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The problem of stochastic synchronization
of neutral-type neural networks with multidelays based on

In recent years, neutral-type systems have been intensively studied due to the cause that many practical processes can be modeled as general neutral-type descriptor systems, such as computer aided design, circuit analysis, chemical process simulation, power systems, real time simulation of mechanical systems, population dynamics, and automatic control (see, e.g., [

On the one hand, time-delays as a source of instability and oscillators always appear in various aspects of neural networks. Recently, the stability of neural networks with time-delays has received lots of attention, such as [

On the other hand, systems with Markov jump parameters, driven by continuous-time Markov chain, have been widely used to model many practical systems where they may experience abrupt changes in their structure and parameters. For example, in paper [

Meanwhile, the stability and synchronization of neutral-type systems which depend on the delays of state and state derivative have attracted a lot of attention (see [

Inspired by the above discussions, in this paper, we are concerned with the analysis issue for the problem of stochastic synchronization of neutral-type neural networks with multidelays and Markovian switching. By using

Consider

For system (

For drive system (

The initial data is given by

For error system (

Each function

For the external input matrix

We now begin with the following concept of stochastic synchronization.

Neutral-type response neural networks (

Now, we describe the problem to solve in this paper as follows.

The following lemmas are useful for obtaining the main result.

Let

For any positive definite matrix

If

Every real eigenvalue of

We are now in a position to derive a condition under which neutral-type multiple time-delay neural networks (

Let Assumptions

Let

Assume also that

We choose the feedback control

Then neutral-type multiple time-delays neural networks (

Fix any

From Lemmas

From control law (

From Assumption

Finally,

Substituting (

Integrating and taking the mathematical expectation on both sides of (

In Theorem

From the analysis of Remark

When the multidelays turn to single delay and the neutral term disappears in the neural networks, respectively, we have the following two special cases of system (

Accordingly we can derive the two corollaries of Theorem

Let Assumptions

Let

Assume also that

We choose the feedback control

Then the response system can synchronize with the drive system of neutral-type.

Let Assumptions

Let

Assume also that

We choose the feedback control

Then the response system can synchronize with the drive system of multiple time-delays.

One example is presented here in order to show the usefulness of our results. Our aim is to examine the stochastic synchronization for the given neutral-type multiple time-delays neural networks with stochastic noise and Markovian switching.

Consider the error system of two-neuron delayed neural network (

The transition rate matrix of Markovian switching is given by

Then we choose

The other parameters are given as follows:

Given

It can be easily verified that

2-state Markov chain.

State trajectory of two systems.

State trajectory of the error system.

Update law.

For Markovian switching with known transition rate, we can choose reasonable parameter

In this paper, we have dealt with the problem of stochastic synchronization of neutral-type neural networks with multidelays and Markovian switching. By using

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant no. 20120075120009 and the Natural Science Foundation of Shanghai under Grant no. 12ZR1440200.