Finite-Time Nonfragile Synchronization of Stochastic Complex Dynamical Networks with Semi-Markov Switching Outer Coupling

1Department of Mathematics, Bharathiar University, Coimbatore 641046, India 2Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea 3Department of Mathematics, Anna University Regional Campus, Coimbatore 641046, India 4Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China 5Department of Applied Mathematics, Kongju National University, Chungcheongnam-do 32588, Republic of Korea


Introduction
During the past twenty years, the investigation of complex dynamical networks (CDNs) that consist of a huge number of interacting dynamical nodes has received a great deal of attention from various science and engineering areas, such as social networks, ecological prey-predator networks, protein networks, power grids, and ecosystems [1,2].It should be mentioned that the analysis of dynamical behaviors of CDNs has become a hot research topic in recent years.Among many dynamical behaviors, synchronization phenomenon is the most important behavior and several interesting and efficient methodologies have been developed in the literature to solve the synchronization problem of various kinds of CDNs; for instance, see [3][4][5][6][7].In [4], the problem of inner synchronization of a CDN has been investigated by considering two different types of guaranteed cost dynamic feedback controllers, where the control gains corresponding to two feedback controllers have different dimensions subject to the topological structure of the CDN.In [5], the problem of outer synchronization between two hybrid-coupled delayed dynamical networks has been discussed by using the aperiodically adaptive intermittent pinning control, where a simple and elegant pinned-node selection scheme is proposed to achieve the required result.It is worth mentioning that the CDN representing real-time systems is generally affected by external noise factors or stochastic disturbances [8].Thus, the consideration of external noise factors in the study of synchronization of CDNs is of great importance in the viewpoints of both theoretical and practical.By taking this Complexity fact into account, in recent years, research communities have eagerly investigated the problems of synchronization of CDNs with external stochastic disturbances, see for example [9][10][11][12].Sakthivel et al. [10] presented some sufficient conditions that ensure the synchronization and solve the state estimation problem of discrete-time stochastic complex networks in the presence of uncertain inner couplings, where the interval matrix approach is employed to characterize the uncertainties encountered in the inner coupling terms.Li et al. [11] developed a new synchronization criterion for a class of discrete-time stochastic complex networks subject to partial mixed impulsive effects, where by using the Lyapunov stability theory and the variation of parameters formula, the required criterion is obtained.
It should be pointed out that the interconnection topology among the nodes of CDNs plays a significant role in the study of synchronization problem.In the existing literature, there have been two reported kinds of interconnection topologies, which are constant/fixed topology [13] and timevarying topology [14].In [14], it has been illustrated that the time-varying interconnection topology is more general than the fixed one.Nevertheless, in most of the real networks, the connectivity of the network topology might be unfixed or randomly changing due to new creation or link failures.In order to tackle these types of issues, it is more appropriate to model the CDNs with randomly switching topologies that are governed by a Markov process.Based on this scenario and following the seminal works reported in [15,16], some interesting and significant results about synchronization of CDNs with Markov jump topologies have been discussed; see [17][18][19].Specifically, in [18], the problem of nonfragile synchronization for a class of discrete-time complex networks subject to Markov jumping switching topology has been investigated in a unified framework that includes the nonfragile H ∞ synchronization control and nonfragile l 2 -l ∞ synchronization control problem as its special cases.However, it is worth pointing out that the aforementioned papers have considered constant transition rates in the Markov process.It should be mentioned that the Markov process might consist of time-varying transition rates when modeling practical systems.Such kind of process is known as semi-Markov process and few interesting results regarding semi-Markov jump systems have been addressed in recent literature [20][21][22][23].Interestingly, in [21], by employing the supplementary variable technique and plant transformation, the state estimation and sliding mode control problems have been investigated for semi-Markovian jump systems in the presence of mismatched uncertainties.Apparently, semi-Markov jump systems are comparatively more general than the traditional Markov jump systems [24].Following the aforementioned seminal works, the concept of semi-Markov process has further been employed in the network topology to obtain the synchronization criteria for CDNs (see [25,26] and the references therein).However, only very few results about the synchronization of CDNs with semi-Markov jump topology have been reported in the literature, which stimulates us to do this present work.
It is worth mentioning that most of the available controller design approaches have a predominant assumption that the designed controller can be implemented accurately.But in some real situations, such an assumption is not always true as the controllers are often very sensitive or fragile to their parameters' variations.Furthermore, it should be noted that a small perturbation in controllers may lead to undesirable oscillatory behavior or even instability [27].Hence, it is desirable as well as necessary to ensure the insensitivity of the controller to certain parameter perturbations without loss of the robust stability and thus, the investigation of nonfragile or resilient controller design that has been capable of tolerating some level of controller parameter gain variations has been enormously increased in recent years [28][29][30][31].To mention a few, in [28], a robust resilient control problem of discretetime Markov jump nonlinear systems has been solved by employing the linear matrix inequality and stochastic analysis techniques; in [29], based on the dissipative theory and the event-triggered sampling scheme, the nonfragile control design problem for a class of network-based singular systems with input time-varying delay and external disturbances has been addressed.Therefore, it is reasonable to consider the nonfragile control design in the study of synchronization of CDNs.It is noteworthy that only few research papers regarding the nonfragile control design for achieving the synchronization of CDNs have been published; see [32,33].On the other hand, it is worth mentioning that most of the existing results based on the classical control theory dealt with the asymptotic property of control system trajectories over an infinite-time interval and did not possess any restriction to the system states.But in many practical problems, it is required that the described system state does not exceed a certain bound during a fixed finite-time interval [34].According to this fact, a great number of interesting results on finite-time control design have been proposed for the synchronization of various CDNs; for instance, see [35][36][37][38].
To the best of our knowledge, however, the problem of robust nonfragile synchronization has not been fully investigated for a class of CDNs over a prescribed finite-time interval.
Motivated by the above analysis, in this paper, we focus on the finite-time nonfragile synchronization problem is investigated for a class of CDNs subject to semi-Markov jump topology and stochastic noises.More precisely, a new delaydependent sufficient condition under which the considered CDNs are synchronized to the target network within a given finite-time interval is developed in terms of linear matrix inequalities by utilizing the Lyapunov stability theory and the stochastic analysis techniques.Subsequently, based on the developed condition, a design algorithm of the proposed nonfragile state feedback controller that can ensure the finitetime stochastic synchronization of the addressed network is presented.Eventually, a numerical example is shown to illustrate the effectiveness of the proposed theoretical results.
The rest of this paper is organized as follows: in Section 2, the problem formulation of the network model under study and the preliminaries required to obtain the main results are given.The finite-time stochastic synchronization criterion for the considered network model is presented in Section 3. A numerical example and its simulations are provided in Section 4. Conclusion of this paper is given in Section 5.

Problem Formulation and Preliminaries
In this paper, we consider a class of complex dynamical networks (CDNs) with semi-Markov jump outer coupling and stochastic noise, which consists of  identical nodes and is defined over the Wiener process 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. Such a network model can be described in the following form: where   () ∈ R  denotes the state vector of the th node;  is a known real constant matrix with suitable dimension; (⋅, ⋅) ∈ R  represents a nonlinear vector-valued function; the constant  > 0 is the coupling strength of network;   (()) are the elements of the outer coupling matrix (()) which describes the network topological structure and is assumed to follow a semi-Markov process () which to be defined later.In particular,   (()) is defined as follows: if there exists a connection between node  and node , then   (()) > 0; otherwise,   (()) = 0. Further, the diagonal elements of the outer coupling matrix are given as follows:   (()) = − ∑  =1, ̸ =   (()); Λ represents the inner coupling matrix and is a positive diagonal matrix with appropriate dimension;   () ∈ R  is the control input of the th node which to be defined later; the function (⋅, ⋅, ⋅) : R × R  × R  → R  is the noise intensity vector-valued function; () is a 1-dimensional Brownian motion defined on the probability space (Ω, F, P) with E{()} = 0, E{ 2 ()} = 1 and E{()()} = 0 for  ̸ = , where E is the mathematical expectation; () is the time-varying delay function satisfying 0 ≤  1 ≤ () ≤  2 < ∞ and τ () ≤  < 1, where  1 ,  2 , and  are known scalars; and   () denotes the initial value of the th node's state and is assumed to be a continuous vector-valued function.
In order to develop the main results, the following assumptions and definition are required.Assumption 3.For each  ∈ S, all the real parts of eigenvalues of (()) are negative except an eigenvalue 0 with multiplicity 1, which means that the reverse of the graph generated by the matrix (()) contains a rooted spanning directed tree for every  ∈ S.

Main Results
Based on the Lyapunov-Krasovskii stability theory, this section aims to develop a new set of delay-dependent sufficient conditions that can guarantee the stochastic synchronization of the considered network model (1) over a finite-time interval.Moreover, based on these conditions, a design of the robust nonfragile state feedback control (3) for the network model under consideration is provided in terms of linear matrix inequalities (LMIs).Theorem 5. Consider the network model ( 1) with Assumptions 1-3.For given positive scalars  1 ,  2 ,  * , , , ,  ∈ [0, 1],  1 ,  2 , symmetric matrix   ( ∈ S), and diagonal matrices  3 ,  4 , the considered network ( 1) is stochastically synchronized in finite-time under the nonfragile controller (3), if there exist symmetric matrices   > 0 ( ∈ S),   > 0 ( = 1, 2, 3),  V > 0 (V = 1, 2) and positive scalars   ( ∈ S),  1 such that the following matrix inequalities hold: where and the rest of elements of Ω  (ℎ) are zero.
Remark 7. It should be mentioned that the constraints in (27) are cannot be solved directly via MATLAB LMI control toolbox due to the existence of the time-varying terms ∑ N =1   (ℎ).To overcome this difficulty, the transition rates   (ℎ) are assumed to be bounded and satisfy  −  ≤   (ℎ) ≤  +  , since they are partially measurable in practice which is mentioned in [26].Moreover, in this case, the following assumptions are made as in [26]: Now, we are able to present the sufficient conditions guaranteeing the stochastic synchronization of the considered network (1) over a finite-time interval in terms of LMIs in which all the elements are either constants or constant matrices according to Remark 7. Thus, we have the following theorem.
Proof.Based on Remark 7, the time-varying element   (ℎ) may take values in the interval [ −  ,  +  ].Then, by using (31) and following the similar lines in the proof of Theorem 6, it is easy to obtain the inequality (32) which completes the proof.Remark 9.It should be noted that, so far in the literature, several control approaches have been proposed for the synchronization problem of several CDNs [3][4][5][6], wherein the interconnection topology among the nodes are assumed to be fixed.However, in practice, this assumption is practically difficult or even impossible.However, yet now, there were no results reported in the existing literature for the synchronization analysis of stochastic CDNs with switching topology.According to this fact, in this paper, finite-time synchronization problem of stochastic CDNs with switching topology is investigated.Furthermore, due to random behavior in the dynamics of stochastic CDNs, it is very difficult to determine the exact fixed control value.Therefore, in this paper, the feedback control gain is considered with uncertain terms, which is more significant to reflect the realistic scenarios.

An Illustrative Example
This section provides an illustrative example to verify the developed theoretical results in the previous section.For the sake of simplicity, consider a class of CDNs in the form of (1) with five identical nodes and the state vector of each node being three-dimensional, that is,  = 5 and  = 3.
Let us select the network matrix and the nonlinear function as It is clear to see that (,   ()) satisfies Assumption 1 with G = diag {0.5, 0.5, 0.5}.In this example, we consider the semi-Markov jump topology with two modes, whose connectivity graph is shown in Figure 1.The inner coupling matrix is assumed to be Λ = diag {0.2, 0.5, 0.7} and the coupling strength is chosen as  = 0.5.The time-varying delay is taken as () = 0.5 + 0.5 sin() from which it can be obtained that  1 = 0,  2 = 1 and  = 0.5.
Then, by solving the LMIs ( 26), (28), and (29) in Theorem 6 along with (32) in Theorem 8 with the aid of MATLAB LMI control toolbox, we can get a set of feasible solutions from which the nonfragile state feedback control gain matrices can be obtained as follows: Here, our aim is to design the nonfragile state feedback controller such that the considered network ( 1) is robustly  Based on these values, simulations are drawn in Figures 2-12.Specifically, the state responses of the first, second, and third nodes together with the isolated node are plotted in Figures 2, 3, and 4, respectively, wherein the dotted line represents the isolated node and the dashed lines denote the five identical nodes.It can easily be observed from these figures that the states of the nodes are exactly synchronized with the states of the isolated node within short period which shows the efficiency of the proposed nonfragile control strategy.Moreover, the corresponding error state responses and the control response curves are given in Figures 5-7 and Figures 8-10, respectively.Further, Figure 11 shows the jumping mode of the semi-Markov switching topology.In    addition, to realize the finite-time synchronization, the time evolution of    ()    () ( = 1, 2, 3, 4, 5), ( ∈ S) is depicted in Figure 12.It can be seen from Figure 12 that the states of the error system do not exceed the prescribed threshold  2 = 2, which means that the synchronization of considered network (1) is achieved within a given finite-time interval.Thus, it can be concluded from the simulations that the designed nonfragile control algorithm effectively works even in the presence of stochastic noise and time-varying coupling delay.

Conclusion
In this paper, we have studied the robust finite-time nonfragile synchronization problem for a class of CDNs with  semi-Markov jump outer coupling, time-varying coupling delay, randomly occurring gain variation and stochastic noise.In particular, we have considered the semi-Markov switching topology to obtain the synchronization criterion.Moreover, we have introduced a stochastic variable satisfying the Bernoulli distribution to represent the random gain variations in the controller design.By employing the Lyapunov-Krasovskii stability theory and some stochastic analysis techniques, we then have developed a new finitetime stochastic synchronization criterion for the considered network in terms of linear matrix inequalities and have presented a design algorithm for the proposed nonfragile state feedback controller to a solution of the obtained set  of linear matrix inequalities.At last, we have provided a numerical example to verify the obtained theoretical results.In addition, it should be pointed out that one of the future research topics would be to investigate the problem of finitetime mixed  ∞ and passivity synchronization of stochastic singular CDNs with semi-Markov switching outer coupling delay and actuator saturation.

Figure 1 :
Figure 1: Connected topology of the considered CDNs.

Figure 2 :
Figure 2: State responses of first node.

Figure 4 :
Figure 4: State responses of third node.

Figure 5 :
Figure 5: Error state responses of first node.

Figure 6 :
Figure 6: Error state responses of second node.

Figure 7 :Figure 8 :
Figure 7: Error state responses of third node.