Robust Impulsive Synchronization of Discrete Dynamical Networks

Advances in Difference Equations Volume 2008, Article ID 184275,17pages doi:10.1155/2008/184275

Research Article

Robust Impulsive Synchronization of Discrete Dynamical Networks

Ming Lei¹and Bin Liu^{1, 2}

1Department of Information and Computation Science, College of Science, Chongqing Jiaotong University, Chongqing 400074, China

2Department of Information Engineering, The Australian National University, ACT 0200, Australia

Correspondence should be addressed to Bin Liu,[email protected] Received 17 June 2007; Revised 13 November 2007; Accepted 11 January 2008 Recommended by Roderick Melnik

We aim to study robust impulsive synchronization problem for uncertain discrete dynamical net- works. For the discrete dynamical networks with unknown but bounded network coupling, we will design some robust impulsive controllers which ensure that the state of a discrete dynamical network asymptotically synchronize with an arbitrarily assigned state of an isolate node of the net- work. Three representative examples are also worked through to illustrate our results.

Copyrightq2008 M. Lei and B. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Since the 1990s, synchronization of chaotic systems has been a current and active research area.

Numerous methods have been developed for chaos synchronizationsee, e.g.,1–9. More re- cently, synchronization of dynamical networks has been reported in the literaturesee, e.g., 10–14. The dynamical networks consist of coupled nodes, which are usually chaotic sys- tems. It has been noticed that when synchronization is applied to the dynamical networks, the network coupling may cause the failure of a synchronization scheme. The network coupling functions may be unknown a priori and may be in form of linear or nonlinear functions. In order to deal with this problem, the robust synchronization for uncertain dynamical networks has become an important research topic. Although robust adaptive synchronization scheme can be used to synchronize nodes of the uncertain dynamical networks where the network coupling is an unknown but bounded nonlinear functionsee, e.g.,14, yet the controller for adaptive synchronization is usually complex. It has been proved in the study of chaotic syn- chronization that impulsive synchronization approach is effective and robust in synchroniza- tion of chaotic systemssee, e.g.,7,8, and has a relatively simple structure. Moreover, since the controller of impulsive synchronization is discontinuous, impulsive synchronization can be

useful for digital secure communization systems9. But up to present, to the best knowledge of the authors, there are not any results about impulsive synchronization of discrete dynamical network.

In this paper, we aim to study the robust impulsive synchronization problem for an un- certain discrete dynamical network. By utilizing the ideas developed in15,16for impulsive systems15–23, we will derive several criteria under which robust impulsive synchronization is achieved for an uncertain discrete dynamical network, with the network coupling functions being unknown but bounded. It will be shown that impulsive synchronization approach of a dynamical network has the same good properties as those in impulsive synchronization of chaotic systems. Moreover, the impulsive controller is also easy to design.

The main contribution of this paper is a proposed new control approach, that is, impul- sive control, for discrete dynamical network or general discrete systemxk1 fk, xk gk, xk. For the classical feedback control,ukis in form ofuk Kk, xk. In this classi- cal control scheme, the control signal is input into the system at all the timek∈N. However, for some practical systems, it is not necessary and in some case is also impossible to input control signal into the system at all the time. In this paper, the classical controluk Kk, xkis replaced by the proposed impulsive controluk _∞

m1δk−tmImk, xk. Thus, the control signal is put into the discrete system just at the impulsive instances{tk, k ∈N}, not at all the time series{k, k∈N}, where functionδtsatisfies

δt

1, t0,

0, t / 0. 1.1

This kind of control scheme will be useful in control theory and applications. For example, it can be used for control and synthesis of the sampled-data control system, and so forth.

The organization of this paper is as follows. InSection 2, we introduce the concept of uniformly positive definite matrix function and some other notations. The robust impulsive synchronization scheme is also formulated for a dynamical network inSection 2. InSection 3, robust impulsive synchronization criteria are established. These criteria can be easily used for the design of a robust feedback controller. For illustration, some representative examples are given inSection 4.Section 5concludes the paper.

2. Problem formulation

LetRⁿdenote then-dimensional Euclidean space. LetR 0,∞,N{1,2, . . .}, and let· stand for the Euclidean norm inRⁿ.

Consider a discrete dynamical network consisting ofNidentical nodesn-dimensional discrete systemswith uncertain network coupling:

xik1 Akxik ϕ

k, xik gi

x1k, x2k, . . . , xNk

, n∈N, i1,2, . . . , N, 2.1 whereAk∈R^n×n, k∈N,ϕ :N×Rⁿ→Rⁿis smooth nonlinear vector-valued function, and g_i:R^m→Rⁿare smooth but unknown network coupling functions, wheremnN.

Clearly, the isolated node of the network is in form of

yk1 Akyk ϕ

k, yk

, k∈N. 2.2

It is assumed that the solution of2.2exists and is unique under any given initial condition y0 y0.

Y y xi

Si Bik

Nk

· · · gi

xi

−

· · · ·Transmission channel of network· · · ·

Figure 1: The impulsive synchronization control for theith nodeSi.

Remark 2.1. When the network achieves synchronization, namely, the statex₁k x₂k

· · ·xNk yk, ask→ ∞, the coupling terms should vanish:giy, y, . . . , y 0.

The robust impulsive synchronization scheme for the discrete network2.1is to design impulsive controllers{Nk, B_ik}such that the state of the following system2.3synchronizes with the state of2.2:

x_in1 Anxin ϕ

n, x_in g_i

x₁n, x2n, . . . , xNn

, n / N_k, xin1 Bik

xin−yn

, nNk, k∈N, i1,2, . . . , N, 2.3 whereΔxiNk1 xiNk1−xiNk. The sequence{Nk}satisfies

i0N0< N1< N2<· · ·, with limk→∞Nk∞;

iifor allk∈N,N_k1−N_k≥2.

Figure 1depicts the entire impulsive synchronization scheme subject to network cou- pling, whereS_istands forith node,Yis the isolated node2.2, andg_iis the uncertain network coupling ofith node,i1,2, . . . , N.

Remark 2.2. It should be noticed that the mathematical modeling of this paper is basically the discrete impulsive systems, in which the impulses occur in a discrete system at some instances.

But they are different from the discrete systems with inputsun, in which the input signals un are input into system at every instancen 1,2, . . . . In this impulsive control discrete system2.3, the input signals are input into system only at some instancesNk,k1,2, . . . . Remark 2.3. The synchronization scheme given by2.1–2.3is some similar to the one used in16,24for impulsive synchronization of continuous dynamical networks, but it is different from that in16,24and it is more significant than that in 16,24because of the following reasons.

iIn the practical networks, the signals, which are used to transmit, receive, and sam- ple, are often in form of discrete signals, not continuous forms. Hence, it is more practically significant to study the synchronization problem of discrete networks than that for continuous networks.

iiThe mathematical modeling is also different from that in16,24. Here, we use the impulsive difference equation discrete impulsive system to depict the impulsive synchro- nization scheme, while in16,24, the impulsive differential equation is used. Although sig- nificant progress has been made in the stability theory of impulsive differential equations, the

corresponding theory for discrete impulsive systems has not been fully developed; see25. It is a new research topic. Hence, the work in this paper is not a trivial extension of the previous work in16,24.

Defining the synchronization error asein xin−yn, then one has an error dynam- ical system of the form

e_in1 Anein ϕ

n, x_in, yn g_i

xn, yn

, n / N_k,

ein1 Bikein, nNk, k∈N, i1,2, . . . , N, 2.4 whereϕt, x i, y ϕt, xi−ϕt, y,gix, y gix1, x2, . . . , xN−giy, y, . . . , y, andBik ∈R^n×n. Clearly, the network2.1 synchronizes robustly with system 2.2by impulsive con- trollers{Nk, B_ik}if and only if the error system2.4is robustly asymptotically stable.

Assumption 2.4. There exist positive constantsr_ij >0, i, j1,2, . . . , N,such that g_i

x₁, x₂, . . . , x_N≤^N

j1

r_ije_j, i1,2, . . . , N. 2.5 Assumption 2.5. Assume that there exists an attractive domainU ⊆ Rⁿ for the isolated node 2.2and for anyxi, y∈U, there exist positive constantsLik>0 such that forn∈Nk, Nk1,

ϕ n, xi

−ϕn, y≤Likxi−y, i1,2, . . . , N, k∈N. 2.6 Remark 2.6. iAssumption 2.4is based ongiy, y, . . . , y 0, fori 1,2, . . . , N, and anyy∈ Rⁿ. Also,Assumption 2.5is based on the fact that the chaotic system is ultimate bounded.

iiIn recent published paper25, by using interval matrix decomposition method and comparing methodfor detail, see25, the robust stability is investigated for interval linear discrete impulsive systems and a class of affine discrete impulsive systems. In this paper, by employing Lyapunov function approach, we focus on the stability of error system2.4, which is a large-scale discrete impulsive system. Based on the stability results of2.4, the impulsive synchronization can be achieved on the isolated node’s attraction domain. Hence, the stability issue studied in this paper is different from that in25.

Definition 2.7. LetX:N→R^n×nbe ann×nmatrix function. Then,Xkis said to be

ia positive definite matrix function if for anyk∈N,Xkis a positive definite matrix;

iia positive definite matrix function bounded from above if it is a positive definite ma- trix function and there exists a positive real numberM >0 such that

λ_max

Xk

≤M, k∈N, 2.7

whereλmax·is the maximum eigenvalue;

iiia uniformly positive definite matrix function if it is a positive definite matrix function and there exists a positive real numberm >0 such that

λmin

Xk

≥m, k∈N, 2.8

whereλmin·is the minimum eigenvalue of matrix·.

Lemma 2.8see15. LetXk ∈ R^n×nbe a positive definite matrix function andYk ∈ R^n×na symmetric matrix. Then, for anyx∈Rⁿ, k∈N, the following inequality holds:

x^TYkx≤λmax

Xk⁻¹Yk

·x^TXkx. 2.9

Proof. It follows from the properties of positive definite matrix.

3. Robustly impulsive synchronization

In this section, we will derive the asymptotical stability criteria for the error system2.4such that the state of the discrete dynamical network synchronizes with an arbitrarily assigned state of an isolated node of the network by the robust impulsive controllers.

Theorem 3.1. Suppose that Assumptions 2.4 and2.5 hold, and assume that there exist uniformly positive definite matrix functions which are bounded from above,P_in,i1,2, . . . , N, and constants >0,γi≥0,αikn≥0, wheren∈Nk, Nk1andβikNk≥0,i, k∈N, such that

ifor alln∈Nk, Nk1, k∈N, the following inequalities hold:

A^TnPin1An2Lik λmax

P_i⁻¹n1A^TnPin1An

· λ_max

P_in1 λmin

Pin1Pin1

λmax

Pin1 1νi

L²_ik

1ν⁻¹_i ^N

j1

r_ij²

I

λ_max

P_i⁻¹n1A^TnPin1An^N

j1

r_ij⁻¹r_ji I

≤α_iknPin;

3.1 iifor allnN_k, k∈N,

λ_max

P_i⁻¹n1

IB_ikT

P_in1IB_ik

P_in1≤β_iknPin; 3.2 iii

∞ j0

lnγ_j−∞, 3.3

where

γj

⎧⎪

⎨

⎪⎩

αkj, ifj ∈

Nk, Nk1 , β_kj, ifjN_k, k∈N,

3.4

andγ01,αkn max1≤i≤N{αikn},β_kNk max1≤i≤N{βikNk}.

Then, for any initial conditionsx_i0 x_i0,y0 y₀ ∈ U, the uncertain discrete dynami- cal network2.1is robust impulsive synchronization with system2.2by the impulsive controllers {Nk, Bik}.

Proof. LetVn Vn, e1, e₂, . . . , e_{N N}

i1e_i^TP_inei. DenoteV_in e^T_iP_inei, i1,2, . . . , N.

Since P_in, i 1,2, . . . , N, are all uniformly positive definite matrix functions and bounded from above, there exist positive constantsa > 0,b > 0 such that the following in- equality holds:

a N

i1

e^T_iei≤Nmin

1≤i≤N

λmin

Pi

^N

i1

e^T_iei≤V ≤Nmax

1≤i≤N

λmax

Pi

^N

i1

e^T_iei≤b N

i1

e^T_iei. 3.5

For anyn∈Nk, Nk1, k∈N, we get

Vin1 ein1^TPin1ein1

Anein ϕgi

_T

Pin1

Anein ϕgi

e_in^TA^TnPin1Anein 2e_in^TA^TnPin1ϕϕ^TP_in1ϕ 2ein^TA^TnPin1gi2ϕ^TA^TnPin1gig_i^TPin1gi.

3.6

ByLemma 2.8, the terms in3.6can be estimated as

ein^TA^TnPin1ϕ

≤ein^TA^TnP_i^1/2n1P_i^1/2n1ϕ ein^TA^TnPin1Anein ϕ^TPin1ϕ

≤L_ik λ_max

P_i⁻¹n1A^TnPin1An

·λ_max

P_in1

· e_in^TP_in1ein e_in^Te_in

≤Lik λmax

P_i⁻¹n1A^TnPin1Anλ_max

P_in1 λmin

Pin1 ·ein^TPin1ein,

3.7

e_in^TA^TnPin1g_i

≤ein^TA^TnPin1gi

≤ e^T_inA^TnPin1Anein·^N

j1

rijejn

≤ λmax

P_i⁻¹n1A^TnPin1An

·^N

j1

rijeinejn

≤ λ_max

P_i⁻¹n1A^TnPin1An

·^N

j1

r_ij 2

e_i^Tnein ⁻¹e^T_jnejn ,

3.8

here, Young’s inequality is used, 2ab≤εa²b²/ε, for anyε >0,

ϕ^TP_in1ϕ≤L²_i

kλ_max

P_in1

e^T_inein,

g^T_iPin1gi≤λmax

Pin1 g_i^Tgi

≤λ_max

P_in1_N

j1

r_ije_jn₂ λmax

Pin1

|e|^Tr_i^Tri|e|

≤λmax

Pin1

λmaxr_i^Trie^T_inein λ_max

P_in1^N

j1

r_ij²e^T_inein,

3.9

whereri ri1, ri2, . . . , riNand|e| e1,e2, . . . ,eN^T, and

ϕ^TPin1gi≤ϕ^TP_i^1/2n1P_i^1/2n1gi

≤ ν_i

2ϕ^TP_in1ϕν_i⁻¹

2 g_i^TP_in1g_i.

3.10

Substituting3.9into3.10and substituting3.7–3.10into3.6, we obtain that

Vin1≤ein^T

⎧⎨

⎩A^TnPin1An 2Lik λmax

P_i⁻¹n1A^TnPin1An

· λ_max

P_in1 λmin

Pin1P_in1 λ_max

P_i⁻¹n1A^TnPin1An^N

j1

r_ijI

λmax

P_in1 1ν_i

L²_ik

1ν⁻¹_i λ_max

r_i^Tr_i I

⎫⎬

⎭e_in

⁻¹ λmax

P_i⁻¹n1A^TnPin1An^N

j1

rije^T_jnejn.

3.11

It follows from3.1that for alln∈Nk, Nk1,

Vn1 ^N

i1

V_in1≤^N

i1

α_ikne^T_inPinein≤α_kn^N

i1

e^T_inPinein α_knVn, 3.12

where for a fixedn,αkn max1≤i≤N{αikn}.

WhennNk, we get

V_in1 e_in1^TP_in1ein1

e_in B_ike_in_T

P_in1

e_in B_ike_in e_in^T

IB_ikT

P_in1 IB_ik

e_in

≤λ_max P⁻¹

IB_ikT

P_in1

IB_ik

e_in^TP_in1ein

≤β_iknein^TP_inein,

3.13

which implies that fornNk,

Vn1 ^N

i1

V_in1≤^N

i1

β_iknein^TP_inein≤β_kn^N

i1

e_in^TP_inein, 3.14

whereβ_kn max1≤i≤N{βikn}.

Hence, for allk∈N,

V N_k1

≤β_k N_k

V N_k

. 3.15

Since

γj

⎧⎪

⎨

⎪⎩

α_kj, ifj∈

N_k, N_k1 ,

β_kj, ifj Nk, k∈N,

3.16

andγ01, then from3.12–3.15, for anyn∈Nk, Nk1, we obtain that

Vn≤ _n−1

j0

γ_j²

V0 e^{2n−1j0 lnγj}V0. 3.17

Denoteen e^T₁n, e^T₂n, . . . , e^T_Nn^T. By3.5, we get en≤

b

ae^n−1j0lnγje0, n∈N. 3.18 Hence, if_∞

j0lnγ_j −∞, then for anye_i0 ∈R^n×n, by3.14, limn→∞ ein 0. Thus, the error system2.4is asymptotically stable. Therefore, the uncertain dynamical network2.1is robust synchronization with system2.2by the impulsive controllers{Nk, Bik}. The proof is complete.

Corollary 3.2. Suppose that Assumptions2.4and2.5hold, and assume that there exist positive con- stantsν_i>0,i∈N, such that the following condition is satisfied:

∞ j0

lnγ_j−∞, 3.19

where

γ_j

⎧⎨

⎩

αikj, ifj∈

Nk, Nk1 IBj, ifjNk, k∈N,,

αikn AnLik

₂

_N

j1

rijrji

AnνiL²_ik

1ν⁻¹_i ^N

j1

r_ij².

3.20

Then, for any initial conditionsx_i0 x_i0,y0 y₀ ∈U, the uncertain discrete dynamical network 2.1is robust impulsive synchronization with system2.2by the impulsive controllers{Nk, B_ik}.

Proof. By the similar proof ofTheorem 3.1, withP_in I,1,i1,2, . . . , N,we obtain that the result holds. The details are omitted here.

Remark 3.3. iByCorollary 3.2, if there does not exist coupling in the network, that is,rij 0, i, j,1,2, . . . , N, then the sufficient condition for the robust synchronization of the network simplifies to

∞ j0

lnγj−∞, whereγj

⎧⎨

⎩

AjL_jk, if j∈

N_k, Nk1

IB_j, ifjN_k, k∈N., 3.21

Hence,Corollary 3.2is the generalization of the results established in20.

iiIfU Rⁿ, then the error system 2.4is globally asymptotically stable; that is, the robust impulsive synchronization can be achieved globally.

In the following, we consider the case in which the parametersrijare not all known, but there exist positive constantsK_1i>0,K_2i>0,K_3i>0,i1,2, . . . , N, such that

N j1

r_ij≤K_1i, N j1

r_ji≤K_2i, N

j1

r_ij² ≤K_3i, i1,2, . . . , N. 3.22 Theorem 3.4. Assume that Assumptions2.4-2.5and conditionsii-iiiofTheorem 3.1hold, while conditioniofTheorem 3.1is changed into the following one:

ifor alln∈Nk, Nk1, k∈N, the following inequalities hold:

A^TnPin1An2Lik λmax

P_i⁻¹n1A^TnPin1An

· λmax

Pin1 λ_min

P_in1Pin1 λ_max

P_in1 1ν_i

L²_i

k

1ν_i⁻¹ K_i3

I λmax

P_i⁻¹n1A^TnPin1An

Ki1⁻¹Ki2 I

≤α_iknPin.

3.23 Then, for any initial conditionsx_i0 x_i0,y0 y₀ ∈U, the uncertain dynamical network2.1is robust impulsive synchronization with system2.2by the impulsive controllers{Nk, B_ik}.

Proof. By the similar proof ofTheorem 3.1, we obtain that the result of this theorem holds. The details are omitted here.

4. Examples and simulations

In this section, three representative examples are given for illustration.

Example 4.1. Consider the entire discrete dynamical network in form of 2.1, where x_i xi1, xi2, xi3^T, and the functionsf,gi,i1,2, . . . , N,satisfy

f k, x_i

⎛

⎜⎝

−3xi1xi2sink²

−xi12x_i2−sinx_i2−cosk xi3sinxi32 sink−1

⎞

⎟⎠,

gjx

⎛

⎜⎝

x_j1−2x_j1,1x_j2,1 0

−xj32x_j1,3−x_j2,3

⎞

⎟⎠,

4.1

wherej1,2, . . . , N−2,andg_N−1x1, x₂, . . . , x_N g_Nx1, x₂, . . . , x_N 0.

Letfk, x i, y fk, xi−fk, y Akeiϕk, e i, wherey y1, y2, y3^T,Ak

%−3 1 0

−1 2 0 0 0 1

&

, andϕk, e i

% 0

−sinxi2siny2

sinxi3−siny3

&

.

It is easy to show thatAk3.6180,ϕk, e i ≤ ei, that is,L_ik 1, for anyx_i, y∈R³, and

gix, ygi

x1, x2, . . . , xN

−giy, y, . . . , y≤√

2ei2√

2ei1√

2ei2, 4.2 wherei1,2, . . . , N−2,and

g_ix, yg_i

x₁, x₂, . . . , x_N

−g_iy, y, . . . , y0, iN−1, N. 4.3 LetN 10, then we obtain that α_ikn ≤ 169.1249. ByCorollary 3.2, we can choose many impulsive control laws{Nk, BNk, k ∈N,}such that the error system is asymptotically stable.

In the following, we takeNk3kandBNk %_{−0.995 0 0}

0 −0.995 0 0 0 −0.995

&

, then

γj

⎧⎨

⎩

α_ikj≤13.0048, ifj / N_k,

IB_j0.005, ifjN_k, k∈N. 4.4

LetS_nn

j1 lnγ_j, then fork∈N,

Sn≤

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

k2 ln 13.0048ln 0.005 −0.1677k, ifn3k,

k2 ln 13.0048ln 0.005 ln 13.0048−0.1677k2.5653, ifn3k1, k2 ln 13.0048ln 0.005 2 ln 13.0048−0.1677k5.1306, ifn3k2,

4.5

which leads to_∞

j1lnγ_j lim_n→∞S_n −∞. Then, byCorollary 3.2, we obtain that the im- pulsive controllers{Nk, B_ik}designed as above can achieve the robust synchronization for this uncertain discrete dynamical network.

0 2 4 6 8 10 12 14 n

−1.5

−1

−0.5 0 0.5 1 1.5 2 2.5

e11–e101

Figure 2: Synchronization errors ofek1,k1,2, . . . ,10.

0 2 4 6 8 10 12 14

n

−3

−2.5

−2

−1.5

−1

−0.5 0 0.5 1

e12–e102

Figure 3: Synchronization errors ofek2,k1,2, . . . ,10.

The numerical simulation is given in Figures2–4. Here, the initial data are given asy0 0.1 0.5 0.4^T,x₁₀ 0.4 0.7 0.6^T,x₂₀ 0.3 0.5 0.4^T,x₃₀ 0.2 0.3 0.2^T,x₄₀ 0.1 0.1 0^T, x₅₀ 0−0.1 −0.2^T,x₆₀ −0.1 −0.3 −0.4^T,x₇₀ −0.2 −0.5 −0.6^T,x₈₀ −0.3 −0.8−0.8^T, x90 −0.4 −1.1 −1^T, andx100 −0.5 −1.4 −1.2^T.

In Figures2–4, one can see that all the trajectories of the error system for this dynamical network asymptotically approach the origin with the designed robust impulsive controller, whereek ek1 ek2 ek3^T,k1,2, . . . ,10.

Example 4.2. Here we consider taking the fold chaotic system as nodes of the discretedynamical network. A single fold chaotic system is in form of

yn1 Ayn ϕ

yn

, n∈N, 4.6

whereyn %_y

1n y2n

&

,A%_{−0.1 1}

0 0

&

,ϕyn % ₀

y1n²−1.7

&

.

0 2 4 6 8 10 12 14 n

−5

−4

−3

−2

−1 0 1

e13–e103

Figure 4: Synchronization errors ofek3,k1,2, . . . ,10.

The entire network is given by x_in1 Ax_in ϕ

x_in g_i

x₁n, x2n, . . . , xNn

, i1,2, . . . , N, 4.7

wherex_i xi1, x_i2^T, and the coupling functionsg_i,i1,2, . . . , N,satisfy

g_ix

−1x²_i11x²_i1,1 2x_i2² −2x²_i1,2

, ₁''≤1, ''₂''≤1, i1,2, . . . , N−1, 4.8

andgNx1, x2, . . . , xN 0.

Letfk, x i, y Ae_iϕk, e i, wherey y1, y₂^T,A_{−0.1 1}

0 0

andϕk, e i

% ₀

x²_i1−y²₁

&

. Letx0 −1.5,0.9^T,y0 −1.5,0.5^T. By simulation, we can estimate the attractive domainUof isolated node:U{y∈R²:y ≤1.5}. Thus, for any initial conditionsx_i0, y₀∈U, it is easy to show thatA1.0050,ϕk, e i ≤3ei, that is,L_ik 3, and

g_ix, yg_i

x₁, x₂, . . . , x_N

−g_iy, y, . . . , y≤4√

1.5e_ie_i1, 4.9 wherei1,2, . . . , N−1,and

g_Nx, yg_N

x₁, x₂, . . . , x_N

−g_Ny, y, . . . , y0. 4.10 LetN 10. ByCorollary 3.2, we obtain thatαikn ≤179.8278. We choose impulsive control law{Nk, BNk, k∈N,}such that the error system is asymptotically stable. In the following, we takeN_k3k,B_{Nk−0.995 0}

0 −0.995

, then

γ_j

⎧⎨

⎩

α_ikj≤13.4100, ifj / N_k,

IBj0.005, ifjNk, k∈N. 4.11

0 2 4 6 8 10 12 14 16 18 20 n

3 2 1 0 1 2 3

e11–e10,1

Figure 5: Synchronization errors ofek1,k1,2, . . . ,10.

0 2 4 6 8 10 12 14 16 18 20

n

−2.5

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2

e21–e10,2

Figure 6: Synchronization errors ofek2,k1,2, . . . ,10.

LetS_nn

j1lnγ_j, then fork∈N,

Sn≤

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

k2 ln 13.4100ln 0.005 −0.1063k, ifn3k,

k2 ln 13.4100ln 0.005 ln 13.4100−0.1063k2.5960, ifn3k1, k2 ln 13.4100ln 0.005 2 ln 13.4100−0.1063k5.1920, ifn3k2,

4.12

which leads to_∞

j1lnγj limn→∞ Sn −∞. Then, byCorollary 3.2, we obtain that the im- pulsive controllers{Nk, B_ik}designed as above can achieve the robust synchronization for this uncertain discrete dynamical network.

The numerical simulation is given in Figures5-6. Here, the initial data are given asy0

−1.5 0.5^T,x₁₀ −1.4 0.7^T,x₂₀ −1.3 0.5^T,x₃₀ 0.1 0.2^T,x₄₀ −0.1 0.1^T,x₅₀ 0.6 − 0.1^T,x₆₀ 1.1 −0.3^T,x₇₀ 1.2 −0.5^T,x₈₀ 1.3 −0.8^T,x₉₀ 1.4 −1.1^T, andx₁₀₀ 1.5 −1.4^T. In Figures5-6, one can see that all the trajectories of the error system for this

dynamical network asymptotically approach the origin with the designed robust impulsive controller, whereek ek1 ek2^T,k1,2, . . . ,10.

Example 4.3. Here we consider taking the chaotic H´enon map as nodes of the discrete dynami- cal network. A single chaotic H´enon map is in form of

yn1 Ayn ϕ

yn

, n∈N, 4.13

whereyn %_y

1n y2n

&

,A%

0 1 0.3 0

&

, andϕyn _1−1.4y²

1

0

. The entire network is given by

xin1 Axin ϕ xin

gi

x1n, x2n, . . . , xNn

, i1,2, . . . , N, 4.14

wherexi xi1, xi2^T, and the coupling functionsgi,i1,2, . . . , N,satisfy

gix

−x²_i11x²_i1,1 2x²_i2−2x_i1,2²

, ''1''≤1, ''2''≤1, i1,2, . . . , N−1, 4.15

andgNx1, x2, . . . , xN 0.

Letfk, x i, y Ae_iϕk, e i, wherey y1, y₂^T, andϕk, e i

%_1.4y2

1−x²_i1 0

&

.

Letx0 0.3,−0.6^T,y0 0.3,−0.1^T. By simulation, we can estimate the attractive domainUof isolated node:U{y∈R² :y ≤3}. Thus, for any initial conditionsx_i0, y₀∈U, it is easy to show thatA1.0000,ϕk, e i ≤8.4ei, that is,L_ik 4.2, and

g_ix, yg_i

x₁, x₂, . . . , x_N

−g_iy, y, . . . , y≤8√

2.1e_ie_i1, 4.16

wherei1,2, . . . , N−1,and

gNx, ygN

x1, x2, . . . , xN

−gNy, y, . . . , y0. 4.17

LetN 10. ByCorollary 3.2, we obtain thatα_ikn ≤248.6386. We choose impulsive control law{Nk, BNk, k∈N,}such that the error system is asymptotically stable. In the following, we takeNk3k,BNk%_−0.996

0 0 −0.996

&

, then

γ_j

⎧⎨

⎩

α_ikj≤15.7683, ifj / N_k,

IBj0.004, ifjNk, k∈N. 4.18

0 2 4 6 8 10 12 14 16 18 20 n

−20

−15

−10

−5 0 5

e11–e10,1

Figure 7: Synchronization errors ofek1,k1,2, . . . ,10.

0 2 4 6 8 10 12 14 16 18 20

n

−1.5

−1

−0.5 0 0.5 1 1.5

e21–e10,2

Figure 8: Synchronization errors ofek2,k1,2, . . . ,10.

LetS_nn

j1lnγ_j, then fork∈N,

Sn≤

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

k2 ln 15.7683ln 0.004 −0.0055k, ifn3k,

k2 ln 15.7683ln 0.004 ln 15.7683−0.0055k2.7580, ifn3k1, k2 ln 15.7683ln 0.004 2 ln 15.7683−0.0055k5.5160, ifn3k2,

4.19

which leads to_∞

j1lnγjlimn→∞Sn−∞. Then, byCorollary 3.2, we obtain that the impul- sive controllers{Nk, Bik} designed as above can achieve the robust synchronization for this uncertain discrete dynamical network.

The numerical simulation is given in Figures7-8. Here, the initial data are given asy₀ 0.3 −0.6^T, andxk0,k1,2, . . . ,10, are the same as inExample 4.2.

In Figures7-8, one can see that all the trajectories of the error system for this dynamical network asymptotically approach the origin with the designed robust impulsive controller, wheree_k ek1 e_k2^T,k1,2, . . . ,10.

5. Conclusions

In this paper, a robust impulsive control method for synchronization of an uncertain discrete dynamical network has been introduced. The controller so designed is robust to uncertain net- work coupling. From the aspect of controller structure and robustness to uncertain network coupling, the developed synchronization scheme is more efficient than those reported in the literature to date. Some simple and effective criteria for achieving robust impulsive synchro- nization have been derived. Because a chaotic system has complex dynamical behaviors and possesses some special features which make the chaotic synchronization very useful to secure communication, it is significative to take discrete chaotic system as nodes in a discrete dynam- ical network. Three examples demonstrate the effectiveness of the theoretical results obtained in this paper.

Acknowledgments

The authors would like to thank the Editor, Professor Roderick Melnik, and the anonymous referees for their helpful comments and suggestions. This work was supported by Scientific Re- search Funds of Chongqing Municipal Education Commissionnos. KJ070404 and KJ070403 and the Australian Research Council Discovery ProjectDP0881391.

References

1 L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990.

2 T. L. Carroll and L. M. Pecora, “Synchronization in chaotic circuits,” IEEE Transactions on Circuits and Systems, vol. 38, no. 4, pp. 453–456, 1991.

3 L. Kocarev and U. Parlitz, “General approach for chaotic synchronization with application to commu- nication,” Physical Review Letters, vol. 74, no. 25, pp. 5028–5031, 1995.

4 C. W. Wu, T. Yang, and L. O. Chua, “On adaptive synchronization and control of nonlinear dynamical systems,” International Journal of Bifurcation and Chaos, vol. 6, no. 3, pp. 455–471, 1996.

5 G. Grassi and S. Mascolo, “Nonlinear observer design to synchronize hyperchaotic systems via a scalar signal,” IEEE Transactions on Circuits and Systems I, vol. 44, no. 10, pp. 1011–1014, 1997.

6 L. Kocarev, G. M. Maggio, M. Ogorzalek, L. M. Pecora, and K. Yao, “Introduction to the special issue,”

IEEE Transactions on Circuits and Systems I, vol. 48, no. 12, pp. 1385–1386, 2001.

7 G. Chen and X. Dong, “On feedback control of chaotic continuous-time systems,” IEEE Transactions on Circuits and Systems I, vol. 40, no. 9, pp. 591–601, 1993.

8 M. E. Yalcin, J. A. K. Suykens, and J. Vandewalle, “Master-slave synchronization of Lur’e systems with time-delay,” International Journal of Bifurcation and Chaos, vol. 11, no. 6, pp. 1707–1772, 2001.

9 T. Yang and L. O. Chua, “Impulsive stabilization for control and synchronization of chaotic systems:

theory and application to secure communication,” IEEE Transactions on Circuits and Systems I, vol. 44, no. 10, pp. 976–988, 1997.

10 X. F. Wang and G. Chen, “Synchronization in small-world dynamical networks,” International Journal of Bifurcation and Chaos, vol. 12, no. 1, pp. 187–192, 2002.

11 X. F. Wang and G. Chen, “Synchronization in scale-free dynamical networks: robustness and fragility,”

IEEE Transactions on Circuits and Systems I, vol. 49, no. 1, pp. 54–62, 2002.

12 J. Jost and M. P. Joy, “Spectral properties and synchronization in coupled map lattices,” Physical Review E, vol. 65, no. 1, part 2, Article ID 016201, 9 pages, 2002.

13 P. M. Grade and C.-K. Hu, “Synchronization chaos in coupled map lattices with small-world interac- tions,” Physical Review E, vol. 62, no. 5, pp. 6409–6413, 2000.

14 Z. Li and G. Chen, “Robust adaptive synchronization of uncertain dynamical networks,” Physics Let- ters A, vol. 324, no. 2-3, pp. 166–178, 2004.

15 B. Liu, X. Liu, and X. Liao, “Stability and robustness of quasi-linear impulsive hybrid systems,” Journal of Mathematical Analysis and Applications, vol. 283, no. 2, pp. 416–430, 2003.

16 B. Liu, X. Liu, G. Chen, and H. Wang, “Robust impulsive synchronization of uncertain dynamical networks,” IEEE Transactions on Circuits and Systems I, vol. 52, no. 7, pp. 1431–1441, 2005.

17 V. Lakshmikantham, D. D. Ba˘ınov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Singapore, 1989.

18 X. Liu, “Stability results for impulsive differential systems with applications to population growth models,” Dynamics and Stability of Systems, vol. 9, no. 2, pp. 163–174, 1994.

19 H. Ye, A. N. Michel, and L. Hou, “Stability analysis of systems with impulse effects,” IEEE Transactions on Automatic Control, vol. 43, no. 12, pp. 1719–1723, 1998.

20 Z. G. Li, C. Y. Wen, and Y. C. Soh, “Analysis and design of impulsive control systems,” IEEE Transac- tions on Automatic Control, vol. 46, no. 6, pp. 894–897, 2001.

21 Z. G. Li, Y. C. Soh, and C. Y. Wen, “Robust stability of a class of hybrid nonlinear systems,” IEEE Transactions on Automatic Control, vol. 46, no. 6, pp. 897–903, 2001.

22 Q. Y. Wang, G. Chen, Q. S. Lu, and F. Hao, “Novel criteria of synchronization stability in complex networks with coupling delays,” Physica A, vol. 378, no. 2, pp. 527–536, 2007.

23 B. Liu and H. J. Marquez, “Quasi-exponential input-to-state stability for discrete-time impulsive hy- brid systems,” International Journal of Control, vol. 80, no. 4, pp. 540–554, 2007.

24 B. Liu, K. L. Teo, and X. Liu, “Global synchronization of dynamical networks with coupling time- delays,” Physics Letters A, vol. 368, no. 1-2, pp. 53–63, 2007.

25 B. Liu and X. Liu, “Robust stability of uncertain discrete impulsive systems,” IEEE Transactions on Circuits and Systems II, vol. 54, no. 5, pp. 455–459, 2007.