# Global Finite-Time and Fixed-Time Synchronization for Discontinuous Complex Dynamical Networks with Semi-Markovian Switching and Mixed Delays

Authors: Lina Yu; Jingchao Zhang; Yunfei Ma; Xinhua Tan; Chunwei Wang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper is concerned with the global finite-time and fixed-time synchronization for a class of discontinuous complex dynamical networks with semi-Markovian switching and mixed time-varying delays. The novel state-feedback controllers, which include integral terms and discontinuous facts, are designed to realize the global synchronization between the drive system and response system. By applying the Lyapunov functional method and matrix inequality analysis technique, the global finite-time and fixed-time synchronization conditions are addressed in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are provided to illustrate the feasibility of the proposed control scheme and the validity of theoretical results.


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## Body

## 1. Introduction

In the past ten years, complex dynamical networks (CDNs) have aroused a great attention owing to their extensive applications in many areas such as the Internet, biological system, the World Wide Web, power grid networks, and so on [1–3]. CDNs consist of many subnetworks, which are called nodes, and each of the nodes has the similar functions. Different connection modes between nodes form different networks. Thus, such a complexity makes that the study of CDNs become a hot topic. Up to now, many significative works with respect to CDNs are reported in the literature, see [4–10] and references therein. It should be mentioned that the analysis of dynamical behaviors of CDNs has become a research hotspot in recent years.The global synchronization, as a dynamical behavior, is the most important. It means the dynamical behavior of the network systems achieve the identical state in time-spatial. Nowadays, the considerable attention is being devoted to the analysis of the global synchronization of CDNs, and some effective synchronization criteria with respect to CDNs have been established in the existing papers [11–15]. In [11], the synchronization problem of complex dynamical networks was addressed under the way of impulsive control. Reference[12] introduced the synchronization problem of complex dynamical networks with switching topology via adaptive control. Also, global synchronization of complex dynamical networks with nonidentical nodes was considered in [13]. Moreover, synchronization of complex dynamical networks with nonidentical nodes was illustrated in Ref. [14], and the obtained result global synchronization for discrete-time stochastic complex networks in Ref. [15] was complex, which has randomly shown nonlinearities and mixed time delays. To the best of our knowledge, most of the existing works with regard to the synchronization of CDNs is taking the continuous networks system into account. But, in practice, it is not applicable to all the synchronization conditions. Thus, the study of discontinuous CDNs has attracted the interests of many scholars. Reference [16] investigated the finite-time synchronization of complex networks with nonidentical discontinuous nodes. Also, exponential synchronization of a discontinuous complex-valued complex dynamical network is realized in [17]. Reference [18] illustrated the synchronization of time-delayed complex dynamical networks with discontinuous coupling.As we all know that, among the different types of synchronization, global finite-time synchronization is one of the most optimal synchronization types, only the maximum synchronization time for global finite-time synchronization can be computed. Moreover, some efforts have been spent on solving the global finite-time synchronization of discontinuous CDNs. The finite-time synchronization of chaotic complex networks with stochastic disturbance was obtained in [19]. In Ref. [20], the finite-time hybrid projective synchronization of the drive-response complex networks with distributed delay was considered by the researchers, based on the adaptive intermittent control. Reference [21] introduces finite-time cluster synchronization of Markovian switching complex networks with stochastic perturbations. In Ref. [22], the finite-time synchronization of Markovian jump complex networks with partially unknown transition rates was introduced by the author. It is also necessary to point out that the settling time of finite-time synchronization heavily depends on the initial conditions which would result in different convergence rates under different initial conditions. But, the initial conditions may be ineffective or invalid in practice application. For overcoming these difficulties, a new concept named fixed-time synchronization is firstly taken into consideration in [23]. Then, the future study on fixed-time synchronization can be found in [24,25]. By means of the sliding mode control technique, the fixed-time synchronization of complex dynamical networks is realized in [24]. In [25], the fixed-time cluster synchronization for complex networks is derived under the pinning control scheme. Furthermore, in [25], accompanying with nonidentical nodes and stochastic noise perturbations, the fixed-time synchronization of complex networks was investigated. Also, the fixed-time synchronization of hybrid coupled networks was addressed in Ref. [26].By adding the Markovian process into the network systems of CDNs, a new network model is developed. Up to now, the study concerning synchronization of CDNs with Markovian switching, especially the global finite-time synchronization, has received wide attention from the scholars, and many efforts have been made for analyzing the synchronization of CDNs with Markovian switching, see [21, 27–30]. Since the sojourn time of the Markovian process obeys exponential distribution, it results in the transition rate to be a constant. However, the sojourn-time of the semi-Markovian process can obey to some other probability distributions, such as Weibull distribution, Gaussian distribution, and so on. Therefore, the investigation for the global finite-time and fixed-time synchronization of CDNs with semi-Markovian switching is of great theoretical value and application value. In [31], the finite-time H∞ synchronization for complex networks with semi-Markov jump topology was addressed. The finite-time H∞ control for linear systems with semi-Markovian switching was discussed in [32–34]. The finite-time nonfragile synchronization of stochastic complex dynamical networks with semi-Markov switching outer coupling was investigated in Ref. [35]. The global fixed-time synchronization of CDNs with semi-Markovian switching is relatively complex. In addition, it is also difficult to apply the semi-Markovian switching to complex dynamic network (CDNs) and use appropriate control methods to derive the corresponding global synchronization conditions. As a result, the global fixed-time synchronization of CDNs with semi-Markovian switching is seldom studied before.Motivated by the abovementioned discussions, we intend to consider the global finite-time and fixed-time synchronization for discontinuous CDNs with semi-Markovian switching and mixed time-varying delays. The main contributions of this paper, different from the existing works, can be summarized as follows:(1)
The novel discontinuous state-feedback controllers, which include integral terms, are designed(2)
The mixed delay term with integral and discontinuous function is handled by the approach of the contraction of inequalities(3)
The mixed delay and semi-Markovian switching are introduced in the construction of the discontinuous CDNs model(4)
The global finite-time and fixed-time synchronization conditions are addressed in terms of LMIsThe rest of this article is arranged as follows. Some preliminaries and model description are described in Section2. In Section 3, we introduce the main results: finite time and fixed-time synchronization with different nonlinear controllers. In Section 4, two examples are presented which are in order to show the correctness of our main results. In Section 5, also the last part, the conclusion of this paper is given.Notation:IN denotes the identity matrix with proper dimension.R represents the sets of real numbers. Rn×n denotes the set of all n×n matrices, and Rn denotes the n-dimensional Euclidean space. x stands for the Euclidean norm of the vector x∈Rn, and x=xTx. D=dijn×n denotes an n×n-dimension matrix D=λmaxDTD, and the superscript T means transposition. For all the matrices, λmin·andλmax· stands for the minimum and the maximum eigenvalue of the matrices, respectively.X<YX>Y, where X and Y are both symmetric matrices, means that X−Y is negative (positive) definite.B=bijn×n and B=bijn×n⋅E stand for mathematical expectation. Ls=1/2L+LT. ΓVxt,rt,t denotes the infinitesimal generator of Vxt,rt,t. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operation.


## 2. Preliminaries and Model Description

### 2.1. Preliminaries

LetΘ,F,Ft≥0,P be the complete probability space with filtration Ft≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0 contain all P-null sets), where Θ is the sample space, F is the algebra of events, and P is the probability measure defined on F. Let rt,t≥0 be a continuous-time semi-Markovian process taking values in a finite state space S=1,2,…,N. The evolution of the semi-Markovian process rt is governed by the following probability transitions:(1)Prt+h=k|rt=s=πskhh+oh,ifs≠k,1+πsshh+oh,ifs=k,where h>0,limh⟶0oh/h=0,πskh≥0s,k∈S,s≠k is the transition rate from mode s to k and for any state or mode, and it satisfies(2)πssh=−∑k=1,k≠sNπskh.Remark 1.
In practice, the transition rateπskh is general bounded, i.e., π¯sk≤πskh≤π¯sk, where π¯sk and π¯sk are real constant scalars. Then, πskh=πsk±Δπsk, where πsk=1/2π¯sk+π¯sk and πsk≤ω¯sk with ω¯sk=1/2π¯sk−π¯sk.Definition 1.
The complex dynamical networks model (17) is said to be synchronized with the networks model (18), if for any initial condition, we have(3)limt⟶tiΕxit−yit=0,Εxit−yit≡0,for t>ti,i=1,2,…,N.Definition 2.
(See [36]). The Filippov set-valued map of fix at x∈Rn is defined as follows:(4)Fx=∩T>0∩μN=0co¯fBx,T\Π,where co¯E is the closure of the convex hull of the set E, Bx,T=y:y−x≤T and μΠ is the Lebesgue measure of set Π.Definition 3.
(See [37]). The set-valued map of F:Rn⟶Rn is said to satisfy the basic conditions in the domain W¯∈Rn, if for any x∈W¯, Fx is upper semicontinuous in W¯.Definition 4.
(See [38]). A function Vx:Rn⟶R is C-regular if it is(5)1regular inRn2positive definite,i.e.,Vx>0forx≠0andV0=03radially unbounded,i.e.,Vx⟶+∞,asx⟶∞Lemma 1 (See [39]).
Given any scalarε and matrix W∈Rn×n, there exists symmetric positive definite matrices U∈Rn×n, such that(6)W+WT≤εW+ε−1U−1WT.Lemma 2 (See [40]).
Letζ1,ζ2,…,ζn≥0,0<s<w,r>1. Then, the following two inequalities hold:(7)∑i=1nζiw1/w≤∑i=1nζis1/s,∑i=1nζir≥n1−r∑i=1nζir.Lemma 3.
LetP∈Rn×n be a symmetric matrix, and let x∈Rn; then, we have the following inequality:(8)λminPxTx≤xTPx≤λmaxPxTx.Lemma 4 (See [41]).
Given constant matricesΦ1, Φ2, and Φ3, where Φ3=Φ3T and Φ2>0, we have(9)Φ1+Φ3TΦ2−1Φ3<0,if and only if(10)Φ1Φ3TΦ3−Φ2≤0.Lemma 5 (See [42]).
Lety1,y2,…,yn∈Rn be any vectors and 0<r^<2 be a real number which satisfies the following condition:(11)y1r^+y2r^+⋯+ynr^≥y12+y22+⋯+yn2r^/2.


### 2.2. Model Description

The model we consider in the present paper is the complex dynamical network model which consists ofN identical nodes with semi-Markovian switching in the probability space Θ,F,Ft≥0,P. The dynamics of ith nodes of drive systems is described by the following differential equations:(12)x˙it=Drtxit+Artfixit+Brtfixit−τt+∑j=1NLijrtΛxjt+c1rt∫t−dttfixisds,with the initial condition xit=ψit,rt,−τ≤t≤0.Also, the corresponding response system is(13)y˙it=Drtyit+Artfiyit+Brtfiyit−τt+∑j=1NLijrtΛyjt+c1rt∫t−dttfyisds+uiti,with the initial condition yit=χit,rt,−τ≤t≤0.Here,rt,t≥0 is the continuous-time semi-Markovian process, and rt describes the evolution of the mode at time t. xit∈Rn, yit∈Rn denotes the state vector of ith network at time t; Drt∈Rn×n is a positive definite diagonal matrix; Art∈Rn×n and Brt∈Rn×n are matrices with real values in mode rt; N is the number of coupled nodes; fixi=f1xi1t,f2xi2t,…,fnxint∈Rn is the nonlinear vector-valued function; and uit∈Rn is the control input of the ith node to be determined later. c1rt represents the distributively delayed connection weight; the diagonal matrix Λ>0 represents the inner-coupling matrix of the complex networks; and Lijn×n stands for the coupling configuration matrix which represents the topological structures of the considered networks. For matrix Lijrt, if there is a connection from node j to ij≠i, then Lijrt>0; otherwise, Lijrt=0j≠i.(14)Lijrt=−∑j=1,j≠iNLijrt,i=1,2,…,N.The functionτt and dt denote the discrete delay and distributed delay and satisfy(15)0<τt≤τ,0<dt≤d,0<τ˙t≤r,0<dt≤ω,where τ>0, d>0 and ω<1,r<1 are some known constants.Throughout this paper, we list the following assumptions:H1:Fixi,i=1,2,…,N satisfy the general conditions defined in Definition 3, where Fixi=∩T>0∩μN=0co¯fBx,TH2: there exists two positive constants αi and βi,i=1,2,…,N, and then, the following inequalities hold:(16)γt−γt˜≤αiyit−xit+βi,where γt∈co¯fixit and γt˜∈co¯fiyit,i=1,2,…,NFor notation simplicity, we replaceDrt, Art, and Brt with Ds,As,andBs, for rt=s∈S. Then, the complex dynamical systems can be rewritten as follows:(17)x˙it=Dsxit+Asco¯fixit+Bsco¯fixit−τt+∑j=1NLijsΛxjt+c1rt∫t−dttco¯fixisds,xit=ψ¯it,−τ≤t≤0,where ψ¯it is a measurable function and ψ¯it∈co¯fiψit.(18)y˙it=Dsyit+Asco¯fiyit+Bsco¯fiyit−τt+∑j=1NLijsΛyjt+c1rt∫t−dttco¯fiyisds+uit,yit=χ¯it,−τ≤t≤0,where χ¯it is a measurable function and χ¯it∈co¯fiχit.The network model (17) is a differential equation with a discontinuous right-hand side, and the traditional definition of the solution is not applicable here. Hence, we introduce the Filippov solution for system (17).Definition 5 (See [43]).
A functionx:−τ,T⟶Rn,T∈0,+∞ is a solution (in the sense of Filippov) of the discontinuous system (17) on −τ,T if(19)ixitis absolutely on−τ,TiiThere exists a measurable functionγt=γ1tγ2t,…,γNtT:−τ,T⟶Rn,such thatγt∈co¯fixit,for a.a.t∈0,TIn this paper, the error dynamics system between drive-response system (17) and (18) attracts us. Hence, let eit=yit−xit be the synchronization error system, and it can be expressed as(20)e˙it=Dseit+Asf˜ieit+Bsf˜ieit−τt+∑j=1NLijsΛeit+c1rt∫t−dttf˜ieisds+uit,eit=ψ¯it−χ¯it,−τ≤t≤0,where f˜ieit=γt−γt˜∈co¯fiyit−co¯fixit, f˜ieit−τt=γt−τt−γt−τt˜∈co¯fiyit−τt−co¯fixit−τt, and eit=ψ¯it−χ¯it∈co¯fiψit−co¯fiχit.In the following, some Lemmas about global finite-time and fixed-time synchronization will be introduced.Lemma 6 (See [44]).
Suppose thatVx:Rn⟶R is C-regular and that xt:0,+∞⟶Rn is absolutely continuous on any compact interval of 0,+∞. Let υt=Vxt, if there exists a continuous function γ:0,+∞⟶R with γσ>0 for σ∈0,+∞, such that(21)ℰυt≤−ℰγυt,for all t≥0 such that υt>0 and υt is differentiable at t and γ· satisfies the condition(22)∫0υ01γσdσ=t1<+∞.Then, we haveυt=0 for t≥t1. In particular, if γσ=Mσv for all σ∈0,+∞, where μ∈0,1 and M>0, then the setting time is estimated as(23)t1=υ1−vM1−v.Lemma 7 (See [23]).
If a continuous radially unbounded functionV:Rn⟶R+ satisfies(24)ℰD+Veit≤−p˜ℰVαeit−q˜ℰVβeit,where p˜>0,q˜>0,0<α<1,β>1, then Vt≡0, with the settling time bounded by(25)Te0≤Tmax=1p˜1−α+1q˜β−1.Lemma 8 (See [45]).
If there exists a continuous radially unbounded functionV:Rn⟶R+ such that(1)
Veit=0⇔eit=0(2)
For somep˜,q˜,α=1−1/2ρ,β=1+1/2ρ,ρ>1, any solution of (20) admits(26)ℰD+Veit≤−p˜ℰVαeit−q˜ℰVβeit,then the origin is global fixed-time stable for (20), and the estimated value of the settling time function satisfies(27)Te0≦Tmax=πρp˜q˜,∀e0∈Rn.


## 2.1. Preliminaries

LetΘ,F,Ft≥0,P be the complete probability space with filtration Ft≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0 contain all P-null sets), where Θ is the sample space, F is the algebra of events, and P is the probability measure defined on F. Let rt,t≥0 be a continuous-time semi-Markovian process taking values in a finite state space S=1,2,…,N. The evolution of the semi-Markovian process rt is governed by the following probability transitions:(1)Prt+h=k|rt=s=πskhh+oh,ifs≠k,1+πsshh+oh,ifs=k,where h>0,limh⟶0oh/h=0,πskh≥0s,k∈S,s≠k is the transition rate from mode s to k and for any state or mode, and it satisfies(2)πssh=−∑k=1,k≠sNπskh.Remark 1.
In practice, the transition rateπskh is general bounded, i.e., π¯sk≤πskh≤π¯sk, where π¯sk and π¯sk are real constant scalars. Then, πskh=πsk±Δπsk, where πsk=1/2π¯sk+π¯sk and πsk≤ω¯sk with ω¯sk=1/2π¯sk−π¯sk.Definition 1.
The complex dynamical networks model (17) is said to be synchronized with the networks model (18), if for any initial condition, we have(3)limt⟶tiΕxit−yit=0,Εxit−yit≡0,for t>ti,i=1,2,…,N.Definition 2.
(See [36]). The Filippov set-valued map of fix at x∈Rn is defined as follows:(4)Fx=∩T>0∩μN=0co¯fBx,T\Π,where co¯E is the closure of the convex hull of the set E, Bx,T=y:y−x≤T and μΠ is the Lebesgue measure of set Π.Definition 3.
(See [37]). The set-valued map of F:Rn⟶Rn is said to satisfy the basic conditions in the domain W¯∈Rn, if for any x∈W¯, Fx is upper semicontinuous in W¯.Definition 4.
(See [38]). A function Vx:Rn⟶R is C-regular if it is(5)1regular inRn2positive definite,i.e.,Vx>0forx≠0andV0=03radially unbounded,i.e.,Vx⟶+∞,asx⟶∞Lemma 1 (See [39]).
Given any scalarε and matrix W∈Rn×n, there exists symmetric positive definite matrices U∈Rn×n, such that(6)W+WT≤εW+ε−1U−1WT.Lemma 2 (See [40]).
Letζ1,ζ2,…,ζn≥0,0<s<w,r>1. Then, the following two inequalities hold:(7)∑i=1nζiw1/w≤∑i=1nζis1/s,∑i=1nζir≥n1−r∑i=1nζir.Lemma 3.
LetP∈Rn×n be a symmetric matrix, and let x∈Rn; then, we have the following inequality:(8)λminPxTx≤xTPx≤λmaxPxTx.Lemma 4 (See [41]).
Given constant matricesΦ1, Φ2, and Φ3, where Φ3=Φ3T and Φ2>0, we have(9)Φ1+Φ3TΦ2−1Φ3<0,if and only if(10)Φ1Φ3TΦ3−Φ2≤0.Lemma 5 (See [42]).
Lety1,y2,…,yn∈Rn be any vectors and 0<r^<2 be a real number which satisfies the following condition:(11)y1r^+y2r^+⋯+ynr^≥y12+y22+⋯+yn2r^/2.


## 2.2. Model Description

The model we consider in the present paper is the complex dynamical network model which consists ofN identical nodes with semi-Markovian switching in the probability space Θ,F,Ft≥0,P. The dynamics of ith nodes of drive systems is described by the following differential equations:(12)x˙it=Drtxit+Artfixit+Brtfixit−τt+∑j=1NLijrtΛxjt+c1rt∫t−dttfixisds,with the initial condition xit=ψit,rt,−τ≤t≤0.Also, the corresponding response system is(13)y˙it=Drtyit+Artfiyit+Brtfiyit−τt+∑j=1NLijrtΛyjt+c1rt∫t−dttfyisds+uiti,with the initial condition yit=χit,rt,−τ≤t≤0.Here,rt,t≥0 is the continuous-time semi-Markovian process, and rt describes the evolution of the mode at time t. xit∈Rn, yit∈Rn denotes the state vector of ith network at time t; Drt∈Rn×n is a positive definite diagonal matrix; Art∈Rn×n and Brt∈Rn×n are matrices with real values in mode rt; N is the number of coupled nodes; fixi=f1xi1t,f2xi2t,…,fnxint∈Rn is the nonlinear vector-valued function; and uit∈Rn is the control input of the ith node to be determined later. c1rt represents the distributively delayed connection weight; the diagonal matrix Λ>0 represents the inner-coupling matrix of the complex networks; and Lijn×n stands for the coupling configuration matrix which represents the topological structures of the considered networks. For matrix Lijrt, if there is a connection from node j to ij≠i, then Lijrt>0; otherwise, Lijrt=0j≠i.(14)Lijrt=−∑j=1,j≠iNLijrt,i=1,2,…,N.The functionτt and dt denote the discrete delay and distributed delay and satisfy(15)0<τt≤τ,0<dt≤d,0<τ˙t≤r,0<dt≤ω,where τ>0, d>0 and ω<1,r<1 are some known constants.Throughout this paper, we list the following assumptions:H1:Fixi,i=1,2,…,N satisfy the general conditions defined in Definition 3, where Fixi=∩T>0∩μN=0co¯fBx,TH2: there exists two positive constants αi and βi,i=1,2,…,N, and then, the following inequalities hold:(16)γt−γt˜≤αiyit−xit+βi,where γt∈co¯fixit and γt˜∈co¯fiyit,i=1,2,…,NFor notation simplicity, we replaceDrt, Art, and Brt with Ds,As,andBs, for rt=s∈S. Then, the complex dynamical systems can be rewritten as follows:(17)x˙it=Dsxit+Asco¯fixit+Bsco¯fixit−τt+∑j=1NLijsΛxjt+c1rt∫t−dttco¯fixisds,xit=ψ¯it,−τ≤t≤0,where ψ¯it is a measurable function and ψ¯it∈co¯fiψit.(18)y˙it=Dsyit+Asco¯fiyit+Bsco¯fiyit−τt+∑j=1NLijsΛyjt+c1rt∫t−dttco¯fiyisds+uit,yit=χ¯it,−τ≤t≤0,where χ¯it is a measurable function and χ¯it∈co¯fiχit.The network model (17) is a differential equation with a discontinuous right-hand side, and the traditional definition of the solution is not applicable here. Hence, we introduce the Filippov solution for system (17).Definition 5 (See [43]).
A functionx:−τ,T⟶Rn,T∈0,+∞ is a solution (in the sense of Filippov) of the discontinuous system (17) on −τ,T if(19)ixitis absolutely on−τ,TiiThere exists a measurable functionγt=γ1tγ2t,…,γNtT:−τ,T⟶Rn,such thatγt∈co¯fixit,for a.a.t∈0,TIn this paper, the error dynamics system between drive-response system (17) and (18) attracts us. Hence, let eit=yit−xit be the synchronization error system, and it can be expressed as(20)e˙it=Dseit+Asf˜ieit+Bsf˜ieit−τt+∑j=1NLijsΛeit+c1rt∫t−dttf˜ieisds+uit,eit=ψ¯it−χ¯it,−τ≤t≤0,where f˜ieit=γt−γt˜∈co¯fiyit−co¯fixit, f˜ieit−τt=γt−τt−γt−τt˜∈co¯fiyit−τt−co¯fixit−τt, and eit=ψ¯it−χ¯it∈co¯fiψit−co¯fiχit.In the following, some Lemmas about global finite-time and fixed-time synchronization will be introduced.Lemma 6 (See [44]).
Suppose thatVx:Rn⟶R is C-regular and that xt:0,+∞⟶Rn is absolutely continuous on any compact interval of 0,+∞. Let υt=Vxt, if there exists a continuous function γ:0,+∞⟶R with γσ>0 for σ∈0,+∞, such that(21)ℰυt≤−ℰγυt,for all t≥0 such that υt>0 and υt is differentiable at t and γ· satisfies the condition(22)∫0υ01γσdσ=t1<+∞.Then, we haveυt=0 for t≥t1. In particular, if γσ=Mσv for all σ∈0,+∞, where μ∈0,1 and M>0, then the setting time is estimated as(23)t1=υ1−vM1−v.Lemma 7 (See [23]).
If a continuous radially unbounded functionV:Rn⟶R+ satisfies(24)ℰD+Veit≤−p˜ℰVαeit−q˜ℰVβeit,where p˜>0,q˜>0,0<α<1,β>1, then Vt≡0, with the settling time bounded by(25)Te0≤Tmax=1p˜1−α+1q˜β−1.Lemma 8 (See [45]).
If there exists a continuous radially unbounded functionV:Rn⟶R+ such that(1)
Veit=0⇔eit=0(2)
For somep˜,q˜,α=1−1/2ρ,β=1+1/2ρ,ρ>1, any solution of (20) admits(26)ℰD+Veit≤−p˜ℰVαeit−q˜ℰVβeit,then the origin is global fixed-time stable for (20), and the estimated value of the settling time function satisfies(27)Te0≦Tmax=πρp˜q˜,∀e0∈Rn.


## 3. Main Results

### 3.1. Global Finite-Time Synchronization

Based on the linear matrix inequalities technique [45] and the Lyapunov functional method [38, 46], this section aims to develop some new conditions which can ensure the synchronization between complex dynamical network models (17) and (18) over a finite-time interval.For the purpose of global finite-time synchronization, a state-feedback controller is designed as follows:(28)uit=−ξirt+κirtsigneit−2ηirteit−κirteiteit2∫t−τtteiTsGeisds1/2−c1rt∫t−dtteisds,in which signeit=signei1t,…,signeintT, eit=ei1t,ei2t,…,eintT, κirt>0 and is a tunable parameter, and ηirt is a parameter to be determined.Theorem 1.
Let the assumptionsH1 and H2 hold. Then, the drive system (17) is synchronized with the response system (18) in finite time, if there exist symmetric positive definite matrices Ps,U, and G, such that(29)D˜+θL^s˜−ηirtIn<0,βiPsAs+PsBs+ΔωPs−Psξirt<0,α^<1,(30)ΩαiPsBs∗r−1G≤0,where D˜=diagD1,D2,…,Dn. L^=lij^n×n, lij^=lij,i≠j, lii^=λmin/θ, θ=Λ, and λmin is the eigenvalue of Λs˜. Δ=maxc1rtβi, α^=maxαi. Ω=∑k=1NπskPk+∑k=1,k≠sNλsk2/4Usk+Pk−PsUsk−1Pk−Ps+αiPsAs+αiAsTPs+G−2Psηirt.Proof.
To develop the finite-time synchronization criterion, we consider a stochastic Lyapunov function as follows:(31)Vt=∑i=1NeiTtPseit+∑i=1N∫t−τtteiTsGeisds.
CalculatingℒVeit,t,r along the trajectory of the error system (20) based on the condition Ito^ formula, we have(32)ℒVt=limΔt⟶0ℰVeit+Δt,rt+Δt,t+Δt|rt=i−Veit,i,tΔt,where Δt is a small positive number. Hence, for every rt=i∈S, it can be deduced that(33)ℒVt=2∑i=1NeiTtPse˙it+∑i=1NeiT∑k=1NπskhPkeit−1−r∑i=1NeiTt−τtGeit−τt+∑i=1NeiTtGeiTt.
Substituting (20) into (33), we have(34)ℒVt≤2∑i=1NeiTtPsDseit+Asfi˜eit+Bsfi˜eit−τt+∑j=1NLijsΛejt+c1rt∫t−dttf˜ieisds+uit+∑i=1NeiTt∑k=1NπskhPkeit+∑i=1NeiTtGeiTt+∑i=1Nr−1eiTt−τtGeit−τt.
Under assumptionH2, we obtain that(35)2eiTtPsAsfi˜eit≤2∑i=1N∑j=1Neitpsijasijfi˜eit≤2∑i=1N∑j=1Neitpsijasijαieit+βi≤2αi∑i=1N∑j=1Neitpsijasijeit+2βi∑i=1N∑j=1Neitpsijasij,and similarly, we have(36)2eiTtPsBsfi˜eit−τt≤2∑i=1N∑j=1Neitpsijbsijfi˜eit−τt≤2αi∑i=1N∑j=1Neitpsijbsijet−τtt+2βi∑i=1N∑j=1Neitpsijasij.
Based on the condition0<dt<ω, we have(37)2c1rt∑i=1NeiTtPs∫t−dttf˜ieisds≤2c1rtα^∑i=1NeiTtPs∫t−dtteisds+2Δω∑i=1NeiTtPs,where Δ=maxc1rtβi.
Combining (33), (35), and (36), we obtain(38)ℒVt≤2∑i=1NeiTtPsDseit+2∑i=1NαieiTtPsAseit+2∑i=1NαieiTtPsBseit−τt+2βi∑i=1NeiTtPsAs+PsBs+2∑i=1N∑j=1NeiTtPsLijsΛejt+2c1rt∑i=1NeiTtPs∫t−dttf˜ieisds+α^∫t−dtteisds+2∑i=1NeiTtPsuit+∑i=1NeiTt∑k=1NπskhPkeit+∑i=1NeiTtGeiTt−1−reiTt−τtGeit−τt.
Then, based on (37), we have the following inequality:(39)ℒVt≤2∑i=1NeiTtPsDseit+2βi∑i=1NeiTtPsAs+PsBs+2∑i=1NαieiTtPsAseit+eiTtPsBseit−τt+2∑i=1N∑j=1NeiTtPsLijsΛejt+2Δω∑i=1NeiTtPs+2c1rtα^−1∑i=1NeiTtPs∫t−dtteisds+2∑i=1NeiTtPsuit+∑i=1NeiTt∑k=1NπskhPkeit+∑i=1NeiTtGeiTt+∑i=1Nr−1eiTt−τtGeit−τt.
Substituting controller (28) into (39),(40)ℒVt≤2∑i=1NeiTtPsDseit+2βi∑i=1NeiTtPsAs+PsBs+2αi∑i=1NeiTtPsAseit+eiTtPsBseit−τt+2∑i=1N∑j=1NeiTtPsLijsΛejt+2Δω∑i=1NeiTtPs+2c1rtα^−1∑i=1NeiTtPs∫t−dtteisds+∑i=1NeiTtGeiTt+r−1eiTt−τtGeit−τt−2∑i=1NeiTtPsξirt+κirt−2eiTtPsηirteit−2∑i=1NeiTtPseiteit2κirt∑i=1N∫t−τtteiTsGeisds1/2+∑i=1NeiTt∑k=1NπskhPkeit.
Noting the condition (29) and Lemma 3, we have(41)ℒVt≤2∑i=1NeiTtPsDseit+2βi∑i=1NeiTtPsAs+PsBs+2αi∑i=1NeiTtPsAseit+eiTtPsBseit−τt+2∑i=1N∑j=1NeiTtPsLijsΛejt+2Δω∑i=1NeiTtPs+∑i=1NeiTtGeiTt+∑i=1NeiTt∑k=1NπskhPkeit+r−1∑i=1NeiTt−τtGeit−τt−4∑i=1NeiTtPsηirteit−2∑i=1NeiTtPsξirt+κirt−2λminPsκirt∑i=1N∫t−τtteiTsGeisds1/2.
In view ofπskh=πsk+Δπsk and Δπsk=−∑k=1,k≠sNΔπsk, according to Lemma 1, we can obtain that(42)∑k=1NπskhPk=∑k=1NπskPk+∑k=1,k≠sNΔπskPk+ΔπssPs,=∑k=1NπskPk+∑k=1,k≠sNΔπskPk−Ps,=∑k=1NπskPk+∑k=1,k≠sN12ΔπskPk−Ps+12ΔπskPk−Ps≤∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Ps.
Then, combining (41) and (42), we get(43)ℒVt≤2∑i=1NeiTtPsDseit+2αi∑i=1NeiTtPsAseit+2βi∑i=1NeiTtPsAs+PsBs+2αi∑i=1NeiTtPsBseit−τt+2∑i=1N∑j=1NeiTtPsLijsΛejt+2Δω∑i=1NeiTtPs+∑i=1NeiTtGeiTt+r−1∑i=1NeiTt−τtGeit−τt+∑i=1NeiTt∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Pseit−4∑i=1NeiTtPsηirteit−2∑i=1NeiTtPsξirt+κirt−2λminPsκirt∑i=1N∫t−τtteiTsGeisds1/2.
By virtue of the condition of the theorem, we get from inequality (43) that(44)ℒVt≤2∑i=1NeiTtPsDseit+2θ∑i,j=1,j≠iNlijeiTtPseit+2∑i=1NλminliieiTtPseit−2∑i=1NηirteiTtPseit+∑i=1NeiTtαiPsAs+αiAsTPseit+∑i=1NeiTtαiPsBs+αiBsTPseit−τt+2∑i=1NeiTtβiPsAs+PsBs+ΔωIn−ξirtIn+∑i=1NeiTtGeiTt+r−1eiTt−τtGeit−τt+∑i=1NeiTt∑k=1NπskPk+∑j=1,j≠iNλsk24Usk+Pk−PsUsk−1Pk−Pseit−2∑i=1NeiTt|Psκirt−4∑i=1NeiTtPsηirteit−2κirtλminPs∑i=1N∫t−τtteiTsGeisds1/2.
According to Lemma3, (44) can be rewritten as(45)ℒVt≤2Pse˜iTtD˜+θL^s˜−ηirtIne˜it+2∑i=1NeiTtβiPsAs+PsBs+ΔωPs−Psξirt+eiTteiTt−τt|ΩαiPsBs∗r−1Geiteit−τt−2λminPsκirt∑i=1NeiTt−2κirtλminPs∑i=1N∫t−τtteiTsGeisds1/2,where e˜it=e1t,e2t,…,entT.
Exploiting condition (29) and (30), we have the following inequality:(46)ℒVt≤−2λminPsκirt∑i=1NeiTt−2κirtλminPs∑i=1N∫t−τtteiTsGeisds1/2.
By means of Lemma2, we obtain that(47)∑i=1Neitw+11/w+1≥∑i=1Neit21/2.
Thus,(48)∑i=1Neit≥∑i=1Neit21/2=∑i=1NeiTteit1/2.
Combining (46), (48), and Lemma 3, we get(49)ℒVt≤−2λminPsλmax1/2Psκirt∑i=1NeiTtPseit1/2−2κirtλminPs∑i=1N∫t−τtteiTsGeisds1/2.
By Lemma5, (49) is equal to(50)ℒVt≤−2ϱ∑i=1NeiTtPseit+∑i=1N∫t−τtteiTsGeisds1/2,that is,(51)ℒVt≤−2ϱVt1/2,where(52)ϱ=minλminPsλmax1/2Psκi,λminPsκi,κirt=diagκi.
As we all know,ℰVt1/2=ℰVt1/2, for any t>0. Thus, we have the following inequality:(53)ℰℒVt≤−2ϱℰVt1/2.
Then, based on Lemma6, the finite-time synchronization of the complex network is finally achieved. At the same time, the settling time is estimated as(54)t1≤1ϱV1/20.
Thus, we can conclude that the drive system (17) synchronizes with the response system (18) in finite time by adopting the controller (28). Also, the settling time t1 is given in (54). The proof of the global finite-time synchronization is finished.Remark 2.
The global finite-time synchronization of discontinuous CDNs with semi-Markovian switching and mixed time-varying delays is considered for the first time in Theorem1. The approach we get the global finite-time synchronization condition is the matrix inequalities but not the linear matrix inequalities because Ω has a nonlinear term ∑k=1,k≠sNPk−PsUsk−1Pk−Ps.


### 3.2. Global Fixed-Time Synchronization

In this part, the global fixed-time synchronization criteria are developed under the following state-feedback controller:(55)uit=−ξirtsigneit−2ηirteit−κirteih¯/mt−δirteiq/pt−c1rt∫t−dtteisds−κirteiteit2∫t−τtteiTsGeisdsh¯+m/2m−δirteiteit2∫t−τtteiTsGeisdsp+q/2p,where ξirt>0 and ηirt>0 are the parameters to be determined. κirt>0 and δirt>0 are tunable parameters. 0<h¯<m,0<p<q.Theorem 2.
Consider network models (17) and (18) with the hypotheses H1 and H2. For any given constants scalars 0<h¯<m,0<p<q, the drive system (17) is said to be synchronized with the response system (18) in fixed-time, if there exist symmetric positive definite matrix Ps and U, and matrices G>0 such that(56)D˜+θL^s˜−ηirtIn<0,βiPsAs+PsBs+ΔωPs−Psξirt<0,α^<1,where D˜=diagD1,D2,…,Dn. L^=lij^n×n, lij^=lij,i≠j, lii^=λmin/θ,θ=Λ, and λmin is the eigenvalue of Λs˜. Δ=maxc1rtβi,α^=maxαi.(57)Ω˜αiPsBsΞ∗r−1G0∗∗−Us≤0,where(58)Ω=∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Ps+αiPsAs+αiAsTPs+G−2Psηirt,Ω˜=∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+αiPsAs+αiAsTPs+G−2Psηirt,Ξ=Ps−P1,…,Ps−Ps−1,Ps−Ps+1,…,Ps−PN,Us=diagUs1,Us2,…,Uss−1,Uss+1,…,UsN.Proof.
Taking the same stochastic Lyapunov functional with Theorem1,(59)Vt=∑i=1NeiTtPseit+∑i=1N∫t−τtteiTsGeisds.
Then, based on (20), (33), and (42), we can obtain the derivative of Veit,t,rt as follows:(60)ℒVt≤2∑i=1NeiTtPsDseit+Asfi˜eit+Bsfi˜eit−τt+∑j=1NLijsΛejt+c1rt∫t−dttf˜ieisds+uit+∑i=1Nr−1eiTt−τtGeit−τt+∑i=1NeiTtGeiTt+∑i=1NeiTt∑k=1NπskhPkeit.
Substituting (55) into (60) and adopting Lemma 3, (60) changes into(61)ℒVt≤2∑i=1NeiTtPsDseit+Asfi˜eit+Bsfi˜eit−τt+∑i=1N∑j=1NLijsΛejt+c1rt∑i=1N∫t−dttf˜ieisds+∑i=1NeiTtGeiTt+∑i=1NeiTt∑k=1NπskhPkeit+r−1∑i=1NeiTt−τtGeit−τt−2c1rt∑i=1NeiTtPs∫t−dtteisds−2∑i=1NeiTtPsξirt−4∑i=1NeiTtPsηirteit−2∑i=1NeiTtPsκirteih¯/mt−2∑i=1NeiTtPsδirteiq/pt−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p.
From (35)–(37), (42), and (61), we have(62)ℒVt≤2∑i=1NeiTtPsDseit+2∑i=1N∑j=1NeiTtPsLijsΛejt+2αi∑i=1NeiTtPsBseit−τt+eiTtPsAseit+2∑i=1NeiTtβiPsAs+PsBs+ΔωPs−Psξirt+2c1rtα^−1∑i=1NeiTtPs∫t−dtteisds+∑i=1NeiTtGeiTt+r−1eiTt−τtGeit−τt+∑i=1NeiTt∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Pseit−2∑i=1N2eiTtPsηirteit+eiTtPiκirteih¯/mt+eiTtPsδirteiq/pt−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p.
Based on the (56), it yields that(63)ℒVt≤2∑i=1NeiTtPsDseit+2αi∑i=1NeiTtPsBseit−τt+2∑i=1N∑j=1NeiTtPsLijsΛejt+2αi∑i=1NeiTtPsAseit+∑i=1NeiTtGeiTt+r−1eiTt−τtGeit−τt+∑i=1NeiTt∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Pseit−2∑i=1N2eiTtPsηirteit+eiTtPiκirteih¯/mt+eiTtPsδirteiq/pt−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p.
Similar to (44), we have the following inequality:(64)ℒVt≤2∑i=1NeiTtPsDseit+2∑i,j=1,j≠iNθlij+∑i=1Nλminlii−ηirtIneiTtPseit+∑i=1NeiTtαiPsAs+αiAsTPseit+∑i=1NeiTtαiPsBs+αiBsTPseit−τt+∑i=1NeiTt∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Pseit+r−1∑i=1NeiTt−τtGeit−τt+∑i=1NeiTtGeiTt−2∑i=1NeiTtPsηirteit−2∑i=1NeiTtPsκirteih¯/mt+eiTtPsδirteiq/pt−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p.
Then, (64) can be rewritten as follows:(65)ℒVt≤2Pse˜iTtD˜+θL^s˜−ηirtIne˜it+eiTteiTt−τt|ΩαiPsBs∗r−1Geiteit−τt−2∑i=1NeiTtPsκirteih¯/mt−2∑i=1NeiTtPsδirteiq/pt−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p,where e˜it=e1t,e2t,…,entT.
Based on the condition (56), we obtain that(66)ℒVt≤−2∑i=1NeiTtPsκirteih¯/mt−2∑i=1NeiTtPsδirteiq/pt−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p+eiTteiTt−τt|ΩαiPsBs∗r−1Geiteit−τt,where(67)Ω=∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Ps+αiPsAs+αiAsTPs+G−2Psηirt.
Because of the nonlinear terms,Pk−PsUsk−1Pk−Ps in Ω is difficult to be handled, and we need to turn it into a linear matrix inequality.
Hence, we define(68)Ω˜=∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+αiPsAs+αiAsTPs+G−2Psηirt.
Then, according to the (56), (30) can be rewritten as follows:(69)ΩαiPsBs∗r−1G=Ω˜αiPsBs∗r−1G+diag∑k=1,k≠sNPk−PsUsk−1Pk−Ps,0≤0.
We note that Lemma4 implies that(70)Ω˜αiPsBsΞ∗r−1G0∗∗−Us≤0.
Hence, from (70), (66) turns into(71)ℒVt≤−2∑i=1NeiTtPsκirteih¯/mt−2∑i=1NeiTtPsδirteiq/pt−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p.
By using Lemmas2 and 3, we can obtain that(72)ℒVt≤−2λminPsλmax−h¯+m/2mPsκirt∑i=1NeiTtPseit−h¯+m/2m−2λminPsλmax−p+q/2pPiδirt∑i=1NeiTtPseit−p+q/2p−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p.
Then, by exploiting Lemmas2 and 5, we obtain(73)ℒVt≤−2κ˜∑i=1NeiTtPseit+∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2δ˜∑i=1NeiTtPseit+∑i=1N∫t−τtteiTsGeisdsp+q/2p.
Also, (73) is equal to(74)ℒVt≤−2κ˜Vth¯+m/2m−2δ˜Vtp+q/2p,where(75)κ˜=minλminPsλmax−h¯+m/2mPsκi,λminPsκi,κirt=diagκi,δ˜=minλminPsλmax−p+q/2pPsnq−p/2pδi,λminPsδi,δirt=diagδi.
Taking the expectation on both sides of (74), we can get the following inequality:(76)ℰℒVt≤−2κ˜ℰVth¯+m/2m−2δ˜ℰVtp+q/2p.
Similar with Theorem1, the differential coefficients of V(t) are given as follows:(77)ℰℒVt≤−2κ˜ℰVth¯+m/2m−2δ˜ℰVtp+q/2p.
Because0<h¯<m,0<q<p,0<h¯+m/2m<1,p+q/2p>1, (77) satisfies Lemma 7, and we are in a position to complete the proof of Theorem 2. In other words, the drive system (17) synchronizes with the response system (18) in fixed-time. Also, the settling time is estimated as(78)Te0≤Tmax≤mκ˜m−h¯+pδ˜q−p.
Thus, the global fixed-time synchronization problem is finally addressed under the controller (55). The proof of this theorem is completed.Corollary 1.
Suppose that the hypothesesH1andH2 hold. For given scalars 0<h¯<m,0<p<q, the drive system (17) synchronizes with the response system (18) in fixed time based on the controller (55), if there exist symmetric positive definite matrices Ps,Usk, and positive definite matrix Q such that(79)di+θli−ηi<0,βiPsAs+PsBs+ΔωPs−Psξirt<0,α^<1,wheredi=λmaxD˜. li=λmaxL^s˜,ηi=λmaxηirtIn. L^=lij^n×n,lij^=lij,i≠j,lii^=λmin/θ,θ=Λ,andλmin is the eigenvalue of Λs˜. Δ=maxc1rtβi, α^=maxαi.(80)Ω˜αiPsBsΞ∗r−1G0∗∗−Us≤0,where(81)Ω=∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Ps+αiPsAs+αiAiTPs+G−2Psηirt,Ω˜=∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+αiPsAs+αiAsTPs+G−2Psηirt,Ξ=Ps−P1,…,Ps−Ps−1,Ps−Ps+1,…,Ps−PN,Ui=diagUs1,Us2,…,Uss−1,Uss+1,…,UsN.Meanwhile, on the condition of Lemma8, the settling time of global fixed-time synchronization is estimated as(82)Te0≤Tmax=πρ2κ˜δ˜,where κ˜ and δ˜ are the same with the parameters in Theorem 2. Also, h¯/m=1−1/2ρ,q/p=1−1/2ρ,ρ>1.Remark 3.
There exist many kinds of chaotic system with discontinuous functions, such as the discontinuous Chua circuit, discontinuous Chen system, and neural network with discontinuous activation functions. In this paper, we choose the discontinuous condition that it is applicable to most of the chaotic systems.Remark 4.
Global fixed-time synchronization is more complex than global finite-time synchronization in the synchronization conditions. For global finite-time synchronization, a term such as−Vαt,0<α<1 is only needed, whereas for fixed-time synchronization, we need −Vh¯/mt,0<h¯<m;−Vq/pt, q>p>0 to realize it. Comparing with the settling time t1 of the global finite-time synchronization in Theorem 1, the settling time Te0 of the global fixed-time synchronization in Theorem 2 is independent of the initial condition e0. When the e0 is so large, the t1 is not reasonable in practice application.


## 3.1. Global Finite-Time Synchronization

Based on the linear matrix inequalities technique [45] and the Lyapunov functional method [38, 46], this section aims to develop some new conditions which can ensure the synchronization between complex dynamical network models (17) and (18) over a finite-time interval.For the purpose of global finite-time synchronization, a state-feedback controller is designed as follows:(28)uit=−ξirt+κirtsigneit−2ηirteit−κirteiteit2∫t−τtteiTsGeisds1/2−c1rt∫t−dtteisds,in which signeit=signei1t,…,signeintT, eit=ei1t,ei2t,…,eintT, κirt>0 and is a tunable parameter, and ηirt is a parameter to be determined.Theorem 1.
Let the assumptionsH1 and H2 hold. Then, the drive system (17) is synchronized with the response system (18) in finite time, if there exist symmetric positive definite matrices Ps,U, and G, such that(29)D˜+θL^s˜−ηirtIn<0,βiPsAs+PsBs+ΔωPs−Psξirt<0,α^<1,(30)ΩαiPsBs∗r−1G≤0,where D˜=diagD1,D2,…,Dn. L^=lij^n×n, lij^=lij,i≠j, lii^=λmin/θ, θ=Λ, and λmin is the eigenvalue of Λs˜. Δ=maxc1rtβi, α^=maxαi. Ω=∑k=1NπskPk+∑k=1,k≠sNλsk2/4Usk+Pk−PsUsk−1Pk−Ps+αiPsAs+αiAsTPs+G−2Psηirt.Proof.
To develop the finite-time synchronization criterion, we consider a stochastic Lyapunov function as follows:(31)Vt=∑i=1NeiTtPseit+∑i=1N∫t−τtteiTsGeisds.
CalculatingℒVeit,t,r along the trajectory of the error system (20) based on the condition Ito^ formula, we have(32)ℒVt=limΔt⟶0ℰVeit+Δt,rt+Δt,t+Δt|rt=i−Veit,i,tΔt,where Δt is a small positive number. Hence, for every rt=i∈S, it can be deduced that(33)ℒVt=2∑i=1NeiTtPse˙it+∑i=1NeiT∑k=1NπskhPkeit−1−r∑i=1NeiTt−τtGeit−τt+∑i=1NeiTtGeiTt.
Substituting (20) into (33), we have(34)ℒVt≤2∑i=1NeiTtPsDseit+Asfi˜eit+Bsfi˜eit−τt+∑j=1NLijsΛejt+c1rt∫t−dttf˜ieisds+uit+∑i=1NeiTt∑k=1NπskhPkeit+∑i=1NeiTtGeiTt+∑i=1Nr−1eiTt−τtGeit−τt.
Under assumptionH2, we obtain that(35)2eiTtPsAsfi˜eit≤2∑i=1N∑j=1Neitpsijasijfi˜eit≤2∑i=1N∑j=1Neitpsijasijαieit+βi≤2αi∑i=1N∑j=1Neitpsijasijeit+2βi∑i=1N∑j=1Neitpsijasij,and similarly, we have(36)2eiTtPsBsfi˜eit−τt≤2∑i=1N∑j=1Neitpsijbsijfi˜eit−τt≤2αi∑i=1N∑j=1Neitpsijbsijet−τtt+2βi∑i=1N∑j=1Neitpsijasij.
Based on the condition0<dt<ω, we have(37)2c1rt∑i=1NeiTtPs∫t−dttf˜ieisds≤2c1rtα^∑i=1NeiTtPs∫t−dtteisds+2Δω∑i=1NeiTtPs,where Δ=maxc1rtβi.
Combining (33), (35), and (36), we obtain(38)ℒVt≤2∑i=1NeiTtPsDseit+2∑i=1NαieiTtPsAseit+2∑i=1NαieiTtPsBseit−τt+2βi∑i=1NeiTtPsAs+PsBs+2∑i=1N∑j=1NeiTtPsLijsΛejt+2c1rt∑i=1NeiTtPs∫t−dttf˜ieisds+α^∫t−dtteisds+2∑i=1NeiTtPsuit+∑i=1NeiTt∑k=1NπskhPkeit+∑i=1NeiTtGeiTt−1−reiTt−τtGeit−τt.
Then, based on (37), we have the following inequality:(39)ℒVt≤2∑i=1NeiTtPsDseit+2βi∑i=1NeiTtPsAs+PsBs+2∑i=1NαieiTtPsAseit+eiTtPsBseit−τt+2∑i=1N∑j=1NeiTtPsLijsΛejt+2Δω∑i=1NeiTtPs+2c1rtα^−1∑i=1NeiTtPs∫t−dtteisds+2∑i=1NeiTtPsuit+∑i=1NeiTt∑k=1NπskhPkeit+∑i=1NeiTtGeiTt+∑i=1Nr−1eiTt−τtGeit−τt.
Substituting controller (28) into (39),(40)ℒVt≤2∑i=1NeiTtPsDseit+2βi∑i=1NeiTtPsAs+PsBs+2αi∑i=1NeiTtPsAseit+eiTtPsBseit−τt+2∑i=1N∑j=1NeiTtPsLijsΛejt+2Δω∑i=1NeiTtPs+2c1rtα^−1∑i=1NeiTtPs∫t−dtteisds+∑i=1NeiTtGeiTt+r−1eiTt−τtGeit−τt−2∑i=1NeiTtPsξirt+κirt−2eiTtPsηirteit−2∑i=1NeiTtPseiteit2κirt∑i=1N∫t−τtteiTsGeisds1/2+∑i=1NeiTt∑k=1NπskhPkeit.
Noting the condition (29) and Lemma 3, we have(41)ℒVt≤2∑i=1NeiTtPsDseit+2βi∑i=1NeiTtPsAs+PsBs+2αi∑i=1NeiTtPsAseit+eiTtPsBseit−τt+2∑i=1N∑j=1NeiTtPsLijsΛejt+2Δω∑i=1NeiTtPs+∑i=1NeiTtGeiTt+∑i=1NeiTt∑k=1NπskhPkeit+r−1∑i=1NeiTt−τtGeit−τt−4∑i=1NeiTtPsηirteit−2∑i=1NeiTtPsξirt+κirt−2λminPsκirt∑i=1N∫t−τtteiTsGeisds1/2.
In view ofπskh=πsk+Δπsk and Δπsk=−∑k=1,k≠sNΔπsk, according to Lemma 1, we can obtain that(42)∑k=1NπskhPk=∑k=1NπskPk+∑k=1,k≠sNΔπskPk+ΔπssPs,=∑k=1NπskPk+∑k=1,k≠sNΔπskPk−Ps,=∑k=1NπskPk+∑k=1,k≠sN12ΔπskPk−Ps+12ΔπskPk−Ps≤∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Ps.
Then, combining (41) and (42), we get(43)ℒVt≤2∑i=1NeiTtPsDseit+2αi∑i=1NeiTtPsAseit+2βi∑i=1NeiTtPsAs+PsBs+2αi∑i=1NeiTtPsBseit−τt+2∑i=1N∑j=1NeiTtPsLijsΛejt+2Δω∑i=1NeiTtPs+∑i=1NeiTtGeiTt+r−1∑i=1NeiTt−τtGeit−τt+∑i=1NeiTt∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Pseit−4∑i=1NeiTtPsηirteit−2∑i=1NeiTtPsξirt+κirt−2λminPsκirt∑i=1N∫t−τtteiTsGeisds1/2.
By virtue of the condition of the theorem, we get from inequality (43) that(44)ℒVt≤2∑i=1NeiTtPsDseit+2θ∑i,j=1,j≠iNlijeiTtPseit+2∑i=1NλminliieiTtPseit−2∑i=1NηirteiTtPseit+∑i=1NeiTtαiPsAs+αiAsTPseit+∑i=1NeiTtαiPsBs+αiBsTPseit−τt+2∑i=1NeiTtβiPsAs+PsBs+ΔωIn−ξirtIn+∑i=1NeiTtGeiTt+r−1eiTt−τtGeit−τt+∑i=1NeiTt∑k=1NπskPk+∑j=1,j≠iNλsk24Usk+Pk−PsUsk−1Pk−Pseit−2∑i=1NeiTt|Psκirt−4∑i=1NeiTtPsηirteit−2κirtλminPs∑i=1N∫t−τtteiTsGeisds1/2.
According to Lemma3, (44) can be rewritten as(45)ℒVt≤2Pse˜iTtD˜+θL^s˜−ηirtIne˜it+2∑i=1NeiTtβiPsAs+PsBs+ΔωPs−Psξirt+eiTteiTt−τt|ΩαiPsBs∗r−1Geiteit−τt−2λminPsκirt∑i=1NeiTt−2κirtλminPs∑i=1N∫t−τtteiTsGeisds1/2,where e˜it=e1t,e2t,…,entT.
Exploiting condition (29) and (30), we have the following inequality:(46)ℒVt≤−2λminPsκirt∑i=1NeiTt−2κirtλminPs∑i=1N∫t−τtteiTsGeisds1/2.
By means of Lemma2, we obtain that(47)∑i=1Neitw+11/w+1≥∑i=1Neit21/2.
Thus,(48)∑i=1Neit≥∑i=1Neit21/2=∑i=1NeiTteit1/2.
Combining (46), (48), and Lemma 3, we get(49)ℒVt≤−2λminPsλmax1/2Psκirt∑i=1NeiTtPseit1/2−2κirtλminPs∑i=1N∫t−τtteiTsGeisds1/2.
By Lemma5, (49) is equal to(50)ℒVt≤−2ϱ∑i=1NeiTtPseit+∑i=1N∫t−τtteiTsGeisds1/2,that is,(51)ℒVt≤−2ϱVt1/2,where(52)ϱ=minλminPsλmax1/2Psκi,λminPsκi,κirt=diagκi.
As we all know,ℰVt1/2=ℰVt1/2, for any t>0. Thus, we have the following inequality:(53)ℰℒVt≤−2ϱℰVt1/2.
Then, based on Lemma6, the finite-time synchronization of the complex network is finally achieved. At the same time, the settling time is estimated as(54)t1≤1ϱV1/20.
Thus, we can conclude that the drive system (17) synchronizes with the response system (18) in finite time by adopting the controller (28). Also, the settling time t1 is given in (54). The proof of the global finite-time synchronization is finished.Remark 2.
The global finite-time synchronization of discontinuous CDNs with semi-Markovian switching and mixed time-varying delays is considered for the first time in Theorem1. The approach we get the global finite-time synchronization condition is the matrix inequalities but not the linear matrix inequalities because Ω has a nonlinear term ∑k=1,k≠sNPk−PsUsk−1Pk−Ps.


## 3.2. Global Fixed-Time Synchronization

In this part, the global fixed-time synchronization criteria are developed under the following state-feedback controller:(55)uit=−ξirtsigneit−2ηirteit−κirteih¯/mt−δirteiq/pt−c1rt∫t−dtteisds−κirteiteit2∫t−τtteiTsGeisdsh¯+m/2m−δirteiteit2∫t−τtteiTsGeisdsp+q/2p,where ξirt>0 and ηirt>0 are the parameters to be determined. κirt>0 and δirt>0 are tunable parameters. 0<h¯<m,0<p<q.Theorem 2.
Consider network models (17) and (18) with the hypotheses H1 and H2. For any given constants scalars 0<h¯<m,0<p<q, the drive system (17) is said to be synchronized with the response system (18) in fixed-time, if there exist symmetric positive definite matrix Ps and U, and matrices G>0 such that(56)D˜+θL^s˜−ηirtIn<0,βiPsAs+PsBs+ΔωPs−Psξirt<0,α^<1,where D˜=diagD1,D2,…,Dn. L^=lij^n×n, lij^=lij,i≠j, lii^=λmin/θ,θ=Λ, and λmin is the eigenvalue of Λs˜. Δ=maxc1rtβi,α^=maxαi.(57)Ω˜αiPsBsΞ∗r−1G0∗∗−Us≤0,where(58)Ω=∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Ps+αiPsAs+αiAsTPs+G−2Psηirt,Ω˜=∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+αiPsAs+αiAsTPs+G−2Psηirt,Ξ=Ps−P1,…,Ps−Ps−1,Ps−Ps+1,…,Ps−PN,Us=diagUs1,Us2,…,Uss−1,Uss+1,…,UsN.Proof.
Taking the same stochastic Lyapunov functional with Theorem1,(59)Vt=∑i=1NeiTtPseit+∑i=1N∫t−τtteiTsGeisds.
Then, based on (20), (33), and (42), we can obtain the derivative of Veit,t,rt as follows:(60)ℒVt≤2∑i=1NeiTtPsDseit+Asfi˜eit+Bsfi˜eit−τt+∑j=1NLijsΛejt+c1rt∫t−dttf˜ieisds+uit+∑i=1Nr−1eiTt−τtGeit−τt+∑i=1NeiTtGeiTt+∑i=1NeiTt∑k=1NπskhPkeit.
Substituting (55) into (60) and adopting Lemma 3, (60) changes into(61)ℒVt≤2∑i=1NeiTtPsDseit+Asfi˜eit+Bsfi˜eit−τt+∑i=1N∑j=1NLijsΛejt+c1rt∑i=1N∫t−dttf˜ieisds+∑i=1NeiTtGeiTt+∑i=1NeiTt∑k=1NπskhPkeit+r−1∑i=1NeiTt−τtGeit−τt−2c1rt∑i=1NeiTtPs∫t−dtteisds−2∑i=1NeiTtPsξirt−4∑i=1NeiTtPsηirteit−2∑i=1NeiTtPsκirteih¯/mt−2∑i=1NeiTtPsδirteiq/pt−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p.
From (35)–(37), (42), and (61), we have(62)ℒVt≤2∑i=1NeiTtPsDseit+2∑i=1N∑j=1NeiTtPsLijsΛejt+2αi∑i=1NeiTtPsBseit−τt+eiTtPsAseit+2∑i=1NeiTtβiPsAs+PsBs+ΔωPs−Psξirt+2c1rtα^−1∑i=1NeiTtPs∫t−dtteisds+∑i=1NeiTtGeiTt+r−1eiTt−τtGeit−τt+∑i=1NeiTt∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Pseit−2∑i=1N2eiTtPsηirteit+eiTtPiκirteih¯/mt+eiTtPsδirteiq/pt−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p.
Based on the (56), it yields that(63)ℒVt≤2∑i=1NeiTtPsDseit+2αi∑i=1NeiTtPsBseit−τt+2∑i=1N∑j=1NeiTtPsLijsΛejt+2αi∑i=1NeiTtPsAseit+∑i=1NeiTtGeiTt+r−1eiTt−τtGeit−τt+∑i=1NeiTt∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Pseit−2∑i=1N2eiTtPsηirteit+eiTtPiκirteih¯/mt+eiTtPsδirteiq/pt−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p.
Similar to (44), we have the following inequality:(64)ℒVt≤2∑i=1NeiTtPsDseit+2∑i,j=1,j≠iNθlij+∑i=1Nλminlii−ηirtIneiTtPseit+∑i=1NeiTtαiPsAs+αiAsTPseit+∑i=1NeiTtαiPsBs+αiBsTPseit−τt+∑i=1NeiTt∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Pseit+r−1∑i=1NeiTt−τtGeit−τt+∑i=1NeiTtGeiTt−2∑i=1NeiTtPsηirteit−2∑i=1NeiTtPsκirteih¯/mt+eiTtPsδirteiq/pt−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p.
Then, (64) can be rewritten as follows:(65)ℒVt≤2Pse˜iTtD˜+θL^s˜−ηirtIne˜it+eiTteiTt−τt|ΩαiPsBs∗r−1Geiteit−τt−2∑i=1NeiTtPsκirteih¯/mt−2∑i=1NeiTtPsδirteiq/pt−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p,where e˜it=e1t,e2t,…,entT.
Based on the condition (56), we obtain that(66)ℒVt≤−2∑i=1NeiTtPsκirteih¯/mt−2∑i=1NeiTtPsδirteiq/pt−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p+eiTteiTt−τt|ΩαiPsBs∗r−1Geiteit−τt,where(67)Ω=∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Ps+αiPsAs+αiAsTPs+G−2Psηirt.
Because of the nonlinear terms,Pk−PsUsk−1Pk−Ps in Ω is difficult to be handled, and we need to turn it into a linear matrix inequality.
Hence, we define(68)Ω˜=∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+αiPsAs+αiAsTPs+G−2Psηirt.
Then, according to the (56), (30) can be rewritten as follows:(69)ΩαiPsBs∗r−1G=Ω˜αiPsBs∗r−1G+diag∑k=1,k≠sNPk−PsUsk−1Pk−Ps,0≤0.
We note that Lemma4 implies that(70)Ω˜αiPsBsΞ∗r−1G0∗∗−Us≤0.
Hence, from (70), (66) turns into(71)ℒVt≤−2∑i=1NeiTtPsκirteih¯/mt−2∑i=1NeiTtPsδirteiq/pt−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p.
By using Lemmas2 and 3, we can obtain that(72)ℒVt≤−2λminPsλmax−h¯+m/2mPsκirt∑i=1NeiTtPseit−h¯+m/2m−2λminPsλmax−p+q/2pPiδirt∑i=1NeiTtPseit−p+q/2p−2λminPsκirt∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2λminPsδirt∑i=1N∫t−τtteiTsGeisdsp+q/2p.
Then, by exploiting Lemmas2 and 5, we obtain(73)ℒVt≤−2κ˜∑i=1NeiTtPseit+∑i=1N∫t−τtteiTsGeisdsh¯+m/2m−2δ˜∑i=1NeiTtPseit+∑i=1N∫t−τtteiTsGeisdsp+q/2p.
Also, (73) is equal to(74)ℒVt≤−2κ˜Vth¯+m/2m−2δ˜Vtp+q/2p,where(75)κ˜=minλminPsλmax−h¯+m/2mPsκi,λminPsκi,κirt=diagκi,δ˜=minλminPsλmax−p+q/2pPsnq−p/2pδi,λminPsδi,δirt=diagδi.
Taking the expectation on both sides of (74), we can get the following inequality:(76)ℰℒVt≤−2κ˜ℰVth¯+m/2m−2δ˜ℰVtp+q/2p.
Similar with Theorem1, the differential coefficients of V(t) are given as follows:(77)ℰℒVt≤−2κ˜ℰVth¯+m/2m−2δ˜ℰVtp+q/2p.
Because0<h¯<m,0<q<p,0<h¯+m/2m<1,p+q/2p>1, (77) satisfies Lemma 7, and we are in a position to complete the proof of Theorem 2. In other words, the drive system (17) synchronizes with the response system (18) in fixed-time. Also, the settling time is estimated as(78)Te0≤Tmax≤mκ˜m−h¯+pδ˜q−p.
Thus, the global fixed-time synchronization problem is finally addressed under the controller (55). The proof of this theorem is completed.Corollary 1.
Suppose that the hypothesesH1andH2 hold. For given scalars 0<h¯<m,0<p<q, the drive system (17) synchronizes with the response system (18) in fixed time based on the controller (55), if there exist symmetric positive definite matrices Ps,Usk, and positive definite matrix Q such that(79)di+θli−ηi<0,βiPsAs+PsBs+ΔωPs−Psξirt<0,α^<1,wheredi=λmaxD˜. li=λmaxL^s˜,ηi=λmaxηirtIn. L^=lij^n×n,lij^=lij,i≠j,lii^=λmin/θ,θ=Λ,andλmin is the eigenvalue of Λs˜. Δ=maxc1rtβi, α^=maxαi.(80)Ω˜αiPsBsΞ∗r−1G0∗∗−Us≤0,where(81)Ω=∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+Pk−PsUsk−1Pk−Ps+αiPsAs+αiAiTPs+G−2Psηirt,Ω˜=∑k=1NπskPk+∑k=1,k≠sNλsk24Usk+αiPsAs+αiAsTPs+G−2Psηirt,Ξ=Ps−P1,…,Ps−Ps−1,Ps−Ps+1,…,Ps−PN,Ui=diagUs1,Us2,…,Uss−1,Uss+1,…,UsN.Meanwhile, on the condition of Lemma8, the settling time of global fixed-time synchronization is estimated as(82)Te0≤Tmax=πρ2κ˜δ˜,where κ˜ and δ˜ are the same with the parameters in Theorem 2. Also, h¯/m=1−1/2ρ,q/p=1−1/2ρ,ρ>1.Remark 3.
There exist many kinds of chaotic system with discontinuous functions, such as the discontinuous Chua circuit, discontinuous Chen system, and neural network with discontinuous activation functions. In this paper, we choose the discontinuous condition that it is applicable to most of the chaotic systems.Remark 4.
Global fixed-time synchronization is more complex than global finite-time synchronization in the synchronization conditions. For global finite-time synchronization, a term such as−Vαt,0<α<1 is only needed, whereas for fixed-time synchronization, we need −Vh¯/mt,0<h¯<m;−Vq/pt, q>p>0 to realize it. Comparing with the settling time t1 of the global finite-time synchronization in Theorem 1, the settling time Te0 of the global fixed-time synchronization in Theorem 2 is independent of the initial condition e0. When the e0 is so large, the t1 is not reasonable in practice application.


## 4. Numerical Examples

In this section, we perform two examples to demonstrate the effectiveness of the results obtained in this paper.Example 1.
consider the complex dynamical networks models in the form of (17) and (18) with two modes and two nodes, and each node is a 2-dimensional dynamical system. The system parameters are described as follows: xit=xi1t,xi2tT,yit=yi1t,yi2tT.(83)D1=1.4001.1,D2=1.5001.0,A1=−1.60.5−0.8−1.0,A2=−2.00.60.8−1.9,B1=−1.2−0.4−0.8−0.6,B2=0.7−0.50.3−0.6.
The coupling matrixT is given as follows:(84)L1=−1.01.01.5−1.5,L2=−0.90.91.2−1.2.
The inner-coupling matrix is(85)Λ=1001.
The scalars we select in this paper are as follows:0≤αi≤0.5,0≤βi≤0.5. c11rt=0.3 and c12rt=0.5. The mixed time-varying delay is assumed to be τt=0.8+0.2cost,dt=0.75+0.25sint, and the initial value is selected as xit=sint,tanhtT and yit=cost,etT. We can see that the upper bound of the delay is τ=1,ω=1,τ˙t≤r=0.5. Also, the activation function is taken as(86)f1i1xit=f1i2xit=0.4t+signt,f2i1xit=f2i2xit=0.1t+0.7signt.
The transition rates of the semi-Markovian jumping system for each mode are given as follows:
For mode 1,(87)π11h∈−0.72,−0.58,π12h∈0.42,0.62.
For mode 2,(88)π21h∈0.84,0.96,π22h∈−0.92,−0.76.
Then, according to the transition rates, we can get the parametersπsk,λsk, where s, k ∈S=1,2.(89)π11=−0.60,π12=0.60,π21=0.90,π22=−0.90,λ11=0.07,λ12=0.10,λ21=0.06,λ22=0.08.
Through simple computations, we have(90)P1=4.48500.31560.31563.6320,P2=5.02680.34620.34624.6459,Q=0.5646000.6024.
For mode 1,(91)ξ1rt=1.1228001.0235,κ1rt=0.5500000.5500,η1rt=1.3142001.3316.
For mode 2,(92)ξ2rt=0.9350001.1526,κ2rt=0.5010000.5010,η2rt=1.3350001.4226,and the parameter V(0) = 8.6528 and the settling time is t∗=3.2475.
Then, Figure1 displays the first state trajectories of the drive-response system with controller (28).The second state trajectories of the drive-response system with controller (28) presented in Figures 2 and 3 depicts the state trajectories of the synchronization error system (20) with the controller (28).
To this, the effectiveness of Theorem1 is proved. The finite-time synchronization problem of discontinuous semi-Markovian switching complex dynamical networks with mixed time-varying delays is realized.Figure 1
The first state trajectory of the drive-response systemsx1it and y1it with controller (28).Figure 2
The second state trajectory of the drive-response systemsx2it and y2it with controller (28).Figure 3
The state trajectories of the synchronization error system (20) with the controller (28).Example 2.
For Theorem2, we take network models in the form of (17) and (18) with two nodes, and each node is a 2-dimensional dynamical system. The system parameters are described as follows: xit=xi1t,xi2tT,yit=yi1t,yi2tT.(93)D1=1.6001.0,D2=1.5001.2,A1=−2.80.5−0.8−2.3,A2=−2.6−0.30.5−2.0,B1=−1.0−0.5−0.60.7,B2=0.8−0.5−0.7−0.6.
The coupling matrixL is given as follows:(94)L1=−1.21.22.0−2.0,L2=−1.31.32.2−2.2.
The inner-coupling matrix we choose is(95)Λ=1001.
The transition rates of the semi-Markovian switching system for each mode are given as follows.
For mode 1,(96)π11h∈−1.02,−0.82,π12h∈0.78,1.10.
For mode 2,(97)π21h∈1.23,1.05,π22h∈−1.32,−1.16.
Then, according to the transition rates, we can get the parametersπsk,λsk, where s, k ∈S=1,2.(98)π11=−0.90,π12=0.90,π21=1.20,π22=−1.20,λ11=0.10,λ12=0.11,λ21=0.09,λ22=0.08.
For Theorem2, the corresponding scalars we choose are m=6,h¯=4,p=2,q=3, m=6,h¯=4,p=2,q=3a0≤αi≤0.5, 0≤βi≤0.5.c11rt=0.3, and c12rt=0.6. The mixed time-varying delays are assumed to be τt=0.75+0.25cost and dt=0.5+0.5sint, and the initial value is selected as xit=sint,etT and yit=cost,tanhtT. We can see that the upper bound of the delay is τ=1,ω=1,τ˙t≤r=0.5. Also, the activation function we select is(99)f1i1xi1t=f1i2xi1t=0.5t+0.8signt,f2i1xi2t=f2i2xi2t=t+0.5signt.
Through simple computations, we have(100)P1=30.03570.23560.235630.8698,P2=11.32420.02620.026211.0633,Q=1.0645001.2315.
Meanwhile, the parameters of the controller we choose are as follows.
For mode 1,(101)ξ1rt=1.3045001.5425,η1rt=2.5235002.5249,κ1rt=1.4000001.4000,δ1rt=1.5125001.5684.
For mode 2,(102)ξ2rt=1.2531001.1626,η2rt=1.5738001.6968,κ2rt=2.2500002.2500,δ2rt=1.6145001.6686,and the settling time is T≤2.8.
Then, Figure4 introduces the first state trajectories of the drive-response system with controller (55). The second state trajectories of drive-response system with controller (55) shown in Figures 5 and 6 display the state trajectories of the synchronization error system (20) with the controller (55).
Hence, the effectiveness of Theorem2 is verified. In other words, the global fixed-time synchronization of discontinuous complex dynamical networks with semi-Markovian switching and mixed delays can be addressed.Figure 4
The first state trajectory of the drive-response systemsx1it and y1it with controller (55).Figure 5
The second state trajectory of the drive-response systemsx2it and y2it with controller (55).Figure 6
The state trajectories of the synchronization error system (20) with the controller (55).


## 5. Conclusions

In this paper, the global finite-time and fixed-time synchronization of discontinuous complex dynamical networks with semi-Markovian switching and mixed time-varying delays are investigated. Two novel state-feedback controllers are designed which include the integral term and mixed delay-term. Based on the linear matrix inequality technique (LMIs), Lyapunov functional method, and the proposed control schemes, some sufficient conditions are established to guarantee the global finite-time and fixed-time synchronization of complex dynamical networks. Then, two numerical examples are provided to confirm the effectiveness of the main results.More complex conditions, such as stochastic disturbance, mixed coupling, and the approach of impulsive control, will be taken into consideration in the future study.



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*Source: 6699198-2021-01-05.xml*