154 Q.J. WU AND H. ZHANG
a symmetrical or irreducible matrix, and the inner connecting matrix Γ is not assumed
to be diagonal or positive definite. Obviously, the network (1) is a generalization of the
model discussed in [5, 10, 27, 28]. As usual, the model (1) is also called as the general
time-varying c omplex delayed dynamical network.
Next we consider an isolated dynamical node in the model (1), which is described by
the following form of n-dimensional differential equations with time-varying delays [3]:
˙x(t) = f(t, x(t), x(t − τ(t))) = Ax(t) + g(t, x(t), x(t − τ(t))), (3)
in which x(t) = (x
1
(t), . . . , x
n
(t))
>
∈ R
n
, A ∈ R
n×n
, and the vector-valued function
g(t, x(t), x(t −τ (t))) = [g
1
(t, x(t), x(t −τ(t))), . . ., g
n
(t, x(t), x(t −τ(t)))]
>
∈ R
n
satisfies
uniform Lipschitz condition with respect to the time t, i. e.,
(A
1
) For any x(t) = (x
1
(t), . . . , x
n
(t))
>
∈ R
n
, y(t) = (y
1
(t), . . . , y
n
(t))
>
∈ R
n
, there
exist constants k
ij
> 0, and l
ij
> 0 satisfying
g
i
(t, x(t), x(t − τ (t))) − g
i
(t, y(t), y(t − τ(t)))
≤
n
X
j=1
k
ij
x
j
(t) − y
j
(t)| + l
ij
|x
j
(t − τ (t)) − y
j
(t − τ (t))
, i = 1, . . . , n.
(4)
Remark 2.3. In general, we are mainly interested in control and synchronization of
complex networks compos ing of coupled dissipative dynamical systems, where the un-
coupled dissipative dynamical system may possess an equilibrium point, a periodic orbit,
or even a chaotic attractors [29]. Therefore, we always assume that all the solutions of
(3) are bounded. And it is not diffic ult to check that the class of systems in the form of
Eqs. (3) – (4) includes almost all the well-known chaotic systems without or with time
delays such as the Lorenz system, Chen system, Chua’s circuit, R¨ossler system as well
as the delayed Hopfied neural networks, delayed Mackey–Glass s ystem or delayed Ikeda
equations and delayed cellular neural networks (CNNs), and so on (see [3, 42] and their
references).
Now we shall address pinning synchronization in complex delayed dynamical network
(1) by feedback controllers. The primary objective here is to design and implement an
appropriate pinning strategy on a small fraction ε (0 < ε 1) of the nodes in network
(1) such that all the states of the controlled network will exponentially synchronize to a
desired synchronization state s(t), which is presented in terms of a weighted average of
all the states in the network. More concretely, this is to say,
lim
t→+∞
kx
i
(t) − s(t)k = 0, s(t) =
N
X
k=1
ξ
k
x
k
(t), i = 1, 2 . . . , N, (5)
where x
i
(t) (i = 1, 2, . . . , N) are the solutions of the delayed dynamical network (1), and
ξ
k
(k = 1, 2, . . . , N) can be chosen such that
P
N
k=1
ξ
k
= 1 and ξ
k
> 0 for the purpose of
practical control strategy [3, 42]. In this case, the controlled network can be described
as
˙x
i
(t) = Ax
i
(t) + g(t, x
i
(t), x
i
(t − τ (t))) + c
N
X
j=1
b
ij
(t)Γ(t)x
j
(t) + U
i
, i = 1, . . . , N.
(6)
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