Finite-time topology identification and stochastic synchronization of complex network with multiple time delays

Author’s Accepted Manuscript Finite-time topology identification and stochastic synchronization of complex network with multiple time delays Hui Zhao, Lixiang Li, Haipeng Peng, Jinghua Xiao, Yixian Yang, Mingwen Zheng www.elsevier.com/locate/neucom

PII: DOI: Reference:

S0925-2312(16)31015-3 http://dx.doi.org/10.1016/j.neucom.2016.09.014 NEUCOM17542

To appear in: Neurocomputing Received date: 25 February 2016 Revised date: 17 June 2016 Accepted date: 8 September 2016 Cite this article as: Hui Zhao, Lixiang Li, Haipeng Peng, Jinghua Xiao, Yixian Yang and Mingwen Zheng, Finite-time topology identification and stochastic synchronization of complex network with multiple time delays, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.09.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Finite-time topology identification and stochastic synchronization of complex network with multiple time delays Hui Zhaoa , Lixiang Lib,∗, Haipeng Pengb , Jinghua Xiaoa , Yixian Yangb , Mingwen Zhenga a State

Key Laboratory of Information Photonics and Optical Communications, School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China b Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China

Abstract This paper investigates issues of finite-time topological identification and stochastic synchronization for two complex networks with multiple time delays. In the paper, we propose two different approaches to identify the topological structure and guarantee stochastic synchronization for complex networks in finite time, which are achieved based on finite-time stability theory and properties of Wiener process. Several useful finite-time synchronization and identification criteria are obtained simultaneously based on adaptive feedback control method. In the final section, numerical examples are examined to illustrate the effectiveness of the analytical results. Keywords: Finite-time synchronization, multiple time delays, topology identification, stochastic noise perturbations

✩ Foundation item: Supported by the National Natural Science Foundation of China (Grant No. 61472045,61170269), the Beijing Natural Science Foundation (Grant No. 4142016), the Scientific Research Project of Beijing Municipal Commission of Education (KZ20150015015, KM201510015009). ∗ Corresponding author. E-mail addresses: li [email protected]

Preprint submitted to Journal of LATEX Templates

September 13, 2016

1. Introduction Complex networks with multiple time delays are universal in real world such as communication network, transportation network and relationship network, etc. [1-2], it means that the complex network can be divided into some sub5

networks based on different multiple time delays and each of them has its own property, we can also call it multi-links network. It is worth mentioning that this multi-links network is not equivalent to a weighted network, multi-links with different link properties cannot be simply merged into a link. Therefore, a weighted network cannot reflect the performance of the multi-links network.

10

Essentially, time-delays are introduced to split the multi-links complex network into sub-networks in order to describe the time-delay property of the networks with multi-links. Taking communication network (transportation network) for example, communication network (transportation network) can be divided into telephone network (airline network), internet network (railway network) and

15

mail network (expressway network) by different time-delays, which is shown in Fig.1. In addition, single link complex network is a special case of multi-links complex network, different from the single-link network in ref.[3] and the mixed time-delays in ref.[4], the research on complex network with multiple time delays is more realistic and representative.

20

In the previous researches, complex network with single delay is investigated to achieve complete synchronization [5], ref.[6] considers distributed synchronization via randomly occurring control. Besides, the different synchronization behaviours including anti-synchronization [7], cluster synchronization [8], projective synchronization [9], robust synchronization [10] and etc. are also studied

25

in the field of systems dynamics behavior. Thus it can be seen that synchronization is a typical collective behavior in the complex network which has very important application merits. With the development of the field of systems and control engineering, different kinds of control techniques are widely used in research for system dynamical behaviour, including continue control [11,13-16]

30

and discontinue control [12,13] etc..

2

7HOHSKRQHQHWZRUN $LUOLQHQHWZRUN IJ IJ 

,QWHUQHWQHWZRUN 5DLOZD\QHWZRUN IJ IJ

&RPPXQLFDWLRQQHWZRUN 7UDQVSRUWDWLRQQHWZRUN 0DLOQHWZRUN ([SUHVVZD\QHWZRUN IJ IJ

Figure 1: Communication network (Transportation network) and its division.

Particularly, since the finite time was introduced, the studies about the finitetime stability and synchronization are of great significance in practical applications. Compared with asymptotic synchronization, finite-time synchronization has lower time complexity and synchronization can be realized in a setting time. 35

Therefore, the finite-time stabilities [18], the finite-time complete synchronization [19,20,46], lag synchronization [21], H∞ synchronization [22] and cluster synchronization [23], etc., have been studied based on different kinds of control techniques in complex networks. Furthermore, the corresponding finite-time synchronization is also investigated in memristor-based neural networks [24,25].

40

However, these researches are given based on the known network models. It is well known that there will be many uncertain factors in the real systems, including stochastic perturbations, uncertain parameters, unknown topology, and etc. In refs.[26-30], researchers consider the influence of stochastic perturbation for the asymptotic synchronization of network, and finite-time stochastic synchro-

45

nization are investigated in refs.[31,32]. Meanwhile, the uncertain parameters or unknown topology as the uncertain factors are investigated in these papers

3

[1,2,33,47-49]. Refs.[37,38] are respectively studied issues of topology identification and others [39,40]. Besides, for the research of uncertain parameters in a finite time, a class of Markovian jump complex networks with partially 50

unknown transition rates is also considered to achieve the finite-time stability or synchronization characteristics of systems [34-36]. Mei et al. [41,42] study the finite-time topological identification and synchronization of drive-response (master-slave) system based on an effective control input and a feedback control with an updated law respectively. Wang et al. [43] investigates finite-time syn-

55

chronization and parameter identification problem for uncertain Lurie systems based on the finite-time stability theory and the adaptive control method. It is often difficult to find a suitable model in the practical application, due to model’s parameters partially known or completely unknown, especially the selection of structure. The issues of parameters or topology identification are

60

urgent and important. It mainly depends on information of the known system to identify and estimate the unknown models, further to achieve the purpose of application. To the best of our knowledge, few works focus on finite-time topology identification and synchronization and the existing researches are not considered the effect of external environment and multiple time delays, which have prac-

65

tical significance in the real system. Motivated by above discussions, in this paper we take fully into account stochastic noise perturbation, uncertain topological structure and multiple delays to build a new dynamic model. In order to overcome these difficulties of the above factors interaction, two approaches are proposed to achieve the topological identification and stochastic synchroniza-

70

tion for complex network in the finite-time. Different from the previous works, the effective controllers are designed, the finite-time topological identification and stochastic synchronization for complex networks with the same topological structure and different topological structure are considered. The former uses the response system with an unknown structure to track the known drive system to

75

achieve the finite-time topological identification, and the topological structure of the drive system can be tracked down and identified based on the adaptive update law of a novel controller in the latter. Several novel identification and 4

synchronization criteria for complex network are then obtained. The paper is organized as follows. In Section 2, the model of complex net80

work and preliminaries are given. Two main results about the finite-time topological structure and synchronization are respectively given in Sections 3 and 4. Two numerical examples are given to show the effectiveness of our results in Section 5. Finally, conclusion and prospect are given in Section 6. 2. Network model and preliminaries

85

In the paper, according to the property of multiple time delays, we consider a model of stochastic complex dynamical network as follows: dxi (t) = [f (xi (t)) + = [f (xi (t)) + +

N  j=1

N m−1  

alij xj (t − τl )]dt + h(xi (t), t)dω(t), l=0 j=1 N N   a0ij xj (t) + a1ij xj (t − τ1 ) + ... j=1 j=1

(1)

am−1 xj (t − τm−1 )]dt + h(xi (t), t)dω(t), ij

where xi (t) = (xi1 (t), xi2 (t), ..., xin (t))T , 1 ≤ i ≤ N is the state vector of the ith node, f : Rn → Rn is a nonlinear function, which describes the dynamics of node i in the absence of interactions with other nodes. τ0 = 0 and τl (l = 1..., m−1) > 90

0 denote different time delays in the sub-networks, which are constant time delays. Al = (alij )N ×N (l = 0, 1..., m − 1) is the l + 1th sub-network’s Laplacian matrix representing the coupling strength and the topological structure of the network. If node i and j are linked by an edge, then alij = alji > 0(i = j);

95

otherwise, alij = alji = 0, and the diagonal elements of matrix Al are defined as  l alii = − N j=1,j=i aij . If there are no isolated nodes in the network, then all of the matrix Al (l = 0, 1, ..., m − 1) is an irreducible real symmetric matrix. In order to introduce our main results in the next section, we state here some necessary properties and notations about stochastic effects as follows, in which h(xi (t), t) ∈ h(x1 (t), x2 (t), ..., xn (t), t) ∈ Rn×n is the noise intensity ma-

100

trix, ω(t) = (ω1 (t), ω2 (t), ..., ωn (t)) ∈ Rn is bounded vector-form Weiner process in which every two elements is statistically independent, then E[ωi (t)] = 5

0, E{[ωi (t)]2 } = 1 and E[ωi (t)ωi (s)] = 0, (s = t), E[·] is the mathematical expectation. Therefore, the forms with noise intensity are described as E[h(xi (t), t)dω(t)] = 0, E[(h(xi (t), t)dω(t))T (h(xi , t)dω(t))] =trace [hT (xi (t), t) 105

h(xi (t), t)]dt. If the complex network (1) is called as drive system, the corresponding response system with a control input which can be characterized by: dyi (t)

= [f (yi (t)) +

N m−1   l=0 j=1

a ˆlij (t)yj (t − τl ) + ui (t)]dt

+h(yi (t), t)dω(t), N N   a ˆ0ij (t)yj (t) + a ˆ1ij (t)yj (t − τ1 ) + ... = [f (yi (t)) + +

N  j=1

j=1

a ˆm−1 (t)yj (t ij

(2)

j=1

− τm−1 ) + ui (t)]dt + h(yi (t), t)dω(t),

where i, j = 1, 2, ..., N , yi (t) = (yi1 (t), yi2 (t), ..., yin (t))T ∈ Rn is the state vector of the ith node of the response network, a ˆlij (t) ∈ Rn , l = 0, 1, ..., m − 1 is the 110

unknown parameters, which can be constantly adjusted to track parameters alij . ui (t) is the controller for node i and the remainder parameters in (2) have the same meanings as those in (1). From the networks (1) and (2), we have the error system which can be described by: dei (t) = dyi (t) − dxi (t), N m−1   (ˆ alij (t)yj (t − τl ) − alij xj (t − τl )) = [F (ei (t)) + l=0 j=1

+ui (t)]dt + h(ei , t)dω(t), N m−1   (alij ej (t − τl ) + (ˆ alij (t) − alij )yj (t − τl )) = [F (ei (t)) +

(3)

l=0 j=1

+ui (t)]dt + h(ei (t), t)dω(t), 115

where ei (t) = yi (t) − xi (t), F (ei (t)) = f (yi (t)) − f (xi (t)) and h(ei (t), t)dω(t) = h(yi (t), t)dω(t) − h(xi (t), t)dω(t). Therefore, the synchronization between systems (1) and (2) is equivalent to stability of the zero solution of system (3). Let C 1,2 (R+ × Rn → R+ ) denote the family of all nonnegative functions V (t, e) on R+ × Rn which are continuous once differentiable in t and twice 6

differentiable in e. For each such V , we define an operator L associated with (3) as L V (t) = Vt (t, e(t)) + Ve (t, e(t)){F (ei (t)) +

N m−1   l=0 j=1

(ˆ alij yj (t − τl ) − alij xj (t − τl ))

+ui (t)} + 12 trace[hT (e(t), t)Vee (t, e(t))h(e(t), t)], where ∂V (t,e(t)) , Ve (t, e(t)) ∂t ∂ 2 V (t,e(t)) = ( ∂ei ej )n×n .

Vt (t, e(t)) = Vee (t, e(t))

= ( ∂V (t,e(t)) , ... ∂V (t,e(t)) ), ∂e1 ∂en

Furthermore, we make the following definition in this paper. Definition 1. ([42]) For the drive system (1) and response system (2), it is said to be topology identification and stochastic synchronization in the finite time if there exists a constant T > 0, such that ˜lij (t) = 0, l = 0, 1, ..., m − 1, lim Eei (t) = lim a

t→T

t→T

and for any i, j = 1, 2, ..., N, t > T , ei (t) = a ˜lij (t) ≡ 0, where a ˜lij (t) = a ˆlij (t) − 120

alij . Therefore, in order to depict finite-time topology identification and stochastic synchronization of systems (1) and (2), the following useful Assumptions and Lemmas are introduced. Assumption 1. ([44]) There exists an nonnegative constant δ such that trace[(h(y(t), t) − h(x(t), t))T (h(y(t), t) − h(x(t), t))] ≤ δeT (t)e(t),

125

where trace(·) stands for the trace of matrix. Assumption 2. ([2]) For the smooth nonlinear function f (·), there exists a positive constant L satisfying the following Lipschitz condition: f (y) − f (x) ≤ L(y − x). Lemma 1. ([45]) Assume that a continuous, positive-definite function V (t) satisfying the following differential inequality: V˙ (t) + αV η (t) ≤ 0, ∀t ≥ t0 , V (t0 ) ≥ 0, 7

where α > 0, 0 < η < 1 are two constants. Then, for any given t0 , V (t) satisfies the following differential inequality: V 1−η (t) ≤ V 1−η (t0 ) − α(1 − η)(t − t0 ), t0 ≤ t ≤ t1 , and V (t) ≡ 0, ∀t ≥ t1 , with t1 given by t1 = t0 +

V 1−η (t0 ) . α(1 − η)

Lemma 2.([45]) Let x1 , x2 , ..., xn ∈ Rn are any vectors and 0 < q < 2 is a real number satisfying: q

x1 q + x2 q + · · · + xn q ≥ (x1 2 + x2 2 + · · · + xn 2 ) 2 . Remark 1. Some works have been done in the finite-time topological identification and synchronization simultaneously for complex network [41,42], they could not consider the effect of external environment and multiple time delays. Therefore, in the next two sections, we consider external environment and 130

multiple time delays, and the issues of finite-time topological identification for complex network with the same and different topological structure are studied respectively. Then proper adaptive feedback controllers are simultaneously designed to guarantee the finite-time stochastic synchronization. 3. Finite-time stochastic synchronization and topology identification

135

for complex networks with the same topological structure In this section, the following Theorem is presented to achieve the finitetime topological identification and stochastic synchronization simultaneously for complex network with multiple time delays. Theorem 1. If Assumptions 1 and 2 are satisfied, and suppose that there

8

140

exists a sufficient large positive constant r, which satisfies ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Ξ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

· · · Am−1

Φ

A1

A2

A1

−IN

0

···

0

A2 .. . .. .

0

−IN

···

0 .. . .. .

Am−1

0

.. ···

.

···

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−IN

where Φ = 2L + 2λmax (A0 ) + 12 δ − 2r + m − 1, and δ is a nonnegative constant, which is defined in Assumption 1. Then the drive system (1) and the response system (2) with the following controller and the adaptive laws will realize finite-time topology identification and stochastic synchronization in the case of multiple time delays: ui (t) = −rei (t) − ksign(ei (t))|ei (t)|β − k

m−1 t β+1 ei (t) ( eTi (s)ei (s)ds) 2 , 2 ei (t) t−τl l=1

and alij (t))|˜ alij (t)|β , a ˆ˙ lij (t) = −eTi (t)yj (t − τl ) − ksign(˜ where i, j = 1, 2, ..., N, l = 0, 1, ..., m − 1, 0 < β < 1, k denotes a tunable constant. a ˜lij (t) = a ˆlij (t) − alij and sign(x) is the sign function which is defined as follows:

⎧ ⎪ ⎪ −1, ⎪ ⎨ sign(x) = 0, ⎪ ⎪ ⎪ ⎩ 1,

if x < 0, if x = 0, if x > 0.

Then, the weight matrices alij of network (1) can be traced and identified by a ˆlij , and the response system (2) can synchronize with the drive system (1) in a finite time: t1 = 145

V (0)1−(β+1)/2 , γ(1 − β+1 2 )

and 9

V (0) =

N i=1

eTi (0)ei (0) +

N m−1

i=1 l=1

0

−τl

eTi (s)ei (s)ds +

N m−1 N (ˆ alij (0) − alij )2 , i=1 j=1 l=0

ˆlij (0) are the initial conditions of ei (t), a ˆlij (t), respectively. where ei (0), a Proof. We choose the Lyapunov function as V (t) = V1 (t) + V2 (t) + V3 (t), where V1 (t) = V2 (t) = V3 (t) =

N 

eTi (t)ei (t),

i=1 N m−1   

t t−τl

i=1 l=1 N m−1 N    i=1 j=1 l=0

eTi (s)ei (s)ds

(˜ alij (t))2 .

By the Itˆ o’s formula, we can calculate L V1 , L V2 and L V3 along the trajectories of the system (3) with controller ui as well as Assumptions 1 and 2,

10

Then we have L V1 (t)

=

N 

i=1 N 

L V2 (t)

=

L V3 (t)

= = =

1 2

N 

trace(hT (ei (t), t)h(ei (t), t)),

i=1 N m−1  

eTi (t)[F (ei (t)) + alij ej (t − τl ) i=1 l=0 j=1 N m−1   (ˆ alij (t) − alij )yj (t − τl ) + ui ] + l=0 j=1 N  trace(hT (ei (t), t)h(ei (t), t)), + 12 i=1 N N  N   2 eTi (t)Lei (t) + 2 eTi (t)a0ij ej (t) i=1 i=1 j=1 N m−1 N N  N    T  ei (t)alij ej (t − τl ) + 2 eTi (t)(ˆ a0ij (t) − a0ij )yj (t) +2 i=1 j=1 l=1 i=1 j=1 N m−1 N  N   T  ei (t)(ˆ alij (t) − alij )yj (t − τl ) + 12 δeTi (t)ei (t) +2 i=1 j=1 l=1 i=1 N N   β+1 eTi (t)ei (t) − 2k |eTi (t)ei (t)| 2 −2r i=1 i=1 N m−1   t β+1 T −2k ( t−τl ei (s)ei (s)ds) 2 ], i=1 l=1 N m−1 N m−1   T   T ei (t)ei (t) − ei (t − τl )ei (t − τl ), i=1 l=1 i=1 l=1 N m−1 N    l 2 a ˜ij (t)a ˜˙ lij (t), i=1 j=1 l=0 N m−1 N    l a ˜ij (t)[−eTi (t)yjl (t − τl ) − ksign(˜ alij (t))|˜ alij (t)|β ], 2 i=1 j=1 l=0 N N m−1 N  N     l a ˜0ij (t)eTi (t)yj (t) − 2 a ˜ij (t)eTi (t)yj (t − τl ) −2 i=1 j=1 i=1 j=1 l=1 N  N  N N m−1    l 0 β+1 |˜ aij (t)| − 2k |˜ aij (t)|β+1 . −2k i=1 j=1 i=1 j=1 l=1

=2

≤

2eTi (t)e˙ i (t) +

(4)

11

150

Then we obtain L V (t) = L V1 (t) + L V2 (t) + L V3 (t), N N N    eTi (t)Lei (t) + 2 eTi (t)a0ij ej (t) ≤2 i=1 N m−1 N   

i=1 j=1

eTi (t)alij ej (t − τl ) − 2r

N 

eTi (t)ei (t) i=1 j=1 l=1 i=1 N N   δeTi (t)ei (t) + (m − 1) eTi (t)ei (t) + 12 i=1 i=1 N m−1   T − ei (t − τl )ei (t − τl ) i=1 l=1 N m−1 N N   T   l β+1 −2k |ei (t)ei (t)| 2 − 2k |˜ aij (t)|β+1 i=1 i=1 j=1 l=0 N m−1   t β+1 −2k ( t−τl eTi (s)ei (s)ds) 2 . i=1 l=1 +2

(5)

Let Al = (alij )N ×N , Aˆl (t) = (ˆ alij (t))N ×N , l = 0, 1, ..., m − 1 and e(t) = [e1 (t), e2 (t), · · · , eN (t)]T , Equation (5) can be written L V (t) ≤

2eT (t)Le(t) + 2eT (t)A0 e(t) + 2

m−1  l=1

eT (t)Al e(t − τl )

+ 12 δeT (t)e(t) − 2eT (t)re(t) + (m − 1)eT (t)e(t) m−1  T β+1 e (t − τl )e(t − τl ) − 2k|eT (t)e(t)| 2 − l=1 m−1 

−2k −2k =

t β+1 ( t−τl eT (s)e(s)ds) 2

l=1 m−1  l=0

|(A˜l (t))2 |

β+1 2

(6)

,

eT (t)(2L + 2λmax (A0 ) + 12 δ − 2r + m − 1)e(t) m−1 m−1  T  T e (t)Al e(t − τl ) − e (t − τl )e(t − τl ) +2 l=1

T

−2k|e (t)e(t)| −2k

m−1  l=0

β+1 2

− 2k

l=1 m−1   l=1

(

t t−τl

eT (s)e(s)ds)

β+1 2

β+1 |(A˜l (t))2 | 2 .

We can obtain from Equation (6) that L V (t) ≤ (eT (t), ..., eT (t − τm−1 ))Ξ(eT (t), ..., eT (t − τm−1 ))T m−1  t β+1 β+1 T 2 ds −γ[|eT (t)e(t)| 2 + t−τl (e (s)e( s)) +

m−1  l=0

l=1

|(A˜l (t))2 |

β+1 2

].

12

(7)

where γ = 2k, Ξ is defined as ⎡ Φ A1 ⎢ ⎢ A −IN 1 ⎢ ⎢ ⎢ A 0 2 ⎢ Ξ =⎢ .. ⎢ ⎢ . ⎢ ⎢ .. ⎢ . ⎣ 0 Am−1 155

· · · Am−1

A2 0

···

0

−IN

···

0 .. . .. .

.. ···

.

···

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−IN

where Φ = 2L + 2λmax (A0 ) + 12 δ − 2r + m − 1. Taking the expectation on both sides of (7), it can be written E[L V (t)]

≤ E{(eT (t), ..., eT (t − τm−1 ))Ξ(eT (t), ..., eT (t − τm−1 ))T m−1  t β+1 β+1 T 2 ds −γ[|eT (t)e(t)| 2 + t−τl (e (s)e(s)) +

m−1  l=0

l=1

|(A˜l (t))2 |

β+1 2

(8)

]}.

According to the conditions of the Theorem 1, we can obtain E[L V (t)]

≤ −γE[|eT (t)e(t)| +

m−1  l=0

β+1 2

+

m−1  

β+1 t (eT (s)e(s)) 2 ds t−τl

l=1

(9)

β+1 |(A˜l (t))2 | 2 ],

≤ −γE[V

β+1 2

(t)].

For any t0 ≥ τ > 0, according to Lemma 2, we can easily obtain E[V

β+1 2

(t0 )] = (E[V (t0 )])

β+1 2

.

Therefore, we obtain E[L V (t)] ≤ −γ(E[V (t)])

β+1 2

.

According to the finite-time stabilization theory of Lemma 1, E[V (t)] converges to zero in a finite time t1 , and t1 is estimated by t1 =

V (0)1−(β+1)/2 γ(1 −

β+1 2 )

.

Hence, the error vector ei (t), i = 1, 2, ..., N will stochastically converge to zero in the finite-time t1 . Meanwhile, under the controller ui (t) and adaptive 13

160

update laws, the response system (2) achieves finite-time topological identification and stochastic synchronization for the drive system (1). This completes the proof. Remark 2. In this section, we give the theoretical derivation about topological identification and synchronization of drive-response network with multiple

165

time delays and stochastic noises. We consider the unknown network as response network and connection weight matrices Aˆl (t)(l = 0, 1, ..., m − 1) are unknown ˆlij (t) = alij , i, j = 1, 2, ..., N ; l = 0, 1, ..., m − 1 is or uncertain. Finally, lim a t→t1

achieved. That is, the weight matrices Al = (alij )N ×N can be successfully tracked and identified by Aˆl (t) = (alij (t))N ×N based on parameter adaptive 170

update rules in a finite-time t1 . Corollary 1. Suppose Assumptions 1 and 2 hold, the weights Aˆl (t), l = 0, 1, ..., m − 1 of the response system (2) are known, that is, Aˆl (t) = Al . And suppose that a positive constant r satisfies: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

· · · Am−1

Φ

A1

A2

A1

−IN

0

···

0

A2 .. . .. .

0

−IN

···

0 .. . .. .

Am−1

0

.. ···

.

···

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−IN

where Φ = 2L + 2λmax(A0 ) + 12 δ − 2r + m − 1. Then the drive system (1) and the response system (2) with the following adaptive controller will realize finite-time stochastic synchronization in the case of multiple time-varying delays: ui (t) = −rei (t) − ksign(ei (t))|ei (t)|β − k

m−1 t β+1 ei (t) ( eTi (s)ei (s)ds) 2 , ei (t)2 t−τl l=1

where i, j = 1, 2, ..., N, l = 0, 1, ..., m − 1, k denotes a tunable constant. Then the response system (2) can synchronize with the derive system (1) in a finite time: t2 =

V (0)1−(β+1)/2 γ(1 − 14

β+1 2 )

,

and

V (0) =

N

eTi (0)ei (0) +

i=1 175

N m−1

i=1 l=1

0

−τl

eTi (s)ei (s)ds,

where ei (0) is the initial conditions of ei (t). Corollary 2. Suppose Assumptions 1 and 2 hold, the delay terms of the coupling matrix Al = 0, l = 1, 2, ..., m−1. The drive system and respond system can be written: dxi (t)

dyi (t) 180

= [f (xi (t)) +

= [f (yi (t)) +

N  j=1

N  j=1

a0ij xj (t)]dt + h(xi (t), t)dω(t),

a ˆ0ij yj (t) + ui (t)]dt + h(yi (t), t)dω(t).

(10)

(11)

There exists positive constant r satisfying 2L + 2λmax (A0 ) + 12 δ − 2r ≤ 0, then the drive system (10) and the response system (11) with the following nonlinear feedback controller (12) and the adaptive laws (13) can realize topology identification and stochastic synchronization in a finite time: ui (t) = −rei (t) − ksign(ei (t))|ei (t)|β ,

(12)

a0ij (t))|˜ a0ij (t)|β , a ˆ˙ 0ij (t) = −eTi (t)yj (t0 ) − ksign(˜

(13)

and

where i, j = 1, 2, ..., N , k denotes a tunable constant, and the finite time t3 = 185

where V (0) =

N  i=1

eTi (0)ei (0) +

V (0)1−(β+1)/2 γ(1 −

N N   i=1 j=1

β+1 2 )

,

(ˆ a0ij (0) − a0ij )2 , and ei (0), a ˆ0ij (0) are the

initial conditions of ei (t), a ˆ0ij (t), respectively. 4. Finite-time stochastic synchronization and topology identification for complex network with different topological structure We consider the following drive and response networks with multiple time 190

delays and noise perturbation: dxi (t) = [f (xi (t)) +

N m−1   l=0 j=1

clij xj (t − τl )]dt + h(xi (t), t)dω(t), 15

(14)

dyi (t)

= [f (yi (t)) +

N m−1   l=0 j=1

dlij yj (t − τl ) + ui (t)]dt + h(yi (t), t)dω(t),

(15)

where Cl = (clij )N ×N and Dl = (dlij )N ×N are the coupling configuration matrices of both networks, clij and dlij are defined as follows: if node i and j are linked by an edge, then clij = clji = 1(i = j) and dlij = dlji = 1(i = j); otherwise, 195

clij = clji = 0 and dlij = dlji = 0, and the diagonal elements of matrix Cl and Dl N N are defined as clii = − j=1,j=i clij and dlii = − j=1,j=i dlij . From systems (14) and (15), the synchronization error with controller ui is given as follows: dei (t)

N m−1 (dlij yj (t − τl ) − clij xj (t − τl )) + ui (t)]dt

= [F (ei (t)) +

j=1 l=0

+h(ei (t), t)dω(t), where dei (t) = dyi (t)−dxi (t), F (ei (t)) = f (yi (t))−f (xi (t)) and h(ei (t), t)dω(t) = 200

h(xi (t), t)dω(t) − h(yi (t), t)dω(t). Theorem 2. Under Assumptions 1 and 2, suppose that a sufficient large positive constant r satisfies: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Ξ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

· · · Dm−1

Φ

D1

D2

D1

−IN

0

···

0

D2 .. . .. .

0

−IN

···

0 .. . .. .

Dm−1

0

.. ···

.

···

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−IN

where Φ = 2L + 2λmax (D0 ) + 12 δ − 2r + m − 1. Then the drive system (14) and the response system (15) with the following 205

controller and the adaptive laws will realize finite-time topology identification

16

and stochastic synchronization in the case of multiple time delays: ui

=−

N m−1   j=1 l=0

dlij yj (t) +

N m−1   j=1 l=0

cˆlij (t)yj (t − τl )

−rei (t) − ksign(ei (t))|ei (t)|β (t) −k eeii(t) 2

t β+1 ( t−τl eTi (s)ei (s)ds) 2 ,

m−1  l=1

clij (t))|˜ clij (t)|β , cˆ˙lij (t) = −eTi (t)yj (t − τl ) − ksign(˜ where c˜lij (t) = cˆlij (t) − clij , (l = 0, 1, ..., m − 1), k denotes a tunable constant. Then, the weight matrices clij of network (14) can be identified with cˆlij (t), and the response system (15) can synchronize with the derive system (14) in a finite time:

V (0)1−(β+1)/2

t4 =

γ(1 −

β+1 2 )

,

and V (0) =

N

eTi (0)ei (0) +

i=1

N m−1

i=1 l=1

0

−τl

eTi (s)ei (s)ds +

N m−1 N (ˆ clij (0) − clij )2 , i=1 j=1 l=0

where ei (0), cˆlij (0) are the initial conditions of ei (t), cˆlij (t) respectively. Proof. Consider the following Lyapunov function: V (t) =

N i=1

eTi (t)ei (t)

+

N m−1

i=1 l=1

t

t−τl

eTi (s)ei (s)ds +

17

N m−1 N (˜ clij (t))2 . i=1 j=1 l=0

Then under Assumptions 1-2 and the controller ui , we obtain L V (t)

=2

N 

eTi (t)e˙ i (t) +

i=1 N m−1  

+

1 2

N  i=1

eTi (t)ei (t) −

i=1 l=1 N m−1 N   

trace(hT (ei (t), t)h(ei (t), t)) N m−1   i=1 l=1

eTi (t − τl )ei (t − τl )

c˜lij (t)c˜˙lij (t), i=1 j=1 l=0 N N m−1    eTi (t)[F (ei ) + dlij ej (t − τl ) 2 i=1 l=0 j=1 N m−1   l c˜ij (t)yj (t − τl ) − rei (t) − ksign(ei (t))|ei (t)|β + l=0 j=1 m−1 N  t  β+1 (t) −k eeii(t) ( t−τl eTi (s)ei (s)ds) 2 ] + 12 δeTi (t)ei (t) 2 i=1 l=1 N m−1 N m−1   T   T ei (t)ei (t) − ei (t − τl )ei (t − τl ) + i=1 l=1 i=1 l=1 N m−1 N    l +2 c˜ij (t)[−eTi yj (t − τl ) − ksign(˜ clij (t))|˜ clij (t)|β ], i=1 j=1 l=0 +2

=

≤2

N 

eTi (t)Lei (t) + 2

i=1 N m−1 N   

N  N  i=1 j=1

eTi (t)d0ij ej (t)

eTi (t)dlij ej (t − τl ) − 2

N 

eTi (t)rei (t) i=1 j=1 l=1 i=1 N N m−1    t β+1 β+1 |eTi (t)ei (t)| 2 − 2k ( t−τl eTi (s)ei (s)ds) 2 −2k i=1 i=1 l=1 N m−1 N m−1   T   T + ei (t)ei (t) − ei (t − τl )ei (t − τl ) i=1 l=1 i=1 l=1 N m−1 N  N    β+1 + 12 δeTi (t)ei (t) − 2k |(˜ clij (t))2 | 2 . i=1 i=1 j=1 l=0 +2

clij (t))N ×N , Dl = (dlij )N ×N , l = 0, 1, ..., m − 1 Let Cl = (clij )N ×N , Cˆl (t) = (ˆ 210

and e(t) = [e1 (t), e2 (t), · · · , eN (t)]T . Similar to the proof of Theorem 1, the

18

above equation can be also written as L V (t) ≤ 2eT (t)Le(t) + 2eT (t)D0 e(t) + 2

m−1 

−2eT (t)re(t) + (m − 1)eT (t)e(t) − + 12 δeT (t)e(t) −2k

m−1  l=0

T

− 2k|e (t)e(t)|

eT (t)Dl e(t − τl )

l=1 m−1 

β+1 2

eT (t − τl )e(t − τl )

l=1 m−1 

− 2k

l=1

t β+1 ( t−τl eT (s)e(s)ds) 2

β+1 |(C˜l )2 | 2 ,

= eT (t)(2L + 2λmax (D0 ) − 2r + m − 1 + 12 δ)e(t) m−1 m−1  T  T e (t)Dl e(t − τl ) − e (t − τl )e(t − τl ) +2 l=1

T

−2k|e (t)e(t)| −2k

m−1  l=0

β+1 2

− 2k

l=1 m−1   l=1

(

t t−τl

eT (s)e(s)ds)

β+1 2

β+1 |(C˜l )2 | 2 ,

T

= (e (t), ..., eT (t − τm−1 ))Ξ(eT (t), ..., eT (t − τm−1 ))T m−1  t β+1 β+1 −γ[|eT (t)e(t)| 2 + ( t−τl eT (s)e(s)ds) 2 ds +

m−1  l=0

l=1

β+1 |(C˜l )2 | 2 ],

where γ = 2k, Ξ is defined as ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Ξ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

· · · Dm−1

Φ

D1

D2

D1

−IN

0

···

0

D2 .. . .. .

0

−IN

···

0 .. . .. .

Dm−1

0

.. ···

.

···

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−IN

where Φ = 2L + 2λmax (D0 ) + 12 δ − 2r + m − 1. Based on the condition of Theorem 2, one obtain that L V (t) ≤ −γ[|eT (t)e(t)| + 215

β+1 2

m−1  l=0

+

t β+1 ( t−τl eT (s)e(s)ds) 2 ds

m−1  l=1

|(C˜l )2 |

where γ = 2k. 19

β+1 2

],

Similarly, take expectations on both sides, we can obtain E[L V (t)]

≤ −γE[|eT (t)e(t)| +

m−1  l=0

|(C˜l ) | 2

β+1 2

β+1 2

+

t β+1 ( t−τl eT (s)e(s)ds) 2 ds

m−1  l=1

],

Similar to the proof of Theorem 1, the following inequality is given as E[L V (t)] ≤ −γ(E[V (t)])

β+1 2

.

Therefore, according to the finite-time stabilization theory of Lemma 1, the error systems E[V (t)] converges to zero in a finite time, and the finite time is estimated by t4 =

V (0)1−(β+1)/2 γ(1 −

β+1 2 )

.

Hence, the error vector ei (t)(i = 1, 2, ..., N ) will stochastically converge to zero in the finite-time t4 . Meanwhile, under the controller ui (t) and adaptive update laws, the drive-response system achieves finite-time topological identifi220

cation and stochastic synchronization. This completes the proof. Remark 3. In the previous researches, some results were obtained in parameters identification and structure identification, including asymptotic topological identification [37-40], exponential asymptotic parameters identification [1] and finite-time parameters identification [41-43]. However, they consider structure

225

identification in the complex network with the same topological structure. To our best knowledge, few papers consider the topology identification between two complex networks with different topological structures in a finite time, let alone take full account of the finite-time topology identification for complex network with multiple time delays and stochastic noise perturbations. Hence, our results

230

have a better universality than previous works. Remark 4. Comparing with drive-response networks (1) and (2), the difference is that the drive-response networks (14) and (15) consider different topology l )N ×N is successfully structures in this section. The weight matrices Cl = (Cij l (t))N ×N of controller parameters. The tracked and identified by Cˆl (t) = (Cij

235

main idea is to design a novel controller ui (t) for the response system, which can 20

weaken even change these different topologies of drive-response networks and in which adaptive update laws can trace and identify the topology structure of the drive system. Corollary 3. Different from feedback control strength r is a positive num240

ber, the adaptive controller to determine r(t). The controller ui (t) and the update law are given as follows: ui (t)

=−

N m−1   j=1 l=0

dlij yj (t − τl ) +

N m−1   j=1 l=0

cˆlij (t)yj (t − τl )

−r(t)ei (t) − ksign(ei (t))|ei (t)|β (t) −k eeii(t) 2

t β+1 ( t−τl eTi (s)ei (s)ds) 2 ,

m−1  l=1

r(t) ˙

r(t))|˜ r (t))|β , = eTi (t)ei (t) − ksign(˜

cˆ˙lij (t)

= −eTi (t)yj (t − τl ) − ksign(˜ clij (t))|˜ clij (t)|β ,

where l = 0, 1, ..., m − 1 and r˜(t) = r(t) − r¯. The proof is similar to that of Theorem 2 with the following Lyapunov function: N N m−1

eTi (t)ei (t)+ V (t) = i=1

i=1 l=1

t

t−τl

eTi (s)ei (s)ds+

N m−1 N N (˜ clij )2 + (r(t)−¯ r )2 , i=1 j=1 l=0

i=1

where i, j = 1, 2, ..., N and r¯ is a sufficiently large positive number. Corollary 4. Suppose Assumptions 1 and 2 hold, and there are not coupling 245

delay in both complex networks (14) and (15), then the network structure cij can be traced and identified by the estimated value cˆij under the following controller and adaptive update laws: ui (t) = −

N

d0ij yj (t) +

j=1

N

cˆ0ij (t)yj (t) − ri ei (t) − ki sign(ei (t))|ei (t)|β ,

j=1

c0ij (t))|˜ c0ij (t)|β , cˆ˙0ij (t) = −eTi (t)yj (t) − ksign(˜ where c˜0ij (t) = cˆ0ij (t) − c0ij . 5. Numerical simulations 250

Example 1. We consider a network with six different sub-networks which have different time-delays. Assuming the network is composed of four nodes, 21

we can easily get the weight configuration matrixes A0 , A1 , A2 , A3 , A4 , A5 , in which matrix A0 has no time-delay, and matrix A5 has the maximal time-delay. The weighted configuration ⎡ −2 1 ⎢ ⎢ ⎢ 1 −2 A0 = ⎢ ⎢ ⎢0 1 ⎣ 1 0 ⎡

−1

⎢ ⎢ ⎢0 A2 = ⎢ ⎢ ⎢1 ⎣ 0 ⎡ 0 ⎢ ⎢ ⎢0 A4 = ⎢ ⎢ ⎢0 ⎣ 0 255

matrixes are described by ⎤ ⎡ 0 1 −3 1 ⎥ ⎢ ⎥ ⎢ ⎢ 1 −1 1 0⎥ ⎥ , A1 = ⎢ ⎥ ⎢ ⎢1 −2 1 ⎥ 0 ⎦ ⎣ 1 −2 1 0 ⎤

0

1

⎤ 1

1

⎥ ⎥ 0⎥ ⎥, ⎥ −1 0 ⎥ ⎦ 0 −1 0

⎤

⎡

0

0

0

0

⎥ ⎢ ⎥ ⎢ ⎥ ⎢0 −1 1 −1 0 1 ⎥ , A3 = ⎢ ⎥ ⎢ ⎢0 1 −2 0 −1 0 ⎥ ⎦ ⎣ 1 0 −1 0 0 1 ⎤ ⎡ 0 0 0 −1 1 0 ⎥ ⎢ ⎥ ⎢ ⎢ 1 −1 0 −2 1 1⎥ ⎥ , A5 = ⎢ ⎥ ⎢ ⎢0 1 −1 0 ⎥ 0 0 ⎦ ⎣ 1 0 −1 0 0 0

0

⎥ ⎥ 0⎥ ⎥, ⎥ 1⎥ ⎦ −1 ⎤ 0 ⎥ ⎥ 0⎥ ⎥. ⎥ 0⎥ ⎦ 0

We have the following drive network and taking Lorenz system as an example of the dynamics of node i in the absence of interactions with other nodes: dxi (t) = [f (xi (t)) +

4 j=1

+

4

a5ij xj (t

a0ij xj (t) +

4

a1ij xj (t − τ1 ) + . . .

j=1

(16)

− τ5 )]dt + h(xi (t), t)dω(t),

i = 1, 2, 3, 4

j=1

where τ1 = 0.05, τ2 = 0.1, τ3 = 0.15, τ4 = 0.2 ⎡ ⎤⎡ x (t) −10 10 0 ⎢ ⎥ ⎢ i1 ⎢ ⎥⎢ f (xi (t)) = ⎢ 28 −1 0 ⎥ ⎢ xi2 (t) ⎣ ⎦⎣ 0 0 − 38 xi3 (t)

and τ5 = 0.25, respectively. ⎤ ⎡ ⎤ 0 ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ + ⎢ −xi1 (t)xi3 (t) ⎥ . ⎦ ⎣ ⎦ xi1 (t)xi2 (t)

For the drive system, the noise intensity function is ⎡ ⎤ ⎡ 0.1xi1 (t) h(xi1 (t)) ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ h(xi (t), t) = ⎢ h(xi2 (t)) ⎥ = ⎢ 0.2xi2 (t) ⎣ ⎦ ⎣ h(xi3 (t)) 0.3xi3 (t) 22

given by ⎤ ⎥ ⎥ ⎥. ⎦

For numerical simulations, we suppose that the only following diagonal el260

ements of couple matrices ai11 , ai22 , ai33 , ai44 (i = 0, 1, 2, 3, 4, 5) will be identified, then the response system is given as follows: dyi (t) = [f (yi (t)) +

4

a ˆ0ij xj (t)

j=1

+

4

+

4

a ˆ1ij xj (t − τ1 ) + . . .

j=1

(17)

a ˆ5ij xj (t − τ5 ) + ui (t)]dt + h(yi (t), t)dω(t),

i = 1, 2, 3, 4

j=1

where

⎡

⎡

⎤ h(yi1 (t))

⎢ ⎢ h(yi , t) = ⎢ h(yi2 (t)) ⎣ h(yi3 (t))

⎤ 0.1yi1 (t)

⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ 0.2yi2 (t) ⎦ ⎣ 0.3yi3 (t)

⎥ ⎥ ⎥. ⎦

According to Theorem 1, we can easily get the parameters adaptive laws as follows:

⎧ ⎪ a ˆ˙ 0 (t) = −eTi (t)yj (t) − ksign(˜ a0ij (t))|˜ a0ij (t)|β , ⎪ ⎪ ⎨ ij .. . ⎪ ⎪ ⎪ ⎩ a ˙ˆ5 (t) = −eT (t)yj (t − τ5 (t)) − ksign(˜ a5ij (t))|˜ a5ij (t)|β , ij i

ˆlij (t) − alij , l = 0, 1, 2, 3, 4, 5 and where the parameters: a ˜lij (t) = a ui (t) = −rei (t) − ksign(ei (t))|ei (t)|β − k

5

β+1 ei (t) t ( eTi (s)ei (s)ds) 2 , 2 ei (t) t−τl l=1

where i = 1, 2, 3, 4, and let L = 71, k = 10, according to Theorem 1, r = 98 is obtained. From Theorem 1, one can conclude that the controlled uncertain or unknown response system (17) is globally finite-time synchronous with the drive system (16) and satisfies ai11 (t)−ai11 ) = lim (ˆ ai22 (t)−ai22 ) = lim (ˆ ai33 (t)−ai33 ) = lim (ˆ ai44 (t)−ai44 ) = 0, lim (ˆ

t→t1 265

t→t1

t→t1

t→t1

where i = 0, 1, 2, 3, 4, 5. In the process of simulations, the initial states of the ith node of drive network and response network are x1 (0) = [20, 20, 20], x2 (0) = [30, 30, 30], x3 (0) = [40, 40, 40], x4 (0) = [50, 50, 50], y1 (0) = [60, 60, 60], y2 (0) = [70, 70, 70], y3 (0) = 23

ei1(t)

50

0

ei2(t)

−50 0 50

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0

−50 0 50 0

i3

e (t)

0.2

−50 −100 0

t Figure 2: The synchronization errors of complex network.

[80, 80, 80],y4(0) = [90, 90, 90], and the initial state a ˆii (0)(i = 1, 2, 3, 4) are all 270

chosen as 1. Fig.2 demonstrates the synchronous errors ei1 (t), ei2 (t), ei3 (t). The synchronization state of drive-response network is given in Fig.3, in which small images are clearly shown the overlap of the drive-response network. Fig.4 gives the identification results of unknown parameters ai11 , ai22 , ai33 , ai44 (i = 0, 1, 2, 3, 4, 5) respectively, from which it is clear to see that the unknown parameters converge

275

to some bounded values and the identification of system parameter is very successful. According to Theorem 1, we compute the convergence time t1 = 2.8863, which agrees with the results of Figs.2-4. Remark 5. Simulation case of topological identification in Fig.4 shows ˆi22 (t), a ˆi33 (t), a ˆi44 (t)(i = 0, 1, 2, 3, 4, 5) unknown time-varying parameters a ˆi11 (t), a

280

respectively converge to constant values in a finite time t1 , which means that unknown parameters successfully trace and identify parameters ai11 , ai22 , ai33 , ai44 based on synchronous property of drive-response networks. Example 2. Considering another two complex networks as drive and response systems, and taking Lorenz system as an example, which is the dynamics

24

xi1(t), yi1(t)

100 3 2 1 0

50 0

0.5

1

1.5

2

2

2 0

i2

1 0 1.98 1.99

−100 0 200

xi3(t), yi3(t)

1.98 1.99 3

i2

x (t), y (t)

−50 0 100

0.5

1

1.5

2

2 24 22

100

20

0

−100 0

2

1.5

1

0.5

18 1.98 1.99

2

t

t

Figure 3: The synchronization state graph of drive-response network (the state of drive network is depicted by black line and response network is depicted by red line).

20

20

10

10

22

ai (t)

11

ai (t)

0 −10

0

−10 −20 −20

−30 0.5

1

1.5

−30 0

2

20

10

10

ai (t)

20

44

0

33

ai (t)

−40 0

−10

−20

−20

0.5

1

1.5

2

−30 0

t

1

1.5

2

0.5

1

1.5

2

0

−10

−30 0

0.5

Figure 4: Identification of parameters ai11 , ai22 , ai33 , ai44 (i = 0, 1, 2, 3, 4, 5) for the coupled complex network

25

ei1(t)

10 0

ei2(t)

−10 0 10

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0

e (t)

−10 0 10 i3

0 −10 0

t

50

i1

x (t), y (t)

Figure 5: The synchronization errors of complex network.

xi3(t), yi3(t)

xi2(t), yi2(t)

i1

0

−50 0 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

−50 0 100

50

0 0

t Figure 6: The synchronization state graph of drive-response network (the state of drive network is depicted by black line and response network is depicted by red line).

26

2

c11(t)

1

1

c011(t)

2

0 0.5

c122(t)

2 0

−2 0 4 2 0

−2 0

1

0.5

t

0.5

1

0.5

1

0.5

1

2 0

−2 0 4

1

0.5

c133(t)

c033(t)

0 0 4

1

0

c22(t)

−1 0 4

1

2 0

−2 0

t

Figure 7: Parameters identification results of the coupled complex network

285

of node i in the absence of interactions with other nodes: dxi (t) = [f (xi (t)) +

3

3

c0ij xj (t) +

j=1

j=1

+ h(xi (t), t)dω(t), dyi (t) = [f (yi (t)) +

3

c1ij xj (t − τ1 )]dt

i = 1, 2, 3,

d0ij yj (t) +

j=1

3

d1ij yj (t − τ1 ) + ui (t)]dt

j=1

+ h(yi (t), t)dω(t),

(18)

(19)

i = 1, 2, 3.

The corresponding connection matrix are as follows: ⎡ ⎤ ⎡ ⎤ −1 1 0 0 0 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ C0 = ⎢ 1 −1 0⎥ , C1 = ⎢0 −1 1 ⎥ , ⎣ ⎦ ⎣ ⎦ 0 0 0 0 1 −1 ⎡ −2 ⎢ ⎢ D0 = ⎢ 1 ⎣ 1

⎤ 1 −1 0

⎡

1

−1

⎥ ⎢ ⎥ ⎢ 0 ⎥ , D1 = ⎢ 0 ⎦ ⎣ −1 1

⎤ 0 −1 1

1

⎥ ⎥ 1 ⎥. ⎦ −2

The parameters cˆij (t) can trace and identify the parameters cij of drive 290

network based on parameters adaptive rules of controller. In simulation, it is 27

assumed that the coupling delay of network is τ1 = 0.1, the remaining parameters are the same as those in Example 1. The initial states of the ith node of drive network and response network are x1 (0) = [20, 20, 20], x2 (0) = [30, 30, 30], x3 (0) = [40, 40, 40], y1 (0) = [10, 10, 10], 295

y2 (0) = [20, 20, 20], y3 (0) = [30, 30, 30], and the initial state cˆij (0)(i, j = 1, 2, 3) are all chosen as 1. According to the inequality of Theorem 2, we can obtain r = 81. Fig.5 demonstrates the synchronous errors ei1 , ei2 , ei3 . The synchronization state of drive-response network is given in Fig.6. Fig.7 gives the identification results of unknown parameters ci11 , ci22 , ci33 (i = 0, 1). According to Theorem 2,

300

we compute the convergence time t4 = 1.5656, which agrees with the results of Figs. 5-7. Remark 6. We fully consider more complex network model with multiple delays and stochastic noise, which increase the difficulty of the synchronization. Compared to existing researches, we overcome these difficulties, design a proper

305

controller and parameter update laws to achieve topological identification and synchronization in a finite time based on drive-response concept. These numerical simulations shows an effectiveness of our propose method. 6. Conclusion and Prospect In this paper, we give two control methods to deal with topological identi-

310

fication and stochastic synchronization of complex network with multiple timedelays in the finite-time. In main results, we take fully account the complex networks with the same and different topological structures, and design two different kinds of effective controllers to obtain effective results of identification and stochastic synchronization in the finite time. Furthermore, based on

315

finite-time control theory, Lyapunov theory and the properties of Wiener process, several useful identification and synchronization criteria are then obtained. Two representative numerical simulations are given to verify the effectiveness of our proposed identification schemes. Following our synchronization performance study, we take the control cost

28

320

into account before synchronization behavior is achieved based on the finite time theory. Impulsive and intermittent control as the discontinuous methods have attracted much interest due to their practical and easy implementation in engineering fields. Impulsive control is activated only at some isolated instants, while intermittent control has a nonzero control width. However, few

325

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