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This paper is concerned with the delay-dependent synchronization criterion for neutral-type stochastic delayed complex networks. Firstly, expectations of stochastic crossterms containing the Itô integral are investigated. In fact, for stochastic delay systems, if we want to obtain the delay-dependent condition with less conservatism, how to deal with expectations of stochastic cross terms properly is of vital importance, and many existing results did not deal with expectations of these stochastic cross terms correctly. Then, based on this, this paper establishes a novel delay-dependent synchronization criterion for neutral-type stochastic delayed complex networks. In the derivation process, the mathematical development avoids bounding stochastic cross terms. Thus, this method shows less conservatism. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed approach.

In the real world, many systems can be described as complex networks such as Internet networks, biological networks, epidemic spreading networks, collaborative networks, social networks, neural networks, and so forth [

Up to now, it has been well realized that in spreading information through complex networks, there always exist time delays caused by the finite speed of information transmission and the limit of bandwidth, which often decrease the quality of the system and even lead to oscillation, divergence, and instability. Accordingly, synchronization problems for many delayed complex networks have been studied in [

On the other hand, in the real world, complex networks are often subject to stochastic disturbances. For example, the signal transfer in a real complex network could be perturbed randomly from the release of probabilistic causes such as neurotransmitters and packet dropouts [

Moreover, for delay systems including delayed complex networks, a very active research topic is to obtain the delay-dependent conditions. The reason is that the delay-dependent condition makes use of the information on the size of time delays, and the delay-dependent condition is generally less conservative than the delay-independent one [

Motivated by the discussion mentioned above, this paper investigates the delay-dependent synchronization problem for neutral-type stochastic delayed complex networks. The main contributions of this paper are summarized as follows. (1) Expectations of stochastic cross terms containing the Itô integral are investigated by stochastic analysis techniques in Lemma

In this paper, we consider the following neutral-type stochastic delayed complex networks consisting of

The initial conditions associated with system (

Let

The neutral-type stochastic delayed complex network (

If a stochastic process

Let

The outer-coupling configuration matrices of the complex networks (

The noise intensity function vector

For all

The Kronecker product has the following properties:

Let

Let

Then, we give the following lemma and corollary which will play a key role in the proof of our main results.

If a stochastic process

Firstly, in order to prove the above results, we will prove that if

Let one consider the following neutral stochastic functional differential equation:

Since

Lemma

Consider the following one-dimensional Langevin equation in [

Consider the following one-dimensional stochastic equation:

Then, we are in the position to present our main result for the synchronization criterion of the neutral-type delayed complex networks with stochastic disturbances.

Under the Assumptions

Firstly, set

We note here that if

If we do not consider the stochastic disturbances in (

For neutral stochastic delay systems, a very active topic is to obtain the delay-dependent condition. For example, [

In Theorem

In this section, we present a simulation example to illustrate the effectiveness of our approach.

Consider the following complex network consisting of three identical nodes:

State error of

State error of

This paper has investigated the problem of delay-dependent synchronization criterion for neutral-type stochastic delayed complex networks. Most important of all, this paper is concerned with expectations of stochastic cross terms containing the Itô integral. By stochastic analysis techniques, we prove that among these stochastic cross terms,

The work of Y. Zhang and B. Song was supported by the National Natural Science Foundation of China under Grants no. 61104221 and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant no. 10KJB120004. The work of J. H. Park was supported by 2012 Yeungnam University Research Grant. The work of Z.-G. Wu was supported by the National Natural Science Foundation of China under Grant no. 61174029.

_{∞}-control of uncertain neutral stochastic systems with time delay

_{∞}control for nonlinear stochastic systems with Markovian jump parameters and interval time-varying delays