Impulsive Pinning Markovian Switching Stochastic Complex Networks with Time-Varying Delay

The synchronization problem of stochastic complex networks withMarkovian switching and time-varying delays is investigated by using impulsive pinning control scheme. The complex network possesses noise perturbations, Markovian switching, and internal and outer time-varying delays. Sufficient conditions for synchronization are obtained by employing the Lyapunov-Krasovskii functional method, Itö’s formula, and the linear matrix inequality (LMI). Numerical examples are also given to demonstrate the validity of the theoretical results.


Introduction and Model Description
Collective behaviors in complex networks and systems have attracted increasing attention in recent years due to their wide applications in physics, mathematics, engineering, biology, and so forth (see [1,2] and references therein).While complex networks are ubiquitously found in nature and in the modern world, such as neural networks, socially interacting animal species, power networks, wireless sensor networks, Internet, and the World Wide Web.
Synchronization in complex dynamical networks is realized via information exchanges among the interconnect nodes [22].The signal traveling along real physical system is usually perturbed randomly by the environmental elements, such as noises, the structures of the interconnections, time delays, and the positions of nodes [9].One popular model is the Markovian switching model driven by continuous-time Markov chains in the sciences and industries (see [23][24][25][26] and references therein).In [23,24], Mao et al. studied stability and controllability of stochastic differential delay equations with Markovian switching, while [25,26] discussed the exponential stability of stochastic delayed neural networks.Liu et al. [26], on the other hand, investigated the synchronization of discrete-time stochastic complex networks with Markovian jumping and mode-dependent mixed time delays.In [16], Wang et al. investigated the mean-square exponential synchronization of stochastic complex networks with Markovian switching and time-varying delays by using the pinning control method, which is described as where   () ( = 1, 2, . . ., ) are the linear state feedback controllers that are defined by and   > 0 ( = 1, 2, . . ., ) are the control gains.
Pinning control has been proved to be effective for the synchronization of complex dynamical networks with continuous state coupling [15,16,27].In many systems, the impulsive effect is a common phenomenon due to instantaneous perturbations at certain moments [27,28].Impulsive control strategies have been widely used to stabilize and synchronize coupled complex dynamical systems, such as signal processing system, computer networks, automatic control systems, and telecommunications [13].In [27], pinning impulsive strategy is proposed for the synchronization of stochastic dynamical networks with nonlinear coupling.Zhou et al. studied synchronization in complex delayed dynamical networks with impulsive effects in [28].And Zhu et al., in [29], investigated the exponential stability of a class of stochastic neural networks with both Markovian jump parameters and mixed fixed time delays.Can the stochastic dynamical network with Markovian switching and timevarying delays be synchronized by impulsive pinning control?This paper is devoted to solving this problem.
In this paper, we study the synchronization of stochastic complex networks with Markovian switching by using the impulse control method.We consider a kind of stochastic complex networks with internal time-varying delayed couplings, Markovian switching, and Wiener processes.By applying the Lyapunov-Krasovskii functional method, Itö's formula and the linear matrix inequality (LMI), some sufficient conditions for synchronization of these networks are derived.Numerical examples are finally given to demonstrate the effectiveness of the proposed impulsive pinning strategy.
Notations.Throughout this paper, R  will denote the dimensional Euclidean space and R × the set of all  ×  real matrices.The superscript  will denote the transpose of a matrix or a vector.And Tr(⋅) stands for the trace of the corresponding matrix. 1  = (1, 1, . . ., 1)  ∈ R  , and   is the -dimensional identity matrix.For square matrices , the notation  > 0 (<0) will mean that  is a positivedefinite (negative-definite) matrix. max () and  min () will denote the greatest and least eigenvalues of a symmetric matrix, respectively.p = max{ 1 ,  2 , . . .,   }, and p = min{ 1 ,  2 , . . .,   }.

𝑖𝑗
. Figure 1 shows the topology structures of the switching networks for 5 nodes.() is the inner time-varying delay satisfying  ≥ () ≥ 0 and   () is the coupling time-varying delay satisfying Mathematical Problems in Engineering Let (),  > 0 be a right-continuous Markov chain on a probability space that takes values in a finite state space  = 1, 2, . . .,  with a generator Γ = [  ] ∈ R × given by for some Δ > 0.Here   ≥ 0 is the transition rate from  to  if  ̸ =  and   = − ∑  ̸ =    , In this paper,  [] is assumed to be symmetric and irreducible, and  [] is assumed to be symmetric, for  = 1, 2, . . ., .
The following assumptions are usual and will be used throughout this paper for establishing the synchronization conditions [9,11,16].

Main Result
In this section, we will deduce our main results.
Remark 6.The stochastic networks studied before are without topological switch, and the time delays are always assumed to be fixed.However, for the sake of applications in the real work, these two points above should be taken into consideration.Of course, it will enhance the difficulties of the investigations on this network.For example, if the network has Markovian switching topology, the structure of the network is fast varying and the Lyapunov function is hard to be determined.By using the Lyapunov-Krasovskii functional, Itö's formula, and LMI, the exponential stability criterion of the pinning impulsive controlled Markovian switching stochastic dynamical network with time-varying delays was obtained.This also showed that the impulsive pinning control is a kind of cheap control strategy for guiding complex dynamical networks to the objective trajectory.
To make Theorem 5 more applicative, we give the following corollaries.
In another case, when we consider the system (3) without Markov switching, that is, where then the solutions  1 (),  2 (), . . ., and   () of system (9) are exponentially stable in mean square.
Furthermore, when we address the system (3) with fixed time delays, that is, () = ,   () =   , the following corollary also can be obtained.
Remark 10.In [29], the exponential stability of a class of stochastic neural networks with both Markovian jump parameters and mixed fixed time delays were investigated.Therefore, we could see our results as a further research about the stochastic dynamic network of [29].

Conclusion
In this paper, we investigated the synchronization problem for stochastic complex networks with Markovian switching and nondelayed and time-varying delayed hybrid couplings.We achieved synchronization by applying an impulse control scheme to a small fraction of the nodes and derived sufficient conditions for stability of synchronization.Finally, we considered some numerical examples that illustrate the theoretical analysis.

Figure 1 :
Figure 1: The topology structures of the switching networks for 5 nodes; (A1) and (A2) the topology structures of the coupling matrix  1 and  2 , respectively; (B1) and (B2) the topology structures of the coupling matrix  1 and  2 , respectively.