H ∞ Cluster Synchronization for a Class of Neutral Complex Dynamical Networks with Markovian Switching

H ∞ cluster synchronization problem for a class of neutral complex dynamical networks (NCDNs) with Markovian switching is investigated in this paper. Both the retarded and neutral delays are considered to be interval mode dependent and time varying. The concept of H ∞ cluster synchronization is proposed to quantify the attenuation level of synchronization error dynamics against the exogenous disturbance of the NCDNs. Based on a novel Lyapunov functional, by employing some integral inequalities and the nature of convex combination, mode delay-range-dependent H ∞ cluster synchronization criteria are derived in the form of linear matrix inequalities which depend not only on the disturbance attenuation but also on the initial values of the NCDNs. Finally, numerical examples are given to demonstrate the feasibility and effectiveness of the proposed theoretical results.


Introduction
During the past decades, the research on the complex dynamical networks (CDNs) has attracted extensive attention of scientific and engineering researchers in all fields domestic and overseas since the pioneering work of Watts and Strogatz [1]. One of the reasons is that the complex networks have extensively existed in many practical applications, such as ecosystems, the Internet, scientific citation web, biological neural networks, and large scale robotic system; see, for example, [2][3][4]. It should be noted that the synchronization phenomena of CDNs have been paid more attention to and intensively have been investigated in various different fields; please refer to [5][6][7][8][9][10] and references therein for more details.
Since time delay inevitably exists and has become an important issue in studying the CDNs, synchronization problems for complex networks with time delays have gained increasing research attention and considerable progress has been made; see, for example, [5][6][7][8][9][10][11][12][13][14][15][16] and references therein for more details. However, in some practical applications, past change rate of the state variables affects the dynamics of nodes in the networks. This kind of complex dynamical network is termed as neutral complex dynamical network (NCDN), which contains delays both in its states and in the derivatives of its states. There are some results about the synchronization design problem for neutral systems [17][18][19][20][21]. In these works, [18,19] had studied the synchronization control for a kind of master-response setup and further extended to the case of neutral-type neural networks with stochastic perturbation. References [17,20] had researched the synchronization problem for a class of complex networks with neutral-type coupling delays. Reference [21] had investigated the robust global exponential synchronization problem for an array of neutraltype neural networks. However, much fewer results have been proposed for neutral complex dynamical networks (NCDNs) compared with the rich results for CDNs with only discrete delays.
Recently, as a special synchronization on CDNs, cluster synchronization has been observed in biological science, distributed computation, and social contact networks. Because most of these networks have the clustering characteristic, many individuals maintain close contact with others in a same cluster, while only a few individuals link with an outside cluster. Hence, the individuals are synchronized inside the same cluster, but there is no synchronization among the clusters. Many researchers have made a lot of progress on the cluster synchronization problem; see, for example, [22][23][24][25][26]. In [24], cluster synchronization criteria are proposed for 2 The Scientific World Journal the coupled Josephson equation by constructing different coupling schemes. Then, in [26], a coupling scheme with cooperative and competitive weigh couplings is used to realize cluster synchronization for connected chaotic networks. In [22], cluster synchronization in an array of hybrid coupled neural networks with delays has been investigated and a new method is proposed to realize cluster synchronization by constructing a special coupling matrix. Besides, in the latest two years, cluster synchronization is considered for an array of coupled stochastic delayed neural networks by using the pinning control strategy in [23]. Linear pinning control schemes are given for cluster mixed synchronization of complex networks with community structure and nonidentical nodes in [25]. However, most of the research results in general complex networks ensure global or asymptotical synchronization, but the external disturbance is always existent, which may cause complex networks to diverge or oscillate. Therefore it is imperative to enhance the anti-interference ability of the system. To our knowledge, not much has been done for ∞ cluster synchronization for continuous-time complex dynamical networks with neutral time delays and Markovian switching. The purpose of this paper is to minimize this gap. In addition, due to the complexity of high-order and large-scale networks, network mode switching is also a universal phenomenon in CDNs of the actual systems, and sometimes the network has finite modes that switch from one to another with certain transition rate; then such switching can be governed by a Markovian chain. The stability and synchronization problem of complex networks and neural networks with Markovian jump parameters and delays are investigated in [15,[27][28][29][30] and references therein. Motivated by the above analysis, the ∞ cluster synchronization problem for a class of NCDNs with Markovian switching and modedependent time-varying delays is investigated in this paper. The addressed NCDNs consist of modes and the networks switch from one mode to another according to a Markovian chain.
In this paper, ∞ cluster synchronization of the NCDNs with Markovian jump parameters is studied for the first time, which is first introduced to quantify the attenuation level of synchronization error dynamics against the exogenous disturbance of NCDNs with Markovian switching. It is assumed that the neutral and retarded delays are interval mode dependent and time varying. By utilizing a new augmented Lyapunov functional, ∞ cluster synchronization criteria, which depend on interval mode-dependent delays, disturbance attenuation lever, and the initial values of NCDNs, are derived based on the Lyapunov stability theory, integral matrix inequalities, and convex combination. All the proposed results are in terms of LMIs that can be solved numerically, which are proved to be effective in numerical examples.
The remainder of the paper is organized as follows. Section 2 presents the problem and preliminaries. Section 3 gives the main results, which are then verified by numerical examples in Section 4. The paper is concluded in Section 5.
Notations. The following notations are used throughout the paper. R denotes the dimensional Euclidean space and R × is the set of all × matrices.
where and are both symmetric matrices, meaning that − is negative (positive) definite. is the identity matrix with proper dimensions. For a symmetric block matrix, we use * to denote the terms introduced by symmetry. E stands for the mathematical expectation, ‖V‖ is the Euclidean norm of vector V, and ‖V‖ = (V V) 1

Problem Statement and Preliminaries
Given a complete probability space {Ω, F, P} where Ω is the sample space, F is the algebra of events and P is the probability measure defined on F. Let { ( ), ≥ 0} be a homogeneous and right-continuous Markov chain taking values in a finite state space = {1, 2, 3, . . . , } with a generator Υ = ( ) × , , ∈ , which is given by where Δ > 0, lim Δ → 0 ( (Δ )/Δ ) = 0, ≥ 0 ( , ∈ , ̸ = ) is the transition rate from mode to and, for any state or mode ∈ , it satisfies Moreover, it is assumed that ( ) is irreducible and available at time .
The nonlinear vector functions, 1 , 2 , and 3 , are assumed to satisfy the following sector-bounded condition [31]: where ( ) 1 and ( ) 2 , = 1, 2, 3, are two constant matrices with ( ) 2 − ( ) 1 ≥ 0. Such a description of nonlinear functions has been exploited in [32][33][34] and is more general than the commonly used Lipschitz conditions, which would be possible to reduce the conservatism of the main results caused by quantifying the nonlinear functions via a matrix inequality technique.
Before moving onto the main results, some definitions and lemmas are introduced below.
Lemma 5 (see [37]). Let be an × matrix in the set (R, ), where R denotes a ring and (R, ) = {the set of matrices with entries R such that the sum of the entries in each row is equal to for some ∈ R}. Then the Furthermore, the matrix can be rewritten explicitly as follows: Proof. From Assumption 1 and Lemma 5, it can be easily obtained that ] The Scientific World Journal This completes the proof.
Definition 8. The neutral complex dynamical networks (3) and (4) are ∞ cluster synchronization with a disturbance attenuation and symmetric positive matrix > 0, if the following condition is satisfied: The index is called disturbance attenuation and used to quantify the attenuation level of synchronization error dynamics against exogenous disturbances. It is noticed that (20) depends not only on the attenuation level but also on the initial values of complex networks.

Main Results
In this section, sufficient conditions are presented to ensure ∞ cluster synchronization for the neutral complex dynamical network (NCDN) (3) and (4).

13
Similarly, we have In addition, according to (8) Then, following the above procedure, we can obtain If (26) is held, integrating the function in (51) from 0 to ∞, then we have By Definition 8, the NCDNs (3) and (4) can reach ∞ cluster synchronization with a disturbance attenuation . This completes the proof.
Remark 13. It should be mentioned that the proposed Lyapunov functional contains some triple-integral terms. Compared with the existing ones, [39,42] have shown that such a Lyapunov functional type is very effective in the reduction of conservatism. Besides, the information on the lower bound of the delay is sufficiently used by introducing the integral terms on Remark 14. ∞ cluster synchronization of the neutral complex dynamical networks with Markovian switching is considered for the first time. The synchronization conditions are in the form of linear matrix inequalities (LMIs), which can be solved by utilizing the LMI toolbox in Matlab. The solvability of derived conditions depends not only on the attenuation level but also on the initial values of the complex networks.
In some special situations, the neutral delay may disappear and be regarded as ( ) ≡ 0, which can be described by the following equality and viewed as a general delayed complex dynamical network with Markovian switching: The following corollary is therefore given to guarantee ∞ cluster synchronization for this case.
Proof. Since ( ) ≡ 0, we choose the Lyapunov functional as follows: Then we follow a similar line as in proof of Theorem 12 and obtain the result.

Numerical Examples
In this section, numerical examples are presented to demonstrate the effectiveness of the developed design on ∞ cluster synchronization.

Example 1.
A four-node NCDN (3) and (4) with Markovian switching between two modes is taken into consideration; that is, = 4 and = 2. The parametric matrices of the NCDN are given as follows: The transition rate matrix is considered as follows: Then, it is easy to verify that The interval mode-dependent time-varying neutral delays and discrete delays are, respectively, assumed to be