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Finite-time synchronization for multi-link complex networks via discontinuous control
Accepted Manuscript Title: Finite-time synchronization for multi-link complex networks via discontinuous control Author: Hui Zhao Lixiang Li Haipeng Peng Jinghua Xiao Yixian Yang Mingwen Zheng Shudong Li PII: DOI: Reference:
S0030-4026(17)30363-7 http://dx.doi.org/doi:10.1016/j.ijleo.2017.03.098 IJLEO 59021
To appear in: Received date: Accepted date:
24-12-2016 21-3-2017
Please cite this article as: Hui Zhao, Lixiang Li, Haipeng Peng, Jinghua Xiao, Yixian Yang, Mingwen Zheng, Shudong Li, Finite-time synchronization for multi-link complex networks via discontinuous control, (2017), http://dx.doi.org/10.1016/j.ijleo.2017.03.098 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 proof before it is published in its final 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.
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Finite-time synchronization for multi-link complex networks via discontinuous control Hui Zhaoa , Lixiang Lib,∗, Haipeng Pengb , Jinghua Xiaoa , Yixian Yangb , Mingwen Zhenga,c , Shudong Lid,e a State
an
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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 c School of Science, Shandong University of Technology, Zibo 255000, China d College of Mathematics and Information Science, Shandong Technology and Business University, Yantai, Shandong 264005,China e School of Computer Science, National University of Defense Technology, 410073 Changsha, China
M
Abstract
This paper is concerned with finite-time synchronization problem for multi-link complex networks with two kinds of discontinuous control approaches, e.g., the
d
intermittent control and the impulsive control. Based on the above two discon-
te
tinuous control methods, comparing with previous continuous control approaches, some less conservative criteria are derived for the finite-time synchronization
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of the multi-link complex networks. We consider the model of multi-link complex network, which is split into some sub-networks based on different time-delays and each sub-networks can be any network forms. Multi-link complex network with different sub-networks may present some interesting dynamical phenomena. Simple intermittent feedback controller and impulsive feedback controller are designed to achieve finite-time synchronization between the drive network and response network. Several novel and useful finite-time synchronization criteria are also derived based on finite-time stability theory, intermittent and impulsive control techniques. Finally, two numerical simulations are provided ∗ Corresponding author. E-mail addresses: li [email protected]
Preprint submitted to Journal of LATEX Templates
December 24, 2016
Page 1 of 31
to illustrate the effectiveness of the theoretical analysis.
Lyapunov stability theory, Adaptive feedback controller
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1. Introduction
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Keywords: Multi-link complex networks, Finite-time synchronization,
Complex dynamical network as a tool of complex dynamical system plays an
us
important role in the field of nonlinear dynamics, and multi-link complex networks [1,2] are general complex dynamical network, which means that there are more than one edge between two nodes based on different properties and many
an
5
complex network with multi-links in the real world, such as human communication network, transportation network and relationship network etc. Single link
M
complex network [3] has been studied by many researchers, which is a special case of multi-link complex network. However, few researchers focus on com10
plex network with multi-links. Therefore, it is necessary to further research in
d
complex dynamical network with multi-links. In this paper, we take the transportation network for example, the vehicle include bus, train and airplane, and
te
the operating speed of these vehicles would be different between the same two places, that is, there causes a certain time delay. We give the corresponding description of multi-link complex network in Figure 1, which describes that the
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15
transportation network can be split into the following three forms of network structure,such as expressway network, railway network and airline network. In 1990, the synchronization behaviour of chaotic system was first observed
by Pecora and Carroll [4]. Since then, synchronization as one of main dynam-
20
ical behaviour has been attracted the attention of the corresponding scientists. What’s more, synchronization as a typical collective behavior has an important potential applications, such as biology, neural networks and secure communication etc. [5,6]. Many types of synchronization and numerous control techniques have been proposed in chaotic systems and different forms of complex networks
25
in afterwards, such as recurrent neural networks, Cohen-Grossberg neural networks and memristive neural networks etc. [8-34]. But few papers are published
2
Page 2 of 31
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/ $LUOLQHQHWZRUN IJ IJ
7UDQVSRUWDWLRQ QHWZRUN
an
/ ([SUHVVZD\QHWZRUN IJ IJ
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/ 5DLOZD\QHWZRUN IJ IJ
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Figure 1: (Color online) Topological structure of the transportation network, which is split into three sub-networks by different time-delays.
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in discontinue control method based on finite-time stability theory. Intermittent control as a discontinuous control technique, which was first introduced
30
te
to linear econometric models in 2000 [7], has attracted much interest in stabilization and synchronization of complex systems [8-23]. Especially, Refs.[8-13]
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investigate the corresponding issues of exponential and asymptotically stability in synchronization of complex networks. Pinning synchronization and cluster synchronization bear important theoretical and realistic significance, which have been studied in Refs.[14-17]. This main characteristic of intermittent control is
35
that control action only occurs in a sequence of mutually disjoint time intervals rather than the entire time. The impulsive control as a discontinue control is different from intermittent control, which is activated only at some isolated instants. Much effort has also been devoted to study the issues of stabilization and synchronization of dynamical networks by using impulsive control in
40
recent years, and many important and interesting results have been obtained (see [24-31]). Especially, Refs.[32-34] investigate the corresponding parameter identification problems based on synchronization performance with impulsive
3
Page 3 of 31
control. After finite-time stability was proposed in 1961 [36]. Many researchers have focused on finite-time stability related control questions until 1972 [37,38]. Recently, several important and interesting results in finite-time synchronization
ip t
45
for chaotic systems and complex networks have been obtained [39-42], includ-
periodically intermittent control and impulsive control etc.
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ing the forms of generalized synchronization, stochastic synchronization, and
50
us
But to our best knowledge, multi-links complex networks model receive few research in finite-time synchronization with intermittent and impulsive control. Inspired by above analyse, in the paper we consider a class of multi-link
an
complex networks model. In order to realize the synchronization goal and simultaneously reduce the control cost, the discontinuous control methods which include intermittent control and impulsive control are directly considered to achieve finite-time synchronization. Compared with different parameter and
M
55
switch rate of intermittent control, we obtain the theoretical significance of intermittent control. Compared with different parameter of impulsive control,
d
we give the corresponding theory verification of impulsive differential equation.
60
te
Finally, several novel synchronization criteria are obtained based on finite-time stability theory and linear matrix inequality (LMI).
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The rest of the paper is organized as follows. In the next section, network model and preliminaries are presented. In Section 3, two different kinds of discontinuous control approaches which include the intermittent control and impulsive control are given to achieve finite-time synchronization for multi-link
65
complex networks. Several effective conditions are obtained based on the Lyapunov function and LMI. In Section 4, two numerical examples are given to demonstrate the effectiveness of the proposed methods. Finally, some conclusions and prospects are drawn in Section 5.
4
Page 4 of 31
2. Network model and preliminaries
by
cr
(1)
us
=
m−1 N ∑ ∑
azij Γxj (t − τz ), z=0 j=1 N N ∑ ∑ g(xi (t)) + a0ij Γxj (t) + a1ij Γ j=1 j=1 N ∑ ×xj (t − τz ) + ... + Γxj (t − τm−1 ), am−1 ij j=1
x˙ i (t) = g(xi (t)) +
ip t
Consider m links complex networks consisting N identical nodes described
70
where i = 1, 2, .., N , xi = (xi1 , xi2 , ..., xin )T ∈ Rn is the state vector of the
an
ith node, g(·) : Rn → Rn is a continuous vector function, which denotes automatic mechanical behavior of node states. τz (z = 0, 1..., m − 1) ≥ 0 de75
note different coupling delays of sub-networks, which τ0 = 0.
Γ ∈ Rn×n
M
is a positive definite diagonal matrix which describes the internal coupling between node i and node j in the complex network, A0 = (a0ij )N ×N and Az = (azij )N ×N (z = 1..., m − 1), respectively, denote the zero sub-network’s
can be random network model in the paper, which can be the same topological
te
80
d
and the zth sub-network’s topological structure and coupling strength, which
structures or different topological structures. If nodes i and j are linked by an
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edge, then azij = azji > 0 (i ̸= j), z = 0, 1, ..., m − 1; otherwise, azij = azji = 0, and the diagonal elements of matrix Az which satisfy the diffusion coupling ∑N condition, are defined as azii = − j=1,j̸=i azij .
85
Remark 1. Compared with the same network topological structures, differ-
ent topological structures show the diversity of the network system may present some interesting dynamical phenomena. To establish the main result of this paper, the following preliminaries are
necessary.
Assumption 1.([43]) Suppose that there exists a constant L > 0 satisfies
the following Lipschitz condition: ∥g(y) − g(x)∥ ≤ L∥y − x∥, 1
90
where x, y ∈ Rn and the norm is defined as ∥x∥ = (xT x) 2 . 5
Page 5 of 31
Remark 2. Notice that if a function g(·) satisfies Lipschitz condition, then it is also satisfies the following assumption:
ip t
the function g(·) ∈ QU AD(Q, ∆, ξ), if there exists a positive definite diagonal matrix Q = diag(q1 , q2 , ..., qn ), a diagonal matrix ∆ = diag(δ1 , δ2 , ..., δn ) and a
cr
scalar ξ > 0, such that (y − x)T Q(g(y) − g(x) − ∆(y − x)) ≤ −ξ(y − x)T (y − x)
us
holds for any x, y ∈ Rn .
Lemma 1. ([44]) For any two vector x, y ∈ Rn and ε > 0, the following inequality holds:
an
2xT y ≤ εxT x + ε−1 y T y.
Lemma 2. ([45]) Let x1 , x2 , ..., xn ∈ Rn be any vectors and 0 < q < 2 is a
M
real number satisfying:
q
∥x1 ∥q + ∥x2 ∥q + · · · + ∥xn ∥q ≥ (∥x1 ∥2 + ∥x2 ∥2 + · · · + ∥xn ∥2 ) 2 .
d
Lemma 3. ([46]) Assume that a continuous, positive-definite function V (t)
te
satisfies the following differential inequality: V˙ (t) ≤ −αV η (t), ∀t ≥ t0 , V (t0 ) ≥ 0,
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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
V 1−η (t0 ) . α(1 − η) Lemma 4. ([41]) Suppose that function V (t) is continuous and nonnegative t1 = t0 +
when t ∈ [0, ∞) and it satisfies the following conditions: V˙ (t) ≤ −αV η (t), V˙ (t) ≤ 0,
lT ≤ t < lT + σT,
lT + σT ≤ t < (l + 1)T, 6
Page 6 of 31
V 1−η (t) ≤ V 1−η (0) − α(1 − η)σt,
0 ≤ t ≤ t3 .
95
cr
3. Main results
ip t
where α, µ > 0, T > 0, η > 0, σ < 1. Then, the following inequality holds:
In the section, we give two kinds of control techniques to obtain finite-time
us
synchronization for multi-link complex networks.
3.1. Finite-time synchronization for multi-link complex networks via intermit-
an
tent control
System (1) is given as the drive (or master) network, and the corresponding response (or slave) network can be written as y˙ i (t) = g(yi (t)) + = g(yi (t)) +
azij Γyj (t − τz ) z=0 j=1 N N ∑ ∑ a0ij Γyj (t) + a1ij Γyj (t j=1 j=1
− τ1 )
(2)
am−1 Γyj (t − τm−1 ) + ui (t), ij
d
N ∑
m−1 N ∑ ∑
M
100
+... +
te
j=1
where yi = (yi1 , yi2 , ..., yin )T ∈ Rn is the response state vector of the ith node.
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And ui (t) is a nonlinear intermittent controller to be designed. In order to achieve finite-time synchronization via intermittent control, the
following intermittent control is defined by β+1 (Q)) 2 [sign(ei (t)) ui (t) = −rei (t) − k (λmax λ (Q) min m−1 ∑ ∫t (t) ×|ei (t)|β + ( ∥eei(t)∥ ( t−τz eTi (s) 2) i z=1
β+1 ×ei (s)ds) 2 ], lT ≤ t < lT + δ, ui (t) = 0, lT + δ ≤ t < (l + 1)T,
105
(3)
where i = 1, 2, ..., N , ei (t) = yi (t) − xi (t), r is a positive constant called control gain and Q is a positive definite diagonal matrix. λmax (Q)(λmin (Q)) denotes the maximum (minimum) eigenvalue of the positive definite diagonal matrix Q,
7
Page 7 of 31
110
cr
and sign(x) is the sign function which is defined as follows: −1, if x < 0, sign(x) = 0, if x = 0, 1, if x > 0.
ip t
δ > 0 is called the control width and the corresponding un-control width is 1−δ,
Define the synchronization error ei (t) = yi (t) − xi (t) based on drive-response
e˙ i (t) = g(yi (t)) − g(xi (t)) + N m−1 ∑ ∑ z=0 j=1
= f (ei (t)) +
z=0 j=1
azij Γyj (t − τz )
an
−
m−1 N ∑ ∑
us
networks (1) and (2), the following error dynamical system is obtained:
azij Γxj (t − τz ) + ui (t), m−1 N ∑ ∑
(4)
− τz ) + ui (t),
M
z=0 j=1
azij Γej (t
lT ≤ t < lT + σT, i = 1, 2, ..., N, and m−1 N ∑ ∑
z=0 j=1
= f (ei (t)) +
z=0 j=1
azij Γyj (t − τz )
azij Γxj (t − τz ),
te
−
m−1 N ∑ ∑
d
e˙ i (t) = g(yi (t)) − g(xi (t)) +
m−1 N ∑ ∑
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z=0 j=1
azij Γej (t
(5) − τz ),
lT + σT ≤ t < (l + 1)T, i = 1, 2, ..., N,
where l ∈ {0, 1, ..., p} is a finite non-negative integer set. In addition, T > 0 is the control period, 0 < δ < T is called the control width and the corresponding
115
un-control width is 1 − δ. Switching rate is defined as σ =
δ T
, in which the
intermittent control degenerate into the continue control method in the case of σ = 1. f (ei (t)) = g(ei (t) + xi (t)) − g(xi (t)). Theorem 1. Consider the drive network (1) and response network (2) with
the intermittent control (3) and suppose Assumption 1 holds. If there exist a 120
positive definite diagonal matrix Q > 0 and a positive definite constant r which
8
Page 8 of 31
Φ
P1
P2
···
Pm−1
P1
−IN
0
···
0
P2 .. . .. .
0
−IN
···
0 .. . .. .
Pm−1
0
.. ···
.
···
≤ 0,
−IN
us
Ξ=
ip t
(6)
cr
satisfy:
m−1 ∑ 2ξIN − + 2∆ + (m − 1)IN + 2P0 + PzT Pz ≤ 0, λmax (Q) z=1
(7)
an
2ξIN where Φ = − λmax (Q) +2(∆−rIN )+(m−1)IN +2P0 , and ∆ = diag(δ1 , δ2 , ..., δN ),
ξ is a positive constant satisfies Assumption 1. IN is an appropriate identity
M
matrix. Then, under the intermittent controller, the synchronization is achieved in a finite time:
1−β
2V 2 (0) t1 ≤ , γσ(1 − β)
V (0) =
N ∑
eTi (0)Qei (0) +
te
i=1
N m−1 ∑ ∑∫
d
where
i=1 z=1
0 −τz
eTi (s)Qei (s)ds,
where ei (0) is the initial conditions of ei .
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Proof. Select the following Lyapunov function as: V (t)
=
N ∑ i=1
125
eTi (t)Qei (t)
+
N m−1 ∑ ∑∫ i=1 z=1
t
eTi (s)Qei (s)ds.
t−τz
When lT ≤ t < lT + σT , the derivative of V (t) is along the trajectories of
9
Page 9 of 31
N N m−1 ∑ ∑ ∑ T ei (t)Qei (t) V˙ (t) = 2 eTi (t)Qe˙ i (t) + i=1 N m−1 ∑ ∑
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te
≤
d
M
an
us
=
− τz )Qei (t − τz ), i=1 z=1 N ∑ T 2 ei (t)Q[f (ei (t)) − ∆ei (t)] i=1 N ∑ N m−1 ∑ ∑ T +2 ei (t)Qazij Γej (t − τz ) i=1 j=1 z=0 N N ∑ ∑ +2 eTi (t)Q∆ei (t) − 2 eTi (t)Qrei (t) i=1 i=1 β+1 N ∑ (λmax (Q)) 2 eTi (t)Qsign(ei (t))|ei (t)|β −2k λmin (Q) i=1 N m−1 ∑ ∑ (λmax (Q)) β+1 2 −2k eTi (t)Q λmin (Q) i=1 z=1 ∫t β+1 (t) T 2 ×( ∥eeii(t)∥ 2 )( t−τ ei (t − τz )ei (t − τz )) z N m−1 ∑ ∑ T + ei (t)Qei (t) i=1 z=1 N m−1 ∑ ∑ T − ei (t − τz )Qei (t − τz ), i=1 z=1 N ∑ 2ξ λmax (Q){− λmax eTi (t)ei (t) (Q) i=1 N N m−1 ∑ ∑ ∑ T +2 eTi (t)(∆ − rIN )ei (t) + ei (t)ei (t) i=1 i=1 z=1 N ∑ N m−1 ∑ ∑ T +2 ei (t)azij Γej (t − τz ) i=1 j=1 z=0 N m−1 ∑ ∑ T − ei (t − τz )ei (t − τz ) i=1 z=1 β+1 N ∑ (λmax (Q)) 2 |eTi (t)ei (t)|β+1 −2k λmin (Q) i=1 N m−1 ∑ ∑ (λmax (Q)) β+1 2 −2k | λmin (Q) i=1 z=1 ∫t β+1 × t−τz (eTi (t − τz )ei (t − τz ))| 2 }.
cr
−
i=1 z=1
eTi (t
ip t
e(t), then we have the following Equation with Assumption 1:
Further, let
e(t) =
[e1 (t), e2 (t), · · · , eN (t)]T ,
Az
=
(azij )N ×N , z = 0, 1, ..., m − 1,
Pz
=
Az ⊗ Γ, z = 0, 1, ..., m − 1.
(8)
10
Page 10 of 31
We obtain
+(m − 1)IN + 2P0 ]e(t) m−1 m−1 ∑ T ∑ T e (t − τz )e(t − τz )} e (t)Pz e(t − τz ) − +2
−2k(λmax (Q))
2
|eT (t)e(t)| m−1 β+1 ∑ ∫ t t−τz
z=1
β+1 2
|eT (s)e(s)|
Ξ=
Φ
P1
P2
···
Pm−1
P1
−IN
0
···
0
P2 .. . .. .
0
−IN
Pm−1
0
,
··· ..
···
0 .. . .. .
.
···
−IN
We can obtain
d
2ξIN where Φ = − λmax (Q) + 2(∆ − rIN ) + (m − 1)IN + 2P0 .
te
130
ds.
an
M
Ξ is defined as
β+1 2
us
β+1 2
cr
z=1
z=1
−2k(λmax (Q))
ip t
2ξIN V˙ (t) ≤ λmax (Q){eT (t)[− λmax (Q) + 2(∆ − rIN )
V˙ (t)
≤ λmax (Q)(eT (t), ..., eT (t − τm−1 ))Ξ
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×(eT (t), ..., eT (t − τm−1 ))T β+1
−γ[|eT (t)Qe(t)| 2 m−1 ∑ ∫t β+1 + (eT (s)Qe(s)) 2 ds], t−τz z=1
where γ = 2k.
Then it follows from the conditions of the Theorem 1 that V˙ (t) ≤ −γV
β+1 2
(t).
11
Page 11 of 31
N N m−1 ∑ ∑ ∑ T ei (t)Qei (t) V˙ (t) = 2 eTi (t)Qe˙ i (t) + i=1 N m−1 ∑ ∑
te
d
M
=
an
us
=
− τz )Qei (t − τz ), i=1 z=1 N ∑ T 2 ei (t)Q[f (ei (t)) i=1 N m−1 ∑ ∑ T +2 ei (t)Qazij Γej (t − τz )] j=1 z=0 N ∑ +(m − 1) eTi (t)Qei (t) i=1 N m−1 ∑ ∑ T − ei (t − τl )Qei (t − τz ), i=1 z=1 N ∑ 2 eTi (t)Q[f (ei (t)) − ∆ei (t)] i=1 N ∑ N m−1 ∑ ∑ T +2 ei (t)Qazij Γej (t − τz ) i=1 j=1 z=0 N ∑ +2 eTi (t)Q∆ei (t) i=1 N ∑ +(m − 1) eTi (t)Qei (t) i=1 N m−1 ∑ ∑ T − ei (t − τz )Qei (t − τz ), i=1 z=1 N ∑ 2ξ eTi (t)ei (t) λmax (Q)[− λmax (Q) i=1 N N ∑ ∑ +2 eTi (t)∆ei (t) + (m − 1) eTi (t)ei (t) i=1 i=1 N ∑ N m−1 ∑ ∑ T +2 ei (t)azij Γej (t − τz ) i=1 j=1 z=0 N m−1 ∑ ∑ T ei (t − τz )ei (t − τz )]. − i=1 z=1
cr
−
i=1 z=1
eTi (t
ip t
When lT + σT ≤ t < (l + 1)T , one obtains
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≤
According to Equation (8), it follows that V˙ (t)
2ξ T T ≤ λmax (Q)[− λmax (Q) e (t)e(t) + 2e (t)∆e(t)
+(m − 1)eT (t)e(t) + 2eT (t)P0 e(t) m−1 ∑ T +2 e (t)Pz e(t − τz )
−
z=1 m−1 ∑
eT (t − τz )e(t − τz )].
z=1
12
Page 12 of 31
According to Lemma 1, the following inequality is established 2
m−1 ∑
e(t)Pz e(t − τz ) ≤
z=1
+
m−1 ∑ z=1 m−1 ∑
eT (t)PzT Pz e(t)
ip t
135
(9)
e(t − τz )eT (t − τz ).
z=1
cr
We have
us
2ξIN V˙ (t) ≤ λmax (Q)eT (t)[− λmax (Q) + 2∆ + (m − 1)IN m−1 ∑ T +2P0 + Pz Pz ]e(t). z=1
an
With the condition of Theorem 1, it is easily obtained
Therefore, we have V˙ (t) ≤ −γV V˙ (t) ≤ 0,
β+1 2
M
V˙ (t) ≤ 0.
lT ≤ t < lT + σT,
(t),
d
lT + σT ≤ t < (l + 1)T.
te
According to Lemma 4, it follows that V
1−β 2
(t) ≤ V
1−β 2
1 (0) − γσ(1 − β)t. 2
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Therefore, the finite-time synchronization for multi-link complex networks
can be achieved via intermittent control. According to Lemma 3, the finite-time is given as:
1−β
t1 ≤
2V 2 (0) . γσ(1 − β)
The proof is completed.
3.2. Finite-time synchronization for multi-link complex networks via impulsive
140
control
The impulsive control as another form of discontinuous control technique which is considered in this subsection, is activated at some isolated instants. Then, we consider the system with impulsive control.
13
Page 13 of 31
System (1) is considered as the drive network, and the corresponding re-
= f (yi (t)) + +... +
N ∑ j=1
m−1 N ∑ ∑
azij Γyj (t − τl ) + ui (t), z=0 j=1 N N ∑ ∑ a0ij Γyj (t) + a1ij Γyj (t − τ1 ) j=1 j=1
am−1 Γyj (t ij
− τm−1 ) + ui (t),
i = 1, 2, ..., N, t ̸= tk , and
cr
y˙ i (t) = f (yi (t)) +
ip t
sponse network is given in the following form of impulsive differential equation:
(10)
us
145
an
− △yi (t) = yi (t+ k ) − yi (tk ) = Bik ei , t = tk , k ∈ ℓ,
yi (t+ 0 ) = yi0 ,
M
where yi = (yi1 , yi2 , ..., yin )T ∈ Rn is the response state vector of the ith node. − yi (t+ k ) = limt→t+ yi (t), yi (tk ) = limt→t− yi (t). ℓ = {1, 2, ..., n, n1 , ..., nk } is a
finite natural number set. For simplicity, it is assumed that yi (t− k ) = yi (tk ), which means yi (t) is left continuous at each tk . The moments of impulse satisfy
d
150
te
t1 < t2 < . . . < tk < tk+1 < . . . and limk→∞ tk = ∞,τk = tk − tk−1 < ∞. ui (t)(i = 1, 2, ..., N ) are nonlinear controllers to be designed.
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According to drive-response networks (1) and (10), the following error dynamical system is obtained: e˙ i (t)
155
= f (ei (t)) +
m−1 N ∑ ∑ z=0 j=1
azij Γej (t − τz )
(11)
+ui (t), t ̸= tk ,
and
e(t+ k)
=
+ y(t+ k ) − x(tk ) = y(tk ) − x(tk ) + Bk e(tk ),
=
(I + Bk )e(tk ), t = tk , k ∈ ℓ,
where ei (t) = yi (t) − xi (t), f (ei (t)) = f (yi (t)) − f (xi (t)), and I is the identical matrix with the same dimension as Bk .
14
Page 14 of 31
In order to achieve finite-time synchronization for multi-link complex net-
ui (t)
= −rei (t) − ksign(ei (t))|ei (t)|β m−1 ∑ ∫t β+1 (t) −k ∥eeii(t)∥ ( t−τz eTi (s)ei (s)ds) 2 , 2 z=1
where i = 1, 2, ..., N , k > 0 is a tunable constant and a real number. Parameter
us
160
(12)
cr
t ̸= tk ,
ip t
works via impulsive control, the controller (12) is designed as follows:
r is designed later.
Theorem 2. Consider the drive-response networks (1) and (10) with con-
d
M
an
troller (12) and Assumption 1 holds. Suppose that a positive constant r satisfies: Φ P1 P2 · · · Pm−1 P −IN 0 ··· 0 1 P 0 −IN · · · 0 2 ≤ 0, Ξ= .. .. . . .. .. .. . . . Pm−1 0 · · · · · · −IN
165
te
where Φ = 2LIN + 2P0 − 2rIN − (m − 1)IN . IN is an identify matrix with appropriate dimension and
Ac ce p
0 < τz ≤ inf k {tk+1 − tk }, z = 1, 2, ..., m − 1, ρk = max(∥I + Bik ∥2 ) < 1, k ∈ ℓ. Then, the synchronization is achieved in a finite time: 2V (0)1−(β+1)/2 , γ(1 − β)
t2 =
and
V (0) =
N ∑
eTi (0)ei (0)
+
N m−1 ∑ ∑∫
−τz
i=1 z=1
i=1
0
eTi (s)ei (s)ds,
where ei (0) is the initial conditions of ei . 170
Proof. Select the following Lyapunov function as: V (t) =
N ∑ i=1
eTi (t)ei (t)
+
N m−1 ∑ ∑∫ i=1 z=1
t
eTi (s)ei (s)ds.
t−τz
15
Page 15 of 31
For t ̸= tk , the derivative of V (t) is along the trajectories of e(t), then we N N m−1 ∑ ∑ ∑ T ei (t)ei (t) V˙ (t) = 2 eTi (t)e˙ i (t) + i=1 z=1
i=1 N m−1 ∑ ∑
eTi (t − τz )ei (t − τz ), i=1 z=1 N N m−1 ∑ ∑ ∑ z 2 eTi (t)[f (ei (t) + aij Γej (t − τz ) i=1 j=1 z=0 (t) −rei (t) − ksign(ei (t))|ei (t)|β − k ∥eeii(t)∥ 2 m−1 ∑ ∫t β+1 × ( t−τz eTi (s)ei (s)ds) 2 ] z=1 N m−1 N m−1 ∑ ∑ T ∑ ∑ T + ei (t)ei (t) − ei (t − τz )ei (t − τz ), i=1 z=1 i=1 z=1 N ∑ N m−1 N ∑ ∑ T ∑ eTi (t)Lei (t) + 2 ei (t)azij Γej (t − τz ) 2 i=1 j=1 z=0 i=1 N N ∑ ∑ +(m − 1) eTi (t)ei (t) − 2r eTi (t)ei (t) i=1 i=1 N m−1 N ∑ ∑ T ∑ β+1 − ei (t − τz )ei (t − τz ) − 2k |eTi (t)ei (t)| 2 i=1 z=1 i=1 N m−1 ∑ ∑ ∫t β+1 T −2k ( t−τz ei (s)ei (s)ds) 2 . i=1 z=1
d
M
≤
an
us
=
cr
−
ip t
have
≤ eT (t)[2LIN + 2P0 − 2rIN − (m − 1)IN ]e(t) m−1 m−1 ∑ T ∑ T +2 e (t)Pz e(t − τz ) − e (t − τz )e(t − τz )
Ac ce p
V˙ (t)
te
According to Equation (8), we have
z=1
−2k|e (t)e(t)| T
β+1 2
−
z=1 m−1 ∑ ∫t 2k ( t−τz z=1
eT (s)e(s)ds)
β+1 2
.
Ξ is defined as
Ξ=
175
Φ
P1
P2
···
Pm−1
P1
−IN
0
···
0
P2 .. . .. .
0
−IN
···
0 .. . .. .
Pm−1
0
.. ···
.
···
,
−IN
where Φ = 2LIN + 2P0 − 2rIN − (m − 1)IN . 16
Page 16 of 31
We can obtain ≤ (eT (t), eT (t − τ1 ), ..., eT (t − τm−1 ))Ξ(eT (t), eT (t − τ1 ), ..., eT (t − τm−1 ))T − γ{|eT (t)e(t)| m−1 ∑ ∫t β+1 ( t−τz eT (s)e(s)ds) 2 }, +
β+1 2
cr
z=1
ip t
V˙ (t)
where γ = 2k.
V˙ (t) ≤ −γV β+1 2 ,
(t).
which implies that
an
For convenience, let 1 − η =
β+1 2
us
Then it follows from the conditions of the Theorem 1 that
V 1−η (t) ≤ V 1−η (tk−1 ) − γ(1 − η)(t − tk−1 ), (13)
When t = tk , one obtains N ∑
d
eTi (t)(I + Bik )T (I + Bik )ei (t) i=1 N m−1 ∑ ∑ ∫t + eT (s)ei (s)ds, t−τl i i=1 l=1 N N m−1 ∑ ∑ ∑ ∫t ρk eTi (t)ei (t) + eT (s)ei (s)ds t−τl i i=1 i=1 l=1
te
V (t+ k) =
M
t ∈ (tk−1 , tk ], k ∈ ℓ.
Ac ce p
≤
Since ρk = max(∥I + Bik ∥2 ) < 1, k ∈ ℓ, we have
180
V (t+ k ) ≤ V (tk ), k ∈ ℓ.
Then, the above equations imply that V 1−η (t) ≤ V 1−η (t+ 0 ) − γ(1 − η)(t − t0 ).
(14)
Then, using the above Equations (13)and (14) imply that V 1−η (t) ≤ V 1−η (t+ 0 ) − γ(1 − η)(t − t0 ). Proof: Using Equation (13), for k = 1 and any t ∈ (t0 , t1 ], it is easy to obtain V 1−η (t) ≤ V 1−η (t+ 0 ) − γ(1 − η)(t − t0 ). 17
Page 17 of 31
and the corresponding inequality is obtained
ip t
V 1−η (t1 ) ≤ V 1−η (t+ 0 ) − γ(1 − η)(t1 − t0 ). From Equation (14), we also yield that
cr
≤ V 1−η (t1 ),
V 1−η (t+ 1)
Similarly, for k = 2 and t ∈ (t1 , t2 ], one has
us
≤ V 1−η (t+ 0 ) − γ(1 − η)(t1 − t0 ).
an
V 1−η (t) ≤ V 1−η (t+ 1 ) − γ(1 − η)(t − t1 ),
≤ V 1−η (t+ 0 ) − γ(1 − η)(t1 − t0 ) −γ(1 − η)(t − t1 ),
V 1−η (t+ 0 ) − γ(1 − η)(t − t0 ).
M
=
In general, for k = m + 1 and t ∈ (tm , tm+1 ], we have ≤ V 1−η (t+ m ) − γ(1 − η)(t − tm ),
d
V 1−η (t)
≤ V 1−η (t+ m−1 ) − γ(1 − η)(tm − tm−1 )
te
−γ(1 − η)(t − tm ),
= V 1−η (t+ m−1 ) − γ(1 − η)(t − tm−1 ),
Ac ce p
≤ V 1−η (t+ m−2 ) − γ(1 − η)(tm−1 − tm−2 ) −γ(1 − η)(t − tm−1 )},
= V 1−η (t+ m−2 ) − γ(1 − η)(t − tm−2 ), .. . ≤ V 1−η (t+ 0 ) − γ(1 − η)(t1 − t0 ) −γ(1 − η)(t − t0 )},
= V 1−η (t+ 0 ) − γ(1 − η)(t − t0 ).
Then, we obtain
V 1−η (t) ≤ V 1−η (t+ 0 ) − γ(1 − η)(t − t0 ). Therefore, from Lemma 3, the finite-time synchronization for multi-link complex networks can be achieved via impulsive control, and the finite-time is given 18
Page 18 of 31
ip t
50
40
30
cr
20
0 20
us
10
20
0
10
0
−20 −10 −20
an
−40
as: t2 ≤
M
Figure 2: The chaotic trajectories of the Lorenz system.
2V (0)1−(β+1)/2 . γ(1 − β)
d
The proof is completed.
te
Remark 3. In Theorem 2, k is a tunable constant, and r is a large enough positive number, which is a feedback control strength. Therefore, we design a strict feedback controller in this section.
Ac ce p
185
Remark 4. In the previous researches, many researchers focused on impul-
sive synchronization [24-31]. However, to the best of our knowledge, few papers have focused on finite-time synchronization for multi-link complex networks via impulsive control. A simple impulsive controller is designed to fill the gap of
190
the existing papers.
4. Numerical simulations In order to show the effectiveness of our proposed method, we give two numerical examples to achieve finite-time intermittent synchronization and impulsive synchronization for multi-link complex networks with 20 identical nodes.
195
Each node is Lorenz system and it has a chaotic attractor in Figure 2.
19
Page 19 of 31
ip t cr us
0 −1
0
0.5
1
1.5
0
0.5
1
1.5
2
0 −0.5
0
0.5
Ac ce p
−0.5
3
3.5
4
2
2.5
3
3.5
4
2 t
2.5
3
3.5
4
d
0
te
ei3
0.5
2.5
M
ei2
0.5
an
ei1
1
1
1.5
Figure 3: The synchronization errors ei1 , ei2 , ei3 (i = 1, 2, ..., 20) between the driving and the response networks via intermittent control with switching rate σ = 0.25.
20
Page 20 of 31
0 −1
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2 t
4
−0.5
ei3
0.5
an
0 −0.5
4
us
0
cr
ei2
0.5
ip t
ei1
1
2.5
3
3.5
4
M
Figure 4: The synchronization errors ei1 , ei2 , ei3 (i = 1, 2, ..., 20) between the driving and the response networks via intermittent control with switching rate
d
σ = 0.5.
0
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.5
1
1.5
2 t
2.5
3
3.5
4
Ac ce p
−1
te
ei1
1
ei2
0.2 0
−0.2
ei3
0.5 0
−0.5
Figure 5: The synchronization errors ei1 , ei2 , ei3 (i = 1, 2, ..., 20) between the driving and the response networks via intermittent control with parameter γ = 10.
21
Page 21 of 31
0 −1
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2 t
4
−0.5
0
an
ei3
0.5
−0.5
cr
0
4
us
ei2
0.5
ip t
ei1
1
2.5
3
3.5
4
M
Figure 6: The synchronization errors ei1 , ei2 , ei3 (i = 1, 2, ..., 20) between the driving and the response networks via intermittent control with parameter γ =
d
20.
te
Consider four links of complex networks as follows x˙ i = f (xi (t), αi ) +
j=1
a0ij Γxj (t) +
a2ij Γxj (t − τ2 ) +
Ac ce p
20 ∑
20 ∑
+
j=1
20 ∑ j=1
20 ∑ j=1
a1ij Γxj (t − τ1 ) (15)
a3ij Γxj (t − τ3 ),
where i = 1, 2, .., 20, Γ = diag{1, 1, 1} and time delays are given as τ1 = 0.05, τ2 = 0.1 and τ3 = 0.15. f (xi (t)) is a Lorenz system, which is defined as follows a(xi2 − xi1 ) f (xi (t)) = bxi1 − xi1 xi3 − xi2 , xi1 xi2 − cxi3
where i = 1, 2, ..., 20, the true values for each parameters are (a, b, c) = (10, 28, 8/3). The weight configuration matrices Al = (alij )20×20 , l = 0, 1, 2, 3 are random matrices, which can be small-world network model, scale-free network model and 200
random network model. 22
Page 22 of 31
0 −1
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2 t
4
−0.5
0
an
ei3
0.5
−0.5
cr
0
4
us
ei2
0.5
ip t
ei1
1
2.5
3
3.5
4
M
Figure 7: The synchronization errors ei1 , ei2 , ei3 (i = 1, 2, ..., 20) between the driving and the response networks via impulsive control with γ = 5.
d
Example 1. We take Q = diag(5, 4, 2) and ∆ = diag(3, 3, 3) such that the
te
Lorenz system satisfies Assumption 1.
Let T = 0.2, β = 0.6 and the initial values of numerical simulations are given as follows: xi (0) = (3 + 2i, 3.5 + 3i, 4 + 4i)T and yi (0) = (2 + 2i, 2 + 3i, 2 + 4i)T , where 1 ≤ i ≤ 20. The synchronization errors ei1 , ei2 , ei3 (i =
Ac ce p 205
1, 2, ..., 20) between the drive-response networks with intermittent control are shown in Figures 3-6, in which sub-networks structure A0 is random network model and its connection probability among nodes is 0.9, A1 and A3 are smallworld network models and the rewiring probability among nodes are 0.5 and 0.6,
210
and A2 is BA scale-free network model and its minimum degree is 8 to show the rich features of different types of structures. Therefore, Theorem 1 is to be verified completely. According to Theorem 1, we compute the convergence time is t1 = 3.1815, which agrees with the results of Figure 3. Remark 5. Through simple computation by the above parameters, we get
215
the setting time as follows: Table 1 gives the settling time when the values of parameter γ = 20 and switch23
Page 23 of 31
0 −1
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2 t
4
−0.5
0
an
ei3
1
−1
4
us
0
cr
ei2
0.5
ip t
ei1
1
2.5
3
3.5
4
M
Figure 8: The synchronization errors ei1 , ei2 , ei3 (i = 1, 2, ..., 20) between the driving and the response networks via impulsive control with γ = 10.
d
Table 1: the setting time is given with fixed parameter γ = 20 (1-D means
te
1-dimension)
1-D
2-D
3-D
0.25
3.3145
3.7276
3.5944
1.6572
1.8640
1.7972
Ac ce p
σ
0.5
ing rate σ are different and Table 2 gives the settling time when the values of parameter σ = 0.5 and switching rate γ are different, which agree with the results of Figures 3,4 and Figures 5,6 respectively. Therefore, Theorem 1 is to
220
be verified completely.
Example 2. An experiment about impulsive control is considered in this
case. Network nodes, time-delays and the initial values of numerical simulations are the same as Example 1. The remainder parameters are given as follows. Let Bik = diag{−0.5, −0.5, −0.5}, τk = 0.2 and L = 71, σ = 0.6. Then, the syn225
chronization errors ei1 (t), ei2 (t), ei3 (t) with different parameter γ are illustrated
24
Page 24 of 31
1-D
2-D
3-D
10
3.3145
3.7276
3.5944
20
1.6572
1.8640
1.7972
cr
γ
ip t
Table 2: the setting time is given with fixed switching rate σ = 0.5
us
Table 3: the setting time is given with fixed parameter β = 0.6 1-D
2-D
3-D
5
2.4022
2.8252
3.1698
10
1.2011
1.4126
1.5849
an
γ
in Figures 7,8, in which sub-networks structure A0 is random network model
M
and its connection probability among nodes is 0.5, A1 and A3 are small-world network models and the rewiring probability among nodes are 0.5 and 0.6, and A2 is BA scale-free network model and its minimum degree is 2. According to
d
Theorem 2, we compute the setting time, which is shown in Table 3.
te
230
5. Conclusion and Prospect
Ac ce p
In the paper, two discontinuous control techniques including intermittent
control and impulsive control are given to achieve finite-time synchronization for multi-link complex networks. Different kinds of sub-networks are selected
235
to show extensive performance in network synchronization, and two nonlinear controllers are designed to guarantee finite-time synchronization between the drive and response networks based on Lyapunov stability theory, LMI and finitetime stability theory. Finally, two examples are given to verify the effectiveness of our theoretical analysis.
240
The memristor which is introduced to the neural network, shows its rich features in the neural system. However, the memory characteristic of memristor is not still fully utilized in the analysis of network stability. Therefore, our further research will turn to focus on the root of memristor component, and try
25
Page 25 of 31
our best to play its important features.
Acknowledgement
ip t
245
The authors thank all the Editor and the anonymous referees for their con-
the quality of this paper.
cr
structive comments and valuable suggestions, which are helpful to improve This paper is supported by the National Natu-
250
us
ral Science Foundation of China (Nos.61472045,61573067), the Beijing City Board of Education Science and technology project (KM201510015009), National key research and development program (2016YFB0800602), the Nation-
an
al Natural Science Foundation of China (No.61672020), Project Funded by China Postdoctoral Science Foundation (2013M542560,2015T81129), A Project
255
M
of Shandong Province Higher Educational Science and Technology Program (No.J16LN61) and Science and Technology Development Plan Project of Yantai (No.2016ZH054).
Ac ce p
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te
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d
Conflict of Interest: The authors declare that they have no conflict of
260
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[38] Cai N, Li W, Jing Y. Finite-time generalized synchronization of chaotic systems with different order[J]. Nonlinear Dynamics, 2011, 64(4):385-393.
[39] Yang H, Jiang B, Zhao J. On Finite-Time Stability of Cyclic Switched Nonlinear Systems[J]. IEEE Transactions on Automatic Control, 2015, 375
60(8):2201-2206.
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[40] Yang X, Cao J. Finite-time stochastic synchronization of complex network-
ip t
s[J]. Applied Mathematical Modelling, 2010, 34(11):3631-3641. [41] Mei J, Jiang M, Wang X, et al. Finite-time synchronization of drive-
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of the Franklin Institute, 2014, 351(5):2691-2710.
cr
response systems via periodically intermittent adaptive control[J]. Journal
[42] Abdurahman A, Jiang H, Teng Z. Finite-time synchronization for
us
memristor-based neural networks with time-varying delays[J]. Neural Networks, 2015, 69:20-28.
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an
[43] Liu X, Chen T. Synchronization analysis for nonlinearly-coupled complex networks with an asymmetrical coupling matrix[J]. Physica A Statistical
M
Mechanics and Its Applications, 2008, 387(16-17):4429-4439. [44] Boyd S P. Linear Matrix Inequalities in System and Control Theory[J]. Proc.ieee Int.conf.robotics Automation, 1994, 85(5):798 - 799.
adaptive synchronization between two chaotic neural networks[J]. Journal of
te
390
d
[45] Mei J, Jiang M, Wang B, et al. Finite-time parameter identification and
the Franklin Institute, 2013, 350(6):1617-1633.
Ac ce p
[46] Tang Y. Terminal sliding mode control for rigid robots[J]. Automatica, 1998, 34(1):51-56.
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S0030-4026(17)30363-7 http://dx.doi.org/doi:10.1016/j.ijleo.2017.03.098 IJLEO 59021
To appear in: Received date: Accepted date:
24-12-2016 21-3-2017
Please cite this article as: Hui Zhao, Lixiang Li, Haipeng Peng, Jinghua Xiao, Yixian Yang, Mingwen Zheng, Shudong Li, Finite-time synchronization for multi-link complex networks via discontinuous control, (2017), http://dx.doi.org/10.1016/j.ijleo.2017.03.098 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 proof before it is published in its final 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.
ip t
Finite-time synchronization for multi-link complex networks via discontinuous control Hui Zhaoa , Lixiang Lib,∗, Haipeng Pengb , Jinghua Xiaoa , Yixian Yangb , Mingwen Zhenga,c , Shudong Lid,e a State
an
us
cr
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 c School of Science, Shandong University of Technology, Zibo 255000, China d College of Mathematics and Information Science, Shandong Technology and Business University, Yantai, Shandong 264005,China e School of Computer Science, National University of Defense Technology, 410073 Changsha, China
M
Abstract
This paper is concerned with finite-time synchronization problem for multi-link complex networks with two kinds of discontinuous control approaches, e.g., the
d
intermittent control and the impulsive control. Based on the above two discon-
te
tinuous control methods, comparing with previous continuous control approaches, some less conservative criteria are derived for the finite-time synchronization
Ac ce p
of the multi-link complex networks. We consider the model of multi-link complex network, which is split into some sub-networks based on different time-delays and each sub-networks can be any network forms. Multi-link complex network with different sub-networks may present some interesting dynamical phenomena. Simple intermittent feedback controller and impulsive feedback controller are designed to achieve finite-time synchronization between the drive network and response network. Several novel and useful finite-time synchronization criteria are also derived based on finite-time stability theory, intermittent and impulsive control techniques. Finally, two numerical simulations are provided ∗ Corresponding author. E-mail addresses: li [email protected]
Preprint submitted to Journal of LATEX Templates
December 24, 2016
Page 1 of 31
to illustrate the effectiveness of the theoretical analysis.
Lyapunov stability theory, Adaptive feedback controller
cr
1. Introduction
ip t
Keywords: Multi-link complex networks, Finite-time synchronization,
Complex dynamical network as a tool of complex dynamical system plays an
us
important role in the field of nonlinear dynamics, and multi-link complex networks [1,2] are general complex dynamical network, which means that there are more than one edge between two nodes based on different properties and many
an
5
complex network with multi-links in the real world, such as human communication network, transportation network and relationship network etc. Single link
M
complex network [3] has been studied by many researchers, which is a special case of multi-link complex network. However, few researchers focus on com10
plex network with multi-links. Therefore, it is necessary to further research in
d
complex dynamical network with multi-links. In this paper, we take the transportation network for example, the vehicle include bus, train and airplane, and
te
the operating speed of these vehicles would be different between the same two places, that is, there causes a certain time delay. We give the corresponding description of multi-link complex network in Figure 1, which describes that the
Ac ce p
15
transportation network can be split into the following three forms of network structure,such as expressway network, railway network and airline network. In 1990, the synchronization behaviour of chaotic system was first observed
by Pecora and Carroll [4]. Since then, synchronization as one of main dynam-
20
ical behaviour has been attracted the attention of the corresponding scientists. What’s more, synchronization as a typical collective behavior has an important potential applications, such as biology, neural networks and secure communication etc. [5,6]. Many types of synchronization and numerous control techniques have been proposed in chaotic systems and different forms of complex networks
25
in afterwards, such as recurrent neural networks, Cohen-Grossberg neural networks and memristive neural networks etc. [8-34]. But few papers are published
2
Page 2 of 31
cr
ip t
/ $LUOLQHQHWZRUN IJ IJ
7UDQVSRUWDWLRQ QHWZRUN
an
/ ([SUHVVZD\QHWZRUN IJ IJ
us
/ 5DLOZD\QHWZRUN IJ IJ
M
Figure 1: (Color online) Topological structure of the transportation network, which is split into three sub-networks by different time-delays.
d
in discontinue control method based on finite-time stability theory. Intermittent control as a discontinuous control technique, which was first introduced
30
te
to linear econometric models in 2000 [7], has attracted much interest in stabilization and synchronization of complex systems [8-23]. Especially, Refs.[8-13]
Ac ce p
investigate the corresponding issues of exponential and asymptotically stability in synchronization of complex networks. Pinning synchronization and cluster synchronization bear important theoretical and realistic significance, which have been studied in Refs.[14-17]. This main characteristic of intermittent control is
35
that control action only occurs in a sequence of mutually disjoint time intervals rather than the entire time. The impulsive control as a discontinue control is different from intermittent control, which is activated only at some isolated instants. Much effort has also been devoted to study the issues of stabilization and synchronization of dynamical networks by using impulsive control in
40
recent years, and many important and interesting results have been obtained (see [24-31]). Especially, Refs.[32-34] investigate the corresponding parameter identification problems based on synchronization performance with impulsive
3
Page 3 of 31
control. After finite-time stability was proposed in 1961 [36]. Many researchers have focused on finite-time stability related control questions until 1972 [37,38]. Recently, several important and interesting results in finite-time synchronization
ip t
45
for chaotic systems and complex networks have been obtained [39-42], includ-
periodically intermittent control and impulsive control etc.
cr
ing the forms of generalized synchronization, stochastic synchronization, and
50
us
But to our best knowledge, multi-links complex networks model receive few research in finite-time synchronization with intermittent and impulsive control. Inspired by above analyse, in the paper we consider a class of multi-link
an
complex networks model. In order to realize the synchronization goal and simultaneously reduce the control cost, the discontinuous control methods which include intermittent control and impulsive control are directly considered to achieve finite-time synchronization. Compared with different parameter and
M
55
switch rate of intermittent control, we obtain the theoretical significance of intermittent control. Compared with different parameter of impulsive control,
d
we give the corresponding theory verification of impulsive differential equation.
60
te
Finally, several novel synchronization criteria are obtained based on finite-time stability theory and linear matrix inequality (LMI).
Ac ce p
The rest of the paper is organized as follows. In the next section, network model and preliminaries are presented. In Section 3, two different kinds of discontinuous control approaches which include the intermittent control and impulsive control are given to achieve finite-time synchronization for multi-link
65
complex networks. Several effective conditions are obtained based on the Lyapunov function and LMI. In Section 4, two numerical examples are given to demonstrate the effectiveness of the proposed methods. Finally, some conclusions and prospects are drawn in Section 5.
4
Page 4 of 31
2. Network model and preliminaries
by
cr
(1)
us
=
m−1 N ∑ ∑
azij Γxj (t − τz ), z=0 j=1 N N ∑ ∑ g(xi (t)) + a0ij Γxj (t) + a1ij Γ j=1 j=1 N ∑ ×xj (t − τz ) + ... + Γxj (t − τm−1 ), am−1 ij j=1
x˙ i (t) = g(xi (t)) +
ip t
Consider m links complex networks consisting N identical nodes described
70
where i = 1, 2, .., N , xi = (xi1 , xi2 , ..., xin )T ∈ Rn is the state vector of the
an
ith node, g(·) : Rn → Rn is a continuous vector function, which denotes automatic mechanical behavior of node states. τz (z = 0, 1..., m − 1) ≥ 0 de75
note different coupling delays of sub-networks, which τ0 = 0.
Γ ∈ Rn×n
M
is a positive definite diagonal matrix which describes the internal coupling between node i and node j in the complex network, A0 = (a0ij )N ×N and Az = (azij )N ×N (z = 1..., m − 1), respectively, denote the zero sub-network’s
can be random network model in the paper, which can be the same topological
te
80
d
and the zth sub-network’s topological structure and coupling strength, which
structures or different topological structures. If nodes i and j are linked by an
Ac ce p
edge, then azij = azji > 0 (i ̸= j), z = 0, 1, ..., m − 1; otherwise, azij = azji = 0, and the diagonal elements of matrix Az which satisfy the diffusion coupling ∑N condition, are defined as azii = − j=1,j̸=i azij .
85
Remark 1. Compared with the same network topological structures, differ-
ent topological structures show the diversity of the network system may present some interesting dynamical phenomena. To establish the main result of this paper, the following preliminaries are
necessary.
Assumption 1.([43]) Suppose that there exists a constant L > 0 satisfies
the following Lipschitz condition: ∥g(y) − g(x)∥ ≤ L∥y − x∥, 1
90
where x, y ∈ Rn and the norm is defined as ∥x∥ = (xT x) 2 . 5
Page 5 of 31
Remark 2. Notice that if a function g(·) satisfies Lipschitz condition, then it is also satisfies the following assumption:
ip t
the function g(·) ∈ QU AD(Q, ∆, ξ), if there exists a positive definite diagonal matrix Q = diag(q1 , q2 , ..., qn ), a diagonal matrix ∆ = diag(δ1 , δ2 , ..., δn ) and a
cr
scalar ξ > 0, such that (y − x)T Q(g(y) − g(x) − ∆(y − x)) ≤ −ξ(y − x)T (y − x)
us
holds for any x, y ∈ Rn .
Lemma 1. ([44]) For any two vector x, y ∈ Rn and ε > 0, the following inequality holds:
an
2xT y ≤ εxT x + ε−1 y T y.
Lemma 2. ([45]) Let x1 , x2 , ..., xn ∈ Rn be any vectors and 0 < q < 2 is a
M
real number satisfying:
q
∥x1 ∥q + ∥x2 ∥q + · · · + ∥xn ∥q ≥ (∥x1 ∥2 + ∥x2 ∥2 + · · · + ∥xn ∥2 ) 2 .
d
Lemma 3. ([46]) Assume that a continuous, positive-definite function V (t)
te
satisfies the following differential inequality: V˙ (t) ≤ −αV η (t), ∀t ≥ t0 , V (t0 ) ≥ 0,
Ac ce p
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
V 1−η (t0 ) . α(1 − η) Lemma 4. ([41]) Suppose that function V (t) is continuous and nonnegative t1 = t0 +
when t ∈ [0, ∞) and it satisfies the following conditions: V˙ (t) ≤ −αV η (t), V˙ (t) ≤ 0,
lT ≤ t < lT + σT,
lT + σT ≤ t < (l + 1)T, 6
Page 6 of 31
V 1−η (t) ≤ V 1−η (0) − α(1 − η)σt,
0 ≤ t ≤ t3 .
95
cr
3. Main results
ip t
where α, µ > 0, T > 0, η > 0, σ < 1. Then, the following inequality holds:
In the section, we give two kinds of control techniques to obtain finite-time
us
synchronization for multi-link complex networks.
3.1. Finite-time synchronization for multi-link complex networks via intermit-
an
tent control
System (1) is given as the drive (or master) network, and the corresponding response (or slave) network can be written as y˙ i (t) = g(yi (t)) + = g(yi (t)) +
azij Γyj (t − τz ) z=0 j=1 N N ∑ ∑ a0ij Γyj (t) + a1ij Γyj (t j=1 j=1
− τ1 )
(2)
am−1 Γyj (t − τm−1 ) + ui (t), ij
d
N ∑
m−1 N ∑ ∑
M
100
+... +
te
j=1
where yi = (yi1 , yi2 , ..., yin )T ∈ Rn is the response state vector of the ith node.
Ac ce p
And ui (t) is a nonlinear intermittent controller to be designed. In order to achieve finite-time synchronization via intermittent control, the
following intermittent control is defined by β+1 (Q)) 2 [sign(ei (t)) ui (t) = −rei (t) − k (λmax λ (Q) min m−1 ∑ ∫t (t) ×|ei (t)|β + ( ∥eei(t)∥ ( t−τz eTi (s) 2) i z=1
β+1 ×ei (s)ds) 2 ], lT ≤ t < lT + δ, ui (t) = 0, lT + δ ≤ t < (l + 1)T,
105
(3)
where i = 1, 2, ..., N , ei (t) = yi (t) − xi (t), r is a positive constant called control gain and Q is a positive definite diagonal matrix. λmax (Q)(λmin (Q)) denotes the maximum (minimum) eigenvalue of the positive definite diagonal matrix Q,
7
Page 7 of 31
110
cr
and sign(x) is the sign function which is defined as follows: −1, if x < 0, sign(x) = 0, if x = 0, 1, if x > 0.
ip t
δ > 0 is called the control width and the corresponding un-control width is 1−δ,
Define the synchronization error ei (t) = yi (t) − xi (t) based on drive-response
e˙ i (t) = g(yi (t)) − g(xi (t)) + N m−1 ∑ ∑ z=0 j=1
= f (ei (t)) +
z=0 j=1
azij Γyj (t − τz )
an
−
m−1 N ∑ ∑
us
networks (1) and (2), the following error dynamical system is obtained:
azij Γxj (t − τz ) + ui (t), m−1 N ∑ ∑
(4)
− τz ) + ui (t),
M
z=0 j=1
azij Γej (t
lT ≤ t < lT + σT, i = 1, 2, ..., N, and m−1 N ∑ ∑
z=0 j=1
= f (ei (t)) +
z=0 j=1
azij Γyj (t − τz )
azij Γxj (t − τz ),
te
−
m−1 N ∑ ∑
d
e˙ i (t) = g(yi (t)) − g(xi (t)) +
m−1 N ∑ ∑
Ac ce p
z=0 j=1
azij Γej (t
(5) − τz ),
lT + σT ≤ t < (l + 1)T, i = 1, 2, ..., N,
where l ∈ {0, 1, ..., p} is a finite non-negative integer set. In addition, T > 0 is the control period, 0 < δ < T is called the control width and the corresponding
115
un-control width is 1 − δ. Switching rate is defined as σ =
δ T
, in which the
intermittent control degenerate into the continue control method in the case of σ = 1. f (ei (t)) = g(ei (t) + xi (t)) − g(xi (t)). Theorem 1. Consider the drive network (1) and response network (2) with
the intermittent control (3) and suppose Assumption 1 holds. If there exist a 120
positive definite diagonal matrix Q > 0 and a positive definite constant r which
8
Page 8 of 31
Φ
P1
P2
···
Pm−1
P1
−IN
0
···
0
P2 .. . .. .
0
−IN
···
0 .. . .. .
Pm−1
0
.. ···
.
···
≤ 0,
−IN
us
Ξ=
ip t
(6)
cr
satisfy:
m−1 ∑ 2ξIN − + 2∆ + (m − 1)IN + 2P0 + PzT Pz ≤ 0, λmax (Q) z=1
(7)
an
2ξIN where Φ = − λmax (Q) +2(∆−rIN )+(m−1)IN +2P0 , and ∆ = diag(δ1 , δ2 , ..., δN ),
ξ is a positive constant satisfies Assumption 1. IN is an appropriate identity
M
matrix. Then, under the intermittent controller, the synchronization is achieved in a finite time:
1−β
2V 2 (0) t1 ≤ , γσ(1 − β)
V (0) =
N ∑
eTi (0)Qei (0) +
te
i=1
N m−1 ∑ ∑∫
d
where
i=1 z=1
0 −τz
eTi (s)Qei (s)ds,
where ei (0) is the initial conditions of ei .
Ac ce p
Proof. Select the following Lyapunov function as: V (t)
=
N ∑ i=1
125
eTi (t)Qei (t)
+
N m−1 ∑ ∑∫ i=1 z=1
t
eTi (s)Qei (s)ds.
t−τz
When lT ≤ t < lT + σT , the derivative of V (t) is along the trajectories of
9
Page 9 of 31
N N m−1 ∑ ∑ ∑ T ei (t)Qei (t) V˙ (t) = 2 eTi (t)Qe˙ i (t) + i=1 N m−1 ∑ ∑
Ac ce p
te
≤
d
M
an
us
=
− τz )Qei (t − τz ), i=1 z=1 N ∑ T 2 ei (t)Q[f (ei (t)) − ∆ei (t)] i=1 N ∑ N m−1 ∑ ∑ T +2 ei (t)Qazij Γej (t − τz ) i=1 j=1 z=0 N N ∑ ∑ +2 eTi (t)Q∆ei (t) − 2 eTi (t)Qrei (t) i=1 i=1 β+1 N ∑ (λmax (Q)) 2 eTi (t)Qsign(ei (t))|ei (t)|β −2k λmin (Q) i=1 N m−1 ∑ ∑ (λmax (Q)) β+1 2 −2k eTi (t)Q λmin (Q) i=1 z=1 ∫t β+1 (t) T 2 ×( ∥eeii(t)∥ 2 )( t−τ ei (t − τz )ei (t − τz )) z N m−1 ∑ ∑ T + ei (t)Qei (t) i=1 z=1 N m−1 ∑ ∑ T − ei (t − τz )Qei (t − τz ), i=1 z=1 N ∑ 2ξ λmax (Q){− λmax eTi (t)ei (t) (Q) i=1 N N m−1 ∑ ∑ ∑ T +2 eTi (t)(∆ − rIN )ei (t) + ei (t)ei (t) i=1 i=1 z=1 N ∑ N m−1 ∑ ∑ T +2 ei (t)azij Γej (t − τz ) i=1 j=1 z=0 N m−1 ∑ ∑ T − ei (t − τz )ei (t − τz ) i=1 z=1 β+1 N ∑ (λmax (Q)) 2 |eTi (t)ei (t)|β+1 −2k λmin (Q) i=1 N m−1 ∑ ∑ (λmax (Q)) β+1 2 −2k | λmin (Q) i=1 z=1 ∫t β+1 × t−τz (eTi (t − τz )ei (t − τz ))| 2 }.
cr
−
i=1 z=1
eTi (t
ip t
e(t), then we have the following Equation with Assumption 1:
Further, let
e(t) =
[e1 (t), e2 (t), · · · , eN (t)]T ,
Az
=
(azij )N ×N , z = 0, 1, ..., m − 1,
Pz
=
Az ⊗ Γ, z = 0, 1, ..., m − 1.
(8)
10
Page 10 of 31
We obtain
+(m − 1)IN + 2P0 ]e(t) m−1 m−1 ∑ T ∑ T e (t − τz )e(t − τz )} e (t)Pz e(t − τz ) − +2
−2k(λmax (Q))
2
|eT (t)e(t)| m−1 β+1 ∑ ∫ t t−τz
z=1
β+1 2
|eT (s)e(s)|
Ξ=
Φ
P1
P2
···
Pm−1
P1
−IN
0
···
0
P2 .. . .. .
0
−IN
Pm−1
0
,
··· ..
···
0 .. . .. .
.
···
−IN
We can obtain
d
2ξIN where Φ = − λmax (Q) + 2(∆ − rIN ) + (m − 1)IN + 2P0 .
te
130
ds.
an
M
Ξ is defined as
β+1 2
us
β+1 2
cr
z=1
z=1
−2k(λmax (Q))
ip t
2ξIN V˙ (t) ≤ λmax (Q){eT (t)[− λmax (Q) + 2(∆ − rIN )
V˙ (t)
≤ λmax (Q)(eT (t), ..., eT (t − τm−1 ))Ξ
Ac ce p
×(eT (t), ..., eT (t − τm−1 ))T β+1
−γ[|eT (t)Qe(t)| 2 m−1 ∑ ∫t β+1 + (eT (s)Qe(s)) 2 ds], t−τz z=1
where γ = 2k.
Then it follows from the conditions of the Theorem 1 that V˙ (t) ≤ −γV
β+1 2
(t).
11
Page 11 of 31
N N m−1 ∑ ∑ ∑ T ei (t)Qei (t) V˙ (t) = 2 eTi (t)Qe˙ i (t) + i=1 N m−1 ∑ ∑
te
d
M
=
an
us
=
− τz )Qei (t − τz ), i=1 z=1 N ∑ T 2 ei (t)Q[f (ei (t)) i=1 N m−1 ∑ ∑ T +2 ei (t)Qazij Γej (t − τz )] j=1 z=0 N ∑ +(m − 1) eTi (t)Qei (t) i=1 N m−1 ∑ ∑ T − ei (t − τl )Qei (t − τz ), i=1 z=1 N ∑ 2 eTi (t)Q[f (ei (t)) − ∆ei (t)] i=1 N ∑ N m−1 ∑ ∑ T +2 ei (t)Qazij Γej (t − τz ) i=1 j=1 z=0 N ∑ +2 eTi (t)Q∆ei (t) i=1 N ∑ +(m − 1) eTi (t)Qei (t) i=1 N m−1 ∑ ∑ T − ei (t − τz )Qei (t − τz ), i=1 z=1 N ∑ 2ξ eTi (t)ei (t) λmax (Q)[− λmax (Q) i=1 N N ∑ ∑ +2 eTi (t)∆ei (t) + (m − 1) eTi (t)ei (t) i=1 i=1 N ∑ N m−1 ∑ ∑ T +2 ei (t)azij Γej (t − τz ) i=1 j=1 z=0 N m−1 ∑ ∑ T ei (t − τz )ei (t − τz )]. − i=1 z=1
cr
−
i=1 z=1
eTi (t
ip t
When lT + σT ≤ t < (l + 1)T , one obtains
Ac ce p
≤
According to Equation (8), it follows that V˙ (t)
2ξ T T ≤ λmax (Q)[− λmax (Q) e (t)e(t) + 2e (t)∆e(t)
+(m − 1)eT (t)e(t) + 2eT (t)P0 e(t) m−1 ∑ T +2 e (t)Pz e(t − τz )
−
z=1 m−1 ∑
eT (t − τz )e(t − τz )].
z=1
12
Page 12 of 31
According to Lemma 1, the following inequality is established 2
m−1 ∑
e(t)Pz e(t − τz ) ≤
z=1
+
m−1 ∑ z=1 m−1 ∑
eT (t)PzT Pz e(t)
ip t
135
(9)
e(t − τz )eT (t − τz ).
z=1
cr
We have
us
2ξIN V˙ (t) ≤ λmax (Q)eT (t)[− λmax (Q) + 2∆ + (m − 1)IN m−1 ∑ T +2P0 + Pz Pz ]e(t). z=1
an
With the condition of Theorem 1, it is easily obtained
Therefore, we have V˙ (t) ≤ −γV V˙ (t) ≤ 0,
β+1 2
M
V˙ (t) ≤ 0.
lT ≤ t < lT + σT,
(t),
d
lT + σT ≤ t < (l + 1)T.
te
According to Lemma 4, it follows that V
1−β 2
(t) ≤ V
1−β 2
1 (0) − γσ(1 − β)t. 2
Ac ce p
Therefore, the finite-time synchronization for multi-link complex networks
can be achieved via intermittent control. According to Lemma 3, the finite-time is given as:
1−β
t1 ≤
2V 2 (0) . γσ(1 − β)
The proof is completed.
3.2. Finite-time synchronization for multi-link complex networks via impulsive
140
control
The impulsive control as another form of discontinuous control technique which is considered in this subsection, is activated at some isolated instants. Then, we consider the system with impulsive control.
13
Page 13 of 31
System (1) is considered as the drive network, and the corresponding re-
= f (yi (t)) + +... +
N ∑ j=1
m−1 N ∑ ∑
azij Γyj (t − τl ) + ui (t), z=0 j=1 N N ∑ ∑ a0ij Γyj (t) + a1ij Γyj (t − τ1 ) j=1 j=1
am−1 Γyj (t ij
− τm−1 ) + ui (t),
i = 1, 2, ..., N, t ̸= tk , and
cr
y˙ i (t) = f (yi (t)) +
ip t
sponse network is given in the following form of impulsive differential equation:
(10)
us
145
an
− △yi (t) = yi (t+ k ) − yi (tk ) = Bik ei , t = tk , k ∈ ℓ,
yi (t+ 0 ) = yi0 ,
M
where yi = (yi1 , yi2 , ..., yin )T ∈ Rn is the response state vector of the ith node. − yi (t+ k ) = limt→t+ yi (t), yi (tk ) = limt→t− yi (t). ℓ = {1, 2, ..., n, n1 , ..., nk } is a
finite natural number set. For simplicity, it is assumed that yi (t− k ) = yi (tk ), which means yi (t) is left continuous at each tk . The moments of impulse satisfy
d
150
te
t1 < t2 < . . . < tk < tk+1 < . . . and limk→∞ tk = ∞,τk = tk − tk−1 < ∞. ui (t)(i = 1, 2, ..., N ) are nonlinear controllers to be designed.
Ac ce p
According to drive-response networks (1) and (10), the following error dynamical system is obtained: e˙ i (t)
155
= f (ei (t)) +
m−1 N ∑ ∑ z=0 j=1
azij Γej (t − τz )
(11)
+ui (t), t ̸= tk ,
and
e(t+ k)
=
+ y(t+ k ) − x(tk ) = y(tk ) − x(tk ) + Bk e(tk ),
=
(I + Bk )e(tk ), t = tk , k ∈ ℓ,
where ei (t) = yi (t) − xi (t), f (ei (t)) = f (yi (t)) − f (xi (t)), and I is the identical matrix with the same dimension as Bk .
14
Page 14 of 31
In order to achieve finite-time synchronization for multi-link complex net-
ui (t)
= −rei (t) − ksign(ei (t))|ei (t)|β m−1 ∑ ∫t β+1 (t) −k ∥eeii(t)∥ ( t−τz eTi (s)ei (s)ds) 2 , 2 z=1
where i = 1, 2, ..., N , k > 0 is a tunable constant and a real number. Parameter
us
160
(12)
cr
t ̸= tk ,
ip t
works via impulsive control, the controller (12) is designed as follows:
r is designed later.
Theorem 2. Consider the drive-response networks (1) and (10) with con-
d
M
an
troller (12) and Assumption 1 holds. Suppose that a positive constant r satisfies: Φ P1 P2 · · · Pm−1 P −IN 0 ··· 0 1 P 0 −IN · · · 0 2 ≤ 0, Ξ= .. .. . . .. .. .. . . . Pm−1 0 · · · · · · −IN
165
te
where Φ = 2LIN + 2P0 − 2rIN − (m − 1)IN . IN is an identify matrix with appropriate dimension and
Ac ce p
0 < τz ≤ inf k {tk+1 − tk }, z = 1, 2, ..., m − 1, ρk = max(∥I + Bik ∥2 ) < 1, k ∈ ℓ. Then, the synchronization is achieved in a finite time: 2V (0)1−(β+1)/2 , γ(1 − β)
t2 =
and
V (0) =
N ∑
eTi (0)ei (0)
+
N m−1 ∑ ∑∫
−τz
i=1 z=1
i=1
0
eTi (s)ei (s)ds,
where ei (0) is the initial conditions of ei . 170
Proof. Select the following Lyapunov function as: V (t) =
N ∑ i=1
eTi (t)ei (t)
+
N m−1 ∑ ∑∫ i=1 z=1
t
eTi (s)ei (s)ds.
t−τz
15
Page 15 of 31
For t ̸= tk , the derivative of V (t) is along the trajectories of e(t), then we N N m−1 ∑ ∑ ∑ T ei (t)ei (t) V˙ (t) = 2 eTi (t)e˙ i (t) + i=1 z=1
i=1 N m−1 ∑ ∑
eTi (t − τz )ei (t − τz ), i=1 z=1 N N m−1 ∑ ∑ ∑ z 2 eTi (t)[f (ei (t) + aij Γej (t − τz ) i=1 j=1 z=0 (t) −rei (t) − ksign(ei (t))|ei (t)|β − k ∥eeii(t)∥ 2 m−1 ∑ ∫t β+1 × ( t−τz eTi (s)ei (s)ds) 2 ] z=1 N m−1 N m−1 ∑ ∑ T ∑ ∑ T + ei (t)ei (t) − ei (t − τz )ei (t − τz ), i=1 z=1 i=1 z=1 N ∑ N m−1 N ∑ ∑ T ∑ eTi (t)Lei (t) + 2 ei (t)azij Γej (t − τz ) 2 i=1 j=1 z=0 i=1 N N ∑ ∑ +(m − 1) eTi (t)ei (t) − 2r eTi (t)ei (t) i=1 i=1 N m−1 N ∑ ∑ T ∑ β+1 − ei (t − τz )ei (t − τz ) − 2k |eTi (t)ei (t)| 2 i=1 z=1 i=1 N m−1 ∑ ∑ ∫t β+1 T −2k ( t−τz ei (s)ei (s)ds) 2 . i=1 z=1
d
M
≤
an
us
=
cr
−
ip t
have
≤ eT (t)[2LIN + 2P0 − 2rIN − (m − 1)IN ]e(t) m−1 m−1 ∑ T ∑ T +2 e (t)Pz e(t − τz ) − e (t − τz )e(t − τz )
Ac ce p
V˙ (t)
te
According to Equation (8), we have
z=1
−2k|e (t)e(t)| T
β+1 2
−
z=1 m−1 ∑ ∫t 2k ( t−τz z=1
eT (s)e(s)ds)
β+1 2
.
Ξ is defined as
Ξ=
175
Φ
P1
P2
···
Pm−1
P1
−IN
0
···
0
P2 .. . .. .
0
−IN
···
0 .. . .. .
Pm−1
0
.. ···
.
···
,
−IN
where Φ = 2LIN + 2P0 − 2rIN − (m − 1)IN . 16
Page 16 of 31
We can obtain ≤ (eT (t), eT (t − τ1 ), ..., eT (t − τm−1 ))Ξ(eT (t), eT (t − τ1 ), ..., eT (t − τm−1 ))T − γ{|eT (t)e(t)| m−1 ∑ ∫t β+1 ( t−τz eT (s)e(s)ds) 2 }, +
β+1 2
cr
z=1
ip t
V˙ (t)
where γ = 2k.
V˙ (t) ≤ −γV β+1 2 ,
(t).
which implies that
an
For convenience, let 1 − η =
β+1 2
us
Then it follows from the conditions of the Theorem 1 that
V 1−η (t) ≤ V 1−η (tk−1 ) − γ(1 − η)(t − tk−1 ), (13)
When t = tk , one obtains N ∑
d
eTi (t)(I + Bik )T (I + Bik )ei (t) i=1 N m−1 ∑ ∑ ∫t + eT (s)ei (s)ds, t−τl i i=1 l=1 N N m−1 ∑ ∑ ∑ ∫t ρk eTi (t)ei (t) + eT (s)ei (s)ds t−τl i i=1 i=1 l=1
te
V (t+ k) =
M
t ∈ (tk−1 , tk ], k ∈ ℓ.
Ac ce p
≤
Since ρk = max(∥I + Bik ∥2 ) < 1, k ∈ ℓ, we have
180
V (t+ k ) ≤ V (tk ), k ∈ ℓ.
Then, the above equations imply that V 1−η (t) ≤ V 1−η (t+ 0 ) − γ(1 − η)(t − t0 ).
(14)
Then, using the above Equations (13)and (14) imply that V 1−η (t) ≤ V 1−η (t+ 0 ) − γ(1 − η)(t − t0 ). Proof: Using Equation (13), for k = 1 and any t ∈ (t0 , t1 ], it is easy to obtain V 1−η (t) ≤ V 1−η (t+ 0 ) − γ(1 − η)(t − t0 ). 17
Page 17 of 31
and the corresponding inequality is obtained
ip t
V 1−η (t1 ) ≤ V 1−η (t+ 0 ) − γ(1 − η)(t1 − t0 ). From Equation (14), we also yield that
cr
≤ V 1−η (t1 ),
V 1−η (t+ 1)
Similarly, for k = 2 and t ∈ (t1 , t2 ], one has
us
≤ V 1−η (t+ 0 ) − γ(1 − η)(t1 − t0 ).
an
V 1−η (t) ≤ V 1−η (t+ 1 ) − γ(1 − η)(t − t1 ),
≤ V 1−η (t+ 0 ) − γ(1 − η)(t1 − t0 ) −γ(1 − η)(t − t1 ),
V 1−η (t+ 0 ) − γ(1 − η)(t − t0 ).
M
=
In general, for k = m + 1 and t ∈ (tm , tm+1 ], we have ≤ V 1−η (t+ m ) − γ(1 − η)(t − tm ),
d
V 1−η (t)
≤ V 1−η (t+ m−1 ) − γ(1 − η)(tm − tm−1 )
te
−γ(1 − η)(t − tm ),
= V 1−η (t+ m−1 ) − γ(1 − η)(t − tm−1 ),
Ac ce p
≤ V 1−η (t+ m−2 ) − γ(1 − η)(tm−1 − tm−2 ) −γ(1 − η)(t − tm−1 )},
= V 1−η (t+ m−2 ) − γ(1 − η)(t − tm−2 ), .. . ≤ V 1−η (t+ 0 ) − γ(1 − η)(t1 − t0 ) −γ(1 − η)(t − t0 )},
= V 1−η (t+ 0 ) − γ(1 − η)(t − t0 ).
Then, we obtain
V 1−η (t) ≤ V 1−η (t+ 0 ) − γ(1 − η)(t − t0 ). Therefore, from Lemma 3, the finite-time synchronization for multi-link complex networks can be achieved via impulsive control, and the finite-time is given 18
Page 18 of 31
ip t
50
40
30
cr
20
0 20
us
10
20
0
10
0
−20 −10 −20
an
−40
as: t2 ≤
M
Figure 2: The chaotic trajectories of the Lorenz system.
2V (0)1−(β+1)/2 . γ(1 − β)
d
The proof is completed.
te
Remark 3. In Theorem 2, k is a tunable constant, and r is a large enough positive number, which is a feedback control strength. Therefore, we design a strict feedback controller in this section.
Ac ce p
185
Remark 4. In the previous researches, many researchers focused on impul-
sive synchronization [24-31]. However, to the best of our knowledge, few papers have focused on finite-time synchronization for multi-link complex networks via impulsive control. A simple impulsive controller is designed to fill the gap of
190
the existing papers.
4. Numerical simulations In order to show the effectiveness of our proposed method, we give two numerical examples to achieve finite-time intermittent synchronization and impulsive synchronization for multi-link complex networks with 20 identical nodes.
195
Each node is Lorenz system and it has a chaotic attractor in Figure 2.
19
Page 19 of 31
ip t cr us
0 −1
0
0.5
1
1.5
0
0.5
1
1.5
2
0 −0.5
0
0.5
Ac ce p
−0.5
3
3.5
4
2
2.5
3
3.5
4
2 t
2.5
3
3.5
4
d
0
te
ei3
0.5
2.5
M
ei2
0.5
an
ei1
1
1
1.5
Figure 3: The synchronization errors ei1 , ei2 , ei3 (i = 1, 2, ..., 20) between the driving and the response networks via intermittent control with switching rate σ = 0.25.
20
Page 20 of 31
0 −1
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2 t
4
−0.5
ei3
0.5
an
0 −0.5
4
us
0
cr
ei2
0.5
ip t
ei1
1
2.5
3
3.5
4
M
Figure 4: The synchronization errors ei1 , ei2 , ei3 (i = 1, 2, ..., 20) between the driving and the response networks via intermittent control with switching rate
d
σ = 0.5.
0
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.5
1
1.5
2 t
2.5
3
3.5
4
Ac ce p
−1
te
ei1
1
ei2
0.2 0
−0.2
ei3
0.5 0
−0.5
Figure 5: The synchronization errors ei1 , ei2 , ei3 (i = 1, 2, ..., 20) between the driving and the response networks via intermittent control with parameter γ = 10.
21
Page 21 of 31
0 −1
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2 t
4
−0.5
0
an
ei3
0.5
−0.5
cr
0
4
us
ei2
0.5
ip t
ei1
1
2.5
3
3.5
4
M
Figure 6: The synchronization errors ei1 , ei2 , ei3 (i = 1, 2, ..., 20) between the driving and the response networks via intermittent control with parameter γ =
d
20.
te
Consider four links of complex networks as follows x˙ i = f (xi (t), αi ) +
j=1
a0ij Γxj (t) +
a2ij Γxj (t − τ2 ) +
Ac ce p
20 ∑
20 ∑
+
j=1
20 ∑ j=1
20 ∑ j=1
a1ij Γxj (t − τ1 ) (15)
a3ij Γxj (t − τ3 ),
where i = 1, 2, .., 20, Γ = diag{1, 1, 1} and time delays are given as τ1 = 0.05, τ2 = 0.1 and τ3 = 0.15. f (xi (t)) is a Lorenz system, which is defined as follows a(xi2 − xi1 ) f (xi (t)) = bxi1 − xi1 xi3 − xi2 , xi1 xi2 − cxi3
where i = 1, 2, ..., 20, the true values for each parameters are (a, b, c) = (10, 28, 8/3). The weight configuration matrices Al = (alij )20×20 , l = 0, 1, 2, 3 are random matrices, which can be small-world network model, scale-free network model and 200
random network model. 22
Page 22 of 31
0 −1
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2 t
4
−0.5
0
an
ei3
0.5
−0.5
cr
0
4
us
ei2
0.5
ip t
ei1
1
2.5
3
3.5
4
M
Figure 7: The synchronization errors ei1 , ei2 , ei3 (i = 1, 2, ..., 20) between the driving and the response networks via impulsive control with γ = 5.
d
Example 1. We take Q = diag(5, 4, 2) and ∆ = diag(3, 3, 3) such that the
te
Lorenz system satisfies Assumption 1.
Let T = 0.2, β = 0.6 and the initial values of numerical simulations are given as follows: xi (0) = (3 + 2i, 3.5 + 3i, 4 + 4i)T and yi (0) = (2 + 2i, 2 + 3i, 2 + 4i)T , where 1 ≤ i ≤ 20. The synchronization errors ei1 , ei2 , ei3 (i =
Ac ce p 205
1, 2, ..., 20) between the drive-response networks with intermittent control are shown in Figures 3-6, in which sub-networks structure A0 is random network model and its connection probability among nodes is 0.9, A1 and A3 are smallworld network models and the rewiring probability among nodes are 0.5 and 0.6,
210
and A2 is BA scale-free network model and its minimum degree is 8 to show the rich features of different types of structures. Therefore, Theorem 1 is to be verified completely. According to Theorem 1, we compute the convergence time is t1 = 3.1815, which agrees with the results of Figure 3. Remark 5. Through simple computation by the above parameters, we get
215
the setting time as follows: Table 1 gives the settling time when the values of parameter γ = 20 and switch23
Page 23 of 31
0 −1
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2
2.5
3
3.5
0
0.5
1
1.5
2 t
4
−0.5
0
an
ei3
1
−1
4
us
0
cr
ei2
0.5
ip t
ei1
1
2.5
3
3.5
4
M
Figure 8: The synchronization errors ei1 , ei2 , ei3 (i = 1, 2, ..., 20) between the driving and the response networks via impulsive control with γ = 10.
d
Table 1: the setting time is given with fixed parameter γ = 20 (1-D means
te
1-dimension)
1-D
2-D
3-D
0.25
3.3145
3.7276
3.5944
1.6572
1.8640
1.7972
Ac ce p
σ
0.5
ing rate σ are different and Table 2 gives the settling time when the values of parameter σ = 0.5 and switching rate γ are different, which agree with the results of Figures 3,4 and Figures 5,6 respectively. Therefore, Theorem 1 is to
220
be verified completely.
Example 2. An experiment about impulsive control is considered in this
case. Network nodes, time-delays and the initial values of numerical simulations are the same as Example 1. The remainder parameters are given as follows. Let Bik = diag{−0.5, −0.5, −0.5}, τk = 0.2 and L = 71, σ = 0.6. Then, the syn225
chronization errors ei1 (t), ei2 (t), ei3 (t) with different parameter γ are illustrated
24
Page 24 of 31
1-D
2-D
3-D
10
3.3145
3.7276
3.5944
20
1.6572
1.8640
1.7972
cr
γ
ip t
Table 2: the setting time is given with fixed switching rate σ = 0.5
us
Table 3: the setting time is given with fixed parameter β = 0.6 1-D
2-D
3-D
5
2.4022
2.8252
3.1698
10
1.2011
1.4126
1.5849
an
γ
in Figures 7,8, in which sub-networks structure A0 is random network model
M
and its connection probability among nodes is 0.5, A1 and A3 are small-world network models and the rewiring probability among nodes are 0.5 and 0.6, and A2 is BA scale-free network model and its minimum degree is 2. According to
d
Theorem 2, we compute the setting time, which is shown in Table 3.
te
230
5. Conclusion and Prospect
Ac ce p
In the paper, two discontinuous control techniques including intermittent
control and impulsive control are given to achieve finite-time synchronization for multi-link complex networks. Different kinds of sub-networks are selected
235
to show extensive performance in network synchronization, and two nonlinear controllers are designed to guarantee finite-time synchronization between the drive and response networks based on Lyapunov stability theory, LMI and finitetime stability theory. Finally, two examples are given to verify the effectiveness of our theoretical analysis.
240
The memristor which is introduced to the neural network, shows its rich features in the neural system. However, the memory characteristic of memristor is not still fully utilized in the analysis of network stability. Therefore, our further research will turn to focus on the root of memristor component, and try
25
Page 25 of 31
our best to play its important features.
Acknowledgement
ip t
245
The authors thank all the Editor and the anonymous referees for their con-
the quality of this paper.
cr
structive comments and valuable suggestions, which are helpful to improve This paper is supported by the National Natu-
250
us
ral Science Foundation of China (Nos.61472045,61573067), the Beijing City Board of Education Science and technology project (KM201510015009), National key research and development program (2016YFB0800602), the Nation-
an
al Natural Science Foundation of China (No.61672020), Project Funded by China Postdoctoral Science Foundation (2013M542560,2015T81129), A Project
255
M
of Shandong Province Higher Educational Science and Technology Program (No.J16LN61) and Science and Technology Development Plan Project of Yantai (No.2016ZH054).
Ac ce p
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te
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d
Conflict of Interest: The authors declare that they have no conflict of
260
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