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

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.

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. Section 2 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. ℝ^𝑛 denotes the 𝑛 dimensional Euclidean space and ℝ^𝑚×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‖=(𝑣^𝑇𝑣)^1/2, while ‖𝐴‖ is spectral norm of matrix 𝐴, ‖𝐴‖=[𝜆_max(𝐴^𝑇𝐴)]^1/2. 𝜆_max(min)(𝐴) is the eigenvalue of matrix 𝐴 with maximum (minimum) real part. The Kronecker product of matrices 𝑃∈ℝ^𝑚×𝑛 and 𝑄∈ℝ^𝑝×𝑞 is a matrix in ℝ^{𝑚𝑝×𝑛𝑞} which is denoted as 𝑃⊗𝑄. Let 𝜍>0 and 𝐶([−𝜍,0],ℝ^𝑛) denote the family of continuous function 𝜑, from [−𝜍,0] to ℝ^𝑛 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,{ℱ_𝑡}_𝑡≥0,𝐏} with a natural filtration {ℱ_𝑡}_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=(𝛾_𝑖𝑗)_𝑀×𝑀, 𝑖,𝑗∈𝑆, which is given by 𝑃𝑟()(𝑡)=𝛾𝑡+Δ𝑡=𝑗∣𝑟=𝑖^𝑖𝑗Δ𝑡+𝑜(Δ𝑡)𝑖≠𝑗,1+𝛾_𝑖𝑖Δ𝑡+𝑜(Δ𝑡)𝑖=𝑗, (1) where Δ𝑡>0, lim_Δ𝑡→0(𝑜(Δ𝑡)/Δ𝑡)=0, and 𝛾_𝑖𝑗≥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 𝛾_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝛾_𝑖𝑗. (2) 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=(𝛾_𝑖𝑗)_𝑀×𝑀 are inaccessible. For instance, in the system with six operation modes, the jump rates matrix Υ may be viewed as 𝛾⁰₁₁𝛾₁₂𝛾⁰₁₃𝛾⁰₁₄𝛾₁₅𝛾⁰₁₆𝛾₂₁𝛾⁰₂₂𝛾⁰₂₃𝛾₂₄𝛾⁰₂₅𝛾₂₆𝛾⁰₃₁𝛾⁰₃₂𝛾₃₃𝛾₃₄𝛾⁰₃₅𝛾⁰₃₆𝛾⁰₄₁𝛾₄₂𝛾⁰₄₃𝛾⁰₄₄𝛾⁰₄₅𝛾₄₆𝛾₅₁𝛾⁰₅₂𝛾⁰₅₃𝛾⁰₅₄𝛾₅₅𝛾⁰₅₆𝛾₆₁𝛾⁰₆₂𝛾⁰₆₃𝛾₆₄𝛾₆₅𝛾⁰₆₆, (3) where 𝛾⁰_𝑖𝑗,𝑖,𝑗∈𝑆 represents the unknown element. Furthermore, let 𝛾^𝑚_𝑖, 𝛾^𝑀_𝑖 be lower and upper bound for the diagonal element 𝛾_𝑖𝑖 or 𝛾⁰_𝑖𝑖, for all 𝑖∈𝑆. For notation clarity, we denote that 𝒮^𝑖=𝒮^𝑖_𝑘⋃𝒮^𝑖_𝑢𝑘, for all 𝑖∈𝑆, and 𝒮^𝑖_𝑘≜𝑗∶𝜋_𝑖𝑗,𝒮isknownfor𝑗∈𝑆_𝑖𝑢𝑘≜𝑗∶𝜋_𝑖𝑗isunknownfor𝑗∈𝑆. (4) If 𝒮^𝑖_𝑘≠Ø, than it is further described as 𝒮^𝑖_𝑘=𝑘^𝑖₁,𝑘^𝑖₂,…,𝑘^𝑖_𝑚,1≤𝑚≤𝑀, (5) where 𝑘^𝑖_𝑗,(𝑗=1,2,…,𝑚) represent the 𝑗th known element of the set 𝒮^𝑖_𝑘 in the 𝑖th row of the transition rate matrix Υ. It should be noted that if 𝒮^𝑖_𝑘=∅, 𝒮^𝑖=𝒮^𝑖_𝑢𝑘 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 of 𝑁 identical nodes with Markovian jump parameters and interval time-varying delays over the space {Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝐏} is investigated in this paper: 𝑥̇_𝑘(𝑡)=^𝑁_𝑙=1𝑔⁽¹⁾_𝑘𝑙(𝑟(𝑡))𝐴(𝑟(𝑡))𝑥_𝑙(𝑡)+^𝑁_𝑙=1𝑔⁽²⁾_𝑘𝑙(𝑟(𝑡))𝐵(𝑟(𝑡))𝑥_𝑙(((𝑡+𝑡−𝑑𝑡,𝑟)))_𝑁𝑙=1𝑔⁽³⁾_𝑘𝑙(𝑟(𝑡))𝐶(𝑟(𝑡))𝑥̇_𝑙(((𝑡(𝑟(𝑡𝑓𝑥𝑡−𝜏𝑡,𝑟)))+𝐷))_𝑘(𝑡)+𝑢_𝑘(𝑡), (6) where 𝑥_𝑘(𝑡)∈ℝ^𝑛, 𝑢_𝑘(𝑡)∈ℝ^𝑛 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))∈ℝ^𝑛×𝑛, 𝐵(𝑟(𝑡))∈ℝ^𝑛×𝑛, and 𝐶(𝑟(𝑡))∈ℝ^𝑛×𝑛 represent the inner-coupling matrices linking between the subsystems in mode 𝑟(𝑡). 𝐺⁽¹⁾(𝑟(𝑡))=[𝑔⁽¹⁾_𝑘𝑙]_𝑁×𝑁, 𝐺⁽²⁾(𝑟(𝑡))=[𝑔⁽²⁾_𝑘𝑙]_𝑁×𝑁, and 𝐺⁽³⁾(𝑟(𝑡))=[𝑔⁽³⁾_𝑘𝑙]_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 𝑔^(𝑚)_𝑘𝑙 is defined as follows: if there exists a connection between 𝑘th and 𝑙th (𝑘≠𝑙) nodes, then 𝑔^(𝑚)_𝑘𝑙(𝑟(𝑡))=𝑔^(𝑚)_𝑙𝑘(𝑟(𝑡))=1; otherwise 𝑔^(𝑚)_𝑘𝑙(𝑟(𝑡))=𝑔^(𝑚)_𝑙𝑘(𝑟(𝑡))=0, and 𝑔^(𝑚)_𝑘𝑘(𝑟(𝑡))=−^𝑁_{𝑙=1,𝑙≠𝑘}𝑔^(𝑚)_𝑘𝑙(𝑟(𝑡))=−^𝑁_{𝑙=1,𝑙≠𝑘}𝑔^(𝑚)_𝑙𝑘(𝑟(𝑡)),𝑚=1,2,3;𝑘=1,2,…,𝑁. (7)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 0≤𝜏_𝑖(𝑡)≤𝜏_𝑖,𝜏̇_𝑖(𝑡)≤𝜈_𝑖<1,0≤𝑑_1𝑖≤𝑑_𝑖(𝑡)≤𝑑_2𝑖,max_𝑖∈𝑆𝑑_1𝑖≤min_𝑗∈𝑆𝑑_2𝑗,𝑑̇_𝑖(𝑡)≤𝜇_𝑖,when𝑟(𝑡)=𝑖, (8) where 𝜏_𝑖, 0≤𝜈_𝑖<1, 𝑑_1𝑖, and 𝑑_2𝑖, 0≤𝜇_𝑖, are real constant scalars.

𝐷(𝑟(𝑡))∈ℝ^𝑛×𝑛 is a parametric matrix with real values in mode 𝑟(𝑡). 𝑓:ℝ^𝑛→ℝ^𝑛 is a continuous vector-valued nonlinear function. It is assumed to satisfy the following sector-bounded condition [36]:𝑓(𝑥)−𝑓𝑦−𝐹₁𝑥−𝑦^𝑇×𝑓(𝑥)−𝑓𝑦−𝐹₂𝑥−𝑦≤0,∀𝑥,𝑦∈ℝ^𝑛, (9) 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.

Let 𝑠(𝑡)∈ℝ^𝑛 be the state trajectory of the unforced node 𝑠(̇𝑡)=𝐷(𝑟(𝑡))𝑓(𝑠(𝑡)); then the synchronization error is defined to be 𝑒_𝑘(𝑡)=𝑥_𝑘(𝑡)−𝑠(𝑡). So the error dynamics of NCDN (6) can be derived as follows: 𝑒_𝑘̇(𝑡)=^𝑁_𝑙=1𝑔⁽¹⁾_𝑘𝑙(𝑟(𝑡))𝐴(𝑟(𝑡))𝑒_𝑙(𝑡)+^𝑁_𝑙=1𝑔⁽²⁾_𝑘𝑙(𝑟(𝑡))𝐵(𝑟(𝑡))𝑒_𝑙(((𝑡+𝑡−𝑑𝑡,𝑟)))_𝑁𝑙=1𝑔⁽³⁾_𝑘𝑙(𝑟(𝑡))𝐶(𝑟(𝑡))𝑒_𝑙̇(((𝑡(𝑟(𝑡𝑓𝑡−𝜏𝑡,𝑟)))+𝐷))_𝑒𝑒_𝑘(𝑡)+𝑢_𝑘(𝑡), (10) where 𝑓_𝑒(𝑒_𝑘(𝑡))=𝑓(𝑥_𝑘(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 times 0=𝑡₀<𝑡₁<⋯<𝑡_𝑠<⋯. Thus the state feedback controller takes the following form: 𝑢_𝑘(𝑡)=𝐾_𝑘𝑒_𝑘𝑡_𝑠,𝑘=1,2,…,𝑁,𝑡_𝑠≤𝑡<𝑡_𝑠+1, (11) where 𝐾_𝑘 is sampled-data feedback controller gain matrix to be determined. 𝑒_𝑘(𝑡_𝑠) is discrete measurement of 𝑒_𝑘(𝑡) at the sampling instant 𝑡_𝑠 satisfying lim_𝑠→+∞𝑡_𝑠=+∞. It is assumed that 𝑡_𝑠+1−𝑡_𝑠=ℎ_𝑠≤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 𝑒_𝑘̇(𝑡)=^𝑁_𝑙=1𝑔⁽¹⁾_𝑘𝑙(𝑟(𝑡))𝐴(𝑟(𝑡))𝑒_𝑙(𝑡)+^𝑁_𝑙=1𝑔⁽²⁾_𝑘𝑙(𝑟(𝑡))𝐵(𝑟(𝑡))𝑒_𝑙(((𝑡+𝑡−𝑑𝑡,𝑟)))_𝑁𝑙=1𝑔⁽³⁾_𝑘𝑙(𝑟(𝑡))𝐶(𝑟(𝑡))𝑒_𝑙̇(((𝑡(𝑟(𝑡𝑓𝑡−𝜏𝑡,𝑟)))+𝐷))_𝑒𝑒_𝑘(𝑡)+𝐾_𝑘𝑒_𝑘(𝑡−ℎ(𝑡)), (12) where ℎ(𝑡)=𝑡−𝑡_𝑠 and 0≤ℎ(𝑡)≤ℎ.

For simplicity of notations, we denote 𝐴(𝑟(𝑡)), 𝐵(𝑟(𝑡)), 𝐶(𝑟(𝑡)), 𝐷(𝑟(𝑡)), 𝐺^(𝑚)(𝑟(𝑡)), (𝑚=1,2,3) by 𝐴_𝑖, 𝐵_𝑖, 𝐶_𝑖, 𝐷_𝑖, 𝐺_𝑖^(𝑚),(𝑚=1,2,3) for 𝑟_𝑡=𝑖∈𝑆. By utilizing the Kronecker product of the matrices, (12) can be written in a more compact form as 𝑒̇(𝑡)=𝔸_𝑖𝑒(𝑡)+𝔹_𝑖𝑒𝑡−𝑑_𝑖(𝑡)+ℂ_𝑖𝑒̇𝑡−𝜏_𝑖(𝑡)+𝔻_𝑖𝐹_𝑒(𝑒(𝑡))+𝕂𝑒(𝑡−ℎ(𝑡)), (13) where 𝔸_𝑖=𝐺_𝑖⁽¹⁾⊗𝐴_𝑖, 𝔹_𝑖=𝐺_𝑖⁽²⁾⊗𝐵_𝑖, ℂ_𝑖=𝐺_𝑖⁽³⁾⊗𝐶_𝑖, 𝔻_𝑖=𝐼_𝑁⊗𝐷_𝑖, and 𝑒(𝑡)=col𝑒₁(𝑡),𝑒₂(𝑡),…,𝑒_𝑁(𝑡,𝑒)𝑡−𝑑_𝑖(𝑡)=col𝑒₁𝑡−𝑑_𝑖(𝑡),𝑒₂𝑡−𝑑_𝑖(𝑡),…,𝑒_𝑁𝑡−𝑑_𝑖(𝑡),𝑒̇𝑡−𝜏_𝑖(𝑡)=col𝑒₁̇𝑡−𝜏_𝑖(𝑡),𝑒₂̇𝑡−𝜏_𝑖(𝑡),…,𝑒_𝑁̇𝑡−𝜏_𝑖(𝑡),𝐹_𝑒(𝑒(𝑡))=col𝑓_𝑒𝑒₁(𝑡),𝑓_𝑒𝑒₂(𝑡),…,𝑓_𝑒𝑒_𝑁(𝑡,𝑒((𝑡𝑒)𝑡−ℎ))=col₁(𝑡−ℎ(𝑡)),𝑒₂(𝑡−ℎ(𝑡)),…,𝑒_𝑁((𝑡,𝐾𝑡−ℎ))𝕂=diag₁,𝐾₂,…,𝐾_𝑁. (14)

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 operator 𝔇:𝐶([−𝜌,0],ℝ^𝑛)→ℝ^𝑛 as 𝔇(𝑥_𝑡)=𝑥(𝑡)−𝐶𝑥(𝑡−𝜏). 𝔇 is said to be stable if the homogeneous difference equation 𝔇𝑥_𝑡𝑥=0,𝑡≥0,₀=𝜓∈𝜙∈𝐶−𝜌,0,ℝ^𝑛∶𝔇𝜙=0 (15) is uniformly asymptotically stable. In this paper, that is, ‖𝐺_𝑖⁽³⁾⊗𝐶_𝑖‖<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 Γ𝑉(𝑒(𝑡),𝑖,𝑡)=lim_Δ𝑡→01[ℰ{𝑉(𝑒()())(𝑡)(𝑡)}(𝑒(𝑡)=𝜕Δ𝑡𝑡+Δ𝑡,𝑟𝑡+Δ𝑡,𝑡+Δ𝑡∣𝑒=𝑒,𝑟=𝑖−𝑉,𝑖,𝑡)]𝑉(𝑒(𝑡))+𝜕𝜕𝑡,𝑖,𝑡𝑉(𝑒(𝑡))𝑒̇(𝑡)+𝜕𝑒,𝑖,𝑡_𝑁𝑗=1𝜋_𝑖𝑗𝑉𝑒(𝑡),𝑗,𝑡. (16)

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), ‖𝑒(𝑡)‖≤𝜅exp−𝜀𝑡−𝑡₀sup_{𝜃∈−𝜍,𝑡0}‖𝑒(𝜃)‖, (17) where 𝜀 is the exponential decay rate; 𝜍=max{ℎ,max_𝑖∈𝑆{𝜏_𝑖},max_𝑖∈𝑆{𝑑_2𝑖}}.

Lemma 4 (see [43]). Given matrices 𝐴, 𝐵, 𝐶, and 𝐷 with appropriate dimensions and scalar 𝛼, by the definition of the Kronecker product, the following properties hold: ()(),()()=()⊗(),()𝛼𝐴⊗𝐵=𝐴⊗𝛼𝐵𝐴+𝐵⊗𝐶=𝐴⊗𝐶+𝐵⊗𝐶,𝐴⊗𝐵)(𝐶⊗𝐷𝐴𝐶𝐵𝐷𝐴⊗𝐵^𝑇=𝐴^𝑇⊗𝐵^𝑇. (18)

Lemma 5. For any constant matrix 𝑄=𝑄^𝑇>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: (a)exp2𝜀ℎ₂(𝑡)−exp2𝜀ℎ₁(𝑡)×2𝜀_{𝑡−ℎ1(𝑡)𝑡−ℎ2(𝑡)}exp{2𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)(𝑠)≥𝑄𝑥𝑑𝑠_{𝑡−ℎ1(𝑡)𝑡−ℎ2(𝑡)}𝑥^𝑇(𝑠)𝑑𝑠𝑄^{𝑡−ℎ1(𝑡)}_{𝑡−ℎ2(𝑡)}𝑥(𝑠),(b)𝜏𝑑𝑠₂₂−𝜏²₁2^−𝜏1_−𝜏2^𝑡_𝑡+𝜃exp{2𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠𝑑𝜃≥exp−2𝜀𝜏₂^−𝜏1_−𝜏2^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑄×𝑑𝑠𝑑𝜃_{−𝜏1−𝜏2𝑡𝑡+𝜃}𝑥(𝑠)𝑑𝑠𝑑𝜃. (19)

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, exp{^−𝜏1_−𝜏2^𝑡_𝑡+𝜃2𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠𝑑𝜃≥exp−2𝜀𝜏₂^−𝜏1_−𝜏2^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠𝑑𝜃; (20)
(b) is thus true by [45].

Lemma 6 (see [46]). For functions 𝜆₁(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 𝑇𝑃𝑇_𝑇𝑄>0, (21) and the following inequality holds: 1𝜆₁(𝑡)𝜂₁𝑃𝜂^𝑇₁+1𝜆₂(𝑡)𝜂₂𝑄𝜂^𝑇₂≥𝜂₁𝜂₂𝑇𝑃𝑇_𝑇𝑄𝜂^𝑇₁𝜂^𝑇₂. (22)

According to Definition 3, 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 scalars 𝛾^𝑚_𝑖, 𝛾^𝑀_𝑖, 𝜀, 𝜏_𝑖, 𝜈_𝑖, ℎ, 𝑑_1𝑖, 𝑑_2𝑖, and 𝜇_𝑖 and constant scalar 𝑑_𝑚𝑖 satisfying 𝑑_1𝑖<𝑑_𝑚𝑖<𝑑_2𝑖, 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 ‖𝐺_𝑖⁽³⁾⊗𝐶_𝑖‖<1, and there exist symmetric positive matrices 𝑃_𝑖>0, 𝑄_1𝑖>0, 𝑄_2𝑖>0, 𝑅_1𝑖>0, (𝑖∈𝑆), 𝑄_𝑗>0, (𝑗=3,4,5), 𝑅_𝑘>0, (𝑘=2,3,4), 𝑇_𝑙>0, (𝑙=1,2,3), 𝑈_𝑚>0, 𝑉_𝑛>0, 𝑊_𝑠>0, (𝑚,𝑛,𝑠=1,2,3,4,5) and matrices 𝑀_𝑘1, 𝑁_𝑘2, (𝑘₁,𝑘₂=1,2,3,4), for any scalar 𝛿>0, and symmetric matrices 𝑋_𝑖, 𝑌_1𝑖, 𝑌_2𝑖, 𝑍_𝑖, (𝑖∈𝑆) such that the following matrix inequalities hold.
When 𝑖∈𝒮^𝑖_𝑘; then _{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝛾_𝑖𝑗𝑄_1𝑗−𝑌_1𝑖−𝛾_𝑖𝑖𝑌_1𝑖≤0,_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝛾_𝑖𝑗𝑄_2𝑗−𝑌_2𝑖−𝛾_𝑖𝑖𝑌_2𝑖≤0,_{𝑗∈𝒮𝑖𝑘,𝑗≠𝑖}𝛾_𝑖𝑗𝑅_1𝑗−𝑍_𝑖−𝛾_𝑖𝑖𝑍_𝑖𝑄≤0,_1𝑗−𝑌_1𝑖≤0,𝑗∈𝒮^𝑖_𝑢𝑘,𝑄_2𝑗−𝑌_2𝑖≤0,𝑗∈𝒮^𝑖_𝑢𝑘,𝑅_1𝑗−𝑍_𝑖≤0,𝑗∈𝒮^𝑖_𝑢𝑘,𝑃_𝑗−𝑋_𝑖≤0,𝑗∈𝒮^𝑖_𝑢𝑘. (23)
When 𝑖∈𝒮^𝑖_𝑢𝑘, then _{𝑗∈𝒮𝑖𝑘}𝛾_𝑖𝑗𝑄_1𝑗−𝑌_1𝑖≤0,_{𝑗∈𝒮𝑖𝑘}𝛾_𝑖𝑗𝑄_2𝑗−𝑌_2𝑖≤0,_{𝑗∈𝒮𝑖𝑘}𝛾_𝑖𝑗𝑅_1𝑗−𝑍_𝑖𝑃≤0,_𝑗−𝑋_𝑖≤0,𝑗∈𝒮^𝑖_𝑢𝑘𝑃,𝑗≠𝑖,_𝑗−𝑋_𝑖≥0,𝑗∈𝒮^𝑖_𝑢𝑘,𝑗=𝑖,−_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝛾^𝑀_𝑖𝑄_1𝑗−𝑌_1𝑖−𝛾^𝑚_𝑖𝑌_1𝑖≤0,−_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝛾^𝑀_𝑖𝑄_2𝑗−𝑌_2𝑖−𝛾^𝑚_𝑖𝑌_2𝑖≤0,−_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝛾^𝑀_𝑖𝑅_1𝑗−𝑍_𝑖−𝛾^𝑚_𝑖𝑍_𝑖≤0, (24)(25)(26)(27) and 𝑈₁𝑀₁𝑀^𝑇₁𝑈₁𝑉>0,¹𝑀₂𝑀^𝑇₂𝑉₁𝑈>0,₂𝑀₃𝑀^𝑇₃𝑈₂𝑉>0,²𝑀₄𝑀^𝑇₄𝑉₂𝑈>0,(i)⁴𝑁₁𝑁^𝑇₁𝑈₄𝑉>0,⁴𝑁₂𝑁^𝑇₂𝑉₄>0,(ii)Ω^𝜀_𝑖0+Ω^𝜀_𝑖1+Λ^𝑇𝑈𝐽Λ<0,(iii)⁵𝑁₃𝑁^𝑇₃𝑈₅𝑉>0,⁵𝑁₄𝑁^𝑇₄𝑉₅>0,(iv)Ω^𝜀_𝑖0+Ω^𝜀_𝑖2+Λ^𝑇𝐽Λ<0, (28)(29)(30) where Ω^𝜀_𝑖0=²⁴_𝑚=1𝐸_𝑚Π_𝑚𝐸^𝑇_𝑚𝐸+✠−𝛿₁𝐸₂₄𝐹₁𝐹₂𝐸∗𝐼𝐸^𝑇1𝑇₂₄−2𝜀𝑑_1𝑖exp2𝜀𝑑_1𝑖𝐸−1¹−𝐸₃𝑉₃𝐸^𝑇₁−𝐸^𝑇₃−exp−2𝜀𝜏_𝑖𝜏_𝑖𝐸₁−𝐸₁₉𝑊₁𝜏_𝑖𝐸^𝑇₁−𝐸^𝑇₁₉−exp−2𝜀ℎℎ𝐸₁−𝐸₂₃𝑊₂ℎ𝐸^𝑇₁−𝐸^𝑇₂₃−exp−2𝜀𝑑_1𝑖𝑑_1𝑖𝐸₁−𝐸₉𝑊₃𝑑_1𝑖𝐸^𝑇₁−𝐸^𝑇₉−exp−2𝜀𝑑_𝑚𝑖𝜚_1𝑖𝐸₁−𝐸₁₂𝑊₄𝜚_1𝑖𝐸^𝑇₁−𝐸^𝑇₁₂−exp−2𝜀𝑑_2𝑖𝜚_2𝑖𝐸₁−𝐸₁₃𝑊₅𝜚_2𝑖𝐸^𝑇₁−𝐸^𝑇₁₃−2𝜀𝜏_𝑖exp2𝜀𝜏_𝑖𝐸−1¹⁸𝐸₁₉−𝐸₁₈×𝑈₁𝑀₁𝑀^𝑇₁𝑈₁𝐸^𝑇₁₈𝐸^𝑇₁₉−𝐸^𝑇₁₈−2𝜀𝜏_𝑖exp2𝜀𝜏_𝑖𝐸−1¹−𝐸₁₆𝐸₁₆−𝐸₁₄×𝑉₁𝑀₂𝑀^𝑇₂𝑉₁𝐸^𝑇₁−𝐸^𝑇₁₆𝐸^𝑇₁₆−𝑒^𝑇₁₄−2𝜀ℎexp2𝜀ℎ𝐸−1²²𝐸₂₃−𝐸₂₂×𝑈₂𝑀₃𝑀^𝑇₃𝑈₂𝐸^𝑇₂₂𝐸^𝑇₂₃−𝐸^𝑇₂₂−2𝜀ℎexp2𝜀ℎ𝐸−1¹−𝐸₂₁𝐸₂₁−𝐸₂₀×𝑉₂𝑀₄𝑀^𝑇₄𝑉₂𝐸^𝑇₁−𝐸^𝑇₂₁𝐸^𝑇₂₁−𝐸^𝑇₂₀, (31) where ✠=𝐸₁𝑃_𝑖𝐺_𝑖⁽²⁾⊗𝐵_𝑖𝐸^𝑇₂+𝐸₁𝑃_𝑖𝐺_𝑖⁽³⁾⊗𝐶_𝑖𝐸₁₇+𝐸₁𝑃_𝑖𝕂𝐸^𝑇₂₁+𝐸₁𝑃_𝑖𝐼_𝑁⊗𝐷_𝑖𝐸^𝑇₂₄+𝐸₂𝐺_𝑖⁽²⁾⊗𝐵_𝑖^𝑇𝑃_𝑖𝐸^𝑇₁+𝐸₁₇𝐺_𝑖⁽³⁾⊗𝐶_𝑖^𝑇𝑃_𝑖𝐸^𝑇₁+𝐸₂₁𝕂^𝑇𝑃_𝑖𝐸^𝑇₁+𝐸₂₄𝐼_𝑁⊗𝐷_𝑖^𝑇𝑃_𝑖𝐸^𝑇₁𝐹₁=𝐼_𝑁⊗𝐹^𝑇₁𝐹₂+𝐹^𝑇₂𝐹₁2,𝐹₂𝐼=−^𝑁⊗𝐹₁^𝑇+𝐼_𝑁⊗𝐹₂^𝑇2,Π₁=𝐺_𝑖⁽¹⁾⊗𝐴_𝑖^𝑇𝑃_𝑖+𝑃_𝑖𝐺_𝑖⁽¹⁾⊗𝐴_𝑖+_{𝑗∈𝒮𝑖𝑘}𝛾_𝑖𝑗𝑃_𝑗−𝑋_𝑖+2𝜀𝑃_𝑖+𝑄_1𝑖+𝑄₃+𝑄₅+𝑅₂+𝜏_𝑖²𝑈₁+ℎ²𝑈₂+𝑑²_1𝑖𝑈₃+𝜚²_1𝑖𝑈₄+𝜚²_2𝑖𝑈₅,Π₂=𝜛𝜀,𝜇_𝑖𝑅_1𝑖,Π₃=exp−2𝜀𝑑_1𝑖𝑅_1𝑖+𝑅₃−𝑅₂,Π₄=exp−2𝜀𝑑_𝑚𝑖𝑅₄−𝑅₃,Π₅=−exp−2𝜀𝑑_2𝑖𝑅₄,Π₆=exp−2𝜀𝑑_1𝑖𝑇₂−𝑇₁,Π₇=exp−2𝜀𝑑_𝑚𝑖𝑇₃−𝑇₂,Π₈=−exp−2𝜀𝑑_2𝑖𝑇₃,Π₉=−2𝜀𝑑_1𝑖exp2𝜀𝑑_1𝑖𝑈−1³,Π₁₄=−exp−2𝜀𝜏_𝑖𝑄₃,Π₁₅=−exp−2𝜀𝜏_𝑖𝑄₄,Π₁₆=−1−𝜈_𝑖exp−2𝜀𝜏_𝑖𝑄_1𝑖,Π₁₇=−1−𝜈_𝑖exp−2𝜀𝜏_𝑖𝑄_2𝑖,Π₂₀=−exp−2𝜀ℎ𝑄₅,Π_𝑚(),Ω=0,𝑚=10,11,12,13,18,19,21,…,24_𝜀𝑖1=−2𝜀𝜚_2𝑖exp2𝜀𝑑_2𝑖−exp2𝜀𝑑_𝑚𝑖𝐸₁₃𝑈₅𝐸^𝑇₁₃−2𝜀𝜚_2𝑖exp2𝜀𝑑_2𝑖−exp2𝜀𝑑_𝑚𝑖𝐸₄−𝐸₅𝑉₅𝐸^𝑇₄−𝐸^𝑇₅−2𝜀𝜚_1𝑖exp2𝜀𝑑_𝑚𝑖−exp2𝜀𝑑_1𝑖𝐸₁₀𝐸₁₂−𝐸₁₀×𝑈₄𝑁₁𝑁^𝑇₁𝑈₄𝐸^𝑇₁₀𝐸^𝑇₁₂−𝐸^𝑇₁₀−2𝜀𝜚_1𝑖exp2𝜀𝑑_𝑚𝑖−exp2𝜀𝑑_1𝑖𝐸₃−𝐸₂𝐸₂−𝐸₄×𝑉₄𝑁₂𝑁^𝑇₂𝑉₄𝐸^𝑇₃−𝐸^𝑇₂𝐸^𝑇₂−𝐸^𝑇₄,Ω^𝜀_𝑖2=−2𝜀𝜚_1𝑖exp2𝜀𝑑_𝑚𝑖−exp2𝜀𝑑_1𝑖𝐸₁₂𝑈₄𝐸^𝑇₁₂−2𝜀𝜚_1𝑖exp2𝜀𝑑_𝑚𝑖−exp2𝜀𝑑_1𝑖×𝐸₃−𝐸₄𝑉₄𝐸^𝑇₃−𝐸^𝑇₄−2𝜀𝜚_2𝑖exp2𝜀𝑑_2𝑖−exp2𝜀𝑑_𝑚𝑖𝐸₁₁𝐸₁₃−𝐸₁₁×𝑈₅𝑁₃𝑁^𝑇₃𝑈₅𝐸^𝑇₁₁𝐸^𝑇₁₃−𝐸^𝑇₁₁−2𝜀𝜚_2𝑖exp2𝜀𝑑_2𝑖−exp2𝜀𝑑_𝑚𝑖𝐸₄−𝐸₂𝐸₂−𝐸₅×𝑉₅𝑁₄𝑁^𝑇₄𝑉₅𝐸^𝑇₄−𝐸^𝑇₂𝐸^𝑇₂−𝐸^𝑇₅,𝐽=𝑄_2𝑖+𝑄₄+𝑇₁+𝜏_𝑖²𝑉₁+ℎ²𝑉₂+𝑑²_1𝑖𝑉₃+𝜚²_1𝑖𝑉₄+𝜚²_2𝑖𝑉₅+𝜏_𝑖⁴4𝑊₁+ℎ⁴4𝑊₂+𝑑⁴_1𝑖4𝑊₃+𝜚²_3𝑖𝑊₄+𝜚²_4𝑖𝑊₅,Λ=𝐺_𝑖⁽¹⁾⊗𝐴_𝑖𝐸^𝑇₁+𝐺_𝑖⁽²⁾⊗𝐵_𝑖𝐸^𝑇₂+𝐺_𝑖⁽³⁾⊗𝐶_𝑖𝐸^𝑇₁₇+𝕂𝐸^𝑇₂₁+𝐼_𝑁⊗𝐷_𝑖𝐸^𝑇₂₄, (32) where 𝐸_𝑖{𝑖=1,2,…,24} are block entry matrices; that is, 𝐸^𝑇₄=[],𝜛000𝐼00000000000000000000𝜀,𝜇_𝑖=𝜇_𝑖−1exp−2𝜀𝑑_1𝑖𝑤𝑖𝑡ℎ𝜇_𝑖𝜇≥1,_𝑖−1exp−2𝜀𝑑_2𝑖𝑤𝑖𝑡ℎ𝜇_𝑖<1,𝜆=max_𝑖∈𝑆𝜆_max𝑃_𝑖+1−exp{−2𝜀𝜍}×2𝜀_5𝑗=3𝜆_max𝑄_𝑗+⁴_𝑘=2𝜆_max𝑅_𝑘+³_𝑙=1𝜆_max𝑇_𝑙+1−exp{−2𝜀𝜍}2𝜀max_𝑖∈𝑆𝜆_max𝑄_1𝑖+max_𝑖∈𝑆𝜆_max𝑄_2𝑖+max_𝑖∈𝑆𝜆_max𝑅_1𝑖+exp{−2𝜀𝜍}+2𝜀𝜍−14𝜀²×⁵_𝑚=1𝜆_max𝑈_𝑚+⁵_𝑛=1𝜆_max𝑉_𝑛+1−exp{−2𝜀𝜍}+2𝜀²𝜍²−2𝜀𝜍8𝜀³×⁵_𝑠=1𝜆_max𝑊_𝑠,𝜆=min_𝑖∈𝑆𝜆_min𝑃_𝑖,𝜚_1𝑖=𝑑_𝑚𝑖−𝑑_1𝑖,𝜚_2𝑖=𝑑_2𝑖−𝑑_𝑚𝑖,𝜚_3𝑖=12𝑑²_𝑚𝑖−𝑑²_1𝑖,𝜚_4𝑖=12𝑑²_2𝑖−𝑑²_𝑚𝑖. (33)

Proof. Construct the following stochastic Lyapunov functional:𝑉(𝑒(𝑡),𝑖,𝑡)=⁷_𝑘=1𝑉_𝑘(𝑒(𝑡),𝑖,𝑡), (34) where 𝑉₁(𝑒(𝑡),𝑖,𝑡)=𝑒^𝑇(𝑡)𝑃_𝑖𝑒(𝑡),𝑉₂(𝑒(𝑡),𝑖,𝑡)=^𝑡_{𝑡−𝜏𝑖(𝑡)}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑄_1𝑖𝑒(𝑠)+𝑑𝑠_{𝑡𝑡−𝜏𝑖(𝑡)}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑄_2𝑖𝑒̇(𝑠)+𝑑𝑠_{𝑡𝑡−𝜏𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑄₃𝑒(𝑠)+𝑑𝑠_{𝑡𝑡−𝜏𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑄₄𝑒̇(𝑠)+𝑑𝑠_{𝑡𝑡−ℎ}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑄₅𝑒(𝑠)𝑉𝑑𝑠,₃(𝑒(𝑡),𝑖,𝑡)=^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑖(𝑡)}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑅_1𝑖𝑒(𝑠)+𝑑𝑠_{𝑡𝑡−𝑑1𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑅₂𝑒(𝑠)+𝑑𝑠_{𝑡−𝑑1𝑖𝑡−𝑑𝑚𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑅₃𝑒(𝑠)𝑑𝑠+^{𝑡−𝑑𝑚𝑖}_{𝑡−𝑑2𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑅₄𝑒(𝑠)𝑉𝑑𝑠,₄(𝑒(𝑡),𝑖,𝑡)=^𝑡_{𝑡−𝑑1𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑇₁𝑒̇(𝑠)+𝑑𝑠_{𝑡−𝑑1𝑖𝑡−𝑑𝑚𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑇₂𝑒̇(𝑠)+𝑑𝑠_{𝑡−𝑑𝑚𝑖𝑡−𝑑2𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑇₃𝑒̇(𝑠)𝑉𝑑𝑠,₅(𝑒(𝑡))=,𝑖,𝑡_{0−𝜏𝑖𝑡𝑡+𝜃}𝜏_𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₁𝑒(𝑠)+𝑑𝑠𝑑𝜃_{0−ℎ𝑡𝑡+𝜃}ℎexp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₂𝑒(𝑠)+𝑑𝑠𝑑𝜃_{0−𝑑1𝑖𝑡𝑡+𝜃}𝑑_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₃𝑒(𝑠)+𝑑𝑠𝑑𝜃_{−𝑑1𝑖−𝑑𝑚𝑖𝑡𝑡+𝜃}𝜚_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₄𝑒(𝑠)+𝑑𝑠𝑑𝜃_{−𝑑𝑚𝑖−𝑑2𝑖𝑡𝑡+𝜃}𝜚_2𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₅𝑒(𝑠)𝑉𝑑𝑠𝑑𝜃,₆(𝑒(𝑡))=,𝑖,𝑡_{0−𝜏𝑖𝑡𝑡+𝜃}𝜏_𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₁𝑒̇(𝑠)+𝑑𝑠𝑑𝜃_{0−ℎ𝑡𝑡+𝜃}ℎexp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₂𝑒̇(𝑠)+𝑑𝑠𝑑𝜃_{0−𝑑1𝑖𝑡𝑡+𝜃}𝑑_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₃𝑒̇(𝑠)+𝑑𝑠𝑑𝜃_{−𝑑1𝑖−𝑑𝑚𝑖𝑡𝑡+𝜃}𝜚_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₄𝑒̇(𝑠)+𝑑𝑠𝑑𝜃_{−𝑑𝑚𝑖−𝑑2𝑖𝑡𝑡+𝜃}𝜚_2𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₅𝑒̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₇(𝑒(𝑡),𝑖,𝑡)=⁰_−𝜏𝑖⁰_𝜃^𝑡_𝑡+𝜆𝜏_𝑖²2exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇×(𝑠)𝑊₁𝑒̇(𝑠)+𝑑𝑠𝑑𝜆𝑑𝜃_{0−ℎ0𝜃𝑡𝑡+𝜆}ℎ²2exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇×(𝑠)𝑊₂𝑒̇(𝑠)+𝑑𝑠𝑑𝜆𝑑𝜃_{0−𝑑1𝑖0𝜃𝑡𝑡+𝜆}𝑑²_1𝑖2exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇×(𝑠)𝑊₃𝑒̇(𝑠)+𝑑𝑠𝑑𝜆𝑑𝜃_{−𝑑1𝑖−𝑑𝑚𝑖0𝜃𝑡𝑡+𝜆}𝜚_3𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇×(𝑠)𝑊₄𝑒̇(𝑠)+𝑑𝑠𝑑𝜆𝑑𝜃_{−𝑑𝑚𝑖−𝑑2𝑖0𝜃𝑡𝑡+𝜆}𝜚_4𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇×(𝑠)𝑊₅𝑒̇(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃. (35)(36)(37)(38)(39)(40)(41)
Taking Γ as its infinitesimal generator along the trajectory of error system (13), we obtain the following from Definition 3 and (34)-(35): Γ𝑉(𝑒(𝑡),𝑖,𝑡)=⁷_𝑘=1Γ𝑉_𝑘(𝑒(𝑡),𝑖,𝑡),Γ𝑉₁(𝑒(𝑡))𝑒,𝑖,𝑡=2^𝑇(𝑡)𝔸^𝑇_𝑖+𝑒^𝑇𝑡−𝑑_𝑖(𝑡)𝔹^𝑇_𝑖+𝑒^𝑇̇𝑡−𝜏_𝑖(𝑡)ℂ^𝑇_𝑖+𝐹^𝑇_𝑒(𝑒(𝑡))𝔻^𝑇_𝑖+𝑒^𝑇(𝑡−ℎ(𝑡))𝕂^𝑇𝑃_𝑖𝑒(𝑡)+𝑒^𝑇(𝑡)_{𝑗∈𝒮𝑖𝑘}𝛾_𝑖𝑗𝑃_𝑗+_{𝑗∈𝒮𝑖𝑢𝑘}𝛾_𝑖𝑗𝑃_𝑗𝑒(𝑡)+2𝜀𝑒^𝑇(𝑡)𝑃_𝑖𝑒(𝑡)−2𝜀𝑉₁(𝑒(𝑡),𝑖,𝑡),Γ𝑉₂(𝑒(𝑡),𝑖,𝑡)=𝑒^𝑇(𝑡)𝑄_1𝑖+𝑄₃+𝑄₅𝑒(𝑡)+𝑒^𝑇̇(𝑡)𝑄_2𝑖+𝑄₄𝑒̇(𝑡)−2𝜀𝑉₂(𝑒(𝑡))−𝜏̇,𝑖,𝑡1−_𝑖(𝑡)exp−2𝜀𝜏_𝑖(𝑡)×𝑒^𝑇𝑡−𝜏_𝑖(𝑡)𝑄_1𝑖𝑒𝑡−𝜏_𝑖(𝑡)−1−𝜏̇_𝑖(𝑡)exp−2𝜀𝜏_𝑖(𝑡×𝑒̇_𝑇)𝑡−𝜏_𝑖(𝑡)𝑄_2𝑖𝑒̇𝑡−𝜏_𝑖(𝑡)−exp−2𝜀𝜏_𝑖𝑒^𝑇𝑡−𝜏_𝑖𝑄₃𝑒𝑡−𝜏_𝑖−exp−2𝜀𝜏_𝑖𝑒^𝑇̇𝑡−𝜏_𝑖𝑄₄𝑒̇𝑡−𝜏_𝑖+_𝑗∈𝑆𝛾_𝑖𝑗^𝑡_{𝑡−𝜏𝑗(𝑡)}{(×𝑒exp2𝜀𝑠−𝑡)}_𝑇(𝑠)𝑄_1𝑗𝑒(𝑠)+𝑒^𝑇̇(𝑠)𝑄_2𝑗𝑒̇(𝑠)𝑑𝑠−exp−2𝜀ℎ𝑒^𝑇𝑡−ℎ𝑄₅𝑒𝑡−ℎ≤𝑒^𝑇(𝑡)𝑄_1𝑖+𝑄₃+𝑄₅𝑒(𝑡)+𝑒^𝑇̇(𝑡)𝑄_2𝑖+𝑄₄𝑒̇(𝑡)−2𝜀𝑉₂(𝑒(𝑡))−𝜏̇,𝑖,𝑡1−_𝑖(𝑡)exp−2𝜀𝜏_𝑖(𝑡)×𝑒^𝑇𝑡−𝜏_𝑖(𝑡)𝑄_1𝑖𝑒𝑡−𝜏_𝑖(𝑡)−1−𝜏̇_𝑖(𝑡)exp−2𝜀𝜏_𝑖(𝑡×𝑒̇_𝑇)𝑡−𝜏_𝑖(𝑡)𝑄_2𝑖𝑒̇𝑡−𝜏_𝑖(𝑡)−exp−2𝜀𝜏_𝑖𝑒^𝑇𝑡−𝜏_𝑖𝑄₃𝑒𝑡−𝜏_𝑖−exp−2𝜀𝜏_𝑖𝑒^𝑇̇𝑡−𝜏_𝑖𝑄₄𝑒̇𝑡−𝜏_𝑖+_𝑗≠𝑖𝛾_𝑖𝑗^𝑡_{𝑡−𝜏𝑗}{(×𝑒exp2𝜀𝑠−𝑡)}_𝑇(𝑠)𝑄_1𝑗𝑒(𝑠)+𝑒^𝑇̇(𝑠)𝑄_2𝑗𝑒̇(𝑠)𝑑𝑠−exp−2𝜀ℎ𝑒^𝑇𝑡−ℎ𝑄₅𝑒𝑡−ℎ,Γ𝑉₃(𝑒(𝑡),𝑖,𝑡)=𝑒^𝑇(𝑡)𝑅₂𝑒(𝑡)+exp−2𝜀𝑑_1𝑖𝑒^𝑇𝑡−𝑑_1𝑖×𝑅_1𝑖+𝑅₃−𝑅₂𝑒𝑡−𝑑_1𝑖+exp−2𝜀𝑑_𝑚𝑖𝑒^𝑇𝑡−𝑑_𝑚𝑖×𝑅₄−𝑅₃𝑒𝑡−𝑑_𝑚𝑖−exp−2𝜀𝑑_2𝑖𝑒^𝑇𝑡−𝑑_2𝑖×𝑅₄𝑒𝑡−𝑑_2𝑖−2𝜀𝑉₃(𝑒(𝑡))−𝑑̇,𝑖,𝑡1−_𝑖(𝑡)exp−2𝜀𝑑_𝑖(𝑡)×𝑒^𝑇𝑡−𝑑_𝑖(𝑡)𝑅_1𝑖𝑒𝑡−𝑑_𝑖(𝑡)+_𝑗∈𝑆𝛾_𝑖𝑗^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑗(𝑡)}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑅_1𝑗𝑒(𝑠)𝑑𝑠≤𝑒^𝑇(𝑡)𝑅₂𝑒(𝑡)+exp−2𝜀𝑑_1𝑖×𝑒^𝑇𝑡−𝑑_1𝑖𝑅_1𝑖+𝑅₃−𝑅₂𝑒𝑡−𝑑_1𝑖+exp−2𝜀𝑑_𝑚𝑖𝑒^𝑇𝑡−𝑑_𝑚𝑖×𝑅₄−𝑅₃𝑒𝑡−𝑑_𝑚𝑖−exp−2𝜀𝑑_2𝑖𝑒^𝑇𝑡−𝑑_2𝑖×𝑅₄𝑒𝑡−𝑑_2𝑖−2𝜀𝑉₃(𝑒(𝑡))−𝑑̇,𝑖,𝑡1−_𝑖(𝑡)exp−2𝜀𝑑_𝑖(𝑡)×𝑒^𝑇𝑡−𝑑_𝑖(𝑡)𝑅_1𝑖𝑒𝑡−𝑑_𝑖(𝑡)+_𝑗≠𝑖𝛾_𝑖𝑗^{𝑡−𝑑1𝑖}_{𝑡−𝑑2𝑗}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑅_1𝑗𝑒(𝑠)𝑑𝑠,Γ𝑉₄(𝑒(𝑡))=𝑒̇,𝑖,𝑡_𝑇(𝑡)𝑇₁𝑒̇(𝑡)+exp−2𝜀𝑑_1𝑖𝑒^𝑇̇𝑡−𝑑_1𝑖𝑇₂−𝑇₁×𝑒̇𝑡−𝑑_1𝑖−2𝜀𝑉₄(𝑒(𝑡),𝑖,𝑡)+exp−2𝜀𝑑_𝑚𝑖𝑒^𝑇̇𝑡−𝑑_𝑚𝑖×𝑇₃−𝑇₂𝑒̇𝑡−𝑑_𝑚𝑖−exp−2𝜀𝑑_2𝑖𝑒^𝑇̇𝑡−𝑑_2𝑖𝑇₃𝑒̇𝑡−𝑑_2𝑖,Γ𝑉₅(𝑒(𝑡),𝑖,𝑡)=𝑒^𝑇(𝑡)𝜏_𝑖²𝑈₁+ℎ²𝑈₂+𝑑²_1𝑖𝑈₃+𝜚²_1𝑖𝑈₄+𝜚²_2𝑖𝑈₅𝑒(𝑡)−2𝜀𝑉₅(𝑒(𝑡),𝑖,𝑡)−^𝑡_{𝑡−𝜏𝑖}𝜏_𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₁𝑒(𝑠)−𝑑𝑠_{𝑡𝑡−ℎ}ℎexp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₂𝑒(𝑠)−𝑑𝑠_{𝑡𝑡−𝑑1𝑖}𝑑_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₃𝑒(𝑠)−𝑑𝑠_{𝑡−𝑑1𝑖𝑡−𝑑𝑚𝑖}𝜚_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₄𝑒(𝑠)−𝑑𝑠_{𝑡−𝑑𝑚𝑖𝑡−𝑑2𝑖}𝜚_2𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₅𝑒(𝑠)𝑑𝑠,Γ𝑉₆(𝑒(𝑡))=𝑒̇,𝑖,𝑡_𝑇(𝑡)𝜏_𝑖²𝑉₁+ℎ²𝑉₂+𝑑²_1𝑖𝑉₃+𝜚²_1𝑖𝑉₄+𝜚²_2𝑖𝑉₅𝑒̇(𝑡)−2𝜀𝑉₆(𝑒(𝑡))−,𝑖,𝑡_{𝑡𝑡−𝜏𝑖}𝜏_𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₁𝑒̇(𝑠)−𝑑𝑠_{𝑡𝑡−ℎ}ℎexp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₂𝑒̇(𝑠)−𝑑𝑠_{𝑡𝑡−𝑑1𝑖}𝑑_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₃𝑒̇(𝑠)−𝑑𝑠_{𝑡−𝑑1𝑖𝑡−𝑑𝑚𝑖}𝜚_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₄𝑒̇(𝑠)−𝑑𝑠_{𝑡−𝑑𝑚𝑖𝑡−𝑑2𝑖}𝜚_2𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₅𝑒̇(𝑠)𝑑𝑠,Γ𝑉₇(𝑒(𝑡))=𝑒̇,𝑖,𝑡_𝑇(𝑡)𝜏_𝑖⁴4𝑊₁+ℎ⁴4𝑊₂+𝑑⁴_1𝑖4𝑊₃+𝜚²_3𝑖𝑊₄+𝜚²_4𝑖𝑊₅𝑒̇(𝑡)−2𝜀𝑉₇(𝑒(𝑡))−,𝑖,𝑡_{0−𝜏𝑖𝑡𝑡+𝜃}𝜏_𝑖²2exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑊₁𝑒̇(𝑠)−𝑑𝑠𝑑𝜃_{0−ℎ𝑡𝑡+𝜃}ℎ²2exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑊₂𝑒̇(𝑠)−𝑑𝑠𝑑𝜃_{0−𝑑1𝑖𝑡𝑡+𝜃}𝑑²_1𝑖2exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑊₃𝑒̇(𝑠)−𝑑𝑠𝑑𝜃_{−𝑑1𝑖−𝑑𝑚𝑖𝑡𝑡+𝜃}𝜚_3𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑊₄𝑒̇(𝑠)−𝑑𝑠𝑑𝜃_{−𝑑𝑚𝑖−𝑑2𝑖𝑡𝑡+𝜃}𝜚_4𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑊₅𝑒̇(𝑠)𝑑𝑠𝑑𝜃. (42)(43)(44)(45)(46)(47)(48)(49)
Define𝜉(𝑡)=col𝑒(𝑡)𝑒𝑡−𝑑_𝑖(𝑡)𝑒𝑡−𝑑_1𝑖𝑒𝑡−𝑑_𝑚𝑖𝑒𝑡−𝑑_2𝑖𝑒̇𝑡−𝑑_1𝑖𝑒̇𝑡−𝑑_𝑚𝑖𝑒̇𝑡−𝑑_{2𝑖𝑡𝑡−𝑑1𝑖}𝑒(𝑠)𝑑𝑠^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑖(𝑡)}𝑒(𝑠)𝑑𝑠^{𝑡−𝑑𝑖(𝑡)}_{𝑡−𝑑2𝑖}𝑒(𝑠)𝑑𝑠^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑚𝑖}𝑒(𝑠)𝑑𝑠^{𝑡−𝑑𝑚𝑖}_{𝑡−𝑑2𝑖}𝑒(𝑠)𝑒𝑑𝑠𝑡−𝜏_𝑖𝑒̇𝑡−𝜏_𝑖𝑒𝑡−𝜏_𝑖(𝑡)𝑒̇𝑡−𝜏_𝑖(𝑡)^𝑡_{𝑡−𝜏𝑖(𝑡)}𝑒(𝑠)𝑑𝑠^𝑡_{𝑡−𝜏𝑖}𝑒(𝑠)𝑒𝑑𝑠𝑡−ℎ𝑒(𝑡−ℎ(𝑡))^𝑡_{(𝑡)𝑡−ℎ}𝑒(𝑠)𝑑𝑠^𝑡_𝑡−ℎ𝑒(𝑠)𝑑𝑠𝐹_𝑒(𝑒(𝑡)). (50)Due to ∑^𝑁_𝑗=1𝛾_𝑖𝑗=0, the following zero equations hold for arbitrary matrices: 𝑋_𝑖=𝑋^𝑇_𝑖,𝑌_1𝑖=𝑌^𝑇_1𝑖,𝑌_2𝑖=𝑌^𝑇_2𝑖,𝑍_𝑖=𝑍^𝑇_𝑖,𝑖∈𝑆; (51) that is, −𝑒^𝑇(𝑡)_{𝑗∈𝒮𝑖𝑘}𝛾_𝑖𝑗𝑋_𝑖+_{𝑗∈𝒮𝑖𝑢𝑘}𝛾_𝑖𝑗𝑋_𝑖𝑒(𝑡)−=0,_{𝑡𝑡−𝜏𝑗}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)×_{𝑗∈𝒮𝑖𝑘}𝛾_𝑖𝑗𝑌_1𝑖+_{𝑗∈𝒮𝑖𝑢𝑘}𝛾_𝑖𝑗𝑌_1𝑖𝑒(𝑠)𝑑𝑠=0,−^𝑡_{𝑡−𝜏𝑗}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)×_{𝑗∈𝒮𝑖𝑘}𝛾_𝑖𝑗𝑌_2𝑖+_{𝑗∈𝒮𝑖𝑢𝑘}𝛾_𝑖𝑗𝑌_2𝑖𝑒̇(𝑠)−𝑑𝑠=0,_{𝑡−𝑑1𝑖𝑡−𝑑2𝑗}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)×_{𝑗∈𝒮𝑖𝑘}𝛾_𝑖𝑗𝑍_𝑖+_{𝑗∈𝒮𝑖𝑢𝑘}𝛾_𝑖𝑗𝑍_𝑖𝑒(𝑠)𝑑𝑠=0. (52) Based on (9), we can obtain the following inequality for any 𝛿>0: 𝛿𝑒^𝑇(𝑡)𝐹^𝑇_𝑒(𝑒(𝑡𝐹))¹𝐹₂𝑒(𝑡)∗𝐼𝐹_𝑒(𝑒(𝑡))=𝛿𝜉^𝑇(𝑡)𝐸₁𝐸₂₄𝐹₁𝐹₂𝐸∗𝐼𝐸^𝑇1𝑇₂₄𝜉(𝑡)≤0. (53) From (42) and (53), we have Γ𝑉(𝑒(𝑡),𝑖,𝑡)≤⁷_𝑘=1Γ𝑉_𝑘(𝑒(𝑡),𝑖,𝑡)−𝛿𝜉^𝑇(𝑡)𝐸₁𝐸₂₄𝐹₁𝐹₂𝐸∗𝐼𝐸^𝑇1𝑇₂₄𝜉(𝑡). (54) From (8) we easily obtain that −1−𝜏̇_𝑖(𝑡)exp−2𝜀𝜏_𝑖(𝑡)≤−1−𝜈_𝑖exp−2𝜀𝜏_𝑖,−1−𝑑̇_𝑖(𝑡)exp−2𝜀𝑑_𝑖(𝑡)≤𝜛𝜀,𝜇_𝑖. (55) Notice (a) of Lemma 5; then, −^𝑡_{𝑡−𝑑1𝑖}𝑑_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₃𝑒(𝑠)𝑑𝑠≤−2𝜀𝑑_1𝑖exp2𝜀𝑑_1𝑖𝜉−1^𝑇(𝑡)𝐸₉𝑈₃𝐸^𝑇₉𝜉(𝑡),−^𝑡_{𝑡−𝑑1𝑖}𝑑_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₃𝑒̇(𝑠)𝑑𝑠≤−2𝜀𝑑_1𝑖exp2𝜀𝑑_1𝑖𝜉−1^𝑇(𝑡)𝐸₁−𝐸₃𝑉₃𝐸^𝑇₁−𝐸^𝑇₃𝜉(𝑡). (56) Notice (b) of Lemma 5; then, −⁰_−𝜏𝑖^𝑡_𝑡+𝜃𝜏_𝑖²2exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑊₁𝑒̇(𝑠)𝑑𝑠𝑑𝜃≤−exp−2𝜀𝜏_𝑖𝜉^𝑇(𝑡)𝜏_𝑖𝐸₁−𝐸₁₉𝑊₁×𝜏_𝑖𝐸^𝑇₁−𝐸^𝑇₁₉𝜉(𝑡),−⁰_−ℎ^𝑡_𝑡+𝜃ℎ²2exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑊₂𝑒̇(𝑠)𝑑𝑠𝑑𝜃≤−exp−2𝜀ℎ𝜉^𝑇(𝑡)ℎ𝐸₁−𝐸₂₃𝑊₂×ℎ_𝑖𝐸^𝑇₁−𝐸^𝑇₂₃𝜉(𝑡),−⁰_−𝑑1𝑖^𝑡_𝑡+𝜃𝑑²_1𝑖2exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑊₃𝑒̇(𝑠)𝑑𝑠𝑑𝜃≤−exp−2𝜀𝑑_1𝑖𝜉^𝑇(𝑡)𝑑_1𝑖𝐸₁−𝐸₉𝑊₃×𝑑_1𝑖𝐸^𝑇₁−𝐸^𝑇₉𝜉(𝑡),−^−𝑑1𝑖_{−𝑑𝑚𝑖}^𝑡_𝑡+𝜃𝜚_3𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑊₄𝑒̇(𝑠)𝑑𝑠𝑑𝜃≤−exp−2𝜀𝑑_𝑚𝑖𝜉^𝑇(𝑡)𝜚_1𝑖𝐸₁−𝐸₁₂𝑊₄×𝜚_1𝑖𝐸^𝑇₁−𝐸^𝑇₁₂𝜉(𝑡),−^{−𝑑𝑚𝑖}_−𝑑2𝑖^𝑡_𝑡+𝜃𝜚_4𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑊₅𝑒̇(𝑠)𝑑𝑠𝑑𝜃≤−exp−2𝜀𝑑_2𝑖𝜉^𝑇(𝑡)𝜚_2𝑖𝐸₁−𝐸₁₃𝑊₅×𝜚_2𝑖𝐸^𝑇₁−𝐸^𝑇₁₃𝜉(𝑡). (57) For 𝑑_𝑖(𝑡)∈[𝑑_1𝑖,𝑑_𝑚𝑖], the following is held from (a) of Lemma 5:−^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑚𝑖}𝜚_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₄𝑒̇(𝑠)𝑑𝑠=−^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑖(𝑡)}+^{𝑡−𝑑𝑖(𝑡)}_{𝑡−𝑑𝑚𝑖}𝜚_1𝑖×exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₄𝑒̇(𝑠)𝑑𝑠≤−2𝜀𝜚_1𝑖exp2𝜀𝑑_𝑖(𝑡)−exp2𝜀𝑑_1𝑖×^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑖(𝑡)}𝑒^𝑇̇(𝑠)𝑑𝑠𝑉₄^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑖(𝑡)}𝑒̇(𝑠)−𝑑𝑠2𝜀𝜚_1𝑖exp2𝜀𝑑_𝑚𝑖{(𝑡×−exp2𝜀𝑑)}_{𝑡−𝑑𝑖(𝑡)𝑡−𝑑𝑚𝑖}𝑒^𝑇̇(𝑠)𝑑𝑠𝑉₄^{𝑡−𝑑𝑖(𝑡)}_{𝑡−𝑑𝑚𝑖}𝑒̇(𝑠)𝑑𝑠=−2𝜀𝜚_1𝑖exp2𝜀𝑑_𝑚𝑖−exp2𝜀𝑑_1𝑖×𝜉^𝑇(𝑡1)𝜆_1𝑖(𝑡)𝐸₃−𝐸₂𝑉₄𝐸^𝑇₃−𝐸^𝑇₂+1𝜆_2𝑖(𝑡)𝐸₂−𝐸₄𝑉₄𝐸^𝑇₂−𝐸^𝑇₄𝜉(𝑡), (58) where 𝜆_1𝑖(𝑡)=exp2𝜀𝑑_𝑖(𝑡)−exp2𝜀𝑑_1𝑖exp2𝜀𝑑_𝑚𝑖−exp2𝜀𝑑_1𝑖,𝜆_2𝑖(𝑡)=exp2𝜀𝑑_𝑚𝑖−exp2𝜀𝑑_𝑖(𝑡)exp2𝜀𝑑_𝑚𝑖−exp2𝜀𝑑_1𝑖. (59)
By Lemma 6, there exists matrix 𝑁₂ with appropriate dimensions such that −^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑚𝑖}𝜚_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₄𝑒̇(𝑠)𝑑𝑠≤−2𝜀𝜚_1𝑖exp2𝜀𝑑_𝑚𝑖−exp2𝜀𝑑_1𝑖𝜉^𝑇(𝑡)×𝐸₃−𝐸₂𝐸₂−𝐸₄𝑉₄𝑁₂𝑁^𝑇₂𝑉₄𝐸^𝑇₃−𝐸^𝑇₂𝐸^𝑇₂−𝐸^𝑇₄𝜉(𝑡),𝑉₄𝑁₂𝑁^𝑇₂𝑉₄>0. (60)
Similarly, considering −∫^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑚𝑖}𝜚_1𝑖exp{2𝜀(𝑠−𝑡)}𝑥^𝑇(𝑠)𝑈₄𝑥(𝑠)𝑑𝑠 and following the same procedure, there exists matrix 𝑁₁ with appropriate dimensions such that −^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑚𝑖}𝜚_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₄𝑒(𝑠)𝑑𝑠≤−2𝜀𝜚_1𝑖exp2𝜀𝑑_𝑚𝑖−exp2𝜀𝑑_1𝑖𝜉^𝑇(𝑡)×𝐸₁₀𝐸₁₂−𝐸₁₀𝑈₄𝑁₁𝑁^𝑇₁𝑈₄𝐸^𝑇₁₀𝐸^𝑇₁₂−𝐸^𝑇₁₀𝜉(𝑡),𝑈₄𝑁₁𝑁^𝑇₁𝑈₄>0. (61)
For 𝜏_𝑖(𝑡)∈[0,𝜏_𝑖], with the same matrix inequalities technique, we obtain the following: −^𝑡_{𝑡−𝜏𝑖}𝜏_𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₁𝑒(𝑠)𝑑𝑠≤−2𝜀𝜏_𝑖exp2𝜀𝜏_𝑖𝜉−1^𝑇(𝑡)𝐸₁₈𝐸₁₉−𝐸₁₈×𝑈₁𝑀₁𝑀^𝑇₁𝑈₁𝐸^𝑇₁₈𝐸^𝑇₁₉−𝐸^𝑇₁₈𝜉(𝑡),𝑈₁𝑀₁𝑀^𝑇₁𝑈₁−>0,_{𝑡𝑡−𝜏𝑖}𝜏_𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₁𝑒̇(𝑠)𝑑𝑠≤−2𝜀𝜏_𝑖exp2𝜀𝜏_𝑖𝜉−1^𝑇(𝑡)𝐸₁−𝐸₁₆𝐸₁₆−𝐸₁₄×𝑉₁𝑀₂𝑀^𝑇₂𝑉₁𝐸^𝑇₁−𝐸^𝑇₁₆𝐸^𝑇₁₆−𝐸^𝑇₁₄𝜉(𝑡),𝑉₁𝑀₂𝑀^𝑇₂𝑉₁>0. (62)
For ℎ(𝑡)∈[0,ℎ], we obtain the following: −^𝑡_𝑡−ℎℎexp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₂𝑒(𝑠)𝑑𝑠≤−2𝜀ℎexp2𝜀ℎ𝜉−1^𝑇(𝑡)𝐸₂₂𝐸₂₃−𝐸₂₂×𝑈₂𝑀₃𝑀^𝑇₃𝑈₂𝐸^𝑇₂₂𝐸^𝑇₂₃−𝐸^𝑇₂₂𝜉(𝑡),𝑈₂𝑀₃𝑀^𝑇₃𝑈₂−>0,_{𝑡𝑡−ℎ}ℎexp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₂𝑒̇(𝑠)𝑑𝑠≤−2𝜀ℎexp2𝜀ℎ𝜉−1^𝑇(𝑡)𝐸₁−𝐸₂₁𝐸₂₁−𝐸₂₀×𝑉₂𝑀₄𝑀^𝑇₄𝑉₂𝐸^𝑇₁−𝐸^𝑇₂₁𝐸^𝑇₂₁−𝐸^𝑇₂₀𝜉(𝑡),𝑉₂𝑀₄𝑀^𝑇₄𝑉₂>0. (63)
Consider −∫^{𝑡−𝑑𝑚𝑖}_{𝑡−𝑑2𝑖}𝜚_2𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₅𝑒(𝑠)𝑑𝑠 and −∫^{𝑡−𝑑𝑚𝑖}_{𝑡−𝑑2𝑖}𝜚_2𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₅𝑒(̇𝑠)𝑑𝑠, which are directly estimated by (a) of Lemma 5; that is, −^{𝑡−𝑑𝑚𝑖}_{𝑡−𝑑2𝑖}𝜚_2𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₅𝑒(𝑠)𝑑𝑠≤−2𝜀𝜚_2𝑖exp2𝜀𝑑_2𝑖−exp2𝜀𝑑_𝑚𝑖𝜉^𝑇(𝑡)𝐸₁₃𝑈₅𝐸^𝑇₁₃𝜉(𝑡),−^{𝑡−𝑑𝑚𝑖}_{𝑡−𝑑2𝑖}𝜚_2𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₅𝑒̇(𝑠)𝑑𝑠≤−2𝜀𝜚_2𝑖exp2𝜀𝑑_2𝑖−exp2𝜀𝑑_𝑚𝑖×𝜉^𝑇(𝑡)𝐸₄−𝐸₅𝑉₅𝐸^𝑇₄−𝐸^𝑇₅𝜉(𝑡). (64)
In addition, with the error system (53), we obtain 𝑒(̇𝑡)=Λ𝜉(𝑡) and 𝑒^𝑇̇(𝑡)𝐽𝑒̇(𝑡)=𝜉^𝑇(𝑡)Λ^𝑇𝐽Λ𝜉(𝑡), (65) where Λ and 𝐽 have been defined in Theorem 7.
Substituting (43)–(52) and (55)–(65) into (54), we obtain Γ𝑉(𝑒(𝑡),𝑖,𝑡)+2𝜀𝑉(𝑒(𝑡),𝑖,𝑡)≤𝜉^𝑇(𝑡)Ω^𝜀_𝑖0+Ω^𝜀_𝑖1+Λ^𝑇𝐽Λ𝜉(𝑡)+℧, (66) where ℧=^𝑡_{𝑡−𝜏𝑗}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)×_𝑗≠𝑖𝛾_𝑖𝑗𝑄_1𝑗−_{𝑗∈𝒮𝑖𝑘}𝛾_𝑖𝑗𝑌_1𝑖−_{𝑗∈𝒮𝑖𝑢𝑘}𝛾_𝑖𝑗𝑌_1𝑖𝑒(𝑠)+𝑑𝑠_{𝑡𝑡−𝜏𝑗}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)×_𝑗≠𝑖𝛾_𝑖𝑗𝑄_2𝑗−_{𝑗∈𝒮𝑖𝑘}𝛾_𝑖𝑗𝑌_2𝑖−_{𝑗∈𝒮𝑖𝑢𝑘}𝛾_𝑖𝑗𝑌_2𝑖𝑒̇(𝑠)+𝑑𝑠_{𝑡−𝑑1𝑖𝑡−𝑑2𝑗}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)×_𝑗≠𝑖𝛾_𝑖𝑗𝑅_1𝑗−_{𝑗∈𝒮𝑖𝑘}𝛾_𝑖𝑗𝑍_𝑖−_{𝑗∈𝒮𝑖𝑢𝑘}𝛾_𝑖𝑗𝑍_𝑖𝑒(𝑠)𝑑𝑠+𝑒^𝑇(𝑡)_{𝑗∈𝒮𝑖𝑢𝑘}𝛾_𝑖𝑗𝑃_𝑗−𝑋_𝑖𝑒(𝑡). (67)
On the other hand, for 𝑑_𝑖(𝑡)∈[𝑑_𝑚𝑖,𝑑_2𝑖], the integral terms −^{𝑡−𝑑𝑚𝑖}_{𝑡−𝑑2𝑖}𝜚_2𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₅𝑒(𝑠)−𝑑𝑠,_{𝑡−𝑑𝑚𝑖𝑡−𝑑2𝑖}𝜚_2𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₅𝑒̇(𝑠)𝑑𝑠 (68) are disposed and estimated by Lemma 6. −^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑚𝑖}𝜚_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑈₄𝑒(𝑠)−𝑑𝑠,_{𝑡−𝑑1𝑖𝑡−𝑑𝑚𝑖}𝜚_1𝑖exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑉₄𝑒̇(𝑠)𝑑𝑠 (69) are directly estimated by (a) of Lemma 5. Therefore, Γ𝑉(𝑒(𝑡),𝑖,𝑡)+2𝜀𝑉(𝑒(𝑡),𝑖,𝑡)≤𝜉^𝑇(𝑡)Ω^𝜀_𝑖0+Ω^𝜀_𝑖2+Λ^𝑇𝐽Λ𝜉(𝑡)+℧. (70)
With (66) and (70), the following inequality is held for 𝑑_𝑖(𝑡)∈[𝑑_1𝑖,𝑑_2𝑖] if (23)–(28) are satisfied: Γ𝑉(𝑒(𝑡),𝑖,𝑡)+2𝜀𝑉(𝑒(𝑡),𝑖,𝑡)<0. (71)
From the stochastic Lyapunov functional (34) and (71), it is held that 𝑉(𝑒(𝑡),𝑖,𝑡)≥min_𝑖∈𝑆𝜆_min𝑃_𝑖‖𝑒(𝑡)‖²=𝜆‖𝑒(𝑡)‖²,𝑉(𝑒(𝑡),𝑖,𝑡)<exp−2𝜀𝑡−𝑡₀𝑉𝑒𝑡₀,𝑟𝑡₀,𝑡₀. (72)
Moreover, we have 𝑉𝑒𝑡₀,𝑟𝑡₀,𝑡₀=⁷_𝑘=1𝑉_𝑘𝑒𝑡₀,𝑟𝑡₀,𝑡₀,_7𝑘=1𝑉_𝑘𝑒𝑡₀,𝑟𝑡₀,𝑡₀≤^𝑡0_𝑡0−𝜍exp2𝜀𝑠−𝑡₀𝑒^𝑇(𝑠)×𝑄_1𝑖+𝑄₃+𝑄₅+𝑅_1𝑖+⁴_𝑘=2𝑅_𝑘𝑒(𝑠)𝑑𝑠+^𝑡0_𝑡0−𝜍exp2𝜀𝑠−𝑡₀𝑒^𝑇̇(𝑠)×𝑄_2𝑖+𝑄₄+³_𝑙=1𝑇_𝑙𝑒̇(𝑠)+𝑑𝑠_{0−𝜍𝑡0𝑡0+𝜃}𝜍exp2𝜀𝑠−𝑡₀𝑒^𝑇×(𝑠)⁵_𝑚=1𝑈_𝑚𝑒(𝑠)+𝑑𝑠𝑑𝜃_{0−𝜍𝑡0𝑡0+𝜃}𝜍exp2𝜀𝑠−𝑡₀𝑒^𝑇̇×(𝑠)⁵_𝑛=1𝑉_𝑛𝑒̇(𝑠)+𝑑𝑠𝑑𝜃_{0−𝜍0𝜃𝑡0𝑡0+𝜆}𝜍²2exp2𝜀𝑠−𝑡₀𝑒^𝑇̇×(𝑠)⁵_𝑠=1𝑊_𝑠𝑒̇(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃+𝑒^𝑇𝑡₀𝑃_𝑖𝑒𝑡₀≤𝜆𝑒𝑡₀²_𝜍. (73)
Then from (72) and (73), it is readily seen that ‖𝑒(𝑡)‖≤𝜆𝜆exp−𝜀𝑡−𝑡₀𝑒𝑡_0𝜍, (74) 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 scalars 𝛾^𝑚_𝑖, 𝛾^𝑀_𝑖, 𝜀, 𝜏_𝑖, 𝜈_𝑖, ℎ, 𝑑_1𝑖, 𝑑_2𝑖, and 𝜇_𝑖 and constant scalar 𝑑_𝑚𝑖 satisfying 𝑑_1𝑖<𝑑_𝑚𝑖<𝑑_2𝑖, 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 ‖𝐺_𝑖⁽³⁾⊗𝐶_𝑖‖<1, and there exist symmetric positive matrices 𝑃>0, 𝑄_𝑗>0, (𝑗=1,2,3,4,5), 𝑅_𝑘>0, (𝑘=1,2,3,4), 𝑇_𝑙>0, (𝑙=1,2,3), 𝑈_𝑚>0, 𝑉_𝑛>0, 𝑊_𝑠>0, (𝑚,𝑛,𝑠=1,2,3,4,5) and matrices 𝑀_𝑘1, 𝑁_𝑘2, (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: −_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝛾^𝑀_𝑖𝑄₁−𝑌₁−𝛾^𝑚_𝑖𝑌₁≤0,−_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝛾^𝑀_𝑖𝑄₂−𝑌₂−𝛾^𝑚_𝑖𝑌₂≤0,−_{𝑗∈𝒮𝑖𝑢𝑘,𝑗≠𝑖}𝛾^𝑀_𝑖𝑅₁−𝑍−𝛾^𝑚_𝑖𝑈𝑍≤0,(i)⁴𝑁₁𝑁^𝑇₁𝑈₄𝑉>0,⁴𝑁₂𝑁^𝑇₂𝑉₄Ω>0,(ii)^𝜀_𝑖0+Ω^𝜀_𝑖1+Λ^𝑇𝐽̃𝑈Λ<0,(iii)⁵𝑁₃𝑁^𝑇₃𝑈₅𝑉>0,⁵𝑁₄𝑁^𝑇₄𝑉₅Ω>0,(iv)^𝜀_𝑖0+Ω^𝜀_𝑖2+Λ^𝑇𝐽Λ̃<0, (75)(76)(77)(78) where 𝐽̃=𝑄₂+𝑄₄+𝑇₁+𝜏_𝑖²𝑉₁+ℎ²𝑉₂+𝑑²_1𝑖𝑉₃+𝜚²_1𝑖𝑉₄+𝜚²_2𝑖𝑉₅+𝜏_𝑖⁴4𝑊₁+ℎ⁴4𝑊₂+𝑑⁴_1𝑖4𝑊₃+𝜚²_3𝑖𝑊₄+𝜚²_4𝑖𝑊₅,Λ=Λ,Ω^𝜀_𝑖1=Ω^𝜀_𝑖1,Ω^𝜀_𝑖2=Ω^𝜀_𝑖2,Ω^𝜀_𝑖0=²⁴_𝑚=1𝐸_𝑚Π_𝑚𝐸^𝑇_𝑚×𝐸+✠−𝛿₁𝐸₂₄𝐹₁𝐹₂𝐸∗𝐼𝐸^𝑇1𝑇₂₄−2𝜀𝑑_1𝑖exp2𝜀𝑑_1𝑖𝐸−1¹−𝐸₃𝑉₃𝐸^𝑇₁−𝐸^𝑇₃−exp−2𝜀𝜏_𝑖𝜏_𝑖𝐸₁−𝐸₁₉𝑊₁𝜏_𝑖𝐸^𝑇₁−𝐸^𝑇₁₉−exp−2𝜀ℎℎ𝐸₁−𝐸₂₃𝑊₂ℎ𝐸^𝑇₁−𝐸^𝑇₂₃−exp−2𝜀𝑑_1𝑖𝑑_1𝑖𝐸₁−𝐸₉𝑊₃𝑑_1𝑖𝐸^𝑇₁−𝐸^𝑇₉−exp−2𝜀𝑑_𝑚𝑖𝜚_1𝑖𝐸₁−𝐸₁₂𝑊₄𝜚_1𝑖𝐸^𝑇₁−𝐸^𝑇₁₂−exp−2𝜀𝑑_2𝑖𝜚_2𝑖𝐸₁−𝐸₁₃𝑊₅𝜚_2𝑖𝐸^𝑇₁−𝐸^𝑇₁₃−2𝜀𝜏_𝑖exp2𝜀𝜏_𝑖𝐸−1¹⁸𝐸₁₉−𝐸₁₈×𝑈₁𝑀₁𝑀^𝑇₁𝑈₁𝐸^𝑇₁₈𝐸^𝑇₁₉−𝐸^𝑇₁₈−2𝜀𝜏_𝑖exp2𝜀𝜏_𝑖𝐸−1¹−𝐸₁₆𝐸₁₆−𝐸₁₄×𝑉₁𝑀₂𝑀^𝑇₂𝑉₁𝐸^𝑇₁−𝐸^𝑇₁₆𝐸^𝑇₁₆−𝑒^𝑇₁₄−2𝜀ℎexp2𝜀ℎ𝐸−1²²𝐸₂₃−𝐸₂₂×𝑈₂𝑀₃𝑀^𝑇₃𝑈₂𝐸^𝑇₂₂𝐸^𝑇₂₃−𝐸^𝑇₂₂−2𝜀ℎexp2𝜀ℎ𝐸−1¹−𝐸₂₁𝐸₂₁−𝐸₂₀×𝑉₂𝑀₄𝑀^𝑇₄𝑉₂𝐸^𝑇₁−𝐸^𝑇₂₁𝐸^𝑇₂₁−𝐸^𝑇₂₀,Π₁=𝐺_𝑖⁽¹⁾⊗𝐴_𝑖^𝑇𝑃+𝑃𝐺_𝑖⁽¹⁾⊗𝐴_𝑖+2𝜀𝑃+𝑄₁+𝑄₃+𝑄₅+𝑅₂+𝜏_𝑖²𝑈₁+ℎ²𝑈₂+𝑑²_1𝑖𝑈₃+𝜚²_1𝑖𝑈₄+𝜚²_2𝑖𝑈₅,Π₂=𝜛𝜀,𝜇_𝑖𝑅₁,Π₃=exp−2𝜀𝑑_1𝑖𝑅₁+𝑅₃−𝑅₂,Π₁₆=−1−𝜈_𝑖exp−2𝜀𝜏_𝑖𝑄₁,Π₁₇=−1−𝜈_𝑖exp−2𝜀𝜏_𝑖𝑄₂,Π₄=Π₄,Π₅=Π₅,Π₆=Π₆,Π₇=Π₇,Π₈=Π₈,Π₉=Π₉,Π₁₄=Π₁₄,Π₁₅=Π₁₅,Π₂₀=Π₂₀,Π_𝑚=0,(𝑚=10,11,12,13,18,19,21,…,24),𝜆=𝜆_max(𝑃)+1−exp{−2𝜀𝜍}×2𝜀_5𝑗=1𝜆_max𝑄_𝑗+⁴_𝑘=1𝜆_max𝑅_𝑘+³_𝑙=1𝜆_max𝑇_𝑙+exp{−2𝜀𝜍}+2𝜀𝜍−14𝜀²×⁵_𝑚=1𝜆_max𝑈_𝑚+⁵_𝑛=1𝜆_max𝑉_𝑛+1−exp{−2𝜀𝜍}+2𝜀²𝜍²−2𝜀𝜍8𝜀³×⁵_𝑠=1𝜆_max𝑊_𝑠. (79)

Other notations are the same as those in Theorem 7.

Proof. Since the elements of transition rates are entirely unknown, we choose the Lyapunov functional as follows: 𝑉(𝑒(𝑡),𝑖,𝑡)=𝑒^𝑇(𝑡)𝑃𝑒(𝑡)+⁷_𝑘=2𝑉_𝑘(𝑒(𝑡),𝑖,𝑡), (80) where 𝑉₂(𝑒(𝑡),𝑖,𝑡)=^𝑡_{𝑡−𝜏𝑖(𝑡)}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑄₁𝑒(𝑠)+𝑑𝑠_{𝑡𝑡−𝜏𝑖(𝑡)}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑄₂𝑒̇(𝑠)+𝑑𝑠_{𝑡𝑡−𝜏𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑄₃𝑒(𝑠)+𝑑𝑠_{𝑡𝑡−𝜏𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑄₄𝑒̇(𝑠)+𝑑𝑠_{𝑡𝑡−ℎ}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑄₅𝑒(𝑠)𝑉𝑑𝑠,₃(𝑒(𝑡),𝑖,𝑡)=^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑖(𝑡)}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑅₁𝑒(𝑠)𝑑𝑠+^𝑡_{𝑡−𝑑1𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑅₂𝑒(𝑠)𝑑𝑠+^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑚𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑅₃𝑒(𝑠)+𝑑𝑠_{𝑡−𝑑𝑚𝑖𝑡−𝑑2𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇(𝑠)𝑅₄𝑒(𝑠)𝑉𝑑𝑠,₄(𝑒(𝑡),𝑖,𝑡)=^𝑡_{𝑡−𝑑1𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑇₁𝑒̇(𝑠)𝑑𝑠+^{𝑡−𝑑1𝑖}_{𝑡−𝑑𝑚𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑇₂𝑒̇(𝑠)𝑑𝑠+^{𝑡−𝑑𝑚𝑖}_{𝑡−𝑑2𝑖}exp{2𝜀(𝑠−𝑡)}𝑒^𝑇̇(𝑠)𝑇₃𝑒̇(𝑠)𝑑𝑠, (81) 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 Theorem 7 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 𝑃_𝑖=diag{𝑃_𝑖⁽¹⁾,𝑃_𝑖⁽²⁾,…,𝑃_𝑖^(𝑁)}>0, 𝑄_1𝑖>0, 𝑄_2𝑖>0, 𝑅_1𝑖>0, (𝑖∈𝑆={1,2,…,𝑀}), 𝑄_𝑗>0, (𝑗=3,4,5), 𝑅_𝑘>0, (𝑘=2,3,4), 𝑇_𝑙>0, (𝑙=1,2,3), 𝑈_𝑚>0, 𝑉_𝑛>0, 𝑊_𝑠>0, (𝑚,𝑛,𝑠=1,2,3,4,5), matrices 𝕐_𝑖=diag{𝑌_𝑖⁽¹⁾,𝑌_𝑖⁽²⁾,…,𝑌_𝑖^(𝑁)} and matrices 𝑀_𝑘1, 𝑁_𝑘2, (𝑘₁,𝑘₂=1,2,3,4), for any scalar 𝛿>0, and symmetric matrices 𝑋_𝑖, 𝑌_1𝑖, 𝑌_2𝑖, 𝑍_𝑖, (𝑖∈𝑆) such that (23)–(28) and the following linear matrix inequalities hold: 𝑈(i)⁴𝑁₁𝑁^𝑇₁𝑈₄𝑉>0,⁴𝑁₂𝑁^𝑇₂𝑉₄Θ>0,(ii)^𝜀𝑖0+Ω^𝜀_𝑖1Λ^𝑇𝑃_𝑖𝑃_𝑖Λ−2𝑃_𝑖𝑈+𝐽<0,(iii)⁵𝑁₃𝑁^𝑇₃𝑈₅𝑉>0,⁵𝑁₄𝑁^𝑇₄𝑉₅Θ>0,(iv)^𝜀𝑖0+Ω^𝜀_𝑖2Λ^𝑇𝑃_𝑖𝑃_𝑖Λ−2𝑃_𝑖+𝐽<0. (82)(83) In addition, the desired sampled-data controllers gain matrices are given by 𝐾_𝑘=𝑃_𝑖^(𝑘)−1𝑌_𝑖^(𝑘),𝑘=1,2,…,𝑁, (84) where Θ^𝜀_𝑖0=²⁴_𝑚=1𝐸_𝑚Π_𝑚𝐸^𝑇_𝑚+✠_Θ−𝛿𝐸₁𝐸₂₄×𝐹₁𝐹₂𝐸∗𝐼𝐸^𝑇1𝑇₂₄−2𝜀𝑑_1𝑖exp2𝜀𝑑_1𝑖×𝐸−1₁−𝐸₃𝑉₃𝐸^𝑇₁−𝐸^𝑇₃−exp−2𝜀𝜏_𝑖𝜏_𝑖𝐸₁−𝐸₁₉𝑊₁𝜏_𝑖𝐸^𝑇₁−𝐸^𝑇₁₉−exp−2𝜀ℎℎ𝐸₁−𝐸₂₃𝑊₂ℎ𝐸^𝑇₁−𝐸^𝑇₂₃−exp−2𝜀𝑑_1𝑖𝑑_1𝑖𝐸₁−𝐸₉𝑊₃𝑑_1𝑖𝐸^𝑇₁−𝐸^𝑇₉−exp−2𝜀𝑑_𝑚𝑖𝜚_1𝑖𝐸₁−𝐸₁₂𝑊₄𝜚_1𝑖𝐸^𝑇₁−𝐸^𝑇₁₂−exp−2𝜀𝑑_2𝑖𝜚_2𝑖𝐸₁−𝐸₁₃𝑊₅𝜚_2𝑖𝐸^𝑇₁−𝐸^𝑇₁₃−2𝜀𝜏_𝑖exp2𝜀𝜏_𝑖𝐸−1¹⁸𝐸₁₉−𝐸₁₈×𝑈₁𝑀₁𝑀^𝑇₁𝑈₁𝐸^𝑇₁₈𝐸^𝑇₁₉−𝐸^𝑇₁₈−2𝜀𝜏_𝑖exp2𝜀𝜏_𝑖𝐸−1¹−𝐸₁₆𝐸₁₆−𝐸₁₄×𝑉₁𝑀₂𝑀^𝑇₂𝑉₁𝐸^𝑇₁−𝐸^𝑇₁₆𝐸^𝑇₁₆−𝑒^𝑇₁₄−2𝜀ℎexp2𝜀ℎ𝐸−1²²𝐸₂₃−𝐸₂₂×𝑈₂𝑀₃𝑀^𝑇₃𝑈₂𝐸^𝑇₂₂𝐸^𝑇₂₃−𝐸^𝑇₂₂−2𝜀ℎexp2𝜀ℎ𝐸−1¹−𝐸₂₁𝐸₂₁−𝐸₂₀×𝑉₂𝑀₄𝑀^𝑇₄𝑉₂𝐸^𝑇₁−𝐸^𝑇₂₁𝐸^𝑇₂₁−𝐸^𝑇₂₀,✠_Θ=𝐸₁𝑃_𝑖𝐺_𝑖⁽²⁾⊗𝐵_𝑖𝐸^𝑇₂+𝐸₁𝑃_𝑖𝐺_𝑖⁽³⁾⊗𝐶_𝑖𝐸₁₇+𝐸₁𝕐_𝑖𝐸^𝑇₂₁+𝐸₁𝑃_𝑖𝐼_𝑁⊗𝐷_𝑖𝐸^𝑇₂₄+𝐸₂𝐺_𝑖⁽²⁾⊗𝐵_𝑖^𝑇𝑃_𝑖𝐸^𝑇₁+𝐸₁₇𝐺_𝑖⁽³⁾⊗𝐶_𝑖^𝑇𝑃_𝑖𝐸^𝑇₁+𝐸₂₁𝕐^𝑇_𝑖𝐸^𝑇₁+𝐸₂₄𝐼_𝑁⊗𝐷_𝑖^𝑇𝑃_𝑖𝐸^𝑇₁. (85)

Other notations are the same as those in Theorem 7.

Proof. Considering (27) and (28), by applying the Schur Complement Lemma, we have (i)𝑈₄𝑁₁𝑁^𝑇₁𝑈₄𝑉>0,⁴𝑁₂𝑁^𝑇₂𝑉₄()Ω>0,ii^𝜀𝑖0+Ω^𝜀_𝑖1Λ^𝑇𝐽()𝑈𝐽Λ−𝐽<0,iii⁵𝑁₃𝑁^𝑇₃𝑈₅𝑉>0,⁵𝑁₄𝑁^𝑇₄𝑉₅()Ω>0,iv^𝜀𝑖0+Ω^𝜀_𝑖2Λ^𝑇𝐽𝐽Λ−𝐽<0. (86)(87)
Define the block diagonal matrix 𝕏=^𝐻_𝑃𝑖𝐽⁻¹, where 𝐻=∑²⁴_𝑚=1𝐸_𝑚𝐸^𝑇_𝑚. Noting that 𝐸^𝑇_𝑖𝐸_𝑗=^0𝑖≠𝑗_{𝐼𝑖=𝑗}, with (ii) of (86) and (iv) of (87), we define 𝕐_𝑖=𝑃_𝑖𝕂 and obtain the following: 𝕏Ω^𝜀_𝑖0+Ω^𝜀_𝑖1Λ^𝑇𝐽𝕏𝐽Λ−𝐽^𝑇=Θ^𝜀_𝑖0+Ω^𝜀_𝑖1Λ^𝑇𝑃_𝑖𝑃_𝑖Λ−𝑃_𝑖𝐽⁻¹𝑃_𝑖𝕏Ω<0,_𝜀𝑖0+Ω^𝜀_𝑖2Λ^𝑇𝐽𝕏𝐽Λ−𝐽^𝑇=Θ^𝜀_𝑖0+Ω^𝜀_𝑖2Λ^𝑇𝑃_𝑖𝑃_𝑖Λ−𝑃_𝑖𝐽⁻¹𝑃_𝑖<0. (88)
Due to 𝐽>0 and (𝑃_𝑖−𝐽)𝐽⁻¹(𝑃_𝑖−𝐽)≥0, we know that −𝑃_𝑖𝐽⁻¹𝑃_𝑖≤−2𝑃_𝑖+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 Definition 3, 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, 𝜏_1𝑖≤𝜏_𝑖(𝑡)≤𝜏_2𝑖; we can choose 𝜏_𝑚𝑖 satisfying 𝜏_1𝑖≤𝜏_𝑚𝑖≤𝜏_2𝑖 and construct the stochastic Lyapunov functional on [𝑡−𝜏_𝑖(𝑡),𝑡−𝜏_1𝑖], [𝑡−𝜏_1𝑖,𝑡], [𝑡−𝜏_2𝑖,𝑡−𝜏_𝑚𝑖], and [𝑡−𝜏_𝑚𝑖,𝑡−𝜏_1𝑖]. 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, 𝑄_𝑗>0, (𝑗=1,2,3,4,5), 𝑅_𝑘>0, (𝑘=1,2,3,4), 𝑇_𝑙>0, (𝑙=1,2,3), 𝑈_𝑚>0, 𝑉_𝑛>0, 𝑊_𝑠>0, (𝑚,𝑛,𝑠=1,2,3,4,5), matrices 𝕐=diag{𝑌⁽¹⁾,𝑌⁽²⁾,…,𝑌^(𝑁)} and matrices 𝑀_𝑘1, 𝑁_𝑘2, (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: 𝑈(i)⁴𝑁₁𝑁^𝑇₁𝑈₄𝑉>0,⁴𝑁₂𝑁^𝑇₂𝑉₄Θ>0,(ii)^𝜀𝑖0+Ω^𝜀_𝑖1Λ^𝑇𝑃𝑈𝑃Λ−2𝑃+𝐽<0,(iii)⁵𝑁₃𝑁^𝑇₃𝑈₅𝑉>0,⁵𝑁₄𝑁^𝑇₄𝑉₅Θ>0,(iv)^𝜀𝑖0+Ω^𝜀_𝑖2Λ^𝑇𝑃𝑃Λ−2𝑃+𝐽<0. (89)(90) In addition, the desired sampled-data controllers gain matrices are given by 𝐾_𝑘=𝑃^(𝑘)−1𝑌^(𝑘),𝑘=1,2,…,𝑁, (91) where Θ^𝜀_𝑖0=²⁴_𝑚=1𝐸_𝑚Π_𝑚𝐸^𝑇_𝑚+✠_Θ−𝛿𝐸₁𝐸₂₄𝐹₁𝐹₂𝐸∗𝐼𝐸^𝑇1𝑇₂₄−2𝜀𝑑_1𝑖exp2𝜀𝑑_1𝑖𝐸−1¹−𝐸₃𝑉₃𝐸^𝑇₁−𝐸^𝑇₃−exp−2𝜀𝜏_𝑖𝜏_𝑖𝐸₁−𝐸₁₉𝑊₁𝜏_𝑖𝐸^𝑇₁−𝐸^𝑇₁₉−exp−2𝜀ℎℎ𝐸₁−𝐸₂₃𝑊₂ℎ𝐸^𝑇₁−𝐸^𝑇₂₃−exp−2𝜀𝑑_1𝑖𝑑_1𝑖𝐸₁−𝐸₉𝑊₃𝑑_1𝑖𝐸^𝑇₁−𝐸^𝑇₉−exp−2𝜀𝑑_𝑚𝑖𝜚_1𝑖𝐸₁−𝐸₁₂𝑊₄𝜚_1𝑖𝐸^𝑇₁−𝐸^𝑇₁₂−exp−2𝜀𝑑_2𝑖𝜚_2𝑖𝐸₁−𝐸₁₃𝑊₅𝜚_2𝑖𝐸^𝑇₁−𝐸^𝑇₁₃−2𝜀𝜏_𝑖exp2𝜀𝜏_𝑖𝐸−1¹⁸𝐸₁₉−𝐸₁₈×𝑈₁𝑀₁𝑀^𝑇₁𝑈₁𝐸^𝑇₁₈𝐸^𝑇₁₉−𝐸^𝑇₁₈−2𝜀𝜏_𝑖exp2𝜀𝜏_𝑖𝐸−1¹−𝐸₁₆𝐸₁₆−𝐸₁₄×𝑉₁𝑀₂𝑀^𝑇₂𝑉₁𝐸^𝑇₁−𝐸^𝑇₁₆𝐸^𝑇₁₆−𝑒^𝑇₁₄−2𝜀ℎexp2𝜀ℎ𝐸−1²²𝐸₂₃−𝐸₂₂×𝑈₂𝑀₃𝑀^𝑇₃𝑈₂𝐸^𝑇₂₂𝐸^𝑇₂₃−𝐸^𝑇₂₂−2𝜀ℎexp2𝜀ℎ𝐸−1¹−𝐸₂₁𝐸₂₁−𝐸₂₀×𝑉₂𝑀₄𝑀^𝑇₄𝑉₂𝐸^𝑇₁−𝐸^𝑇₂₁𝐸^𝑇₂₁−𝐸^𝑇₂₀,✠_Θ=𝐸₁𝑃𝐺_𝑖⁽²⁾⊗𝐵_𝑖𝐸^𝑇₂+𝐸₁𝑃𝐺_𝑖⁽³⁾⊗𝐶_𝑖𝐸₁₇+𝐸₁𝕐_𝑖𝐸^𝑇₂₁+𝐸₁𝑃𝐼_𝑁⊗𝐷_𝑖𝐸^𝑇₂₄+𝐸₂𝐺_𝑖⁽²⁾⊗𝐵_𝑖^𝑇𝑃𝐸^𝑇₁+𝐸₁₇𝐺_𝑖⁽³⁾⊗𝐶_𝑖^𝑇𝑃𝐸^𝑇₁+𝐸₂₁𝕐^𝑇_𝑖𝐸^𝑇₁+𝐸₂₄𝐼_𝑁⊗𝐷_𝑖^𝑇𝑃𝐸^𝑇₁. (92)

Other notations are the same as those in Corollary 8.

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: 𝐴₁=−0.50−0.350.10−0.50,𝐴₂=,𝐵−0.300.090.20−0.40₁=0.20−0.150.50−0.50,𝐵₂=,𝐶0.310.23−0.120.17₁=0.280.02−0.060.11,𝐶₂=,𝐷0.510.240.02−0.44₁=0.2000−0.15,𝐷₂=,𝐺0.300−0.100.23₁₍₁₎=−2111−2111−2,𝐺₂⁽¹⁾=,𝐺0001−1011−2₁₍₂₎=−2110−1110−1,𝐺₂⁽²⁾=,𝐺−21100011−2₁₍₃₎=0001−2110−1,𝐺₂⁽³⁾=.−1011−21000 (93) The partially known transition rate matrix is considered as in the following two cases: Υ₁=𝛾−22₀₂₁𝛾⁰₂₂,Υ₂=𝛾⁰₁₁𝛾⁰₁₂𝛾⁰₂₁𝛾⁰₂₂. (94) Furthermore, the nonlinear function 𝑓(𝑥_𝑘(𝑡)) is given by 𝑓𝑥_𝑘(𝑡)=0.4𝑥_𝑘1(𝑡)−tanh0.1𝑥_𝑘1(𝑡)0.1𝑥_𝑘2(𝑡)^𝑇. (95) Then, it is easy to verify that 𝐹₁=0.4000.1,𝐹₂=0.3000.1. (96)
The interval mode-dependent time-varying neutral delays and discrete delays are, respectively, assumed to be 𝜏₁(𝑡)=0.5(1+sin𝑡),𝜏₂(𝑡)(),𝑑=0.61+cos𝑡₁(𝑡)=0.41+sin⁴(𝑡),𝑑₂(𝑡)=0.81+cos⁴(𝑡). (97) They are governed by the Markov process {𝑟(𝑡),𝑡≥0} and shown in Figures 1 and 2. It can be readily obtained that 𝜏₁=1,𝜏₂=1.2,𝜈₁=0.5,𝜈₂𝑑=0.6,₁₁=0.4,𝑑₂₁=0.8,𝑑₁₂𝑑=0.8,₂₂=1.6,𝜇₁=3√35,𝜇₂=3√3.10 (98) Given decay rate 𝜀=0.3 and the maximum value of sampling period ℎ=0.2, we choose 𝑑_𝑚1=0.5 and 𝑑_𝑚2=1.0 and seek to solve the sampled-data controllers.
For the case of Υ₁, with the above parameters, by Theorem 9, we can obtain the sampled-data controllers as follows: 𝐾₁=,𝐾−0.7246−0.0345−0.0253−0.6358₂=,𝐾−0.5149−0.0326−0.0117−0.5048₃=.−0.3557−0.0263−0.0046−0.4356 (99) 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 of Υ₂, the sampled-data controllers can be readily obtained by Corollary 12.

(a)

(b)

Figure 1 

Time-varying neutral delay 𝜏𝑖(𝑡) at mode 1 and mode 2.

(a)

(b)

Figure 2 

Time-varying discrete delay 𝑑𝑖(𝑡) at mode 1 and mode 2.

Figure 3 

State trajectories of the error system in Example 1.

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 𝐺⁽¹⁾=𝐺⁽²⁾=−1010−1111−2,𝐺⁽³⁾=0. (100) In fact, the NCDN in consideration here has become a common CDN.
The nonlinear function 𝑓(𝑥_𝑘(𝑡)) is taken as𝑓𝑥_𝑘(𝑡)=−0.5𝑥_𝑘1(𝑡)+tanh0.2𝑥_𝑘1(𝑡)+0.2𝑥_𝑘2(𝑡)0.95𝑥_𝑘2(𝑡)−tanh0.75𝑥_𝑘2(𝑡)^𝑇, (101)(101) can be easily verified that 𝐹₁=−0.50.200.95,𝐹₂=−0.30.200.2. (102)
The time-varying delay is chosen as 𝑑(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,  𝑑_𝑚=0.18 satisfying 𝑑₁<𝑑_𝑚<𝑑₂ and consider the following two cases.
Case  1. The inner-coupling matrices are given as 𝐴=0 and 𝐵=^0.50_00.5. 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 period ℎ with 𝑑_𝑚=0.18 and 𝜀=0.

Table 2 

Maximum value of sampling period ℎ with 𝑑_𝑚=0.18 and 𝜀=0.

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.

Acknowledgments

This work was supported in part by the National Key Scientific Research Project (61233003), the National Natural Science Foundation of China (60935001, 61174061, and 61074033), the Doctoral Fund of the Ministry of Education of China (20093402110019), the Anhui Provincial Natural Science Foundation (11040606M143), the Fundamental Research Funds for the Central Universities, and the Program for New Century Excellent Talents in University.