#### Abstract

In this paper, we propose and explore the synchronization examination for fuzzy stochastic complex networks’ Markovian jumping parameters portrayed by Takagi-Sugeno (T-S) fuzzy model with mixed time-varying coupling delays via impulsive control. The hybrid coupling includes time-varying discrete and distributed delays. Based on appropriate Lyapunov–Krasovskii functional (LKF) approach, Newton–Leibniz formula, and Jensen’s inequality, the stochastic examination systems and Kronecker product to create delay-dependent synchronization criteria that guarantee stochastically synchronous of the proposed T-S fuzzy stochastic complex networks with mixed time-varying delays. Adequate conditions for the synchronization criteria for the frameworks are established in terms of linear matrix inequalities (LMIs). At long last, numerical examples and simulations are given to demonstrate the correctness of the hypothetical outcomes.

#### 1. Introduction

Complex systems [1–5] have obtained significant consideration in recent years. They are generally utilized in different areas of engineering, biology, material science, mathematics, human science monetary science, and shale-gas extraction [6–9]. It is commonly realized that complex systems consist of a sizeable amount of nodes and edges, and every node and edge have its very own correspondent interpretation; along these lines, numerous frameworks as a general rule can be described by complex frameworks, for example, the networking net, traffic systems, neural systems, epidemic spreading, scientific citations, WWW, informal organizations, power matrices, global economic markets, etc [2–12]. Different logical models were expected to portray diverse complex systems, small world systems [13] and scale-free systems [14], and so forth. Complex systems normally show unpredictable and entrancing dynamical lead including synchronization [15], consensus [16], running, and so forth.

The synchronization and control of complex structures have been generally inspected in different areas of planning as well as innovation because of its numerous hidden reality purposes. Time delays may cause unfortunate powerful system practices, for example, wavering and instability [17]. Synchronization of complex networks has found extensive applications in fields such as data ferrying with unmanned aerial vehicles, image encryption, secure communication, biological systems, chemical reaction, nonlinear oscillation, chaos generator design, and swarm intelligence [18, 19]. As a rule, the fundamental thought of drive-response synchronization is to construct the controlled slave system to reach the same behavior with the autonomous drive system. The synchronization phenomena are common and essential in actual-world networks, along with synchronization at the net, synchronization transfer of digital or analog signals in communique networks, and synchronization related to organic neural networks. Along these lines, various scientists have looked at the synchronization of complex frameworks with delays and a lot of significant comments have been acquired [20, 21].

In practice, a real system is often affected by external random perturbations, i.e., the system has stochastic effects which can also lead to complex dynamic behaviours together with oscillation, divergence, chaos, instability, or different terrible performance. On the other hand, within the real world, complex networks are often challenging to environmental disturbances, and therefore, the stochastic modeling issue has been of vital importance. Especially, the framework models are commonly upset by outer disturbances which can be treated as stochastic problems of the structure. Array of networks scalar Wiener processes, is chatted on [22], which internal that each node is influenced by the noise.

In recent years, Markovian jump complex networks (MJCNs) have received increasing attention inside the area of control. Markovian jump systems may be defined through a set of linear systems that switch transfer from one to another at various times with a certain transition rate [23]. Dynamic structures can be exhibited by an uncommon class of hybrid system, for a model Markovian jump complex dynamical systems. Markovian chains are utilized for the jumping of finite nodes of the complex dynamical systems which may jump from one to another at various times. The frameworks with Markovian jump parameters, which move by ceaseless-time Markov chains, have been extensively used to show various genuine frameworks, where they may rehearse unexpected modification in their network and parameters. The jump structures have the benefit of showing the powerful frameworks branch of knowledge to unexpected variety in their systems, for example, part disappointments or fixes, abrupt ecological unsettling influence, and working at various purposes of a nonlinear systems [24–27]. In addition, in Markovian jump complex networks, the jumps in operation modes are governed by the Markov process or Markov chain.

Recently, many control techniques have been introduced to structure valuable controllers, such as adaptive, intermittent, impulsive, pinning, sampling data, sliding mode, and so on. Impulsive controller is a marvel that has been taken into discussion when modeling the complex networks. Impulsive frameworks normally are an extensive variation of developmental procedures wherein components, all of a sudden, change at explicit depictions of time [28–30]. Scientifically, these impulsive frameworks are generally portrayed by the relating impulsive differential conditions, which are the model of impulsive control frameworks [31–37]. We all know that various organic wonders incorporating blasting mood models in science and ideal control models in financial aspects show motivation impacts.

In the previous decades, Takagi-Sugeno (T-S) fuzzy version, primarily based on the dead control technique, was generally used to appropriate numerous complex nonlinear systems, and as of late it has turned out to be dynamic because of its wide designing applications [38, 39]. T-S fuzzy frameworks are nonlinear frameworks depicted by a set of IF-THEN standards. The T-S fuzzy model is a powerful one to break down, and its nonlinear elements are found everywhere in sign handling, synthetic procedures, and mechanical technology frameworks [40–42]. Furthermore, the fuzzy logic control procedure, as of late, has turned into a functioning exploration subject due to its potential applications in various fields [43–51]. On the other hand, to the creators’ information, up until this point, the issue of synchronization considering T-S fuzzy complicated systems by Markovian jumping as well as stochastic coupling postponement has not been tended to in the writing.

Motivated by the examinations mentioned above, this article intends to explore about the synchronization investigation for Markovian jumping parameters with stochastic discrete and distributed time-varying delays by means of impulsive control. We constructed a novel Lyapunov–Krasovskii functional (LKF) and derived delay-dependent synchronization criteria for drive-response Markovian jump complex systems with mixed time varying delays in terms of linear matrix inequalities (LMIs). We employed the properties of Kronecker product approach and stochastic assessment techniques to derive the proposed results. All the derived conditions arrived are communicated as LMIs whose feasibility could be successfully investigated by utilizing mathematically effective MATLAB LMI Control toolbox. Finally, a numerical model is given to show the adequacy of the proposed outcomes.

*Notations 1. *Throughout this article, and indicate, individually, the arrangement of nonnegative real numbers and the -dimensional Euclidean space. The superscript “” represents matrix transposition and characters (individually, ) indicate is semidefinite (individually, positive definite). represents the set of positive integers and stands for block-diagonal matrix. The asterisk “” in a symmetric matrix is utilized to mean term that is induced by symmetry. and 0 represent the identity matrix and a zero matrix, respectively. The Kronecker product of grids as well as in is signified as . represents the mathematical expectation administrator. indicates the Euclidean standard of a vector and its induced standard of a matrix. Let be a complete probability space which relates to an increasing family of algebras , where denotes the sample space, represents algebra of subsets on the sample space, and is the probability measure defined on . Let be a right continuous Markovian chain on the probability space taking values in the finite space with the initial model . Along generator , let be the transition rate matrix with transition probabilitywhere , and , is the progress rate mode to if andBesides, the transition probabilities matrix is indicated as follows:

#### 2. Problem Description and Preliminaries

In this paper, we consider the following stochastic impulsive complex networks with coupling delays consisting of identical nodes with drive-response Markovian jumping parameters by a T-S fuzzy model. The system dynamics can be captured by a set of fuzzy rules which can characterize local correlations in the state space. The subsequent rules are as follows. Drive system rule l: **IF****and****and** … **and**, THEN Response system rule l: **IF****and****and** … **and**, THENwhere . is the number of fuzzy **IF**-**THEN** rules. are fuzzy sets that are characterized by the membership functions; are premise variables; is the state vector of drive system (4) associated with th node at time ; is the state vector of response system (5) associated with th node at time ; , , , and are known constant matrices with appropriate dimensions; , , and are constants denoting the coupling strengths; , and , , and and are unknown but sector-bounded nonlinear functions of drive and response framework (4) and (5) individually, where including signifies discrete and distributed time-varying delays, which are accepted to fulfillingrespectively, where , , and are consistent and are self-determining of the Markovian process ; furthermore, is the annoyance force function vector; is an -dimensional Brownian motion determined on a complete probability space satisfyingwhere is the constant external input. denotes the impulse gain matrix at the moment of time . The discrete set satisfies . , and , denote the left-hand and right-hand limits at of drive and response system, respectively. and are continuously vector-valued initial condition defined on of the drive and response system. We assume that and are right continuous, i.e., and . are constant diagonal inner-coupling matrices among the subsystems; , , and continue the external combination arrangement matrices characterizing the topological structure of the complex systems in which is characterized as follows: if there is association within node and , at that point ; otherwise, , , and the diagonal components of matrices are characterized by

Utilizing the singleton fuzzifier, product fuzzy inference, and weighted average defuzzifier [52], the yield of above fuzzy drive framework is gathered as follows: Drive system rule l: **IF****and****and** … **and**, THENwherein which is the grade of membership of in and . Then, it can be seen that for and for all , we assumefor all , where can be regarded as the normalized weight of the **IF**-**THEN** rules.

Compared to the fuzzy drive framework (4), the response framework is represented by the subsequent rules. Response system rule l: **IF****and****and** … **and**, THEN

From the drive system (9) and response system (12), the error system can be expressed as follows:where , , , , .

For comfort, every conceivable estimation of is meant by , in the sequel. At the point, we havewhere , , , and for any are known constant frameworks of suitable measurements.

For simplicity, let

By employing the Kronecker product “” of matrices, system (10) can be written in a compact structure as follows:where means the identity matrix.

The following assumptions, definitions, and lemmas are useful in the proof of our main results.

*Assumption 1. *For all , the nonlinear functions and are assumed to satisfy the following sector-bounded conditions:where , , and and , , and are real constant matrices with , and .

*Assumption 2. *There exist constant matrices with appropriate dimensions so that the following linear growth condition [53]. satisfiesfor all , , and .

*Definition 1. *(see [54]). Drive system (9) and response system (12) are said to be stochastically synchronous if error system (17) is stochastically stable, that is, for any initial condition defined on the interval and , the following condition is satisfied:

*Definition 2. *(see [55]). The function whenever(1)The function is continuous on various of the sets and , , .(2) is locally Lipschitzian in , .(3)From each , and , there occur finite limitswhere is convinced.

*Definition 3. *(see [56]). We introduce stochastic positive functional as ; the weak infinitesimal operator of satisfies

Lemma 1. *(see [57]). Let represent the Kronecker product for which the following properties hold:*

Lemma 2. *(see [58]). Assume that matrices and a positive semidefinite matrix are given. If and , then*

Lemma 3. *(see [59]). For a positive matrix , scalar and vector function such that the integrations concerned are well defined; then, the following inequality holds:*

Lemma 4. *(see [60]). Let , , where and where with , . If and each row sum of is zero, next*

#### 3. Main Results

Here, some LMI conditions will be created to guarantee that drive framework (9) and response framework (12) are stochastically synchronous by utilizing stochastic Lyapunov–Krasovskii functionals. To introduce the main results of this segment, for simplicity, we denote

Theorem 1. *Given some positive scalars , and , assume that Assumptions 1 and 2 hold. Then, the T-S fuzzy stochastic complex networks (9) and (12) are stochastically synchronous, if there exist positive definite matrices , , , , , , , , , matrices , , , and positive scalars , , , such that the following linear matrix inequalities (LMIs) hold:where*

*Proof. *For the sake of notational convenience, we now letwhereChoose the following Lyapunov–Krasovskii functional for systemwherewithObserving that , based on the properties of the Kronecker product, for any matrix with appropriate dimension, from Lemma 1, we haveThe stochastic derivative of along the trajectories of the network (17) becomes