# Pinning Cluster Synchronization in Linear Hybrid Coupled Delayed Dynamical Networks

Authors: Anping Bao; Ting Wang; Shumin Fei; Xiaomin Tian
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

The problem on cluster synchronization will be investigated for a class of delayed dynamical networks based on pinning control strategy. Through utilizing the combined convex technique and Kronecker product, two sufficient conditions can be derived to ensure the desired synchronization when the designed feedback controller is employed to each cluster. Moreover, the inner coupling matrices are unnecessarily restricted to be diagonal and the controller design can be converted into solving a series of linear matrix inequalities (LMIs), which greatly improve the present methods. Finally, two numerical examples are provided to demonstrate the effectiveness and reduced conservatism.


---

## Body

## 1. Introduction

In past decade, the synchronization of various chaotic systems has received considerable attention since the pioneering works have appeared [1, 2]. Presently, it is widely known that many benefits of having synchronization can be existent. In particular, the synchronization in language emergence and development results can come up with the common vocabulary and agents’ synchronization in organization management can improve their work efficiency. Thus recently, the synchronization has been widely studied owing to its great potential applications. Furthermore, since chaos synchronization in arrays of coupled dynamical networks was initially studied [3], various coupled networks have received the attention because they can exhibit some interesting phenomena [4, 5], and many elegant results have been reported [6–32]. In particular, in [6, 7, 33], time-delay is unavoidable and delayed neural networks (DNNs) are verified to exhibit some complex and unpredictable behaviors, such as periodic oscillations, bifurcation, and chaotic attractors; then, the impulsive and adaptive synchronization has also been studied [8–11], and some uneasy-to-test results have been presented. Most recently, through using Kronecker product, the global synchronization has been studied and elegant criteria have been obtained in terms of LMIs [12–20, 23]. Yet it is worth noting that, in the above works, some most developed techniques were not utilized and the addressed networks seemed to be of simple forms. Thus, researchers have used some effective tools to give less conservative results ensuring the synchronization for more general coupled DNNs [23].In 1992, as the truth that the effective coupling among neurons varies temporally in a rather short time scale has been found [34], some researchers have mentioned that the degree of synchronization among pairs of neurons changed both temporally and by the choice of pairs. Therefore, the cluster synchronization has been imposed to various dynamical networks [21, 22, 24–31, 35]. However, due to the existence of embedding of invariant synchronization manifolds, it may occur that the system can reach different clustering patterns from the different initial conditions [24–29]. Thus, together with pinning control, some suitable methods have appeared but have been independent of initial states [30]. In [30], the pinning control strategy has been used to realize the cluster synchronization for stochastic coupled DNNs, in which the upper bound of delay variation was less than 1. Later, some effective techniques were used to overcome the shortcoming during tackling the delayed dynamical networks [31, 35]. Yet though these results above were elegant, there still exist some points waiting for the improvements. Firstly, most works above have not contained lower bound of delay variation and, in fact, its information can play an important role in reducing the conservatism. Secondly, in [30, 31], the inner coupling matrices had to be diagonal, which unavoidably limits the application areas. Thirdly, as for delay τ
(
t
)
∈
[
τ
0
,
τ
m
], since the triple integral LKF terms such as τ
m
2
-
τ
0
2
/
2
∫
-
τ
m
-
τ
0
∫
ϱ
0
∫
t
+
θ
t
x
˙
T
(
s
)
Q
x
˙
(
s
)
d
s
d
θ
d
ϱ were firstly put forward [35], it has been used and improved owing to the fact that it could help reduce the conservatism greatly [36]. Yet the authors noticed that some important terms have been ignored when estimating its derivative [35, 36], which also induces the conservatism. Therefore, the tighter estimation should be given. Overall, as for the pinning cluster synchronization of coupled networks, the mentioned points above have not been considered, which remains important and motivates this work.Inspired by the above discussions, this paper aims to study the problem on cluster synchronization for a class of coupled time-delay networks with linear hybrid coupling by means of pinning control. Through choosing two augmented Lyapunov-Krasovskii functionals (LKFs) and using the combined convex technique, some novel sufficient conditions are presented via Kronecker product and LMIs, whose feasibility can be easily checked by resorting to Matlab LMI Toolbox. In particular, we will give the tighter upper bounds on time derivative of LKF terms. The efficiency and less conservatism can be verified on the basis of two numerical examples.Notations. R
n
×
m is the set of all n
×
m real matrices; I
m represents the m
×
m identity matrix and 0
m
·
n denotes the m
×
n zero matrix; A
⊗
B represents Kronecker product of matrices A and B.


## 2. Problem Formulations and Preliminaries

Firstly, suppose the nodes are coupled with statesx
i
t, i
∈
{
1
,
…
,
N
}; we consider the dynamical networks with each node being an n-dimensional DNN with linear hybrid coupling as(1)
x
˙
i
t
=
-
C
x
i
t
+
A
f
¯
x
i
t
+
B
f
¯
x
i
t
-
τ
t
+
I
t
+
u
i
t
+
∑
j
=
1
N
l
i
j
1
G
x
j
t
+
∑
j
=
1
N
l
i
j
2
H
x
j
t
-
τ
t
+
∑
j
=
1
N
l
i
j
3
K
∫
t
-
τ
t
t
x
j
s
d
s
,where x
i
(
t
)
=
[
x
i
1
(
t
)
,
x
i
2
(
t
)
,
…
,
x
i
n
(
t
)
]
T are the state vectors; here C
=
d
i
a
g
(
c
1
,
c
2
,
…
,
c
n
)
>
0, A
=
[
a
i
j
]
n
×
n, B
=
[
b
i
j
]
n
×
n, and f
¯
(
x
i
(
·
)
)
=
[
f
¯
1
x
i
1
·
,
…
,
f
¯
n
x
i
n
·
]
T are the activation functions; also here we assume G
=
[
g
i
j
]
n
×
n, H
=
[
h
i
j
]
n
×
n, K
=
[
k
i
j
]
n
×
n denote the inner coupling matrices, u
i
(
t
) is the control input, and I
(
t
)
∈
R
n is the input vector.Remark 1.
In system (1), the hybrid coupling is utilized in model (1) and it should be emphasized that the inner coupling matrices G, H, and K are not necessarily restricted to be of diagonal form, which can represent more general cases than the ones in [30, 31].Suppose that networks (1) will be controlled onto some desired inhomogeneous state as {
x
1
(
t
)
,
…
,
x
m
1
(
t
)
}
→
s
1
(
t
), {
x
m
1
+
1
(
t
)
,
…
,
x
m
2
(
t
)
}
→
s
2
(
t
)
,
…
,
{
x
m
k
-
1
+
1
(
t
)
,
…
,
x
m
k
(
t
)
}
→
s
k
(
t
); that is, M
=
{
{
s
1
(
t
)
,
…
,
s
1
(
t
)
}
,
{
s
2
(
t
)
,
…
,
s
2
(
t
)
}
,
…
,
{
s
k
(
t
)
,
…
,
s
k
(
t
)
}
}
∈
R
n
×
N is the desired cluster synchronization pattern under the pinning control, where x
i
(
t
)
→
s
l
(
t
) means that l
i
m
t
→
+
∞
x
i
(
t
)
-
s
l
(
t
)
=
0 for i
∈
{
m
l
-
1
,
…
,
m
l
} with m
0
=
0 and l
∈
{
1
,
…
,
k
}. The function s
l
(
t
) is defined as(2)
s
˙
l
t
=
-
C
s
l
t
+
A
f
¯
s
l
t
+
B
f
¯
s
l
t
-
τ
t
+
I
t
,
l
=
1
,
…
,
k
.For the dynamical networks described by (1), the following assumptions are utilized.(A
1) Here τ
(
t
) denotes the interval time-varying delay satisfying(3)
0
≤
τ
0
≤
τ
t
≤
τ
m
,
μ
0
≤
τ
˙
t
≤
μ
m
<
+
∞
.Moreover, we give the denotations as τ
¯
m
=
τ
m
-
τ
0, μ
¯
m
=
μ
m
-
μ
0, and δ
m
=
τ
m
2
-
τ
0
2.(A
2) For ν
=
1,2
,
3 and the configuration matrices(4)
L
ν
=
L
11
ν
L
12
ν
⋯
L
1
k
ν
L
21
ν
L
22
ν
⋯
L
2
k
ν
⋮
⋮
⋱
⋮
L
k
1
ν
L
k
2
ν
⋯
L
k
k
νwith L
i
i
ν
∈
R
(
m
i
-
m
i
-
1
)
×
(
m
i
-
m
i
-
1
) and L
i
j
ν
∈
R
(
m
i
-
m
i
-
1
)
×
(
m
j
-
m
j
-
1
), i
,
j
∈
{
1
,
…
,
k
}, assume that every matrix L
i
i
ν
=
[
l
i
i
,
g
h
ν
] for i
∈
{
1
,
…
,
k
} satisfies l
i
i
,
g
g
ν
≥
0, l
i
i
,
g
g
ν
=
-
∑
h
=
1
,
h
≠
g
m
i
-
m
i
-
1
l
i
i
,
g
h
ν, and the sums of all rows in every L
i
j
ν
(
i
≠
j
) are zeros.(A
3) There exist constants σ
i
-
,
σ
i
+
∈
R such that the bounded functions f
¯
i
(
·
) satisfy (5)
σ
i
-
≤
f
¯
i
α
-
f
¯
i
β
α
-
β
≤
σ
i
+
,
∀
α
,
β
∈
R
,
i
=
1
,
…
,
n
.Here we set Σ
+
=
d
i
a
g
(
σ
1
+
,
…
,
σ
n
+
), Σ
-
=
d
i
a
g
(
σ
1
-
,
…
,
σ
n
-
), and(6)
Σ
1
=
d
i
a
g
σ
1
+
σ
1
-
,
…
,
σ
n
+
σ
n
-
,
Σ
2
=
d
i
a
g
σ
1
+
+
σ
1
-
2
,
…
,
σ
n
+
+
σ
n
-
2
.In this paper, we consider one special case that the accurate information on time-delay is available. Without loss of generality, to achieve the goal of cluster synchronization in this work, we will apply the pinning control strategy on the nodes setJ
=
{
m
1
,
m
2
,
…
,
m
k
} and adopt the following pinning controller as (7)
u
i
t
=
-
l
i
G
x
i
t
-
s
l
t
-
r
i
H
x
i
t
-
τ
t
-
s
l
t
-
τ
t
,
i
=
m
l
,
l
=
1
,
…
,
k
,
0
,
i
≠
m
l
.Let e
i
(
t
)
=
x
i
(
t
)
-
s
l
(
t
); one can check that ∑
j
=
1
N
l
i
j
1
G
x
j
(
·
)
=
∑
j
=
1
N
l
i
j
1
G
e
j
(
·
), ∑
j
=
1
N
l
i
j
2
H
x
j
(
·
)
=
∑
j
=
1
N
l
i
j
2
H
e
j
(
·
), and ∑
j
=
1
N
l
i
j
3
K
x
j
(
·
)
=
∑
j
=
1
N
l
i
j
3
K
e
j
(
·
). Then, combining (1) and (2) with (7) yields (8)
e
˙
i
t
=
-
C
+
l
¯
i
G
e
i
t
+
A
f
e
i
t
+
B
f
e
i
t
-
τ
t
-
r
¯
i
H
e
i
t
-
τ
t
+
∑
j
=
1
N
l
i
j
1
G
e
j
t
+
∑
j
=
1
N
l
i
j
2
H
e
j
t
-
τ
t
+
∑
j
=
1
N
l
i
j
3
K
∫
t
-
τ
t
t
e
j
s
d
s
,where f
(
e
i
(
·
)
)
=
f
¯
(
(
e
i
(
·
)
+
s
l
(
·
)
)
)
-
f
¯
(
s
l
(
·
)
) and(9)
l
¯
i
=
l
i
,
i
=
m
l
,
l
=
1,2
,
…
,
k
,
0
,
i
≠
m
l
,
r
¯
i
=
r
i
,
i
=
m
l
,
l
=
1,2
,
…
,
k
,
0
,
i
≠
m
l
.Then, we can easily check that the functions f
(
·
) satisfy assumption (A
3) and we set(10)
Ξ
=
d
i
a
g
0
,
…
,
0
,
l
m
1
_
,
…
,
0
,
…
,
0
,
l
m
k
_
,
Θ
=
d
i
a
g
0
,
…
,
0
,
r
m
1
_
,
…
,
0
,
…
,
0
,
r
m
k
_
.In what follows, some useful basic definition and denotations will be introduced.Definition 2 (see [30]).
The dynamical network (1) with N nodes is said to achieve the cluster synchronization, if the N nodes are split into k clusters G
1
,
G
2
,
…
,
G
k as {
G
1
=
(
1
,
…
,
m
1
)
,
G
2
=
(
m
1
+
1
,
…
,
m
2
)
,
…
,
G
k
=
(
m
k
-
1
+
1
,
…
,
m
k
)
} such that the nodes synchronize with each other in the same cluster; namely, for the states x
i
(
t
) and x
j
(
t
) of the arbitrary nodes i and j in the same cluster G
l
(
l
=
1
,
…
,
k
), l
i
m
t
→
+
∞
x
i
(
t
)
-
x
j
(
t
)
=
0 holds, in which · stands for the Euclidean norm.Denotation 1.
Denote the3
N
n
×
3
N
n constant matrix E as (11)
E
=
E
1
E
N
+
1
E
2
N
+
1
_
E
2
E
N
+
2
E
2
N
+
2
_
⋯
E
N
E
2
N
E
3
N
_
,in which the 3
N
n
×
n matrix E
i
(
i
=
1
,
…
,
3
N
) can be expressed as follows:(12)
E
i
T
=
0
n
0
n
⋯
I
n
⋯
0
n
0
nwith the identity matrix I
n denoting the ith one in the matrix vector E
i.Denotation 2.
Denote(13)
e
T
·
=
e
1
T
·
e
2
T
·
⋯
e
N
T
·
,
e
˙
T
t
=
e
˙
1
T
t
e
˙
2
T
t
⋯
e
˙
N
T
t
.


## 3. Pinning Cluster Synchronization

Prior to addressing the main results, the following lemmas will be useful in the proof.Lemma 3 (see [35]).
For any constant matrixX
∈
R
n
×
n, X
=
X
T
≥
0, two scalars h
2
≥
h
1
≥
0, such that the following integrations are well defined; then(14)
-
h
2
-
h
1
∫
t
-
h
2
t
-
h
1
x
T
s
X
x
s
d
s
≤
-
∫
t
-
h
2
t
-
h
1
x
s
d
s
T
X
∫
-
h
2
-
h
1
x
s
d
s
;
-
h
2
2
-
h
1
2
2
∫
-
h
2
-
h
1
∫
t
+
θ
t
x
T
s
X
x
s
d
s
d
θ
≤
-
∫
-
h
2
-
h
1
∫
t
+
θ
t
x
s
d
s
d
θ
T
X
∫
-
h
2
-
h
1
∫
t
+
θ
t
x
s
d
s
d
θ
.Lemma 4 (see [37]).
For any vectorsζ
1 and ζ
2, constant matrices R and S, and real scalars α
≥
0 and β
≥
0 satisfying that R
S
∗
R
≥
0 and α
+
β
=
1, the inequality holds:(15)
-
1
α
ζ
1
T
R
ζ
1
-
1
β
ζ
2
T
R
ζ
2
≤
-
ζ
1
ζ
2
T
R
S
∗
R
ζ
1
ζ
2
.Lemma 5 (see [23]).
Suppose thatΩ, Ξ
1, and Ξ
2 are the constant matrices of appropriate dimensions, α
∈
[
0,1
]; then Ω
+
α
Ξ
1
+
(
1
-
α
)
Ξ
2
<
0 holds, if and only if Ω
+
Ξ
1
<
0, Ω
+
Ξ
2
<
0
.Lemma 6 (see [37]).
For the symmetric appropriately dimensional matricesR
>
0
,
Ξ, and matrix Γ, the two following statements are equivalent: (i) Ξ
-
Γ
T
R
Γ
<
0; (ii) there exists a matrix of appropriate dimension Π such that (16)
Ξ
+
Γ
T
Π
T
+
Π
Γ
Π
R
-
1
∗
-
R
-
1
<
0
o
r
Ξ
+
Γ
T
Π
T
+
Π
Γ
Π
∗
-
R
<
0
.Now, together with the pinning control strategy, two less conservative criteria will be presented for the cluster synchronization based on Kronecker product and LMI approach.Theorem 7.
Suppose that assumptions(
A
1
)
–
(
A
3
) are true; then, the controlled dynamical networks (1) can achieve the desired cluster synchronization, if there exist two 6
N
n
×
N
n matrices Π
i
(
i
=
1,2
) making Π
=
Π
1
Π
2, n
×
n constant matrices P
>
0, P
i
>
0
(
i
=
1,2
,
3,4
), Q
j
>
0, R
j
>
0, S
j
(
j
=
1,2
,
3
), and M
l
,
N
l
>
0
(
l
=
1,2
) guaranteeing R
j
S
j
∗
R
j
≥
0 and n
×
n diagonal matrices E
>
0, F
>
0, U
>
0, and V
>
0 such that the LMIs in (17)-(18) hold: (17)
Φ
+
I
E
I
N
⊗
Υ
i
Π
T
+
Π
I
E
I
N
⊗
Υ
i
T
Π
1
Π
2
∗
-
I
N
⊗
R
1
-
I
N
⊗
S
1
∗
∗
-
I
N
⊗
R
1
<
0
,
i
=
1,2
,
(18)
Ω
+
μ
¯
m
Γ
1
P
1
0
0
P
3
Γ
1
T
<
0
,
Ω
+
μ
¯
m
Γ
2
P
4
Γ
2
T
<
0
, where E is expressed in Denotation 1 and (19)
Φ
=
Φ
11
Φ
12
Φ
13
Φ
14
0
Φ
16
∗
Φ
22
0
0
0
Φ
26
∗
∗
Φ
33
0
0
Φ
36
∗
∗
∗
Φ
44
Φ
45
Φ
46
∗
∗
∗
∗
Φ
55
0
∗
∗
∗
∗
∗
Φ
66
,
Ω
=
Ω
11
Q
3
M
1
B
0
Ω
15
M
1
B
Ω
17
∗
Ω
22
Ω
23
S
3
0
0
0
∗
∗
Ω
33
Ω
34
0
V
Σ
2
0
∗
∗
∗
Ω
44
0
0
0
∗
∗
∗
∗
Ω
55
0
Ω
57
∗
∗
∗
∗
∗
Ω
66
B
T
M
2
T
∗
∗
∗
∗
∗
∗
Ω
77
,
I
T
=
I
N
n
0
N
n
·
3
N
n
0
N
n
·
2
N
n
0
N
n
·
3
N
n
I
N
n
0
N
n
·
2
N
n
0
N
n
·
4
N
n
I
N
n
0
N
n
·
N
n
,
Υ
1
=
τ
¯
m
I
n
0
n
-
I
n
0
n
0
n
-
I
n
,
Υ
2
=
0
n
τ
¯
m
I
n
-
I
n
0
n
0
n
-
I
n
,
Γ
1
=
0
5
n
·
n
0
3
n
·
n
I
n
I
n
0
n
0
3
n
·
n
, and Γ
2
=
0
n
·
3
n
I
n
0
n
·
3
n
T with (20)
Φ
11
=
L
1
⊗
M
2
G
+
L
1
⊗
M
2
G
T
-
2
Ξ
⊗
M
1
G
-
I
N
⊗
τ
0
2
Q
1
,
Φ
12
=
L
2
T
-
Θ
⊗
M
1
H
,
Φ
13
=
I
N
⊗
τ
0
Q
1
+
L
3
⊗
M
1
K
,
Φ
14
=
L
3
⊗
M
1
K
,
Φ
16
=
L
1
T
-
Ξ
⊗
M
2
G
T
,
Φ
22
=
-
I
N
⊗
N
2
,
Φ
26
=
L
2
T
-
Θ
⊗
M
2
H
T
,
Φ
33
=
-
I
N
⊗
Q
1
+
Q
2
,
Φ
36
=
L
3
⊗
M
2
K
T
,
Φ
44
=
-
I
N
⊗
R
2
,
Φ
45
=
-
I
N
⊗
S
2
,
Φ
46
=
L
3
⊗
M
2
K
T
,
Φ
55
=
-
I
N
⊗
R
2
,
Φ
66
=
-
I
N
⊗
N
1
,
Ω
11
=
P
2
+
τ
0
2
Q
2
+
τ
¯
m
2
R
2
-
Q
3
-
U
Σ
1
-
M
1
C
-
C
M
1
T
,
Ω
15
=
M
1
A
+
U
Σ
2
,
Ω
17
=
P
-
C
M
2
T
-
M
1
+
F
Σ
+
-
E
Σ
-
,
Ω
22
=
-
P
2
-
Q
3
-
R
3
+
P
3
,
Ω
23
=
R
3
-
S
3
,
Ω
33
=
μ
0
-
1
P
3
+
1
-
μ
m
P
4
-
2
R
3
+
S
3
+
S
3
T
-
V
Σ
1
+
N
2
,
Ω
34
=
R
3
-
S
3
,
Ω
44
=
-
P
4
-
R
3
,
Ω
55
=
-
U
+
P
1
,
Ω
57
=
E
T
-
F
T
+
A
T
M
2
T
,
Ω
66
=
-
V
+
μ
0
-
1
P
1
,
Ω
77
=
-
M
2
-
M
2
T
+
N
1
+
τ
0
2
Q
3
+
τ
¯
m
2
R
3
+
τ
0
4
4
Q
1
+
δ
m
2
4
R
1
.In what follows, based on Theorem7, we can consider the pinning cluster synchronization for the dynamical networks composed of N time-delay Lur’e systems [32]:(21)
x
˙
i
t
=
-
C
x
i
t
+
A
f
¯
W
T
x
i
t
+
∑
j
=
1
N
l
i
j
1
G
x
j
t
+
∑
j
=
1
N
l
i
j
2
H
x
j
t
-
τ
t
+
u
i
t
,where C
=
[
a
i
j
]
n
×
n, A
=
[
c
i
j
]
n
×
n
1, and W
=
[
w
i
j
]
n
×
n
1
=
[
W
1
,
W
2
,
…
,
W
n
1
] are constant matrices; here f
¯
(
W
T
x
(
t
)
) denotes the nonlinear function satisfying (A
3) and f
¯
(
0
)
=
0. Then, by using the pinning controller (7), we can derive the following theorem.Theorem 8.
Suppose that assumptions(
A
1
)
–
(
A
3
) are true; then, the controlled dynamical networks (21) can achieve the desired cluster synchronization, if there exist two 6
N
n
×
N
n matrices Π
i
(
i
=
1,2
) making Π
=
Π
1
Π
2, n
×
n constant matrices P
>
0, P
i
>
0, Q
i
>
0, R
i
>
0, S
i
(
i
=
1,2
,
3
), and M
l
,
N
l
>
0
(
l
=
1,2
) guaranteeing R
i
S
i
∗
R
i
≥
0, and n
1
×
n
1 diagonal matrices E
>
0, F
>
0, and U
>
0 such that the LMIs in (22) hold: (22)
Φ
+
I
E
I
N
⊗
Υ
i
Π
T
+
Π
I
E
I
N
⊗
Υ
i
T
Π
1
Π
2
∗
-
I
N
⊗
R
1
-
I
N
⊗
S
1
∗
∗
-
I
N
⊗
R
1
<
0
,
i
=
1,2
,
Ω
+
μ
¯
m
Γ
P
2
Γ
T
<
0
,
Ω
+
μ
¯
m
Γ
P
3
Γ
T
<
0
,where E is expressed in Denotation 1, I is expressed in Theorem 7, and (23)
Φ
=
Φ
11
Φ
12
Φ
13
0
0
Φ
16
∗
Φ
22
0
0
0
Φ
26
∗
∗
Φ
33
0
0
0
∗
∗
∗
Φ
44
Φ
45
0
∗
∗
∗
∗
Φ
55
0
∗
∗
∗
∗
∗
Φ
66
,
Ω
=
Ω
11
Q
3
0
0
Ω
15
Ω
16
∗
Ω
22
Ω
23
S
3
0
0
∗
∗
Ω
33
R
3
-
S
3
0
0
∗
∗
∗
-
P
3
-
R
3
0
0
∗
∗
∗
∗
-
U
Ω
56
∗
∗
∗
∗
∗
Ω
66
,
Υ
1
=
τ
¯
m
I
n
0
n
-
I
n
0
n
0
n
-
I
n
,
Υ
2
=
0
n
τ
¯
m
I
n
-
I
n
0
n
0
n
-
I
n
,
Γ
=
0
3
n
·
n
I
n
0
2
n
·
nwith(24)
Φ
11
=
L
1
⊗
M
2
G
+
L
1
⊗
M
2
G
T
-
2
Ξ
⊗
M
1
G
-
I
N
⊗
τ
0
2
Q
1
,
Φ
12
=
L
2
T
-
Θ
⊗
M
1
H
,
Φ
13
=
I
N
⊗
τ
0
Q
1
,
Φ
16
=
L
1
T
-
Ξ
⊗
M
2
G
T
,
Φ
22
=
-
I
N
⊗
N
2
,
Φ
26
=
L
2
T
-
Θ
⊗
M
2
H
T
,
Φ
33
=
-
I
N
⊗
Q
1
+
Q
2
,
Φ
44
=
-
I
N
⊗
R
2
,
Φ
45
=
-
I
N
⊗
S
2
,
Φ
55
=
-
I
N
⊗
R
2
,
Φ
66
=
-
I
N
⊗
N
1
,
Ω
11
=
P
1
+
τ
0
2
Q
2
+
τ
¯
m
2
R
2
-
Q
3
-
W
U
Σ
1
W
T
-
M
1
C
-
C
M
1
T
,
Ω
15
=
M
1
A
+
W
U
Σ
2
,
Ω
16
=
-
M
1
+
P
-
C
M
2
T
+
F
Σ
+
W
T
-
E
Σ
-
W
T
,
Ω
22
=
-
P
1
-
Q
3
-
R
3
+
P
2
,
Ω
23
=
R
3
-
S
3
,
Ω
33
=
μ
0
-
1
P
2
+
1
-
μ
m
P
3
-
2
R
3
+
S
3
+
S
3
T
-
V
Σ
1
+
N
2
,
Ω
56
=
E
T
-
F
T
+
A
T
M
2
T
,
Ω
66
=
N
1
+
τ
0
2
Q
3
+
τ
¯
m
2
R
3
+
τ
0
4
4
Q
1
+
δ
m
2
4
R
1
-
M
2
-
M
2
T
.


## 4. Numerical Examples

Two numerical examples will be provided to illustrate the derived results with some typical cases.Example 1.
Consider one 2-dimensional delayed dynamical network (1) described by(25)
x
˙
i
t
=
-
C
x
i
t
+
A
f
¯
x
i
t
+
B
f
¯
x
i
t
-
τ
t
+
u
i
t
+
∑
j
=
1
4
l
i
j
1
G
x
j
t
+
∑
j
=
1
4
l
i
j
1
H
x
j
t
-
τ
t
+
∑
j
=
1
4
l
i
j
1
K
∫
t
-
τ
t
t
x
j
s
d
s
,
i
=
1,2
,
3,4
, with the following parameters: (26)
C
=
1.25
0
0
1.25
,
A
=
1.8
-
0.15
-
5.2
-
3.5
,
B
=
-
1.7
-
0.12
-
0.26
-
2.5
,
G
=
2.5
0.2
0.2
0.5
,
H
=
0.5
0.25
0.25
0.5
,
K
=
0.5
0.25
0.25
0.5
,
f
¯
x
i
=
tanh
⁡
x
i
1
tanh
⁡
x
i
2
.Then, through adopting the pinning controller in (7), (27)
u
i
t
=
-
1.5
G
x
i
t
-
s
l
t
-
0.25
H
x
i
t
-
τ
t
-
s
l
t
-
τ
t
,
i
=
m
l
,
l
=
1,2
,
0
,
i
≠
m
l
,with Ξ
=
d
i
a
g
(
0,1.5,0
,
1.5
) and Θ
=
d
i
a
g
(
0,0.25,0
,
0.25
), we assume that the desired cluster synchronization states of DNNs (25) are s
1
(
t
) and s
2
(
t
), which can satisfy(28)
s
˙
i
t
=
-
C
s
i
t
+
A
f
¯
s
i
t
+
B
f
¯
s
i
t
-
τ
t
,
i
=
1,2
,with different initial conditions. In order to reduce the number of controllers and realize the cluster synchronization, we can use the controlled networks sets {
m
1
,
m
2
}
=
{
2,4
} and, respectively, choose the configuration matrices as L
1
=
L
11
1
L
12
1
L
21
1
L
22
1 with (29)
L
11
1
=
-
1.5
1.5
1.5
-
1.5
,
L
12
1
=
-
1
1
-
1
1
,
L
21
1
=
0.5
-
0.5
0.5
-
0.5
,
L
22
1
=
-
2
2
2
-
2
.In what follows, two cases will be given to illustrate the efficiency and reduced conservatism of our results.Case 1.
Givenτ
0
=
0, choose three inner coupling matrices of diagonal form as (30)
G
=
0.5
0
0
0.5
,
H
=
0.25
0
0
0.25
,
K
=
0.25
0
0
0.25
.Then, through, respectively, setting μ
0
=
-
0.2
,
-
0.1, and unavailable μ
0, we can compute the corresponding maximum allowable upper bounds (MAUBs) in Table 1 based on Theorem 7 and Remark A.2 by resorting to Matlab LMI Toolbox.
In Table1, the term “—” means that the corresponding value is unavailable. Based on the MAUBs in Table 1, one can verify that our results can be less conservative than some existent ones. In particular, as the inner coupling matrices G, H, and K are not diagonal, our theorems still can be applicable while [30, 31] fail.Table 1
The calculated MAUBsτ
m
a
x by setting τ
0
=
0,  μ
0
=
-
0.2
,
-
0.1, and unavailable μ
0.
Methods
μ
m
0.4
0.6
0.8
1.2
Li and Cao [30]
-
0.2
1.112
1.035
1.004
—
Wang et al. [31]
1.154
1.074
1.048
0.987
Theorem7
1.243
1.126
1.100
1.022
Li and Cao [30]
-
0.1
1.112
1.035
1.004
—
Wang et al. [31]
1.165
1.084
1.077
0.996
Theorem7
1.263
1.132
1.127
1.065
Li and Cao [30]
—
1.112
1.035
1.004
—
Wang et al. [31]
1.127
1.046
1.035
0.945
Theorem8
1.195
1.094
1.067
0.996Case 2.
Choosingτ
(
t
)
=
0.2
+
0.8
sin
2
⁡
(
2
t
) and the inner coupling matrices G, H, and K as Case 1, one can derive τ
0
=
0.2, τ
m
=
1.0, μ
0
=
-
1.6, and μ
m
=
1.6. Owing to the fact that μ
m
>
1 and G, H, and K are not diagonal, the methods in [30, 31] fail to verify the synchronization. Yet, by resorting to Matlab LMI Toolbox, Theorem 7 can guarantee the pinning cluster synchronization and the feasible solution to LMIs in (17)-(18) can be obtained as follows: (31)
P
1
=
0.562
0.121
0.121
0.532
,
P
2
=
0.343
0.022
0.022
0.346
,
P
3
=
1.206
0.202
0.202
0.987
,
P
4
=
1.633
0.193
0.193
0.857
,
Q
1
=
0.002
0.001
0.001
0.001
,
Q
2
=
0.005
0.001
0.001
0.012
,
Q
3
=
1.214
0.314
0.314
0.685
,
R
1
=
1.073
0.511
0.511
1.034
,
R
2
=
1.428
0.544
0.544
1.428
,
R
3
=
1.224
0.124
0.124
0.985
,
S
1
=
0.335
0.050
0.132
0.325
,
S
2
=
0.544
0.224
0.327
0.675
,
S
3
=
0.220
0.091
0.068
0.202
,
M
1
=
1.304
-
0.101
-
0.114
0.132
,
M
2
=
1.142
1.102
0.786
0.132
,
N
1
=
0.562
0.121
0.121
0.867
,
N
2
=
0.445
0.342
0.342
0.766
,
P
=
0.154
0.105
0.105
0.445
,
E
=
0.202
I
2
,
F
=
0.175
I
2
,
U
=
0.006
I
2
,
V
=
0.010
I
2
.Example 2.
In this example, we consider the well-known Chua’s circuit to illustrate our synchronization results, which can be expressed as(32)
x
˙
1
=
α
x
2
-
m
1
x
1
+
h
x
1
,
x
˙
2
=
x
1
-
x
2
+
x
3
,
x
˙
3
=
-
β
x
2with h
(
x
1
)
=
1
/
2
m
1
-
m
0
x
1
+
c
-
x
1
-
c and the parameters m
0
=
-
1
/
7, m
1
=
2
/
7, α
=
9, β
=
14.286, and c
=
1. Then, the circuit model can be represented as the Lur’e system: (33)
x
˙
t
=
-
18
7
9
0
1
-
1
1
0
-
14.286
0
x
t
+
27
7
0
0
0
0
0
0
0
0
f
¯
W
T
x
twith W
=
1
0
0
T and f
¯
1
(
ξ
)
=
1
/
2
ξ
+
1
-
ξ
-
1 belonging to the sector [
0,1
]. Now we consider the cluster synchronization of the dynamical networks with each node being a 2-dimensional system (33) with linear coupling as (34)
x
˙
i
t
=
-
C
x
i
t
+
A
f
¯
W
T
x
i
t
+
∑
j
=
1
4
l
i
j
1
G
x
j
t
+
∑
j
=
1
4
l
i
j
2
H
x
j
t
-
τ
t
+
u
i
t
.In order to reduce the number of controllers and realize the cluster synchronization, we adopt the pinning controller (7) as (35)
u
i
t
=
-
1.5
G
x
i
t
-
s
l
t
,
i
=
m
l
,
l
=
1,2
,
0
,
i
≠
m
l and the desired cluster synchronization states of (A.15) satisfy (36)
s
˙
i
t
=
-
C
s
i
t
+
A
f
¯
W
T
s
i
t
,
i
=
1,2with the initial conditions s
1
(
0
)
=
0.4
-
0.5 and s
2
(
0
)
=
-
0.2
0.3. Now through setting Ξ
=
d
i
a
g
(
0,1.5,0
,
1.5
), G
=
1.5
0.2
0.2
1.5, and the configuration matrices as L
1
=
L
11
1
L
12
1
L
21
1
L
22
1 with (37)
L
11
1
=
-
1.5
1.5
1.5
-
1.5
,
L
12
1
=
-
1
1
-
1
1
,
L
21
1
=
0
0
0
0
,
L
22
1
=
-
2
2
2
-
2
,together with Theorem 8 and LMI in Matlab Toolbox, we can easily verify that network (34) can achieve the desired cluster synchronization, which can be further supported by the synchronization error states in Figure 1.Figure 1
Phase and state trajectories of the error states.


## 5. Conclusions

This paper has investigated the problem on pinning cluster synchronization for delayed dynamical networks with linearly hybrid coupling. Two novel conditions have been derived by employing the Lyapunov-Krasovskii stability theory. It is worth pointing out that some most recently developed techniques such as combined convex technique and triple integral LKF terms have been employed, which can help extend the application areas. The synchronization criteria are presented in the forms of LMIs, which can be checked easily by referring to Matlab LMI Toolbox. Finally, two numerical examples can illustrate the less conservatism of our theorems based on some comparing results.



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*Source: 6021497-2016-06-09.xml*