Almost Sure Asymptotical Adaptive Synchronization for Neutral-Type Neural Networks with Stochastic Perturbation and Markovian Switching

The problem of almost sure (a.s.) asymptotic adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching is researched. Firstly, we proposed a new criterion of a.s. asymptotic stability for a general neutral-type stochastic differential equation which extends the existing results. Secondly, based upon this stability criterion, by making use of Lyapunov functional method and designing an adaptive controller, we obtained a condition of a.s. asymptotic adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching. 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
As it is well known, the stability and synchronization of neural networks can be applied to create chemical and biological systems, secure communication systems, information science, image processing, and so on.In recent years, different control methods are derived to achieve different synchronization, such as randomly occurring control [1], sampled-data control [2,3], passivity analysis [4], impulsive control [5][6][7][8], and adaptive control [9].
By utilizing adaptive control method, the parameters of the system need to be estimated and the control law needs to be updated when the neural networks evolve.In the past decade, much attention has been devoted to the research of the adaptive synchronization for neural networks.In [9,10], the adaptive lag synchronization of unknown chaotic neural networks is considered.Adaptive synchronization problem of delayed neural networks with stochastic perturbation is studied in [11].Besides these, there are many literatures to study adaptive synchronization problems (see, e.g., [12,13] and the references therein).
Recently, the stability and synchronization of neutraltype systems, specially neutral-type neural networks, which depend on the derivative of the state and the delay state have attracted a lot of attention (see, e.g., [14][15][16][17][18][19] and the references therein) due to the fact that some physical systems in the real world can be described by neutral-type models (see [20]).However, the adaptive control was not investigated in [14][15][16][17], and the neutral term of derivative of the delay state was not taken into account in the neural networks proposed in [9][10][11][12][13].Zhou et al. in [18] did not study the almost sure (a.s.) synchronization for neutral-type neural networks.Zhu et al. in [19] did not research the synchronization problem for neural networks with Markovian switching parameters.From the authors' best knowledge, so far the almost surely adaptive synchronization problem for neutral-type neural networks with stochastic perturbation and Markovian switching parameters has not been fully investigated yet.This motivates our current work.
In this paper, the problem of almost sure (a.s.) asymptotic adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching is researched.By making use of Lyapunov functional method and designing an adaptive controller, we obtained a condition of a.s.asymptotic adaptive synchronization for neutral-type 2 Mathematical Problems in Engineering neural networks with stochastic perturbation and Markovian switching.Finally, we introduced a numerical example to illustrate the effectiveness of the method and result obtained in this paper.The main contributions of this paper are as follows.
(1) A new model for a class of neutral-type neural networks with stochastic perturbation and Markovian switching is given; it is more general than other models.
(2) A new criterion of a.s.asymptotic stability for a general neutral-type stochastic differential equation is proposed which extends the existing results.
The notations are quite standard.Throughout this paper, R + , R  , and R × denote the set of nonnegative real numbers, -dimensional Euclidean space, and the set of all  ×  real matrices, respectively.The superscript  denotes matrix transposition, trace(⋅) denotes the trace of the corresponding matrix, and  denotes the identity matrix.| ⋅ | stands for the Euclidean norm in R  .diag{⋅ ⋅ ⋅ } stands for the block diagonal matrix.Let (Ω, F, P) be a complete probability space with a filtration {F  } ≥0 satisfying the usual conditions (i.e., the filtration contains all -null sets and is increasing and right continuous).For  > 0, denote by

Problem Formulation and Preliminaries
Let {()} ≥0 be a right-continuous Markov chain on the probability space taking values in a finite state space  = {1, 2, . . ., } with generator Γ = (  ) × given by where  > 0 and   ≥ 0 is the transition rate from  to  if  ̸ =  while Consider the following neutral-type neural networks called drive system and represented by the compact form as follows:
The initial condition of system (5) is given in the following form: for any   ∈ L 2 F 0 ([−, 0]; R  ).Let () = () − () be the synchronization error vector.From the drive system and the response system, the error system can be written as follows: where The initial condition of system ( 7) is given in the following form: with (0) = 0.The primary object here is to deal with the adaptive synchronization problem of the drive system (3) and the response system (5) and derive sufficient conditions such that the response system (5) synchronizes with the drive system (3).
To prove our main results, the following assumptions are needed.
Assumption 3.For the external input matrix   ( ∈ ), there exists positive constant   ∈ (0, 1), such that where  = max ∈   and (  ) is the spectral radius of matrix   .
The following concept is necessary in this paper.
If the error system ( 7) is almost surely asymptotically stable, then the drive system (3) and the response system ( 5) are said to be almost surely asymptotically synchronization.
Consider the more general neutral-type stochastic delay differential equation (NSDDE) with Markovian switching: where () is an -dimensional Brownian motion defined on the probability space (Ω, F, P) but independent of the Markov chain {()} ≥0 and are all Borel-measurable functions.
For NSDDE (13), the following hypothesis is needed.
Then, we present some preliminary lemmas which play an important role in the proof of the main results.
Now we cite the convergence theorem of nonnegative semimartingales (see [22], Theorem 7 on page 139) which is a useful lemma.

Main Results
In this section, we give some criteria of adaptive synchronization for the drive system (3) and the response system (5).First, we establish a general result which can be applied widely.
The proof of this theorem is given in the Appendix.
Remark 13.From the proof of Theorem 11, we can see that if condition (H1) is substituted by (H1)  , then the conclusion (R2) is also true.
Remark 15.In this section, a numerical example will be given to support the main results obtained in this paper.

Numerical Examples
In this section, a numerical example will be given to support the main results obtained in this paper.
To illustrate the effectiveness of the result in this paper, we depict the evolution figures of the systems as Figures 1, 2, 3, and 4. Figure 1 shows the two-state Markov chain in the systems.Figure 2 shows that the drive system (3) synchronizes the response system (5) from the moment of  = 7.It can be seen from Figure 3 that the state of the error system (7) tends to zero from  = 7, which also describes the synchronization of the drive system (3) and the response system (5).The update law of the adaptive control gain () is depicted in Figure 4. Figure 4 shows us that the update law of the control gain () no longer varies after the response system (5) synchronizes with the drive system (3).

Conclusions
In this paper, we have proposed a new criterion of a.s.asymptotic stability for a general neutral-type stochastic differential equation which extends the existing results.Based upon this new stability criterion, we have obtained a condition of a.s.asymptotic adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian   switching by making use of Lyapunov functional method and designing an adaptive controller.The synchronization condition is expressed as linear matrix inequality which can be easily solved by Matlab.Finally, we have employed a numerical example to illustrate the effectiveness of the method and result obtained in this paper.In the future, we will consider the condition of a.s.asymptotic adaptive synchronization for neutral-type neural networks with timevarying delay by making use of M-matrix method.

Appendix
Proof.The proof of (R1) is the same as [16] and is omitted here.To prove (R2), we will divide it into five steps.We change  into  in subsequence for simplicity.

Figure 1 :
Figure 1: The varying curve of Markov chain with 2 states.

Figure 2 :
Figure 2: The dynamic trajectory of the drive system and the response system.

Figure 3 :
Figure 3: The trajectory of the error state.

Figure 4 :
Figure 4: The dynamic curve of the update law of the gain ().