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synchronization in complex networks. An effective distributed adaptive strategy to tune the coupling weights of a network is designed based on local information ...
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 8, AUGUST 2012

G(t) = (Gij (t))N2N . If there is an edge between node vi and node vj at time t, then the element Gij (t) = Gji (t) > 0, which is the time-varying weight associated with the edge eij = eji ; otherwise, Gij (t) = Gji (t) = 0 (j 6= i). The corresponding time-varying com-

Distributed Adaptive Control of Synchronization in Complex Networks Wenwu Yu, Pietro DeLellis, Guanrong Chen, Mario di Bernardo, and Jürgen Kurths

plex dynamical network model [3]–[5] can be written as

Abstract—This technical note studies distributed adaptive control of synchronization in complex networks. An effective distributed adaptive strategy to tune the coupling weights of a network is designed based on local information of node dynamics. The analysis is then extended to the case where only a small fraction of coupling weights can be adjusted. A general criterion is derived and it is found that synchronization can be reached if the subgraph consisting of the edges and nodes corresponding to the updated coupling weights is connected. Finally, simulation examples are given to illustrate the theoretical analysis. Index Terms—Complex network, distributed adaptive law, pinning control, synchronization.

I. INTRODUCTION Complex networks are ubiquitous in science and engineering [1], [2]. Examples include the Internet, the World Wide Web, wireless communication networks, electric power grids, biological networks, social networks, and scientific citation networks [1]. Much research effort has been devoted to collecting data on real networks and characterizing some of their basic features such as degree-distribution, average distance, and other macroscopic observables. At the same time, scientists and engineers have been studying large-scale dynamical systems consisting of many interacting agents communicating over a complex network. It has been noticed that such systems can be modeled by a linearly and diffusively coupled dynamical network consisting of nodes, with each node being an -dimensional dynamical system. Let G = (V E ) be a weighted undirected graph of order , with the set of nodes V = f 1 2 . . . N g, the set of undirected edges E  V 2 V , and a weighted time-varying adjacency matrix

; ;G

N n v ;v ; ;v

2153

N

Manuscript received April 07, 2010; revised December 24, 2010, July 07, 2011, and December 09, 2011; accepted December 22, 2011. Date of publication January 09, 2012; date of current version July 19, 2012. This work was supported in part by the National Natural Science Foundation of China under Grant 61104145, the Natural Science Foundation of Jiangsu Province of China under Grant BK2011581, the Research Fund for the Doctoral Program of Higher Education of China under Grant 20110092120024, the Fundamental Research Funds for the Central Universities of China, the Hong Kong Research Grants Council under GRF Grant CityU1114/11, the DAAD Scholarship, Ref. 423, and the German Science Foundation (DFG) in the IRTG 1470. Recommended by Associate Editor A. Loria. W. Yu is with the Department of Mathematics, Southeast University, Nanjing 210096, China and also with the School of Electrical and Computer Engineering, RMIT University, Melbourne Vic. 3001, Australia (e-mail: [email protected]). G. Chen is with the Department of Electronic Engineering, City University of Hong Kong, Hong Kong (e-mail: [email protected]). P. DeLellis is with the Department of Systems and Computer Engineering, University of Naples Federico II, Naples 80125, Italy (e-mail: [email protected]). M. di Bernardo is with the Department of Systems and Computer Engineering, University of Naples Federico II, Naples 80125, Italy and also with the Department of Engineering Mathematics, University of Bristol BS8 1TR, U.K. (e-mail: [email protected]). J. Kurths is with the Research Domain IV, Institute for Climate Impact Research, Potsdam, Germany and also with the Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3FX, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2012.2183190

x_ i (t) = f (xi (t);t) + c

N

j =1;j6=i

Gij (t)0(xj (t) 0 xi (t))

(1)

i = 1; . . . ; N , where xi = (xi1 ; nxi2 ; . . .+; xin )Tn 2 n is the 2 is a nonlinear state vector of the ith node, f : 0 ! vector function, c is the overall constant coupling strength, and 0 = diag( 1 ; . . . ; n ) 2 n2n is a positive semi-definite inner coupling matrix with j > 0 if two nodes can communicate through the j th state, and j = 0 otherwise.

An important subject for investigation is the emergent behavior in a network of dynamical systems and in particular the emergence of their coordinated motion. Indeed, synchronization and consensus are common features in complex networked systems and are frequently encountered in nature and technology, e.g., [1], [2], [6]–[14]. Synchronization has been the subject of considerable research effort in a diverse range of disciplines, from biology [15], [16] to opto-electronics [17] and cryptography [18], to name just a few. In control theory, much ongoing research is focused on consensus problems [19], [20]. Flocking, for example, is a typical consensus problem, in which a group of mobile agents has to align their velocity vectors and simultaneously stabilize the inter-agent distances under decentralized nearest-neighbor interaction rules (see, for instance, [21], [22]). Another common example of consensus is the rendezvous problem, in which a group of agents meets in a certain manner at a selected point in space, each using only local information about the positions of its nearest neighbors (see [23], [24]). Recently, in order to guarantee stability in complex networks, adaptive strategies to appropriately tune the strengths of the interconnections among network nodes have been proposed [25]–[27]. Mathematically, these strengths are represented by the non-null elements of the time-varying Laplacian matrix ( ) of the network (1). In particular, as in [28] and [29], different adaptive laws for network synchronization were proposed and shown numerically to be effective. In [5] and [30], moreover, the stability of two possible classes of adaptive strategies, named respectively edge-based and vertex-based adaptation laws, were investigated via a Lyapunov theoretic approach. The aim of this technical note is to extend the stability analysis of the edge-based strategy presented in [5], by making weaker assumptions on the node dynamics. The derived sufficient conditions for stability can be applied to explain the effectiveness of the approach for a much wider range of node dynamics. Moreover, the stability analysis is further extended to the case where it is feasible to control only a subset of the coupling weights, providing an estimation for the minimum number of weights to be controlled in order to synchronize the network. All this constitutes the main contribution of the technical note.

Lt

II. PRELIMINARIES Firstly, notice that network (1) can be rewritten as

x_ i (t) = f (xi (t);t) 0 c i

; ;n

for = 1 . . . , where matrix defined by

L

N

j =1

Lij (t)0xj (t)

Lij is the (ij )th element of the Laplacian

Lij (t) = 0Gij (t); i 6= j; Lii (t) = 0

0018-9286/$31.00 © 2012 IEEE

(2)

N j =1;j6=i

Lij (t)

(3)

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 8, AUGUST 2012

which ensures that N j =1 Lij (t) = 0 for all t  0. The control objective here is to find some adaptive laws acting on Lij (t) under a given and fixed network topology such that the solutions of the controlled network (2) can achieve synchronization in the sense that

t0 ! 1 kxi (t) 0 xj (t)k = 0; i; j = 1; 2; . . . ; N: lim

L_ ij (t) = 0 ij (xi 0 xj )T 0(xi 0 xj ); Lij (0) = Lji (0)  0

(4)

Here, it is emphasized that the network structure is fixed and only the coupling weights can be time-varying, but if there is no connection between nodes i and j , then Lij (t) = Lji (t) = 0 for all t. Assumption 1: There exist a constant diagonal matrix 1 = diag(1 ; . . . ; n ) and an " > 0 such that

x 0 y)T (f (x; t) 0 f (y; t)) 0 (x 0 y)T 1(x 0 y)  0"(x 0 y)T (x 0 y); x; y 2 n ; t 2 + :

for i = 1; 2; . . . ; N , where ei = xi 0 x . Theorem 1: Suppose that Assumption 1 holds, 0 is positive definite, and f belongs to C 0 . Then, network (2) is synchronized under the following distributed adaptive law:

(

(5)

i; j ) 2 E , where ij

= ji are positive constants. Proof: Consider the Lyapunov function candidate

(

V (x(t)) = 12

(3)

x

1

T L = 12

V_

2

N N i=1 j =1

i=1

eiT (t)ei (t) +

i=1 N i=1

eiT (t)e_ i (t) +

N

N

c (L (t) + c )2 ij ij i=1 j =1;j 6=i ij

(9)

4

N

N

c (L (t) + c )L_ (t) ij ij ij i=1 j =1;j 6=i ij 2

eiT (t) f (xi (t); t) 0 N1 N

0c

j =1 c N

N

j =1

f (xj (t); t)

Lij (t)0ej (t) N

02

Lij (t) + cij )(xi 0 xj )T 0(xi 0 xj )

(

i=1 j =1;j 6=i N eiT (t) f (xi (t); t) + f (x; t) 0 N1 f (xj (t); t) = i=1 j =1 N 0 c Lij (t)0ej (t) 0 f (x; t) j =1 N N T 0 2c (Lij (t) + cij )(ei 0 ej ) 0(ei 0 ej ): i=1 j =1;j 6=i N

xT Lx : =0;x6=0 xT x N,

N =

=

min

For any  = (1 ; . . . ; N )T

N

where cij = cji are nonnegative constants, x = (x1T ; . . . ; xTN )T , and cij = 0 if and only if Lij = 0. The derivative of V along the trajectories of (7) gives

Assumption 1 is the so-called QUAD condition (or one-sided Lipschitz) on vector fields [31], [32]. In what follows, assume 1 = H 0, where H is a diagonal matrix and 0 is the inner coupling matrix defined in (1). Note that this assumption is mild. For example, all linear and piecewise-linear time-invariant continuous functions satisfy this condition. In addition, the condition is satisfied if @fi =@xj (i; j = 1; 2; . . . ; n) are uniformly bounded and 0 is positive definite, which includes many well-known systems. Without loss of generality, only connected networks are considered throughout the technical note; otherwise, one may consider the synchronization on each connected component of the network separately. A well-known useful result is the following. Lemma 1: ([31], pp. 279–288): (1) The Lapacian matrix L in the undirected network G is positive semi-definite. It has a simple eigenvalue 0 and all the other eigenvalues are positive if and only if the network is connected. (2) The smallest nonzero eigenvalue 2 (L) of the Laplacian matrix L satisfies

2 (L) =

(8)

N eT (t) = 0, one has i=1 i N N eiT (t) f (x; t) 0 N1 f (xj (t); t) i=1 j =1

Since

Gij (i 0 j )2 :

(10)

:

=0

From Assumption 1, it follows that III. DISTRIBUTED ADAPTIVE CONTROL OF COMPLEX NETWORKS

N

Following from [5], a distributed adaptive law on the weights Lij (t) with i 6= j is proposed in this section, which results in corresponding adaptive laws on Gij since Lij = 0Gij ; i 6= j . N xj . Then, one has  = 1=N Let x j =1

i=1

x_ (t) = N1

N

j =1

f (xj (t); t):

(6)

Subtracting (6) from (2) yields the following error dynamical network:

e_ i (t) = f (xi (t); t) 0 N1

N j =1

f (xj (t); t) 0 c

N j =1

Lij (t)0ej (t)

(7)

eiT (t) [f (xi (t); t) 0 f (x; t)]

 0"

N i=1

eiT (t)ei (t) +

N i=1

eiT (t)H 0ei (t):

(11)

Define the Laplacian matrix 6 = (~ ij )N 2N , where ~ij = 0cij ; i 6= j ; ~ii = 0 Nj=1 ~ij . In view of Lemma 1, one obtains

c

N

j 6=i

N

i=1 j =1;j 6=i =

Lij (t) + cij )(ei 0 ej )T 0(ei 0 ej )

(

02c

N N i=1 j =1

Lij (t)eiT 0ej + 2c

N N i=1 j =1

~ij (t)eiT 0ej :

(12)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 8, AUGUST 2012

and there exists < K < 1 such that t0 ! 61 kf x; t k  K < 1, 8x 2 n , where g 0! n can be any n-dimensional function

Therefore, combining (10)–(12) and by using Lemma 1, one has

V_

 0" 0c

N

eTi (t)ei (t) +

i=1 N N

N i=1

i=1 j =1 T = e (t) 0 "(IN In )

(13)

Let 3 be the diagonal matrix associated with 6, that is, there exists a unitary matrix P = (p1 ; . . . ; pNp) such that P T 6P = 3. Let y (t) = T In )e(t). Since p1 = 1= N (1; . . . ; 1)T , one has y1 = (p1T

(P In )e(t) = 0. Then, it follows that

V_

 eT t 0" IN In IN H 0 c P In P T In e t eT t 0" IN In IN H e t 0 cyT t y t : ( )[

)+ (

(

(

)(3

( )[

=

0

)] ( )

)+ (

( )(3

By Lemma 1 and since

0)

0)(

(

0)]

( )

0) ( )

V_

 eT t 0" IN In IN H 0 c2 yT t IN y t eT t 0" IN In IN H T 0 c2 e t P In IN

eT t 0" IN In IN H 0 c2 IN e t : ( )[

(

(6)

=

( )[

=

( )[

)+ (

( )(

(

(6)

( )(

)(

)+ (

(6)(

0)]



e(t)

0) ( )

)+ (

(

(14)

is positive definite, one has y T (t)(3

T 0)y (t)  2 (6)y (t)(IN 0)y (t). Hence

 0"eT

0)

tet

( ) ( )

(15)

(16)

T (t))T . where e(t) = (e1T (t), ; e2T (t); . . . ; eN Thus, it follows that function (9) is non-increasing, and so each term of (9) is bounded. Consequently, both the error vector e and each of the coupling gains Lij are bounded. Since Lij is monotonically decreasing [see (8)], one can conclude that each gain asymptotically converges to some finite negative value. The convergence of all Lij implies, from (8) and from the fact that 0 is positive definite, that the error vector asymptotically approaches zero. Remark 1: It should be emphasized that the assumptions of Theorem 1 are weak and relaxed compared to those imposed by [5] and [30], where node vector fields are required to satisfy the QUAD condition for some 1, ", such that 1 0 "I < 0. This is equivalent to require that the node vector fields are contracting, see [31]. Here, the assumption 1 0 "I < 0 has been completely removed. This relaxation implies that the stability of the adaptive strategy is proved for a much wider range of systems, not included by [5] and [30], such as unstable linear systems, C 0 systems in Lur’e form, Lipschitz vector fields, and C 1 systems with bounded Jacobians. Furthermore, differing from [5] and [30], the analysis here is applicable also to non-autonomous systems. To extend the stability analysis to the case where the network nodes are connected only through a subset of the state vectors, i.e., 0  0, a further assumption on f is needed. Assumption 2: The vector field f (x; t) can be written as f (x; t) = f~(x; t) + g(t);

of t. Now, we are ready to state and prove the following theorem. Theorem 2: Suppose that 0 is positive semi-definite, the vector field f belongs to class C 1 , and Assumptions 1 and 2 hold. Then, network (2) is synchronized under the distributed adaptive law (8). Proof: Proceeding as in the proof of Theorem 1, it follows from (16) that e and each Lij are bounded. By some algebraic manipulations, we have

V

N

=

i=1

0c

e_ iT (t)

N

1

N

N j =1

f (xi (t); t) 0 f (xj (t); t)]

[

Lij (t)0ej (t) j =1 N N 1 + [ f_(xi (t); t) 0 f_(xj (t); t)] eiT (t) N j =1 i=1 N N 0 c L_ ij (t)0ej (t) 0 c Lij (t)0e_ j (t) j =1 j =1 N N _ 0 2c (L ij (t) + cij )(ei 0 ej )T 0(ei 0 ej ) i=1 j =1;j6=i N N T (Lij (t) + cij )(e _i 0 e 0c _ j ) 0(ei 0 ej ): i=1 j =1;j6=i

Since f is continuously differentiable, V is continuous. The boundedness of e implies that (xi 0 xj ) is bounded for any pair (i; j ). Thus, from the continuity of f and f_, and from Assumption 2, also N [f (xi (t); t) 0 f (xj (t); t)] and N [f_(xi (t); t) 0 f_(xj (t); t)] j =1 j =1 are bounded as well, implying that V is itself bounded. Therefore, V_ is uniformly continuous. So, as shown in [33] (pp. 199–208), the error goes to zero asymptotically, and, from (8), all Lij converge to a finite value. Remark 2: As a simple example of a system satisfying Assumption 2, consider the linear time-varying system 

By choosing cij sufficiently large such that c2 (6) > maxj (hj ), one obtains (IN H 0) 0 c2 (6)(IN 0)  0. Therefore

V_

)

(17)

e(t) T In )e(t) 0)(P 0)]

0)] ( )

~(

:

~ij (t)eiT 0ej

IN H 0) 0 c(6 0) e(t):

lim

0

eiT (t)H 0ei (t)

+ (

2155

x_ (t) = A(t)x(t) + B (t)u(t) with a bounded dynamical matrix A(t), for any control matrix B (t) and control signal u(t). It should be noted that Assumption 2 is not implied by Assumption 1. As an example, the vector field f (x; t) = 0 xt, > 0, satisfies Assumption 1 but not Assumption 2. A. Edge Pinning Synchronization In Theorem 1, all the coupling weights are adjusted according to the distributed adaptive law (8). Here, only a minority of the coupling weights is updated to achieve synchronization. This procedure is denoted as edge pinning synchronization. For pinning control on nodes, the reader is referred to [34]. Assume that the pinning strategy is applied on a small fraction  (0 <  < 1) of the coupling weights in network (2). The total number of selected edges will be l = bN c, the integer part of the real number N . ~ ) be a subgraph of G , with a set of l selected undiLet G~ = (V ; E~; G ~ ~ = (G ~ ) rected edges E  E and a weighted adjacency matrix G ij N2N . ~ ~ Let L be the Laplacian matrix corresponding to G. Here, consider the case where only the weights on the edges in E~ are adapted. Thus

L_ ij (t) = L_ ji (t) = 0 ij (xi 0 xj )T 0(xi 0 xj );

i; j ) 2 E~ (18)

(

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 8, AUGUST 2012

where

ij

(i; j ) 2 E~.

=

ji

are positive constants, and

Lij (0)

= L (0), for ji

Theorem 3: Suppose that Assumption 1 holds, 0 is positive definite and f belongs to class C 0 . Under the distributed adaptive law (18), network (2) is synchronized if there are positive constants  ~ij such that

[I (H 0 0 "I )] 0 c2 (L)(I 0) < 0 N

where L

n

= (L ) 2 js N

N

N

(19)

is a Laplacian matrix defined as

0~ij 0 Lij (t) =

6

k=1;k=i

Lij (0)

if (i; j ) 2 E~ if i = j

Lik

(20)

otherwise.

Proof: Proceeding as in Theorem 3, one can obtain inequality (22). Since G~ is connected, one has V_  0. Thus, from the same arguments used in the proof of Theorem 2, the result follows.

IV. SIMULATION EXAMPLES In this section, some simulation examples are provided to validate the theoretical analysis. In the simulation examples, scale-free and ER random complex networks are considered, assuming that connectivity is guaranteed in the designed networks by choosing an appropriate number of connections. Here, it is noted that in the following examples the node dynamics satisfy the assumptions of the theorems, while the requirements for stability derived in [5], [30] are not fulfilled, for example in the case of Lorenz dynamics to be discussed below.

Proof: Consider the Lyapunov function candidate

V (x(t)) = 1 2

N i=1

e (t)ei (t) + T i

j

c 2 2 ij (Lij (t) + ~ij )

2E~

A. A Network of Chua’s Circuits (21)

~ = ~ are positive constants to be determined, = 1; 2; . . . ; l, and E~  E~ is the subset of edges of the subgraph E~ starting from node i.

where

ij

i

ji

i

The derivative of V along the trajectories of (21) gives

V_

=

N i=1

eiT (t) f (xi (t); t) 0 f (x; t) 0 c

0c j

2E~

N j =1

Lij (t)0ej (t)

(L (t) + ~ )(e 0 e ) 0(e 0 e ) ij

ij

j

i

T

j

i

 eT (t)[0"(IN In ) + (IN H 0) 0 c(L 0)] e(t): (22) The proof can be completed by following the similar steps as in Theorem 1. Remark 3: In Theorem 3, a general criterion is given for reaching synchronization by using the designed adaptive law (18). Clearly, one can solve (19) by using an LMI approach. However, sometimes the condition in (19) is difficult to apply to a large-scale network. However, a sufficient condition for (19) is that the network is connected. In fact, one can write

[I (H 0 0 "I )] 0 c2 (L)(I 0)  [I (H 0 0 "I )] 0 c2 (6~ )(I 0) (23) ~ )  2 (L), where 6~ is the Laplacian matrix defined as because 2 (6 0~ if (i; j ) 2 E~ (24) 6~ (t) = ~ ~ if i = j 2E otherwise. 0 ~ ) 6= 0 and it can be made arbitrarily large If G~ is connected, then 2 (6 ~ , (i; j ) 2 E~. Thereby choosing sufficiently large positive constants  N

Consider a network of Chua’s circuits [35]. Denote the state of the = (pi ; qi ; ri ). Then, the individual node dynamics are described by

i-th circuit by xi

N

n

N

(25)

where l(pi ) = bpi + 0:5(a 0 b)(jpi + 1j 0 jpi 0 1j). System (25) is chaotic when  = 10, = 18, a = 04=3, and b = 03=4. In what follows, consider an ER random network [36] of N = 100 nodes, generated with probability p = 0:15, which contains about pN (N 0 1)=2  742 edges. For connected nodes i and j , Gij (0) = Gji (0) = 1 (i 6= j ); i; j = 1; 2; . . . ; 100, and the coupling strength c = 0:2. A simulation-based analysis on the random network is performed by using a random edge pinning scheme, where one randomly selects m = bpN (N 0 1)=2c edges, with  = 0:35, only pinning a small fraction of the network edges (m = 259). As displayed in Figs. 1(a) and 1(b), synchronization is asymptotically achieved, while the coupling gains converge to steady-state values, which are slightly higher than the initial value 1. It is worth noting that, for the selected parameters, Chua’s circuit is QUAD for any 1, " such that 1 0 "I > 22:15 I. Hence, since 0 > 0, and as the pinning edges form a connected subgraph, the assumptions of Theorem 3 are satisfied. The numerical simulations therefore agrees with the stability analysis.

N

n

ij

ij

p_ i =  (0pi + qi 0 l(p1 )) q_i = pi 0 qi + ri r_i = 0 qi

ij

j

ij

fore, one finally gets (19). As in Section III, the analysis can be extended to the case where the network nodes are connected only through a subset of the state vectors, that is, 0  0. Theorem 4: Suppose that 0 is positive semi-definite, f belongs to class C 1 , and Assumptions 1 and 2 hold. Then, if the subgraph G~ is connected, network (2) can be synchronized under the distributed adaptive law (18).

B. A Network of Lorenz Oscillators Consider a network of 100 Lorenz oscillators coupled via a scalefree like topology [37]. In this case, the state of each node i has three components, xi = (pi ; qi ; ri )T , and the individual node dynamics are described by f (xi ) = (10(qi 0 pi ); 0pi ri +28ri 0 qi ; pi qi 0 8=3ri )T . Assume that the network nodes are coupled through the decentralized adaptive strategy (8) only on the first two state variables, that is

1 0 0 0= 0 1 0 : 0 0 0 The simulation starts from null initial coupling gains and initial states taken randomly from a normal distribution with standard deviation 40, and ij = ji = 0:1; 8(i; j ) 2 E . As depicted in Fig. 2, synchronization is asymptotically achieved and the adaptive gains asymptotically converge to constant values.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 8, AUGUST 2012

Fig. 1. Decentralized adaptive control of a network of 100 Chua’s circuits coup ;q ;r ; i pled through all the state vector (a) Orbits of the states x . (b) Orbits of the adaptive weights G ; i; j 2 E . ; ; ;

=( ( ) ~

1 2 . . . 100

) =

2157

Fig. 2. Decentralized adaptive control of a network of 100 Lorenz systems coupled through the first two state variables. (a) Orbits of the state variable ; ; ; . (b) Orbits of the adaptive weights G ; i; j 2 E . p;i

= 1 2 . . . 100

( )

REFERENCES In fact, the Lorenz system is QUAD [38] and, differing from Chua’s circuit, clearly continuously differentiable. Moreover, 0  0. So all the assumptions of Theorem 2 are satisfied. The stability of a network of the Lorenz system in the presence of adaptive couplings was never proved analytically before. In fact, in [5], the matrix (1 0 "I ) was required to be negative definite. This condition is not satisfied by any continuously differentiable chaotic system like the Lorenz system discussed here. In this technical note, such a difficulty has been overcome.

V. CONCLUSION In this technical note, a distributed adaptive control scheme for synchronization in complex networks has been presented and analyzed, which adaptively tunes the coupling weights of the network towards reaching synchronization. Some of the strict sufficient conditions for synchronization required by similar schemes in the present literature have been significantly weakened or completely relaxed. An effective pinning control scheme on the underlying network edges for updating a small fraction of coupling weights has also been developed. A general criterion has been given to guarantee synchronization if the subgraph, consisting of the edges corresponding to the updated coupling weights and the nodes of the initial graph, is connected. The established theory is illustrated by numerical examples, confirming the validity and effectiveness of the proposed adaptive synchronization strategy for complex networks of chaotic oscillators, which cannot be handled by other similar methods available in the literature.

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