# Synchronization Analysis for Stochastic T-S Fuzzy Complex Networks with Markovian Jumping Parameters and Mixed Time-Varying Delays via Impulsive Control

Authors: M. Syed Ali; M. Usha; Quanxin Zhu; Saravanan Shanmugam
Journal: Mathematical Problems in Engineering
 (2020)

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## 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.


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## Body

## 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,ℝ+ and ℝn indicate, individually, the arrangement of nonnegative real numbers and the n-dimensional Euclidean space. The superscript “T” represents matrix transposition and characters ℬ>A (individually, ℬ≤A) indicate A−ℬ is semidefinite (individually, positive definite). N+ represents the set of positive integers and diag⋯ stands for block-diagonal matrix. The asterisk “∗” in a symmetric matrix is utilized to mean term that is induced by symmetry. I and 0 represent the identity matrix and a zero matrix, respectively. The Kronecker product of grids Q∈ℝm×n as well as ℛ∈ℝp×q in ℝmp×nq is signified as Q⊗ℛ. E⋅ represents the mathematical expectation administrator. ⋅ indicates the Euclidean standard of a vector and its induced standard of a matrix. Let Ω,ℱ,ℱtt≥0,P be a complete probability space which relates to an increasing family ℱtt≥0 of σ algebras ℱtt≥0⊂ℱ, where Ω denotes the sample space, ℱ represents σ algebra of subsets on the sample space, and P is the probability measure defined on ℱ. Let ϑtt≥0 be a right continuous Markovian chain on the probability space Ω,ℱ,ℱtt≥0,P taking values in the finite space S=1,2,…,n with the initial model ϑ0=ϑ0. Along generator Θ=θκjn×n, let κ,j∈S be the transition rate matrix with transition probability(1)Prϑt+Δ=jϑt=κ=θκjΔ+oΔ,ifκ≠j,1+θκκΔ+oΔ,ifκ=j,where Δ>0, and θκj≥0, is the progress rate mode κ to j if κ≠j and(2)θκκ=−∑j=1,j≠κθκj.
Besides, the transition probabilities matrixΘ is indicated as follows:(3)Θ=θ11θ12⋯θ1nθ21θ22⋯θ2n⋮⋮⋱⋮θn1θn2⋯θnn.


## 2. Problem Description and Preliminaries

In this paper, we consider the following stochastic impulsive complex networks with coupling delays consisting ofN 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:IFΛ1tisχ1landΛ2tisχ2land … andΛgtisχgl,THEN(4)dxit=Alϑtxit+ℬlϑtfxit+Clϑtgxit−τt+Dlϑt∫t−ϱtthxisds+c1∑j=1NG1lijΓ1ϑtxit+c2∑j=1NG2lijΓ2ϑtxit−τt+c3∑j=1NG3lijΓ3ϑt∫t−ϱttxjsds+ℐitdt+ρlit,xit,xit−τt,∫t−ϱttxisdsdωt,t≠tk,t≥t0,xitk=Jlϑtxitk−,i=1,2,…,N,t=tk,k∈N+xit=ϕit,t∈−maxτ,ϱ,0.Response system rule l:IFΛ1tisχ1landΛ2tisχ2land … andΛgtisχgl,THEN(5)dy^it=Alϑty^it+ℬlϑtfy^it+Clϑtgy^it−τt+Dlϑt∫t−ϱtthy^isds+c1∑j=1NG1lijΓ1ϑty^iy^isds+c2∑j=1NG2lijΓ2ϑty^it−τt+c3∑j=1NG3lijΓ3ϑt∫t−ϱtty^jsds+ℐitdt+ρlit,y^it,y^it−τt,∫t−ϱtty^isdsdωt,t≠tk,t≥t0,y^itk=Jlϑty^itk−,i=1,2,…,N,t=tk,k∈N+y^it=φit,t∈−maxτ,ϱ,0,where l∈S=1,2,…,υ. υ is the number of fuzzy IF-THEN rules. χ1l,…,χgl are fuzzy sets that are characterized by the membership functions; Λt=Λ1t,…,ΛgtT are premise variables; xit=xi1t,xi2t,…,xintT∈ℝn is the state vector of drive system (4) associated with ith node at time t; y^it=y^i1t,y^i2t,…,y^intT∈ℝn is the state vector of response system (5) associated with ith node at time t; Alϑt, ℬlϑt, Clϑt, and Dlϑt are known constant matrices with appropriate dimensions; c1>0, c2>0, and c3>0 are constants denoting the coupling strengths; fxit=colf1xi1t,…,fnxintT, gxit=colg1xi1t,…,gnxintT and hxit=colh1xi1t,…,hnxintT, fy^it=colf1y^i1t,…,fny^intT, and gy^it=colg1y^i1t,…,gny^intT and hy^it=colh1y^i1t,…,hny^intT are unknown but sector-bounded nonlinear functions of drive and response framework (4) and (5) individually, where 0<τt including 0<ϱt signifies discrete and distributed time-varying delays, which are accepted to fulfilling(6)0≤τt≤τ,τ˙t≤μ<1,0≤ϱt≤ϱ,respectively, where τ, ϱ, and μ are consistent and are self-determining of the Markovian process ϑtt≥0; furthermore, ρli⋅,⋅,⋅,⋅:ℝ×ℝn×ℝn⟶ℝn is the annoyance force function vector; ωt=ω1t,ω2t,…,ωntT is an n-dimensional Brownian motion determined on a complete probability space Ω,ℱ,ℱtt≥0,P satisfying(7)Edωt=0,Edωt2=dt,where ℐit is the constant external input. Jlϑt denotes the impulse gain matrix at the moment of time tk. The discrete set tk satisfies 0=t0<t1<⋯<tk<⋯limk⟶∞tk=+∞. xitk−, y^itk− and xitk+, y^itk+ denote the left-hand and right-hand limits at tk of drive and response system, respectively. ϕit and φit are continuously vector-valued initial condition defined on −maxτ,ϱ,0 of the drive and response system. We assume that xitk and y^itk are right continuous, i.e., xitk+=xitk and y^itk+=y^itk. Γaϑt=diagqa1ϑt,qa2ϑt,…,qanϑta=1,2,3 are constant diagonal inner-coupling matrices among the subsystems; G1l=G1lijN×N, G2l=G2lijN×N, and G3l=G3lijN×N continue the external combination arrangement matrices characterizing the topological structure of the complex systems in which Gqlijq=1,2,3 is characterized as follows: if there is association within node i and ji≠j, at that point Gqlij=Gqlji=1; otherwise, Gqlij=Gqlji=0, i≠j, and the diagonal components of matrices Gql are characterized by(8)G1lii=∑j=1,j≠iNG1lij=−∑j=1,j≠iNG1lij,G2lii=∑j=1,j≠iNG2lij=−∑j=1,j≠iNG2lij,G3lii=∑j=1,j≠iNG3lij=−∑j=1,j≠iNG3lij,l=1,2,…,υ,i,j=1,2,…,N.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:IFΛ1tisχ1landΛ2tisχ2land … andΛgtisχgl,THEN(9)dxit=∑l=1υδlΛtAlϑtxit+ℬlϑtfxit+Clϑtgxit−τt+Dlϑt∫t−ϱtthxisds+c1∑j=1NG1lijΓ1ϑtxit+ℐit+c2∑j=1NG2lijΓ2ϑtxit−τt+c3∑j=1NG3lijΓ3ϑt∫t−ϱttxjsdsdt+ρlit,xit,xit−τt,∫t−ϱttxisdsdωt,t≠tk,t≥t0,xitk,xitk=∑l=1υδlΛtJlϑtxitk−,i=1,2,…,N,t=tk,k∈N+,xit=ϕit,t∈−maxτ,ϱ,0,where(10)δlΛt=∏j=1υϖljΛjt∑l=1υ∏j=1gϖljΛjt,l=1,2,…,υ,in which ϖljΛjt is the grade of membership of Λjt in χgl and Λt=Λ1t,…,Λgt. Then, it can be seen that for l=1,2,…,υ and for all t, we assume(11)δlΛt≥0,∑l=1υδlΛt=1,l=1,2,…,υ,for all t∈ℛ+, where δlΛt≥0 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:IFΛ1tisχ1landΛ2tisχ2land … andΛgtisχgl,THEN(12)dy^it=∑l=1υδlΛtAlϑty^it+ℬlϑtfy^it+Clϑtgy^it−τt+Dlϑt∫t−ϱtthy^isds+c1∑j=1NG1lijΓ1ϑty^it+Iit+c2∑j=1NG2lijΓ2ϑty^it−τt+c3∑j=1NG3lijΓ3ϑt∫t−ϱtty^jsdsdt+ρlit,y^it,y^it−τt,∫t−ϱtty^isdsdωt,t≠tk,t≥t0,y^itk=∑l=1υδlΛtJlϑty^itk−,i=1,2,…,N,t=tk,k∈N+,y^it=φit,t∈−maxτ,ϱ,0.From the drive system (9) and response system (12), the error system can be expressed as follows:(13)deit=∑l=1υδlΛtAlϑteit+ℬlϑtfeit+Clϑtgeit−τt+Dlϑt∫t−ϱttheisds+c1∑j=1NG1lijΓ1ϑteit+c2∑j=1NG2lijΓ2ϑteit−τt+c3∑j=1NG3lijΓ3ϑt∫t−ϱttejsdsdt+ρlit,eit,eit−τt,∫t−ϱtteisdsdωt,t≠tk,t≥t0,eitk=∑l=1υδlΛtJlϑteitk−,i=1,2,…,N,t=tk,k∈N+,eit=ϕit−φit,t∈−maxτ,ϱ,0,where feit=fxit−fy^it, geit−τt=gxit−τt−gy^it−τt, ∫t−ϱttheisds=∫t−ϱtthxisds−∫t−ϱtthy^isds, eit−τt=xit−τt−y^it−τt, ρlit,eit,eit−τt,∫t−ϱtteisds=ρlit,xit,xit−τt∫t−ϱttxisds−ρlit,y^it,y^it−τt,∫t−ϱtty^isds.For comfort, every conceivable estimation ofϑt is meant by κ, κ∈S in the sequel. At the point, we have(14)Alκ=Alϑt,ℬlκ=ℬlϑt,Clκ=Clϑt,Dlκ=Dlϑt,Γ1ϑt=Γ1κ,Γ2ϑt=Γ2κ,where Alκ, ℬlκ, Clκ, and Dlκ for any κ∈S are known constant frameworks of suitable measurements.(15)deit=∑l=1υδlΛtAlκeit+ℬlκfeit+Clκgeit−τt+Dlκ∫t−ϱttheisds+c1∑j=1NG1lijΓ1κeit+c2∑j=1NG2lijΓ2κeit−τt+c3∑j=1NG3lijΓ3κ∫t−ϱttejsdsdt+ρlit,eit,eit−τt,∫t−ϱtteisdsdωt,t≠tk,t≥t0,eitk=∑l=1υδlΛtJlκeitk−,i=1,2,…N,t=tk,k∈N+,eit=ϕit−φit,t∈−maxτ,ϱ,0.For simplicity, let(16)et=e1Tt,e2Tt,…,eNTtT,Fet=fTe1t,fTe2t,…,fTeNtT,Get−τt=gTe1t−τt,gTe2t−τt,…,gTeNt−τtT,Het=hTe1t,hTe2t,…,hTeNtT,ρlt=ρl1Tt,e1t,e1t−τt,∫tϱtte1sds,…,ρlNTt,eNt,eNt−τt,∫tϱtteNsdsT.By employing the Kronecker product “⊗” of matrices, system (10) can be written in a compact structure as follows:(17)det=∑l=1υδlΛtIW⊗Alκet+IW⊗ℬlκFet+IW⊗ClκGet−τt+IW⊗Dlκ∫tϱttHesds+c1G1l⊗Γ1κet+c2G2l⊗Γ2κet−τt+c3G3l⊗Γ3κ∫t−ϱttesdsdt+ρ˜ltdωt,t≠tk,etk=∑l=1υδlΛtIW⊗Jlκetk−,t=tk,k∈N+,where IW means the W×W identity matrix.The following assumptions, definitions, and lemmas are useful in the proof of our main results.Assumption 1.
For allu,v∈ℝn, the nonlinear functions f⋅,g⋅ and h⋅ are assumed to satisfy the following sector-bounded conditions:(18)fu−fv−X1u−vTfu−fv−X2u−v≤0,gu−gv−Y1u−vTgu−gv−Y2u−v≤0,hu−hv−ℋ1u−vThu−hv−ℋ2u−v≤0,where X1, Y1, and ℋ1 and X2, Y2, and ℋ2 are real constant matrices with X2−X1≥0, Y2−Y1≥0 and ℋ2−ℋ1≥0.Assumption 2.
There exist constant matricesΥlii=1,2,3,l=1,2,…,υ with appropriate dimensions so that the following linear growth condition [53]. satisfies(19)ρlit,p1,q1,r1−ρljt,p2,q2,r2T×ρlit,p1,q1,r1−ρljt,p2,q2,r2≤Υl1p1−p22+Υl2q1−q22+Υl3r1−r22,for all i,j=1,2,…,N, l=1,2,…,υ, ρlit,p1,q1,r1:ℝ×ℝn×ℝn⟶ℝn and pi,qi,rii=1,2∈ℝn.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 et=Φt defined on the interval −maxτ,ϱ,0 and e0∈S, the following condition is satisfied:(20)limT⟶∞E∫0Tes2dsΦt,ϑ0<∞.Definition 2.
(see [55]). The function V:t0,∞×ℝn×S⟶ℝ+∈Ψ0 whenever(1)
The functionV is continuous on various of the sets tk−1,tk×ℝn×S and ∀t≥t0, Vt,0,κ≡0, κ∈S.(2)
Vt,e1,κ is locally Lipschitzian in e1∈ℝn, κ∈S.(3)
From eachk=1,2,…, and κ,j∈S, there occur finite limits(21)limt,w,j⟶tk−,e1,κVtk−,e1,κ,limt,w,j⟶tk+,e1,jVtk+,e1,j,where Vtk+,e1,j=Vtk,e1,j is convinced.Definition 3.
(see [56]). We introduce stochastic positive functional as Vet,t,ϑt=κ,t>0; the weak infinitesimal operator ℒ of Vet,t,κ satisfies(22)ℒVet,t,κ≜limΔ⟶0+1ΔsupEVet+Δ,t+Δ,κet,t,κ−Vet,t,κ.Lemma 1.
(see [57]). Let ⊗ represent the Kronecker product for which the following properties hold:(23)1βX⊗Y=X⊗βY;2X+Y⊗Z=X⊗Z+Y⊗Z;3X⊗YZ⊗W=XZ⊗YW;4X⊗YT=XT⊗YT.Lemma 2.
(see [58]). Assume that matrices Wll=1r∈ℝn×m and a positive semidefinite matrix ℱ∈ℝn×n are given. If ∑l=1rδlΛt=1 and 0≤δlΛt≤1, then(24)∑l=1rδlΛtWlTℱ∑l=1rδlΛtWl≤∑l=1rδlΛtWlTℱWl.Lemma 3.
(see [59]). For a positive matrix O>0, scalar β>βt>0 and vector function w:0,β⟶ℝn such that the integrations concerned are well defined; then, the following inequality holds:(25)∫0βtwsdsTO∫0βtwsds≤βt∫0βtwTsOwsds.Lemma 4.
(see [60]). Let ℳ=αijW×W, S∈ℝn×n, e=e1T,e2T,…,eNTT where ei=ei1,ei2,…,einT∈ℝn and z=z1T,z2T,…,zNTT where zi=zi1,zi2,…,zinT∈ℝn with ek, zk∈ℝnk=1,2,…,N. If ℳ=ℳT and each row sum of ℳ is zero, next(26)eTℳ⊗Sz=−∑1≤i<j≤Nαijei−ejTSzi−zj.


## 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(27)X^=X1TX2+X2TX1,Xˇ=X1T+X2T,Y^=Y1TY2+Y2TY1,Yˇ=Y1T+Y2T,X^=ℋ1Tℋ2+ℋ2Tℋ1,ℋˇ=ℋ1T+ℋ2T.Theorem 1.
Given some positive scalarsτ>0, ϱ>0 and μ>0, 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 Pκ>0κ∈S, Q1>0, T1>0, T2>0, Q>0, ℛ>0, T>0, Z>0, Z1>0, matrices Sil, i=1,2,3,4, l=1,2,…,υ, and positive scalars ϵ, α, β, γ such that the following linear matrix inequalities (LMIs) hold:(28)Pκ<ϵI,κ∈S,(29)Ψlij=Ψl11Ψl12Ψl13Ψl14Ψl15Ψl160Ψl180Ψl110Ψl111Ψl11200∗Ψl22Ψl2300Ψl260Ψl2800Ψl211Ψl21200∗∗Ψl3300000000000∗∗∗Ψl440000000000∗∗∗∗Ψl55000000000∗∗∗∗∗Ψl66Ψl67Ψl68Ψl6900000∗∗∗∗∗∗Ψl770000000∗∗∗∗∗∗∗Ψl88000000∗∗∗∗∗∗∗∗Ψl9900000∗∗∗∗∗∗∗∗∗Ψl10100000∗∗∗∗∗∗∗∗∗∗Ψl1111000∗∗∗∗∗∗∗∗∗∗∗Ψl121200∗∗∗∗∗∗∗∗∗∗∗∗Ψl13130∗∗∗∗∗∗∗∗∗∗∗∗∗Ψl1414<0,(30)JlκTPjJlκ−Pκ≤0,hereϑtk=j,where(31)Ψl11=S3lAlκ+AlκTS3lT+c1WG1lS3lΓ1κ+Γ1κTS3l+PκAlκ+AlκTPκ+c1WG1lPκΓ1κ+Γ1κTPκ+ϱZ1+S1l+S1lT−αX^+εΥ1Υ1T−βY^−γX^+Q1+τT1+∑j=1nθκjPj,Ψl12=−S3l+AlκTS4lT+S2lT+c1WG1lS4lTΓ1κT,Ψl13=S3lℬlκ+αXˇ+Pκℬlκ,Ψl14=βYˇ,Ψl15=γℋˇ,Ψl16=c2WG2lS3lΓ2κ+PκΓ2κ−S1l,Ψl18=S3lClκ+PκClκ,Ψl110=c3WG3lS3lΓ3κ+PκΓ3κ,Ψl111=−S1l,Ψl112=S3lDlκ+PκDlκ,Ψl22=τ2Z−S4l−S4lT,Ψl23=S4lℬlκ,Ψl26=c2S4lWG2lΓ2κ−S2l,Ψl28=S4lClκ,Ψl210=c3S4lWG3lΓ3κ,Ψl211=−S2l,Ψl212=S4lDlκ,Ψl33=Q−2αI,Ψl44=T−2βI+τT2,Ψl55=ϱ2ℛ−2γI,Ψl66=−1−μQ1−αX^−βY^−γX^+εΥ2Υ2T,Ψl67=αXˇ,Ψl68=βyˇ,Ψl69=γℋˇ,Ψl77=−2αI−1−μQ,Ψl88=−2βI−1−μT,Ψl99=−2γI,Ψl1010=−1ϱZ1+εΥ3Υ3T,Ψl1111=−Z,Ψl1212=−ℛ,Ψl1313=−1τT1,Ψl1414=−1τT2.Proof.
For the sake of notational convenience, we now let(32)det=∑l=1υδlΛtztdt+∑l=1υδlΛtρ˜ltdωt,etk=∑l=1υδlΛtIW⊗Jlκetk−,k∈N+,where(33)zt=IW⊗Alκet+IW⊗ℬlκFet+IW⊗ClκGet−τt+IW⊗Dlκ∫t−ϱttHesds+c1G1l⊗Γ1κet+c2G2l⊗Γ2κet−τt+c3G3l⊗Γ3κ∫t−ϱttesds,ρ˜lt=ρlt,et,et−τt,∫t−ϱttesds.
Choose the following Lyapunov–Krasovskii functional for system(34)Vet,t,κ=∑o=19Voet,t,κ,where(35)V1et,t,κ=eTtℳ⊗Pκet,V2et,t,κ=∫t−τtteTsℳ⊗Q1esds,V3et,t,κ=∫t−τtt∫θteTsℳ⊗T1esdsdθ,V4et,t,κ=∫t−τttFTesℳ⊗QFesds,V5et,t,κ=∫t−τttGTesℳ⊗TGesds,V6et,t,κ=∫−τt0∫t+θtGTesℳ⊗T2Gesdsdθ,V7et,t,κ=ϱ∫−ϱ0∫t+θtHTesℳ⊗ℛHesdsdθ,V8et,t,κ=τ∫−τ0∫t+θtzTsℳ⊗Zzsdsdθ,V9et,t,κ=∫−ϱ0∫t+θteTsℳ⊗Z1esdsdθ.with(36)ℳ=W−1−1⋯−1−1W−1⋯−1⋯⋯⋯⋯−1−1⋯W−1W×W.
Observing thatℳGil=Gilℳ=WGili=1,2,3,l=1,2,…,, based on the properties of the Kronecker product, for any matrix ℋ with appropriate dimension, from Lemma 1, we have(37)ℳ⊗ℋGil⊗Γiκ=ℳGil⊗ℋΓiκ=WGil⊗ℋΓiκ.
The stochastic derivative ofV along the trajectories of the network (17) becomes(38)ℒVet,t,κ=∑o=19ℒVoet,t,κ,ℒV1et,t,κ=2eTtℳ⊗Pκ∑l=1υδlΛtzt+∑l=1υδlΛtρ˜ltTℳ⊗Pκ∑l=1υδlΛtρ˜lt+eTt∑j=1nθκjℳ⊗Pjet,(39)=2eTtℳ⊗Pκ∑l=1υδlΛtIW⊗AlκeΛt+IW⊗ℬlκFet+IW⊗ClκGet−τt+IW⊗Dlκ∫t−ϱttHesds+c1G1l⊗Γ1κet+c2G2l⊗Γ2κet−τt+c3G3l⊗Γ3κ∫t−ϱttesds+∑l=1υδlΛtρlt,et,et−τΛtρl,∫t−ϱttesdsT×ℳ⊗Pκ∑l=1υδlΛtρlt,et,et−τt,∫t−ϱttesds+eTt∑j=1nθκjℳ⊗Pjet,(40)ℒV2et,t,κ≤eTtℳ⊗Q1et−1−μeTt−τtℳ⊗Q1et−τt,(41)ℒV3et,t,κ≤τeTtℳ⊗T1et−∫t−τtteTsℳ⊗T1esds,(42)ℒV4et,t,κ≤FTetℳ⊗QFet−1−μFet−τtℳ⊗QFet−τt,(43)ℒV5et,t,κ≤GTetℳ⊗TGet−1−μGTet−τtℳ⊗TGet−τt,(44)ℒV6et,t,κ≤τGTetℳ⊗T2Get−∫t−τttGTesℳ⊗T2Gesds,(45)ℒV7et,t,κ=ϱ2HTetℳ⊗ℛHet−ϱ∫t−ϱtHTetℳ⊗ℛHesds,(46)ℒV8et,t,κ=τ2zTtℳ⊗Zzt−τ∫t−τtzTsℳ⊗Zzsds,(47)ℒV9et,t,κ=ϱeTtℳ⊗Z1et−∫t−ϱteTsℳ⊗Z1esds.
By Lemma3,(48)−∫t−τtteTsℳ⊗T1esds≤−1τ∫t−τttesdsTℳ⊗T1∫t−τttesds,(49)−∫t−τttGTesℳ⊗T2Gesds≤−1τ∫t−τttGesdsTℳ⊗T2∫t−τttGesds,(50)−ϱ∫t−ϱtHTesℳ⊗ℛHesds≤−∫t−ϱttHesdsTℳ⊗ℛ∫t−ϱttHesds,(51)−τ∫t−τtzTsℳ⊗Zzsds≤−∫t−τttzsdsTℳ⊗Z∫t−τttzsds,(52)−∫t−ϱtzTsℳ⊗Z1zsds≤−1ϱ∫t−ϱttzsdsTℳ⊗Z1∫t−ϱttzsds.
By Lemma2, we get(53)∑l=1υδlΛtρlt,et,et−τt,∫t−ϱttesdsTℳ⊗Pκ∑l=1κδlΛtρlt,et,et−τt,∫t−ϱttesds≤∑l=1υδlΛtρlTt,et,et−τt,∫t−ϱttesdsℳ⊗Pκρlt,et,et−τt,∫t−ϱttesds.
As indicated by the Newton–Leibniz formula, the following equation holds:(54)2∑l=1υδlΛteTtℳ⊗S1l+zTtℳ⊗S2let−et−τt−∫t−τttzsds−∫t−τttρ˜lsdωs=0.
Also, the following condition holds for any matricesS3l and S4ll=1,2,…,υ.(55)2∑l=1υδlΛteTtℳ⊗S3l+zTtℳ⊗S4lIW⊗Alκet+IW⊗ℬlκFet+IW⊗ClκGet−τt+IW⊗Dlκ∫t−ϱttHesds+c1G1l⊗Γ1κet+c2G2l⊗Γ2κet−τt+c3G3l⊗Γ3κ∫t−ϱttesds−zt=0.
Noting thatEdωt=0 and combining (39)–(55), using Lemma 1 and Lemma 4, we have(56)ℒVet,t,κ≤∑l=1υδlΛt∑i=1N−1∑j=i+1N2eit−ejΛtTPκAlκeit−ejt+eit−ejtTPκℬlκ×feit−fejt+eit−ejtTPκClκgeit−τt−gejt−τt+eit−ejtTPκDlκ∫t−ϱttheis−hejsds+eit−ejtTc1WG1lΓ1κPκ×eit−ejt+eit−ejtTc2WG2lΓ2κPκeit−τt−ejt−τt+eit−ejt×c3WG3lΓ3κPκ∫t−ϱtteis−ejsds+ρlit,eit,eit−τt,∫t−ϱtteisds−ρljt,ejt,ejt−τt,∫t−ϱtteisdsTPκρlit,eit,eit−τt,∫t−ϱtteisds−ρljt,ejt,ejt−τt,∫t−ϱtteisds+eit−ejtTQ1eit−ejt−1−μ×eit−τt−ejt−τtTQ1eit−τt−ejt−τt+τeit−ejtTT1eit−ejt−1τ∫t−τtteis−ejsdsTT1∫t−τtteis−ejsds+feit−fejtTQ×feit−fejt−1−μfeit−τt−fejt−τtTQfeit−τt−fejt−τt+geit−gejtTTgeit−gejt−1−μ×geit−τt−gejt−τtTTeit−τt−gejt−τt+ϱ2heit−hejtTℛheit−hejt−∫t−ϱttheis−hejsdsT×ℛ∫t−ϱttheis−hejsds+τ2zit−zjtTZzit−zjt−∫t−τttzis−zjsdsTZ∫t−τttzis−zjsds+eit−ejtTS1l×eit−ejt−eit−ejtTS1leit−τt−ejt−τt−eit−ejtT×S1l∫t−τttzis−zjsds+zit−zjtTS2leit−ejt−zit−zjtS2l×eit−τt−ejt−τt−zit−zjtTS2l∫t−τttzis−zjsds+eit−ejtTS3lAlκeit−ejt+eit−ejtTS3lℬlκfeit−fejt+eit−ejtTS3lClκgeit−τt−gejt−τt+eit−ejtTS3lDlκ×∫t−ϱttheis−hejsds+eit−ejtTc1WG1lS3lΓ1κeit−ejt+eit−ejtTc2WG2lS3lΓ2κeit−τt−ejt−τt+eit−fejtT×c3WG3lS3lΓ3κ∫t−ϱtteis−ejsds−eit−ejtTS3lzit−zjt+zit−zjtTS4lAlκeit−ejt+zit−zjtTS4lℬlκfeit−fejt+zit−zjtTS4lClκgeit−τt−gejt−τt+zit−zjtTS4lDlκ×∫t−ϱttheis−hejsds+zit−zjtTc1WG1lΓ1κS4leit−ejt+eit−ejtTc2WG2lS3lΓ2κeit−τt−ejt−τt+eit−ejtT×c3WG3lS3lΓ3κ∫t−ϱtteis−ejsds−eit−ejtTS3lzit−zjt+zit−zjtTS4lAlκeit−ejt+zit−zjtTS4lℬlκfeit−fejt+zit−zjtTS4lClκgeit−τt−gejt−τt+zit−zjtTS4lDlκ×∫t−ϱttheis−hejsds+zit−zjtTc1WG1lΓ1κS4leit−ejt+zit−zjtTS4lc2WG2lΓ2κeit−τt−ejt−τt+zit−zjtTS4lc3×WG3lΓ3κ∫t−ϱtteis−ejsds−zit−zjtTS4lzit−zjt+τgeit−gejtTTgeit−gejt−1τ∫t−τttgeis−gejsdsTT2×∫t−τttgeis−gejsds+ϱeit−ejtTZ1eit−ejt−1ϱ∫t−ϱtteis−ejsdsTZ1∫t−ϱtteis−ejsds+eit−ejtT∑j=1nθκjPjeit−ejt.
Assumption1 ensures that the following inequality holds:(57)αeit−ejtfeit−fejtTX1TX2+X2TX1−X1T+X2T∗2Ieit−ejtfeit−fejt≤0,i.e.,(58)αeit−ejtfeit−fejtTX^−Xˇ∗2Ieit−ejtfeit−fejt≤0,1≤i<j≤N,αeit−τt−ejt−τtfeit−τt−fejt−τtTX^−Xˇ∗2Ieit−τt−ejt−τtfeit−τt−fejt−τt≤0,and similarly, one has(59)βeit−ejtgeit−gejtTY^Yˇ∗2Ieit−ejtgeit−gejt≤0,βeit−τt−ejt−τtgeit−τt−gejt−τtTY^−Yˇ∗2Ieit−τt−ejt−τtgeit−τt−gejt−τt≤0,γeit−ejtheit−hejtTX^−ℋˇ∗2Ieit−ejtheit−hejt≤0,γeit−τt−ejt−τtheit−τt−hejt−τtTX^−ℋˇ∗2Ieit−τt−ejt−τtgeit−τt−gejt−τt≥0.
From Assumption2 and condition (28), we get(60)ρlit,eit,eit−τt,∫t−ϱtteisds−ρljt,ejt,ejt−τt,∫t−ϱttejsdsT×Pκρlit,eit,eit−τt,∫t−ϱtteisds−ρljt,ejt,ejt−τt,∫t−ϱttejsds≤ϵeit−ejtTΥ1TΥ1eit−ejt+eit−τt−ejt−τtTΥ2TΥ2eit−τt−ejt−τt+∫t−ϱtteis−ejsdsTΥ3TΥ3∫t−ϱtteis−ejsds.
Combining equations (56)–(60), with detailed mathematical expectation, we have(61)EℒVet,t,κ≤E∑l=1υδlΛt∑i=1N−1∑j=i+1NΩijTtΨlijΩijt,where(62)Ωijt=eit−ejtTzit−zjtTfeit−fejtTgeit−gejtTheit−hejtTeit−τt−ejt−τtTfeit−τt−fejt−τtTgeit−τt−gejt−τtTheit−τt−hejt−τtT∫t−ϱtteis−ejsdsT∫t−τttzis−zjsdsT∫t−ϱttheis−hejsdsT∫t−τtteis−ejsdsT∫t−τttheis−hejsdsTT.
AndΨlij is defined in equation (29). It follows that(63)EℒVet,t,κ≤∑l=1υδlΛt−ηet2,for some η>0. Now, using Dynkin’s formula, we have that for all T≥0,(64)EVeT,T,κ−EVe0,0,κ≤−ηE∑l=1υδlΛt∫0Tes2ds.
Hence,(65)E∑l=1υδlΛt∫0Tes2ds≤η−1EVe0,0,κ,which implies that (20) holds. Therefore, from Definition 1, we have that drive system (9) and response system (12) are stochastically synchronous.
On the other hand, for the impulsive jumping time pointt=tk, by condition (30), we have(66)Vetk,tk,j−Vetk−,tk−,κ=eTtkℳ⊗Pjetk−eTtk−ℳ⊗Pκetk−+∫tk−τtktkeTsℳ⊗Q1esds+∫tk−τtktk∫θtkeTsℳ⊗T1esdsdθ+∫tk−τtktkFTesℳ⊗QFesds+∫tk−τtktkGTesℳ⊗TGesds+∫−τtk0∫tk+θtGTesℳ⊗T2Gesdsdθ+ϱ∫−ϱ0∫tk+θtkHTesℳ⊗ℛHesdsdθ+τ∫−τ0∫tk+θtkzTsℳ⊗Zzsdsdθ+∫−ϱ0∫tk+θtkeTsℳ⊗Z1esdsdθ−∫tk−−τtk−tk−eTsℳ⊗Q1esds−∫tk−−τtk−tk−∫θtk−eTsℳ⊗T1esdsdθ−∫tk−−τtk−tk−FTesℳ⊗QFesds−∫tk−−τtk−tk−GTesℳ⊗TGesds−∫−τtk−tk−∫tk−+θtk−GTesℳ⊗T2Gesdsdθ−ϱ∫−ϱ0∫tk−+θtk−HTesℳ⊗ℛHesdsdθ−τ∫−τ0∫tk−+θtk−zTsℳ⊗Zzsdsdθ,−∫−ϱ0∫tk−+θtk−eTsℳ⊗Z1esdsdθ=eTtkℳ⊗Pjetk−etk−ℳ⊗Pκetk−.
By Lemma4, we have(67)Vetk,tk,j−Vetk−,tk−,κ=eitk−−ejtk−TJlκTPjJlκeitk−−ejtk−−eitk−−ejtk−TPκeitk−−ejtk−=eitk−−ejtk−TJlκTPjJlκ−Pκeitk−−ejtk−≤0,which gives(68)Vetk,tk,j≤Vetk−,tk−,κ.
This completes the proof of Theorem1.Remark 1.
If there is no effect of the stochastic terms and impulsive control, Markovian jump is not considered, and it can be seen that the aforementioned system (17) is reduced to(69)det=IW⊗Aet+IW⊗ℬFet+IW⊗CGet−τt+IW⊗D∫t−ϱttHesds+c1G1⊗Γ1et+c2G2⊗Γ2et−τt+c3G3⊗Γ3∫t−ϱttesds.
Using the same method in Theorem1, we can get the following results.Corollary 1.
Given some positive scalarsτ>0, ϱ>0, μ>0, assume that Assumption 1 holds. Then, the complex networks (69) is asymptotically stable, if there exist positive definite matrices P1>0Q1>0, T1>0, T2>0, Q>0, ℛ>0, T>0, Z1>0, any matrix S1, and positive scalars α, β, γ such that the following linear matrix inequalities (LMIs) hold:(70)Σ=Σlm13×13<0,where(71)Σ11=P1A+ATP1+Q1+τT1+c1WG1P1Γ1+Γ1TP1+ϱZ1−αX^−βY^−γX^,Σ12=P1ℬ+αXˇ,Σ13=βYˇ,Σ14=γℋˇ,Σ15=c2WG2Γ2P1+c2WG2Γ2TS1T,Σ17=P1C,Σ19=c3WG3lP1Γ3+c3WG3Γ3TS1T,Σ110=P1D,Σ113=ATS1+Wc1G1Γ1TS1T,Σ22=Q−2αI,Σ213=ℬTS1,Σ33=T−2βI+τT2,Σ44=ϱ2ℛ−2γI,Σ55=−1−μQ1−αX^−βY^−γX^,Σ56=αXˇ,Σ57=βYˇ,Σ58=βℋˇ,Σ66=−2αI−1−μQ,Σ77=−2βI−1−μT,Σ713=CTS1,Σ88=−2γI,Σ99=−1ϱZ1,Σ1010=−ℛ,Σ1013=DTS1,Σ1111=−1τT1,Σ1212=−1τT2,Σ1313=−S1−S1T.Proof.
Consider the equivalent LKF in Theorem1. Eliminating the stochastic term and following the equivalent argument of Theorem 1, we can see that the following condition is satisfied for any matrix S1 with suitable measurement:(72)2e˙TtIW⊗S1−e˙t−IW⊗Aet+IW⊗ℬFet+IW⊗CGet−τt+IW⊗D∫t−ϱttHesds+c1G1⊗Γ1et+c2G2⊗Γ2et−τt+c3G3⊗Γ3∫t−ϱttesds=0,which implies that(73)EℒVet,t≤EΩijtTΣΩijt,where(74)Ωijt=eit−ejtTfeit−fejtTgeit−gejtTheit−hejtTeit−τt−ejt−τtTfeit−τt−fejt−τtTgeit−τt−gejt−τtTheit−τt−hejt−τtT∫t−ϱtteis−ejsdsT∫t−ϱttheis−hejsdsT∫t−τtteis−ejsdsT∫t−τttgeis−gejsdsTe˙it−e˙jtTT.
At that point, similar to the proof of Theorem1, we can get that framework (69) is asymptotically stable. This complete the proof.Remark 2.
In the past few years, the fuzzy logic theory has been seen to be effective in dealing with different types of complex nonlinear systems, which has received a great deal of attention in the literature [3, 61–64]. The T-S fuzzy model is one of the most popular models in the fuzzy systems. In this type of fuzzy models, a nonlinear system is represented by a set of local linear models smoothly connected by nonlinear membership functions, which has a convenient and simple dynamic structure, so the existing results for linear systems theory can be readily extended for the class of nonlinear systems. In this article, a general T-S fuzzy stochastic complex network with mixed time-varying delays is introduced.Remark 3.
In practice, due to the limitations of equipment and the disturbances of environment, the perfect information is rarely available. Thus, it is practical meaning to consider more constraints for fuzzy complex networks. However, in [65, 66], the sampled-data synchronization problem of complex networks with T-S fuzzy nodes has been investigated, but the mixed time-varying delays are not considered, which may be restrictive in many practical applications. Hence, it is very interesting to consider mixed time-varying coupling delays and not consider sampled data in fuzzy complex networks.Remark 4.
In order to show the reduced conservatism of our stability criteria, we consider the following system as a special case of system (17) which reduces to a delayed complex network described by(75)det=∑l=1υδlΛtIW⊗Alet+IW⊗ℬlFet+IW⊗ClGet−τt+IW⊗Dl∫t−ϱttHesds+G1l⊗Γ1et+G2l⊗Γ2et−τtdt+ρ˜ltdωt.
The proof follows similar ideas in to Theorem1 and Corollary 1. Now, we present the result in the following corollary.Corollary 2.
Given some positive scalarsτ>0, ϱ>0, and μ>0, 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 P1>0, Q1>0, T1>0, T2>0, Q>0, ℛ>0, T>0, Z>0, Z1>0, matrices Sil, i=1,2,3,4, l=1,2,…,υ, and positive scalars ϵ, α, β, γ such that the following linear matrix inequalities (LMIs) hold:(76)<P1ϵI,κ∈S,Ψ˜=Ψ˜14×14<0,where(77)Ψ˜l11=S3lAl+AlTS3lT+WG1lS3lΓ1+Γ1TS3l+P1Al+AlTP1+WG1lP1Γ1+Γ1TP1+ϱZ1+S1l+S1lT−αX^+ϵΥ1Υ1T−βY^−γX^+Q1+τT1,Ψ˜l12=−S3l+AlTS4lT+S2lT+WG1lS4lTΓ1T,Ψ˜l13=S3lℬl+αXˇ+P1ℬl,Ψ˜l14=βYˇ,Ψ˜l15=γℋˇ,Ψ˜l16=WG2lS3lΓ2+P1Γ2−S1l,Ψ˜l18=S3lCl+P1Cl,Ψ˜l111=−S1l,Ψ˜l112=S3lDl+P1Dl,Ψ˜l22=τ2Z−S4l−S4lT,Ψ˜l23=S4lℬl,Ψ˜l26=S4lWG2lΓ2−S2l,Ψ˜l28=S4lCl,Ψ˜l211=−S2l,Ψ˜l212=S4lDl,Ψ˜l33=Q−2αI,Ψ˜l44=T−2βI+τT2,Ψ˜l55=ϱ2ℛ−2γI,Ψ˜l66=−1−μQ1−αX^−βY^−γX^+ϵΥ2Υ2T,Ψ˜l67=αXˇ,Ψ˜l68=βYˇ,Ψ˜l69=γℋˇ,Ψ˜l77=−2αI−1−μQ,Ψ˜l88=−2βI−1−μT,Ψ˜l99=−2γI,Ψ˜l1010=−1ϱZ1+ϵΥ3Υ3T,Ψ˜l1111=−Z,Ψ˜l1212=−ℛ,Ψ˜l1313=−1τT1,Ψ˜l1414=−1τT2.


## 4. Numerical Examples

In this section, numerical models are displayed to show the adequacy of the results determined previously.Example 1.
We consider the following stochastic T-S fuzzy complex systems with mixed time-varying delays and Markovian jumping parameters with 3 nodes andκ=1,2, l=1,2.Fuzzy rule 1:IFΛ1t=x11tisχ11x11tTHEN(78)dxit=A11xit+ℬ11fxit+C11gxit−τt+D11∫t−ϱtthxisds+c1G11ijΓ11xit+c2G21ijΓ21xit−τt+c3G31ijΓ31∫t−ϱttxisdsdt+ρ1it,xit,xit−τt,∫t−ϱttxisdsdωt,t≠tk,t≥t0,xitk=J11xitk−,i=1,2,…,N,t=tk,Fuzzy rule 2:IFΛ1t=x11tisχ12x11tTHEN(79)dxit=A21xit+ℬ21fxit+C21gxit−τt+D21∫t−ϱtthxisds+c1G11ijΓ11xit+c2G22ijΓ21xit−τt+c3G32ijΓ31∫t−ϱttxisdsdt+ρ2it,xitxit−τt,∫t−ϱttxisdsdωt,t≠tk,t≥t0,xitk=J21xitk−,i=1,2,…,N,t=tk.Fuzzy Rule 1:IFΛ1t=x11tisχ11x11tTHEN(80)dxit=A12xit+ℬ12fxit+C12gxit−τt+D12∫t−ϱtthxisds+c1G12ijΓ12xit+c2G21ijΓ22xit−τt+c3G31ijΓ32∫t−ϱttxisdsdt+ρ1it,xit,xit−τt,∫t−ϱttxisdsdωt,t≠tk,t≥t0.xitk=J12xitk−,i=1,2,….,N,t=tk,Fuzzy rule 2:IFΛ1t=x11tisχ12x11tTHEN(81)dxit=A22xit+ℬ22fxt+C22gxit−τt+D22∫t−tthxisds+c1G12ijΓ12xit+c2G22ijΓ22xit−τt+c3G32ijΓ32∫t−ϱttxisdsdt+ρ2it,xit,xit−τt,∫t−ϱttxisdsdωt,t≠tk,t≥t0,xitk=J22xitk−,i=1,2,…,N,t=tk,where xt=xi1t,xi2tTi=1,2 and effective relevant parameters are given as follows.Mode 1:(82)A11=1.5001.5,ℬ11=−0.01−0.080.060.019,C11=0.030.020.060.021,D11=−0.025−0.010.30.075,A21=−10.20−1,ℬ21=1.8−0.12−5.23.5,C21=0.930.621.060.021,D21=−3.025−1.01−0.33.075,Γ11=0.8000.8,Γ21=1001,Γ31=0.5000.5,J11=0.4000.4,J21=0.4000.4,G11=−0.20.10.10.1−0.20.10.10.1−0.2,G21=−0.30.150.150.15−0.30.150.150.15−0.3,G31=−1010−1111−2.Mode 2:(83)A12=2002,ℬ12=0.15−1.080.752.019,C12=0.231.020.561.021,D12=−0.025−1.01−1.30.75,A22=1.650.201.65,ℬ22=1.15−1.080.851.019,C22=−1.230.021.560.021,D22=−1.025−0.01−2.31.75,Γ12=0.5000.5,Γ22=Γ32=1001,J12=0.4000.4,J22=0.4000.4,G12=−2200−2220−2,G22=−2100−2110−2,G32=−1103−1−201−1.
The transition rate matrix with two operation modes is(84)Θ=−364−7.
Letϱ=1, τ=1.5, μ=0.5, W=3, c1=c2=c3=1. The nonlinear vector-valued functions are(85)fxit=tanh0.2xi1t,−tanh0.75xi2tT,gxit=hxit=0.2xi1k−tanh0.1xi1t,0.1xi2tT.
The other parameters are chosen as follows:(86)X1=0.30.200.2,Y1=0.1000.1,ℋ1=0.1000.1,X2=0.50.200.95,Y2=0.2000.1,ℋ2=0.2000.1,Σ1=Σ2=Σ3=0.03000.03.
Solving LMIs (28)–(30) in Theorem 1, we can obtain the feasible solutions as follows:(87)P1=10−40.3928−0.4520−0.45200.7102,P2=10−40.3381−0.4489−0.44890.7351,Q1=10−3−0.1152−0.1070−0.10700.1014,T1=10−40.1915−0.1051−0.10510.1581,T2=0.0013−0.0001−0.00010.0014,Q=10−3−0.09330.08440.0844−0.1099,ℛ=0.0016−0.0004−0.00040.0020,T=10−3−0.52210.11130.1113−0.6436,Z=10−60.75170.95570.95570.1196,Z1=0.1156−0.0845−0.08450.1132,S11=10−4−0.10430.02050.0205−0.1511,S21=10−6−0.2608−0.0113−0.01130.1344,S31=10−4−0.10800.06950.0695−0.1348,S41=10−50.09740.04300.0430−0.1378,S12=10−2−0.1749−0.0102−0.0102−0.2199,S22=10−6−0.7636−0.0203−0.02030.4242,S32=10−4−0.31250.42070.4207−0.5928,S42=10−50.1445−0.1174−0.11740.0773,ρ=6.5704,α=2.0583,β=9.7893,γ=0.0037.
The maximum allowable upper boundsμ obtained for different values of τ are listed in Table 1.
For example simulation, we examine the initial conditions ofxit=−2,0.15 with mode i=1,2 depicted in Figure 1, the state response from which we can see that all the components converge to zero. Figures 2 and 3 illustrate the synchronization error of system (25) and consider the networks (9) and (12). State response of the errors eit is illustrated by Figure 4. Additionally, Figure 5 shows the jumping between models during simulation. From the simulations, one can find that the T-S fuzzy Markovian jumping complex networks are stochastically synchronous.Table 1
Maximum allowable bound forμ with different values of τ.
τ0.50.60.70.80.90.101.0μ1.34561.27891.00180.73560.63450.58900.2359Figure 1
State responses of the system in Example1.Figure 2
Synchronization error of system (25).Figure 3
Synchronization error of the system in Example1.Figure 4
State responses of the error system.Figure 5
Jumping between modes during simulation.Example 2.
Consider the stochastic complex networks (75) with the following matrix parameters:(88)A1=−0.50.200.95,ℬ1=1001,C1=1001,D1=1001,G11=−0.20.10.10.1−0.20.10.10.1−0.2,G21=−0.30.10.20.1−0.30.20.10.2−0.3,Γ1=Γ1=0.5000.5,X1=disg00,X2=disg0.20.75,Y1=disg0.20.1,Y2=disg0.10.1,ℋ1=disg0.20.1,ℋ2=disg0.10.1.
Letϱ=1, τ=1.5, μ=0.5, W=3. Solve Example 2 using LMIs in Corollary 2. The maximum value of upper bound τ compared with the results in [67] is listed in Table 2. One can know clearly that the results obtained by Corollary 2 can provide large admissible upper bounds than the stability criteria in [67].Table 2
The corresponding allowable upper bound of time delayτ for Example 2.
Method[67]Corollary2Example21.50012.5670


## 5. Conclusion

In this study, we obtained a novel design of a synchronization analysis for T-S fuzzy complex networks’ Markovian jumping parameters with stochastic discrete and distributed time-varying delays by means of impulsive control. By using Kronecker product and stochastic examination hypothesis, constructing an appropriate Lyapunov–Krasovskii functional (LKF), and employing Newton–Leibniz formulation, we show that the synchronization problem of drive-response framework is resolvable in terms of linear matrix inequalities (LMIs) which are feasible. At last, the effectiveness of the theoretical results were illustrated with numerical examples and their simulations. In the further examinations, we will look through the networks with generalizability to deal with some other problems on fault tolerant control and stochastic synchronization of Markovian switching for complex dynamical networks.



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