The exponential synchronization issue for stochastic neural networks (SNNs) with mixed time delays and Markovian jump parameters using sampled-data controller is investigated. Based on a novel Lyapunov-Krasovskii functional, stochastic analysis theory, and linear matrix inequality (LMI) approach, we derived some novel sufficient conditions that guarantee that the master systems exponentially synchronize with the slave systems. The design method of the desired sampled-data controller is also proposed. To reflect the most dynamical behaviors of the system, both Markovian jump parameters and stochastic disturbance are considered, where stochastic disturbances are given in the form of a Brownian motion. The results obtained in this paper are a little conservative comparing the previous results in the literature. Finally, two numerical examples are given to illustrate the effectiveness of the proposed methods.
Neural networks, such as Hopfield neural networks, cellular neural networks, the Cohen-Grossberg neural networks, and bidirectional associative neural networks, are very important nonlinear circuit networks and, in the past few decades, have been extensively studied due to their potential applications in classification, signal and image processing, parallel computing, associate memories, optimization, cryptography, and so forth; see [
It has been widely reported that a neural network sometimes has finite modes that switch from one mode to another at different times; such a switching (jumping) signal between different neural network models can be governed by a Markovian chain; see [
It is well known that noise disturbance widely exists in biological networks due to environmental uncertainties, which is a major source of instability and can lead to poor performances in neural networks. Such systems are described by stochastic differential systems which have been used efficiently in modeling many practical problems that arise in the fields of engineering, physics, and science as well. Therefore, the theory of stochastic differential equation is also attracting much attention in recent years and many results have been reported in the literature [
On the other hand, as the rapid development of computer hardware, the sampled-data control technology has shown superiority over other control approaches because it is difficult to guarantee that the state variables transmitted to controllers are continuous in many real-world applications. In [
Motivated by the above discussion, in this paper we study the delay-dependent exponential synchronization of neural networks with stochastic perturbation, discrete and distributed time-varying delays, and Markovian jump parameters. Here, it should be mentioned that our results are delay dependent, which depend on not only the upper bounds of time delays but also their lower bounds. Moreover, the derivatives of time delays are not necessarily zero or smaller than one since several free matrices are introduced in our results. By constructing an appropriate Lyapunov-Krasovskii functional based on delay partitioning, several improved delay-dependent criteria are developed to achieve the exponential synchronization in mean square in terms of linear matrix inequalities. Two numerical examples are also provided to demonstrate the advantage of the theoretical results.
The rest of this paper is organized as follows. In Section
Let
The probability defined in (
Fix a probability space
Throughout this paper, we make the following assumptions. There exist positive constants Each activation function
where
In the earlier literature, the activation functions
In this paper, we consider system (
In order to investigate the problem of exponential synchronization between systems (
The control signal is assumed to be generated by using a zero-order-hold function with a sequence of hold times
By substituting (
For convenience, in the following, each possible value of
The first purpose of this paper is to design a controller with the form (
As mentioned earlier, it is often the case in practice that the neural network is disturbed by environmental noises that affect the stability of the equilibrium. Motivated by this we express a stochastic system whose consequent parts are a set of stochastic uncertain recurrent neural networks with mixed time delays:
And a slave system for (
The mode-independent state feedback controller is made as the form of (
We impose the following assumption:
Our second purpose of this paper is to find a feedback gain matric
To state our main results, the following definition and lemmas are first introduced, which are essential for the proof in the sequel.
Master system and slave system are said to be exponentially synchronous if error system is exponentially stable; that is, for any initial condition
For any constant matrix
Given one positive definite matrix
For any constant matric
To present the main results of this section, we denote
Here, some LMI conditions will be developed to ensure that master system (
Under Assumptions
are satisfied, where
Denote
Let
Note that, when
Therefore,
Inspired by the free-weighting matric approach [
From (
Furthermore, according to error system (
On the other hand, we have from (
Similarly, for any appropriately dimensioned diagonal matrices
Adding the left-hand sides of (
where
According to Lemma
From (
Thus, we can show from (
From the definition of
Define a new function
Due to the fact that
On the other hand, from (
By using Dynkin's formula, for
Consequently, by changing the integration sequence, the following inequalities hold:
After substituting (
So,
Consequently, according to the Lyapunov-Krasovskii stability theory and Definition
In this section, some sufficient conditions of exponential synchronization for stochastic error system (
Under Assumptions
Let
To analyze the stability of error system (
Let
So, we have that
It is easy to see from (
Taking the mathematical expectation on both sides of (
Applying Lemma
Similarly, it can be seen that there exist matrices
Furthermore, according to the definition of
Taking the mathematical expectation on both sides of (
Applying Lemma
Following the similar line of the proof of Theorem
Thus, the master system and the slave system are exponentially synchronized; the sampled-data feedback control gain is given by
In this section, two numerical examples are given to demonstrate the effectiveness of the theoretical results.
Consider the second-order master system (
It can be verified that
It is assumed that
The chaotic behaviors of the master system (
Choosing
Thus, the corresponding gain matrix in (
Under the obtained gain matrix in (
Chaotic behavior of master system (
Chaotic behavior of slave system (
Control input
State responses of error system (
In the following, we consider the second-order stochastic master system (
For the two operating conditions (modes), the associated data are
It can be verified that
In this example,
Figures
By using Matlab LMI Control Toolbox to solve the LMIs given in Theorem
Thus, the corresponding gain matrix in (
Under the above given gain matrix, Figures
Chaotic behavior of master system (
Chaotic behavior of slave system (
Control input
State responses of error system (
In this paper, the exponential synchronization issue for stochastic neural networks (SNNs) with mixed time delays and Markovian jump parameters under sampled-data control has been addressed. New delay-dependent conditions have been presented in terms of LMIs to ensure the exponential stability of the considered error systems, and, thus, the master systems exponentially synchronize with the slave systems. The results obtained in this paper are a little conservative comparing the previous results in the literature. The methods of this paper can be applied to other classes of neural networks such as complex neural networks and impulsive neural networks.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Science and Technology Major Project of China (2011ZX05020-006) and the Natural Science Foundation of Hebei Province of China (A2011203103).