Some Further Results on Fixed-Time Synchronization of Neural Networks with Stochastic Perturbations

In this paper, fixed-time (FXT) synchronization issue of a type of neural networks (NNs) with stochastic perturbations is considered. First, we obtained some novel sufficient criteria to guarantee the FXT synchronization of considered networks via introducing two types of controllers and employing some inequality techniques. Lastly, our theoretical results are verified via giving two numerical examples with their Matlab simulations.


Introduction
In the last decades, the various types of neural networks (NNs) including Hopfield NNs, cellular NNs, convolution NNs, Cohen-Grossberg NNs, BAM NNs and so on [1] [2] [3], have been introduced and broadly investigated due to their important applications in great number of fields ranging from speech recognition [4] to image encryption [5], from secure communication [6] to robotic manipulators [7], etc. As pointed out in [8], stability of NNs is prerequisite in some applications. As a result, the stability analysis of NNs has been investigated extensively by many scholars [2] [3] [8] [9] [10] [11] [12]. Duo to the security reasons or just to improve system performance in many practical applications, it is desirable that the systems solution trajectories converge to equilibrium as fast as possible [13]. Compared to the classical Lyapunov stability such as asymptotical stability or exponential stability, finite-time (FNT) stability allows the solution of is called the settling time. Thus, FNT stability and stabilization of NNs have been investigated widely in the past two decades [10] [11] [12] [13] [14].
One of the important tasks in FNT stability is to estimate the settling time (ST) ( ) 0 T x , and it is desirable to obtain smaller upper-bound of ( ) 0 T x . However, in some cases, it is inconvenient to accurately estimate it due to its heavily dependence to the initial values of the system. So it is to better obtain FNT stability with a ST irreverent to initial conditions of the system. This issue was firstly studied by Polyakov [15] via defining a so-called fixed-time (FXT) stability which its ST is independent to systems initial conditions. Presently, FXT stability receives a hot research attention from many scholars since it has awesome applications in multi-agent systems [16], power systems [17], complex networks [18] and so on.
In addition, the synchronization of chaotic nonlinear system has received great attention in the past thirty years due to the fact that synchronization is unique in nature and plays a crucial role in many fields including biology, climatology and sociology, etc. [19]. As mentioned in [9], synchronization in neuronal systems can produce a lot of physiological mechanisms of brain functions such as attention, learning, memory formation and so on. Thus, to understand these brain functions deeply, it is an important task to study synchronization behaviors of NNs. For this reason, considerable efforts have been devoted to study the synchronization of NNs [5] [6] [20]- [25]. Especially, due to the advantage of faster convergence rate, better robustness and disturbance rejection properties, FNT and FXT synchronization of various NNs have studied recently. For example, in [9], the authors investigated the FXT synchronization of coupled discontinuous NNs by introducing new FXT stability results for dynamical systems. In [11], the authors considered FNT stabilization issue of a class of delayed memristive NNs with discontinuous right-hand side by designing two types of discontinuous controllers. In [21], the authors concerned the FXT synchronization of a class of memristor-based NNs with impulsive effects. In [26], the authors studied the FNT synchronization of a type of complex-valued NNs with distributed delays. In [27], the authors studied the FXT synchronization problem of a type of quaternion-valued NN with time delays.
However, it is worthy to note that most of the above mentioned results have only considered cases without stochastic perturbations. As depicted in [28], noises are frequently encountered in both nature and man-made systems. For instance, synaptic transmission in the real nervous systems can be seen as a noisy process which is caused by random fluctuations due to the release of neurotransmitters and other probabilistic effects. Besides, for many natural renewable energy resources such as wind or solar radiation, their availability is somewhat subject to stochastic fluctuations [19]. Therefore, recently many scholars paid their attention to study the synchronization of NNs with stochastic perturba-tions, and till now there are many excellent results on the complete synchronization, lag synchronization, projective synchronization and FNT synchronization of stochastic NNs with or without time-delays. But, up to now, there are very few results on the FXT synchronization of stochastic NNs.
Inspired by what mentioned above, in this paper, we considered the FXT synchronization of a type of NNs with stochastic perturbations via using some

Problem Formulation and Preliminaries
Consider a class of n-dimensional stochastic NNs depicted by the following equation represents the state vector of the network  In the paper, we assume that the following assumptions are satisfied for the system (1). Assumption 1 The neuron activation functions i h in system (1) satisfy the Lipschitz condition. That is, for each i there exists a positive constant i L such that be the synchronization error between drive-response systems (1) and (2), then the error dynamical system can be derived as follows: De t Ag e t u t t t e t t σ ω . Furthermore, to obtain our main results, we give some related properties of stochastic perturbation, which can be found in [29].
Denote by ( ) . Now consider the following general stochastic nonlinear system: For convenience, we denote by ( ) ( ) 0 , z t z t z = the solution of stochastic nonlinear system (4) satisfy the initial value ( ) 0 0 z z = . Also, in order to get our main results in this part, we state here some needed definitions and lemmas as follows.
Definition 1 (FXT stable in probability [30]). The zero solution of stochastic system (4) is called to be FNT stable in probability, if the following conditions hold true.
1) FNT attractiveness in probability. That is, for any initial conditions ; 2) Stability in probability: For every pair of scalers 0 1 Definition 2 (FXT stable in probability [30]). The zero solution 0 z = of system (4) is said to be globally FXT stable in probability, if the following statements are satisfied for all the initial states 0 T z ω is independent on the initial state 0 z of (4) and its upper bound is bounded by a positive constant max T . That is, where M is positive constant. If the following inequality is satisfied for all then the zero solution 0 z = of system (4) is globally stochastically FXT stable in probability and the its ST function T z ω can be estimated as Lemma 2 [20]. For system (4), if there exists a positive definite function and positive numbers , , , , then the zero solution of system (4) is globally stochastically FXT stable in probability, and its ST ( ) T z ω can be estimated as is a positive definite function, and it satisfies the following conditions.
2) For any solution ( ) z t of system (4), following inequality hold true for some , 0, Then the zero solution of system (4) can achieve FXT stability, and its corresponding ST ( )  (4) is FXT stable and its settling-time , the zero solution of system (4) is FXT stable and its I z ρ ν stands for the incomplete beta function ratio for the zero solution of system (4) Then the zero solution of system (4) is FXT stable and its corresponding ST can be estimated as Lemma 7 [21]. If 1 2 , , ,  [22]. Let v and z be any two column vectors in m R , then the following matrix inequality is satisfied for any positive definite matrix

Main Results
In this section, based on the FXT stability results introduced in above section, we will derive some sufficient criteria for the FXT synchronization between the drive-response systems (1) and (2). To this, first we design the controller ( ) u t in response system (2) as follows:   following results can be derived.
Theorem 1. Suppose that the Assumptions 1 and 2 are satisfied, if the control gain matrix Λ satisfy the following matrix inequality ( ) where 1 Q is an arbitrary n n × positive matrix. Then the drive-response networks (1) and (2)  Proof. First, we construct the following Lyapunov function By Lemma 8 and Assumption 1, we obtain the following inequality: Let ( ) , and using the well-known Schur complement equivalence [32] to 0 Π > , which is defined in (15) In view of (22), (21) and (22), we can have Therefore, we can conclude from the Lemmas 3 and 4 that the origin of error system (3)   Proof. Similar to proof of Theorem 1, we know that the inequality (24) is satisfied under the conditions of Corollary 1. Thus from Lemma 6, we can obtain that the conclusions of Corollary 1 hold true. The proof is completed.  In the following, we will realize the fixed time synchronization between the systems (1) and (2) via designing a simplified controller given as follows where parameters , , p ρ ξ and q are the same as defined in controller (10). Theorem 2. Suppose that 2 Q is an arbitrary n n × positive matrix and the Assumptions 1 and 2 hold true, then the drive-response systems (1) and (2) will realize FXT synchronization in probability via controller (24  Proof. We again chose the Lyapunov function as ( ) ( ) ( ) Then, under controller (24), we have , .
Introducing (21) and (22) to (26) yields According to Lemma 4, the drive-response networks (1) and (2) will achieve FXT synchronization in probability. In addition, its ST 8 max T can estimate through following analysis. The proof of Theorem 2 is completed.  When 2 p q + = in the controller (24), we have a following result from Theorem 2 and Lemma 4. Journal of Applied Mathematics and Physics Corollary 2. Suppose that 2 p q + = in controller (10) and the Assumptions 1 and 2 are satisfied, then the drive-response systems (1) and (2) will achieve FXT synchronization in probability via controller (24)  Proof. From the proof of Theorem 2, we know that the inequality (28) is satisfied. Thus according to the Lemma 4, the drive-response networks (1) and (2) can be FXT synchronized in probability. In addition, its ST 9 max T can estimate through following analysis.
1) If 0 2 c ab < <  , then from Lemma 5, we can get that

Numerical Examples and Simulations
In this section, the following two numerical examples are provided to illustrate the effectiveness of the established theoretical results in above sections. Example 1. For n = 3, consider the FXT synchronization between drive-response systems (1) and (2) with the following system parameters: Set the initial values of system (1) in Example 1 as ( ) , then the numerical simulation of system (1) with above parameters are illustrated in Figure 1, which shows that it has a chaotic attractor.
It is not difficult check that the Assumptions 1 and 2 are satisfied with ( ) . Then, from Theorem 1 and Corollary 1, the derive system (1) is FXT stochastic synchronized to response system (2) under the controller (10). The time evolution of synchronization errors between systems (1) and (2) for above two different set of parameters are shown in Figure 2 and Figure 3, respectively, where the initial conditions of response systems (2) (1) and (2) under the controller (24) with the following system parameters: ( ) ( ) Now set the initial values of system (1)  (1) with above parameters are illustrated in Figure 4, which shows that it also has a chaotic attractor.
It is not difficult check that ( ) . Then, all the conditions of Theorem 2 and Corollary 2 are satisfied for case (i) and case (ii). Therefore, from Theorem 2 and Corollary 2, the derive system (1) is FXT stochastic synchronized to response system (2) under the controller (24). The time evolution of synchronization errors between systems (1) and (2) for above two different set of parameters are shown in Figure 5 and Figure 6 respectively, where the initial conditions of response systems (2)

Conclusions
In this paper, first, some recently developed new results on the FXT stability of deterministic dynamical systems are extended to stochastic dynamical systems. First, some earlier results on FXT stability of deterministic nonlinear systems are extended to the stochastic nonlinear systems. Then, based on these results, some simple sufficient conditions insuring the FXT synchronization of considered networks are derived by introducing two types of FXT controllers and utilizing some inequality techniques. Finally, our theoretical results are illustrated via giving two numerical examples with their Matlab simulations. Recently, the FXT stability and synchronization of impulsive neural networks have been studied. However, there are very few works on the FXT synchronization issue of the stochastic neural networks with impulsive effects; this issue may be somewhat challenging since we have to deal with the effects of caused by impulsive term and stochastic perturbations at the same time, and it will be one of our future studying directions.