systems (2.3) and (2.5) have the same equilibrium, that is z� ¼ y� ¼ 0. Thus, to prove thestability of equilibrium point x� of system (2.1), it is sufficient to prove the stability of the trivial solution of (2.3)or (2.5).
Remark 2.2. So far as delay is concerned, in (2.1), time delay function ð1� qijÞt ! þ1 as qij – 1; t ! þ1, so thatmodel (2.1) is different from delayed neural network models in [3–20]. In addition, model (2.5) is different frommodels in [11]. The coefficients of model in [11] are bounded time-varying functions, but the coefficients of model(2.5) which contain et are unbounded time-varying functions. Thus, those stability results [3–20] cannot be directlyapplied to (2.1) and (2.5).
For the system
_xðtÞ ¼ f ðt; xðtÞ; xtÞ; ð2:7Þ
where x 2 Rn; xtðhÞ ¼ xðt þ hÞ; h 2 ½�s;0�; s > 0. f 2 CðRþ � Rn � C;RnÞ. C denotes a Banach space with uniform convergencetopology structure knk ¼max�s6s60knðsÞk, continuous function nðtÞ 2 Cð½�s;0�;RnÞ. t0 2 Rþ. xðt; t0; nÞ is the solution of (2.7)from initial function nðtÞ.
Definition 2.3 [35]. The zero solution of system (2.7) is said to be stable (uniform stable), if 8e > 0; 9dðt0; eÞ > 0 ð9dðeÞ > 0Þ,when knðtÞk < d; t P t0, there is kxðt; t0; nÞk < e. The zero solution of system (2.7) is said to be attractive (uniform attractive),if 9rðt0Þ > 0 ðr > 0Þ; 8g > 0; 9Tðt0;gÞ > 0 ð9TðgÞ > 0Þ, when knðtÞk < r; t P t0 þ T , there is kxðt; t0; nÞk < g. If the zerosolution of system (2.7) is stable and attractive, then it is asymptotic stable; If the zero solution of system (2.7) is uniformstable and uniform attractive, then it is uniform asymptotic stability.
Definition 2.4 [35]. The zero solution of system (2.7) is said to be global asymptotic stability, if it is stable and for arbitraryinitial function nðtÞ, satisfies
limt!1kxðt; t0; nÞk ¼ 0:
Lemma 2.5 [35]. In the region GH ¼ fðt; xÞ; t P t0; kxk < Hg, if there exists a positive definite functional Vðt; xÞ with infinitesmall upper bound, such that
dVdtjð2:7Þ < 0; x – 0;
holds, then the zero solution of system (2.7) is uniformly asymptotically stable.
Lemma 2.6 [8]. For any a; b 2 Rn and e > 0, the following matrix inequality
2aT b 6 eaT Xaþ e�1bT X�1b
holds, in which X 2 Rn�n is any matrix with X > 0.
Lemma 2.7 (Schur complement [36]). The linear matrix inequalityQðxÞ SðxÞSTðxÞ RðxÞ
� �> 0, where QðxÞ ¼ QTðxÞ; RðxÞ ¼ RTðxÞ, is
equivalent to
RðxÞ > 0 and QðxÞ � SðxÞR�1ðxÞSTðxÞ > 0:
3. Main results
Theorem 3.1. The origin of system (2.3) is globally asymptotically stable, if there exist positive diagonal matricesM ¼ diagðm1;m2; . . . ;mnÞ; Ni ¼ diagðni1;ni2; . . . ;ninÞ and a constant b > 0, such that the following inequality holds:
MAþ AT M � 2MDL�1 þXn
i¼1
ðbNiQ�1i þ b�1MWiN
�1i WT
i MÞ < 0; ð3:1Þ
where D ¼ diagðd1; d2; . . . ; dnÞ; A ¼ ðaijÞn�n; L ¼ diagðl1; l2; . . . ; lnÞ; Wi is an n� n square matrix, whose ith row is composed ofðbi1; bi2; . . . ; binÞ and other rows are all zeros, i; j ¼ 1;2; . . . ;n and Q�1
i ¼ diagðq�1i1 ; q
�1i2 ; . . . ; q�1
in Þ; i ¼ 1;2; . . . ;n.
460 L. Zhou et al. / Applied Mathematics and Computation 229 (2014) 457–466
Proof. Consider the following Lyapunov functional:
VðzðtÞÞ ¼Xn
i¼1
2mi
Z ziðtÞ
0giðsÞdsþ
Xn
i¼1
Xn
j¼1
bqij
Z t
qijtnijg2
j ðzjðsÞÞds; ð3:2Þ
where mi > 0; nij > 0; i; j ¼ 1;2; . . . ;n and b > 0.In (3.2), by (2.4), if zðtÞ ¼ 0, i.e. ziðtÞ ¼ 0; i ¼ 1;2; . . . ;n, then gðziðtÞÞ ¼ 0; i ¼ 1;2; . . . ;n, thus VðzðtÞÞ ¼ 0. And here we
show VðzðtÞÞ > 0 as zðtÞ– 0.In fact, by zðtÞ– 0, there exists at least one index i such that ziðtÞ – 0. By the integral mean value theorem,R ziðtÞ
0 giðsÞds ¼ giðhiÞziðtÞ, where hi is a number between 0 and ziðtÞ. From (2.2) and (2.4), When ziðtÞ > 0, we gethi > 0; giðhiÞP 0; giðhiÞziðtÞP 0; When ziðtÞ < 0, we have hi < 0; giðhÞ 6 0, and giðhÞziðtÞP 0. Thus, we obtainR ziðtÞ
0 giðsÞds P 0, andPn
i¼12miR ziðtÞ
0 giðsÞds P 0; zðtÞ– 0. Further, we will prove thatPn
i¼12miR ziðtÞ
0 giðsÞds ¼ 0 as zðtÞ– 0does not hold. Assume
Pni¼12mi
R ziðtÞ0 giðsÞds ¼ 0 as zðtÞ – 0, there must be numbers hi; i ¼ 1;2; . . . ;n thatPn
i¼12miR ziðtÞ
0 giðsÞds ¼Pn
i¼12migiðhiÞziðtÞ ¼ 0, where hi is a number between 0 and ziðtÞ. Thus, we obtain giðhiÞ ¼ 0 orziðtÞ ¼ 0 for i ¼ 1;2; . . . ;n. When ziðtÞ ¼ 0; i ¼ 1;2; . . . ;n, we get zðtÞ ¼ 0, this contradicts with zðtÞ – 0. WhengiðhiÞ ¼ 0; i ¼ 1;2; . . . ;n, we have giðhiÞ ¼ fiðhi þ x�i Þ � fiðx�i Þ ¼ 0, i.e. fiðhi þ x�i Þ ¼ fiðx�i Þ; i ¼ 1;2; . . . ;n, by (2.2), then fiðhi þ x�i Þis a constant functions for hi 2 ½0; ziðtÞ� or hi 2 ½ziðtÞ;0�, this contradicts with a nonlinear activation functionfiðxiðtÞÞ; i ¼ 1;2; . . . ;n. Thus,
Pni¼12mi
R ziðtÞ0 giðsÞds > 0 as zðtÞ – 0.
That is to say, the first term of VðzðtÞÞ is positive definite. Clearly, the second term of VðzðtÞÞP 0. That isVðzðtÞÞ > 0; zðtÞ – 0. Thus (3.2) is positive definite.
By calculating the time derivative of VðzðtÞÞ along the trajectory of system (2.3), we get
_VðzðtÞÞ ¼ 2Xn
i¼1
migiðziðtÞÞ _ziðtÞ þXn
i¼1
Xn
j¼1
bnij
qij½g2
j ðzjðtÞÞ � g2j ðzjðqijtÞÞqij�
¼ 2Xn
i¼1
migiðziðtÞÞ �diziðtÞ þXn
j¼1
aijgjðzjðtÞÞ þXn
j¼1
bijgjðzjðqijtÞÞ" #
þXn
i¼1
Xn
j¼1
bnij
qij½g2
j ðzjðtÞÞ � g2j ðzjðqijtÞÞqij�
¼ �2gðzðtÞÞT MDzðtÞ þ 2gðzðtÞÞT MAgðzðtÞÞ þ 2Xn
i¼1
migiðziðtÞÞ½bi1; bi2; . . . ; bin�gðzð�qitÞÞ
þXn
i¼1
½bgTðzðtÞÞNiQ�1i gðzðtÞÞ � bgTðzð�qitÞÞNigðzð�qitÞÞ�; ð3:3Þ
where gðzð�qitÞÞ ¼ ðg1ðz1ðqi1tÞÞ; g2ðz2ðqi2tÞÞ; . . . ; gnðznðqintÞÞÞT ; Ni ¼ diagðni1;ni2; . . . ;ninÞ and Q�1i ¼ diagðq�1
i1 ; q�1i2 ; . . . ; q�1
in Þ;i ¼ 1;2; . . . ;n.
By Lemma 2.2, note that the following condition holds:
2Xn
i¼1
migiðziðtÞÞ½bi1; bi2; . . . ; bin�gðzð�qitÞÞ ¼Xn
i¼1
2gTðzðtÞÞMWigðzð�qitÞÞ
6 b�1gTðzðtÞÞXn
i¼1
MWiN�1i WT
i M
!gðzðtÞÞ þ b
Xn
i¼1
gTðzð�qitÞÞNigðzð�qitÞÞ: ð3:4Þ
Substituting (3.4) into (3.3) yields
_VðzðtÞ; tÞ 6 �2gTðzðtÞÞMDzðtÞ þ 2gTðzðtÞÞMAgðzðtÞÞ þ b�1gTðzðtÞÞXn
i¼1
MWiN�1i WT
i M
" #gðzðtÞÞ
þ bXn
i¼1
gTðzð�qitÞÞNigðzð�qitÞÞ þXn
i¼1
½bgTðzðtÞÞNiQ�1i gðzðtÞÞ � bgTðzð�qitÞNigðzð�qitÞ�
¼ �2gTðzðtÞÞMDzðtÞ þ 2gTðzðtÞÞMAgðzðtÞÞ þ gTðzðtÞÞ b�1Xn
i¼1
MWiN�1i WT
i M þ bNiQ�1i
" #gðzðtÞÞ
þ 2gTðzðtÞÞMDL�1gðzðtÞÞ � 2gTðzðtÞÞMDL�1gðzðtÞÞ: ð3:5Þ
From (2.4), we obtain
�Xn
i¼1
liziðtÞgiðziðtÞÞ 6 �Xn
i¼1
g2i ðziðtÞÞ;
that is
�2gTðzðtÞÞzðtÞ 6 �2gTðzðtÞÞL�1gðzðtÞÞ:
L. Zhou et al. / Applied Mathematics and Computation 229 (2014) 457–466 461
Thus,
�2gTðzðtÞÞMDzðtÞ 6 �2gTðzðtÞÞMDL�1gðzðtÞÞ;
one gets
�2gTðzðtÞÞMDzðtÞ þ 2gTðzðtÞÞMDL�1gðzðtÞÞ 6 0: ð3:6Þ
Let gðzðtÞÞ – 0, which implies that zðtÞ – 0. From (3.6), (3.5) can be rewritten as follows:
_VðzðtÞÞ 6 gTðzðtÞÞ �2MDL�1 þMAþ AT M þ b�1Xn
i¼1
MWiN�1i WT
i M þ bNiQ�1i
" #gðzðtÞÞ:
Thus, if (3.1) holds, then _VðzðtÞÞ < 0.Now, consider the case where gðzðtÞÞ ¼ 0 and zðtÞ– 0. Then, we have
_VðzðtÞÞ ¼ �Xn
i¼1
Xn
j¼1
bnijg2j ðzjðqijtÞÞ ¼ �
Xn
i¼1
bgTðzð�qitÞÞNigðzð�qiðtÞÞ;
if there exist at least one index i such that gðzð�qitÞÞ– 0, we obtain _VðzðtÞÞ < 0. Assume that gðzð�qitÞÞ ¼ 0 for all i. Sincegðzð�qitÞÞ ¼ ðg1ðz1ðqi1tÞÞ; g2ðz2ðqi2tÞÞ; . . . ; gnðznðqintÞÞÞT , we get gjðzjðqijtÞÞ ¼ 0; i; j ¼ 1;2; . . . ;n, i.e. fjðzjðqijðtÞÞþx�j Þ � fjðx�j Þ ¼ fjðxjðqijtÞÞ � fjðx�j Þ ¼ 0; i; j ¼ 1;2; . . . ;n, so then
fjðxjðqijtÞÞ ¼ fjðx�j Þ; i; j ¼ 1;2; . . . ;n: ð3:7Þ
By zðtÞ – 0, we have zð�qitÞ – 0, there exist at least one index j such that zjðqijtÞ– 0, that is xjðqijtÞ – x�j . Whenx�j – 0; xjðqijtÞ ¼ 0, we get
fjðx�j Þ– 0; fjðxjðqijtÞÞ ¼ 0 ð3:8Þ
for one j. The contradiction in (3.7) and (3.8) means that the assumption gðzð�qitÞÞ ¼ 0 for all i is violated, i.e. there exist atleast one index i such that gðzð�qitÞÞ– 0.
Thus, we have proven that _VðzðtÞÞ < 0 for every zðtÞ– 0.Now, let zðtÞ ¼ 0, which implies that gðzðtÞÞ ¼ 0, then
_VðzðtÞÞ ¼ �Xn
i¼1
Xn
j¼1
bnijg2j ðzjðqijtÞÞ ¼ �
Xn
i¼1
bgTðzð�qitÞÞNigðzð�qiðtÞÞ;
if there exist at least one index i such that gðzð�qitÞÞ – 0, we obtain _VðzðtÞÞ < 0, and _VðzðtÞÞ ¼ 0 if and only ifgðzð�qitÞÞ ¼ 0; i ¼ 1;2; . . . ;n.
Hence, we have shown that _VðzðtÞÞ ¼ 0 if and only if zðtÞ ¼ gðzðtÞÞ ¼ gðzð�qitÞÞ ¼ 0; i ¼ 1;2; . . . ;n, otherwise _VðzðtÞÞ < 0,that is _VðzðtÞÞ is negative definite. Moreover, VðzðtÞÞ is radially unbounded since VðzðtÞÞ ! 1 as kzðtÞk ! 1. Hence, fromCorollary 3.2 in [34], we can conclude that if condition (3.1) holds, then the origin of system (2.3) is globally asymptoticallystable. And thus the equilibrium point x ¼ x� of system (2.1) is globally asymptotically stable. This completes the proof. h
A sufficient conditions of delay-dependent global asymptotic stability for (2.3) is obtained in Theorem 3.1. In the follow-ing, we give a sufficient condition of delay-independent uniform asymptotic stability for (2.5).
Theorem 3.2. The origin point of system (2.5) is uniformly asymptotically stable if there exist positive diagonal matricesM ¼ diagðm1;m2; . . . ;mnÞ; Ni ¼ diagðni1; ni2; . . . ;ninÞ and a constant b > 0, such that the following inequality holds:
MAþ AT M � 2MDL�1 þXn
i¼1
ðbNi þ b�1MWiN�1i WT
i MÞ < 0; ð3:9Þ
where D ¼ diagðd1; d2; . . . ; dnÞ;A ¼ ðaijÞn�n; L ¼ diagðl1; l2; . . . ; lnÞ, and Wi is an n� n square matrix, whose ith row is composed ofðbi1; bi2; . . . ; binÞ and other rows are all zeros, i; j ¼ 1;2; . . . ;n.
Proof. Consider the following Lyapunov functional:
VðyðtÞ; tÞ ¼ 2Xn
i¼1
e�tmi
Z yiðtÞ
0giðsÞdsþ
Xn
i¼1
Xn
j¼1
Z t
t�sij
bnijg2j ðyjðsÞÞds; ð3:10Þ
where mi > 0; nij > 0; i; j ¼ 1;2; . . . ;n and b > 0.Since VðyðtÞ; tÞP WðyðtÞÞ, where WðyðtÞÞ ¼
Pni¼1Pn
j¼1
R tt�sij
bnijg2j ðyjðsÞÞds is positive definite, and Vð0; tÞ � 0, thus
VðyðtÞ; tÞ is positive definite. And VðyðtÞ; tÞ 6 UðyðtÞÞ, where UðyðtÞÞ ¼ 2Pn
i¼1miR yiðtÞ
0 giðsÞdsþPn
i¼1Pn
j¼1
R tt�sij
bnijg2j ðyjðsÞÞds
is positive definite. Thus, VðyðtÞ; tÞ is a functional with infinite small upper bound.There time derivative of functional (3.10) along the trajectories of system (2.5) is as follows:
462 L. Zhou et al. / Applied Mathematics and Computation 229 (2014) 457–466
_VðyðtÞ; tÞ ¼ �2Xn
i¼1
e�tmi
Z yiðtÞ
0giðsÞdsþ 2
Xn
i¼1
e�tmigiðyiðtÞÞ _yiðtÞ þXn
i¼1
Xn
j¼1
nijb½g2j ðyjðtÞÞ � g2
j ðyjðt � sijÞÞ�
6 2Xn
i¼1
e�tmigiðyiðtÞÞ _yiðtÞ þXn
i¼1
Xn
j¼1
nijb½g2j ðyjðtÞÞ � g2
j ðyjðt � sijÞÞ�
¼ 2Xn
i¼1
e�tmigiðyiðtÞÞ et �diyiðtÞ þXn
j¼1
aijgjðyjðtÞÞ þXn
j¼1
bijgjðyjðt � sijÞÞ" #( )
þXn
i¼1
Xn
j¼1
nijb½g2j ðyjðtÞÞ
� g2j ðyjðt � sijÞÞ�
¼ �2gðyðtÞÞT MDyðtÞ þ 2gðyðtÞÞT MAgðyðtÞÞ þ 2Xn
i¼1
migiðyiðtÞÞ½bi1; bi2; . . . ; bin�gðyðt � �siÞÞ
þXn
i¼1
½bgTðyðtÞÞNigðyðtÞÞ � bgTðyðt � �siÞÞNigðyðt � �siÞÞ�; ð3:11Þ
where gðyðt � �siÞÞ ¼ ðg1ðy1ðt � si1ÞÞ; g2ðy2ðt � si2ÞÞ; . . . ; gnðynðt � sinÞÞT . Ni ¼ diagðni1;ni2; . . . ; ninÞ; i ¼ 1;2; . . . ;n.Note that the following condition holds:
2Xn
i¼1
migðyiðtÞÞ½bi1; bi2; . . . ; bin�gðyðt � �siÞÞ ¼ 2Xn
i¼1
gTðyðtÞÞMWigðyðt � �siÞÞ 6 b�1gTðyðtÞÞXn
i¼1
MWiN�1i WT
i M
!gðyðtÞÞ
þ bXn
i¼1
gTðyðt � �siÞÞNigðyðt � �siÞÞ: ð3:12Þ
Substituting (3.12) into (3.11) yields
_VðyðtÞ; tÞ 6 �2gTðyðtÞÞMDyðtÞ þ 2gTðyðtÞÞMAgðyðtÞÞ þ b�1gTðyðtÞÞXn
i¼1
MWiN�1i WT
i M
" #gðyðtÞÞ
þ bXn
i¼1
gTðyðt � �siÞÞNigðyðt � �siÞÞ þXn
i¼1
½bgTðyðtÞÞNigðyðtÞÞ � bgTðyðt � �siÞÞNigðyðt � �siÞÞ�
¼ �2gTðyðtÞÞMDyðtÞ þ 2gTðyðtÞÞMAgðyðtÞÞ þ gTðyðtÞÞ b�1Xn
i¼1
MWiN�1i WT
i M þ bNi
" #gðyðtÞÞ
þ 2gTðyðtÞÞMDL�1gðyðtÞÞ � 2gTðyðtÞÞMDL�1gðyðtÞÞ: ð3:13Þ
According to (2.6), the following inequality holds:
�2gTðyðtÞÞMDyðtÞ þ 2gTðyðtÞÞMDL�1gðyðtÞÞ 6 0: ð3:14Þ
From (3.14), (3.13) can be rewritten as follows:
_VðyðtÞ; tÞ 6 gTðyðtÞÞ �2MDL�1 þMAþ AT M þXn
i¼1
ðb�1MWiN�1WT
i M þ bNiÞ" #
gðyðtÞÞ:
Similar to the discussion of Theorem 3.1, we obtain that if condition (3.9) holds, then _VðyðtÞ; tÞ < 0 for any yðtÞÞ– 0._VðyðtÞ; tÞ ¼ 0 if and only if yðtÞ ¼ gðyðtÞÞ ¼ gðyðt � �siÞÞ ¼ 0; i ¼ 1;2; . . . ;n. By Lemma 2.5, the origin point of system (2.5) isuniformly asymptotically stable. This completes the proof. h
When proportional delay factors qij ¼ qj in (2.1), i; j ¼ 1;2; . . . ;n, the following results are obtained.
Corollary 3.3. The origin point of (2.3) with qij ¼ qj is globally asymptotically stable if there exist positive diagonal matricesM ¼ diagðm1;m2; . . . ;mnÞ; N ¼ diagðn1;n2; . . . ;nnÞ and a constant b > 0, such that the following inequality holds:
MAþ AT M � 2MDL�1 þ b�1MBN�1Q�1BT M þ bN < 0; ð3:15Þ
where B ¼ ðbijÞn�n; Q�1 ¼ diagðq�11 ; q�1
2 ; . . . ; q�1n Þ and others are the same as those defined in Theorem 3.1.
Proof. Consider the following radically unbounded and positive definite Lyapunov functional:
VðzðtÞ; tÞ ¼Xn
i¼1
2mi
Z ziðtÞ
0gðsÞdsþ
Xn
j¼1
bqj
Z t
qjtnjg2ðzjðsÞÞds; ð3:16Þ
L. Zhou et al. / Applied Mathematics and Computation 229 (2014) 457–466 463
where mi > 0; nj > 0; i; j ¼ 1;2; . . . ;n; b > 0. Similar to the proof of Theorem 3.1, we can conclude if condition (3.15) holds,then the origin point of (2.3) is globally asymptotically stable. The details are omitted. h
Corollary 3.4. The origin point of (2.5) with qij ¼ qj is uniformly asymptotically stable if there exist positive diagonal matricesM ¼ diagðm1;m2; . . . ;mnÞ; N ¼ diagðn1;n2; . . . ; nnÞ and a constant b > 0, such that the following inequality holds:
MAþ AT M � 2MDL�1 þ b�1MBN�1BT M þ bN < 0; ð3:17Þ
where B ¼ ðbijÞn�n and others are the same as those defined in Theorem 3.2.
Proof. Consider the following positive definite Lyapunov functional:
VðyðtÞ; tÞ ¼Xn
i¼1
2e�tmi
Z yiðtÞ
0giðsÞdsþ
Xn
j¼1
Z t
t�sj
bnjg2j ðyjðsÞÞds; ð3:18Þ
where mi > 0; nj > 0; i; j ¼ 1;2; . . . ;n; b > 0. Similar to the proof of Theorem 3.2, we can conclude if condition (3.17) holds,then the origin point of system (2.5) is uniformly asymptotically stable. The details are omitted. h
Remark 3.5. From Lemma 2.7, Theorem 3.2 and Corollary 3.4 can be expressed in the form of linear matrix inequality as
MAþ AT M � 2MDL�1 þXn
i¼1
bNi MW1 � � � MWn
WT1M �bN1 � � � 0
..
. ... . .
. ...
WTnM 0 � � � �bNn
0BBBBBBB@
1CCCCCCCA< 0
and
MAþ AT M � 2MDL�1 þ bN MB
BT M �bN
!< 0:
LMIs can be easily calculated by using the Matlab toolbox. Therefore, the conditions in Theorem 3.2 and Corollary 3.4 areeasy to be verified.
Remark 3.6. From 0 < qij 6 1, one obtains q�1ij P 1; i; j ¼ 1;2; . . . ;n. By Q�1
i ¼ diagðq�1i1 ; q
�1i2 ; . . . ; q�1
in Þ; i ¼ 1;2; . . . ;n, we havebNiQ
�1i � bNi ¼ bNiðQ�1
i � IÞ > 0, where I denotes the identity matrix with appropriate dimensions. If condition (3.1) holds,then condition (3.9) holds. That is to say, Theorem 3.1 implies Theorem 3.2. Likewise, Corollary 3.3 implies Corollary 3.4.
Remark 3.7. It should be noted that the results in [32,33] are only suitable for model (2.1) with qij ¼ qj, i.e., for the followingcase of unequal delays:
_xiðtÞ ¼ �dixiðtÞ þXn
j¼1
aijf ðxjðtÞÞ þXn
j¼1
bijf ðxjðqjtÞ þ Ii: ð3:19Þ
Obviously, from the viewpoint of time delay, model (3.19) is the special cases of model (2.1). Therefore, our results havemuch wider application fields than those in [32,33]. we should also note that Theorems 3.1 and 3.2 can also be applied tomodel (3.19) with some conservativeness, while the results in [32,33] cannot be applied to model (2.1) at all.
Remark 3.8. The results in [32,33] did not consider the sign of entries in connection weight matrix, therefore, the neuron’sexcitatory and inhibitory effect on neural networks was not considered. In contrast, our results consider the sign entries inconnection weight matrix, and the neuron’s excitatory and inhibitory effect on neural network is considered.
4. Numerical examples
Example 4.1. Consider the neural network model (2.1), where
D ¼8 00 6
� �; A ¼
�2 11 �1
� �; B ¼
�0:5 �1�0:3 �0:6
� �; q ¼
0:5 0:80:8 0:5
� �; I ¼
00
� �;
464 L. Zhou et al. / Applied Mathematics and Computation 229 (2014) 457–466
the activation function is described by f ðxiÞ ¼ 0:5ðjxi þ 1j � jxi � 1jÞði ¼ 1;2Þ with li ¼ 1; i ¼ 1;2. Clearly, f ðxiÞ; i ¼ 1;2 satisfycondition (2.2) above.
By some simple calculations, we obtain Q1 ¼ diagð0:5;0:8Þ;Q2 ¼ diagð0:8;0:5Þ, and L�1 ¼ diagð1;1Þ. And
W1 ¼�0:5 �1
0 0
� �; W2 ¼
0 0�0:3 �0:6
� �; Q�1
1 ¼ diagð2;1:25Þ; Q�12 ¼ ð1:25;2Þ;
Take M ¼ diagð2;3Þ; N1 ¼ diagð3;4Þ; N2 ¼ diagð1;3Þ and b ¼ 1. Applying Theorem 3.1, we have
� MAþ AT M � 2MDL�1 þX2
i¼1
ðbNiQ�1i þ b�1MWiN
�1i WT
i MÞ" #
¼13:7500 �5:0000�5:0000 13:9900
� �> 0;
Therefore, the concerned neural network is globally asymptotically stable, ð0;0Þ is an equilibrium point of the system andit is globally asymptotically stable (see Fig.1). By Remarks 2.2 and 3.7, the results in [3–20,32,33] are not suitable for thisexample.
Example 4.2. Consider the neural network model (2.1), where
D ¼3 00 4
� �; A ¼
�2 11 1
� �; B ¼
2 �10 �1
� �; I ¼
00
� �
and q ¼ ðqijÞ2�2in which 0 < qij < 1. The activation functions f ðxiÞ ¼ tanhðxiÞwith li ¼ 1; i ¼ 1;2 satisfy condition (2.2) above.
And
W1 ¼2 �10 0
� �; W2 ¼
0 00 �1
� �; L�1 ¼ diagð1;1Þ:
Take M ¼ N1 ¼ N2 ¼ diagð1;1Þ and b ¼ 1. By some simple calculations, we obtain
� MAþ AT M � 2MDL�1 þX2
i¼1
ðbNi þ b�1MWiN�1i WT
i MÞ" #
¼3 �2�2 3
� �> 0:
Therefore, by Theorem 3.2, the system is uniformly asymptotically stable. ð0;0Þ is an equilibrium point of the system and it isuniformly asymptotically stable (see Fig. 2), where taking qij ¼ 0:5; i; j ¼ 1;2. By Remark 2.1, the results in [3–20] are notsuitable for this example. On the other hand, it can be easily checked that
d1 �X2
j¼1
p1
pjðjaj1j þ jbj1jÞL1 ¼ 4� p1
p1� 4� p1
p2� 1 ¼ � p1
p2< 0 ð4:1Þ
for any positive constants pi > 0; i ¼ 1;2. The criteria in [32] are not satisfied, so those in [32] are not applicable to thisexample.
In addition, in Example 4.2, take qij ¼ qj ¼ 0:5; i; j ¼ 1;2, the system is uniformly asymptotically stable.
−4 −3 −2 −1 0 1 2 3 4−6
−4
−2
0
2
4
6
X1
X 2
Fig. 1. Phase trajectories of Example 4.1.
−3 −2 −1 0 1 2 3 4−6
−4
−2
0
2
4
6
8
X1
X 2
Fig. 2. Phase trajectories of Example 4.2.
L. Zhou et al. / Applied Mathematics and Computation 229 (2014) 457–466 465
From qj ¼ 0:5, we get sj ¼ � log 0:5 ¼ 0:6931, it can be easily checked that
d1 � 1�X2
j¼1
ðjaj1jL1 þ jbj1jL1es1 Þ ¼ �4:9998 < 0: ð4:2Þ
Remark 3.5 in [33] are not satisfied, so those in [33] are not applicable to this example.
5. Conclusions
New sufficient conditions are derived for asymptotic stability of equilibrium point for cellular neural networks with mul-tiple proportional delays, which is different from the existing ones and has wider application fields. It can be shown that thederived criteria are less conservative than previously existing results through the numerical example. The obtained resultscan be expressed in the form of linear matrix inequality and be easy to be verified by utilizing Matlab LMI toolbox. Theseresults may be applied to establish a QoS routing algorithm based on the neural networks with proportional delays.
References
[1] L.O. Chua, L. Yang, Cellular neural networks: theory and applications, IEEE Trans. Circuits Syst. I 35 (10) (1988) 1257–1290.[2] V. Singh, Global asymptotic stability of cellular neural networks with unequal delays: LMI approach, Electron. Lett. 40 (9) (2004) 548–549.[3] Q. Zhang, X.P. Wei, J. Xu, Delay-depend exponential stability of cellular neural networks with time-varying delays, Chaos Solitons Fractals 23 (4) (2005)
1363–1369.[4] X.X. Liao, J. Wang, Z. Zeng, Global asymptotic stability and global exponential stability of delayed cellular neural networks, IEEE Trans. Circuits Syst. II
52 (7) (2005) 403–409.[5] Y. He, M. Wu, J. She, An improved global asymptotic stability criterion for delayed cellular neural networks, IEEE Trans. Neural Networks 17 (1) (2006)
250–252.[6] L. Zhou, G. Hu, Global exponential periodicity and stability of cellular neural networks with variable and distributed delays, Appl. Math. Comput. 195
(2) (2008) 402–411.[7] K. Ma, L. Yu, W. Zhang, Global exponential stability of cellular neural networks with time-varying discrete and distributed delays, Neurocomputing 72
(10–12) (2009) 2705–2709.[8] M.C. Tan, Global asympotic stability of fuzzy cellular neural networks with unbounded distributed delays, Neural Process. Lett. 31 (2) (2010) 147–157.[9] P. Balasubramaniam, M. Syedali, S. Arik, Global asymptotic stability of stochastic fuzzy cellular neural networks with multi time-varying delays, Expert
Syst. Appl. 37 (12) (2010) 7737–7744.[10] Z. Feng, J. Lam, Stability and dissipativity analysis of distributed delay cellular neural networks, IEEE Trans. Neural Networks 22 (6) (2011) 981–997.[11] Y. Kao, C. Gao, Global exponential stability analysis for cellular neural networks with variable coefficients and delays, Neural Comput. Appl. 17 (3)
(2008) 291–296.[12] L. Hu, H. Gao, W. Zheng, Novel stability of cellular neural networks with interval time-varying delay, Neural Networks 21 (10) (2008) 1458–1463.[13] C. Huang, J. Cao, Almost sure exponential stability of stochastic cellular neural networks with unbounded distributed delays, Neurcomputing 72 (13–
15) (2009) 3352–3356.[14] H. Zhang, Z. Wang, Global asymptotic stability of delayed cellular neural networks, IEEE Trans. Neural Networks 18 (3) (2007) 947–950.[15] R.S. Gan, C.H. Lien, G. Hsieh, Global exponential stability for uncertain cellular neural networks with time-varying delays via LMI approach, Chaos
Solitons Fractals 32 (2007) 1258–1267.[16] L. Li, L. Huang, Equilibrium analysis for improved signal range model of delayed cellular neural networks, Neural Process. Lett. 31 (3) (2010) 1771–
1794.[17] T.J. Su, M. Huang, C. Hou, Y. Lin, Cellular neural networks for gray image noise cancellation based on a hybrid linear matrix inequality and particle
swarm optimization approach, Neural Process. Lett. 32 (2) (2010) 147–165.
466 L. Zhou et al. / Applied Mathematics and Computation 229 (2014) 457–466
[18] Neyir Ozcan, A new sufficient condition for global robust stability of delayed neural networks, Neural Process. Lett. 34 (3) (2011) 305–316.[19] H. Zhang, Z. Wang, D. Liu, Global asymptotic stability of recurrent neural networks with multiple time-varying delays, IEEE Trans. Neural Networks 19
(5) (2008) 855–873.[20] W. Han, Y. Liu, L.S. Wang, Global exponential stability of delayed fuzzy cellular neural networks with Markovian jumping parameters, Neural Comput.
Appl. 21 (1) (2012) 67–72.[21] H. He, L. Yan, J. Tu, Guaranteed cost stabilization of time-varying delay cellular neural networks via Riccati inequality approach, Neural Process. Lett. 35
(2) (2012) 151–158.[22] H. He, L. Yan, J. Tu, Guaranteed cost stabilization of cellular neural networks with time-varying delay, Asian J. Control (2012), http://dx.doi.org/
10.1002/asjc.631.[23] J. Tu, H. He, Guaranteed cost synchronization of chaotic cellular neural networks with time-varying delay, Neural Comput. 24 (1) (2012) 217–233.[24] J. Tu, H. He, P. Xiong, Guaranteed cost synchronous control of time-varying delay cellular neural networks, Neural Comput. Appl. 22 (1) (2013) 103–
110.[25] T. Kato, J.B. Mcleod, The functional-differential equation, Bull. Am. Math. Soc. 77 (2) (1971) 891–937.[26] L. Fox, D.F. Mayers, On a functional differential equational, J. Inst. Math. Appl. 8 (3) (1971) 271–307.[27] A. Iserles, The asymptotic behavior of certain difference equation with proportional delays, Comput. Math. Appl. 28 (1–3) (1994) 141–152.[28] A. Iserles, On neural functional-differential equation with proportional delays, J. Math. Anal. Appl. 207 (1) (1997) 73–95.[29] Y.K. Liu, Asymptotic behavior of functional differential equations with proportional time delays, Eur. J. Appl. Math. 7 (1) (1996) 11–30.[30] B. Van, C. Marshall, G.C. Wake, Holomorphic solutions to pantograph type equations with neural fixed points, J. Math. Anal. Appl. 295 (2) (2004) 557–
569.[31] M.C. Tan, Feedback stabilization of linear systems with proportional delay, China Inform. Control 35 (6) (2006) 690–694.[32] Y. Zhang, L. Zhou, Exponential stability of a class of cellular neural networks with multi-pantograph delays, Acta Electron. Sin. 40 (6) (2012) 1159–
1163.[33] L. Zhou, Delay-dependent exponential stability of cellular neural networks with multi-proportional delays, Neural Process. Lett. 38 (3) (2013) 347–359.[34] H.K. Khalil, Nonlinear Systems, Macmillan, New York, 1988.[35] X.X. Liao, Theory and Application of Stability for Dynamical Systems, National Defence Industry Press, Beijing, 2000.[36] S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994.