Stability analysis of impulsive parabolic complex networks

Chaos, Solitons & Fractals 44 (2011) 1020–1034

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Stability analysis of impulsive parabolic complex networks Jin-Liang Wang ⇑, Huai-Ning Wu Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University, XueYuan Road, No. 37, HaiDian District, Beijing 100191, China

a r t i c l e

i n f o

Article history: Received 30 March 2011 Accepted 27 August 2011 Available online 29 September 2011

a b s t r a c t In the present paper, two kinds of impulsive parabolic complex networks (IPCNs) are considered. In the first one, all nodes have the same time-varying delay. In the second one, different nodes have different time-varying delays. Using the Lyapunov functional method combined with the inequality techniques, some global exponential stability criteria are derived for the IPCNs. Furthermore, several robust global exponential stability conditions are proposed to take uncertainties in the parameters of the IPCNs into account. Finally, numerical simulations are presented to illustrate the effectiveness of the results obtained here. Crown Copyright Ó 2011 Published by Elsevier Ltd. All rights reserved.

1. Introduction In the real world, complex networks are ubiquitous, and have been considered as a fundamental tool to understand dynamical behavior and the response of real systems such as food webs, communication networks, social networks, power grids, cellular networks, World Wide Web, metabolic systems, disease transmission networks, and many others [1]. In recent years, the topology and dynamical behavior of complex networks have been extensively studied by the researchers. In particular, the synchronization of complex networks has received much attention, and many interesting results on synchronization were derived for various complex networks such as time invariant, time-varying, discrete, and impulsive network models [2–15]. Synchronization is timekeeping which requires the coordination of events to operate a system in unison, and timekeeping technologies such as the GPS satellites and network time protocol (NTP) provide real-time access to a close approximation to the UTC timescale. Synchronization is an important concept in the field of computer science, cryptography, multimedia, neuroscience, telecommunication, etc. [2]. To our knowledge, in most existing works on the complex networks (see also the above mentioned references), it is assumed that the node state is only dependent on the time. Food webs are among the most well-known examples of complex networks and hold a central place in ecology to study the dynamics of animal populations. They consist of top species, intermediate species, and basal species. A food web can be characterized by a model of complex network, in which a node represents a species. Species are usually inhomogeneously distributed in a bounded habitat, then it is important and interesting to investigate their spatial density in order to better protect and control their population. In such a case, the node state will represent the spatial density of the species. Obviously, the spatial density of the species is seriously dependent on the time and space. Therefore, it is essential to consider the node state varying both temporally and spatially. Moreover, impulsive phenomena widely exist in food webs. Due to pollution, habitat loss, competition with other species, predator, diseases, hunting, and natural disasters or other reasons, sharp change of the density of species often occurs in a very short period of time. Hence, in order to describe more accurately the dynamics changes of species, impulsive effects should also be considered.

⇑ Corresponding author. Tel.: +86 1082317332; fax: +86 10 82317332. E-mail address: [email protected] (J.-L. Wang). 0960-0779/$ - see front matter Crown Copyright Ó 2011 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2011.08.005

J.-L. Wang, H.-N. Wu / Chaos, Solitons & Fractals 44 (2011) 1020–1034

1021

Recently, many results have been obtained for food webs [16–33]. Wang [18] investigated the asymptotic behavior of solutions for a time-delayed Lotka–Volterra N-species mutualism reaction–diffusion system with homogeneous Neumann boundary condition. In [21], Pao studied the asymptotic behavior of time-dependent solutions of a three-species reaction–diffusion system in a bounded domain under a Neumann boundary condition. Ko and Ahn [25] considered a diffusive ratio-dependent simple food chain model. Some sufficient conditions for the existence and non-existence of coexistence states were obtained. Kim and Lin [26] studied the blowup properties of solutions for a parabolic system with homogeneous Dirichlet boundary condition. In [27], Aly et al. investigated the conditions of the existence and stability properties of the equilibrium solutions in a reaction–diffusion model in which predator mortality is increasing with the predator abundance. Du and Shi [30] studied the effects of a protection zone for the prey on a diffusive predator–prey model with Holling type II response and no-flux boundary condition. Nevertheless, in most existing works (see also the above mentioned references), the impulsive effect has not been considered. Motivated by the above discussions, in this paper, we propose two IPCN models. By constructing suitable Lyapunov functionals and utilizing some inequality techniques, Lu [34] analyzed the global exponential stability and periodicity of a class of reaction–diffusion delayed recurrent neural networks with constant delays and Dirichlet boundary conditions. In this paper, we use similar methods to obtain criterions for the global exponential stability of the proposed network models with timevarying delays and Dirichlet boundary conditions. Furthermore, we also study the robustness of the global exponential stability under perturbations of the parameters of the IPCNs. The rest of this paper is organized as follows. In Section 2, our mathematical models of complex networks are presented and some preliminaries are given. The main result of this paper are given in Section 3. In Section 4, numerical examples are provided to illustrate the effectiveness of the proposed results. Concluding remarks are collected in Section 5. 2. Network model and preliminaries Let N ¼ f1; 2; 3; . . .g; R ¼ ð1; þ1Þ; Rþ ¼ ½0; þ1Þ and Rn be the n-dimensional Euclidean space, I and X denote, respectively, a subset of R and a subset of Rl ðl 2 NÞ; PC½I; X :¼ f/ : I ! Xj/ðtþ Þ ¼ /ðtÞ for t 2 I, /(t) exists for t 2 I, /(t) = /(t) for all but points t k 2 I; k 2 Ng; mesX denotes the Lebesgue measure of X, In denotes the n  n real identity matrix. For any t 2 R; eðx; tÞ ¼ ðe1 ðx; tÞ; e2 ðx; tÞ; . . . ; en ðx; tÞÞT 2 L2 ðX; Rn Þ; keðx; tÞk2 denotes

Z X n

keðx; tÞk2 ¼

X i¼1

!12 e2i ðx; tÞ dx

for any t 2 ½s; 0; Hðx; tÞ ¼ ðh1 ðx; tÞ; h2 ðx; tÞ; . . . ; hn ðx; tÞÞT 2 L2 ðX; Rn Þ, we define

kHks ¼ sup kHðx; tÞk2 : s6t60

In this paper, we consider two IPCNs consisting of N nonidentical nodes with diffusive and delay coupling. The first is described by the following equations

8 N P > @wi ðx;tÞ > > < @t ¼ di Dwi ðx; tÞ þ pi wi ðx; tÞ þ Gij wj ðx; t  sðtÞÞ; j¼1

> > > :

t – tk ;



 

wi ðx; t k Þ ¼ ð1 þ cik Þwi x; tk ;

ð1Þ

8k 2 N

where i = 1, 2, . . . , N. s(t) is the time-varying delay with 0 6 s(t) 6 s. In network (1), all nodes in the whole network have the same time-varying delay. In the following, a more general network is analyzed,

8 N P > @wi ðx;tÞ > > < @t ¼ di Dwi ðx; tÞ þ pi wi ðx; tÞ þ Gij wj ðx; t  sj ðtÞÞ; j¼1

> > > :

t – tk ;   wi ðx; t k Þ ¼ ð1 þ cik Þwi x; tk ;

ð2Þ

8k 2 N

where i = 1, 2, . . . , N. sj(t) is the time-varying delay with 0 6 sj(t) 6 s, j = 1, 2, . . . , N. wi ðx; tÞ 2 R represents the state of node i; ðx; tÞ 2 X  Rþ ; X ¼ fx ¼ ðx1 ; x2 ; . . . ; xm ÞT jjxk j < lk ; k ¼ 1; 2; . . . ; mg is the P @2 bounded domain with smooth boundary @ X; mesX > 0; di > 0; pi ; cik 2 R; D ¼ m k¼1 @x2 is the Laplace diffusion operator k on X,G = (Gij)NN represents the topological structure of network and coupling strength between nodes, where Gij is defined P as follows: if there is a connection from node i to node j (i – j), then Gij – 0; otherwise, Gij = 0 (i – j), and Gii ¼  Nj¼1 Gij . The j–i fixed moments tk satisfy 0 ¼ t 0 < t 1 < t 2 <    ; limk!þ1 t k ¼ þ1; k 2 N. Let w(x, t) = (w1(x, t), w2(x, t), . . . , wN(x, t))T. The initial value and boundary value conditions associated with the networks (1) and (2) are given in form

1022

J.-L. Wang, H.-N. Wu / Chaos, Solitons & Fractals 44 (2011) 1020–1034

ðx; tÞ 2 X  ½s; 0;

wðx; tÞ ¼ Uðx; tÞ;

ð3Þ

Uðx; tÞ ¼ ð/1 ðx; tÞ; /2 ðx; tÞ; . . . ; /N ðx; tÞÞT ; wðx; tÞ ¼ 0; ðx; tÞ 2 @ X  ½s; þ1Þ;

ð4Þ

where /i(x, t) is a bounded and continuous function on X  [s, 0] for all i = 1, 2, . . . , N. Let w(x, t, U) be the state trajectory of networks (1) and (2) from the above initial value and boundary value conditions. In food webs, Gij(i – j) denotes the impact strength of the jth species on the ith species. Gij, Gji(i – j) are dependent on the relationship between the species i and j. Specifically, we have Gij < 0, Gji > 0 if species i and j are the prey and predator, respectively; we have Gij < 0, Gji < 0 if species i and j are competitors; we have Gij = Gji = 0 if there is no relationship between species i and j. Remark 1. In IPCNs (1) and (2), the coupling configurations are not restricted to the symmetric, irreducible connections and the non-negative off-diagonal links. Moreover, each dynamical node may have different node dynamics. Next, we introduce some useful definitions and lemmas. Definition 2.1. The IPCN (1) (IPCN (2)) is said to be globally exponentially stable if there exist constants  > 0 and M P 1 such that for any two solutions w(x, t, W), w(x, t, U) of IPCN (1) (IPCN (2)) with initial functions W, U, respectively, it holds that

kwðx; t; WÞ  wðx; t; UÞk2 6 MkW  Uks et for all t P 0. Definition 2.2. Letting V : Rþ ! R, we define the upper right derivative of V(t) as follows

Dþ VðtÞ ¼ lim sup h!0

þ

Vðt þ hÞ  VðtÞ : h

Lemma 2.1 (See [34]). Let X be a cube jxkj < lk (k = 1, 2, . . . , m) and let h(x) be a real-valued function belonging to C1(X) which vanishes on the boundary oX of X, i.e., h(x)joX = 0. Then

Z

X

2

2

h ðxÞ dx 6 lk

Z  X

@h @xk

2

dx;

where x = (x1, x2, . . . , xm)T. Lemma 2.2. Let 0 6 si(t) 6 s, i = 1, 2, . . . , n. If there exist real numbers

r > 0; #^1 P 0; #^2 P 0; . . . ; #^n P 0 and # such that

8 þ > D uðtÞ 6 #uðtÞ þ #^1 uðt  s1 ðtÞÞ þ #^2 uðt  s2 ðtÞÞ > > > > < þ    þ #^ uðt  s ðtÞÞ; n

> > > > > :

n

t P 0;   uðtk Þ 6 ru tk ; k 2 N;

8 þ > D v ðtÞ > #v ðtÞ þ #^1 v ðt  s1 ðtÞÞ þ #^2 v ðt  s2 ðtÞÞ > > > > < þ    þ #^ v ðt  s ðtÞÞ; n

> > > > > :

n

t P 0;   v ðtk Þ ¼ rv tk ; k 2 N;

then u(t) 6 v(t), for s 6 t 6 0 implies

uðtÞ 6 v ðtÞ;

for t P 0

where uðtÞ; v ðtÞ 2 PC½½s; þ1Þ; R. Proof. Firstly, we prove that

uðtÞ 6 v ðtÞ;

t 2 ½0; t1 Þ:

ð5Þ

If (5) is not true, by using the continuity of u(t), v(t) for t 2 [0, t1) and u(t) 6 v(t) for t 2 [s, 0], then there must exist a t 2 ½0; t1 Þ such that

uðtÞ ¼ v ðtÞ;

Dþ uðtÞ P Dþ v ðtÞ;

uðtÞ 6 v ðtÞ;

t 6 t:

ð6Þ

J.-L. Wang, H.-N. Wu / Chaos, Solitons & Fractals 44 (2011) 1020–1034

1023

In view of (6), and the fact that #^1 ; #^2 ; . . . ; #^n P 0; we have

Dþ uðtÞ 6 #uðtÞ þ #^1 uðt  s1 ðtÞÞ þ #^2 uðt  s2 ðtÞÞ þ    þ #^n uðt  sn ðtÞÞ 6 #v ðtÞ þ #^1 v ðt  s1 ðtÞÞ þ #^2 v ðt  s2 ðtÞÞ þ    þ #^n v ðt  sn ðtÞÞ < Dþ v ðtÞ; which contradicts the first inequality in (6). So, (5) holds. For any s 2 N, we suppose

uðtÞ 6 v ðtÞ;

for t 2 ½0; ts Þ:

Then, we have

  uðts Þ 6 ruðt s Þ 6 rv ts ¼ v ðt s Þ: Thus, u(t) 6 v(t) for t 2 [s, ts]. Employing the similar process of the proof of (5), we can get that u(t) 6 v(t), for t 2 [ts, ts+1). By mathematical induction, it is easy to conclude that u(t) 6 v(t) for t P 0. This completes the proof. h Lemma 2.3 [35]. Let ftk gk2N be an increasing sequence of real numbers and consider the impulsive differential system

8 _ ¼ AðtÞyðtÞ þ f ðt; yðtÞÞ; t – t k ; > < yðtÞ   Dyðt k Þ ¼ yðt k Þ  yðtk Þ ¼ Bk y tk ; k 2 N; > : yðt 0 Þ ¼ y0 ;

ð7Þ

under the following assumptions: (i) A(t) is an n  n matrix, piecewise continuous from Rþ to Rnn with discontinuities of the first kind at t = tk and A(t) is right continuous at t = tk. (ii) f : Rþ  Rn ! Rn is continuous in ½tk1 ; tk Þ  Rn and for every y 2 Rn ; limðt;mÞ!ðtk ;yÞ f ðt; mÞ exists for t < tk. (iii) for every k, Bk is an n  n matrix, and det (In + Bk) – 0. Let y(t) be any solution of (7) existing on [t0, +1). Then y(t) satisfies the integral equation for t P t0,

yðtÞ ¼ Wðt; t 0 Þy0 þ

Z

t

Wðt; sÞf ðs; yðsÞÞ ds; t0

where

8 U k ðt; sÞ; for t; s 2 ½tk1 ; tk Þ; > >   > > > U ðt; tk ÞðIn þ Bk ÞU k tk ; s ; for tk1 6 s < tk 6 t < t kþ1 ; > < kþ1 !   iQ þ1   Wðt; sÞ ¼  > ðIn þ Bi ÞU i ti ; s ; ðt; t Þ ðI þ B ÞU t ; t U > n j j j1 kþ1 k j > > j¼k > > : for t i1 6 s < t i < t k 6 t < tkþ1 ; is the Cauchy matrix [36] of the impulsive differential system

8 _ ¼ AðtÞyðtÞ; t – tk ; > < yðtÞ     Dyðt k Þ ¼ yðt k Þ  y t k ¼ Bk y t k ; > : yðt 0 Þ ¼ y0

k 2 N;

and Uk(t, s) is the standard fundamental solution matrix of the linear differential system

_ yðtÞ ¼ AðtÞyðtÞ;

t k1 6 t < tk :

3. Main results In this section, we shall investigate the global exponential stability and robust global exponential stability of IPCNs (1) and (2). 3.1. Stability analysis Suppose w(x, t, W) and w(x, t, U) are two arbitrary solutions of IPCN (1) with initial conditions W, U, define z(x, t) = w(x, t, W)w(x, t, U), Wz = W  U, then the dynamics of the difference vector z(x, t) = (z1(x, t), z2(x, t), . . . , zN(x, t))T is governed by the following equations

1024

J.-L. Wang, H.-N. Wu / Chaos, Solitons & Fractals 44 (2011) 1020–1034

8 N > < @zi ðx;tÞ ¼ di Dzi ðx; tÞ þ p zi ðx; tÞ þ P Gij zj ðx; t  sðtÞÞ;

t – tk ;

i

@t

> :

  zi ðx; t k Þ ¼ ð1 þ cik Þzi x; tk ;

ð8Þ

j¼1

k 2 N;

where i = 1, 2, . . . , N. Theorem 3.1. If there exists a positive constant b < 1 such that

j1 þ cik j 6 b; S1 þ

2 ln b

q

þ

i ¼ 1; 2; . . . ; N; S2 b2

k 2 N;

ð9Þ

< 0;

ð10Þ

where

( N X

q ¼ supftk  tk1 ; k 2 Ng; S2 ¼ max

i¼1;2;...;N

( S1 ¼ max



i¼1;2;...;N

m X

2di

k¼1

lk

2

þ 2pi þ

N X

) jGji j ;

j¼1

) jGij j ;

j¼1

then IPCN (1) is globally exponentially stable in the following sense: kt

kwðx; t; WÞ  wðx; t; UÞk2 6 b1 kW  Uks e 2 ;

t P 0;

where k > 0 is a unique solution of

k þ S1 þ

2 ln b

q

S2

þ eks

b2

¼ 0:

ð11Þ

b ks S2 Proof. Firstly, let hðkÞ ¼ k þ S1 þ 2 ln h(+1) > 0 and h0 (k) > 0. q þ e b2 . From (10), we have h(0) < 0. Moreover, it is obvious that P R Using the continuity and the monotonicity of h(k), (11) has a unique solution k > 0. Let V 1 ðtÞ ¼ Ni¼1 X z2i ðx; tÞ dx. Then the upper right derivative of V1(t) is given for all t – tk by

" Z # Z Z Z N N N X X X @zi ðx; tÞ dx ¼ 2 zi ðx; tÞ 2 di zi ðx; tÞDzi ðx; tÞdx þ pi z2i ðx; tÞ dx þ zi ðx; tÞ Gij zj ðx; t  sðtÞÞ dx @t X X X X i¼1 i¼1 j¼1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffisZ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# " Z Z Z N N X X 2 di zi ðx; tÞDzi ðx; tÞdx þ pi z2i ðx; tÞ dx þ jGij j z2i ðx; tÞdx z2j ðx; t  sðtÞÞ dx 6

Dþ V 1 ðtÞ ¼

X

i¼1

6

N X

" 2di

Z

X

zi ðx; tÞDzi ðx; tÞ dx þ 2pi þ

X

i¼1

X

j¼1

N X

jGij j

!Z

X

j¼1

z2i ðx; tÞ dx

X

þ

N X j¼1

jGij j

Z X

# z2j ðx; t

From Green’s theorem and the boundary condition, we have

Z

zi ðx; tÞDzi ðx; tÞ dx ¼ 

X

2 m Z  X @zi ðx; tÞ dx: @xk X k¼1

According to Lemma 2.1, we can obtain

2 m Z  m Z X X z2i ðx; tÞ @zi ðx; tÞ dx P dx: 2 @x k X X lk k¼1 k¼1 Therefore, þ

D V 1 ðtÞ 6

N X

" 

m X 2di

i¼1

¼

N X



k¼1

N Z X i¼1

lk

m X 2di

i¼1

6 S1

2

k¼1

X

2 lk

þ 2pi þ

N X

!Z

jGij j

X

j¼1

þ 2pi þ

N X

z2i ðx; tÞ dx þ S2

jGij j

X

j¼1 N Z X i¼1

!Z

X

z2i ðx; tÞ dx

þ

z2i ðx; tÞ dx þ

N X

jGij j

X

j¼1 N X N X i¼1

Z

j¼1

jGji j

# z2j ðx; t Z X

 sðtÞÞdx

z2i ðx; t  sðtÞÞ dx

z2i ðx; t  sðtÞÞ dx ¼ S1 V 1 ðtÞ þ S2 V 1 ðt  sðtÞÞ:

 sðtÞÞ dx :

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J.-L. Wang, H.-N. Wu / Chaos, Solitons & Fractals 44 (2011) 1020–1034

Furthermore,

V 1 ðt k Þ ¼

Z N N Z X X     ð1 þ cik Þ2 z2i ðx; t k Þ dx 6 b2 z2i x; t k dx ¼ b2 V 1 t k ; X

i¼1

i¼1

k 2 N:

X

For any e > 0, let g(t) be a unique solution of impulsive delay system

8 _ þ  þ S2 gðt  sðtÞÞ; t – t k ; t P 0; > < gðtÞ ¼ S1 gðtÞ   gðtk Þ ¼ b2 g tk ; k 2 N; > : gðtÞ ¼ kWz ðx; tÞk22 ; s 6 t 6 0: Since V 1 ðtÞ ¼ kWz ðx; tÞk22 ¼ gðtÞ for s 6 t 6 0. According to Lemma 2.2, we have

gðtÞ P V 1 ðtÞ P 0; t P 0: By utilizing Lemma 2.3, we have

gðtÞ ¼ Wðt; 0Þgð0Þ þ

Z

t

Wðt; sÞðS2 gðs  sðsÞÞ þ Þ ds;

tP0

0

where W(t, s), t, s P 0 is the Cauchy matrix of linear system



-_ ðtÞ ¼ S1 -ðtÞ; t – tk ; -ðtk Þ ¼ b2 -ðtk Þ; k 2 N:

According to the representation of the Cauchy matrix, we get the following estimate since 0 < b < 1 and q P tk  tk1

!

Y

S1 ðtsÞ

Wðt; sÞ ¼ e

2

b

2 ln b

6 eS1 ðtsÞ b2ð q 1Þ ¼ eðS1 þ ts

2 ln b

q  q ÞðtsÞ

2 ln b

b2ð q 1Þ ¼ b2 eðS1 þ ts

q ÞðtsÞ

;

t P s P 0:

s
Let c ¼ b2 kWz ðx; tÞk2s . Accordingly

gðtÞ 6 b2 eðS1 þ 6 ceðS1 þ

2 ln b

q Þt

2 ln b

q Þt

kWz ðx; 0Þk22 þ

þ

Z

t

2 ln b

eðS1 þ

Z

q ÞðtsÞ

0

t

eðS1 þ

2 ln b

q ÞðtsÞ

0



S2 2

b



S2



b

b2

gðs  sðsÞÞ þ 2

gðs  sðsÞÞ þ

 b

2



ds;

t P 0:



ds ð12Þ

S2 b Since ; k > 0; S1  2 ln q  b2 > 0 and 0 < b < 1, we have

gðtÞ 6

kWz ðx; tÞk22



 ; < cekt þ  2 S2 b S1  2 ln q  b2 b

b2

s 6 t 6 0:

ð13Þ

In the following, we shall prove that



gðtÞ < cekt þ 

S1 

2 ln b

q

 ;  bS22 b2

t P 0:

ð14Þ

If this is not true, according to the estimate (13) and gðtÞ 2 PC½½s; þ1Þ; Rþ , then there obviously exists a t⁄ > 0 such that 



gðt Þ P cekt þ 

 2 S2 b S1  2 ln q  b2 b

gðtÞ < cekt þ 



 ; S2 2 b S1  2 ln q  b2 b

ð15Þ

t < t :

ð16Þ

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J.-L. Wang, H.-N. Wu / Chaos, Solitons & Fractals 44 (2011) 1020–1034

According to (12) and (16), we get   Z t 2 ln b  2 ln b  S2  gðt Þ 6 ceðS1 þ q Þt þ eðS1 þ q Þðt sÞ 2 gðs  sðsÞÞ þ 2 ds b b 0 8 2 0 1 3 9 Z t < = 2 ln b  2 ln    ðS1 þ q Þt ðS1 þ q bÞs 4 S2 @ kðssðsÞÞ  A þ 2 5 ds cþ þ e ce þ
¼ cekt þ 



 : 2 S2 b S1  2 ln q  b2 b

This contradicts (15), and so the estimate (14) holds. Letting

0 6 V 1 ðtÞ 6 gðtÞ 6 cekt ;

 ? 0, we have

t P 0:

It implies the conclusion and the proof is completed.

h

Theorem 3.2. If there exists a constant b P 1 such that

j1 þ cik j 6 b; S1 þ

2 ln b

q

i ¼ 1; 2; . . . ; N;

k 2 N;

ð17Þ

2

þ b S2 < 0;

ð18Þ

where

q ¼ infftk  tk1 ; k 2 Ng; S2 ¼ max

( N X

i¼1;2;...;N

( S1 ¼ max



i¼1;2;...;N

m X 2di 2

k¼1

lk

N X

þ 2pi þ

)

) jGji j

j¼1

jGij j ;

j¼1

then IPCN(1) is globally exponentially stable in the following sense: kt

kwðx; t; WÞ  wðx; t; UÞk2 6 bkW  Uks e 2 ;

tP0

where k > 0 is a unique solution of

k þ S1 þ

2 ln b

q

þ eks b2 S2 ¼ 0:

ð19Þ

b ks 2 ^ ^ ^ ^0 Proof. Let hðkÞ ¼ k þ S1 þ 2 ln q þ e b S2 . From (18), we have hð0Þ < 0. Moreover, it is obvious that hðþ1Þ > 0 and h ðkÞ > 0. ^ Using the continuity and the monotonicity of hðkÞ, (19) has a unique solution k > 0. Take the same Lyapunov functional V1(t) as in Theorem 3.1. By the proof of Theorem 3.1, we can get the following estimate since b P 1 and q 6 tk  tk1

Y

Wðt; sÞ ¼ eS1 ðtsÞ

!

b2

2 ln b 2 ln b

6 eS1 ðtsÞ b2ð q þ1Þ ¼ eðS1 þ ts

q  q ÞðtsÞ

b2ð q þ1Þ ¼ b2 eðS1 þ ts

2 ln b

q ÞðtsÞ

;

t P s P 0:

s
^ ¼ b2 kWz ðx; tÞk2s . Accordingly Let c 2 ln b

gðtÞ 6 b2 eðS1 þ ^eðS1 þ 6c

q Þt

2 ln b

q Þt

kWz ðx; 0Þk22 þ

þ

Z

t

Z

eðS1 þ

2 ln b

q ÞðtsÞ

ðb2 S2 gðs  sðsÞÞ þ b2 Þ ds

0

2 ln b

eðS1 þ

t

q ÞðtsÞ

ðb2 S2 gðs  sðsÞÞ þ b2 Þ ds;

t P 0:

ð20Þ

0 2 b Since ; k > 0; S1  2 ln q  b S2 > 0 and b P 1, we have

gðtÞ 6 b2 kWz ðx; tÞk22 < c^ekt þ 

b2

2 b S1  2 ln q  b S2

;

s 6 t 6 0:

ð21Þ

J.-L. Wang, H.-N. Wu / Chaos, Solitons & Fractals 44 (2011) 1020–1034

1027

In the following, we shall prove that

b2

gðtÞ < c^ekt þ 

S1 

2 ln b

q

 b 2 S2

;

t P 0:

ð22Þ

If this is not true, according to the estimate (21) and gðtÞ 2 PC½½s; þ1Þ; Rþ , then there obviously exists a t⁄ > 0 such that

b2



gðt Þ P c^ekt þ 

2 b S1  2 ln q  b S2

b2

gðtÞ < c^ekt þ 

2 b S1  2 ln q  b S2

;

;

ð23Þ

t < t :

ð24Þ

According to (20) and (24), we can get Z t 2 ln b  2 ln b  gðt Þ 6 c^eðS1 þ q Þt þ eðS1 þ q Þðt sÞ ðb2 S2 gðs  sðsÞÞ þ b2 Þ ds 0 8 0 1 2 3 9 Z t < = 2 ln b  b  b2  b2 25 ðS1 þ q Þt ðS1 þ2 ln Þs 4 2 kðssðsÞÞ @ A q ^ ^ þ  þ b ds cþ e b S2  ce þ
b2



^ekt þ  ¼c

2 b S1  2 ln q  b S2

:

This contradicts (23), and so the estimate (22) holds. Letting

^ekt ; 0 6 V 1 ðtÞ 6 gðtÞ 6 c

 ? 0, we have

t P 0:

It implies the conclusion and the proof is completed.

h

Remark 2. Note that Theorems 3.1 and 3.2 give criterions of global exponential stability of IPCN (1) as well as an estimation of the exponential rate of convergence through Eqs. (11) and (19). In the following, we shall discuss the global exponential stability of IPCN (2). Suppose w(x, t, W) and w(x, t, U) are two arbitrary solutions of IPCN (2) with initial conditions W, U. Let z(x, t) = w(x, t, W)  w(x, t, U) and Wz = W  U, then the dynamics of the difference vector z(x, t) = (z1(x, t), z2(x, t), . . . , zN(x, t))T is governed by the following equations

8 N > < @zi ðx;tÞ ¼ d Dz ðx; tÞ þ p z ðx; tÞ þ P G z ðx; t  s ðtÞÞ; t – t ; i i ij j j k i i @t j¼1 > : zi ðx; t k Þ ¼ ð1 þ cik Þzi ðx; tk Þ; 8k 2 N;

ð25Þ

where i = 1, 2, . . . , N. Theorem 3.3. Let s_ j ðtÞ 6 r < 1. The IPCN (2) is globally exponentially stable if there exist positive constants ki, e, l and E > 1 such that

2ki  þ 2ki pi 

m X 2ki di k¼1

2 lk

þ ki

N X j¼1

jGij j þ

N X kj jGji je2s 6 0; 1r j¼1

ð26Þ

l 6 infftk  tk1 ; k 2 Ng;

ð27Þ

maxfbk ; k 2 Ng 6 E < e2el ;

ð28Þ

where

bk ¼ max fð1 þ cik Þ2 ; 1g; i¼1;2;...;N

i; j ¼ 1; 2; . . . ; N:

1028

J.-L. Wang, H.-N. Wu / Chaos, Solitons & Fractals 44 (2011) 1020–1034

Proof. Define the following Lyapunov functional for the system (25)

V 2 ðtÞ ¼

N X

( ki e

2t

Z

z2i ðx; tÞ dx

X

i¼1

) Z t Z N e2s X 2 2s þ jGij j zj ðx; sÞe dx ds : 1  r j¼1 tsj ðtÞ X

Then the upper right derivative of V2(t) is given for all t – tk by þ

D V 2 ðtÞ ¼

N X

( ki e

2t

2

Z X

i¼1 2s



e 1r N X

6

z2i ðx; tÞ dx þ

N X

Z

2zi ðx; tÞ

X

ð1  s_ j ðtÞÞjGij je2sj ðtÞ

j¼1

Z X

(

ki e2t ð2 þ 2pi Þ

Z X

i¼1

Z N @zi ðx; tÞ e2s X dx þ jGij j z2j ðx; tÞ dx @t 1  r j¼1 X )

z2j ðx; t  sj ðtÞÞdx

z2i ðx; tÞ dx þ 2di

Z

zi ðx; tÞDzi ðx; tÞ dx þ 2

X

Z X

zi ðx; tÞ

N X

Gij zj ðx; t  sj ðtÞÞ dx

j¼1

) Z Z N N X e2s X 2 2 jGij j zj ðx; tÞ dx  jGij j zj ðx; t  sj ðtÞÞ dx þ 1  r j¼1 X X j¼1 ( !Z Z N m N X X X 2ki di 6 e2t 2ki  þ 2ki pi  z2i ðx; tÞ dx þ 2ki zi ðx; tÞ Gij zj ðx; t  sj ðtÞÞ dx 2 X X lk i¼1 j¼1 k¼1 ) Z Z N N N X X ki e2s X 2 2 þ jGij j zj ðx; tÞ dx  ki jGij j zj ðx; t  sj ðtÞÞ dx 6 e2t ð2ki  þ 2ki pi 1  r j¼1 X X j¼1 i¼1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z Z Z Z m N N X X 2ki di ki e2s X 2 2 2 Þ z ðx; tÞ dx þ 2k jG j z ðx; tÞ dx z ðx; t  s ðtÞÞ dx þ jG j z2j ðx; tÞ dx  i ij j ij i i j 2 1  r j¼1 X X X X lk j¼1 k¼1 ) ( Z Z N N m N X X X X 2ki di ki jGij j z2j ðx; t  sj ðtÞÞ dx 6 e2t ð2ki  þ 2ki pi  þ ki jGij jÞ z2i ðx; tÞ dx 2 X X lk j¼1 i¼1 j¼1 k¼1 ) (" ) #Z Z N N m N N X X X X ki e2s X 2ki di kj jGji je2s 2 2t 2 jGij j zj ðx; tÞdx ¼ e 2ki  þ 2ki pi  þ ki jGij j þ zi ðx; tÞ dx : þ 2 1  r j¼1 1r X X lk i¼1 j¼1 j¼1 k¼1 It follows from (26) that

Dþ V 2 ðtÞ 6 0; t–tk ; k 2 N:

ð29Þ

On the other hand,

V 2 ðt k Þ ¼

N X

( 2t k

ki e

X

i¼1

¼

N X

Z

( ki e2tk

i¼1

Z

z2i ðx; t k Þ dx

) Z tk Z N e2s X 2 2 s þ jGij j zj ðx; sÞe dx ds 1  r j¼1 t k sj ðtk Þ X

) Z Z tk N     e2s X ð1 þ cik Þ2 z2i x; t k dx þ jGij j z2j ðx; sÞe2s dx ds 6 bk V 2 tk : 1  r j¼1 X t k sj ðt k Þ X

It follows from (29) and (30) that

V 2 ðtÞ 6 V 2 ð0Þ

k1 Y

bi 6 V 2 ð0ÞEk1 ;

t 2 ½tk1 ; tk Þ;

k 2 N:

i¼1

Since

l 6 infftk  tk1 ; k 2 Ng, one has k  1 6 tk1 l , which implies lnE

Ek1 6 e l t ;

t 2 ½tk1 ; t k Þ;

Hence, we have lnE

V 2 ðtÞ 6 V 2 ð0Þe l t ; Furthermore,

t P 0:

k 2 N:

ð30Þ

J.-L. Wang, H.-N. Wu / Chaos, Solitons & Fractals 44 (2011) 1020–1034

) Z 0 Z N e2s X jGij j z2j ðx; sÞe2s dx ds 1  r j¼1 sj ð0Þ X X i¼1 " ( )# N se2s X kj jGji j kWz k2s ; 6 max fki g þ max i¼1;2;...;N i¼1;2;...;N 1  r j¼1

V 2 ð0Þ ¼

(Z

1029

N X

ki

z2i ðx; 0Þ dx þ

V 2 ðtÞ P min fki ge2t kzðx; tÞk22 : i¼1;2;...;N

Let

kþ ¼ max fki g;

k ¼ min fki g; i¼1;2;...;N

(

j ¼ max

i¼1;2;...;N

se

i¼1;2;...;N

2s

N X

1r

)

kj jGji j ;

j¼1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kþ þ j : M¼ k

Then M P 1, and we can obtain lnE

kzðx; tÞk2 6 Með2l Þt kWz ks : Namely, lnE

kwðx; t; WÞ  wðx; t; UÞk2 6 MkW  Uks eð2lÞt : This completes the proof of Theorem 3.3.

h

According to the Theorem 3.3, we can easily obtain the following corollary. Corollary 3.1. Let s_ j ðtÞ 6 r < 1. The IPCN (2) is globally exponentially stable if there exist positive constants ki and

2ki  þ 2ki pi 

m X 2ki di 2

k¼1

lk

max fð1 þ cik Þ2 g 6 1;

i¼1;2;...;N

þ ki

N X j¼1

N X kj jGji je2s jGij j þ 6 0; 1r j¼1

k2N

 such that ð31Þ

ð32Þ

where i, j = 1, 2, . . . , N. 3.2. Robust stability analysis Due to possible external perturbations and modeling errors, the parameters di, pi, Gij may be known only with a given precision. This is why we study now the robustness of the property of global exponential stability of the IPCNs. We will only consider bounded regions of parameters, that is,

; di 6 di 6 d i

i pi 6 pi 6 p

and Gij 6 Gij 6 Gij

8i; j 2 f1; 2; . . . ; Ng;

ð33Þ

where di ; di ; pi ; pi ; Gij and Gij are constants. Definition 2.2. The IPCN (1) (IPCN (2)) is robustly globally exponentially stable in the region of parameter D delimited by ; p; p the constants di ; d i i  i ; Gij and Gij , if the IPCN(1) (IPCN (2)) is globally exponentially stable for any set of parameters satisfying (33). Corollary 3.2 and Corollary 3.3 give sufficient conditions to ensure IPCN (1) is robustly exponentially stable in some region of the form (33). Corollary 3.2. If there exists a positive constant b < 1 such that

j1 þ cik j 6 b;

i ¼ 1; 2; . . . ; N;

k 2 N;

2 ln b b S2 b þ 2 < 0; S1 þ q b where

q ¼ supftk  tk1 ; k 2 Ng; i; j ¼ 1; 2; . . . ; N;

ð34Þ ð35Þ

1030

J.-L. Wang, H.-N. Wu / Chaos, Solitons & Fractals 44 (2011) 1020–1034

( b S 1 ¼ max



i¼1;2;...;N

b S 2 ¼ max

m X 2di 2

k¼1

( N X

i¼1;2;...;N

lk

N X

þ 2pi þ

) b ij ; G

j¼1

)

n o b ij ¼ max Gij ; Gij ; G

b ji ; G

j¼1

then IPCN (1) with the parameter ranges defined by (33) is robustly globally exponentially stable in the following sense: kt

kwðx; t; WÞ  wðx; t; UÞk2 6 b1 kW  Uks e 2 ;

tP0

where k > 0 is a unique solution of

kþb S1 þ

2 ln b

þ eks

q

b S2 b2

¼ 0:

ð36Þ

Proof. It is easy to check that if (34) and (35) are satisfied, then any IPCN(1) whose parameters are in the region defined by (33) satisfies the relations (9) and (10) of Theorem 3.1, and thus is globally exponentially stable. This completes the proof. h Corollary 3.3. If there exists a positive constant b P 1 such that

j1 þ cik j 6 b;

i ¼ 1; 2; . . . ; N;

k 2 N;

ð37Þ

2 ln b b þ b2 b S 2 < 0; S1 þ

ð38Þ

q

where

q ¼ infftk  tk1 ; k 2 Ng; i; j ¼ 1; 2; . . . ; N; (

b S 1 ¼ max



i¼1;2;...;N

b S 2 ¼ max

m X 2di 2

k¼1

( N X

i¼1;2;...;N

l )k

i þ þ 2p

N X

) b ij ; G

j¼1

n o b ij ¼ max Gij ; Gij G

b ji ; G

j¼1

then IPCN (1)with the parameter ranges defined by (33) is robustly globally exponentially stable in the following sense: kt

kwðx; t; WÞ  wðx; t; UÞk2 6 bkW  Uks e 2 ;

tP0

where k > 0 is a unique solution of

2 ln b S 2 ¼ 0: kþb S1 þ þ eks b2 b

ð39Þ

q

Proof. It is easy to check that if (37) and (38) are satisfied, then any IPCN(1) whose parameters are in the region defined by (33) satisfies the relations (17) and (18) of Theorem 3.2, and thus is globally exponentially stable. This completes the proof. h By the similar proof of Theorem 3.3 and Corollary 3.1, we can obtain the following conclusions. Here we omit their proof to avoid the repetition. Theorem 3.4. Let s_ j ðtÞ 6 r < 1. The IPCN (2) with the parameter ranges defined by (33) is robustly globally exponentially stable if there exist positive constants ki, e, l and E > 1 such that

i  2ki  þ 2ki p

m X 2ki di k¼1

2 lk

þ ki

N X j¼1

b ij þ G

N b ji e2s X kj G 6 0; 1r j¼1

ð40Þ

l 6 infftk  tk1 ; k 2 Ng;

ð41Þ

maxfbk ; k 2 Ng 6 E < e2el ;

ð42Þ

where

bk ¼ max fð1 þ cik Þ2 ; 1g; i¼1;2;...;N n o b ij ¼ max Gij ; Gij ; i; j ¼ 1; 2; . . . ; N: G

1031

J.-L. Wang, H.-N. Wu / Chaos, Solitons & Fractals 44 (2011) 1020–1034

0.25

i

||w ||2, i= 1, 2, 3

0.2

0.15

0.1

0.05

0

0

0.1

0.2

t

0.3

0.4

0.5

Fig. 1. The change processes of kw1(x, t)k2, kw2(x, t)k2 and kw3(x, t)k2 for case 1 of Example 1.

0.25

||wi||2, i= 1, 2, 3

0.2

0.15

0.1

0.05

0

0

0.1

0.2

t

0.3

0.4

0.5

Fig. 2. The change processes of kw1(x, t)k2, kw2(x, t)k2 and kw3(x, t)k2 for case 2 of Example 1.

Corollary 3.4. Let s_ j ðtÞ 6 r < 1. The IPCN (2) with the parameter ranges defined by (33) is robustly globally exponentially stable if there exist positive constants ki and  such that

i  2ki  þ 2ki p

m X 2ki di k¼1

2 lk

max fð1 þ cik Þ2 g 6 1;

i¼1;2;...;N

þ ki

N X

b ij þ G

j¼1

N b ji e2s X kj G 6 0; 1r j¼1

k 2 N;

ð43Þ

ð44Þ

n o b ij ¼ max Gij ; Gij ; i; j ¼ 1; 2; . . . ; N. where G 4. Example In this section, we give two examples to show the effectiveness of the proposed theoretical results. Example 1. Consider a complex network model, in which each node is a 1-dimensional dynamical system described by

@wi ðx; tÞ @ 2 wi ðx; tÞ ¼ di þ pi wi ðx; tÞ; @t @x2

i ¼ 1; 2; 3:

1032

J.-L. Wang, H.-N. Wu / Chaos, Solitons & Fractals 44 (2011) 1020–1034

0.45 0.4 0.35

i

||w ||2, i= 1, 2, 3

0.3 0.25 0.2 0.15 0.1 0.05 0

0

0.1

0.2

t

0.3

0.4

0.5

Fig. 3. The change processes of kw1(x, t)k2, kw2(x, t)k2 and kw3(x,t)k2 for case 1 of Example 2.

0.45 0.4 0.35

i

||w ||2, i= 1, 2, 3

0.3 0.25 0.2 0.15 0.1 0.05 0

0

0.1

0.2

t

0.3

0.4

0.5

Fig. 4. The change processes of kw1(x, t)k2, kw2(x, t)k2 and kw3(x, t)k2 for case 2 of Example 2.

Take di ¼ 0:4; pi ¼ 0:1i; cik ¼ 0:5; tk ¼ 0:1k; k 2 N; x 2 ð0:5; 0:5Þ; l ¼ 0:5. The matrix G is chosen as

0

0:2 B @ 0:08

0:09 0:18

0:08 0:07

1 0:11 C 0:1 A: 0:15

Case 1: Set s(t) = 0.15  0.15et, s = 0.15. We can find constants b = 0.5, q = 0.1 satisfying the conditions (9) and (10). According to Theorem 3.1, IPCN (1) with above given parameters is globally exponentially stable. The simulation results are shown in Fig. 1. t Case 2: Set sj ðtÞ ¼ 0:2  0:4e ; s ¼ 0:2; r ¼ 0:2. We can find constants k1 = k2 = k3 = 1 and  = 0.25 satisfying the condi1þj tions (31) and (32). From Corollary 3.1, IPCN (2) with above given parameters is globally exponentially stable. The simulation results are shown in Fig. 2. Example 2. Consider a complex network model, in which each node is a 1-dimensional dynamical system described by

@wi ðx; tÞ @ 2 wi ðx; tÞ ¼ di þ pi wi ðx; tÞ; @t @x2

i ¼ 1; 2; 3:

Take cik ¼ 0:3; t k ¼ 0:1k; k 2 N; x 2 ð0:5; 0:5Þ; l ¼ 0:5. The quantities di, pi and Gij may be intervalized as follows:

1033

J.-L. Wang, H.-N. Wu / Chaos, Solitons & Fractals 44 (2011) 1020–1034

0:6 6 di 6 0:8;

0:1i 6 pi 6 0:2i;

6 G21 6 0:06; 6 0:04;

0:15 6 G11 6 0:11;

0:1 6 G22 6 0:14;

0:11 6 G33 6 0:14;

0:04 6 G12 6 0:06;

0:06 6 G23 6 0:04;

i ¼ 1; 2; 3:

0:07 6 G13 6 0:09;

0:09 6 G31 6 0:07;

0:08

0:05 6 G32 ð45Þ

Case 1: Set s(t) = 0.15  0.15et, s = 0.15. We can find constants b = 0.7, q = 0.1 satisfying the conditions (34) and (35). By Corollary 3.2, we know that IPCN (1) with the parameter ranges defined by (45) is robustly globally exponentially stable. The simulation results are shown in Fig. 3. t Case 2: Set sj ðtÞ ¼ 0:2  0:4e ; s ¼ 0:2; r ¼ 0:2. We can find constants k1 = k2 = k3 = 1 and  = 0.3 satisfying the conditions 1þj (43) and (44). Thus, we can conclude from Corollary 3.4 that IPCN (2) with the parameter ranges defined by (45) is robustly globally exponentially stable. The simulation results are shown in Fig. 4. 5. Conclusion Two IPCN models have been introduced, in which the node states are dependent on the time and space variables. The global exponential stability and robust global exponential stability of the proposed network models have been taken into consideration in this paper, and some sufficient conditions have been established. 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