Pinning Cluster Synchronization in Linear Hybrid Coupled Delayed Dynamical Networks

The problem on cluster synchronization will be investigated for a class of delayed dynamical networks based on pinning control strategy. Through utilizing the combined convex technique and Kronecker product, two sufficient conditions can be derived to ensure the desired synchronizationwhen the designed feedback controller is employed to each cluster.Moreover, the inner coupling matrices are unnecessarily restricted to be diagonal and the controller design can be converted into solving a series of linear matrix inequalities (LMIs), which greatly improve the present methods. Finally, two numerical examples are provided to demonstrate the effectiveness and reduced conservatism.


Introduction
In past decade, the synchronization of various chaotic systems has received considerable attention since the pioneering works have appeared [1,2].Presently, it is widely known that many benefits of having synchronization can be existent.In particular, the synchronization in language emergence and development results can come up with the common vocabulary and agents' synchronization in organization management can improve their work efficiency.Thus recently, the synchronization has been widely studied owing to its great potential applications.Furthermore, since chaos synchronization in arrays of coupled dynamical networks was initially studied [3], various coupled networks have received the attention because they can exhibit some interesting phenomena [4,5], and many elegant results have been reported .In particular, in [6,7,33], time-delay is unavoidable and delayed neural networks (DNNs) are verified to exhibit some complex and unpredictable behaviors, such as periodic oscillations, bifurcation, and chaotic attractors; then, the impulsive and adaptive synchronization has also been studied [8][9][10][11], and some uneasy-to-test results have been presented.Most recently, through using Kronecker product, the global synchronization has been studied and elegant criteria have been obtained in terms of LMIs [12][13][14][15][16][17][18][19][20]23].Yet it is worth noting that, in the above works, some most developed techniques were not utilized and the addressed networks seemed to be of simple forms.Thus, researchers have used some effective tools to give less conservative results ensuring the synchronization for more general coupled DNNs [23].
In 1992, as the truth that the effective coupling among neurons varies temporally in a rather short time scale has been found [34], some researchers have mentioned that the degree of synchronization among pairs of neurons changed both temporally and by the choice of pairs.Therefore, the cluster synchronization has been imposed to various dynamical networks [21,22,[24][25][26][27][28][29][30][31]35].However, due to the existence of embedding of invariant synchronization manifolds, it may occur that the system can reach different clustering patterns from the different initial conditions [24][25][26][27][28][29].Thus, together with pinning control, some suitable methods have appeared but have been independent of initial states [30].In [30], the pinning control strategy has been used to realize the cluster synchronization for stochastic coupled DNNs, in which the upper bound of delay variation was less than 1.Later, some effective techniques were used to overcome 2 Mathematical Problems in Engineering the shortcoming during tackling the delayed dynamical networks [31,35].Yet though these results above were elegant, there still exist some points waiting for the improvements.Firstly, most works above have not contained lower bound of delay variation and, in fact, its information can play an important role in reducing the conservatism.Secondly, in [30,31], the inner coupling matrices had to be diagonal, which unavoidably limits the application areas.Thirdly, as for delay () ∈ [ 0 ,   ], since the triple integral LKF terms such as ẋ  () ẋ ()   were firstly put forward [35], it has been used and improved owing to the fact that it could help reduce the conservatism greatly [36].Yet the authors noticed that some important terms have been ignored when estimating its derivative [35,36], which also induces the conservatism.Therefore, the tighter estimation should be given.Overall, as for the pinning cluster synchronization of coupled networks, the mentioned points above have not been considered, which remains important and motivates this work.
Inspired by the above discussions, this paper aims to study the problem on cluster synchronization for a class of coupled time-delay networks with linear hybrid coupling by means of pinning control.Through choosing two augmented Lyapunov-Krasovskii functionals (LKFs) and using the combined convex technique, some novel sufficient conditions are presented via Kronecker product and LMIs, whose feasibility can be easily checked by resorting to Matlab LMI Toolbox.In particular, we will give the tighter upper bounds on time derivative of LKF terms.The efficiency and less conservatism can be verified on the basis of two numerical examples.
Notations.R × is the set of all  ×  real matrices;   represents the  ×  identity matrix and 0 ⋅ denotes the  ×  zero matrix;  ⊗  represents Kronecker product of matrices  and .

Problem Formulations and Preliminaries
Firstly, suppose the nodes are coupled with states   (),  ∈ {1, . . ., }; we consider the dynamical networks with each node being an -dimensional DNN with linear hybrid coupling as where Remark 1.In system (1), the hybrid coupling is utilized in model ( 1) and it should be emphasized that the inner coupling matrices , , and  are not necessarily restricted to be of diagonal form, which can represent more general cases than the ones in [30,31].
In what follows, some useful basic definition and denotations will be introduced.

Pinning Cluster Synchronization
Prior to addressing the main results, the following lemmas will be useful in the proof.

Numerical Examples
Two numerical examples will be provided to illustrate the derived results with some typical cases.
Then, through, respectively, setting  0 = −0.2,−0.1, and unavailable  0 , we can compute the corresponding maximum allowable upper bounds (MAUBs) in Table 1 based on Theorem 7 and Remark A.2 by resorting to Matlab LMI Toolbox.
In Table 1, the term "-" means that the corresponding value is unavailable.Based on the MAUBs in Table 1, one can verify that our results can be less conservative than some existent ones.In particular, as the inner coupling matrices , , and  are not diagonal, our theorems still can be applicable while [30,31] fail.
Case 2. Choosing () = 0.2 + 0.8 sin 2 (2) and the inner coupling matrices , , and  as Case 1, one can derive  0 = 0.2,   = 1.0,  0 = −1.6, and   = 1.6.Owing to the fact that   > 1 and , , and  are not diagonal, the methods in [30,31] fail to verify the synchronization.Yet, by resorting to Matlab LMI Toolbox, Theorem 7 can guarantee the pinning cluster synchronization and the feasible solution to LMIs in ( 17)-( 18) can be obtained as follows: Example 2. In this example, we consider the well-known Chua's circuit to illustrate our synchronization results, which can be expressed as In order to reduce the number of controllers and realize the cluster synchronization, we adopt the pinning controller (7) as together with Theorem 8 and LMI in Matlab Toolbox, we can easily verify that network (34) can achieve the desired cluster synchronization, which can be further supported by the synchronization error states in Figure 1.

Mathematical Problems in Engineering
Then, the time derivative of   (()) ( = 1, 2) along system (8) can be directly computed as

Case 1 .
Given  0 = 0, choose three inner coupling matrices of diagonal form as

4 Figure 1 :
Figure 1: Phase and state trajectories of the error states.