Global Robust Exponential Dissipativity for Interval Recurrent Neural Networks with Infinity Distributed Delays

Abstract

This paper is concerned with the robust dissipativity problem for interval recurrent neural networks (IRNNs) with general activation functions, and continuous time-varying delay, and infinity distributed time delay. By employing a new differential inequality, constructing two different kinds of Lyapunov functions, and abandoning the limitation on activation functions being bounded, monotonous and differentiable, several sufficient conditions are established to guarantee the global robust exponential dissipativity for the addressed IRNNs in terms of linear matrix inequalities (LMIs) which can be easily checked by LMI Control Toolbox in MATLAB. Furthermore, the specific estimation of positive invariant and global exponential attractive sets of the addressed system is also derived. Compared with the previous literatures, the results obtained in this paper are shown to improve and extend the earlier global dissipativity conclusions. Finally, two numerical examples are provided to demonstrate the potential effectiveness of the proposed results.

1. Introduction

Neural networks have been a subject of intense research activities over the past few decades due to their wide applications in many areas such as signal processing, pattern recognition, associative memories, parallel computation, and optimization solution. Therefore, increasing attention has been paid to the problem of stability analysis of neural networks with time-varying delays, and recently a lot of research works have been reported for delayed neural networks and system (see [1–17] and references therein).

It is well known that the stability problem is central to analysis of a dynamical system on an equilibrium point. However, from a practical point of view, it is not always the case that the neural network trajectories will approach a single equilibrium point that is the equilibrium point will be unstable. It is also possible that there is no equilibrium point in some situations, especially for interval recurrent neural networks with infinity distributed delays. But what we can know is that the orbits of the neural networks will always enter into a bounded region and stay there from then on. Therefore, the concept of dissipativity (or called Lagrange stability) has been introduced in [18]. Actually, the concept of dissipativity in dynamical systems is a generalization of the Lyapunov stability. The global Lyapunov stability especially can be viewed as a special case of global dissipativity by regarding an equilibrium point as an attractive set [19–21]. Generally speaking, the goal of study on globally dissipative for neural networks is to determine globally attractive sets. Therefore, many initial findings on the global dissipativity [18, 22–30] or Lagrange stability [31–36] analysis of neural networks have been reported. At present, global dissipativity theory has been shown to be an appealing and efficient approach for dealing with the problems such as stability theory, chaos and synchronization theory, system norm estimation, and robust control of neural networks without uncertainties [18, 22, 23, 29, 30] or of those with uncertainties [24–28, 37, 38].

As it is well known, the use of constant fixed delays in models of delayed feedback provides a good approximation in simple circuits consisting of a small number of cells. However, neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. Thus there will be a distribution of conduction velocities along these pathways and a distribution of propagation delays. In these circumstances, the signal propagation is not instantaneous and cannot be modeled with discrete delays. A more appropriate way is to incorporate continuously distributed delays [11, 16, 29, 34, 35, 37, 39–41]. However, these distributed delays are usually unbounded, and this fact motivates our work.

Recently, much attention has also been paid to the robust questions of interval neural networks [7–12, 39, 40, 42, 43]. In [7], global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type has been considered. Bao et al. have investigated the robust stability problem of interval fuzzy Cohen-Grossberg neural networks with piecewise constant argument of generalized type [9]. The global robust passivity analysis for stochastic fuzzy interval neural networks with time-varying delays has been studied in [42]. Balasubramaniam et al. have studied the robust stability for Markovian jumping interval neural networks with discrete and distributed time-varying delays [11]. Xu et al. have studied the stochastic exponential robust stability problem of interval neural networks with reaction-diffusion terms and mixed delays [12]. And the stationary oscillation problem of interval neural networks with discrete and distributed time-varying delays under impulsive perturbations has been studied by using the LMI approach in [40]. Moreover, there are some works on global dissipativity for interval neural networks with time delays such as time-varying delays [24, 25, 27, 38], mixed time-varying delays [26, 37]. Despite the existence of many reported results in the literature, there are still needs for more in-depth and comprehensive investigations. For example, in almost all the existing results, the activation functions of the neural networks are limited to be sigmoid functions, piecewise linear monotone nondecreasing functions with bounded ranges. Moreover, in these recent publications, time-varying delays [44] terms are required to be continuously differentiable, and the derivative is bounded and smaller than one.

Although there has been published a rich literature on dissipativity problem for neural networks, to the best of our knowledge, few authors pay attention to the dissipativity problem for interval neural networks with both discrete and infinity distributed delays. The problem for global robust exponential dissipativity of IRNNs with mixed time-varying delays and general activation functions, particularly made on it by means of LMIs [2, 6, 27, 36, 38, 40], especially remains open. Hence, this gives the motivation of our present investigation. It is worth pointing out that the proposed results are nontrivial because of (1) establishing a generalized differential inequality which is aimed dealing with the infinity distributed delay appearing in IRNNs; (2) proposing a new Lyapunov-Krasovskii functional that should be used for the general activation function skillfully; (3) proving a lemma to handle the appropriate matrices deformation so as to make use of linear matrix inequalities.

In this paper, we focus on the problem of global robust exponential dissipativity for a class of interval recurrent neural networks with general activation functions and mixed delays, which consists of time-varying and infinite distributed delay. For the sake of comparison, Lyapunov function and Lyapunov-Krasovskii functional are constructed, respectively, which can be used to handle the interval uncertain terms masterly by the LMI approach. What is more, when the LMIs-based uncertain parameters are feasible, several sufficient conditions are established to guarantee the global robust exponential dissipativity for the addressed IRNNs. And the specific estimation of positive invariant and global exponential attractive sets is also put forward. The purpose of this paper is trebling. First, we demonstrate two important Lemmas which play a vital role in the later theorems. Second, we tackle the problem of global robust exponential dissipativity for IRNNs with both time-varying and infinite distributed delays based on the general activation functions. Third, the results are performed in LMIs, which will be efficiently solved by the MATLAB LMI Toolbox [45] and compared with those presented in [26, 36, 37]. The rest of this paper is organized as follows. In the next section, some preliminaries, including some definitions, assumptions, and significant lemmas, will be described. Section 3 will state the main results. Section 4 will present two illustrative examples to verify the main results, and finally a summery will be given in Section 5.

Notations. Throughout this paper, 𝐼 represents the unit matrix; in 𝑅⁺=[0,∞), the symbols 𝑅^𝑛 and 𝑅^𝑛×𝑚 stand, respectively, for the 𝑛-dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices. 𝐴^𝑇 and 𝐴⁻¹ denote the matrix transpose and matrix inverse. 𝐴>0 or 𝐴<0 denotes that the matrix 𝐴 is a symmetric and positive definite or negative definite matrix. Meanwhile, 𝐴<𝐵 indicates 𝐴−𝐵<0 and ‖∗‖ is the Euclidean vector norm. When 𝑥 is a variable, ‖𝑥‖=∑^𝑛_𝑖=1|𝑥_𝑖|. [•]^† denotes the floor function and Λ={1,2,…,𝑛}. Moreover, in symmetric block matrices, we use an asterisk “*” to represent a term that is induced by symmetry and diag{⋯} stands for a block-diagonal matrix.

2. Preliminaries

The interval recurrent neural networks with infinity distributed delays are described by the following equation group: 𝑥̇(𝑡)=−𝐷𝑥(𝑡)+𝐴𝑔(𝑥(𝑡))+𝐵𝑔(𝑥(𝑡−𝜏(𝑡)))+𝐶^𝑡_−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠+𝑈, (1) where 𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^𝑇 is the neuron state vector of the neural network; 𝑈=(𝑈₁,…,𝑈_𝑛)^𝑇 is an external input; 𝜏(𝑡) is the transmission delay of the neural networks, which is time varying and satisfies 0≤𝜏(𝑡)≤𝜏, where 𝜏 is a positive constant; 𝑔(𝑥(⋅))=(𝑔₁(𝑥₁(⋅)),…,𝑔_𝑛(𝑥_𝑛(⋅)))^𝑇 represents the neuron activation function, and ℎ(⋅)=diag{ℎ₁(⋅),…,ℎ_𝑛(⋅)} represents the delay kernel function. The matrices 𝐷= diag{𝑑₁,…,𝑑_𝑛}, 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛, and 𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛 are some unknown diagonal matrix, connection weight matrix, the delayed weight matrix, and the distributively delayed connection weight matrix, respectively, satisfying 𝐷∈𝔻,𝐴∈𝔸,𝐵∈𝔹,𝐶∈ℂ, (2) where 𝔻=[𝐷,𝐷]∶={diag{𝑑₁,…,𝑑_𝑛}∣0<𝑑_𝑖≤𝑑_𝑖≤𝑑_𝑖,𝑖∈Λ},𝔸=[𝐴,𝐴]∶= {(𝑎_𝑖𝑗)_𝑛×𝑛∣𝑎_𝑖≤𝑎_𝑖≤𝑎_𝑖, 𝑖,𝑗∈Λ},𝔹=[𝐵,𝐵]∶={(𝑏_𝑖𝑗)_𝑛×𝑛∣𝑏_𝑖≤𝑏_𝑖≤𝑏_𝑖,𝑖,𝑗∈Λ}, ℂ=[𝐶,𝐶]∶={(𝑐_𝑖𝑗)_𝑛×𝑛∣𝑐_𝑖≤𝑐_𝑖≤𝑐_𝑖,𝑖,𝑗∈Λ} with 𝐷=diag{𝑑₁,…,𝑑_𝑛},𝐷=diag{𝑑₁,…,𝑑_𝑛},𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐵=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐵=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐶= (𝑎_𝑖𝑗)_𝑛×𝑛, 𝐶=(𝑎_𝑖𝑗)_𝑛×𝑛.

In addition, let Ω∶=diag𝛿₁₁,…,𝛿_1𝑛,…,𝛿_𝑛1,…,𝛿_𝑛𝑛∈𝑅^𝑛2×𝑛2∣𝛿_𝑖𝑗,𝐷≤1,𝑖,𝑗∈Λ₀=𝐷+𝐷2,𝐴₀=𝐴+𝐴2,𝐵₀=𝐵+𝐵2,𝐶₀=𝐶+𝐶2,𝛼_{𝑖𝑗𝑛×𝑛}=𝐷−𝐷2,𝛽_{𝑖𝑗𝑛×𝑛}=𝐴−𝐴2,𝛾_{𝑖𝑗𝑛×𝑛}=𝐵−𝐵2,𝜗_{𝑖𝑗𝑛×𝑛}=𝐶−𝐶2,𝑀₁=√𝛼₁₁𝑒₁,…,√𝛼_1𝑛𝑒₁,…,√𝛼_𝑛1𝑒_𝑛,…,√𝛼_𝑛𝑛𝑒_{𝑛𝑛×𝑛2},𝑀₂=𝛽₁₁𝑒₁,…,𝛽_1𝑛𝑒₁,…,𝛽_𝑛1𝑒_𝑛,…,𝛽_𝑛𝑛𝑒_{𝑛𝑛×𝑛2},𝑀₃=√𝛾₁₁𝑒₁,…,√𝛾_1𝑛𝑒₁,…,√𝛾_𝑛1𝑒_𝑛,…,√𝛾_𝑛𝑛𝑒_{𝑛𝑛×𝑛2},𝑀₄=𝜗₁₁𝑒₁,…,𝜗_1𝑛𝑒₁,…,𝜗_𝑛1𝑒_𝑛,…,𝜗_𝑛𝑛𝑒_{𝑛𝑛×𝑛2},𝐽₁=√𝛼₁₁𝑒₁,…,√𝛼_1𝑛𝑒_𝑛,…,√𝛼_𝑛1𝑒₁,…,√𝛼_𝑛𝑛𝑒_{𝑛𝑛2×𝑛},𝐽₂=𝛽₁₁𝑒₁,…,𝛽_1𝑛𝑒_𝑛,…,𝛽_𝑛1𝑒₁,…,𝛽_𝑛𝑛𝑒_{𝑛𝑛2×𝑛},𝐽₃=√𝛾₁₁𝑒₁,…,√𝛾_1𝑛𝑒_𝑛,…,√𝛾_𝑛1𝑒₁,…,√𝛾_𝑛𝑛𝑒_{𝑛𝑛2×𝑛},𝐽₄=𝜗₁₁𝑒₁,…,𝜗_1𝑛𝑒_𝑛,…,𝜗_𝑛1𝑒₁,…,𝜗_𝑛𝑛𝑒_{𝑛𝑛2×𝑛}, (3) where 𝑒_𝑖∈𝑅^𝑛 denotes the column vector with 𝑖th element to be 1 and others to be 0.

By some simple calculations, one can transform system (1) into the following form: 𝑥̇(𝑡)=−𝐷₀+𝑀₁Ω₁𝐽₁𝑥(𝑡)+𝐴₀+𝑀₂Ω₂𝐽₂𝑔(𝑥(𝑡+𝐵))₀+𝑀₃Ω₃𝐽₃𝑔(𝑥((𝑡+𝐶𝑡−𝜏)))₀+𝑀₄Ω₄𝐽₄^𝑡_−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠+𝑈, (4) Or, equivalently, 𝑥̇(𝑡)=−𝐷₀𝑥(𝑡)+𝐴₀𝑔(𝑥(𝑡))+𝐵₀𝑔(𝑥(𝑡−𝜏(𝑡)))+𝐶₀^𝑡_−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠+𝕄Ψ(𝑡)+𝑈, (5) where 𝕄=𝑀₁,𝑀₂,𝑀₃,𝑀_{4𝑛×4𝑛2},Ω_𝑖Ψ(𝑡)=Ω∈Ω,𝑖=1,2,3,4,₁𝐽₁0000Ω₂𝐽₂0000Ω₃𝐽₃0000Ω₄𝐽₄×𝑥(𝑡)𝑔(𝑥(𝑡𝑔(𝑥((𝑡∫))𝑡−𝜏)))_𝑡−∞ℎ()𝑔(𝑥(𝑠.𝑡−𝑠))𝑑𝑠 (6)

In this paper, the system (1) is supplemented the initial condition given by 𝑥(𝑠)=𝜙(𝑠),𝑠∈(−∞,0], and 𝑖∈Λ, where 𝜙(⋅)∈𝒞, 𝒞 denotes real-valued continuous functions defined on (−∞,0]. Here, it is assumed that, for any initial condition 𝜙(⋅)∈𝒞, there exists at least one solution of model (1). As usual, we will also assume that 𝑔(0)=0 for all 𝑡∈𝑅⁺ in this paper.

For further discussion, the following assumptions and lemmas are needed.

(A1) The activation function 𝑔 satisfies 𝑔(0)=0, and 𝑙⁻_𝑗≤𝑔_𝑗(𝑥)−𝑔_𝑗𝑦𝑥−𝑦≤𝑙⁺_𝑗, (7) for all 𝑥≠𝑦,𝑥,𝑦∈𝑅, where 𝑙⁺_𝑗 and 𝑙⁻_𝑗,𝑗∈Λ, are some real constants.

(A2) The delay kernels ℎ_𝑗(𝑡),𝑗∈Λ are some real value nonnegative continuous functions defined in (−∞,0] and satisfy ℎ_𝑗(𝑡)≤𝕙(𝑡),𝑗∈Λ, ^∞₀ℎ_𝑗(𝑡)𝑑𝑡=1,^∞₀𝕙(𝑡)𝑑𝑡=𝕙,_∞0𝕙(𝑡)𝑒^𝜚𝑡𝑑𝑡=̇𝕙^⋆<∞, (8) in which 𝕙(𝑡) corresponds to some nonnegative function defined in (−∞,0]; constants 𝜚,𝕙, and 𝕙^⋆ are some positive numbers.

Next, we first introduce the definitions of global robust exponential dissipativity for interval recurrent neural networks (1) or (5) and then state the notation of the upper right Dini derivative and some preliminary lemmas, which are needed to prove our main results.

Definition 1 (see [19]). If there exists a compact set Ω∈ℛ^𝑛 such that, for all 𝑠∈[−∞,𝑡₀], for all 𝑥(𝑠)∈ℛ^𝑛⧵Ω, lim_𝑡→+∞𝜌(𝑥(𝑡),Ω)=0; then Ω is said to be a globally attractive set of (1) or (5), where ℛ^𝑛⧵Ω is the complement set of Ω. A set Ω is called positive invariant set of (1) or (5), if, for all 𝑠∈[−∞,𝑡₀], for all 𝑥(𝑠)∈ℛ^𝑛⧵Ω implies 𝑥(𝑡)∈Ω for 𝑡≥𝑡₀.

Definition 2 (see [19]). The neural network defined by (1) or (5) is called a globally exponentially dissipative system, if there exists a radially unbounded and positive definite Lyapunov function 𝑉(𝑥(𝑡)), which satisfies 𝑉(𝑥(𝑡))≥‖𝑥‖^𝛼, where 𝛼>0 is a constant, and constants 𝜁>0,𝛽>0, such that for 𝑉(𝑥(𝑡₀))>𝜁,𝑉(𝑥(𝑡))>𝜁,𝑡≥𝑡₀, the inequality 𝑉(𝑥(𝑡))−𝜁≤(𝑉(𝑥(𝑡₀))−𝜁)exp{−𝛽(𝑡−𝑡₀)} always holds. And {𝑥∣𝑉(𝑥(𝑡))≤𝜁} is said to be a globally exponentially attractive set of (1) or (5), where 𝑉(𝑥(𝑡₀))≥𝑉(𝑥(𝑡₀)) and 𝑉(𝑥(𝑡₀)) is a constant.

Definition 3. The neural network defined by (1) or (5) is a globally robustly exponentially dissipative system if the system is globally exponentially dissipative for all 𝐷∈𝔻,𝐴∈𝔸,𝐵∈𝔹, and 𝐶∈ℂ.

Definition 4. For any function 𝑓(𝑡), we define its right-hand derivative as 𝐷⁺𝑓(𝑡)=lim_𝑠→0+𝑓(𝑡+𝑠)−𝑓(𝑡)𝑠. (9)

Lemma 5 (see [40]). For any vectors 𝑎,𝑏∈𝑅^𝑛, the inequality ±2𝑎^𝑇𝑏≤𝑎^𝑇𝑋𝑎+𝑏^𝑇𝑋⁻¹𝑏 (10) holds, in which 𝑋 is any 𝑛×𝑛 matrix with 𝑋>0.

Lemma 6 (Schur complement [13]). For a given matrix 𝑆=^𝑆11𝑆12_{𝑆𝑇12𝑆22}, with 𝑆₁₁=𝑆^𝑇₁₁, 𝑆₂₂=𝑆^𝑇₂₂, then the following conditions are equivalent: (1)𝑆<0, (2)𝑆₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0, (3)𝑆₁₁<0,𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0.

The following two lemmas will be used for deriving our main results.

Lemma 7. Let 𝑝,𝑞,𝑟, and 𝜏 denote nonnegative constants, and function 𝑓∈𝐶(𝑅,𝑅⁺) satisfies the scalar differential inequality 𝐷⁺𝑓(𝑡)≤−𝑝𝑓(𝑡)+𝑞sup_{𝑡−𝜏≤𝑠≤𝑡}𝑓(𝑠)+𝑟^𝜎₀𝑘(𝑠)𝑓(𝑡−𝑠)𝑑𝑠,𝑡≥𝑡₀, (11) where 0<𝜎≤+∞, 𝑘(⋅)∈𝐶([0,𝜎],𝑅⁺) satisfies ∫^𝜎₀𝑘(𝑠)𝑒^𝜂0𝑠𝑑𝑠<∞ for some positive constant 𝜂₀>0 in the case when 𝜎=+∞. Moreover, when 𝜎=+∞, the interval [𝑡−𝜎,𝑡] is understood to be replaced by (−∞,𝑡]. Assume that 𝑝>𝑞+𝑟^𝜎₀𝑘(𝑠)𝑑𝑠. (12) Then 𝑓(𝑡)≤𝑓(𝑡₀)exp(−𝜆(𝑡−𝑡₀)) for all 𝑡≥𝑡₀, where 𝑓(𝑡₀)=sup_{𝑡0−max{𝜎,𝜏}≤𝑠≤𝑡0}𝑓(𝑠), and 𝜆∈(0,𝜂₀) satisfies the inequality 𝜆<𝑝−𝑞𝑒^𝜆𝜏−𝑟^𝜎₀𝑘(𝑠)𝑒^𝜆𝑠𝑑𝑠. (13)

Proof. We first note that condition (11) implies that there exists a scalar 𝜆∈(0,𝜂₀) such that inequality (12) holds.
Consider the following equation: 𝑄𝛾=𝑝−𝑞𝑒^𝛾𝜏−𝑟^𝜎₀𝑘(𝑠)𝑒^𝛾𝑠𝑑𝑠−𝛾. (14) Because 𝑄(0)=𝑝−𝑞−𝑟∫^𝜎₀𝑘(𝑠)𝑑𝑠>0 and 𝑄^′(𝛾)=−𝑞𝜏𝑒^𝛾𝜏−𝑟∫^𝜎₀𝑠𝑘(𝑠)𝑒^𝛾𝑠𝑑𝑠−1<0, we follow that 𝑄(𝛾) is a strictly monotone decreasing function. Meanwhile, we also notice that there always exists a positive constant 𝜂₀ such that 𝑄(𝜂₀)=𝑝−𝑞𝜏𝑒^𝜂0𝜏−𝑟∫^𝜎₀𝑘(𝑠)𝑒^𝜂0𝑠𝑑𝑠−𝜂₀<0. Therefore, by view of the mean value theorem, there is a constant 0<𝜆^∗<𝜂₀ such that 𝑄(𝜆^∗)=0. Correspondingly, there exists a constant 0<𝜆<𝜆^∗<𝜂₀ such that 𝑄(𝜆)>0, namely, 𝜆<𝑝−𝑞𝑒^𝜆𝜏−𝑟∫^𝜎₀𝑘(𝑠)𝑒^𝜆𝑠𝑑𝑠.
Next, we will show 𝑓(𝑡)≤𝑓(𝑡₀)exp(−𝜆(𝑡−𝑡₀)) for all 𝑡≥𝑡₀. In order to do this, let Ψ(𝑡)=𝑓(𝑡)𝑒^{𝜆(𝑡−𝑡0)},𝑡≥𝑡₀,𝑓(𝑡),𝑡₀−max{𝜎,𝜏}≤𝑡≤𝑡₀. (15) Now we only need to show that Ψ(𝑡)≤𝑓(𝑡₀),𝑡≥𝑡₀. It is clear that Ψ(𝑡)≤𝑓(𝑡₀) for 𝑡₀−max{𝜎,𝜏}≤𝑡≤𝑡₀ by the definition of 𝑓. Next, we can prove that Ψ(𝑡)≤𝑓(𝑡₀) for 𝑡≥𝑡₀. Suppose, on the contrary, that there exist some 𝑡∈[𝑡₀,+∞) such Ψ(𝑡)>𝑓(𝑡₀). Let 𝑡^∗=inf{𝑡∈ [𝑡₀,+∞),Ψ(𝑡)>𝑓(𝑡₀)}; then Ψ𝑡^∗=𝑓𝑡₀,Ψ(𝑡)≤𝑓𝑡₀,𝑡∈𝑡₀−max{𝜎,𝜏},𝑡^∗,𝐷⁺Ψ𝑡^∗≥0. (16) Suppose that 𝑓(𝜃_𝑡∗)=sup_{𝑡∗−𝜏≤𝑠≤𝑡∗}𝑓(𝑠),𝜃_𝑡∗∈[𝑡^∗−𝜏,𝑡^∗]. Calculating the upper right Dini derivative 𝐷⁺Ψ(𝑡) along the solution of (5), by (12) and (13), we get 𝐷⁺Ψ(𝑡)|_𝑡=𝑡∗=𝐷⁺𝑓𝑡^∗𝑒^{𝜆(𝑡∗−𝑡0)}+𝜆𝑓𝑡^∗𝑒^{𝜆(𝑡∗−𝑡0)}≤−𝑝𝑓𝑡^∗+𝑞𝑓𝜃_𝑡∗+𝑟^𝜎₀𝑘(𝑠)𝑓𝑡^∗−𝑠𝑑𝑠𝑒^{𝜆(𝑡∗−𝑡0)}+𝜆𝑓𝑡^∗𝑒^{𝜆(𝑡∗−𝑡0)}≤𝜆−𝑝𝑓𝑡^∗+𝑞𝑓𝜃_𝑡∗+𝑟^𝜎₀𝑘(𝑠)𝑓𝑡^∗−𝑠𝑑𝑠𝑒^{𝜆(𝑡∗−𝑡0)}<−𝑞𝑒^𝜆𝜏𝑓𝑡^∗𝑒^{𝜆(𝑡∗−𝑡0)}+𝑞𝑒^{𝜆(𝑡∗−𝜃𝑡∗)}𝑓𝜃_𝑡∗𝑒^{𝜆(𝜃𝑡∗−𝑡0)}−𝑟𝑓𝑡^∗𝑒^{𝜆(𝑡∗−𝑡0)𝜎}₀𝑘(𝑠)𝑒^𝜆𝑠𝑑𝑠+𝑟^𝜎₀𝑘(𝑠)𝑒^𝜆𝑠𝑓𝑡^∗−𝑠𝑒^{𝜆(𝑡∗−𝑠−𝑡0)}𝑑𝑠≤−𝑞𝑒^𝜆𝜏Ψ𝑡^∗+𝑞𝑒^𝜆𝜏Ψ𝜃_𝑡∗−𝑟Ψ𝑡^∗𝜎₀𝑘(𝑠)𝑒^𝜆𝑠𝑑𝑠+𝑟^𝜎₀𝑘(𝑠)𝑒^𝜆𝑠Ψ𝑡^∗−𝑠𝑑𝑠≤−𝑞𝑒^𝜆𝜏Ψ𝑡^∗+𝑞𝑒^𝜆𝜏𝑓𝑡₀−𝑟Ψ𝑡^∗𝜎₀𝑘(𝑠)𝑒^𝜆𝑠𝑑𝑠+𝑟𝑓𝑡₀^𝜎₀𝑘(𝑠)𝑒^𝜆𝑠𝑑𝑠=0, (17) which contradicts (13). So we have proven Ψ(𝑡)≤𝑓(𝑡₀) for all 𝑡≥𝑡₀, that is; 𝑓(𝑡)≤𝑓(𝑡₀)exp(−𝜆(𝑡−𝑡₀)) for all 𝑡≥𝑡₀, where 𝜆 satisfies (12). This completes the proof.

Remark 8. It should be noted that it is only require that the function 𝑘(⋅) satisfies the assumption: 𝑘(⋅)∈𝐶([0,𝜎],𝑅⁺) is integrable if 𝜎<+∞.

Lemma 9. Given constant matrices 𝐴₁, 𝐴₂, 𝐴₃, 𝐴₄, 𝐵₁, 𝐵₂, 𝐵₃, and 𝐵₄∈𝑅^𝑛×𝑛 and appropriate reversible matrices 𝑋₁,𝑋₂,𝑋₃, and 𝑋₄, let Σ₁=𝐴₁𝐵₁𝑋₁⁻¹𝐴₁𝐵₁^𝑇+𝐴₂𝐵₂𝑋₂⁻¹𝐴₂𝐵₂^𝑇+𝐴₃𝐵₃𝑋₃⁻¹𝐴₃𝐵₃^𝑇+𝐴₄𝐵₄𝑋₄⁻¹𝐴₄𝐵₄^𝑇,Σ₂=𝐴₁𝐴₂𝐴₃𝐴₄𝐵₁𝐵₂𝐵₃𝐵₄𝑋₁⁻¹0000𝑋₂⁻¹0000𝑋₃⁻¹0000𝑋₄⁻¹×𝐴₁𝐴₂𝐴₃𝐴₄𝐵₁𝐵₂𝐵₃𝐵₄^𝑇. (18) Then Σ₁=Σ₂.

Proof. Firstly, we discuss Σ₁, and Σ₁=𝐴₁𝑋₁⁻¹𝐴^𝑇₁𝐴₁𝑋₁⁻¹𝐵^𝑇₁𝐵₁𝑋₁⁻¹𝐴^𝑇₁𝐵₁𝑋₁⁻¹𝐵^𝑇₁+𝐴₂𝑋₂⁻¹𝐴^𝑇₂𝐴₂𝑋₂⁻¹𝐵^𝑇₂𝐵₂𝑋₂⁻¹𝐴^𝑇₂𝐵₂𝑋₂⁻¹𝐵^𝑇₂+𝐴₃𝑋₃⁻¹𝐴^𝑇₃𝐴₃𝑋₃⁻¹𝐵^𝑇₃𝐵₃𝑋₃⁻¹𝐴^𝑇₃𝐵₃𝑋₃⁻¹𝐵^𝑇₃+𝐴₄𝑋₄⁻¹𝐴^𝑇₄𝐴₄𝑋₄⁻¹𝐵^𝑇₄𝐵₄𝑋₄⁻¹𝐴^𝑇₄𝐵₄𝑋₄⁻¹𝐵^𝑇₄=Δ₁₁Δ₁₂∗Δ₂₂, (19) where Δ₁₁=𝐴₁𝑋₁⁻¹𝐴^𝑇₁+𝐴₂𝑋₂⁻¹𝐴^𝑇₂+𝐴₃𝑋₃⁻¹𝐴^𝑇₃+𝐴₄𝑋₄⁻¹𝐴^𝑇₄; Δ₁₂=𝐴₁𝑋₁⁻¹𝐵^𝑇₁+𝐴₂𝑋₂⁻¹𝐵^𝑇₂+𝐴₃𝑋₃⁻¹𝐵^𝑇₃+ 𝐴₄𝑋₄⁻¹𝐵^𝑇₄; Δ₂₂=𝐵₁𝑋₁⁻¹𝐵^𝑇₁+𝐵₂𝑋₂⁻¹𝐵^𝑇₂+𝐵₃𝑋₃⁻¹𝐵^𝑇₃+𝐵₄𝑋₄⁻¹𝐵^𝑇₄.
Calculating Σ₂, we obtain Σ₂=𝐴₁𝑋₁⁻¹𝐴₂𝑋₂⁻¹𝐴₃𝑋₃⁻¹𝐴₄𝑋₄⁻¹𝐵₁𝑋₁⁻¹𝐵₂𝑋₂⁻¹𝐵₃𝑋₃⁻¹𝐵₄𝑋₄⁻¹×𝐴^𝑇₁𝐵^𝑇₁𝐴^𝑇₂𝐵^𝑇₂𝐴^𝑇₃𝐵^𝑇₃𝐴^𝑇₄𝐵^𝑇₄=Δ₁₁Δ₁₂∗Δ₂₂. (20) Comparing the above equations, we can know Σ₁=Σ₂. The proof is finished.

Lemma 10 (see [46]). The following inequality is true: 0≤^𝑥𝑖₀^(𝑡)𝑔_𝑖(𝑠)−𝑙⁻_𝑖𝑠𝑑𝑠≤𝑔_𝑖𝑥_𝑖(𝑡)−𝑙⁻_𝑖𝑥_𝑖(𝑡)𝑥_𝑖(𝑡). (21)

3. Main Results

In this section, we shall derive some sufficient conditions for globally robust exponentially dissipative for the IRNNs (1) or (5) with general activation functions, continuous time-varying delay, and infinity distributed time delay by means of applying the lemmas in Section 2 repeatedly.

3.1. Results Using Lyapunov Functions

In this part, sufficient conditions for global robust exponential dissipativity of (1) or (5) are got by using Lyapunov functions and inequality techniques.

Theorem 11. Assume that Assumptions (A1)-(A2) hold; if there exist three constants 𝛽_𝑖>0,𝑖=1,2, and 3, seven positive diagonal matrices 𝑄₁,𝑄₂, and 𝑄₃∈𝑅^𝑛×𝑛,𝑄₄,𝑄₅,𝑄₆, and 𝑄₇∈𝑅^𝑛2×𝑛2, and two positive definite matrices 𝑃,𝑄₈∈𝑅^𝑛×𝑛 such that the following inequalities hold: Π₁𝑃𝐴₀𝑃𝐵₀𝑃𝐶₀𝑃𝕄𝑃∗𝑄₁0000∗∗𝑄₂000∗∗∗𝑄₃00∗∗∗∗Π₂0∗∗∗∗∗𝑄₈𝛽≥0,₂𝑃≥𝑊𝑄₂𝑊+𝑊𝐽^𝑇₃𝑄₆𝐽₃𝛽𝑊,₃𝑃≥𝑊𝑄₃𝑊+𝑊𝐽^𝑇₄𝑄₇𝐽₄𝛽𝑊,₁>𝛽₂+𝛽₃𝕙, (22) where Π₁=𝑃𝐷₀+𝐷₀𝑃−𝑊𝑄₁𝑊−𝐽^𝑇₁𝑄₄𝐽₁−𝑊𝐽^𝑇₂𝑄₅𝐽₂𝑊−𝛽₁𝑃,𝑊=diag{𝑤₁,…,𝑤_𝑛}, 𝑤_𝑗= max{|𝑙⁻_𝑗|,|𝑙⁺_𝑗|}, Π₂=𝑄₄0000𝑄₅0000𝑄₆0000𝑄₇. (23) Then the neural network defined by (1) or (5) is a globally robust exponentially dissipative system, and the set Φ={𝑥∈𝑅^𝑛∣𝑥^𝑇(𝑡)𝑃𝑥(𝑡)≤𝑈^𝑇𝑄₈𝑈/(𝛽₁−𝛽₂−𝛽₃𝕙)} is a positive invariant and globally exponential attractive set.

Proof. Now, we consider the following Lyapunov function: 𝑉(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑃𝑥(𝑡). (24)
Calculating the derivative of 𝑉(𝑥(𝑡)) along the trajectories of (5), we can obtain 𝑑𝑉(𝑥(𝑡))𝑑𝑡₍₂₎≤2𝑥^𝑇(𝑡)𝑃−𝐷₀𝑥(𝑡)+𝐴₀𝑔(𝑥(𝑡))+𝐵₀𝑔(𝑥(𝑡−𝜏(𝑡)))+𝐶₀^𝑡_−∞ℎ()𝑔(𝑥(𝑠(𝑡)𝑡−𝑠))𝑑𝑠+𝕄Ψ+𝑈 (25)
From Assumption (A1) and Lemma 5, we know that there exist three positive diagonal matrices 𝑄₁,𝑄₂, and 𝑄₃∈𝑅^𝑛×𝑛 and a positive definite matrix 𝑄₈∈𝑅^𝑛×𝑛 such that the following inequalities hold: 2𝑥^𝑇(𝑡)𝑃𝐴₀𝑔(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝑃𝐴₀𝑄₁⁻¹𝐴^𝑇₀𝑃𝑥(𝑡)+𝑔^𝑇(𝑥(𝑡))𝑄₁𝑔(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝑃𝐴₀𝑄₁⁻¹𝐴^𝑇₀𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑊𝑄₁𝑊𝑥(𝑡),2𝑥^𝑇(𝑡)𝑃𝐵₀𝑔(𝑥(𝑡−𝜏(𝑡)))≤𝑥^𝑇(𝑡)𝑃𝐵₀𝑄₂⁻¹𝐵^𝑇₀𝑃𝑥(𝑡)+𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑄₂𝑔(𝑥(𝑡−𝜏(𝑡)))≤𝑥^𝑇(𝑡)𝑃𝐵₀𝑄₂⁻¹𝐵^𝑇₀𝑃𝑥(𝑡)+𝑥^𝑇(𝑡−𝜏(𝑡))𝑊𝑄₂𝑊𝑥(𝑡−𝜏(𝑡)),2𝑥^𝑇(𝑡)𝑃𝐶₀^𝑡_−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠≤𝑥^𝑇(𝑡)𝑃𝐶₀𝑄₃⁻¹𝐶^𝑇₀𝑃𝑥(𝑡)+^𝑡_−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠^𝑇×𝑄₃^𝑡_−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠, (26)(27)(28) and by well-known Cauchy-Schwarz inequality and Assumption (A2), we get ^𝑡_−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠^𝑇𝑄₃^𝑡_−∞ℎ()𝑔(𝑥(𝑠=𝑡−𝑠))𝑑𝑠_𝑛𝑗=1𝑞_𝑗⁽³⁾₀^+∞ℎ_𝑗(𝑢)𝑔_𝑗𝑥_𝑗(𝑡−𝑢)𝑑𝑢²≤^𝑛_𝑗=1𝑞_𝑗⁽³⁾₀^+∞ℎ_𝑗(𝑢)𝑑𝑢₀^+∞ℎ_𝑗(𝑢)𝑔²_𝑗𝑥_𝑗(≤𝑡−𝑢)𝑑𝑢_𝑛𝑗=1𝑞_𝑗⁽³⁾₀^+∞𝕙(𝑢)𝑤²_𝑗𝑥²_𝑗(𝑡−𝑢)𝑑𝑢=₀^+∞𝕙(𝑢)^𝑛_𝑗=1𝑞_𝑗⁽³⁾𝑤²_𝑗𝑥²_𝑗()=𝑡−𝑢𝑑𝑢_0+∞𝕙(𝑠)𝑥^𝑇(𝑡−𝑠)𝑊𝑄₃𝑊𝑥(𝑡−𝑠)𝑑𝑠, (29) which implies that 2𝑥^𝑇(𝑡)𝑃𝐶₀^𝑡_−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠≤𝑥^𝑇(𝑡)𝑃𝐶₀𝑄₃⁻¹𝐶^𝑇₀(𝑡)+𝑃𝑥_0+∞𝕙(𝑠)𝑥^𝑇(𝑡−𝑠)𝑊𝑄₃𝑊𝑥(𝑡−𝑠)𝑑𝑠,2𝑥^𝑇(𝑡)𝑃𝑈≤𝑥^𝑇(𝑡)𝑃𝑄₈⁻¹𝑃𝑥(𝑡)+𝑈^𝑇𝑄₈𝑈. (30)(31) In view of the definition of Ω, we have the following inequality: Ψ^𝑇(𝑡)Ψ(𝑡)≤𝑥^𝑇(𝑡)𝐽^𝑇₁𝐽₁𝑥(𝑡)+𝑔^𝑇(𝑥(𝑡))𝐽^𝑇₂𝐽₂𝑔(𝑥(𝑡))+𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝐽^𝑇₃𝐽₃𝑔(𝑥((𝑡+𝑡−𝜏)))_𝑡−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠^𝑇𝐽^𝑇₄𝐽₄×^𝑡_−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠. (32) Considering Lemma 5 and (31), we derive 2𝑥^𝑇(𝑡)𝑃𝕄Ψ(𝑡)≤𝑥^𝑇(𝑡)𝑃𝕄Π₂⁻¹𝕄^𝑇𝑃𝑥(𝑡)+Ψ^𝑇(𝑡)Π₂Ψ(𝑡)≤𝑥^𝑇(𝑡)𝑃𝕄Π₂⁻¹𝕄^𝑇𝑃+𝐽^𝑇₁𝑄₄𝐽₁+𝑊𝐽^𝑇₂𝑄₅𝐽₂𝑊𝑥(𝑡)+𝑥^𝑇(𝑡−𝜏(𝑡))𝑊𝐽^𝑇₃𝑄₆𝐽₃𝑊𝑥((𝑡+𝑡−𝜏))_0+∞𝕙(𝑠)𝑥^𝑇(𝑡−𝑠)𝑊𝐽^𝑇₄𝑄₇𝐽₄𝑊𝑥(𝑡−𝑠)𝑑𝑠. (33)
Now, adding the terms on the right of (26)–(30) and (32) to (22), considering conditions (17), and making use of Lemma 6, we can obtain that 𝑑𝑉(𝑥(𝑡))𝑑𝑡₍₂₎≤𝑥^𝑇(𝑡)−𝑃𝐷₀−𝐷₀𝑃+𝑃𝐴₀𝑄₁⁻¹𝐴^𝑇₀𝑃+𝑊𝑄₁𝑊+𝑃𝐵₀𝑄₂⁻¹𝐵^𝑇₀𝑃+𝑃𝐶₀𝑄₃⁻¹𝐶^𝑇₀𝑃+𝑃𝑄₈⁻¹𝑃+𝑃𝕄Π₂⁻¹𝕄^𝑇𝑃+𝐽^𝑇₁𝑄₄𝐽₁+𝑊𝐽^𝑇₂𝑄₅𝐽₂𝑊𝑥(𝑡)+𝑥^𝑇(𝑡−𝜏(𝑡))𝑊𝑄₂𝑊+𝑊𝐽^𝑇₃𝑄₆𝐽₃𝑊𝑥((𝑡+𝑡−𝜏))_0+∞𝕙(𝑠)𝑥^𝑇(𝑡−𝑠)𝑊𝑄₃𝑊+𝑊𝐽^𝑇₄𝑄₇𝐽₄𝑊,𝑥(𝑡−𝑠)𝑑𝑠+𝑈^𝑇𝑄₈𝑈≤−𝛽₁𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝛽₂𝑥^𝑇(𝑡−𝜏(𝑡))𝑃𝑥(𝑡−𝜏(𝑡))+𝛽₃₀^+∞𝕙(𝑠)𝑥^𝑇(𝑡−𝑠)𝑃𝑥(𝑡−𝑠)𝑑𝑠+𝑈^𝑇𝑄₈𝑈≤−𝛽₁𝑉(𝑥(𝑡))+𝛽₂𝑉𝑥^𝑇(𝑡−𝜏(𝑡))+𝛽₃₀^+∞𝕙(𝑠)𝑉(𝑥(𝑡−𝑠))𝑑𝑠+𝑈^𝑇𝑄₈𝑈. (34) Transforming (33) into the following inequality, we get 𝑑𝑉(𝑥(𝑡))−𝜂𝑑𝑡₍₂₎≤−𝛽₁𝑉(𝑥(𝑡))−𝜂+𝛽₂sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑥(𝑠))−𝜂+𝛽₃₀^+∞𝕙(𝑠)𝑉(𝑥(𝑡−𝑠))−𝜂𝑑𝑠,𝑡≥𝑡₀, (35) where 𝜂=𝑈^𝑇𝐻𝑈/(𝛽₁−𝛽₂−𝛽₃𝕙).
From formula (34), we can know that it satisfies the (11) of Lemma 7. Meanwhile, noticing Assumption (A2), it can be deduced that 𝛽₁>𝛽₂+𝛽₃𝕙⇔𝛽₁>𝛽₂+𝛽₃∫₀^+∞𝕙(𝑠)𝑑𝑠. So (12) of Lemma 7 is also satisfied. From this, when 𝑉(𝑥(𝑡))>𝜂,sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑥(𝑠))>𝜂, and sup_{−∞≤𝑠≤𝑡}𝑉(𝑥(𝑠))>𝜂, according to Lemma 7, we are able to derive 𝑉(𝑥(𝑡))−𝜂≤𝑉(𝑥(𝑡))−𝜂exp−𝜆𝑡−𝑡₀, (36) where 𝑉(𝑥(𝑡))=sup_{−∞≤𝑠≤𝑡}𝑉(𝑥(𝑠)),𝜆∈(0,𝜚) satisfies 𝜆<𝛽₁−𝛽₂𝑒^𝜆𝜏−𝛽₃₀^+∞𝕙(𝑠)𝑒^𝜆𝑠𝑑𝑠. (37) Simultaneously, judging by [1], it is easy to prove that there exists a constant 𝛼 such that 𝑉(𝑥(𝑡))≥‖𝑥‖^𝛼. In terms of Definitions 1, 2, and 3, we know that the neural network defined by (1) or (5) is a globally robust exponentially dissipative system, and Φ={𝑥∈𝑅^𝑛∣𝑥^𝑇(𝑡)𝑃𝑥(𝑡)≤ 𝑈^𝑇𝐻𝑈/(𝛽₁−𝛽₂−𝛽₃𝕙)} is a positive invariant and globally exponentially attractive set of system (1) or (5). Hence, the proof of Theorem 11 is completed.

Remark 12. It should be noted that the exponential convergence rate 𝜆 of IRNNs (1) or (5) is also derived in (35). Moreover, one may find that condition 𝛽₁−𝛽₂−𝛽₃𝕙>0 implies that there exists constant 𝜆∈(0,𝜚) such that (1) or (5) holds for any given 𝜏>0.

When 𝐷,𝐴,𝐵, and 𝐶 are some known constant matrices, we have the following simple result.

Corollary 13. Assume that Assumptions (A1)-(A2) hold; then the neural network defined by (1) or (5) is a globally exponential dissipative system if there exist three constants 𝛽_𝑖>0,𝑖=1,2, and 3, three positive diagonal matrices 𝑄₁,𝑄₂, and 𝑄₃∈𝑅^𝑛×𝑛, and two positive definite matrices 𝑃,𝐻∈𝑅^𝑛×𝑛 such that the following inequalities hold: Π₁𝑃𝐴𝑃𝐵𝑃𝐶𝑃∗𝑄₁000∗∗𝑄₂00∗∗∗𝑄₃0𝛽∗∗∗∗𝐻≥0,₂𝑃≥𝑊𝑄₂𝛽𝑊,₃𝑃≥𝑊𝑄₃𝛽𝑊,₁>𝛽₂−𝛽₃𝕙, (38) where Π₁=𝑃𝐷+𝐷𝑃−𝑊𝑄₁𝑊−𝑃𝐻⁻¹𝑃−𝛽₁𝑃, 𝑊=diag{𝑤₁,…,𝑤_𝑛}, 𝑤_𝑗=max{|𝑙⁻_𝑗|,|𝑙⁺_𝑗|}, and the set Φ={𝑥∈𝑅^𝑛∣𝑥^𝑇(𝑡)𝑃𝑥(𝑡)≤𝑈^𝑇𝐻𝑈/(𝛽₁−𝛽₂−𝛽₃𝕙)} is a positive invariant and globally exponential attractive set.

In the IRNNs system (1), when getting rid of the term of infinite distributed delay ∫^𝑡_−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠, we get Corollary 14 based on Theorem 11.

Corollary 14. Assume that Assumption (A1) holds, if there exist three constants 𝛽_𝑖>0,𝑖=1,2, five positive diagonal matrices 𝑄₁,𝑄₂∈𝑅^𝑛×𝑛,𝑄₃,𝑄₄, and 𝑄₅∈𝑅^𝑛2×𝑛2, and two positive definite matrices 𝑃,𝑄₆∈𝑅^𝑛×𝑛 such that the following inequalities hold: Π₁𝑃𝐴₀𝑃𝐵₀𝑃𝕄𝑃∗𝑄₁000∗∗𝑄₂00∗∗∗Π₃0∗∗∗∗𝑄₆𝛽≥0,₂𝑃≥𝑊𝑄₂𝑊+𝑊𝐽^𝑇₃𝑄₅𝐽₃𝛽𝑊,₁>𝛽₂, (39) where Π₁=𝑃𝐷₀+𝐷₀𝑃−𝑊𝑄₁𝑊−𝐽^𝑇₁𝑄₃𝐽₁−𝑊𝐽^𝑇₂𝑄₄𝐽₂𝑊−𝑃𝑄₆⁻¹𝑃−𝛽₁𝑃, 𝑊=diag{𝑤₁,…,𝑤_𝑛}, 𝑤_𝑗=max{|𝑙⁻_𝑗|,|𝑙⁺_𝑗|}, Π₃=𝑄₃000𝑄₄000𝑄₅. (40) Then the neural network defined by (1) or (5) is a globally robust exponentially dissipative system, and the set Φ={𝑥∈𝑅^𝑛∣𝑥^𝑇(𝑡)𝑃𝑥(𝑡)≤𝑈^𝑇𝑄₈𝑈/(𝛽₁−𝛽₂)} is a positive invariant and globally exponential attractive set.

Proof. In front of the course of proof is almost parallel to that of Theorem 11, except for inequality (27) in Theorem 11, here no longer say. In the end, we can also obtain 𝑑𝑉(𝑥(𝑡))−𝜂𝑑𝑡₍₁₎≤−𝛽₁𝑉(𝑥(𝑡))−𝜂+𝛽₂𝑉(𝑥(𝑡))−𝜂,𝑡≥𝑡₀, (41) where 𝜂=𝑈^𝑇𝐻𝑈/(𝛽₁−𝛽₂),𝑉(𝑥(𝑡))=sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑥(𝑠)).
It is noticed that 𝛽₁>𝛽₂; hence according to the famous Halanay Inequality [47], when 𝑉(𝑥(𝑡))>𝜂 and sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑥(𝑠))>𝜂, we are able to derive 𝑉(𝑥(𝑡))−𝜂≤𝑉(𝑥(𝑡))−𝜂exp−𝜆𝑡−𝑡₀, (42) where 𝜆 is the unique positive root of 𝜆=𝛽₁−𝛽₂𝑒^𝜆𝜏. Similarly, it is obtained that Ω= {𝑥∈𝑅^𝑛∣𝑥^𝑇(𝑡)𝑃𝑥(𝑡)≤𝑈^𝑇𝐻𝑈/(𝛽₁−𝛽₂)} is a positive invariant and globally exponential attractive set of system (1). Hence, the proof is gained.

3.2. Results Using Lyapunov-Krasovskii Functionals

In this part, sufficient conditions for global robust exponential dissipativity of (1) or (5) are obtained by using Lyapunov-Krasovskii functional and inequality techniques.

Theorem 15. Assume that Assumptions (A1)-(A2) hold, if there exist three constants 𝛽_𝑖>0,𝑖=1,2, and 3, eight positive diagonal matrices 𝑄,𝑅₁,𝑅₂, and 𝑅₃∈𝑅^𝑛×𝑛, 𝑅₄,𝑅₅,𝑅₆, and 𝑅₇∈𝑅^𝑛2×𝑛2, and two positive definite matrices 𝑃,𝑅₈∈𝑅^𝑛×𝑛 such that the following inequalities hold: Γ<0,𝑊𝑅₂𝑊+𝑊𝐽^𝑇₃𝑅₆𝐽₃𝑊≤𝛽₂𝑃,𝑊𝑅₃𝑊+𝑊𝐽^𝑇₄𝑅₇𝐽₄𝑊≤𝛽₃𝛽𝑃,₂+𝛽₃𝕙<1+𝛽₁,ΘΓ=¹¹Θ₁₂Θ₁₃Θ₁₄Θ₁₅Θ₁₆∗Θ₂₂𝑄𝐵₀𝑄𝐶₀𝑄𝕄𝑄∗∗−𝑅₂000∗∗∗−𝑅₃00∗∗∗∗−Π0∗∗∗∗∗−𝑅₈, (43) where Θ₁₁=𝑃+𝑄(𝐹−𝐿)−𝑃𝐷₀−𝐷₀𝑃+2𝐿𝑄𝐷₀+𝑊𝑅₁𝑊+𝐽^𝑇₁𝑅₄𝐽₁+𝑊𝐽^𝑇₂𝑅₅𝐽₂𝑊, Θ₁₂=𝑃𝐴₀− 𝐿𝑄𝐴₀−𝐷₀𝑄, Θ₁₃=𝑃𝐵₀−𝐿𝑄𝐵₀, Θ₁₄=𝑃𝐶₀−𝐿𝑄𝐶₀, Θ₁₅=𝑃𝕄−𝐿𝑄𝕄, Θ₁₆=𝑃−𝐿𝑄, Θ₂₂=𝑄𝐴₀+ 𝐴^𝑇₀𝑄−𝑅₁, 𝑊=diag{𝑤₁,𝑤₂,…,𝑤_𝑛},𝑤_𝑗=max{|𝑙⁻_𝑗|,|𝑙⁺_𝑗|}, for all 𝑗∈Λ, 𝐿=diag{𝑙⁻₁,𝑙⁻₂, …,𝑙⁻_𝑛},𝐹=diag{𝑙⁺₁,𝑙⁺₂,…,𝑙⁺_𝑛}, 𝑅Π=⁴0000𝑅₅0000𝑅₆0000𝑅₇. (44) Then the neural network defined by (1) or (5) is a globally robust exponentially dissipative system, and the set Ω=𝑥∈𝑅^𝑛∣𝑥^𝑇(𝑡)(𝑡)≤𝑈𝑃𝑥^𝑇𝑅₈𝑈1+𝛽₁−𝛽₂−𝛽₃𝕙 (45) is a positive invariant and globally exponential attractive set of system (1).

Proof. Now, we consider another Lyapunov functional 𝑉(𝑥(𝑡))=𝑥^𝑇𝑃𝑥(𝑡)+2^𝑛_𝑖=1𝑞_𝑖^𝑥𝑖₀^(𝑡)𝑔_𝑖(𝑠)−𝑙⁻_𝑖𝑠𝑑𝑠. (46)
Calculating the derivative of 𝑉(𝑥(𝑡)) along the trajectories of (5) and using Lemma 10, we can obtain 𝑑𝑉(𝑥(𝑡))𝑑𝑡₍₂₎≤2𝑥^𝑇(𝑡)𝑃−𝐷₀𝑥(𝑡)+𝐴₀𝑔(𝑥(𝑡))+𝕄Ψ(𝑡)+𝐵₀𝑔(𝑥(𝑡−𝜏(𝑡)))+𝐶₀^𝑡_−∞ℎ()𝑔(𝑥(𝑠𝑔(𝑥(𝑡(𝑡)𝑡−𝑠))𝑑𝑠+𝑈+2))−𝐿𝑥^𝑇𝑄×−𝐷₀𝑥(𝑡)+𝐴₀𝑔(𝑥(𝑡(𝑡)))+𝕄Ψ+𝐵₀𝑔(𝑥(𝑡−𝜏(𝑡)))+𝐶₀^𝑡_−∞ℎ()𝑔(𝑥(𝑠𝑥𝑡−𝑠))𝑑𝑠+𝑈=2^𝑇(𝑡)𝑃+𝑔^𝑇(𝑥(𝑡))𝑄−𝑥^𝑇(𝑡)×𝐿𝑄−𝐷₀𝑥(𝑡)+𝐴₀𝑔(𝑥(𝑡𝑥))+2^𝑇(𝑡)𝑃+𝑔^𝑇(𝑥(𝑡))𝑄−𝑥^𝑇(𝑡)𝐿𝑄×𝐵₀𝑔(𝑥((𝑡𝑥𝑡−𝜏)))+2^𝑇(𝑡)𝑃+𝑔^𝑇(𝑥(𝑡))𝑄−𝑥^𝑇(𝑡)𝐿𝑄𝐶₀×^𝑡_−∞ℎ()𝑔(𝑥(𝑠𝑥𝑡−𝑠))𝑑𝑠+2^𝑇(𝑡)𝑃+𝑔^𝑇(𝑥(𝑡))𝑄−𝑥^𝑇(𝑡)(𝑡)𝑥𝐿𝑄𝕄Ψ+2^𝑇(𝑡)𝑃+𝑔^𝑇(𝑥(𝑡))𝑄−𝑥^𝑇(𝑡)𝐿𝑄𝑈. (47) From Assumption (A1), for given positive diagonal matrix 𝑅₁ we have 2𝑥^𝑇(𝑡)𝑃+𝑔^𝑇(𝑥(𝑡))𝑄−𝑥^𝑇(𝑡)×𝐿𝑄−𝐷₀𝑥(𝑡)+𝐴₀𝑔(𝑥(𝑡))≤2𝑥^𝑇(𝑡)−𝑃𝐷₀+𝐿𝑄𝐷₀𝑥(𝑡)+2𝑔^𝑇(𝑥(𝑡))𝑄𝐴₀𝑔(𝑥(𝑡))+2𝑥^𝑇(𝑡)𝑃𝐴₀−𝐿𝑄𝐴₀−𝑄𝐷₀𝑔(𝑥(𝑡))+𝑥^𝑇(𝑡)𝑊𝑅₁𝑊𝑥(𝑡)−𝑔^𝑇(𝑥(𝑡))𝑅₁𝑔(𝑥(𝑡=))𝑥(𝑡)𝑔(𝑥(𝑡))^𝑇Ξ₁𝑥(𝑡)𝑔(𝑥(𝑡,)) (48) where Ξ₁=Ξ₁₁𝑃𝐴₀−𝐿𝑄𝐴₀−𝐷₀𝑄∗𝑄𝐴₀+𝐴^𝑇₀𝑄−𝑅₁, (49)Ξ₁₁=−𝑃𝐷₀−𝐷₀𝑃+2𝐿𝑄𝐷₀+𝑊𝑅₁𝑊.
By using Assumption (A1) and Lemma 5, we know that there exist six positive diagonal matrices 𝑅₂,𝑅₃∈𝑅^𝑛×𝑛,𝑄₄,𝑄₅,𝑄₆, and 𝑄₇∈𝑅^𝑛2×𝑛2 and a positive definite matrix 𝑅₈ such that the following inequalities hold: 2𝑥^𝑇(𝑡)𝑃𝐵₀+𝑔^𝑇(𝑥(𝑡))𝑄𝐵₀−𝑥^𝑇(𝑡)𝐿𝑄𝐵₀𝑔(𝑥((𝑡≤𝑥𝑡−𝜏)))_𝑇(𝑡)𝑃𝐵₀+𝑔^𝑇(𝑥(𝑡))𝑄𝐵₀−𝑥^𝑇(𝑡)𝐿𝑄𝐵₀𝑅₂⁻¹×𝑥^𝑇(𝑡)𝑃𝐵₀+𝑔^𝑇(𝑥(𝑡))𝑄𝐵₀−𝑥^𝑇(𝑡)𝐿𝑄𝐵₀^𝑇+𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑅₂𝑔(𝑥((𝑡≤𝑡−𝜏)))𝑥(𝑡)𝑔(𝑥(𝑡))^𝑇(𝑃−𝐿𝑄)𝐵₀𝑄𝐵₀𝑅₂⁻¹(𝑃−𝐿𝑄)𝐵₀𝑄𝐵₀^𝑇×𝑥(𝑡)𝑔(𝑥(𝑡))+𝑥^𝑇(𝑡−𝜏(𝑡))𝑊𝑅₂((𝑡,2𝑥𝑊𝑥𝑡−𝜏))_𝑇(𝑡)𝑃+𝑔^𝑇(𝑥(𝑡))𝑄−𝑥^𝑇(𝑡)𝐿𝑄𝑈≤𝑈^𝑇𝑅₈𝑈+𝑥^𝑇(𝑡)𝑃+𝑔^𝑇(𝑥(𝑡))𝑄−𝑥^𝑇(𝑡)𝐿𝑄𝑅₈⁻¹×𝑥^𝑇(𝑡)𝑃+𝑔^𝑇(𝑥(𝑡))𝑄−𝑥^𝑇(𝑡)𝐿𝑄^𝑇=𝑥(𝑡)𝑔(𝑥(𝑡))^𝑇𝑄𝑅𝑃−𝐿𝑄₈₋₁×𝑄𝑃−𝐿𝑄^𝑇𝑥(𝑡)𝑔(𝑥(𝑡))+𝑈^𝑇𝑅₈2𝑥𝑈,_𝑇(𝑡)𝑃𝐶₀+𝑔^𝑇(𝑥(𝑡))𝑄𝐶₀−𝑥^𝑇(𝑡)𝐿𝑄𝐶₀×^𝑡_−∞ℎ()𝑔(𝑥(𝑠≤𝑥𝑡−𝑠))𝑑𝑠_𝑇(𝑡)𝑃𝐶₀+𝑔^𝑇(𝑥(𝑡))𝑄𝐶₀−𝑥^𝑇(𝑡)𝐿𝑄𝐶₀𝑅₃⁻¹×𝑥^𝑇(𝑡)𝑃𝐶₀+𝑔^𝑇(𝑥(𝑡))𝑄𝐶₀−𝑥^𝑇(𝑡)𝐿𝑄𝐶₀^𝑇+^𝑡_−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠^𝑇𝑅₃×^𝑡_−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠. (50) Similarly, by well-known Cauchy-Schwarz inequality and Assumption (A2), we get ^𝑡_−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠^𝑇𝑅₃^𝑡_−∞ℎ()𝑔(𝑥(𝑠≤𝑡−𝑠))𝑑𝑠_0+∞𝕙(𝑠)𝑥^𝑇(𝑡−𝑠)𝑊𝑅₃𝑊𝑥(𝑡−𝑠)𝑑𝑠, (51) which implies that 2𝑥^𝑇(𝑡)𝑃𝐶₀+𝑔^𝑇(𝑥(𝑡))𝑄𝐶₀−𝑥^𝑇(𝑡)𝐿𝑄𝐶₀×^𝑡_−∞ℎ()𝑔(𝑥(𝑠≤𝑡−𝑠))𝑑𝑠𝑥(𝑡)𝑔(𝑥(𝑡))^𝑇(𝑃−𝐿𝑄)𝐶₀𝑄𝐶₀𝑅₃⁻¹×(𝑃−𝐿𝑄)𝐶₀𝑄𝐶₀^𝑇𝑥(𝑡)𝑔(𝑥(𝑡))+₀^+∞𝕙(𝑠)𝑥^𝑇(𝑡−𝑠)𝑊𝑅₃()2𝑥𝑊𝑥𝑡−𝑠𝑑𝑠,_𝑇(𝑡)𝑃+𝑔^𝑇(𝑥(𝑡))𝑄−𝑥^𝑇(𝑡)(𝑡)≤𝑥𝐿𝑄𝕄Ψ_𝑇(𝑡)𝑃+𝑔^𝑇(𝑥(𝑡))𝑄−𝑥^𝑇(𝑡)𝐿𝑄𝕄^𝑇Π⁻¹×𝑥^𝑇(𝑡)𝑃+𝑔^𝑇(𝑥(𝑡))𝑄−𝑥^𝑇(𝑡)𝕄(𝑡)𝐿𝑄+Ψ^𝑇(𝑡)≤𝑥(𝑡)𝑔(𝑥(𝑡ΠΨ))^𝑇()𝕄Π𝑃−𝐿𝑄𝑄𝕄⁻¹×(𝑃−𝐿𝑄)𝕄𝑄𝕄^𝑇𝑥(𝑡)𝑔(𝑥(𝑡))+𝑥^𝑇(𝑡)𝐽^𝑇₁𝑅₄𝐽₁𝑥(𝑡)+𝑔^𝑇(𝑥(𝑡))𝐽^𝑇₂𝑅₅𝐽₂𝑔(𝑥(𝑡))+𝑔^𝑇(𝑥(𝑡−𝜏(𝑡)))𝐽^𝑇₃𝑅₆𝐽₃𝑔(𝑥((𝑡+𝑡−𝜏)))_𝑡−∞ℎ(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠^𝑇𝐽^𝑇₄𝑄₇𝐽₄×^𝑡_−∞ℎ()𝑔(𝑥(𝑠≤𝑔(𝑥(𝑡𝑡−𝑠))𝑑𝑠𝑥(𝑡)))^𝑇()𝕄Π𝑃−𝐿𝑄𝑄𝕄⁻¹(𝑃−𝐿𝑄)𝕄𝑄𝕄^𝑇×𝑥(𝑡)𝑔(𝑥(𝑡))+𝑥^𝑇(𝑡)𝐽^𝑇₁𝑅₄𝐽₁+𝑊𝐽^𝑇₂𝑅₅𝐽₂𝑊𝑥(𝑡)+𝑥^𝑇(𝑡−𝜏(𝑡))𝑊𝐽^𝑇₃𝑅₆𝐽₃𝑊𝑥((𝑡+𝑡−𝜏))_0+∞𝕙(𝑠)𝑥^𝑇(𝑡−𝑠)𝑊𝐽^𝑇₄𝑄₇𝐽₄𝑊𝑥(𝑡−𝑠)𝑑𝑠. (52)(53) Here, the processing method of the term Ψ(𝑡)^𝑇ΠΨ(𝑡) is similar to the technique of Theorem 11.
Substituting (47)–(52) in (46) and using Lemma 9, we also have 𝑑𝑉(𝑥(𝑡))𝑑𝑡₍₁₎≤𝑥(𝑡)𝑔(𝑥(𝑡))^𝑇Ξ₁+Ξ₂𝑥(𝑡)𝑔(𝑥(𝑡))+𝑥^𝑇(𝑡−𝜏(𝑡))𝑊𝑅₂𝑊+𝑊𝐽^𝑇₃𝑅₆𝐽₃𝑊𝑥((𝑡+𝑡−𝜏))_0+∞𝕙(𝑠)𝑥^𝑇(𝑡−𝑠)𝑊𝑅₃𝑊+𝑊𝐽^𝑇₄𝑅₇𝐽₄𝑊×𝑥(𝑡−𝑠)𝑑𝑠+𝑈^𝑇𝑅₈𝑈, (54) where Ξ₁=Ξ₁₁+𝐽^𝑇₁𝑅₄𝐽₁+𝑊𝐽^𝑇₂𝑅₅𝐽₂𝑊𝑃𝐴₀−𝐿𝑄𝐴₀−𝐷₀𝑄∗𝑄𝐴₀+𝐴^𝑇₀𝑄−𝑅₁Ξ₂,=(𝑃−𝐿𝑄)𝐵₀(𝑃−𝐿𝑄)𝐶₀(𝑃−𝐿𝑄)𝕄𝑃−𝐿𝑄𝑄𝐵₀𝑄𝐶₀×𝑅𝑄𝕄𝑄₂₋₁0000𝑅₃⁻¹0000Π⁻¹0000𝑅₈⁻¹×(𝑃−𝐿𝑄)𝐵₀(𝑃−𝐿𝑄)𝐶₀(𝑃−𝐿𝑄)𝕄𝑃−𝐿𝑄𝑄𝐵₀𝑄𝐶₀𝑄𝕄𝑄^𝑇. (55)(56)
Following from Γ<0, there exists 0<𝛽₁≪1 such that Θ₁₁Θ₁₂Θ₁₃Θ₁₄Θ₁₅Θ₁₆∗Θ₂₂𝑄𝐵₀𝑄𝐶₀𝑄𝕄𝑄∗∗−𝑅₂000∗∗∗−𝑅₃00∗∗∗∗−Π0∗∗∗∗∗−𝑅₈<0, (57) where Θ₁₁=(1+𝛽₁)(𝑃+𝑄(𝐹−𝐿))−𝑃𝐷₀−𝐷₀𝑃+2𝐿𝑄𝐷₀+𝑊𝑅₁𝑊+𝐽^𝑇₁𝑅₄𝐽₁. In the light of Lemma 6, one gets Θ₁₁Θ₁₂∗Θ₂₂+Ξ₂<0. (58)
Meanwhile, it is noticed that Θ₁₁Θ₁₂∗Θ₂₂=Ξ₁+1+𝛽₁(𝑃+𝑄(𝐹−𝐿))000. (59)
Therefore, it can be deduced that Ξ₁+Ξ₂<−1+𝛽₁(𝑃+𝑄(𝐹−𝐿))000. (60) Combining the inequalities 𝑊𝑅₂𝑊+𝑊𝐽^𝑇₃𝑅₆𝐽₃𝑊≤𝛽₂𝑃, 𝑊𝑅₃𝑊+𝑊𝐽^𝑇₄𝑅₇𝐽₄𝑊≤𝛽₃𝑃 and formulas (53) and (54), we can derive 𝑑𝑉(𝑥(𝑡))𝑑𝑡₍₂₎≤−1+𝛽₁𝑥^𝑇(𝑡)(𝑃+𝑄(𝐹−𝐿))𝑥(𝑡)+𝛽₂𝑥^𝑇(𝑡−𝜏(𝑡))𝑃𝑥(𝑡−𝜏(𝑡))+𝑈^𝑇𝑅₈𝑈+𝛽₃₀^+∞𝕙(𝑠)𝑥^𝑇(𝑡−𝑠)𝑃𝑥(𝑡−𝑠)𝑑𝑠,𝑡≥𝑡₀. (61)
From Assumption (A1) and formula (43), one also gets 𝑥^𝑇(𝑡)𝑃𝑥(𝑡)≤𝑉(𝑥(𝑡))≤𝑥^𝑇(𝑡)(𝑃+𝑄(𝐹−𝐿))𝑥(𝑡). (62)
Noticing 𝑥^𝑇(𝑡−𝜏(𝑡))𝑃𝑥(𝑡−𝜏(𝑡))≤𝑉(𝑡−𝜏(𝑡))≤sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑥(𝑠)) and according to (60) and (61), we obtain 𝑑𝑉(𝑥(𝑡))𝑑𝑡₍₂₎≤−1+𝛽₁𝑉(𝑥(𝑡))+𝛽₂sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑥(𝑠))+𝛽₃₀^+∞𝕙(𝑠)𝑉(𝑥(𝑡−𝑠))𝑑𝑠+𝑈^𝑇𝑅₈𝑈,𝑡≥𝑡₀. (63) Transforming (62) into the following inequality, we get 𝑑𝑉(𝑥(𝑡))−𝜂𝑑𝑡₍₂₎≤−1+𝛽₁𝑉(𝑥(𝑡))−𝜂+𝛽₂sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑥(𝑠))−𝜂+𝛽₃₀^+∞𝕙(𝑠)𝑉(𝑥(𝑡−𝑠))−𝜂𝑑𝑠,𝑡≥𝑡₀, (64) where 𝜂=𝑈^𝑇𝑅₈𝑈/(1+𝛽₁−𝛽₂−𝛽₃𝕙).
From formula (63), we can know that it satisfies (11) of Lemma 7. Meanwhile, noticing Assumption (A2), it can be deduced that 1+𝛽₁>𝛽₂+𝛽₃𝕙⇔1+𝛽₁>𝛽₂+𝛽₃∫₀^+∞𝕙(𝑠)𝑑𝑠. So (12) of Lemma 7 is also satisfied. From this, according to Lemma 7, when 𝑉(𝑥(𝑡))>𝜂, sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑥(𝑠))>𝜂, and sup_{−∞≤𝑠≤𝑡}𝑉(𝑥(𝑠))>𝜂, we are able to derive 𝑉(𝑥(𝑡))−𝜂≤𝑉(𝑥(𝑡))−𝜂exp−𝜆₀𝑡−𝑡₀, (65) where 𝑉(𝑥(𝑡))=sup_{−∞≤𝑠≤𝑡}𝑉(𝑥(𝑠)), and 𝜆₀∈(0,𝜚) satisfies 𝜆₀<1+𝛽₁−𝛽₂𝑒^𝜆0𝜏−𝛽₃₀^+∞𝕙(𝑠)𝑒^𝜆0𝑠𝑑𝑠. (66) Simultaneously, judging by [1], it is easy to prove that there exists a constant 𝛼 such that 𝑉(𝑥(𝑡))≥‖𝑥‖^𝛼. In terms of Definitions 1, 2, and 3, we know that the neural network defined by (1) or (5) is a globally robust exponentially dissipative system. And noticing 𝑉(𝑥(𝑡))≥𝑥^𝑇(𝑡)𝑃𝑥(𝑡), it is said that Ω={𝑥∈𝑅^𝑛∣𝑥^𝑇(𝑡)𝑃𝑥(𝑡)≤𝑈^𝑇𝑅₈𝑈/(1+𝛽₁−𝛽₂−𝛽₃𝕙)} is a positive invariant and globally exponential attractive set of system (1) or (5). Hence, the proof of Theorem 15 is completed.

Remark 16. For the dissipativity or Lagrange condition given in [22, 25, 26, 28, 31–35, 37], the time delays are constant delays or time-varying delays that are differentiable such that their derivatives are not greater than one or finite. Note that in this paper we do not impose those restrictions on our time-varying delays, which means that our presented results have wider application range.
In the special condition that 𝐶=0 and 𝐷,  𝐴, and 𝐵 are some known constant matrices, the model (1) can be rewritten as follows: 𝑥̇(𝑡)=−𝐷𝑥(𝑡)+𝐴𝑔(𝑥(𝑡))+𝐵𝑔(𝑥(𝑡−𝜏(𝑡)))+𝑈. (67) The other conditions are similar to the model (1). Through Theorem 15, the following corollary can be obtained.

Corollary 17. Assume that Assumptions (A1) holds, if there exist three constants 𝛽_𝑖>0,𝑖=1,2, three positive diagonal matrices 𝑄,𝑅₁, and 𝑅₂∈𝑅^𝑛×𝑛, and two positive definite matrices 𝑃,𝐻∈𝑅^𝑛×𝑛 such that the following inequalities hold: ΘΓ=¹¹Θ₁₂Θ₁₃Θ₁₄∗Θ₂₂𝑄𝐵𝑄∗∗−𝑅₂0∗∗∗−𝐻<0,𝑊𝑅₂𝑊≤𝛽₂𝛽𝑃,₂<1+𝛽₁, (68) where Θ₁₁=𝑃+𝑄(𝐹−𝐿)−𝑃𝐷−𝐷𝑃+2𝐿𝑄𝐷+𝑊𝑅₁𝑊, Θ₁₂=𝑃𝐴−𝐿𝑄𝐴−𝐷𝑄, Θ₁₃=𝑃𝐵−𝐿𝑄𝐵, Θ₁₄=𝑃−𝐿𝑄, Θ₂₂=𝑄𝐴+𝐴^𝑇𝑄−𝑅₁, 𝑊=diag{𝑤₁,…,𝑤_𝑛},𝑤_𝑗=max{|𝑙⁻₁|,|𝑙⁺₁|}, for all 𝑗∈Λ, 𝐿=diag{𝑙⁻₁,𝑙⁻₂,…,𝑙⁻_𝑛}, 𝐹=diag{𝑙⁺₁,𝑙⁺₂,…,𝑙⁺_𝑛}, then the neural network (66) is a globally exponential dissipative system, and the set Ω=𝑥∈𝑅^𝑛∣𝑥^𝑇(𝑡)(𝑡)≤𝑈𝑃𝑥^𝑇𝐻𝑈1+𝛽₁−𝛽₂ (69) is a positive invariant and globally exponentially attractive set of system (66).

Proof. The course of proof is almost parallel to that of Corollary 14 and Theorem 15.

Remark 18. When 𝛽₁>0,𝛽₂=1 in Corollary 17, its result will turn into that of [36] right away.

Remark 19. To the best of our knowledge, few authors have discussed the dissipativity analysis of interval neural networks with general activation functions and infinity distributed delay, and there are few results made on it by LMIs [2, 6, 27, 36, 38, 40]. So, the results of this paper are novel and meaningful. Meanwhile, different from [26, 36, 37], some of the more general results are considered in this paper.

Remark 20. It is not also difficult to find that Lemmas 7 and 9 play a vital roles in the whole paper, especially Lemma 7. The full-text results are obtained based on the case of 𝜎=+∞ in Lemma 7. In addition, it is needed to point out that Lemma 7 is also suitable for the RNNs (1) with finite distributed delays ∫^𝑡_{𝑡−𝜎(𝑡)}𝑓(𝑥(𝑠))𝑑𝑠. Consequently, the results of this text are also suitable for the case of finite distributed delay. So, the conclusions of this paper are more general and valuable than the literature [26, 37].

Remark 21. In this paper, the main results are obtained mainly by means of constructing two different kinds of 𝑉(𝑥(𝑡)), of which the difference is depended on the form of general activation function 𝑔(𝑥(𝑡)). Generally speaking, Theorem 11 is suitable for the case that 𝑔(𝑥(𝑡)) is unbounded continuous function and Lurie-type function, and Theorem 15 is adapted that 𝑔(𝑥(𝑡)) is bounded function and Lipschitz-type function. After a short while, in the part of Example 1, about this will be given elaborate illustration.

4. Illustrative Examples

In this section, two numerical examples are presented to demonstrate the usefulness of the developed methods on the globally robust dissipativity by comparing with the previous results of [19, 26, 28].

Example 1. Consider the interval neural networks model (1) with the following parameters: 𝜏(𝑡)=0.3+0.5[sin(𝑡)]^†,𝑈=(11)^𝑇,𝐷=7.8009.6,𝐷=,6.2008.4𝐴=1.60.10.91.2,𝐴=,0−0.50.10𝐵=20.4−0.31.6,𝐵=,0.4−0.2−1.10.4𝐶=0.20.801.2,𝐶=.−1.40.2−0.80 (70) And the delay kernel ℎ(𝑠) is elected as ℎ_𝑖(𝑠)=𝑒^−𝑠 for 𝑠∈[0,+∞),𝑖=1,2, and 3.
In this case, by simple calculation, it can be obtained that 𝐷₀=𝐷+𝐷2=,𝐴7009₀=𝐴+𝐴2=,𝐵0.8−0.20.50.6₀=𝐵+𝐵2=,𝐶1.20.1−0.71₀=𝐶+𝐶2=,𝛼−0.60.5−0.40.6_𝑖𝑗2×2=𝐷−𝐷2=,𝛽0.8000.6_𝑖𝑗2×2=𝐴−𝐴2=,𝛾0.80.30.40.6_𝑖𝑗2×2=𝐵−𝐵2=,𝜐0.80.30.40.6_𝑖𝑗2×2=𝐶−𝐶2=,𝑀0.80.30.40.6₁=√√0.8000000,𝑀0.6₂=𝑀₃=𝑀₄=√0.8√√0.300000.4√,𝐽0.6₁=√0√0.800000,𝐽0.6₂=𝐽₃=𝐽₄=√√0.800√0.300√0.4.0.6 (71) Clearly, 𝕙=1,𝜏=0.8, and we choose 𝜚=0.8<1. In addition, let 𝑔(𝑥(𝑡))=(1/4)(|𝑥+1|−|𝑥−1|); the activation function satisfies Assumption (A1) with 𝐿=diag{−0.5,−0.5}, 𝐹=𝑊=diag{0.5,0.5}. In this case, we choose 𝛽₁=1,𝛽₂=0.5, and 𝛽₃=0.4. Obviously, it satisfies the condition 𝛽₂+𝛽₃𝕙<1+𝛽₁. Note that 𝕄=[𝑀₁,𝑀₂,𝑀₃,𝑀₄], and solving the LMIs in Theorem 15 using the Matlab LMI Control Toolbox, we obtain the feasible solutions as follows: ,𝑅𝑃=32.071.411.4151.68,𝑄=27.350045.68₁=432.200910.0,𝑅₂=,𝑅4.878003.413₃=,𝑅1.099001.311₄=192.5𝐼_4×4,𝑅₈=186.35𝐼_2×2,𝑅₅=,𝑅55.50000153.70000231.60000155.9₆=,𝑅2.73000018.6000012.900007.13₇=.4.84000023.31000014.9600008.63 (72) Hence, the above results show that all the conditions stated in Theorem 15 have been satisfied, and the networks (70) is a globally robust exponentially dissipative system. Moreover, by calculating the eigenvalues of 𝑃, we gain that the set Ω={𝑥∈𝑅^𝑛∣31.97𝑥²₁+51.78𝑥²₂≤ 𝑈^𝑇𝑅₈𝑈/(1+𝛽₁−𝛽₂−𝛽₃𝕙)=186.35/1.1=169.41} is a positive invariant and globally exponential attractive set of (1).
On the other hand, we can conclude that system (70) is globally robust exponentially dissipative by Theorem 11.
All other things are being equal; if the activation function 𝑔(𝑥) is replaced by 𝑔(𝑥(𝑡))= (1/2)tanh(𝑥(𝑡)), then it is easy to check that the LMIs in Theorem 15 do not have feasible solution in MATLAB. Therefore, Theorem 15 is ineffective in the case. However, it can be deduced that the LMIs in Theorem 11 are feasible, and 𝑃=3.907−0.044−0.0445.761,𝑄₁=,𝑄24.130044.59₂=1.601001.293,𝑄₃=,𝑄0.946001.010₄=14.672𝐼_4×4,𝑄₈=13.912𝐼_2×2,𝑄₅=,𝑄29.5000076.30000111.2000074.8₆=,𝑄1.13700003.46400002.62600001.923₇=.1.50100003.46400002.62600001.923 (73) Thereby, by Theorem 11, we obtain that system (70) with 𝑔(𝑥(𝑡))=(1/2)tanh(𝑥(𝑡)) is globally robust exponentially dissipative, which implies that Theorem 11 can be applied to the case not covered in Theorem 15.

Remark 22. By virtue of Theorem 3.4 in [28], we have 𝐴+𝐴^𝑇+(‖𝐵^∗‖_∞+‖𝐵^∗‖₁+ ‖𝐶^∗‖_∞+‖𝐶^∗‖₁)𝐼=^10.71_19.9>0, which does not satisfy the condition of Theorem 3.4, so the conclusion of [28] is not feeble in this case. It implies that the proposed results in this paper improve and generalize [28].

Remark 23. It should be noted that the exponentially dissipative rate 𝜆 in Example 1 is also obtained, and it satisfies 𝜆∈(0,0.8) and 𝜆<1−0.5𝑒^0.8𝜆−0.4∫₀^+∞𝑒^{(𝜆−1)𝑠}𝑑𝑠.

Remark 24. In order to imitate the dynamic behavior of system (1), we choose some parameters in (70) randomly as follows: ,.𝐷=6.4008,𝐴=10.80.51𝐵=1.50.1−0.81,𝐶=−10.5−0.61 (74) Figures 1 and 2 depict the state trajectories of system (1) with parameters (70), respectively. Figure 3 depicts the phase plots of system (1) with parameters (70) in the case of 𝑔(𝑥(𝑡))=(1/2)(|𝑥+1|−|𝑥−1|). These numerical results show that system (1) with parameters (70) is globally robustive exponentially dissipative.

Figure 1 

The state trajectories of 𝑥1 in system (70) with arbitrary initial condition.

Figure 2 

The state trajectories of 𝑥2 in system (70) with arbitrary initial condition.

Figure 3 

The phase plots of system (70) with random initials.

Example 2. Considering the following two-neuron RNNs with time-varying delay: 𝑥̇(𝑡)=−𝐷𝑥(𝑡)+𝐴𝑔(𝑥(𝑡))+𝐵𝑔(𝑥(𝑡−𝜏(𝑡)))+𝑈, (75) where 𝜏(𝑡)=0.3+0.5[sin(𝑡)]^†,𝑈=(1,1)^𝑇, ,.𝐷=3.5006.2,𝐴=8−242𝐵=2−548 (76) Clearly, 𝜏=0.8. In addition, If the activation function is chosen as 𝑔(𝑥(𝑡))=(1/5)(𝑥(𝑡)+tanh(𝑥(𝑡))), it is obvious that the activation function 𝑔(⋅) satisfies Assumption (A1) with 𝐿=0, 𝐹=𝑊=^0.40_00.4.
Under the circumstance, we choose 𝛽₁=0.5,𝛽₂=1. Obviously, it satisfies the condition 𝛽₂<1+𝛽₁. Then by using the MATLAB LMI Control Toolbox, the solutions are derived as follows: ,𝑅𝑃=5.26320.25790.25799.0456,𝑄=1.0160001.7402₁=,𝑅170.000800115.4947₂=,.73.560900143.4947𝐻=79.0877−0.3404−0.340478.1530 (77) Calculating the eigenvalues of 𝑃, we get the eigenvalues of it are 5.2457 and 9.0631. Therefore, following from Corollary 17, we gain that the set Ω={𝑥∈𝑅^𝑛∣5.2457𝑥²₁+9.0631𝑥²₂≤ 𝑈^𝑇𝐻𝑈/(1+𝛽₁−𝛽₂)=156.5599/0.5=313.1198} is a positive invariant and globally exponential attractive set of system (75). However, according to Theorem 4.2 of [19], we gain 𝑄⁽¹⁾=𝐴+𝐴^𝑇2+𝐼_2×2−𝑑+diag¹𝑤₁𝑑,−²𝑤₂𝐵2𝐵2^𝑇−𝐼_2×2=.0.2511−2.51−12.52412−10−2.540−1 (78) By calculating we get the four eigenvalues of 𝑄⁽¹⁾ which are −14.2006, −2.6268, 0.0778, and 2.4997, from which we are able to know that 𝑄⁽¹⁾ is not negative definite. So the conclusion in [19] can not be applied to determine the positive invariant and globally exponential attractive sets of (75) and further ensure the dissipation of system (75).
Meanwhile, noting the conditions of Example 2 also which satisfies Corollary 4 in [26], we verify the effectiveness of its conclusion by means of solving the LMI in equation (34) of [26]. It is easy to check that the linear matrix inequality (34) in [26] has not got feasible solution. Hence, for this example, our results in this paper are less conservative than those in [19, 26].

5. Conclusion

In this paper, we have studied the globally robust exponentially dissipative for interval neural networks with general activation functions and infinity distributed delays. To the authors best knowledge, few scholars have investigated the globally robust exponentially dissipative for interval neural networks even without infinity distributed delays [26, 37]. By employing the inequality techniques including a novelty delay differential inequality and some LMIs, we have established some sufficient conditions to ensure globally robust exponentially dissipative for interval neural networks with both time-varying delays and infinity distributed delays. In addition, a series of positive invariant and globally exponentially attractive sets of system (1) or (5) are also obtained. Those obtained results improve and complement some recent works (e.g., in [19, 25, 26, 28, 36, 37]). These criteria are stated in LMIs, so that their verification and applications are straightforward and convenient. The results obtained in this paper have also been justified by numerical examples using computer simulations.

For the further work, we intend to generalize the techniques introduced in this paper to the stochastic neural networks based on memristor [17, 48, 49] or impulsive stochastic reaction-diffusion [12, 14] and fuzzy neural networks [4, 15, 16, 50].

Acknowledgments

The authors would like to thank the editor and the anonymous reviewers for their helpful comments and suggestions, which have greatly improved the presentation of the paper.