Stochastic Synchronization of Neutral-Type Neural Networks with Multidelays Based on M-Matrix

The problem of stochastic synchronization of neutral-type neural networks with multidelays based on M-matrix is researched. Firstly, we designed a control law of stochastic synchronization of the neural-type and multiple time-delays neural network. Secondly, by making use of Lyapunov functional and M-matrix method, we obtained a criterion under which the drive and response neutral-type multiple time-delays neural networks with stochastic disturbance and Markovian switching are stochastic synchronization. The synchronization condition is expressed as linear matrix inequality which can be easily solved by MATLAB. Finally, we introduced a numerical example to illustrate the effectiveness of the method and result obtained in this paper.


Introduction
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., [1][2][3][4][5][6] and the references therein).For example, in [1], the author studied the stability and asymptotic properties of a class of neutral-type functional differential equations based on the pattern equation method.In [2], the author investigated the asymptotic stability properties of neutraltype systems in Hilbert space.
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 [7,8], and many methods, such as the linear matrix inequality (LMI) approach and -matrix approach, have been adopted by the scholars; see, for example, [9,10].
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 [11], a general model of an array of N linearly coupled delayed neural networks with Markovian jumping hybrid coupling is researched.In paper [12], the author researched the feedback control problem for a class of linear systems with Markovian jump parameters.The stabilization of stochastic delayed neural networks with Markovian switching was discussed in paper [13][14][15][16].
Meanwhile, the stability and synchronization of neutraltype systems which depend on the delays of state and state derivative have attracted a lot of attention (see [17][18][19][20][21][22] and the references therein) due to the fact that some physical systems in the real world can be described by neutral-type models.Besides the above these, according to [23,24], matrix approach can not only design feed controller and trace all information of Markovian switching parameters but also has lower complexity than that of LMIs technology.
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 matrix approach and the stochastic analysis method, some synchronization criteria are obtained to ensure the stochastic synchronization for the neutral-type neural networks with multidelays.A numerical example is provided to illustrate the effectiveness of the results obtained in this paper.The main contributions of this paper are twofold: (1) Stochastic synchronization for a new class of neutral-type neural networks with multidelays and Markovian switching is considered.
(2) The theory results which are obtained by -method approach are more practical than that of LMIs technology.
For drive system (1) and response system (3), we can obtain the error system as follows: where () = (()) − (()) and The initial data is given by {( For error system (4), we impose the following assumptions.
Assumption 3.For the external input matrix   ( ∈ S), there exists positive constant   ∈ (0, 1), such that where  = max ∈S   and (  ) is the spectral radius of matrix   .
We now begin with the following concept of stochastic synchronization.
Definition 4. Neutral-type response neural networks (3) are said to be stochastic synchronized with drive neural network (1) if, for any where (;  0 , ()) is the solution of system (4) for the initial condition ().
Now, we describe the problem to solve in this paper as follows.
Target Description.For neutral-type and multiple time-delays neural networks ( 1) and ( 3) with stochastic disturbance and Markovian switching, by using Lyapunov functional, general Itô's formula, and -matrix method, this paper will design a control law and obtain some criteria of stochastic synchronization.
The following lemmas are useful for obtaining the main result.
(ii) Every real eigenvalue of  is positive.

Main Results
We are now in a position to derive a condition under which neutral-type multiple time-delay neural networks (3) with stochastic disturbance and Markovian switching are stochastic synchronized with drive system (1).We have the following result.Assume also that where  = min ∈S min 1≤≤    ,  = max ∈S max 1≤≤    ,  = min ∈S   ,   =  2 , and   = (Φ 2 +   )/(1 − τ).
We choose the feedback control   with the update law as with where   > 0 ( = 1, 2, . . ., ) are arbitrary constants.Then neutral-type multiple time-delays neural networks (3) with stochastic disturbance and Markovian switching can be stochastic synchronized with drive system (1).
Remark 10.From the analysis of Remark 9, we can also obtain that if feedback control update law (12) has been designed, because  := − diag{, , . . .,  ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟  }−Γ, using -matrix method, Markovian rate generator Γ can be checked.This also reflects that all information of Markovian switching parameters is traced.
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 (4).Assume also that
We choose the feedback control   with the update law as with where   > 0 ( = 1, 2, . . ., ) are arbitrary constants.Then the response system can synchronize with the drive system of multiple time-delays.

Numerical Simulation
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 timedelays neural networks with stochastic noise and Markovian switching.
The transition rate matrix of Markovian switching is given by (37) Then we choose Φ = 0.25,  = 0.25, and   = 0 so that Assumptions 1 and 2 can be satisfied.
The other parameters are given as follows: It can be easily verified that  is -matrix.We then derived  = 1.0078 and  = 0.0509 > 0. Hence it follows from Theorem 8 that drive system (1) and response system (3) are stochastic synchronization.Figures 1-4 show the 2state Markov chain, the state trajectory of the drive system and response system, the state trajectory and evolution of the error system, and the update law of the control gain matrix, respectively.We can see from Figure 3 that the system state tends to zero with the increase of , which verifies the synchronization of the drive system and system.Remark 13.For Markovian switching with known transition rate, we can choose reasonable parameter  to realize  := − diag{, , . . .,  ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟  } − Γ to become -matrix.In simulation results, choosing  = 0.5, it is easily observed that  is matrix.Furthermore, we can get  and .From the above result, we can see that the analysis of -matrix approach in Remarks 9 and 10 is reasonable.

Conclusion
In this paper, we have dealt with the problem of stochastic synchronization of neutral-type neural networks with multidelays and Markovian switching.By using -matrix  approach and the stochastic analysis method, some synchronization criteria are obtained to ensure the stochastic synchronization for the neutral-type neural networks with multidelays.A numerical example is provided to illustrate the effectiveness of the result obtained in this paper.

Figure 3 :
Figure 3: State trajectory of the error system.