Distributed Synchronization of Coupled Time-Delay Neural Networks Based on Randomly Occurring Control

In this paper, the distributed synchronization of stochastic coupled neural networks with time-varying delay is concerned via randomly occurring control. We use two Bernoulli stochastic variables to describe the occurrence of distributed adaptive control and updating law according to certain probabilities. The distributed adaptive control and updating law for each vertex in the network depend on the state information on each vertex’s neighborhood. Based on Lyapunov stability theory, Itô differential equations, etc., by constructing the appropriate Lyapunov functional, we study and obtain sufficient conditions for the distributed synchronization of such networks in mean square.


Introduction
With the rapid development of the information age, complex networks have received extensive attention as a frontier of interdisciplinary and challenging research fields. In real life, complex networks are everywhere, from the huge Internet to WWW [1], from large power networks to transportation networks, from neural networks to the metabolism of living things and social networks. It is no exaggeration to say that we live in a world full of various complex networks.
the theoretical foundation of stochastic differential equations. In the past two hundred years, scientists have studied and analyzed stochastic phenomena, explored stochastic laws, and established a complete theoretical system of stochastic systems. Since then, the theory of stochastic systems based on stochastic analysis and stochastic differential equations has received full attention, and a large number of research results have been obtained which have permeated into many research fields and been widely applied. Therefore, the stochastic system control theory has become a hot research topic today and has important research value.
In the past ten years, many scholars have obtained a large number of constructive results on the study of synchronization of complex networks, and have proposed many effective methods. In [7], Barahona and Pecora proposed master stability functions to characterize the synchronization of complex networks. In [8], an improved simulated annealing method was used to detect optimal synchronization networks. Researchers have further studied the synchronization of complex networks with time-delay coupling [9] [10], weighted coupling matrices [11], linearly coupled form [12], adaptive updating weights [13], randomly oc- curring nonlinearities [14], stochastic coupling [15]. In [16], an adaptive strategy is used to study the synchronization of general complex networks when network topology is slowly time-varying. However, synchronization of complex networks may not be guaranteed if control is not introduced [17]. Researchers have developed many control-based methods to synchronize complex dynamical networks. For example, in [18], the problem of pinning control of complex dynamical networks is studied. When synchronizing and controlling networks, pinning control has been shown to be a simple but effective technique for stabilization and synchronization [17] [19] [20] [21] [22]. In [23] [24] [25] [26], some studies have discovered how to effectively control complex networks with a small number of nodes. The distributed synchronization of networks has attracted more and more attention from researchers in various fields. Vertex i in the network synchronizes the system state based not only on the state of vertex i, but also on the states of its neighboring vertices according to the given complex network topology [27]. In [28], an adaptive technique was proposed to synchronize Complex networks, where only neighborhood information was used to design updating law. Literature [29] has designed a distributed adaptive controller, the main feature of which is that nodes can more effectively use the local information on its neighbors without the need for global information on the entire networks. The actual implementation of the controller is always disturbed by various uncertainties caused by internal and external environments [30] [31].
Such disturbances widely exist in control implementation and system design, and are due to random abrupt changes [14] [30]. The distributed controller designed in [29] takes stochastic disturbances into consideration and proposes a distributed controller that occurs randomly. The first reason is that the signals in the network system cannot be completely transmitted or cannot be controlled, just as in the cases of packet dropouts, random failures and repairs of actuators [30] [31]. Another reason is positive. With consideration of economic or system life, control will be suspended from time to time [31]. Therefore, control activation and network systems may occur in a probabilistic or switching manner, and vary randomly in terms of their types and/or intensity [14]  In summary, randomness and time-delay are the most common phenomena in real systems, such as power and biological systems in actual engineering systems are typical time-delayed stochastic systems. In the past, the research on synchronization of complex networks mainly focused on the fixed controller. In reference [29], considering that the controlled system is interfered by random mutations, a method of randomly occurring control is proposed, and the synchronization behavior of complex networks is studied, but it does not take into account the impact of time-delay factors, so its model is somewhat conservative.

Model Description and Preliminaries
Notations: Let  resents a sample space, F is called an σ-algebra, and P is a probability measure.
In this paper, we consider the following model of a complex network stochastic system with coupled time-varying delays, which can be expressed as: Additionally, from the Gershgorin disk theorem, all the eigenvalues of the Laplacian matrix L corre- In order to achieve the synchronization of the stochastic complex network in (1), controllers are added to each vertex.
For the ith vertex, ( ) i u t is designed as is the control strength of vertex i.
In (3), is a Bernoulli stochastic variable that describes the following random events for (2): where the probability of event The distributed controller (3) in this paper takes stochastic disturbances into account. The distributed controller plemented and it can model control failure in a stochastic way. In short, randomly occurring distributed control can effectively use the information of neighboring points to simulate real-world disturbances. (3) is updated according to the following randomly occurring distributed updating law: where 0 α > and ( ) t ξ is a Bernoulli stochastic variable representing the following random events for (6): Similarly, Let ( ) It can be seen from the above model that the network studied in this paper has the following characteristics: 1) The activation of the controller and the updating law of control gain both occur in a probabilistic manner, and the distributed synchronization of stochastic complex networks is studied by considering randomly occurring control and updating law.
2) Considering the effect of coupled time-delays, the model is more general. In order to get the main results, the following definitions, assumptions and lemmas are needed.
be the solution of the stochastic complex network with coupled time-varying delay in (1) or (2), if they satisfy the following condition: then the stochastic complex network is said to achieve synchronization in mean square.
Lemma 1 (Itô formula) [35]: Let ( ) x t be an Itô process on 0 t ≥ with the stochastic differential , V x t t is again an Itô process with the stochastic differential given by holds for all , n x y ∈  and 0 t ≥ .
Assumption 2 can describe many real-world systems very well. It has been widely employed or discussed in [14] [35]. ( ) represents the derivative of ( ) t τ to t.

Main Results
In this section, we study the distributed synchronization problem of the stochastic complex network in (2) coupled with time-varying delay via randomly occurring control and updating law and obtain sufficient conditions for the distributed synchronization of such networks in mean square.
Theorem 1: Suppose that Assumptions 1-3 hold, then the stochastic complex network in (2) under the distributed adaptive controller (3) and updating law (6) will be globally synchronized in mean square, if the following conditions are satisfied: Proof: It should be noted that the stochastic dynamical network in (2) with is a special stochastic system with Markovian switching. Therefore, the existence and uniqueness of solutions to (2) can be transformed into the existence and uniqueness of solutions to a stochastic system with Markovian switching. The proof of the existence and uniqueness of solutions to a stochastic system with Markovian switching can be found in [32]. Also, the proof of the existence and uniqueness of solutions to (2) is shown in Supporting Information.
Consider the following Lyapunov function where b is a positive constant. According to (2) and (3) we can easily obtain By the Lemma 1 (Itô formula), the stochastic derivative of ( ) V t can be obtained as and according to (20), the Itô differential operator L is given as Note that and take expectations of ( ) Then, using Assumptions 1 and 2, (22), and (24), we have Besides, by using Lemma 2 and Lemma 3, one obtains that . And substitute (26), (27) and (28) into (25), one gets that Note that condition (15) in Theorem 1 yields Therefore, from (26), (29), (30) and (31) where μ is a positive constant. Thus, the distributed synchronization of the stochastic complex network with coupled time-varying delay in (2) via randomly occurring control and updating law can be achieved in mean square. When the delay in the stochastic complex network in (2) is a constant delay, the following simpler conditions can be obtained.
Corollary 1: Suppose that Assumptions 1 and 2 hold, then the stochastic complex network in (2) under the distributed adaptive controller (3) and updating law (6) will be globally synchronized in mean square, if the following conditions are satisfied:   proof of Theorem 1, and will not be described in detail here.

Conclusion
In this paper, we have studied the distributed synchronization problem of stochastic complex networks with coupled time-varying delays via randomly occurring control and updating law. The activation of the controller and the updating law of control gain both occur in a probabilistic manner. According to the Lyapunov stability theory, Itô differential equations, etc., by constructing the appropriate Lyapunov functional, the sufficient conditions for the distributed synchronization of such networks in mean square are obtained.