LMI-based criteria for globally robust stability of delayed Cohen – Grossberg neural networks W. Wang and J. Cao Abstract: The issue of globally robust asymptotic stability with norm-bounded parameter uncertainties is studied for delayed Cohen –Grossberg neural networks. By constructing a suitable Lyapunov functional, several sufficient conditions are obtained guaranteeing the global robust convergence of the equilibrium point. The obtained conditions are given in the form of matrix and linear matrix inequalities that can be checked numerically and very efficiently by resorting to the recently developed interior-point method. Finally, an illustrative numerical example is provided to demonstrate the effectiveness of the obtained results.
1
Introduction
The research of the dynamical behaviour of recurrently connected neural networks with or without delays is an important topic in the neural network theory and has been widely investigated for the sake of theoretical interest as well as application considerations. It has been extensively applied to optimisation, pattern recognition, signal processing, associative memories and so on. Such applications rely on the existence of the equilibrium and usually require the equilibrium point of the designed network to be stable. Therefore the analysis of dynamic behaviours of the network is a necessary step for practical design and application of the networks. Because of the finite switching speed of the amplifier and the inherent communication time of neurons, time delays unavoidably exist in the interaction between the neurons, which may influence the stability of the networks. Marcus and Westervelt [1] first introduced a single time delay into the connection terms of the Hopfield neural networks, and many results on the neural network with single time delay can be found in the work of Rong [2], Arik [3], Singh [4] and so on. Later, Baldi and Atiya [5] constructed a network with delayed interactions in which different delays are introduced for the communications among different neurons. Since Ye et al. [6] introduced different delays into the Cohen – Grossberg neural networks (CGNNs), neural networks with multiple time delays have been extensively investigated [7 –9]. However, there might also be some uncertainties such as parameter perturbations, which can lead to oscillations and chaos because of the existence of modelling errors, external disturbance and parameter fluctuation. Thus, it is necessary to study the robust stability of the delayed neural networks (DNNs). Liao and Yu [10] have extended the model of the DNNs to the so-called interval DNNs and have obtained a sufficient condition for the existence # IEE, 2006 IEE Proceedings online no. 20050197 doi:10.1049/ip-cta:20050197 Paper first received 24th May and in revised form 16th October 2005 The authors are with the Department of Mathematics, Southeast University, Nanjing 210096, People’s Republic of China E-mail:
[email protected] IEE Proc.-Control Theory Appl., Vol. 153, No. 4, July 2006
and robust stability of the unique equilibrium. Liao et al. [11], Arik [12] and Cao and Wang [13] studied the robust stability of the interval DNNs. In Singh [4], Baldi and Atiya [5], Ye et al. [6] and Li et al. [7] several new criteria on the robust stability of a DNN with parametric uncertainties are obtained by using a linear matrix inequality (LMI) approach. However, in practice, delays in many electronic neural networks vary sometimes violently with time. Therefore we deal with the problem of robust stability analysis for CGNNs with multiple time-varying delays. The parameter uncertainties are expressed in terms of structural matrices and assumed to be norm bounded. We all know that CGNNs are quite general and include several well-known neural networks as special cases (Hopfield neural networks, cellular neural networks). Recently, the dynamic behaviours of CGNNs with delays have been extensively investigated by Rong [2], Wang and Zou [9], Lu and Chen [14] and Rong [15]. In Rong [2] several sufficient conditions guaranteeing the global robust convergence of the equilibrium point are derived on the basis of the Lyapunov stability theory and LMI technique. In Wang and Zou [9], several criteria are given on global asymptotic stability of a unique equilibrium for CGNNs with multiple time delays without assuming the symmetry of the interconnections. However, few authors consider the robust stability for CGNNs with multiple time-varying delays. The purpose of this paper is to derive new criteria for global robust asymptotic stability (GRAS) of the CGNN by constructing suitable Lyapunov functionals. The results are given in the form of LMI(s), which can be performed efficiently via numerical algorithms such as the interior point algorithms [16, 17]. In this paper, we further extend the results of Rong [2] to a significantly larger class of neural system. Consider the neural network model with time-varying delays described by the following state equations " n X dxi ðtÞ ¼ ai ðxi ðtÞÞ bi ðxi ðtÞÞ wð0Þ ij sj ðxj ðtÞÞ dt j¼1 # r X n X ðkÞ wij sj ðxj ðt tk ðtÞÞÞ þ Ji
ð1Þ
k¼1 j¼1
397
2
or in vector form "
dxðtÞ ¼ aðxðtÞÞ bðxðtÞÞ W0 SðxðtÞÞ dt # r X Wk Sðxðt tk ðtÞÞÞ þ J
ð2Þ
k¼1
where n 2 is the number of neurons in the networks; x(t) ¼ [x1(t), x2(t), . . . , xn(t)]T is the state vector of the neural networks; a(x(t)) ¼ diagfa1(x1(t)), a2(x2(t)), . . . , an(xn(t))g, with the diagonal element composed of the gain functions and ai(xi(t)) is assumed to be positive; b(x(t)) ¼ [b1(x1(t)), b2(x2(t)), . . . , bn(xn(t))]T with bi(.) modelling self-inhibition of the neuron. The activation functions sj(xj) show how neurons respond to each other, J ¼ (J1 , J2 , . . . , S(x(t)) ¼ [s1(x1), s2(x2), . . . , sn(xn)]T; Jn)T [ Rn denotes the constant input from outside the and Wk ¼ (w(k) k ¼ 1, system. W0 ¼ (w(0) ij )nn ij )nn , 2, . . . , r, are the connection weight matrix and delayed connection matrices, respectively. We do not limit the interconnecting structure to be symmetry. The initial condition associated with (1) or (2) is given in the form xi ðsÞ ¼ fi ðsÞ [ Cð½t; 0; RÞ;
i ¼ 1; 2; . . . ; n
ð3Þ
and 0 tk(t) t. We mainly consider the following uncertain neural networks " dxðtÞ ¼ aðxðtÞÞ bðxðtÞÞ ðW0 þ DW0 ÞSðxðtÞÞ dt # r X ðWk þ DWk ÞSðxðt tk ðtÞÞÞ þ J ð4Þ k¼1
where DWk ¼ (Dw(k) ij )nn , k ¼ 0, 1, 2, . . . , r, represent the parameter uncertainties in the matrices Wk , k ¼ 0, 1, 2, . . . , r, respectively. We assume that DWk are of the following form DWk ¼ HFEk ;
k ¼ 0; 1; 2; . . . ; r
ð5Þ
where H, Ek , k ¼ 0, 1, 2, . . . , r, are known constant matrices of appropriate dimensions and F is an unknown matrix bounded by FT F I
ð6Þ
in which I is the identity matrix of appropriate dimension, and if both (5) and (6) hold, DWk , k ¼ 0, 1, 2, . . . , r, are said to be admissible. Throughout this paper, we give the following assumptions † (H1) For each i [ f1, 2, . . . , ng, bi(xi) [ C 1(R, R) and b i0 gi . 0 † (H2) For each i [ f1, 2, . . . , ng, si(xi) is bounded and monotonically non-decreasing on R † (H3) For each i [ f1, 2, . . . , ng, si: R ! R is Lipschitz continuous with a Lipschitz constant li , that is, there exists constant li such that jsi(a) 2 si(b)j lija 2 bj, 8a, b [ R. Notations: Rn is the n-dimensional Euclidean space; Rmn denotes the set of m n real matrix and ‘ ’ denotes the symmetric part. For real symmetric matrix A, A 0 (A . 0) means that the matrix A is positive semi-definite (positive definite). 398
Preliminaries
As the activation functions si(xi), i ¼ 1, 2, . . . , n, are bounded, we can prove that there exists at least one equilibrium point for system (1) or (2) (refer to Proposition 3.1 in Wang and Zou [9]). Assume that model (1) or (2) has an equilibrium point x ¼ [x1, x2, . . . , xn]T for a given J. To simplify the proofs, we will shift the equilibrium point x of (1) to the origin. Let y(t) ¼ x(t) 2 x , y(t 2 tk(t)) ¼ x(t 2 tk(t)) 2 x , then model (1) can be transformed into the following form " n X dyi ðtÞ ¼ ai ð yi ðtÞÞ bi ð yi ðtÞÞ wð0Þ ij gj ð yj ðtÞÞ dt j¼1 # r X n X ðkÞ wij gj ð yj ðt tk ðtÞÞÞ ð7Þ k¼1 j¼1
where ai( yi(t)) ¼ ai( yi(t) þ xi ), bi( yi(t)) ¼ bi( yi(t) þ xi ) 2 bi(xi ), gj( yj(t)) ¼ sj( yj(t) þ xj ) 2 sj(xj ), and if we let yðtÞ ¼ ½ y1 ðtÞ; y2 ðtÞ; . . . ; yn ðtÞT Að yðtÞÞ ¼ diagfa1 ð y1 ðtÞÞ; a2 ð y2 ðtÞÞ; . . . ; an ð yn ðtÞÞg Bð yðtÞÞ ¼ ½b1 ð y1 Þ; b2 ð y2 Þ; . . . ; bn ð yn ÞT gð yðtÞÞ ¼ ½g1 ð y1 Þ; g2 ð y2 Þ; . . . ; gn ð yn ÞT uncertain system (4) can be transformed into the following equation " dyðtÞ ¼ Að yðtÞÞ Bð yðtÞÞ ðW0 þ DW0 Þgð yðtÞÞ dt # r X ðWk þ DWk Þgð yðt tk ðtÞÞÞ ð8Þ k¼1
Clearly, according to the assumption (H2) and (H3), gi(.) is monotonically non-decreasing with the relation yi ðtÞgi ð yi ðtÞÞ 0;
gi2 ð yi ðtÞÞ li yi ðtÞgi ð yi ðtÞÞ
8yi [ R; i ¼ 1; 2; . . . ; n
ð9Þ
and from the assumption (H1), we have yi ðtÞbi ð yi ðtÞÞ gi y2i ðtÞ
ð10Þ
Definition 1 [2]: The equilibrium point of system (1) or (2) is said to be globally robustly stable with respect to the perturbation Dw(k) ij , k ¼ 0, 1, . . . , r, if the equilibrium point of system (4) is globally asymptotically stable. Lemma 1: (Scalar complement [18]): The following LMI QðxÞ SðxÞ ,0 S T ðxÞ RðxÞ where Q(x) ¼ QT(x), R(x) ¼ RT(x). S(x) depends on x and is equivalent to each of the following conditions (1) Q(x) , 0, R(x) 2 ST(x)Q21(x)S(x) , 0 (2) R(x) , 0, Q(x) 2 S(x)R21(x)ST(x) , 0. Lemma 2 [19]: For a given scalar 1 . 0 and matrices D, E, F of appropriate dimensions with FTF I, the inequality DFE þ ETFTDT 1DDT þ 121ETE is always satisfied. IEE Proc.-Control Theory Appl., Vol. 153, No. 4, July 2006
3
Main results
by (9) and (10)
In this section, we will investigate the global robust stability of the equilibrium point of system (4). Theorem 1: Under the assumptions (H1)–(H3) and the delayed functions t k0 (t) hk , 1, k ¼ 1, 2, . . . , r. If there exist positive definite symmetric matrix Qk , k ¼ 1, 2, . . . , r, and positive definite diagonal matrix P ¼ diagfp1 , p2 , . . . , png . 0, D ¼ diagfd1 , d2 , . . . , dng . 0 and scalar 1 . 0 such that 0 PW1 2P G PW0 B M DW1 B B B ð1 t01 ðtÞÞQ1 þ 1ET1 E1 B V¼B . .. .. .. B .. . . . B B @ 0 0 1 PWr PH C DWr DH C C C 0 0 C C,0 .. .. C . . C C T 0 ð1 tr ðtÞÞQr þ 1Er Er 0 A 0 ½e=ðr þ 1Þ I where G ¼ diagfg1 , g2 , . . . , gng, P L ¼ diagfl1 , l2 , . . . , lng, M ¼ 22DL21G þ DW0 þ W T0 D þ rk¼1 Qk þ 1ET0 E0 , then for every input J, the equilibrium point of the uncertain delayed Cohen–Grossberg neural networks (DCGNNs) in (4) is globally asymptotically stable for all admissible uncertainties.
V ð yðtÞÞ ¼ 2
ð yi ðtÞ pi 0
i¼1
þ
r ðt X k¼1
s ds þ 2 ai ðsÞ
n X
ð yi ðtÞ di
i¼1
0
gi ðsÞ ds ai ðsÞ
Calculating the derivative of V along the trajectory of (8), we obtain " n n X X ð0Þ V_ ð yðtÞÞ ¼ 2 pi yi ðtÞ bi ð yi ðtÞÞ ðwð0Þ ij þ Dwij Þ j¼1 r X n X ðkÞ ðwðkÞ ij þ Dwij Þgj ð yj ðt tk ðtÞÞÞ
#
k¼1 j¼1
2
n X
" di gi ð yi ðtÞÞ bi ð yi ðtÞÞ
i¼1
n X ð0Þ ðwð0Þ ij þ Dwij Þ j¼1
r X n X ðkÞ gj ð yj ðtÞÞ ðwðkÞ ij þ Dwij Þgj ð yj ðt tk ðtÞÞÞ k¼1 j¼1
þ
r X
gT ð yðtÞÞQk gð yðtÞÞ
k¼1
r X
ð1 t0k ðtÞÞgT ð yðt tk ðtÞÞÞQk gð yðt tk ðtÞÞÞ
k¼1
IEE Proc.-Control Theory Appl., Vol. 153, No. 4, July 2006
yT ðtÞPðWk þ DWk Þgð yðt tk ðtÞÞÞ
k¼1
2gT ð yðtÞÞDGyðtÞ þ 2gT ð yðtÞÞDðW0 þ DW0 Þgð yðtÞÞ þ2
r X
gT ð yðtÞÞDðWk þ DWk Þgð yðt tk ðtÞÞÞ
k¼1
þ
r X
gT ð yðtÞÞQk gð yðtÞÞ
k¼1
r X ð1 t0k ðtÞÞgT ð yðt tk ðtÞÞÞQk gð yðt tk ðtÞÞÞ k¼1
2yT ðtÞP GyðtÞ þ 2yT ðtÞPðW0 þ DW0 Þgð yðtÞÞ þ2
r X
yT ðtÞPðWk þ DWk Þgð yðt tk ðtÞÞÞ
k¼1
2gT ð yðtÞÞDL1 Ggð yðtÞÞ þ 2gT ð yðtÞÞDðW0 þ DW0 Þgð yðtÞÞ þ2
r X
gT ð yðtÞÞDðWk þ DWk Þgð yðt tk ðtÞÞÞ
k¼1
þ
r X
gT ð yðtÞÞQk gð yðtÞÞ
k¼1
r X ð1 t0k ðtÞÞgT ð yðt tk ðtÞÞÞQk gð yðt tk ðtÞÞÞ
¼ ½ y ðtÞ; gT ð yðtÞÞ; gT ð yðt t1 ðtÞÞÞ; . . . ;
ttk ðtÞ
gj ð yj ðtÞÞ
r X
T
gT ð yðsÞÞQk gð yðsÞÞ ds
i¼1
þ2
k¼1
Proof: Consider the following Lyapunov functional n X
V_ ð yðtÞÞ 2yT ðtÞP GyðtÞ þ 2yT ðtÞPðW0 þ DW0 Þgð yðtÞÞ
#
gT ð yðt tr ðtÞÞÞ 1 0 yðtÞ C B gð yðtÞÞ C B C B gð yðt t1 ðtÞÞÞ C Sk B C B .. C B A @ . 0 B B B B k S ¼B B B @
gð yðt tr ðtÞÞÞ 2P G
PðW0 þ DW0 Þ
Y
DðW1 þ DW1 Þ
.. .
.. .
ð1 t01 ðtÞÞQ1 .. . 0
.. .
1 PðWr þ DWr Þ DðWr þ DWr Þ C C C C 0 C .. C A .
ð1 t0r ðtÞÞQr
PðW1 þ DW1 Þ
ð11Þ
T T 21 where 0 þ DW0) þ (W 0 þ DW 0 )D þ Pr Y ¼ 22DL G þ D(W k ˙ k¼1 Qk . Hence, if S , 0, then V( y(t)) 0 and all the solutions of (8) approach the set S ¼ fy(t): V˙( y(t)) ¼ 0g. Furthermore, V˙( y(t)) ¼ 0, if and only if
399
g( y(t)) ¼ g( y(t 2 t1(t))) ¼ . . . ¼ g( y(t 2 tk(t))) ¼ 0. (8) reduces to
So,
dyi ðtÞ ¼ ai ð yi ðtÞÞbi ð yi ðtÞÞ ¼ ai ð yi ðtÞÞb0i yi ðtÞ dt Since ai( yi(t)) . 0 and b i0 gi . 0, we know that the trivial solution is globally asymptotically stable. For more detailed proof, the reader is referred to Zhou and Zhou ([20], Theorem 1). So, to prove the theorem, it suffices to prove that Sk , 0. Next, we will provePthat V , 0 implies Sk , 0. From (11), Sk ¼ L þ L 0 þ rk¼1 Lk , where 0 PW0 2P G r P B B 2DL1 G þ DW0 þ W T0 D þ Qk B k¼1 B L¼B B . .. B . @ . .
PWr
DW1
DWr
.. .
0 .. . ð1 tr0 ðtÞÞQr
PDW0
0
0
DDW0 þ DW T0 D 0 .. .
0 0 .. .
0C C C 0C .. C .. C . .A
0
0
0
t10 ðtÞÞQ1
0
B DW T P B 0 B 0 L0 ¼ B B . B . @ . 0
B B B B B B B B B @
C C C C C C C A
2PG
PW0 M
PW1 DW1
.. .
.. .
Q1 þ 1ET1 E1 .. .
.. .
0 0
1
11 bbT þ 1gT0 g0
ð12Þ
and 0
0 B 0 B B . B . B . B T Lk ¼ B B DW k P B B 0 B . B . @ . 0
.. .
PDWk DDWk .. .
DW Tk D 0 .. .
.. .
0 0 .. .
0 0C C .. C .. C . .C C 0C C C 0C .C .. C . .. A
0
0 .. .
0 .. .
Qr þ 1ETr Er
0 e I rþ1
0
0
C C C C C C,0 C C C C A
where G ¼ diagfg1 , g2 , . . . , gng, P L ¼ diagfl1 , l2 , . . . , lng, r M ¼ 22DL21G þ DW0 þ W T0 D þ k¼1 Qk þ 1ET0 E0 , then for every input J, the equilibrium point of the uncertain DCGNNs in (4) with multiple constant delays is globally asymptotically stable for all admissible uncertainties. Clearly, the global robust stability is degenerated as global asymptotic stability when DWk ¼ 0. Thus, from (11) we can obtain the following condition for the asymptotic stability of system (1).
0 B B B B L¼B B B @
T
¼ bF gk þ ðbF gk Þ
11 bbT þ 1gTk gk
PH DH
Corollary 2: Under the assumptions (H1) –(H3) and the delayed functions t k0 (t) hk , 1, k ¼ 1, 2, . . . , r. If there exist positive definite symmetric matrix Qk , k ¼ 1, 2, . . . , r, and positive definite diagonal matrix P ¼ diagf p1 , p2 , . . . , png . 0, D ¼ diagfd1 , d2 , . . . , dng . 0 such that the following condition holds
1
0 0 .. .
1
PWr DWr
0
¼ bF g0 þ ðbFg0 ÞT
ð13Þ
1
PH B DH C B C B C b ¼ B 0 C; gk ¼ ð0 0 Ek 0Þ; k ¼ 0; 1; . . . ; r B .. C @ . A 0
and Ek is the (k þ 2)th element of gk . The inequalities (12) and (13) can be obtained by using Lemma 2 combined (6). Pr with conditions (5) and Lk L þ (r þ 1)121bbTþ Therefore Sk ¼ L þ L0 þ k¼1 Pr B 1 k¼1 gTk gk ¼ V , 0. This completes the proof. 400
0
1
PW1
.. . 0
where 0
Corollary 1: Under the assumptions (H1) – (H3), if there exist positive definite symmetric matrix Qk , k ¼ 1, 2, . . . , r, and positive definite diagonal matrix P ¼ diagf p1 , p2 , . . . , png . 0, D ¼ diagfd1 , d2 , . . . , dng . 0 and scalar 1 . 0 such that
ð1
0
If the transmission delays tk(t) ¼ tk ¼ constant, k ¼ 1, 2, . . . , r, we can obtain the following result.
2P G
PW0 2DL1 G þ DW0 þ W T0 D þ
.. .
.. .
PW1
PWr
DW1
DWr
ð1 t01 ðtÞÞQ1 .. . 0
0 .. .. . . ð1 t0r ðtÞÞQr
r P
Qk
k¼1
1 C C C C C,0 C C A
where G ¼ diagfg1 , g2 , . . . , gng, L ¼ diagfl1 , l2 , . . . , lng, then for every input J, system (1) has a unique equilibrium point x that is globally asymptotically stable, dependent of the delays. When r ¼ 1 and t(t) ¼ t, then model (4) reduces to the following differential equation that is investigated by IEE Proc.-Control Theory Appl., Vol. 153, No. 4, July 2006
Rong [2] using a different Lyapunov functional
where 0
dxðtÞ ¼ aðxðtÞÞ½bðxðtÞÞ ðW0 þ DW0 ÞSðxðtÞÞ dt ðW1 þ DW1 ÞSðxðt tÞÞ þ J
ð14Þ
In theorem 1, let r ¼ 1, then we can obtain the following corollary. Corollary 3: Under the assumptions (H1) – (H3), the parameter uncertainties DW0 and DW1 satisfy conditions (5) and (6). If there exist positive definite symmetric matrix Q1 , positive definite diagonal matrix P ¼ diagfp1 , p2 , . . . , png . 0, D ¼ diagfd1 , d2 , . . . , dng . 0 and scalar 1 . 0 such that the following LMI holds 0 2PG PW0 1 B 2DL G þ DW0 þ W T0 D þ Q1 þ 1ET0 E0 B B @ 1
PH
DW1
DH C C C,0 0 A
Q1 þ 1ET1 E1 0
0 0 1
0
x1 ðtÞ B C bðxðtÞÞ ¼ @ x2 ðtÞ A x3 ðtÞ 0 0:1063 B W0 ¼ @ 0:1589 0:5475 0 0:2031 B W1 ¼ @ 0:1707 0:0767
0
0
1
C 4 þ cosðx2 ðtÞÞ 0 A 0 2 sinðx3 ðtÞÞ
1 0:9370 0:1655 C 0:9141 0:2899 A 0:4478 0:0202 1 0:1132 0:0702 C 0:0888 0:0355 A 0:4425 0:0944
H ¼ ð0:2397; 0:2823; 0:2976ÞT E0 ¼ 0:0212 0:0238 0:1008 E1 ¼ 0:0742 0:1082 0:0131
PW1
B aðxðtÞÞ ¼ @
3 þ sinðx1 ðtÞÞ
ð15Þ
121I
where G ¼ diagfg1 , g2 , . . . , gng, L ¼ diagfl1 , l2 , . . . , lng, then for every input J, the equilibrium point of the uncertain DCGNNs in (14) is globally asymptotically stable for all admissible uncertainties. Remark 1: Chen and Rong [21] investigated the robust stability of DCGNNs without considering the signs of the connection matrices. But in our paper, the disadvantage is overcome. Hwang et al. [22] and Wang and Zou [23] always assumed 0 , ai ai(x) a¯i , that is, the criteria in ¯ these literatures are dependent of the amplification functions ai(x). However, in our paper, ai(x) is harmless to the stability of the DNNs. So our results are less conservative and restrictive than those given in the literatures. Remark 2: If the conditions given in this paper are not satisfied, then model (4) may not be stable. In this case, we can design a suitable controller to stabilise the system considered, and this will be considered in the future works. That is, we can further consider the following system " dxðtÞ ¼ aðxðtÞÞ bðxðtÞÞ ðW0 þ DW0 ÞSðxðtÞÞ dt # r X ðWk þ DWk ÞSðxðt tk ðtÞÞÞ þ J þ u
and si(xi) ¼ (1/2)(jxi þ 1j 2 jxi 2 1j). But it can be verified that the robust stability condition of Corollary 4 in Rong [2] is not satisfied, so the robust stability condition in Rong [2] fails to show whether (16) is asymptotically stable or not. However, by Corollary 3, LMI (15) has a feasible solution 0
0:4746 0 B P¼@ 0 0:4706 0 0
0:8939
0
0 0
1 C A
0:5081
0:2999 0:1259
1
B C 1:1381 0:2006 A Q1 ¼ @ 0:2999 0:1259 0:2006 0:9959 0 1 0:8711 0 0 B C D¼@ 0 0:5960 0 A 1 ¼ 1:3373 0 0 0:9610 Therefore we can prove that the uncertain DCGNNs have a unique equilibrium point that is globally robust, asymptotically stable.
k¼1
We will design u such that the whole system is stabilised. 4
Numerical example
In this section, we will provide an example to show the effectiveness of the results obtained in the previous sections. Considering the following differential equations dxðtÞ ¼ aðxðtÞÞ½bðxðtÞÞ ðW0 þ DW0 ÞSðxðtÞÞ dt ðW1 þ DW1 ÞSðxðt 1ÞÞ þ J IEE Proc.-Control Theory Appl., Vol. 153, No. 4, July 2006
ð16Þ
Fig. 1 Dynamical convergence of (16) 401
As a special case, when F ¼ I, from [DW0 DW1] ¼ HF[E0 E1], we have 0 1 0:0051 0:0057 0:0242 B C DW0 ¼ @ 0:0060 0:0067 0:0285 A 0
0:0063 0:0071
0:0300
0:0259
0:0031
0:0178
B 0:0305 DW1 ¼ @ 0:0209 0:0221 0:0322
1
C 0:0037 A 0:0039
So we can give the simulation for the delayed differential (16) that is shown in Fig. 1. For numerical simulation, let J ¼ [0.6, 0.2, 0.4]T and the delay parameter t ¼ 1, with the initial state x(t) ¼ [0.8, 0.7, 0.8]T for t [ [21, 0]; Fig. 1 shows the time response of state variables of x1(t), x2(t), and x3(t). 5
Conclusions
In this paper, GRAS is studied for DCGNNs with multiple time-varying delays. Several new sufficient criteria have been derived to ensure the global stability for the DNNs by using the Lyapunov method, LMIs and matrices inequality technique. The results obtained improve and extend the earlier works and the criteria are easy to check and apply. An illustrative example has also been presented to demonstrate both the theoretical importance and practical significance of the results obtained. 6
Acknowledgment
This work was jointly supported by the National Natural Science Foundation of China under grants nos. 60574043 and 60373067 and the Natural Science Foundation of Jiangsu Province, China under grant no. BK2003053. 7
References
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IEE Proc.-Control Theory Appl., Vol. 153, No. 4, July 2006
