Function Projective Synchronization of Complex Networks With Distributed Delays via Hybrid Feedback Control

Due to its applications, some types of synchronization of complex networks have been intensively investigated. In particular, as a more general type of synchronization, function projective synchronization (FPS) has been investigated for complex networks with time delay or with time-varying delay. In this paper, we investigate FPS of complex networks with distributed delays. It is proven that, FPS of such networks can be realized via hybrid feedback control. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.

FPS means synchronizing the driver and response systems up to a scaling function. Hence it is a more general type of chaotic synchronization, and encompasses complete synchronization and projective synchronization. For a complex network, FPS means with a desired scaling function all the nodes are synchronized to an equilibrium point or a periodic orbit. Because the scaling function is hard to predict, FPS has application in secure communication [16]. Hence FPS has attracted a lot of research interests [17]- [20]. In particular, FPS was investigated for complex networks with time delay [19] and those with time-varying delay [20].
The associate editor coordinating the review of this manuscript and approving it for publication was Hocine Cherifi .
Complex networks usually have propagation delays, which have been observed in lasers, neuron models, electronic circuits and so on. Many phenomena in the real world indicate that the current state of a node is affected by those of its neighbors in the previous period. Therefore complex networks with distributed delays were introduced into the model system [21], [22]. In order to serve the practical life better, complex networks with distributed time delays are worthy of serious investigation. Some types of synchronization of such networks have been investigated [23]- [25]. However, to our best knowledge, up to now there are no works concerning FPS for these networks.
In this paper we investigate the problem of FPS for general complex networks with distributed delays. In Section II we introduce the network model, and prove that FPS of such networks can be realized via hybrid feedback control. In Section III, two numerical examples are provided to illustrate the effectiveness of our method.

II. FPS OF COMPLEX NETWORKS WITH DISTRIBUTED DELAYS
Consider a general complex network with N identical nodes v 1 , v 2 , · · · , v N , which are linearly coupled. The network topology is represented by the matrix G = (g ij ) ∈ R N ×N , where for i = j, g ij = 0 if v i is connected to v j , and 0 otherwise; And g ii , 1 ≤ i ≤ N , is defined as − N j=1,j =i g ij .
∈ R n will denote the state of v i , and the behavior of nodes can be described by a continuously differentiable vector function f : R n → R n . Let u i (t) ∈ R n be the control input. Then a controlled network with distributed delays can be described by the following system of integro-differential equations: (1) (1) is said to achieve function projective synchronization if there exists a continuously differentiable scaling function α(t) such that where s(t) ∈ R n is an equilibrium point, a periodic orbit, or an orbit of a chaotic attractor, which satisfiesṡ(t) = f (s(t)).
Based on the following result presented in [26], we are able to prove that, FPS of network (1) can be realized via a hybrid feedback control.
Lemma 1: For any vectors x, y ∈ R n and positive definite matrix Q ∈ R n×n , the following matrix inequality holds: Theorem 2: For any given initial conditions x i (0), d i (0), and s(0), FPS of network (1) can be realized by the control law: and k i > 0 is an any constant.
Combining with (2) − (4), we havė Construct the Lyapunov function where d * ∈ R + is to be determined. Consequently, we havė Let Q = G I n (the Kronecker product of G and I n ), and h(t) = (h 1    where ρ(M ) is the greatest eigenvalue of a symmetric matrix M . Hence, if d * = ρ QQ T 2 + 3 2 , thenV (t) ≤ −e T (t)e(t). Therefore, the error system (5) is asymptotically stable from the Lyapunov stability theory. This completes the proof.

III. COMPUTER SIMULATIONS
In this section, two simulation examples will be employed to illustrate the theoretical result obtained in the previous section.
Simulation 1: Consider the following single Lorenz system: where a = 10, b = 28, and c = 8 3 . The chaotic attractor of the system is depited in Figure 1.
We choose the coupling configuration matrix as Then the controlled network with distributed delays is described as i = 1, 2, 3. Choose K (x) = e −x as the kernel. By using Theorem 2 the controllers can be designed as In this numerical simulation, we take the initial states as  Figure 3 displays that e(t) → 0 when t → +∞. That is, FPS takes place rapidly in network (6).  Simulation 2: Consider the following single Chen system: where a = 35, b = 3, and c = 28. Figure 4 depicts the chaotic attractor of the Chen system.

IV. CONCLUSION
FPS schemes for complex networks with distributed delays are investigated in this paper. A hybrid feedback control method is presented to realize FPS in such networks. Finally, two numerical simulations are used to demonstrate the effectiveness of our method.
Recently, quantized techniques were shown to be an effective technique to realize synchronization of complex networks [27]- [32]. On the other hand, finite-time control [30]- [32] is more practical for engineering applications. In future works, we will investigate the problem of FPS of complex networks with distributed delays via finite-time quantized control.
XIULIANG QIU received the M.S. degree in applied mathematics from Xiamen University, Xiamen, China, in 2007. Since 2007, he has been a member of the Chengyi University College, Jimei University, where he is currently an Associate Professor. His research interests include graph theory, complex networks, and nonlinear system theory.
WENSHUI LIN (Member, IEEE) received the Ph.D. degree in applied mathematics from Xiamen University, Xiamen, China, in 2007. Since 2007, he has been a member of the School of Informatics, Xiamen University. He is currently an Associate Professor with Xiamen University, where he is also a member of the Fujian Key Laboratory of Sensing and Computing for Smart City. His research interests include graph theory, algorithms, and machine learning.
YIMING ZHENG received the B.S. degree in computer science from Xiamen University, Xiamen, China, in 2017, where he is currently pursuing the M.S. degree in computer science with the School of Informatics. His research interests include algorithms and artificial intelligence.