"通过固定控制将网络推向所需轨道：基于钉扎控制的同步稳定性分析"

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he development of a novel approach for analyzing pinning synchronization stability in complex delayed dynamical networks with directed coupling. The concept of pinning control, which involves driving a network to a desired orbit by fixing a small fraction of its nodes, is integral to the study. The study conducted by Quanjun Wu and Hua Zhang in the cybernetics journal focused on deriving simple and generic criteria for pinning such coupled networks analytically. This work represents a significant advancement in the field, as it provides a more accessible and practical method for achieving synchronization in complex network systems. Pinning control has been recognized as a powerful strategy for achieving synchronization in complex networks, where each node's dynamics are interconnected and influenced by the behavior of its neighbors. By fixing a small number of nodes in the network to specific values, one can effectively drive the entire network towards a desired state or orbit. The key contribution of this research is the development of analytical criteria for pinning synchronization stability in complex delayed dynamical networks with directed coupling. These criteria provide a clear and systematic way to determine the optimal nodes that need to be fixed in order to achieve synchronization within the network. By applying the principles of pinning control, researchers and practitioners can effectively manage and manipulate the dynamics of complex network systems. This has wide-ranging implications across various fields, including communication networks, social networks, biological systems, and more. Overall, the study presented in the cybernetics journal represents a significant step forward in the understanding and application of pinning control in network synchronization. The analytical criteria derived by Wu and Zhang provide a solid foundation for further research and practical implementation of pinning strategies in complex network systems.

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

(t), . . . , x

(t))

∈ R

, A ∈ R

n×n

, and the vector-valued function

g(t, x(t), x(t −τ (t))) = [g

(t, x(t), x(t −τ(t))), . . ., g

(t, x(t), x(t −τ(t)))]

∈ R

satisﬁes

uniform Lipschitz condition with respect to the time t, i. e.,

) For any x(t) = (x

(t), . . . , x

(t))

∈ R

, y(t) = (y

(t), . . . , y

(t))

∈ R

, there

exist constants k

> 0, and l

> 0 satisfying



(t, x(t), x(t − τ (t))) − g

(t, y(t), y(t − τ(t)))



≤

j=1





(t) − y

(t)| + l

(t − τ (t)) − y

(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→+∞

(t) − s(t)k = 0, s(t) =

k=1

(t), i = 1, 2 . . . , N, (5)

where x

(t) (i = 1, 2, . . . , N) are the solutions of the delayed dynamical network (1), and

(k = 1, 2, . . . , N) can be chosen such that

k=1

= 1 and ξ

> 0 for the purpose of

practical control strategy [3, 42]. In this case, the controlled network can be described

˙x

(t) = Ax

(t) + g(t, x

(t), x

(t − τ (t))) + c

j=1

(t)Γ(t)x

(t) + U

, i = 1, . . . , N.

(6)

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