# Exponential Synchronization for Neutral Complex Dynamical Networks with Interval Mode-Dependent Delays and Sampled Data

Authors: Xinghua Liu; Hongsheng Xi
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The exponential synchronization and sampled-data controller problem for a class of neutral complex dynamical networks (NCDNs) with Markovian jump parameters, partially unknown transition rates and delays, is investigated in this paper. Both the discrete and neutral delays are considered to be interval mode dependent and time varying, while the sampling period is assumed to be time varying and bounded. Based on a new augmented stochastic Lyapunov functional, the delay-range-dependent
and rate-dependent exponential stability conditions for the closed-loop error system are obtained by the Lyapunov-Krasovskii
stability theory and reciprocally convex lemma. Then according to the proposed exponential stability conditions, the sampled-data
synchronization controllers are designed in terms of the solution to linear matrix inequalities that can be solved effectively by
using Matlab. Finally, numerical examples are given to demonstrate the feasibility and effectiveness of the proposed methods.


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## Body

## 1. Introduction

It is well known that many practical systems can be described as complex networks such as Internet networks, biological networks, epidemic spreading networks, collaborative networks, social networks and neural networks [1–4]. Thus, during the past decades, the research on the dynamics of complex dynamical networks (CDNs) has attracted extensive attention of scientific and engineering researchers in all fields domestic and overseas since the pioneering work of Watts and Strogatz [5]. As one of the most significant and important collective behaviors in CDNs, synchronization has received more attention; please refer to [6–10] and references therein for more details.Since time delay inevitably exists and has become an important issue in studying the CDNs, synchronization problems for complex networks with time delays have gained increasing research attention, and considerable progress has been made; see, for example, [6–17] and references therein for more details. However, in some practical applications of communication networks, signal transmission channels often involve network-induced delays, packet dropouts, bit errors, environment disturbances, and so on, which will cause the error of transmission from one system to another. Then past change rate of the state variables affects the dynamics of nodes in the networks. This kind of complex dynamical network is termed as neutral complex dynamical network (NCDN), which contains delays both in its states and the derivatives of its states. There are some results about the synchronization design problem for neutral systems [18–23]. In these works, [19, 20] had studied the synchronization control for a kind of master-response setup and were further extended to the case of neutral-type neural networks with stochastic perturbation. The authors of [18, 22] had researched the synchronization problem for a class of complex networks with neutral-type coupling delays. The authors in [21] had studied the global asymptotic stability of neural networks of neutral type with mixed delays, which include constant delay in the leakage term, time-varying delays, and continuously distributed delays. The authors in [23] had investigated the robust global exponential synchronization problem for an array of neutral-type neural networks. However, much fewer results have been proposed for neutral complex dynamical networks (NCDNs) compared with the rich results for CDNs with only discrete delays.On the other hand, network mode switching is also a universal phenomenon in CDNs of the actual systems, and sometimes the network has finite modes that switch from one to another with certain transition rate; then such switching can be governed by a Markovian chain. The stability and synchronization problems of complex networks and neural networks with Markovian jump parameters and delays are investigated in [16, 24–30], and references therein. The authors in [24] have established sufficient global exponential stability conditions on Markovian jump neural networks with impulse control and time varying delays. The authors in [27] have studied synchronization in an array of coupled neural networks with Markovian jumping and random coupling strength.  Particularly, Ma et al. [25] have considered the stability and synchronization problems for Markovian jump delayed neural networks with partly unknown transition probabilities, which have not been fully investigated and need to propose more good results. Besides, sampled-data systems have attracted great attention because the digital signal processing methods require better reliability, accuracy, and stable performance with the rapid development in digital measurement and intelligent instrument. There are some important and essential results which have been reported in the literature [31–35]. What is worth mentioning is that the sampled-data synchronization control problem has been investigated for a class of general complex networks with time-varying coupling delays in [34, 35], where conditions have been presented to ensure the exponential stability of the closed-loop error system, and the desired sampled data feedback controllers have been designed. However, few results are available for neutral complex dynamical networks (NCDNs) with sampled data. To the best of the authors' knowledge, the NCDNs are difficult to treat and they are very challenging, especially in the presence of Markovian jump parameters, mode-dependent time-varying delays, and sampled data. Motivated by the above analysis, the exponential synchronization and sampled-data controller problem for a class of NCDNs with Markovian jump parameters and mode-dependent time-varying delay is investigated in this paper. The addressed NCDNs consist of M modes, and the networks switch from one mode to another according to a Markovian chain with partially known transition rate.In this paper, the synchronization and sampled-data controller problem is studied for NCDNs with Markovian jump parameters and partially known transition rates. The sampling period considered here is assumed to be time varying and bounded, while the neutral and discrete delays are interval mode-dependent and time varying. Firstly, by constructing a new augmented stochastic Lyapunov functional, exponential stability conditions are derived based on the Lyapunov stability theory and reciprocally convex lemma. Then the design method of the desired sampled-data controllers is solved on the basis of the obtained conditions. Moreover, all the derived results are in terms of LMIs that can be solved numerically, which are proved to be less conservative than the existing results.The remainder of the paper is organized as follows. Section2 presents the problem and preliminaries. Section 3 gives the main results, which are then verified by numerical examples in Section 4. Section 5 concludes the paper.Notations. The following notations are used throughout the paper. ℝn denotes the n dimensional Euclidean space and ℝm×n is the set of all m×n matrices. X<Y (X>Y), where X and Y are both symmetric matrices, which means that X-Y is negative (positive) definite. I is the identity matrix with proper dimensions. For a symmetric block matrix, we use * to denote the terms introduced by symmetry. ℰ stands for the mathematical expectation; ∥v∥ is the Euclidean norm of vector v, ∥v∥=(vTv)1/2, while ∥A∥ is spectral norm of matrix A, ∥A∥=[λmax(ATA)]1/2. λmax(min)(A) is the eigenvalue of matrix A with maximum (minimum) real part. The Kronecker product of matrices P∈ℝm×n and Q∈ℝp×q is a matrix in ℝmp×nq which is denoted as P⊗Q. Let ς>0 and C([-ς,0],ℝn) denote the family of continuous function φ, from [-ς,0] to ℝn with the norm |φ|=sup-ς≤θ≤0∥φ(θ)∥. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.


## 2. Problem Statement and Preliminaries

Given a complete probability space{Ω,ℱ,{ℱt}t≥0,P} with a natural filtration {ℱt}t≥0 satisfying the usual conditions, where Ω is the sample space, ℱ is the algebra of events and P is the probability measure defined on ℱ. Let {r(t),t≥0} be a right-continuous Markov chain taking values in a finite state space S={1,2,3,…,M} with a generator Υ=(γij)M×M, i,j∈S, which is given by
(1)P(r(t+Δt)=j∣r(t)=i)={γijΔt+o(Δt)i≠j,1+γiiΔt+o(Δt)i=j,
where Δt>0, limΔt→0(o(Δt)/Δt)=0, and γij≥0(i,j∈S,i≠j) is the transition rate from mode i to j, and for any state or mode i∈S, it satisfies
(2)γii=-∑j=1,j≠iN‍γij.
It is assumed that r(t) is irreducible and available at time t, but the transition rates of the Markov chain are partially known in this paper, which means that some elements in matrix Υ=(γij)M×M are inaccessible. For instance, in the system with six operation modes, the jump rates matrix Υ may be viewed as
(3)[γ110γ12γ130γ140γ15γ160γ21γ220γ230γ24γ250γ26γ310γ320γ33γ34γ350γ360γ410γ42γ430γ440γ450γ46γ51γ520γ530γ540γ55γ560γ61γ620γ630γ64γ65γ660],
where γij0,i,j∈S represents the unknown element. Furthermore, let γim, γiM be lower and upper bound for the diagonal element γii or γii0, for all i∈S. For notation clarity, we denote that 𝒮i=𝒮ki⋃‍𝒮uki, for all i∈S, and
(4)𝒮ki≜{j:πijisknownforj∈S},𝒮uki≜{j:πijisunknownforj∈S}.
If 𝒮ki≠⌀, than it is further described as
(5)𝒮ki={k1i,k2i,…,kmi},1≤m≤M,
where kji,(j=1,2,…,m) represent the jth known element of the set 𝒮ki in the ith row of the transition rate matrix Υ. It should be noted that if 𝒮ki=∅, 𝒮i=𝒮uki which means that any information between the ith mode and the other M-1 modes is not accessible, then MJSs with M modes can be regarded as ones with M-1 modes.The following neutral complex dynamical network (NCDN) consisting ofN identical nodes with Markovian jump parameters and interval time-varying delays over the space {Ω,ℱ,{ℱt}t≥0,P} is investigated in this paper:
(6)x˙k(t)=∑l=1N‍gkl(1)(r(t))A(r(t))xl(t)+∑l=1N‍gkl(2)(r(t))B(r(t))xl(t-d(t,r(t)))+∑l=1N‍gkl(3)(r(t))C(r(t))x˙l(t-τ(t,r(t)))+D(r(t))f(xk(t))+uk(t),
where xk(t)∈ℝn, uk(t)∈ℝn are state variable and the control input of the node k∈{1,2,…,N}, respectively. r(t) describes the evolution of the mode. A(r(t))∈ℝn×n, B(r(t))∈ℝn×n, and C(r(t))∈ℝn×n represent the inner-coupling matrices linking between the subsystems in mode r(t). G(1)(r(t))=[gkl(1)]N×N, G(2)(r(t))=[gkl(2)]N×N,  and G(3)(r(t))=[gkl(3)]N×N are the coupling configuration matrices of the networks representing the coupling strength and the topological structure of the NCDNs in mode r(t), in which gkl(m) is defined as follows: if there exists a connection between kth and lth (k≠l) nodes, then gkl(m)(r(t))=glk(m)(r(t))=1; otherwise gkl(m)(r(t))=glk(m)(r(t))=0, and
(7)gkk(m)(r(t))=-∑l=1,l≠kN‍gkl(m)(r(t))=-∑l=1,l≠kN‍glk(m)(r(t)),m=1,2,3;k=1,2,…,N.τ(t,r(t)) and d(t,r(t)) denote the mode-dependent time-varying neutral delay and retarded delay, respectively. They are assumed to satisfy
(8)0≤τi(t)≤τi¯,τ˙i(t)≤νi<1,0≤d1i≤di(t)≤d2i,maxi∈S{d1i}≤minj∈S{d2j},d˙i(t)≤μi,whenr(t)=i,
where τi¯, 0≤νi<1, d1i, and d2i, 0≤μi, are real constant scalars.D
(
r
(
t
)
)
∈
ℝ
n
×
n is a parametric matrix with real values in mode r(t). f:ℝn→ℝn is a continuous vector-valued nonlinear function. It is assumed to satisfy the following sector-bounded condition [36]:(9)[f(x)-f(y)-F1(x-y)]T×[f(x)-f(y)-F2(x-y)]≤0,∀x,y∈ℝn,
where F1, F2 are two constant matrices. Such a description of nonlinear functions has been exploited in [37–39] and is more general than the commonly used Lipschitz conditions. The sector-bounded condition would be possible to reduce the conservatism of the main results caused by quantifying the nonlinear functions via a matrix inequality technique.Lets(t)∈ℝn be the state trajectory of the unforced node s˙(t)=D(r(t))f(s(t)); then the synchronization error is defined to be ek(t)=xk(t)-s(t). So the error dynamics of NCDN (6) can be derived as follows:
(10)e˙k(t)=∑l=1N‍gkl(1)(r(t))A(r(t))el(t)+∑l=1N‍gkl(2)(r(t))B(r(t))el(t-d(t,r(t)))+∑l=1N‍gkl(3)(r(t))C(r(t))e˙l(t-τ(t,r(t)))+D(r(t))fe(ek(t))+uk(t),
where fe(ek(t))=f(xk(t))-f(s(t)),  k=1,2,…,N.The control signal is assumed to be generated by using a zero-order-hold (ZOH) function with a sequence of hold times0=t0<t1<⋯<ts<⋯. Thus the state feedback controller takes the following form:
(11)uk(t)=Kkek(ts),k=1,2,…,N,ts≤t<ts+1,
where Kk is sampled-data feedback controller gain matrix to be determined. ek(ts) is discrete measurement of ek(t) at the sampling instant ts satisfying lims→+∞ts=+∞. It is assumed that ts+1-ts=hs≤h¯ for any integer s≥0, where h¯>0 represents the largest sampling interval.With (10) and (11), the error dynamics of NCDN (6) can be replaced as
(12)e˙k(t)=∑l=1N‍gkl(1)(r(t))A(r(t))el(t)+∑l=1N‍gkl(2)(r(t))B(r(t))el(t-d(t,r(t)))+∑l=1N‍gkl(3)(r(t))C(r(t))e˙l(t-τ(t,r(t)))+D(r(t))fe(ek(t))+Kkek(t-h(t)),
where h(t)=t-ts and 0≤h(t)≤h¯.For simplicity of notations, we denoteA(r(t)), B(r(t)), C(r(t)), D(r(t)), G(m)(r(t)), (m=1,2,3) by Ai, Bi, Ci, Di, Gi(m),(m=1,2,3) for rt=i∈S. By utilizing the Kronecker product of the matrices, (12) can be written in a more compact form as
(13)e˙(t)=𝔸ie(t)+𝔹ie(t-di(t))+ℂie˙(t-τi(t))+𝔻iFe(e(t))+𝕂e(t-h(t)),
where 𝔸i=Gi(1)⊗Ai, 𝔹i=Gi(2)⊗Bi, ℂi=Gi(3)⊗Ci, 𝔻i=IN⊗Di, and
(14)e(t)=col{e1(t),e2(t),…,eN(t)},e(t-di(t))=col{e1(t-di(t)),e2(t-di(t)),…,eN(t-di(t))},e˙(t-τi(t))=col{e˙1(t-τi(t)),e˙2(t-τi(t)),…,e˙N(t-τi(t))},Fe(e(t))=col{{(eN(t))}fe(e1(t)),fe(e2(t)),…,fe(eN(t))},e(t-h(t))=col{e1(t-h(t)),e2(t-h(t)),…,eN(t-h(t))},𝕂=diag{K1,K2,…,KN}.The purpose of this paper is to design a serial of sampled-data controllers (11) to ensure the exponential synchronization of NCDN (6). Before proceeding with the main results, we present the following definitions and lemmas.Definition 1 (see [40]).
Define operator𝔇:C([-ρ,0],ℝn)→ℝn as 𝔇(xt)=x(t)-Cx(t-τ). 𝔇 is said to be stable if the homogeneous difference equation
(15)𝔇(xt)=0,t≥0,x0=ψ∈{ϕ∈C([-ρ,0],ℝn):𝔇ϕ=0}
is uniformly asymptotically stable. In this paper, that is, ∥Gi(3)⊗Ci∥<1.Definition 2 (see [41]).
Define the stochastic Lyapunov-Krasovskii function of the error system (13) as V(e(t),r(t)=i, t>0)=V(e(t),i,t) where its infinitesimal generator is defined as
(16)ΓV(e(t),i,t)=limΔt→01Δt[ℰ{V(e(t+Δt),r(t+Δt),t+Δt)∣e(t)=e,r(t)=i}-V(e(t),i,t)]=∂∂tV(e(t),i,t)+∂∂eV(e(t),i,t)e˙(t)+∑j=1N‍πijV(e(t),j,t).Definition 3.
The NCDN (6) is said to be exponentially synchronized if the error system (13) is exponentially stable; that is, [42] there exist two constants ε>0 and κ≥1 such that for all, e(t),
(17)∥e(t)∥≤κexp{-ε(t-t0)}supθ∈[-ς,t0]∥e(θ)∥,
where ε is the exponential decay rate; ς=max{h¯,maxi∈S{τi¯},maxi∈S{d2i}}.Lemma 4 (see [43]).
Given matricesA, B, C, and D with appropriate dimensions and scalar α, by the definition of the Kronecker product, the following properties hold:
(18)(αA)⊗B=A⊗(αB),(A+B)⊗C=A⊗C+B⊗C,(A⊗B)(C⊗D)=(AC)⊗(BD),(A⊗B)T=AT⊗BT.Lemma 5.
For any constant matrixQ=QT>0, continuous functions 0≤h1(t)≤h2(t), constant scalars 0≤τ1<τ2, and constant ε>0 such that the following integrations are well defined:
(19)(a)exp{2εh2(t)}-exp{2εh1(t)}2ε×∫t-h2(t)t-h1(t)‍exp{2ε(s-t)}xT(s)Qx(s)ds≥[∫t-h2(t)t-h1(t)‍xT(s)ds]Q[∫t-h2(t)t-h1(t)‍x(s)ds],(b)τ22-τ122∫-τ2-τ1‍∫t+θt‍exp{2ε(s-t)}xT(s)Qx(s)dsdθ≥exp{-2ετ2}[∫-τ2-τ1‍∫t+θt‍xT(s)dsdθ]Q×[∫-τ2-τ1‍∫t+θt‍x(s)dsdθ].Proof.
(a) is directly obtained from [44]. In addition, from -τ2≤θ≤-τ1 and t+θ≤s≤t, it is held that -τ2≤θ≤s-t≤0. Then,
(20)∫-τ2-τ1‍∫t+θt‍exp{2ε(s-t)}xT(s)Qx(s)dsdθ≥exp{-2ετ2}∫-τ2-τ1‍∫t+θt‍xT(s)Qx(s)dsdθ;
(b) is thus true by [45].Lemma 6 (see [46]).
For functionsλ1(t), λ2(t)∈[0,1], λ1(t)+λ2(t)=1, and η1=0 with λ1(t)=0 and η2=0 with λ2(t)=0 and matrices P>0, Q>0, then there exists matrix T such that
(21)[PTTTQ]>0,
and the following inequality holds:
(22)1λ1(t)η1Pη1T+1λ2(t)η2Qη2T≥[η1η2][PTTTQ][η1Tη2T].According to Definition3, the aim will be achieved if we obtain the gain matrices 𝕂 such that the error system (13) is exponentially stable. So we give the main results as follows.


## 3. Main Results

In this section, sufficient conditions are presented to ensure that the error system (13) is  exponentially stable. Then, we propose a design method of the sampled-data controllers for NCDN (6).


### 3.1. Exponential Stability for the Error System

Theorem 7.
For given scalarsγim, γiM, ε, τi¯, νi, h¯, d1i, d2i, and μi and constant scalar dmi satisfying d1i<dmi<d2i, the error system (13) with partially known transition rates and sector-bounded condition (9) is exponentially stable with decay rate ε and κ=λ¯/λ_ if ∥Gi(3)⊗Ci∥<1, and there exist symmetric positive matrices Pi>0, Q1i>0, Q2i>0, R1i>0, (i∈S), Qj>0, (j=3,4,5), Rk>0, (k=2,3,4), Tl>0, (l=1,2,3), Um>0, Vn>0, Ws>0, (m,n,s=1,2,3,4,5) and matrices Mk1, Nk2, (k1,k2=1,2,3,4), for any scalar δ>0, and symmetric matrices Xi, Y1i, Y2i, Zi, (i∈S) such that the following matrix inequalities hold.
Wheni∈𝒮ki; then
(23)∑j∈𝒮ki,j≠i‍γij(Q1j-Y1i)-γiiY1i≤0,∑j∈𝒮ki,j≠i‍γij(Q2j-Y2i)-γiiY2i≤0,∑j∈𝒮ki,j≠i‍γij(R1j-Zi)-γiiZi≤0,Q1j-Y1i≤0,j∈𝒮uki,Q2j-Y2i≤0,j∈𝒮uki,R1j-Zi≤0,j∈𝒮uki,Pj-Xi≤0,j∈𝒮uki.
Wheni∈𝒮uki, then
(24)∑j∈𝒮ki‍γij(Q1j-Y1i)≤0,∑j∈𝒮ki‍γij(Q2j-Y2i)≤0,∑j∈𝒮ki‍γij(R1j-Zi)≤0,(25)Pj-Xi≤0,j∈𝒮uki,j≠i,Pj-Xi≥0,j∈𝒮uki,j=i,(26)-∑j∈𝒮uki,j≠i‍γiM(Q1j-Y1i)-γimY1i≤0,-∑j∈𝒮uki,j≠i‍γiM(Q2j-Y2i)-γimY2i≤0,(27)-∑j∈𝒮uki,j≠i‍γiM(R1j-Zi)-γimZi≤0,
and
(28)[U1M1M1TU1]>0,[V1M2M2TV1]>0,[U2M3M3TU2]>0,[V2M4M4TV2]>0,(29)(i)[U4N1N1TU4]>0,[V4N2N2TV4]>0,(ii)Ωi0ε+Ωi1ε+ΛTJΛ<0,(30)(iii)[U5N3N3TU5]>0,[V5N4N4TV5]>0,(iv)Ωi0ε+Ωi2ε+ΛTJΛ<0,
where
(31)Ωi0ε=∑m=124‍EmΠmEmT+✠-δ[E1E24][F1^F2^*I][E1TE24T]-2εd1iexp{2εd1i}-1(E1-E3)V3(E1T-E3T)-exp{-2ετi¯}(τi¯E1-E19)W1(τi¯E1T-E19T)-exp{-2εh¯}(h¯E1-E23)W2(h¯E1T-E23T)-exp{-2εd1i}(d1iE1-E9)W3(d1iE1T-E9T)-exp{-2εdmi}(ϱ1iE1-E12)W4(ϱ1iE1T-E12T)-exp{-2εd2i}(ϱ2iE1-E13)W5(ϱ2iE1T-E13T)-2ετi¯exp{2ετi¯}-1[E18E19-E18]×[U1M1M1TU1][E18TE19T-E18T]-2ετi¯exp{2ετi¯}-1[E1-E16E16-E14]×[V1M2M2TV1][E1T-E16TE16T-e14T]-2εh¯exp{2εh¯}-1[E22E23-E22]×[U2M3M3TU2][E22TE23T-E22T]-2εh¯exp{2εh¯}-1[E1-E21E21-E20]×[V2M4M4TV2][E1T-E21TE21T-E20T],
where
(32)✠=E1Pi(Gi(2)⊗Bi)E2T+E1Pi(Gi(3)⊗Ci)E17+E1Pi𝕂E21T+E1Pi(IN⊗Di)E24T+E2(Gi(2)⊗Bi)TPiE1T+E17(Gi(3)⊗Ci)TPiE1T+E21𝕂TPiE1T+E24(IN⊗Di)TPiE1TF1^=IN⊗(F1TF2+F2TF1)2,F2^=-(IN⊗F1)T+(IN⊗F2)T2,Π1=(Gi(1)⊗Ai)TPi+Pi(Gi(1)⊗Ai)+∑j∈𝒮ki‍γij(Pj-Xi)+2εPi+Q1i+Q3+Q5+R2+τi¯2U1+h¯2U2+d1i2U3+ϱ1i2U4+ϱ2i2U5,Π2=ϖ(ε,μi)R1i,Π3=exp{-2εd1i}(R1i+R3-R2),Π4=exp{-2εdmi}(R4-R3),Π5=-exp{-2εd2i}R4,Π6=exp{-2εd1i}(T2-T1),Π7=exp{-2εdmi}(T3-T2),Π8=-exp{-2εd2i}T3,Π9=-2εd1iexp{2εd1i}-1U3,Π14=-exp{-2ετi¯}Q3,Π15=-exp{-2ετi¯}Q4,Π16=-(1-νi)exp{-2ετi¯}Q1i,Π17=-(1-νi)exp{-2ετi¯}Q2i,Π20=-exp{-2εh¯}Q5,Πm=0,(m=10,11,12,13,18,19,21,…,24),Ωi1ε=-2εϱ2iexp{2εd2i}-exp{2εdmi}E13U5E13T-2εϱ2iexp{2εd2i}-exp{2εdmi}(E4-E5)V5(E4T-E5T)-2εϱ1iexp{2εdmi}-exp{2εd1i}[E10E12-E10]×[U4N1N1TU4][E10TE12T-E10T]-2εϱ1iexp{2εdmi}-exp{2εd1i}[E3-E2E2-E4]×[V4N2N2TV4][E3T-E2TE2T-E4T],Ωi2ε=-2εϱ1iexp{2εdmi}-exp{2εd1i}E12U4E12T-2εϱ1iexp{2εdmi}-exp{2εd1i}×(E3-E4)V4(E3T-E4T)-2εϱ2iexp{2εd2i}-exp{2εdmi}[E11E13-E11]×[U5N3N3TU5][E11TE13T-E11T]-2εϱ2iexp{2εd2i}-exp{2εdmi}[E4-E2E2-E5]×[V5N4N4TV5][E4T-E2TE2T-E5T],J=Q2i+Q4+T1+τi¯2V1+h¯2V2+d1i2V3+ϱ1i2V4+ϱ2i2V5+τi¯44W1+h¯44W2+d1i44W3+ϱ3i2W4+ϱ4i2W5,Λ=(Gi(1)⊗Ai)E1T+(Gi(2)⊗Bi)E2T+(Gi(3)⊗Ci)E17T+𝕂E21T+(IN⊗Di)E24T,
where Ei{i=1,2,…,24} are block entry matrices; that is,
(33)E4T=[000I00000000000000000000],ϖ(ε,μi)={(μi-1)exp{-2εd1i}withμi≥1,(μi-1)exp{-2εd2i}withμi<1,λ¯=maxi∈S{λmax(Pi)}+1-exp{-2ες}2ε×{∑j=35‍λmax(Qj)+∑k=24‍λmax(Rk)+∑l=13‍λmax(Tl)}+1-exp{-2ες}2ε{maxi∈S{λmax(Q1i)}+maxi∈S{λmax(Q2i)}+maxi∈S{λmax(R1i)}}+exp{-2ες}+2ες-14ε2×{∑m=15‍λmax(Um)+∑n=15‍λmax(Vn)}+1-exp{-2ες}+2ε2ς2-2ες8ε3×{∑s=15‍λmax(Ws)},λ_=mini∈S{λmin(Pi)},ϱ1i=dmi-d1i,ϱ2i=d2i-dmi,ϱ3i=12(dmi2-d1i2),ϱ4i=12(d2i2-dmi2).Proof.
Construct the following stochastic Lyapunov functional:(34)V(e(t),i,t)=∑k=17‍Vk(e(t),i,t),
where
(35)V1(e(t),i,t)=eT(t)Pie(t),(36)V2(e(t),i,t)=∫t-τi(t)t‍exp{2ε(s-t)}eT(s)Q1ie(s)ds+∫t-τi(t)t‍exp{2ε(s-t)}e˙T(s)Q2ie˙(s)ds+∫t-τi¯t‍exp{2ε(s-t)}eT(s)Q3e(s)ds+∫t-τi¯t‍exp{2ε(s-t)}e˙T(s)Q4e˙(s)ds+∫t-h¯t‍exp{2ε(s-t)}eT(s)Q5e(s)ds,(37)V3(e(t),i,t)=∫t-di(t)t-d1i‍exp{2ε(s-t)}eT(s)R1ie(s)ds+∫t-d1it‍exp{2ε(s-t)}eT(s)R2e(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}eT(s)R3e(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}eT(s)R4e(s)ds,(38)V4(e(t),i,t)=∫t-d1it‍exp{2ε(s-t)}e˙T(s)T1e˙(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}e˙T(s)T2e˙(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}e˙T(s)T3e˙(s)ds,(39)V5(e(t),i,t)=∫-τi¯0‍∫t+θt‍τi¯exp{2ε(s-t)}eT(s)U1e(s)dsdθ+∫-h¯0‍∫t+θt‍h¯exp{2ε(s-t)}eT(s)U2e(s)dsdθ+∫-d1i0‍∫t+θt‍d1iexp{2ε(s-t)}eT(s)U3e(s)dsdθ+∫-dmi-d1i‍∫t+θt‍ϱ1iexp{2ε(s-t)}eT(s)U4e(s)dsdθ+∫-d2i-dmi‍∫t+θt‍ϱ2iexp{2ε(s-t)}eT(s)U5e(s)dsdθ,(40)V6(e(t),i,t)=∫-τi¯0‍∫t+θt‍τi¯exp{2ε(s-t)}e˙T(s)V1e˙(s)dsdθ+∫-h¯0‍∫t+θt‍h¯exp{2ε(s-t)}e˙T(s)V2e˙(s)dsdθ+∫-d1i0‍∫t+θt‍d1iexp{2ε(s-t)}e˙T(s)V3e˙(s)dsdθ+∫-dmi-d1i‍∫t+θt‍ϱ1iexp{2ε(s-t)}e˙T(s)V4e˙(s)dsdθ+∫-d2i-dmi‍∫t+θt‍ϱ2iexp{2ε(s-t)}e˙T(s)V5e˙(s)dsdθ,(41)V7(e(t),i,t)=∫-τi¯0‍∫θ0‍∫t+λt‍τi¯22exp{2ε(s-t)}e˙T×(s)W1e˙(s)dsdλdθ+∫-h¯0‍∫θ0‍∫t+λt‍h¯22exp{2ε(s-t)}e˙T×(s)W2e˙(s)dsdλdθ+∫-d1i0‍∫θ0‍∫t+λt‍d1i22exp{2ε(s-t)}e˙T×(s)W3e˙(s)dsdλdθ+∫-dmi-d1i‍∫θ0‍∫t+λt‍ϱ3iexp{2ε(s-t)}e˙T×(s)W4e˙(s)dsdλdθ+∫-d2i-dmi‍∫θ0‍∫t+λt‍ϱ4iexp{2ε(s-t)}e˙T×(s)W5e˙(s)dsdλdθ.
TakingΓ as its infinitesimal generator along the trajectory of error system (13), we obtain the following from Definition 3 and (34)-(35):
(42)ΓV(e(t),i,t)=∑k=17‍ΓVk(e(t),i,t),(43)ΓV1(e(t),i,t)=2[eT(t)𝔸iT+eT(t-di(t))𝔹iT+e˙T(t-τi(t))ℂiT+FeT(e(t))𝔻iT+eT(t-h(t))𝕂T]Pie(t)+eT(t)[∑j∈𝒮ki‍γijPj+∑j∈𝒮uki‍γijPj]e(t)+2εeT(t)Pie(t)-2εV1(e(t),i,t),(44)ΓV2(e(t),i,t)=eT(t)[Q1i+Q3+Q5]e(t)+e˙T(t)[Q2i+Q4]e˙(t)-2εV2(e(t),i,t)-(1-τ˙i(t))exp{-2ετi(t)}×eT(t-τi(t))Q1ie(t-τi(t))-(1-τ˙i(t))exp{-2ετi(t)}×e˙T(t-τi(t))Q2ie˙(t-τi(t))-exp{-2ετi¯}eT(t-τi¯)Q3e(t-τi¯)-exp{-2ετi¯}e˙T(t-τi¯)Q4e˙(t-τi¯)+∑j∈S‍γij∫t-τj(t)t‍exp{2ε(s-t)}×[eT(s)Q1je(s)+e˙T(s)Q2je˙(s)]ds-exp{-2εh¯}eT(t-h¯)Q5e(t-h¯)≤eT(t)[Q1i+Q3+Q5]e(t)+e˙T(t)[Q2i+Q4]e˙(t)-2εV2(e(t),i,t)-(1-τ˙i(t))exp{-2ετi(t)}×eT(t-τi(t))Q1ie(t-τi(t))-(1-τ˙i(t))exp{-2ετi(t)}×e˙T(t-τi(t))Q2ie˙(t-τi(t))-exp{-2ετi¯}eT(t-τi¯)Q3e(t-τi¯)-exp{-2ετi¯}e˙T(t-τi¯)Q4e˙(t-τi¯)+∑j≠i‍γij∫t-τj¯t‍exp{2ε(s-t)}×[eT(s)Q1je(s)+e˙T(s)Q2je˙(s)]ds-exp{-2εh¯}eT(t-h¯)Q5e(t-h¯),(45)ΓV3(e(t),i,t)=eT(t)R2e(t)+exp{-2εd1i}eT(t-d1i)×[R1i+R3-R2]e(t-d1i)+exp{-2εdmi}eT(t-dmi)×[R4-R3]e(t-dmi)-exp{-2εd2i}eT(t-d2i)×R4e(t-d2i)-2εV3(e(t),i,t)-(1-d˙i(t))exp{-2εdi(t)}×eT(t-di(t))R1ie(t-di(t))+∑j∈S‍γij∫t-dj(t)t-d1i‍exp{2ε(s-t)}eT(s)R1je(s)ds≤eT(t)R2e(t)+exp{-2εd1i}×eT(t-d1i)[R1i+R3-R2]e(t-d1i)+exp{-2εdmi}eT(t-dmi)×[R4-R3]e(t-dmi)-exp{-2εd2i}eT(t-d2i)×R4e(t-d2i)-2εV3(e(t),i,t)-(1-d˙i(t))exp{-2εdi(t)}×eT(t-di(t))R1ie(t-di(t))+∑j≠i‍γij∫t-d2jt-d1i‍exp{2ε(s-t)}eT(s)R1je(s)ds,(46)ΓV4(e(t),i,t)=e˙T(t)T1e˙(t)+exp{-2εd1i}e˙T(t-d1i)[T2-T1]×e˙(t-d1i)-2εV4(e(t),i,t)+exp{-2εdmi}e˙T(t-dmi)×[T3-T2]e˙(t-dmi)-exp{-2εd2i}e˙T(t-d2i)T3e˙(t-d2i),(47)ΓV5(e(t),i,t)=eT(t)[τi¯2U1+h¯2U2+d1i2U3+ϱ1i2U4+ϱ2i2U5{h¯2}]e(t)-2εV5(e(t),i,t)-∫t-τi¯t‍τi¯exp{2ε(s-t)}eT(s)U1e(s)ds-∫t-h¯t‍h¯exp{2ε(s-t)}eT(s)U2e(s)ds-∫t-d1it‍d1iexp{2ε(s-t)}eT(s)U3e(s)ds-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}eT(s)U4e(s)ds-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}eT(s)U5e(s)ds,(48)ΓV6(e(t),i,t)=e˙T(t)[τi¯2V1+h¯2V2+d1i2V3+ϱ1i2V4+ϱ2i2V5]e˙(t)-2εV6(e(t),i,t)-∫t-τi¯t‍τi¯exp{2ε(s-t)}e˙T(s)V1e˙(s)ds-∫t-h¯t‍h¯exp{2ε(s-t)}e˙T(s)V2e˙(s)ds-∫t-d1it‍d1iexp{2ε(s-t)}e˙T(s)V3e˙(s)ds-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}e˙T(s)V4e˙(s)ds-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}e˙T(s)V5e˙(s)ds,(49)ΓV7(e(t),i,t)=e˙T(t)[τi¯44W1+h¯44W2+d1i44W3+ϱ3i2W4+ϱ4i2W5]e˙(t)-2εV7(e(t),i,t)-∫-τi¯0‍∫t+θt‍τi¯22exp{2ε(s-t)}e˙T(s)W1e˙(s)dsdθ-∫-h¯0‍∫t+θt‍h¯22exp{2ε(s-t)}e˙T(s)W2e˙(s)dsdθ-∫-d1i0‍∫t+θt‍d1i22exp{2ε(s-t)}e˙T(s)W3e˙(s)dsdθ-∫-dmi-d1i‍∫t+θt‍ϱ3iexp{2ε(s-t)}e˙T(s)W4e˙(s)dsdθ-∫-d2i-dmi‍∫t+θt‍ϱ4iexp{2ε(s-t)}e˙T(s)W5e˙(s)dsdθ.
Define(50)ξ(t)=col{{∫t-h¯t‍}e(t)e(t-di(t))e(t-d1i)e(t-dmi)e(t-d2i)e˙(t-d1i)e˙(t-dmi)e˙(t-d2i)∫t-d1it‍e(s)ds∫t-di(t)t-d1i‍e(s)ds∫t-d2it-di(t)e(s)ds‍∫t-dmit-d1i‍e(s)ds∫t-d2it-dmi‍e(s)dse(t-τi¯)e˙(t-τi¯)e(t-τi(t))e˙(t-τi(t))∫t-τi(t)t‍e(s)ds∫t-τi¯t‍e(s)dse(t-h¯)e(t-h(t))∫t-h(t)t‍e(s)ds∫t-h¯t‍e(s)dsFe(e(t))}.Due to ∑j=1N‍γij=0, the following zero equations hold for arbitrary matrices:
(51)Xi=XiT,Y1i=Y1iT,Y2i=Y2iT,Zi=ZiT,i∈S;
that is,
(52)-eT(t)[∑j∈𝒮ki‍γijXi+∑j∈𝒮uki‍γijXi]e(t)=0,-∫t-τj¯t‍exp{2ε(s-t)}eT(s)×[∑j∈𝒮ki‍γijY1i+∑j∈𝒮uki‍γijY1i]e(s)ds=0,-∫t-τj¯t‍exp{2ε(s-t)}e˙T(s)×[∑j∈𝒮ki‍γijY2i+∑j∈𝒮uki‍γijY2i]e˙(s)ds=0,-∫t-d2jt-d1i‍exp{2ε(s-t)}eT(s)×[∑j∈𝒮ki‍γijZi+∑j∈𝒮uki‍γijZi]e(s)ds=0.
Based on (9), we can obtain the following inequality for any δ>0:
(53)δ[eT(t)FeT(e(t))][F1^F2^*I][e(t)Fe(e(t))]=δξT(t)[E1E24][F1^F2^*I][E1TE24T]ξ(t)≤0.
From (42) and (53), we have
(54)ΓV(e(t),i,t)≤∑k=17‍ΓVk(e(t),i,t)-δξT(t)[E1E24][F1^F2^*I][E1TE24T]ξ(t).
From (8) we easily obtain that
(55)-(1-τ˙i(t))exp{-2ετi(t)}≤-(1-νi)exp{-2ετi¯},-(1-d˙i(t))exp{-2εdi(t)}≤ϖ(ε,μi).
Notice (a) of Lemma 5; then,
(56)-∫t-d1it‍d1iexp{2ε(s-t)}eT(s)U3e(s)ds≤-2εd1iexp{2εd1i}-1ξT(t)E9U3E9Tξ(t),-∫t-d1it‍d1iexp{2ε(s-t)}e˙T(s)V3e˙(s)ds≤-2εd1iexp{2εd1i}-1ξT(t)(E1-E3)V3(E1T-E3T)ξ(t).
Notice (b) of Lemma 5; then,
(57)-∫-τi¯0‍∫t+θt‍τi¯22exp{2ε(s-t)}e˙T(s)W1e˙(s)dsdθ≤-exp{-2ετi¯}ξT(t)(τi¯E1-E19)W1×(τi¯E1T-E19T)ξ(t),-∫-h¯0‍∫t+θt‍h¯22exp{2ε(s-t)}e˙T(s)W2e˙(s)dsdθ≤-exp{-2εh¯}ξT(t)(h¯E1-E23)W2×(hi¯E1T-E23T)ξ(t),-∫-d1i0‍∫t+θt‍d1i22exp{2ε(s-t)}e˙T(s)W3e˙(s)dsdθ≤-exp{-2εd1i}ξT(t)(d1iE1-E9)W3×(d1iE1T-E9T)ξ(t),-∫-dmi-d1i‍∫t+θt‍ϱ3iexp{2ε(s-t)}e˙T(s)W4e˙(s)dsdθ≤-exp{-2εdmi}ξT(t)(ϱ1iE1-E12)W4×(ϱ1iE1T-E12T)ξ(t),-∫-d2i-dmi‍∫t+θt‍ϱ4iexp{2ε(s-t)}e˙T(s)W5e˙(s)dsdθ≤-exp{-2εd2i}ξT(t)(ϱ2iE1-E13)W5×(ϱ2iE1T-E13T)ξ(t).
For di(t)∈[d1i,dmi], the following is held from (a) of Lemma 5:(58)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}e˙T(s)V4e˙(s)ds=-{∫t-di(t)t-d1i‍+∫t-dmit-di(t)‍}ϱ1i×exp{2ε(s-t)}e˙T(s)V4e˙(s)ds≤-2εϱ1iexp{2εdi(t)}-exp{2εd1i}×[∫t-di(t)t-d1i‍e˙T(s)ds]V4[∫t-di(t)t-d1i‍e˙(s)ds]-2εϱ1iexp{2εdmi}-exp{2εd(t)}×[∫t-dmit-di(t)‍e˙T(s)ds]V4[∫t-dmit-di(t)‍e˙(s)ds]=-2εϱ1iexp{2εdmi}-exp{2εd1i}×ξT(t)[1λ1i(t)(E3-E2)V4(E3T-E2T)+1λ2i(t)(E2-E4)V4(E2T-E4T)]ξ(t),
where
(59)λ1i(t)=exp{2εdi(t)}-exp{2εd1i}exp{2εdmi}-exp{2εd1i},λ2i(t)=exp{2εdmi}-exp{2εdi(t)}exp{2εdmi}-exp{2εd1i}.
By Lemma6, there exists matrix N2 with appropriate dimensions such that
(60)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}e˙T(s)V4e˙(s)ds≤-2εϱ1iexp{2εdmi}-exp{2εd1i}ξT(t)×[E3-E2E2-E4][V4N2N2TV4][E3T-E2TE2T-E4T]ξ(t),[V4N2N2TV4]>0.
Similarly, considering-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}xT(s)U4x(s)ds and following the same procedure, there exists matrix N1 with appropriate dimensions such that
(61)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}eT(s)U4e(s)ds≤-2εϱ1iexp{2εdmi}-exp{2εd1i}ξT(t)×[E10E12-E10][U4N1N1TU4][E10TE12T-E10T]ξ(t),[U4N1N1TU4]>0.
Forτi(t)∈[0,τi¯], with the same matrix inequalities technique, we obtain the following:
(62)-∫t-τi¯t‍τi¯exp{2ε(s-t)}eT(s)U1e(s)ds≤-2ετi¯exp{2ετi¯}-1ξT(t)[E18E19-E18]×[U1M1M1TU1][E18TE19T-E18T]ξ(t),[U1M1M1TU1]>0,-∫t-τi¯t‍τi¯exp{2ε(s-t)}e˙T(s)V1e˙(s)ds≤-2ετi¯exp{2ετi¯}-1ξT(t)[E1-E16E16-E14]×[V1M2M2TV1][E1T-E16TE16T-E14T]ξ(t),[V1M2M2TV1]>0.
Forh(t)∈[0,h¯], we obtain the following:
(63)-∫t-h¯t‍h¯exp{2ε(s-t)}eT(s)U2e(s)ds≤-2εh¯exp{2εh¯}-1ξT(t)[E22E23-E22]×[U2M3M3TU2][E22TE23T-E22T]ξ(t),[U2M3M3TU2]>0,-∫t-h¯t‍h¯exp{2ε(s-t)}e˙T(s)V2e˙(s)ds≤-2εh¯exp{2εh¯}-1ξT(t)[E1-E21E21-E20]×[V2M4M4TV2][E1T-E21TE21T-E20T]ξ(t),[V2M4M4TV2]>0.
Consider-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}eT(s)U5e(s)ds and -∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}e˙T(s)V5e˙(s)ds, which are directly estimated by (a) of Lemma 5; that is,
(64)-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}eT(s)U5e(s)ds≤-2εϱ2iexp{2εd2i}-exp{2εdmi}ξT(t)E13U5E13Tξ(t),-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}e˙T(s)V5e˙(s)ds≤-2εϱ2iexp{2εd2i}-exp{2εdmi}×ξT(t)(E4-E5)V5(E4T-E5T)ξ(t).
In addition, with the error system (53), we obtain e˙(t)=Λξ(t) and
(65)e˙T(t)Je˙(t)=ξT(t)ΛTJΛξ(t),
where Λ and J have been defined in Theorem 7.
Substituting (43)–(52) and (55)–(65) into (54), we obtain
(66)ΓV(e(t),i,t)+2εV(e(t),i,t)≤ξT(t)(Ωi0ε+Ωi1ε+ΛTJΛ)ξ(t)+℧,
where
(67)℧=∫t-τj¯t‍exp{2ε(s-t)}eT(s)×{∑j≠i‍γijQ1j-∑j∈𝒮ki‍γijY1i-∑j∈𝒮uki‍γijY1i}e(s)ds+∫t-τj¯t‍exp{2ε(s-t)}e˙T(s)×{∑j≠i‍γijQ2j-∑j∈𝒮ki‍γijY2i-∑j∈𝒮uki‍γijY2i}e˙(s)ds+∫t-d2jt-d1i‍exp{2ε(s-t)}eT(s)×{∑j≠i‍γijR1j-∑j∈𝒮ki‍γijZi-∑j∈𝒮uki‍γijZi}e(s)ds+eT(t){∑j∈𝒮uki‍γij(Pj-Xi)}e(t).
On the other hand, fordi(t)∈[dmi,d2i], the integral terms
(68)-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}eT(s)U5e(s)ds,-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}e˙T(s)V5e˙(s)ds
are disposed and estimated by Lemma 6.
(69)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}eT(s)U4e(s)ds,-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}e˙T(s)V4e˙(s)ds
are directly estimated by (a) of Lemma 5. Therefore,
(70)ΓV(e(t),i,t)+2εV(e(t),i,t)≤ξT(t)(Ωi0ε+Ωi2ε+ΛTJΛ)ξ(t)+℧.
With (66) and (70), the following inequality is held for di(t)∈[d1i,d2i] if (23)–(28) are satisfied:
(71)ΓV(e(t),i,t)+2εV(e(t),i,t)<0.
From the stochastic Lyapunov functional (34) and (71), it is held that
(72)V(e(t),i,t)≥mini∈S{λmin(Pi)}∥e(t)∥2=λ_∥e(t)∥2,V(e(t),i,t)<exp{-2ε(t-t0)}V(e(t0),r(t0),t0).
Moreover, we have(73)V(e(t0),r(t0),t0)=∑k=17‍Vk(e(t0),r(t0),t0),∑k=17‍Vk(e(t0),r(t0),t0)≤∫t0-ςt0‍exp{2ε(s-t0)}eT(s)×{Q1i+Q3+Q5+R1i+∑k=24‍Rk}e(s)ds+∫t0-ςt0‍exp{2ε(s-t0)}e˙T(s)×{Q2i+Q4+∑l=13‍Tl}e˙(s)ds+∫-ς0‍∫t0+θt0‍ςexp{2ε(s-t0)}eT×(s)(∑m=15‍Um)e(s)dsdθ+∫-ς0‍∫t0+θt0‍ςexp{2ε(s-t0)}e˙T×(s)(∑n=15‍Vn)e˙(s)dsdθ+∫-ς0‍∫θ0‍∫t0+λt0‍ς22exp{2ε(s-t0)}e˙T×(s)(∑s=15‍Ws)e˙(s)dsdλdθ+eT(t0)Pie(t0)≤λ¯∥e(t0)∥ς2.
Then from (72) and (73), it is readily seen that
(74)∥e(t)∥≤λ¯λ_exp{-ε(t-t0)}∥e(t0)∥ς,
where κ=λ¯/λ_≥1.
Therefore, the error system (13) with partially known transition rates and sector-bounded condition (9) is exponentially stable with a decay rate ε. This completes the proof.In some situations, due to the complexity of NCDN (6), the information on the transition rates may be completely unknown, which can be viewed as switched complex dynamical networks with arbitrary switching. The following corollary is therefore given to guarantee the exponential stability for this case.Corollary 8.
For given scalarsγim, γiM, ε, τi¯, νi, h¯, d1i, d2i, and μi and constant scalar dmi satisfying d1i<dmi<d2i, the error system (13) with entirely unknown transition rates and sector-bounded condition (9) is exponentially stable with decay rate ε and κ=λ¯/λ_ if ∥Gi(3)⊗Ci∥<1, and there exist symmetric positive matrices P>0, Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4), Tl>0, (l=1,2,3), Um>0, Vn>0, Ws>0, (m,n,s=1,2,3,4,5) and matrices Mk1, Nk2, (k1,k2=1,2,3,4), for any scalar δ>0, and symmetric matrices Y1, Y2, Z such that (28) and the following matrix inequalities hold:
(75)-∑j∈𝒮uki,j≠i‍γiM(Q1-Y1)-γimY1≤0,-∑j∈𝒮uki,j≠i‍γiM(Q2-Y2)-γimY2≤0,(76)-∑j∈𝒮uki,j≠i‍γiM(R1-Z)-γimZ≤0,(77)(i)[U4N1N1TU4]>0,[V4N2N2TV4]>0,(ii)Ω~i0ε+Ω~i1ε+Λ~TJ~Λ~<0,(78)(iii)[U5N3N3TU5]>0,[V5N4N4TV5]>0,(iv)Ω~i0ε+Ω~i2ε+Λ~TJ~Λ~<0,
where
(79)J~=Q2+Q4+T1+τi¯2V1+h¯2V2+d1i2V3+ϱ1i2V4+ϱ2i2V5+τi¯44W1+h¯44W2+d1i44W3+ϱ3i2W4+ϱ4i2W5,Λ~=Λ,Ω~i1ε=Ωi1ε,Ω~i2ε=Ωi2ε,Ω~i0ε=∑m=124‍EmΠ~mEmT+✠-δ×[E1E24][F1^F2^*I][E1TE24T]-2εd1iexp{2εd1i}-1(E1-E3)V3(E1T-E3T)-exp{-2ετi¯}(τi¯E1-E19)W1(τi¯E1T-E19T)-exp{-2εh¯}(h¯E1-E23)W2(h¯E1T-E23T)-exp{-2εd1i}(d1iE1-E9)W3(d1iE1T-E9T)-exp{-2εdmi}(ϱ1iE1-E12)W4(ϱ1iE1T-E12T)-exp{-2εd2i}(ϱ2iE1-E13)W5(ϱ2iE1T-E13T)-2ετi¯exp{2ετi¯}-1[E18E19-E18]×[U1M1M1TU1][E18TE19T-E18T]-2ετi¯exp{2ετi¯}-1[E1-E16E16-E14]×[V1M2M2TV1][E1T-E16TE16T-e14T]-2εh¯exp{2εh¯}-1[E22E23-E22]×[U2M3M3TU2][E22TE23T-E22T]-2εh¯exp{2εh¯}-1[E1-E21E21-E20]×[V2M4M4TV2][E1T-E21TE21T-E20T],Π~1=(Gi(1)⊗Ai)TP+P(Gi(1)⊗Ai)+2εP+Q1+Q3+Q5+R2+τi¯2U1+h¯2U2+d1i2U3+ϱ1i2U4+ϱ2i2U5,Π~2=ϖ(ε,μi)R1,Π~3=exp{-2εd1i}(R1+R3-R2),Π~16=-(1-νi)exp{-2ετi¯}Q1,Π~17=-(1-νi)exp{-2ετi¯}Q2,Π~4=Π4,Π~5=Π5,Π~6=Π6,Π~7=Π7,Π~8=Π8,Π~9=Π9,Π~14=Π14,Π~15=Π15,Π~20=Π20,Π~m=0,(m=10,11,12,13,18,19,21,…,24),λ¯=λmax(P)+1-exp{-2ες}2ε×{∑j=15‍λmax(Qj)+∑k=14‍λmax(Rk)+∑l=13‍λmax(Tl)}+exp{-2ες}+2ες-14ε2×{∑m=15‍λmax(Um)+∑n=15‍λmax(Vn)}+1-exp{-2ες}+2ε2ς2-2ες8ε3×{∑s=15‍λmax(Ws)}.Other notations are the same as those in Theorem7.Proof.
Since the elements of transition rates are entirely unknown, we choose the Lyapunov functional as follows:(80)V(e(t),i,t)=eT(t)Pe(t)+∑k=27‍Vk(e(t),i,t),
where
(81)V2(e(t),i,t)=∫t-τi(t)t‍exp{2ε(s-t)}eT(s)Q1e(s)ds+∫t-τi(t)t‍exp{2ε(s-t)}e˙T(s)Q2e˙(s)ds+∫t-τi¯t‍exp{2ε(s-t)}eT(s)Q3e(s)ds+∫t-τi¯t‍exp{2ε(s-t)}e˙T(s)Q4e˙(s)ds+∫t-h¯t‍exp{2ε(s-t)}eT(s)Q5e(s)ds,V3(e(t),i,t)=∫t-di(t)t-d1i‍exp{2ε(s-t)}eT(s)R1e(s)ds+∫t-d1it‍exp{2ε(s-t)}eT(s)R2e(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}eT(s)R3e(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}eT(s)R4e(s)ds,V4(e(t),i,t)=∫t-d1it‍exp{2ε(s-t)}e˙T(s)T1e˙(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}e˙T(s)T2e˙(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}e˙T(s)T3e˙(s)ds,
and V5(e(t),i,t), V6(e(t),i,t), V7(e(t),i,t) are the same as (39), (40), (41).
Then we follow a similar line as in proof of Theorem7 and obtain the result.


### 3.2. The Sampled-Data Controllers for NCDN

In this subsection, we give the design method of the desired sampled-data controllers to ensure the NCDN (6) exponentially synchronized on the basis of Theorem 7.Theorem 9.
The NCDN (6) with partially known transition rates and sector-bounded condition (9) is exponentially synchronized by controllers of the form (11) if there exist symmetric positive definite matrices Pi=diag{Pi(1),Pi(2),…,Pi(N)}>0, Q1i>0, Q2i>0, R1i>0, (i∈S={1,2,…,M}), Qj>0, (j=3,4,5), Rk>0, (k=2,3,4), Tl>0, (l=1,2,3), Um>0, Vn>0, Ws>0, (m,n,s=1,2,3,4,5), matrices 𝕐i=diag{Yi(1),Yi(2),…,Yi(N)} and matrices Mk1, Nk2, (k1,k2=1,2,3,4), for any scalar δ>0, and symmetric matrices Xi, Y1i, Y2i, Zi, (i∈S) such that (23)–(28) and the following linear matrix inequalities hold:
(82)(i)[U4N1N1TU4]>0,[V4N2N2TV4]>0,(ii)[Θi0ε+Ωi1εΛTPiPiΛ-2Pi+J]<0,(83)(iii)[U5N3N3TU5]>0,[V5N4N4TV5]>0,(iv)[Θi0ε+Ωi2εΛTPiPiΛ-2Pi+J]<0.
In addition, the desired sampled-data controllers gain matrices are given by
(84)Kk=(Pi(k))-1Yi(k),k=1,2,…,N,
where
(85)Θi0ε=∑m=124‍EmΠmEmT+✠Θ-δ[E1E24]×[F1^F2^*I][E1TE24T]-2εd1iexp{2εd1i}-1×(E1-E3)V3(E1T-E3T)-exp{-2ετi¯}(τi¯E1-E19)W1(τi¯E1T-E19T)-exp{-2εh¯}(h¯E1-E23)W2(h¯E1T-E23T)-exp{-2εd1i}(d1iE1-E9)W3(d1iE1T-E9T)-exp{-2εdmi}(ϱ1iE1-E12)W4(ϱ1iE1T-E12T)-exp{-2εd2i}(ϱ2iE1-E13)W5(ϱ2iE1T-E13T)-2ετi¯exp{2ετi¯}-1[E18E19-E18]×[U1M1M1TU1][E18TE19T-E18T]-2ετi¯exp{2ετi¯}-1[E1-E16E16-E14]×[V1M2M2TV1][E1T-E16TE16T-e14T]-2εh¯exp{2εh¯}-1[E22E23-E22]×[U2M3M3TU2][E22TE23T-E22T]-2εh¯exp{2εh¯}-1[E1-E21E21-E20]×[V2M4M4TV2][E1T-E21TE21T-E20T],✠Θ=E1Pi(Gi(2)⊗Bi)E2T+E1Pi(Gi(3)⊗Ci)E17+E1𝕐iE21T+E1Pi(IN⊗Di)E24T+E2(Gi(2)⊗Bi)TPiE1T+E17(Gi(3)⊗Ci)TPiE1T+E21𝕐iTE1T+E24(IN⊗Di)TPiE1T.Other notations are the same as those in Theorem7.Proof.
Considering (27) and (28), by applying the Schur Complement Lemma, we have
(86)(i)[U4N1N1TU4]>0,[V4N2N2TV4]>0,(ii)[Ωi0ε+Ωi1εΛTJJΛ-J]<0,(87)(iii)[U5N3N3TU5]>0,[V5N4N4TV5]>0,(iv)[Ωi0ε+Ωi2εΛTJJΛ-J]<0.
Define the block diagonal matrix𝕏=[HPiJ-1], where H=∑m=124‍EmEmT. Noting that EiTEj={0i≠jIi=j, with (ii) of (86) and (iv) of (87), we define 𝕐i=Pi𝕂 and obtain the following:
(88)𝕏[Ωi0ε+Ωi1εΛTJJΛ-J]𝕏T=[Θi0ε+Ωi1εΛTPiPiΛ-PiJ-1Pi]<0,𝕏[Ωi0ε+Ωi2εΛTJJΛ-J]𝕏T=[Θi0ε+Ωi2εΛTPiPiΛ-PiJ-1Pi]<0.
Due toJ>0 and (Pi-J)J-1(Pi-J)≥0, we know that -PiJ-1Pi≤-2Pi+J. Thus (88) are held if (ii) of (82) and (iv) of (83) are satisfied, which implies that (ii) of (27) and (iv) of (28) are held.
According to Definition3, by the sampled-data controllers gain matrices (84), the NCDN (6) with partially known transition rates and sector-bounded condition (9) is exponentially synchronized. This completes the proof.Remark 10.
It should be mentioned that the sampled-data synchronization problem has been solved for NCDN (6) in Theorem 9, and the desired controllers can be obtained when LMIs (23)–(26), (82), and (83) are feasible. In this paper, we construct the novel stochastic Lyapunov functional (34) containing some triple-integral terms; which is very effective in the reduction of conservatism [45]. Besides, we apply Lemma 5 to the corresponding terms, the method by using reciprocally convex lemma [46] can achieve less conservative results. Thus the obtained delay-range-dependent and decay rate-dependent stability condition for NCDN (6) in Theorem 7 is less conservative than the previous ones, which can achieve larger maximum value of sampling period h¯ and will be verified in Section 4.Remark 11.
It should be pointed out that the given results can be extended to more general neutral delay in the form of NCDN (6). For instance, τ1i≤τi(t)≤τ2i; we can choose τmi satisfying τ1i≤τmi≤τ2i and construct the stochastic Lyapunov functional on [t-τi(t),t-τ1i], [t-τ1i,t], [t-τ2i,t-τmi], and [t-τmi,t-τ1i]. Following the same thought, the time delay h(t) in the control input can also be extended to be more general.Corollary 12.
The NCDN (6) with completely unknown transition rates and sector-bounded condition (9) is exponentially synchronized by controllers of the form (11) if there exist symmetric positive definite matrices P=diag{P(1),P(2),…,P(N)}>0, Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4), Tl>0, (l=1,2,3), Um>0, Vn>0, Ws>0, (m,n,s=1,2,3,4,5), matrices 𝕐=diag{Y(1),Y(2),…,Y(N)} and matrices Mk1, Nk2, (k1,k2=1,2,3,4), for any scalar δ>0, and symmetric matrices Y1, Y2, Z such that (28), (75), (76), and the following linear matrix inequalities hold:
(89)(i)[U4N1N1TU4]>0,[V4N2N2TV4]>0,(ii)[Θ~i0ε+Ω~i1εΛTPPΛ-2P+J]<0,(90)(iii)[U5N3N3TU5]>0,[V5N4N4TV5]>0,(iv)[Θ~i0ε+Ω~i2εΛTPPΛ-2P+J]<0.
In addition, the desired sampled-data controllers gain matrices are given by
(91)Kk=(P(k))-1Y(k),k=1,2,…,N,
where
(92)Θ~i0ε=∑m=124‍EmΠ~mEmT+✠~Θ-δ[E1E24][F1^F2^*I][E1TE24T]-2εd1iexp{2εd1i}-1(E1-E3)V3(E1T-E3T)-exp{-2ετi¯}(τi¯E1-E19)W1(τi¯E1T-E19T)-exp{-2εh¯}(h¯E1-E23)W2(h¯E1T-E23T)-exp{-2εd1i}(d1iE1-E9)W3(d1iE1T-E9T)-exp{-2εdmi}(ϱ1iE1-E12)W4(ϱ1iE1T-E12T)-exp{-2εd2i}(ϱ2iE1-E13)W5(ϱ2iE1T-E13T)-2ετi¯exp{2ετi¯}-1[E18E19-E18]×[U1M1M1TU1][E18TE19T-E18T]-2ετi¯exp{2ετi¯}-1[E1-E16E16-E14]×[V1M2M2TV1][E1T-E16TE16T-e14T]-2εh¯exp{2εh¯}-1[E22E23-E22]×[U2M3M3TU2][E22TE23T-E22T]-2εh¯exp{2εh¯}-1[E1-E21E21-E20]×[V2M4M4TV2][E1T-E21TE21T-E20T],✠~Θ=E1P(Gi(2)⊗Bi)E2T+E1P(Gi(3)⊗Ci)E17+E1𝕐iE21T+E1P(IN⊗Di)E24T+E2(Gi(2)⊗Bi)TPE1T+E17(Gi(3)⊗Ci)TPE1T+E21𝕐iTE1T+E24(IN⊗Di)TPE1T.Other notations are the same as those in Corollary8.


## 3.1. Exponential Stability for the Error System

Theorem 7.
For given scalarsγim, γiM, ε, τi¯, νi, h¯, d1i, d2i, and μi and constant scalar dmi satisfying d1i<dmi<d2i, the error system (13) with partially known transition rates and sector-bounded condition (9) is exponentially stable with decay rate ε and κ=λ¯/λ_ if ∥Gi(3)⊗Ci∥<1, and there exist symmetric positive matrices Pi>0, Q1i>0, Q2i>0, R1i>0, (i∈S), Qj>0, (j=3,4,5), Rk>0, (k=2,3,4), Tl>0, (l=1,2,3), Um>0, Vn>0, Ws>0, (m,n,s=1,2,3,4,5) and matrices Mk1, Nk2, (k1,k2=1,2,3,4), for any scalar δ>0, and symmetric matrices Xi, Y1i, Y2i, Zi, (i∈S) such that the following matrix inequalities hold.
Wheni∈𝒮ki; then
(23)∑j∈𝒮ki,j≠i‍γij(Q1j-Y1i)-γiiY1i≤0,∑j∈𝒮ki,j≠i‍γij(Q2j-Y2i)-γiiY2i≤0,∑j∈𝒮ki,j≠i‍γij(R1j-Zi)-γiiZi≤0,Q1j-Y1i≤0,j∈𝒮uki,Q2j-Y2i≤0,j∈𝒮uki,R1j-Zi≤0,j∈𝒮uki,Pj-Xi≤0,j∈𝒮uki.
Wheni∈𝒮uki, then
(24)∑j∈𝒮ki‍γij(Q1j-Y1i)≤0,∑j∈𝒮ki‍γij(Q2j-Y2i)≤0,∑j∈𝒮ki‍γij(R1j-Zi)≤0,(25)Pj-Xi≤0,j∈𝒮uki,j≠i,Pj-Xi≥0,j∈𝒮uki,j=i,(26)-∑j∈𝒮uki,j≠i‍γiM(Q1j-Y1i)-γimY1i≤0,-∑j∈𝒮uki,j≠i‍γiM(Q2j-Y2i)-γimY2i≤0,(27)-∑j∈𝒮uki,j≠i‍γiM(R1j-Zi)-γimZi≤0,
and
(28)[U1M1M1TU1]>0,[V1M2M2TV1]>0,[U2M3M3TU2]>0,[V2M4M4TV2]>0,(29)(i)[U4N1N1TU4]>0,[V4N2N2TV4]>0,(ii)Ωi0ε+Ωi1ε+ΛTJΛ<0,(30)(iii)[U5N3N3TU5]>0,[V5N4N4TV5]>0,(iv)Ωi0ε+Ωi2ε+ΛTJΛ<0,
where
(31)Ωi0ε=∑m=124‍EmΠmEmT+✠-δ[E1E24][F1^F2^*I][E1TE24T]-2εd1iexp{2εd1i}-1(E1-E3)V3(E1T-E3T)-exp{-2ετi¯}(τi¯E1-E19)W1(τi¯E1T-E19T)-exp{-2εh¯}(h¯E1-E23)W2(h¯E1T-E23T)-exp{-2εd1i}(d1iE1-E9)W3(d1iE1T-E9T)-exp{-2εdmi}(ϱ1iE1-E12)W4(ϱ1iE1T-E12T)-exp{-2εd2i}(ϱ2iE1-E13)W5(ϱ2iE1T-E13T)-2ετi¯exp{2ετi¯}-1[E18E19-E18]×[U1M1M1TU1][E18TE19T-E18T]-2ετi¯exp{2ετi¯}-1[E1-E16E16-E14]×[V1M2M2TV1][E1T-E16TE16T-e14T]-2εh¯exp{2εh¯}-1[E22E23-E22]×[U2M3M3TU2][E22TE23T-E22T]-2εh¯exp{2εh¯}-1[E1-E21E21-E20]×[V2M4M4TV2][E1T-E21TE21T-E20T],
where
(32)✠=E1Pi(Gi(2)⊗Bi)E2T+E1Pi(Gi(3)⊗Ci)E17+E1Pi𝕂E21T+E1Pi(IN⊗Di)E24T+E2(Gi(2)⊗Bi)TPiE1T+E17(Gi(3)⊗Ci)TPiE1T+E21𝕂TPiE1T+E24(IN⊗Di)TPiE1TF1^=IN⊗(F1TF2+F2TF1)2,F2^=-(IN⊗F1)T+(IN⊗F2)T2,Π1=(Gi(1)⊗Ai)TPi+Pi(Gi(1)⊗Ai)+∑j∈𝒮ki‍γij(Pj-Xi)+2εPi+Q1i+Q3+Q5+R2+τi¯2U1+h¯2U2+d1i2U3+ϱ1i2U4+ϱ2i2U5,Π2=ϖ(ε,μi)R1i,Π3=exp{-2εd1i}(R1i+R3-R2),Π4=exp{-2εdmi}(R4-R3),Π5=-exp{-2εd2i}R4,Π6=exp{-2εd1i}(T2-T1),Π7=exp{-2εdmi}(T3-T2),Π8=-exp{-2εd2i}T3,Π9=-2εd1iexp{2εd1i}-1U3,Π14=-exp{-2ετi¯}Q3,Π15=-exp{-2ετi¯}Q4,Π16=-(1-νi)exp{-2ετi¯}Q1i,Π17=-(1-νi)exp{-2ετi¯}Q2i,Π20=-exp{-2εh¯}Q5,Πm=0,(m=10,11,12,13,18,19,21,…,24),Ωi1ε=-2εϱ2iexp{2εd2i}-exp{2εdmi}E13U5E13T-2εϱ2iexp{2εd2i}-exp{2εdmi}(E4-E5)V5(E4T-E5T)-2εϱ1iexp{2εdmi}-exp{2εd1i}[E10E12-E10]×[U4N1N1TU4][E10TE12T-E10T]-2εϱ1iexp{2εdmi}-exp{2εd1i}[E3-E2E2-E4]×[V4N2N2TV4][E3T-E2TE2T-E4T],Ωi2ε=-2εϱ1iexp{2εdmi}-exp{2εd1i}E12U4E12T-2εϱ1iexp{2εdmi}-exp{2εd1i}×(E3-E4)V4(E3T-E4T)-2εϱ2iexp{2εd2i}-exp{2εdmi}[E11E13-E11]×[U5N3N3TU5][E11TE13T-E11T]-2εϱ2iexp{2εd2i}-exp{2εdmi}[E4-E2E2-E5]×[V5N4N4TV5][E4T-E2TE2T-E5T],J=Q2i+Q4+T1+τi¯2V1+h¯2V2+d1i2V3+ϱ1i2V4+ϱ2i2V5+τi¯44W1+h¯44W2+d1i44W3+ϱ3i2W4+ϱ4i2W5,Λ=(Gi(1)⊗Ai)E1T+(Gi(2)⊗Bi)E2T+(Gi(3)⊗Ci)E17T+𝕂E21T+(IN⊗Di)E24T,
where Ei{i=1,2,…,24} are block entry matrices; that is,
(33)E4T=[000I00000000000000000000],ϖ(ε,μi)={(μi-1)exp{-2εd1i}withμi≥1,(μi-1)exp{-2εd2i}withμi<1,λ¯=maxi∈S{λmax(Pi)}+1-exp{-2ες}2ε×{∑j=35‍λmax(Qj)+∑k=24‍λmax(Rk)+∑l=13‍λmax(Tl)}+1-exp{-2ες}2ε{maxi∈S{λmax(Q1i)}+maxi∈S{λmax(Q2i)}+maxi∈S{λmax(R1i)}}+exp{-2ες}+2ες-14ε2×{∑m=15‍λmax(Um)+∑n=15‍λmax(Vn)}+1-exp{-2ες}+2ε2ς2-2ες8ε3×{∑s=15‍λmax(Ws)},λ_=mini∈S{λmin(Pi)},ϱ1i=dmi-d1i,ϱ2i=d2i-dmi,ϱ3i=12(dmi2-d1i2),ϱ4i=12(d2i2-dmi2).Proof.
Construct the following stochastic Lyapunov functional:(34)V(e(t),i,t)=∑k=17‍Vk(e(t),i,t),
where
(35)V1(e(t),i,t)=eT(t)Pie(t),(36)V2(e(t),i,t)=∫t-τi(t)t‍exp{2ε(s-t)}eT(s)Q1ie(s)ds+∫t-τi(t)t‍exp{2ε(s-t)}e˙T(s)Q2ie˙(s)ds+∫t-τi¯t‍exp{2ε(s-t)}eT(s)Q3e(s)ds+∫t-τi¯t‍exp{2ε(s-t)}e˙T(s)Q4e˙(s)ds+∫t-h¯t‍exp{2ε(s-t)}eT(s)Q5e(s)ds,(37)V3(e(t),i,t)=∫t-di(t)t-d1i‍exp{2ε(s-t)}eT(s)R1ie(s)ds+∫t-d1it‍exp{2ε(s-t)}eT(s)R2e(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}eT(s)R3e(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}eT(s)R4e(s)ds,(38)V4(e(t),i,t)=∫t-d1it‍exp{2ε(s-t)}e˙T(s)T1e˙(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}e˙T(s)T2e˙(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}e˙T(s)T3e˙(s)ds,(39)V5(e(t),i,t)=∫-τi¯0‍∫t+θt‍τi¯exp{2ε(s-t)}eT(s)U1e(s)dsdθ+∫-h¯0‍∫t+θt‍h¯exp{2ε(s-t)}eT(s)U2e(s)dsdθ+∫-d1i0‍∫t+θt‍d1iexp{2ε(s-t)}eT(s)U3e(s)dsdθ+∫-dmi-d1i‍∫t+θt‍ϱ1iexp{2ε(s-t)}eT(s)U4e(s)dsdθ+∫-d2i-dmi‍∫t+θt‍ϱ2iexp{2ε(s-t)}eT(s)U5e(s)dsdθ,(40)V6(e(t),i,t)=∫-τi¯0‍∫t+θt‍τi¯exp{2ε(s-t)}e˙T(s)V1e˙(s)dsdθ+∫-h¯0‍∫t+θt‍h¯exp{2ε(s-t)}e˙T(s)V2e˙(s)dsdθ+∫-d1i0‍∫t+θt‍d1iexp{2ε(s-t)}e˙T(s)V3e˙(s)dsdθ+∫-dmi-d1i‍∫t+θt‍ϱ1iexp{2ε(s-t)}e˙T(s)V4e˙(s)dsdθ+∫-d2i-dmi‍∫t+θt‍ϱ2iexp{2ε(s-t)}e˙T(s)V5e˙(s)dsdθ,(41)V7(e(t),i,t)=∫-τi¯0‍∫θ0‍∫t+λt‍τi¯22exp{2ε(s-t)}e˙T×(s)W1e˙(s)dsdλdθ+∫-h¯0‍∫θ0‍∫t+λt‍h¯22exp{2ε(s-t)}e˙T×(s)W2e˙(s)dsdλdθ+∫-d1i0‍∫θ0‍∫t+λt‍d1i22exp{2ε(s-t)}e˙T×(s)W3e˙(s)dsdλdθ+∫-dmi-d1i‍∫θ0‍∫t+λt‍ϱ3iexp{2ε(s-t)}e˙T×(s)W4e˙(s)dsdλdθ+∫-d2i-dmi‍∫θ0‍∫t+λt‍ϱ4iexp{2ε(s-t)}e˙T×(s)W5e˙(s)dsdλdθ.
TakingΓ as its infinitesimal generator along the trajectory of error system (13), we obtain the following from Definition 3 and (34)-(35):
(42)ΓV(e(t),i,t)=∑k=17‍ΓVk(e(t),i,t),(43)ΓV1(e(t),i,t)=2[eT(t)𝔸iT+eT(t-di(t))𝔹iT+e˙T(t-τi(t))ℂiT+FeT(e(t))𝔻iT+eT(t-h(t))𝕂T]Pie(t)+eT(t)[∑j∈𝒮ki‍γijPj+∑j∈𝒮uki‍γijPj]e(t)+2εeT(t)Pie(t)-2εV1(e(t),i,t),(44)ΓV2(e(t),i,t)=eT(t)[Q1i+Q3+Q5]e(t)+e˙T(t)[Q2i+Q4]e˙(t)-2εV2(e(t),i,t)-(1-τ˙i(t))exp{-2ετi(t)}×eT(t-τi(t))Q1ie(t-τi(t))-(1-τ˙i(t))exp{-2ετi(t)}×e˙T(t-τi(t))Q2ie˙(t-τi(t))-exp{-2ετi¯}eT(t-τi¯)Q3e(t-τi¯)-exp{-2ετi¯}e˙T(t-τi¯)Q4e˙(t-τi¯)+∑j∈S‍γij∫t-τj(t)t‍exp{2ε(s-t)}×[eT(s)Q1je(s)+e˙T(s)Q2je˙(s)]ds-exp{-2εh¯}eT(t-h¯)Q5e(t-h¯)≤eT(t)[Q1i+Q3+Q5]e(t)+e˙T(t)[Q2i+Q4]e˙(t)-2εV2(e(t),i,t)-(1-τ˙i(t))exp{-2ετi(t)}×eT(t-τi(t))Q1ie(t-τi(t))-(1-τ˙i(t))exp{-2ετi(t)}×e˙T(t-τi(t))Q2ie˙(t-τi(t))-exp{-2ετi¯}eT(t-τi¯)Q3e(t-τi¯)-exp{-2ετi¯}e˙T(t-τi¯)Q4e˙(t-τi¯)+∑j≠i‍γij∫t-τj¯t‍exp{2ε(s-t)}×[eT(s)Q1je(s)+e˙T(s)Q2je˙(s)]ds-exp{-2εh¯}eT(t-h¯)Q5e(t-h¯),(45)ΓV3(e(t),i,t)=eT(t)R2e(t)+exp{-2εd1i}eT(t-d1i)×[R1i+R3-R2]e(t-d1i)+exp{-2εdmi}eT(t-dmi)×[R4-R3]e(t-dmi)-exp{-2εd2i}eT(t-d2i)×R4e(t-d2i)-2εV3(e(t),i,t)-(1-d˙i(t))exp{-2εdi(t)}×eT(t-di(t))R1ie(t-di(t))+∑j∈S‍γij∫t-dj(t)t-d1i‍exp{2ε(s-t)}eT(s)R1je(s)ds≤eT(t)R2e(t)+exp{-2εd1i}×eT(t-d1i)[R1i+R3-R2]e(t-d1i)+exp{-2εdmi}eT(t-dmi)×[R4-R3]e(t-dmi)-exp{-2εd2i}eT(t-d2i)×R4e(t-d2i)-2εV3(e(t),i,t)-(1-d˙i(t))exp{-2εdi(t)}×eT(t-di(t))R1ie(t-di(t))+∑j≠i‍γij∫t-d2jt-d1i‍exp{2ε(s-t)}eT(s)R1je(s)ds,(46)ΓV4(e(t),i,t)=e˙T(t)T1e˙(t)+exp{-2εd1i}e˙T(t-d1i)[T2-T1]×e˙(t-d1i)-2εV4(e(t),i,t)+exp{-2εdmi}e˙T(t-dmi)×[T3-T2]e˙(t-dmi)-exp{-2εd2i}e˙T(t-d2i)T3e˙(t-d2i),(47)ΓV5(e(t),i,t)=eT(t)[τi¯2U1+h¯2U2+d1i2U3+ϱ1i2U4+ϱ2i2U5{h¯2}]e(t)-2εV5(e(t),i,t)-∫t-τi¯t‍τi¯exp{2ε(s-t)}eT(s)U1e(s)ds-∫t-h¯t‍h¯exp{2ε(s-t)}eT(s)U2e(s)ds-∫t-d1it‍d1iexp{2ε(s-t)}eT(s)U3e(s)ds-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}eT(s)U4e(s)ds-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}eT(s)U5e(s)ds,(48)ΓV6(e(t),i,t)=e˙T(t)[τi¯2V1+h¯2V2+d1i2V3+ϱ1i2V4+ϱ2i2V5]e˙(t)-2εV6(e(t),i,t)-∫t-τi¯t‍τi¯exp{2ε(s-t)}e˙T(s)V1e˙(s)ds-∫t-h¯t‍h¯exp{2ε(s-t)}e˙T(s)V2e˙(s)ds-∫t-d1it‍d1iexp{2ε(s-t)}e˙T(s)V3e˙(s)ds-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}e˙T(s)V4e˙(s)ds-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}e˙T(s)V5e˙(s)ds,(49)ΓV7(e(t),i,t)=e˙T(t)[τi¯44W1+h¯44W2+d1i44W3+ϱ3i2W4+ϱ4i2W5]e˙(t)-2εV7(e(t),i,t)-∫-τi¯0‍∫t+θt‍τi¯22exp{2ε(s-t)}e˙T(s)W1e˙(s)dsdθ-∫-h¯0‍∫t+θt‍h¯22exp{2ε(s-t)}e˙T(s)W2e˙(s)dsdθ-∫-d1i0‍∫t+θt‍d1i22exp{2ε(s-t)}e˙T(s)W3e˙(s)dsdθ-∫-dmi-d1i‍∫t+θt‍ϱ3iexp{2ε(s-t)}e˙T(s)W4e˙(s)dsdθ-∫-d2i-dmi‍∫t+θt‍ϱ4iexp{2ε(s-t)}e˙T(s)W5e˙(s)dsdθ.
Define(50)ξ(t)=col{{∫t-h¯t‍}e(t)e(t-di(t))e(t-d1i)e(t-dmi)e(t-d2i)e˙(t-d1i)e˙(t-dmi)e˙(t-d2i)∫t-d1it‍e(s)ds∫t-di(t)t-d1i‍e(s)ds∫t-d2it-di(t)e(s)ds‍∫t-dmit-d1i‍e(s)ds∫t-d2it-dmi‍e(s)dse(t-τi¯)e˙(t-τi¯)e(t-τi(t))e˙(t-τi(t))∫t-τi(t)t‍e(s)ds∫t-τi¯t‍e(s)dse(t-h¯)e(t-h(t))∫t-h(t)t‍e(s)ds∫t-h¯t‍e(s)dsFe(e(t))}.Due to ∑j=1N‍γij=0, the following zero equations hold for arbitrary matrices:
(51)Xi=XiT,Y1i=Y1iT,Y2i=Y2iT,Zi=ZiT,i∈S;
that is,
(52)-eT(t)[∑j∈𝒮ki‍γijXi+∑j∈𝒮uki‍γijXi]e(t)=0,-∫t-τj¯t‍exp{2ε(s-t)}eT(s)×[∑j∈𝒮ki‍γijY1i+∑j∈𝒮uki‍γijY1i]e(s)ds=0,-∫t-τj¯t‍exp{2ε(s-t)}e˙T(s)×[∑j∈𝒮ki‍γijY2i+∑j∈𝒮uki‍γijY2i]e˙(s)ds=0,-∫t-d2jt-d1i‍exp{2ε(s-t)}eT(s)×[∑j∈𝒮ki‍γijZi+∑j∈𝒮uki‍γijZi]e(s)ds=0.
Based on (9), we can obtain the following inequality for any δ>0:
(53)δ[eT(t)FeT(e(t))][F1^F2^*I][e(t)Fe(e(t))]=δξT(t)[E1E24][F1^F2^*I][E1TE24T]ξ(t)≤0.
From (42) and (53), we have
(54)ΓV(e(t),i,t)≤∑k=17‍ΓVk(e(t),i,t)-δξT(t)[E1E24][F1^F2^*I][E1TE24T]ξ(t).
From (8) we easily obtain that
(55)-(1-τ˙i(t))exp{-2ετi(t)}≤-(1-νi)exp{-2ετi¯},-(1-d˙i(t))exp{-2εdi(t)}≤ϖ(ε,μi).
Notice (a) of Lemma 5; then,
(56)-∫t-d1it‍d1iexp{2ε(s-t)}eT(s)U3e(s)ds≤-2εd1iexp{2εd1i}-1ξT(t)E9U3E9Tξ(t),-∫t-d1it‍d1iexp{2ε(s-t)}e˙T(s)V3e˙(s)ds≤-2εd1iexp{2εd1i}-1ξT(t)(E1-E3)V3(E1T-E3T)ξ(t).
Notice (b) of Lemma 5; then,
(57)-∫-τi¯0‍∫t+θt‍τi¯22exp{2ε(s-t)}e˙T(s)W1e˙(s)dsdθ≤-exp{-2ετi¯}ξT(t)(τi¯E1-E19)W1×(τi¯E1T-E19T)ξ(t),-∫-h¯0‍∫t+θt‍h¯22exp{2ε(s-t)}e˙T(s)W2e˙(s)dsdθ≤-exp{-2εh¯}ξT(t)(h¯E1-E23)W2×(hi¯E1T-E23T)ξ(t),-∫-d1i0‍∫t+θt‍d1i22exp{2ε(s-t)}e˙T(s)W3e˙(s)dsdθ≤-exp{-2εd1i}ξT(t)(d1iE1-E9)W3×(d1iE1T-E9T)ξ(t),-∫-dmi-d1i‍∫t+θt‍ϱ3iexp{2ε(s-t)}e˙T(s)W4e˙(s)dsdθ≤-exp{-2εdmi}ξT(t)(ϱ1iE1-E12)W4×(ϱ1iE1T-E12T)ξ(t),-∫-d2i-dmi‍∫t+θt‍ϱ4iexp{2ε(s-t)}e˙T(s)W5e˙(s)dsdθ≤-exp{-2εd2i}ξT(t)(ϱ2iE1-E13)W5×(ϱ2iE1T-E13T)ξ(t).
For di(t)∈[d1i,dmi], the following is held from (a) of Lemma 5:(58)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}e˙T(s)V4e˙(s)ds=-{∫t-di(t)t-d1i‍+∫t-dmit-di(t)‍}ϱ1i×exp{2ε(s-t)}e˙T(s)V4e˙(s)ds≤-2εϱ1iexp{2εdi(t)}-exp{2εd1i}×[∫t-di(t)t-d1i‍e˙T(s)ds]V4[∫t-di(t)t-d1i‍e˙(s)ds]-2εϱ1iexp{2εdmi}-exp{2εd(t)}×[∫t-dmit-di(t)‍e˙T(s)ds]V4[∫t-dmit-di(t)‍e˙(s)ds]=-2εϱ1iexp{2εdmi}-exp{2εd1i}×ξT(t)[1λ1i(t)(E3-E2)V4(E3T-E2T)+1λ2i(t)(E2-E4)V4(E2T-E4T)]ξ(t),
where
(59)λ1i(t)=exp{2εdi(t)}-exp{2εd1i}exp{2εdmi}-exp{2εd1i},λ2i(t)=exp{2εdmi}-exp{2εdi(t)}exp{2εdmi}-exp{2εd1i}.
By Lemma6, there exists matrix N2 with appropriate dimensions such that
(60)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}e˙T(s)V4e˙(s)ds≤-2εϱ1iexp{2εdmi}-exp{2εd1i}ξT(t)×[E3-E2E2-E4][V4N2N2TV4][E3T-E2TE2T-E4T]ξ(t),[V4N2N2TV4]>0.
Similarly, considering-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}xT(s)U4x(s)ds and following the same procedure, there exists matrix N1 with appropriate dimensions such that
(61)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}eT(s)U4e(s)ds≤-2εϱ1iexp{2εdmi}-exp{2εd1i}ξT(t)×[E10E12-E10][U4N1N1TU4][E10TE12T-E10T]ξ(t),[U4N1N1TU4]>0.
Forτi(t)∈[0,τi¯], with the same matrix inequalities technique, we obtain the following:
(62)-∫t-τi¯t‍τi¯exp{2ε(s-t)}eT(s)U1e(s)ds≤-2ετi¯exp{2ετi¯}-1ξT(t)[E18E19-E18]×[U1M1M1TU1][E18TE19T-E18T]ξ(t),[U1M1M1TU1]>0,-∫t-τi¯t‍τi¯exp{2ε(s-t)}e˙T(s)V1e˙(s)ds≤-2ετi¯exp{2ετi¯}-1ξT(t)[E1-E16E16-E14]×[V1M2M2TV1][E1T-E16TE16T-E14T]ξ(t),[V1M2M2TV1]>0.
Forh(t)∈[0,h¯], we obtain the following:
(63)-∫t-h¯t‍h¯exp{2ε(s-t)}eT(s)U2e(s)ds≤-2εh¯exp{2εh¯}-1ξT(t)[E22E23-E22]×[U2M3M3TU2][E22TE23T-E22T]ξ(t),[U2M3M3TU2]>0,-∫t-h¯t‍h¯exp{2ε(s-t)}e˙T(s)V2e˙(s)ds≤-2εh¯exp{2εh¯}-1ξT(t)[E1-E21E21-E20]×[V2M4M4TV2][E1T-E21TE21T-E20T]ξ(t),[V2M4M4TV2]>0.
Consider-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}eT(s)U5e(s)ds and -∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}e˙T(s)V5e˙(s)ds, which are directly estimated by (a) of Lemma 5; that is,
(64)-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}eT(s)U5e(s)ds≤-2εϱ2iexp{2εd2i}-exp{2εdmi}ξT(t)E13U5E13Tξ(t),-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}e˙T(s)V5e˙(s)ds≤-2εϱ2iexp{2εd2i}-exp{2εdmi}×ξT(t)(E4-E5)V5(E4T-E5T)ξ(t).
In addition, with the error system (53), we obtain e˙(t)=Λξ(t) and
(65)e˙T(t)Je˙(t)=ξT(t)ΛTJΛξ(t),
where Λ and J have been defined in Theorem 7.
Substituting (43)–(52) and (55)–(65) into (54), we obtain
(66)ΓV(e(t),i,t)+2εV(e(t),i,t)≤ξT(t)(Ωi0ε+Ωi1ε+ΛTJΛ)ξ(t)+℧,
where
(67)℧=∫t-τj¯t‍exp{2ε(s-t)}eT(s)×{∑j≠i‍γijQ1j-∑j∈𝒮ki‍γijY1i-∑j∈𝒮uki‍γijY1i}e(s)ds+∫t-τj¯t‍exp{2ε(s-t)}e˙T(s)×{∑j≠i‍γijQ2j-∑j∈𝒮ki‍γijY2i-∑j∈𝒮uki‍γijY2i}e˙(s)ds+∫t-d2jt-d1i‍exp{2ε(s-t)}eT(s)×{∑j≠i‍γijR1j-∑j∈𝒮ki‍γijZi-∑j∈𝒮uki‍γijZi}e(s)ds+eT(t){∑j∈𝒮uki‍γij(Pj-Xi)}e(t).
On the other hand, fordi(t)∈[dmi,d2i], the integral terms
(68)-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}eT(s)U5e(s)ds,-∫t-d2it-dmi‍ϱ2iexp{2ε(s-t)}e˙T(s)V5e˙(s)ds
are disposed and estimated by Lemma 6.
(69)-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}eT(s)U4e(s)ds,-∫t-dmit-d1i‍ϱ1iexp{2ε(s-t)}e˙T(s)V4e˙(s)ds
are directly estimated by (a) of Lemma 5. Therefore,
(70)ΓV(e(t),i,t)+2εV(e(t),i,t)≤ξT(t)(Ωi0ε+Ωi2ε+ΛTJΛ)ξ(t)+℧.
With (66) and (70), the following inequality is held for di(t)∈[d1i,d2i] if (23)–(28) are satisfied:
(71)ΓV(e(t),i,t)+2εV(e(t),i,t)<0.
From the stochastic Lyapunov functional (34) and (71), it is held that
(72)V(e(t),i,t)≥mini∈S{λmin(Pi)}∥e(t)∥2=λ_∥e(t)∥2,V(e(t),i,t)<exp{-2ε(t-t0)}V(e(t0),r(t0),t0).
Moreover, we have(73)V(e(t0),r(t0),t0)=∑k=17‍Vk(e(t0),r(t0),t0),∑k=17‍Vk(e(t0),r(t0),t0)≤∫t0-ςt0‍exp{2ε(s-t0)}eT(s)×{Q1i+Q3+Q5+R1i+∑k=24‍Rk}e(s)ds+∫t0-ςt0‍exp{2ε(s-t0)}e˙T(s)×{Q2i+Q4+∑l=13‍Tl}e˙(s)ds+∫-ς0‍∫t0+θt0‍ςexp{2ε(s-t0)}eT×(s)(∑m=15‍Um)e(s)dsdθ+∫-ς0‍∫t0+θt0‍ςexp{2ε(s-t0)}e˙T×(s)(∑n=15‍Vn)e˙(s)dsdθ+∫-ς0‍∫θ0‍∫t0+λt0‍ς22exp{2ε(s-t0)}e˙T×(s)(∑s=15‍Ws)e˙(s)dsdλdθ+eT(t0)Pie(t0)≤λ¯∥e(t0)∥ς2.
Then from (72) and (73), it is readily seen that
(74)∥e(t)∥≤λ¯λ_exp{-ε(t-t0)}∥e(t0)∥ς,
where κ=λ¯/λ_≥1.
Therefore, the error system (13) with partially known transition rates and sector-bounded condition (9) is exponentially stable with a decay rate ε. This completes the proof.In some situations, due to the complexity of NCDN (6), the information on the transition rates may be completely unknown, which can be viewed as switched complex dynamical networks with arbitrary switching. The following corollary is therefore given to guarantee the exponential stability for this case.Corollary 8.
For given scalarsγim, γiM, ε, τi¯, νi, h¯, d1i, d2i, and μi and constant scalar dmi satisfying d1i<dmi<d2i, the error system (13) with entirely unknown transition rates and sector-bounded condition (9) is exponentially stable with decay rate ε and κ=λ¯/λ_ if ∥Gi(3)⊗Ci∥<1, and there exist symmetric positive matrices P>0, Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4), Tl>0, (l=1,2,3), Um>0, Vn>0, Ws>0, (m,n,s=1,2,3,4,5) and matrices Mk1, Nk2, (k1,k2=1,2,3,4), for any scalar δ>0, and symmetric matrices Y1, Y2, Z such that (28) and the following matrix inequalities hold:
(75)-∑j∈𝒮uki,j≠i‍γiM(Q1-Y1)-γimY1≤0,-∑j∈𝒮uki,j≠i‍γiM(Q2-Y2)-γimY2≤0,(76)-∑j∈𝒮uki,j≠i‍γiM(R1-Z)-γimZ≤0,(77)(i)[U4N1N1TU4]>0,[V4N2N2TV4]>0,(ii)Ω~i0ε+Ω~i1ε+Λ~TJ~Λ~<0,(78)(iii)[U5N3N3TU5]>0,[V5N4N4TV5]>0,(iv)Ω~i0ε+Ω~i2ε+Λ~TJ~Λ~<0,
where
(79)J~=Q2+Q4+T1+τi¯2V1+h¯2V2+d1i2V3+ϱ1i2V4+ϱ2i2V5+τi¯44W1+h¯44W2+d1i44W3+ϱ3i2W4+ϱ4i2W5,Λ~=Λ,Ω~i1ε=Ωi1ε,Ω~i2ε=Ωi2ε,Ω~i0ε=∑m=124‍EmΠ~mEmT+✠-δ×[E1E24][F1^F2^*I][E1TE24T]-2εd1iexp{2εd1i}-1(E1-E3)V3(E1T-E3T)-exp{-2ετi¯}(τi¯E1-E19)W1(τi¯E1T-E19T)-exp{-2εh¯}(h¯E1-E23)W2(h¯E1T-E23T)-exp{-2εd1i}(d1iE1-E9)W3(d1iE1T-E9T)-exp{-2εdmi}(ϱ1iE1-E12)W4(ϱ1iE1T-E12T)-exp{-2εd2i}(ϱ2iE1-E13)W5(ϱ2iE1T-E13T)-2ετi¯exp{2ετi¯}-1[E18E19-E18]×[U1M1M1TU1][E18TE19T-E18T]-2ετi¯exp{2ετi¯}-1[E1-E16E16-E14]×[V1M2M2TV1][E1T-E16TE16T-e14T]-2εh¯exp{2εh¯}-1[E22E23-E22]×[U2M3M3TU2][E22TE23T-E22T]-2εh¯exp{2εh¯}-1[E1-E21E21-E20]×[V2M4M4TV2][E1T-E21TE21T-E20T],Π~1=(Gi(1)⊗Ai)TP+P(Gi(1)⊗Ai)+2εP+Q1+Q3+Q5+R2+τi¯2U1+h¯2U2+d1i2U3+ϱ1i2U4+ϱ2i2U5,Π~2=ϖ(ε,μi)R1,Π~3=exp{-2εd1i}(R1+R3-R2),Π~16=-(1-νi)exp{-2ετi¯}Q1,Π~17=-(1-νi)exp{-2ετi¯}Q2,Π~4=Π4,Π~5=Π5,Π~6=Π6,Π~7=Π7,Π~8=Π8,Π~9=Π9,Π~14=Π14,Π~15=Π15,Π~20=Π20,Π~m=0,(m=10,11,12,13,18,19,21,…,24),λ¯=λmax(P)+1-exp{-2ες}2ε×{∑j=15‍λmax(Qj)+∑k=14‍λmax(Rk)+∑l=13‍λmax(Tl)}+exp{-2ες}+2ες-14ε2×{∑m=15‍λmax(Um)+∑n=15‍λmax(Vn)}+1-exp{-2ες}+2ε2ς2-2ες8ε3×{∑s=15‍λmax(Ws)}.Other notations are the same as those in Theorem7.Proof.
Since the elements of transition rates are entirely unknown, we choose the Lyapunov functional as follows:(80)V(e(t),i,t)=eT(t)Pe(t)+∑k=27‍Vk(e(t),i,t),
where
(81)V2(e(t),i,t)=∫t-τi(t)t‍exp{2ε(s-t)}eT(s)Q1e(s)ds+∫t-τi(t)t‍exp{2ε(s-t)}e˙T(s)Q2e˙(s)ds+∫t-τi¯t‍exp{2ε(s-t)}eT(s)Q3e(s)ds+∫t-τi¯t‍exp{2ε(s-t)}e˙T(s)Q4e˙(s)ds+∫t-h¯t‍exp{2ε(s-t)}eT(s)Q5e(s)ds,V3(e(t),i,t)=∫t-di(t)t-d1i‍exp{2ε(s-t)}eT(s)R1e(s)ds+∫t-d1it‍exp{2ε(s-t)}eT(s)R2e(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}eT(s)R3e(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}eT(s)R4e(s)ds,V4(e(t),i,t)=∫t-d1it‍exp{2ε(s-t)}e˙T(s)T1e˙(s)ds+∫t-dmit-d1i‍exp{2ε(s-t)}e˙T(s)T2e˙(s)ds+∫t-d2it-dmi‍exp{2ε(s-t)}e˙T(s)T3e˙(s)ds,
and V5(e(t),i,t), V6(e(t),i,t), V7(e(t),i,t) are the same as (39), (40), (41).
Then we follow a similar line as in proof of Theorem7 and obtain the result.


## 3.2. The Sampled-Data Controllers for NCDN

In this subsection, we give the design method of the desired sampled-data controllers to ensure the NCDN (6) exponentially synchronized on the basis of Theorem 7.Theorem 9.
The NCDN (6) with partially known transition rates and sector-bounded condition (9) is exponentially synchronized by controllers of the form (11) if there exist symmetric positive definite matrices Pi=diag{Pi(1),Pi(2),…,Pi(N)}>0, Q1i>0, Q2i>0, R1i>0, (i∈S={1,2,…,M}), Qj>0, (j=3,4,5), Rk>0, (k=2,3,4), Tl>0, (l=1,2,3), Um>0, Vn>0, Ws>0, (m,n,s=1,2,3,4,5), matrices 𝕐i=diag{Yi(1),Yi(2),…,Yi(N)} and matrices Mk1, Nk2, (k1,k2=1,2,3,4), for any scalar δ>0, and symmetric matrices Xi, Y1i, Y2i, Zi, (i∈S) such that (23)–(28) and the following linear matrix inequalities hold:
(82)(i)[U4N1N1TU4]>0,[V4N2N2TV4]>0,(ii)[Θi0ε+Ωi1εΛTPiPiΛ-2Pi+J]<0,(83)(iii)[U5N3N3TU5]>0,[V5N4N4TV5]>0,(iv)[Θi0ε+Ωi2εΛTPiPiΛ-2Pi+J]<0.
In addition, the desired sampled-data controllers gain matrices are given by
(84)Kk=(Pi(k))-1Yi(k),k=1,2,…,N,
where
(85)Θi0ε=∑m=124‍EmΠmEmT+✠Θ-δ[E1E24]×[F1^F2^*I][E1TE24T]-2εd1iexp{2εd1i}-1×(E1-E3)V3(E1T-E3T)-exp{-2ετi¯}(τi¯E1-E19)W1(τi¯E1T-E19T)-exp{-2εh¯}(h¯E1-E23)W2(h¯E1T-E23T)-exp{-2εd1i}(d1iE1-E9)W3(d1iE1T-E9T)-exp{-2εdmi}(ϱ1iE1-E12)W4(ϱ1iE1T-E12T)-exp{-2εd2i}(ϱ2iE1-E13)W5(ϱ2iE1T-E13T)-2ετi¯exp{2ετi¯}-1[E18E19-E18]×[U1M1M1TU1][E18TE19T-E18T]-2ετi¯exp{2ετi¯}-1[E1-E16E16-E14]×[V1M2M2TV1][E1T-E16TE16T-e14T]-2εh¯exp{2εh¯}-1[E22E23-E22]×[U2M3M3TU2][E22TE23T-E22T]-2εh¯exp{2εh¯}-1[E1-E21E21-E20]×[V2M4M4TV2][E1T-E21TE21T-E20T],✠Θ=E1Pi(Gi(2)⊗Bi)E2T+E1Pi(Gi(3)⊗Ci)E17+E1𝕐iE21T+E1Pi(IN⊗Di)E24T+E2(Gi(2)⊗Bi)TPiE1T+E17(Gi(3)⊗Ci)TPiE1T+E21𝕐iTE1T+E24(IN⊗Di)TPiE1T.Other notations are the same as those in Theorem7.Proof.
Considering (27) and (28), by applying the Schur Complement Lemma, we have
(86)(i)[U4N1N1TU4]>0,[V4N2N2TV4]>0,(ii)[Ωi0ε+Ωi1εΛTJJΛ-J]<0,(87)(iii)[U5N3N3TU5]>0,[V5N4N4TV5]>0,(iv)[Ωi0ε+Ωi2εΛTJJΛ-J]<0.
Define the block diagonal matrix𝕏=[HPiJ-1], where H=∑m=124‍EmEmT. Noting that EiTEj={0i≠jIi=j, with (ii) of (86) and (iv) of (87), we define 𝕐i=Pi𝕂 and obtain the following:
(88)𝕏[Ωi0ε+Ωi1εΛTJJΛ-J]𝕏T=[Θi0ε+Ωi1εΛTPiPiΛ-PiJ-1Pi]<0,𝕏[Ωi0ε+Ωi2εΛTJJΛ-J]𝕏T=[Θi0ε+Ωi2εΛTPiPiΛ-PiJ-1Pi]<0.
Due toJ>0 and (Pi-J)J-1(Pi-J)≥0, we know that -PiJ-1Pi≤-2Pi+J. Thus (88) are held if (ii) of (82) and (iv) of (83) are satisfied, which implies that (ii) of (27) and (iv) of (28) are held.
According to Definition3, by the sampled-data controllers gain matrices (84), the NCDN (6) with partially known transition rates and sector-bounded condition (9) is exponentially synchronized. This completes the proof.Remark 10.
It should be mentioned that the sampled-data synchronization problem has been solved for NCDN (6) in Theorem 9, and the desired controllers can be obtained when LMIs (23)–(26), (82), and (83) are feasible. In this paper, we construct the novel stochastic Lyapunov functional (34) containing some triple-integral terms; which is very effective in the reduction of conservatism [45]. Besides, we apply Lemma 5 to the corresponding terms, the method by using reciprocally convex lemma [46] can achieve less conservative results. Thus the obtained delay-range-dependent and decay rate-dependent stability condition for NCDN (6) in Theorem 7 is less conservative than the previous ones, which can achieve larger maximum value of sampling period h¯ and will be verified in Section 4.Remark 11.
It should be pointed out that the given results can be extended to more general neutral delay in the form of NCDN (6). For instance, τ1i≤τi(t)≤τ2i; we can choose τmi satisfying τ1i≤τmi≤τ2i and construct the stochastic Lyapunov functional on [t-τi(t),t-τ1i], [t-τ1i,t], [t-τ2i,t-τmi], and [t-τmi,t-τ1i]. Following the same thought, the time delay h(t) in the control input can also be extended to be more general.Corollary 12.
The NCDN (6) with completely unknown transition rates and sector-bounded condition (9) is exponentially synchronized by controllers of the form (11) if there exist symmetric positive definite matrices P=diag{P(1),P(2),…,P(N)}>0, Qj>0, (j=1,2,3,4,5), Rk>0, (k=1,2,3,4), Tl>0, (l=1,2,3), Um>0, Vn>0, Ws>0, (m,n,s=1,2,3,4,5), matrices 𝕐=diag{Y(1),Y(2),…,Y(N)} and matrices Mk1, Nk2, (k1,k2=1,2,3,4), for any scalar δ>0, and symmetric matrices Y1, Y2, Z such that (28), (75), (76), and the following linear matrix inequalities hold:
(89)(i)[U4N1N1TU4]>0,[V4N2N2TV4]>0,(ii)[Θ~i0ε+Ω~i1εΛTPPΛ-2P+J]<0,(90)(iii)[U5N3N3TU5]>0,[V5N4N4TV5]>0,(iv)[Θ~i0ε+Ω~i2εΛTPPΛ-2P+J]<0.
In addition, the desired sampled-data controllers gain matrices are given by
(91)Kk=(P(k))-1Y(k),k=1,2,…,N,
where
(92)Θ~i0ε=∑m=124‍EmΠ~mEmT+✠~Θ-δ[E1E24][F1^F2^*I][E1TE24T]-2εd1iexp{2εd1i}-1(E1-E3)V3(E1T-E3T)-exp{-2ετi¯}(τi¯E1-E19)W1(τi¯E1T-E19T)-exp{-2εh¯}(h¯E1-E23)W2(h¯E1T-E23T)-exp{-2εd1i}(d1iE1-E9)W3(d1iE1T-E9T)-exp{-2εdmi}(ϱ1iE1-E12)W4(ϱ1iE1T-E12T)-exp{-2εd2i}(ϱ2iE1-E13)W5(ϱ2iE1T-E13T)-2ετi¯exp{2ετi¯}-1[E18E19-E18]×[U1M1M1TU1][E18TE19T-E18T]-2ετi¯exp{2ετi¯}-1[E1-E16E16-E14]×[V1M2M2TV1][E1T-E16TE16T-e14T]-2εh¯exp{2εh¯}-1[E22E23-E22]×[U2M3M3TU2][E22TE23T-E22T]-2εh¯exp{2εh¯}-1[E1-E21E21-E20]×[V2M4M4TV2][E1T-E21TE21T-E20T],✠~Θ=E1P(Gi(2)⊗Bi)E2T+E1P(Gi(3)⊗Ci)E17+E1𝕐iE21T+E1P(IN⊗Di)E24T+E2(Gi(2)⊗Bi)TPE1T+E17(Gi(3)⊗Ci)TPE1T+E21𝕐iTE1T+E24(IN⊗Di)TPE1T.Other notations are the same as those in Corollary8.


## 4. Numerical Examples

In this section, numerical examples are used to illustrate the effectiveness of the results derived above.Example 1.
As shown in the example, a three-node NCDN (6) with Markovian switching between two modes is taken into consideration; that is, N=3 and M=2. The parametric matrices of the NCDN are given as follows:
(93)A1=[-0.50-0.350.10-0.50],A2=[-0.300.090.20-0.40],B1=[0.20-0.150.50-0.50],B2=[0.310.23-0.120.17],C1=[0.280.02-0.060.11],C2=[0.510.240.02-0.44],D1=[0.2000-0.15],D2=[0.300-0.100.23],G1(1)=[-2111-2111-2],G2(1)=[0001-1011-2],G1(2)=[-2110-1110-1],G2(2)=[-21100011-2],G1(3)=[0001-2110-1],G2(3)=[-1011-21000].
The partially known transition rate matrix is considered as in the following two cases:
(94)Υ1=[-22γ210γ220],Υ2=[γ110γ120γ210γ220].
Furthermore, the nonlinear function f(xk(t)) is given by
(95)f(xk(t))=[0.4xk1(t)-tanh(0.1xk1(t))0.1xk2(t)]T.
Then, it is easy to verify that
(96)F1=[0.4000.1],F2=[0.3000.1].
The interval mode-dependent time-varying neutral delays and discrete delays are, respectively, assumed to be(97)τ1(t)=0.5(1+sint),τ2(t)=0.6(1+cost),d1(t)=0.4(1+sin4(t)),d2(t)=0.8(1+cos4(t)).
They are governed by the Markov process {r(t),t≥0} and shown in Figures 1 and 2. It can be readily obtained that
(98)τ1¯=1,τ2¯=1.2,ν1=0.5,ν2=0.6,d11=0.4,d21=0.8,d12=0.8,d22=1.6,μ1=335,μ2=3310.
Given decay rate ε=0.3 and the maximum value of sampling period h¯=0.2, we choose dm1=0.5 and dm2=1.0 and seek to solve the sampled-data controllers.
For the case ofΥ1, with the above parameters, by Theorem 9, we can obtain the sampled-data controllers as follows:
(99)K1=[-0.7246-0.0345-0.0253-0.6358],K2=[-0.5149-0.0326-0.0117-0.5048],K3=[-0.3557-0.0263-0.0046-0.4356].
Furthermore, the state trajectories of the error system (13) are given in Figure 3, where x1(0)=[0.20.1]T, x2(0)=[0.3-0.2]T, x3(0)=[-0.10.1]T, and s0=[0.10.2]T.
For another case ofΥ2, the sampled-data controllers can be readily obtained by Corollary 12.Time-varying neutral delayτi(t) at mode 1 and mode 2.
(a)
(b)Time-varying discrete delaydi(t) at mode 1 and mode 2.
(a)
(b)Figure 3
State trajectories of the error system in Example1.Example 2.
Consider NCDN (6) with three nodes but only one mode; that is, N=3 and M=1. So the subscript of the mode i is omitted here. The outer-coupling matrices are assumed to be
(100)G(1)=G(2)=[-1010-1111-2],G(3)=0.
In fact, the NCDN in consideration here has become a common CDN.
The nonlinear functionf(xk(t)) is taken as(101)f(xk(t))=[-0.5xk1(t)+tanh(0.2xk1(t))+0.2xk2(t)0.95xk2(t)-tanh(0.75xk2(t))]T,(101) can be easily verified that
(102)F1=[-0.50.200.95],F2=[-0.30.200.2].
The time-varying delay is chosen asd(t)=0.2+0.05sin(10t); then we know that  d1=0.15, d2=0.25, and μ=0.5. Besides, we choose D=I2,  dm=0.18 satisfying d1<dm<d2 and consider the following two cases.
Case  1. The inner-coupling matrices are given as A=0 and B=[0.5000.5]. According to the above parameters, the maximum value of sampling period h¯, which satisfies LMIs (23)–(26), (82), and (83) in Theorem 9, can be calculated by solving a quasiconvex optimization problem. The degraded NCDN has also been considered in [34, 35]. The results on the maximum value of sampling period h¯ are compared in Table 1, where it can be seen that our method is less conservative than [34, 35].
Case  2. The inner-coupling matrices are given as A=diag{0.3,0.3} and B=diag{0.4,0.4}. The maximum value of sampling period h¯ also can be solved by Theorem 9. The results are listed in Table 2. In addition, the proposed method in [34] is not applicable here. From Table 2, it also can be seen that our method is less conservative than existing ones.Table 1
Maximum value of sampling periodh¯ with dm=0.18 and ε=0.
Methods
Sampled-data controllers
Li et al. [34]
h
¯
=
0.5409
K
1
=
[
-
0.4856
-
0.1435
-
0.0987
-
1.5662
], K2=[-0.4857-0.1435-0.0987-1.5664], K3=[-0.2435-0.1036-0.0063-1.0367]
Wu et al. [35]
h
¯
=
0.5573
K
1
=
[
-
0.4201
-
0.1614
0.0001
-
1.1698
], K2=[-0.4201-0.16140.0001-1.1698], K3=[0.1221-0.2073-0.0024-1.0093]
Theorem9
h
¯
=
0.5764
K
1
=
[
-
0.4096
-
0.1523
0.0012
-
1.1577
], K2=[-0.4096-0.15230.0012-1.1577], K3=[0.1176-0.2135-0.0046-0.9438]Table 2
Maximum value of sampling periodh¯ with dm=0.18 and ε=0.
Methods
Sampled-data controllers
Li et al. [34]
—
—
Wu et al. [35]
h
¯
=
0.0500
K
1
=
[
-
0.4330
-
0.2074
-
0.0631
-
1.5627
], K2=[-0.4330-0.2071-0.0631-1.5627], K3=[-0.1631-0.1744-0.0355-1.0803]
Theorem9
h
¯
=
0.0617
K
1
=
[
-
0.4252
-
0.1836
0.0094
-
1.1847
], K2=[-0.4252-0.18450.0092-1.1847], K3=[-0.1466-0.1746-0.0243-1.0125]


## 5. Conclusions

In this paper, the sampled-data synchronization problem has been solved for a class of NCDNs with Markovian jump parameters and partially known transition rates. The discrete and neutral delays are considered to be interval mode-dependent and time varying, while the sampling period here is bounded and time-varying. With a novel stochastic Lyapunov functional, the delay-range-dependent and rate-dependent exponential synchronization conditions have been proposed by Lyapunov theory and better technique of matrix inequalities. Then the sampled-data controllers have been designed on the basis of the obtained conditions. These theoretical results are successfully verified through numerical examples, whose simulation results are less conservative than the previous results. Finally, the main contributions of this paper can be summarized as follows. (i) To achieve exponential synchronization for NCDN (6), the desired sampled-data feedback controllers have been designed in terms of the solution to certain LMIs. The proposed results are expressed in a new representation, which are theoretically and numerically proved to be less conservative than some existing ones. (ii) The constructed stochastic Lyapunov functional contains some triple-integral terms, which are very effective in the reduction of conservativeness and have not appeared in the context of NCDNs. (iii) The bound of the delay is fully utilized in this paper; that is, improved bounding technique is used to reduce the conservativeness. (iv) The reciprocally convex lemma is used to derive the delay-range-dependent and rate-dependent stability conditions, which can well reduce the conservativeness of the investigated systems.



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