Stochastic synchronization of complex networks via aperiodically intermittent noise
Introduction
Complex networks comprise a large number of nodes and there is a complex relationship and topological structure between these nodes. Some nodes may affect the function of other nodes by transmitting coupled information. A small number of nodes often connect to a large number of other nodes, while most nodes have a few connections with the others. Complex networks are widely applied in the fields of the Internet, transport network and power grid [1], [2], [3], [4]. These applications are largely dependent on the asymptotic behavior of complex networks, especially on the synchronization.
Some feedback control approaches, such as adaptive control, pinning control and event-triggered control, have been proposed [5], [6], [7], [8], [9], [10], [11] in order to achieve synchronization. These control approaches can be classified into two classes according to controlled time and nodes. One class is that parts of the nodes are selected from all and then are controlled throughout the whole operation interval, while the other class is that the networks are controlled only in a portion of time intervals. The intermittent control falls into the second class [12], [13], [14]. Here we make use of aperiodically intermittent control to synchronize complex networks. It means that the networks are controlled only in some intervals, while the feedback controller is removed from the networks and the networks are not controlled in other intervals. Intermittent control offers considerable definite advantage over continuous control to the networks, as it is an effective way to reduce controlled time and computational load of the controller.
White noise is ubiquitous and common in nature. White noise is also used to suppress the divergence of solution and change the convergent speed of solution. Indeed, considerable effects of white noise on the system have been studied in the past decades [15], [16]. In view of this practical consideration, we utilize aperiodically intermittent white noise to synchronize complex networks and characterize it as Brown motion. As a result, the networks are viewed as stochastic systems when controlled by noise. That is, the networks switch between stochastic and deterministic systems because of intermittent noise. Stochastic system differs greatly from deterministic one. In stochastic system there is a diffusion part, which is a martingale. It is known that handling martingale is challenging due to the complicated structure of the networks. Finding a way to overcome this difficulty under intermittent noise is the first motivation of this study.
In real-time control engineering applications, it is common that the observation time of states lags behind the arrival time of the feedback controller [17], [18], [19], [20], [21], [22], and there exist time delays between them. Delay observations on the noise states were made in [23] and stochastic stabilization of differential system with time delays was discussed first. Note that complex networks are significantly different from differential systems. There is a more complicated topological structure for complex networks. In addition, both aperiodically intermittent noise and time delays simultaneously exist in the networks, which makes the structure even more complicated. To our best knowledge, stochastic synchronization of complex networks via intermittent noise with time delays has not been fully investigated, which is another motivation of the present work.
Intermittent and adaptive control was developed to synchronize the coupled networks in [24], and in this case the networks were deterministic systems. Continuous noise and intermittent control were employed to synchronize the networks in [25], [26], where the intermittent control was imposed on the drift part. The nonlinear systems were stabilized by intermittent white noise and almost sure stability was obtained in [27]. More recently, almost sure stabilization of hybrid differential equations controlled by continuous Brown motion were investigated in [28], [29], [30]. Compared with these existing work, there are some new features and advantages of the approaches proposed in this paper. We develop the intermittent noise method to synchronize the complex networks with delay observations. Under this circumstance, the networks are stochastic systems and the controller is put on the diffusion part when controlled by noise. It is well known that circumstance noise exists widely in nature, as a result, the applications of stochastic systems are much wider than those of deterministic systems in real world. In addition, using intermittent noise to study synchronization problem in complex networks is more effective than continuous noise to reduce operation cost.
One of the novelties of this work is that we consider both intermittent noise and time delays in the networks, which are more useful in practice, but more difficult to handle in theory. We introduce the auxiliary networks via intermittent noise without time delays to overcome this difficulty by comparison principle. Another difficulty is that the intermittent Brown motion turns out to be a martingale. As far as we know, although some approaches have been developed to handle martingale process in complex networks, they are difficult to be used due to the strong conditions imposed. Therefore, they are not quite applicable for practical situations. Here we make use of some ingenious integral inequalities and Borel–Cantelli Lemma to deal with the martingale. Moreover we need to solve a transcendental equation and find the upper bound of time delays, and even there may be no solution to this equation. We show the existence of the solution of the transcendental equation under certain conditions and also derive the solution by iterating process.
Summarizing the discussion above, we focus our attentions on stochastic synchronization of complex networks in this paper. The contributions of this paper are as follows. First, we make delay observations on the noise states and derive the least upper bound of time delays by a transcendental equation. Second, we obtain stochastic exponential synchronization of complex networks via aperiodically intermittent noise with time delays and the noise intensity. Finally, we derive the sufficient criteria for stochastic synchronization of complex networks via continuous noise with time delays and via intermittent noise without time delays as two special cases.
The rest of the paper is organized as follows. The equations characterizing complex networks are given and some preliminaries are presented in Section 2. A sufficient criterion is established to ensure exponential synchronization of complex networks in Section 3. A verification example is provided in Section 4 to illustrate the effectiveness of the proposed new design techniques, and the paper is concluded in Section 5.
Notation. Let be an identity matrix. For a real matrix () means that it is positive (negative) definite. The complete probability space is the sign with a filtration and satisfies the usual conditions. For denote the continuous function family with the norm . Denote as the family of all -measurable with valued random variables and satisfying . denotes the Kronecker product.
Section snippets
Networks formulation and preliminaries
Complex networks are common in engineering and manufacturing systems. They are often related to information technology, machinery and physical systems. Synchronized networks have been widely used in real world applications, see for example [31], [32] and the references therein. Unsynchronized networks need to be synchronized via feedback controller before being applied. In this paper we consider unsynchronized complex networks as follows,where
Aperiodically intermittent controller and synchronization
The main results are presented in this section. We consider small sampling moment synchronization of networks (3) for . Exponentially synchronization between complex (3) and the isolated node (2) is equivalent to the stabilization of error system (6). It is challenging to obtain stochastic exponential synchronization of networks (3) and the upper bound of time delays. The proof of the main theorem is new and it depends heavily on the following three lemmas. Lemma 3.1 Let Assumptions 2.1 and 2.2
Verification example
It is more difficult to achieve synchronization between chaotic system and the isolated node than that between non-chaotic system and the isolated node. Here a Lorenz chaotic system with 50 nodes is considered [37], [38],where with initial values and the isolated nodeWe take and .is the error system. The
Conclusion
In this paper, the problem of exponential synchronization has been investigated for a class of complex networks via aperiodically intermittent noise with time delays. An aperiodically intermittent controller is designed to synchronize the networks. By incorporating the upper bound of time delays and intermittent rate, the networks are guaranteed to be exponentially synchronized to the isolated node. The synchronization between auxiliary networks and the isolated node has also been achieved. A
Declaration of Competing Interest
We declare that we have no financial or personal relationship with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature kind in any product, service or company that could be construed as influencing the position presented in ,or the review of, the manuscript entitled.
Acknowledgments
This work is partially supported by Fundamental Research Funds for the Central Universities (Grant no. 2018B19914).
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