Exponential Synchronization for Neutral Complex Dynamical Networks with Interval Mode-Dependent Delays and Sampled Data

The exponential synchronization and sampled-data controller problem for a class of neutral complex dynamical networks (NCDNs) withMarkovian jump parameters, partially unknown transition rates and delays, is investigated in this paper. Both the discrete and neutral delays are considered to be interval mode dependent and time varying, while the sampling period is assumed to be time varying and bounded. Based on a new augmented stochastic Lyapunov functional, the delay-range-dependent and rate-dependent exponential stability conditions for the closed-loop error system are obtained by the Lyapunov-Krasovskii stability theory and reciprocally convex lemma. Then according to the proposed exponential stability conditions, the sampled-data synchronization controllers are designed in terms of the solution to linear matrix inequalities that can be solved effectively by using Matlab. Finally, numerical examples are given to demonstrate the feasibility and effectiveness of the proposed methods.


Introduction
It is well known that many practical systems can be described as complex networks such as Internet networks, biological networks, epidemic spreading networks, collaborative networks, social networks and neural networks [1][2][3][4].Thus, during the past decades, the research on the dynamics of 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 [5].As one of the most significant and important collective behaviors in CDNs, synchronization has received more attention; please refer to [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, [6][7][8][9][10][11][12][13][14][15][16][17] and references therein for more details.However, in some practical applications of communication networks, signal transmission channels often involve network-induced delays, packet dropouts, bit errors, environment disturbances, and so on, which will cause the error of transmission from one system to another.Then 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 the derivatives of its states.There are some results about the synchronization design problem for neutral systems [18][19][20][21][22][23].In these works, [19,20] had studied the synchronization control for a kind of master-response setup and were further extended to the case of neutral-type neural networks with stochastic perturbation.The authors of [18,22] had researched the synchronization problem for a class of complex networks with neutral-type coupling delays.The authors in [21] had studied the global asymptotic stability of neural networks of neutral type with mixed delays, which include constant delay in the leakage term, time-varying delays, and continuously distributed delays.The authors in [23] had investigated the robust global exponential synchronization problem for an array of neutraltype neural networks.However, much fewer results have been Mathematical Problems in Engineering proposed for neutral complex dynamical networks (NCDNs) compared with the rich results for CDNs with only discrete delays.
On the other hand, 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 problems of complex networks and neural networks with Markovian jump parameters and delays are investigated in [16,[24][25][26][27][28][29][30], and references therein.The authors in [24] have established sufficient global exponential stability conditions on Markovian jump neural networks with impulse control and time varying delays.The authors in [27] have studied synchronization in an array of coupled neural networks with Markovian jumping and random coupling strength.Particularly, Ma et al. [25] have considered the stability and synchronization problems for Markovian jump delayed neural networks with partly unknown transition probabilities, which have not been fully investigated and need to propose more good results.Besides, sampled-data systems have attracted great attention because the digital signal processing methods require better reliability, accuracy, and stable performance with the rapid development in digital measurement and intelligent instrument.There are some important and essential results which have been reported in the literature [31][32][33][34][35]. What is worth mentioning is that the sampled-data synchronization control problem has been investigated for a class of general complex networks with time-varying coupling delays in [34,35], where conditions have been presented to ensure the exponential stability of the closed-loop error system, and the desired sampled data feedback controllers have been designed.However, few results are available for neutral complex dynamical networks (NCDNs) with sampled data.To the best of the authors' knowledge, the NCDNs are difficult to treat and they are very challenging, especially in the presence of Markovian jump parameters, mode-dependent time-varying delays, and sampled data.Motivated by the above analysis, the exponential synchronization and sampled-data controller problem for a class of NCDNs with Markovian jump parameters and modedependent time-varying delay 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 with partially known transition rate.
In this paper, the synchronization and sampled-data controller problem is studied for NCDNs with Markovian jump parameters and partially known transition rates.The sampling period considered here is assumed to be time varying and bounded, while the neutral and discrete delays are interval mode-dependent and time varying.Firstly, by constructing a new augmented stochastic Lyapunov functional, exponential stability conditions are derived based on the Lyapunov stability theory and reciprocally convex lemma.Then the design method of the desired sampled-data controllers is solved on the basis of the obtained conditions.Moreover, all the derived results are in terms of LMIs that can be solved numerically, which are proved to be less conservative than the existing results.
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. Section 5 concludes the paper.

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, which means 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, ‖V‖ = (V  V) 1/2 , while ‖‖ is spectral norm of matrix , ‖‖ = [ max (  )] 1/2 . max(min) () is the eigenvalue of matrix  with maximum (minimum) real part.The Kronecker product of matrices  ∈ R × and  ∈ R × is a matrix in R × which is denoted as  ⊗ .Let  > 0 and ([−, 0], R  ) denote the family of continuous function , from [−, 0] to R  with the norm || = sup −≤≤0 ‖()‖.Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

Problem Statement and Preliminaries
Given a complete probability space {Ω, F, {F  } ≥0 , P} with a natural filtration {F  } ≥0 satisfying the usual conditions, 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 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, and   ≥ 0 (,  ∈ ,  ̸ = ) is the transition rate from mode  to , and for any state or mode  ∈ , it satisfies It is assumed that () is irreducible and available at time , but the transition rates of the Markov chain are partially known in this paper, which means that some elements in matrix Υ = (  ) × are inaccessible.For instance, in the system with six operation modes, the jump rates matrix Υ may be viewed as where  0  , ,  ∈  represents the unknown element.Furthermore, let    ,    be lower and upper bound for the diagonal element   or  0  , for all  ∈ .For notation clarity, we denote that S  = S   ⋃ S   , for all  ∈ , and If S   ̸ = ⌀, than it is further described as where    , ( = 1, 2, . . ., ) represent the th known element of the set S   in the th row of the transition rate matrix Υ.It should be noted that if S   = 0, S  = S   which means that any information between the th mode and the other  − 1 modes is not accessible, then MJSs with  modes can be regarded as ones with  − 1 modes.
Lemma 6 (see [46]).For functions and  1 = 0 with  1 () = 0 and  2 = 0 with  2 () = 0 and matrices  > 0,  > 0, then there exists matrix  such that and the following inequality holds: According to Definition 3, the aim will be achieved if we obtain the gain matrices K such that the error system ( 13) is exponentially stable.So we give the main results as follows.

Main Results
In this section, sufficient conditions are presented to ensure that the error system ( 13) is exponentially stable.Then, we propose a design method of the sampled-data controllers for NCDN (6).
Therefore, the error system (13) with partially known transition rates and sector-bounded condition ( 9) is exponentially stable with a decay rate .This completes the proof.
In some situations, due to the complexity of NCDN ( 6), the information on the transition rates may be completely unknown, which can be viewed as switched complex dynamical networks with arbitrary switching.The following corollary is therefore given to guarantee the exponential stability for this case.
Then we follow a similar line as in proof of Theorem 7 and obtain the result.

The Sampled-Data Controllers for NCDN.
In this subsection, we give the design method of the desired sampleddata controllers to ensure the NCDN (6) exponentially synchronized on the basis of Theorem 7.
According to Definition 3, by the sampled-data controllers gain matrices (84), the NCDN (6) with partially known transition rates and sector-bounded condition ( 9) is exponentially synchronized.This completes the proof.
Remark 10.It should be mentioned that the sampled-data synchronization problem has been solved for NCDN (6) in Theorem 9, and the desired controllers can be obtained when LMIs ( 23)-( 26), (82), and (83) are feasible.In this paper, we construct the novel stochastic Lyapunov functional (34) containing some triple-integral terms; which is very effective in the reduction of conservatism [45].Besides, we apply Lemma 5 to the corresponding terms, the method by using reciprocally convex lemma [46] can achieve less conservative results.Thus the obtained delay-range-dependent and decay rate-dependent stability condition for NCDN (6) in Theorem 7 is less conservative than the previous ones, which can achieve larger maximum value of sampling period ℎ and will be verified in Section 4.

Numerical Examples
In this section, numerical examples are used to illustrate the effectiveness of the results derived above.
Example 1.As shown in the example, a three-node NCDN (6) with Markovian switching between two modes is taken into consideration; that is,  = 3 and  = 2.The parametric matrices of the NCDN are given as follows: The partially known transition rate matrix is considered as in the following two cases: Furthermore, the nonlinear function (  ()) is given by Mathematical Problems in Engineering Then, it is easy to verify that The interval mode-dependent time-varying neutral delays and discrete delays are, respectively, assumed to be They are governed by the Markov process {(),  ≥ 0} and shown in Figures 1 and 2. It can be readily obtained that Given decay rate  = 0.3 and the maximum value of sampling period ℎ = 0.2, we choose  1 = 0.5 and  2 = 1.0 and seek to solve the sampled-data controllers.
For the case of Υ 1 , with the above parameters, by Theorem 9, we can obtain the sampled-data controllers as follows: Furthermore, the state trajectories of the error system ( 13) are given in Figure 3, where For another case of Υ 2 , the sampled-data controllers can be readily obtained by Corollary 12.
Case 1.The inner-coupling matrices are given as  = 0 and  = [ 0.5 0 0 0.5 ].According to the above parameters, the maximum value of sampling period ℎ, which satisfies LMIs ( 23)-( 26), (82), and (83) in Theorem 9, can be calculated by solving a quasiconvex optimization problem.The degraded NCDN has also been considered in [34,35].The results on the maximum value of sampling period ℎ are compared in Table 1, where it can be seen that our method is less conservative than [34,35].
Case 2. The inner-coupling matrices are given as  = diag{0.3,0.3} and  = diag{0.4,0.4}.The maximum value of sampling period ℎ also can be solved by Theorem 9.The results are listed in Table 2.In addition, the proposed method in [34] is not applicable here.From Table 2, it also can be seen that our method is less conservative than existing ones.

Conclusions
In this paper, the sampled-data synchronization problem has been solved for a class of NCDNs with Markovian jump parameters and partially known transition rates.The discrete and neutral delays are considered to be interval modedependent and time varying, while the sampling period here is bounded and time-varying.With a novel stochastic Lyapunov functional, the delay-range-dependent and rate-dependent exponential synchronization conditions have been proposed by Lyapunov theory and better technique of matrix inequalities.Then the sampled-data controllers have been designed on the basis of the obtained conditions.These theoretical results are successfully verified through numerical examples, whose simulation results are less conservative than the previous results.Finally, the main contributions of this   paper can be summarized as follows.(i) To achieve exponential synchronization for NCDN (6), the desired sampleddata feedback controllers have been designed in terms of the solution to certain LMIs.The proposed results are expressed in a new representation, which are theoretically and numerically proved to be less conservative than some existing ones.
(ii) The constructed stochastic Lyapunov functional contains some triple-integral terms, which are very effective in the reduction of conservativeness and have not appeared in the context of NCDNs.(iii) The bound of the delay is fully utilized in this paper; that is, improved bounding technique is used to reduce the conservativeness.(iv) The reciprocally convex lemma is used to derive the delay-range-dependent and ratedependent stability conditions, which can well reduce the conservativeness of the investigated systems.

20 Mathematical Problems in Engineering Table 2 :Figure 3 :
Figure 3: State trajectories of the error system in Example 1.