Global Robust Exponential Stability Analysis for Interval Neural Networks with Mixed Delays

Abstract

A class of interval neural networks with time-varying delays and distributed delays is investigated. By employing H-matrix and M-matrix theory, homeomorphism techniques, Lyapunov functional method, and linear matrix inequality approach, sufficient conditions for the existence, uniqueness, and global robust exponential stability of the equilibrium point to the neural networks are established and some previously published results are improved and generalized. Finally, some numerical examples are given to illustrate the effectiveness of the theoretical results.

1. Introduction

In recent years, great attention has been paid to the neural networks due to their applications in many areas such as signal processing, associative memory, pattern recognition, parallel computation, and optimization. It should be pointed out that the successful applications heavily rely on the dynamic behaviors of neural networks. Stability, as one of the most important properties for neural networks, is crucially required when designing neural networks. For example, in order to solve problems in the fields of optimization, neural control, and signal processing, neural networks have to be designed such that there is only one equilibrium point, and it is globally asymptotically stable so as to avoid the risk of having spurious equilibria and local minima.

We should point out that neural networks have recently been implemented on electronic chips. In electronic implementation of neural networks, time delays are unavoidably encountered during the processing and transmission of signals, which can cause oscillation and instability of a neural network. On the other hand, there exist inevitably some uncertainties caused by the existence of modeling errors, external disturbance, and parameter fluctuation, which would lead to complex dynamic behaviors. Thus, a good neural network should have robustness against such uncertainties. If the uncertainties of a system are due to the deviations and perturbations of parameters and if these deviations and perturbations are assumed to be bounded, then this system is called an interval system. Recently, global robust stability of interval neural networks with time delays are widely investigated (see [1–22] and references therein). In particular, Faydasicok and Arik [3, 4] proposed two criteria for the global asymptotical robust stability to a class of neural networks with constant delays by utilizing the Lyapunov stability theorems and homeomorphism theorem. The obtained conditions are independent of time delays and only rely on the network parameters of the neural system. Employing Lyapunov-Krasovskii functionals, Balasubramaniam et al. [10, 11] derived two passivity criteria for interval neural networks with time-varying delays in terms of linear matrix inequalities (LMI), which are dependent on the size of the time delays. In practice, to achieve fast response, it is often expected that the designed neural networks can converge fast enough. Thus, it is not only theoretically interesting but also practically important to establish some sufficient conditions for global robust exponential stability of neural networks. In [8], Zhao and Zhu established some sufficient conditions for the global robust exponential stability for interval neural networks with constant delays. In [18], Wang et al. obtained some criteria for the global robust exponential stability for interval Cohen-Grossberg neural networks with time-varying delays using LMI, matrix inequality, matrix norm, and Halanay inequality techniques. In [15–17], employing homeomorphism techniques, Lyapunov method, H-matrix and M-matrix theory, and LMI approach, Shao et al. established some sufficient conditions for the existence, uniqueness, and global robust exponential stability of the equilibrium point for interval Hopfield neural networks with time-varying delays. Recently, the stability of neural networks with time-varying delays has been extensively investigated and various sufficient conditions have been established for the global asymptotic and exponential stability in [10–27]. Generally, 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. It is desired to model them by introducing continuously distributed delays over a certain duration of time such that the distant past has less influence compared to the recent behavior of the state (see [28–30]). However, the distributed delays were not taken into account in [15–17] and most of the above references. To the best of our knowledge, there are fewer robust stability results about the interval neural networks with both the time-varying delays and the distributed delays (see [21, 22]).

Motivated by the works of [15–17] and the discussions above, the purpose of this paper is to present some new sufficient conditions for the global robust exponential stability of neural networks with time-varying and distributed delays. The obtained results can be easily checked. Comparisons are made with some previous works by some remarks and numerical examples, which show that our results effectually improve and generalize some existing works. The neural network can be described by the following differential equations: 𝑥̇_𝑖(𝑡)=−𝑑_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗^𝑡_𝑡−𝜎𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑥_𝑗(𝑠)d𝑠+𝐽_𝑖,𝑖=1,2,…,𝑛, (1.1) or equivalently 𝑥̇(𝑡)=−𝐷𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+𝐶^𝑡_𝑡−𝜎𝐾(𝑡−𝑠)𝑓(𝑥(𝑠))d𝑠+𝐽, (1.2) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^T∈ℝ^𝑛 denotes the state vector associated with the neurons, 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑛) is a positive diagonal matrix, and 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛, and 𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛 are the interconnection weight matrix and the time-varying delayed interconnection weight matrix and the distributed delayed interconnection weight matrix, respectively. 𝑓(𝑥)=(𝑓₁(𝑥₁),𝑓₂(𝑥₂),…,𝑓_𝑛(𝑥_𝑛))^T∈ℝ^𝑛, and 𝑓_𝑗(𝑥_𝑗) denotes the activation function, 𝜏_𝑗(𝑡) denotes the time-varying delay associated with the 𝑗th neuron, 𝑓(𝑥(𝑡−𝜏(𝑡)))=(𝑓₁(𝑥₁(𝑡−𝜏₁(𝑡))),𝑓₂(𝑥₂(𝑡−𝜏₂(𝑡))),…,𝑓_𝑛(𝑥_𝑛(𝑡−𝜏_𝑛(𝑡))))^T, 𝐾(𝑡)=diag(𝑘₁(𝑡),𝑘₂(𝑡),…,𝑘_𝑛(𝑡)), 𝑘_𝑗(𝑡)>0 represents the delay kernel function, which is a real-valued continuous function defined in [0,𝜎] satisfying ∫^𝜎₀𝑘_𝑗(𝑠)d𝑠=1,𝑗=1,2,…,𝑛. 𝐽=(𝐽₁,𝐽₂,…,𝐽_𝑛)^T is the constant input vector. The coefficients 𝑑_𝑖, 𝑎_𝑖𝑗, 𝑏_𝑖𝑗, and 𝑐_𝑖𝑗 can be intervalised as follows: 𝐷_𝐼=𝐷,𝐷=𝐷=diag𝑑_𝑖∶0<𝑑_𝑖≤𝑑_𝑖≤𝑑_𝑖,𝐴,𝑖=1,2,…,𝑛_𝐼=𝐴,𝐴=𝐴=𝑎_{𝑖𝑗𝑛×𝑛}∶𝑎_𝑖𝑗≤𝑎_𝑖𝑗≤𝑎_𝑖𝑗,𝐵,𝑖,𝑗=1,2,…,𝑛_𝐼=𝐵,𝐵=𝐵=𝑏_{𝑖𝑗𝑛×𝑛}∶𝑏_𝑖𝑗≤𝑏_𝑖𝑗≤𝑏_𝑖𝑗,𝐶,𝑖,𝑗=1,2,…,𝑛_𝐼=𝐶,𝐶=𝐶=𝑐_{𝑖𝑗𝑛×𝑛}∶𝑐_𝑖𝑗≤𝑐_𝑖𝑗≤𝑐_𝑖𝑗,𝑖,𝑗=1,2,…,𝑛, (1.3) where 𝐷=diag(𝑑_𝑖), 𝐷=diag(𝑑_𝑖), 𝑋=(𝑥_𝑖𝑗)_𝑛×𝑛, 𝑋=(𝑥_𝑖𝑗)_𝑛×𝑛, and 𝑋=𝐴,𝐵,𝐶. Denote 𝐵^∗=(𝐵+𝐵)/2 and 𝐵_∗=(𝐵−𝐵)/2. Clearly, 𝐵_∗ is a nonnegative matrix and the interval matrix [𝐵,𝐵]=[𝐵^∗−𝐵_∗,𝐵^∗+𝐵_∗]. Consequently, 𝐵=𝐵^∗+Δ𝐵, Δ𝐵∈[−𝐵_∗,𝐵_∗]. 𝐶^∗ and 𝐶_∗ are defined correspondingly.

Throughout this paper, we make the following assumptions.(H1)𝑓_𝑖(𝑥)(𝑖=1,2,…,𝑛) are Lipschitz continuous and monotonically nondecreasing, that is, there exist constants 𝑙_𝑖>0 such that 0≤𝑓_𝑖(𝑥)−𝑓_𝑖𝑦𝑥−𝑦≤𝑙_𝑖𝑥−𝑦²,∀𝑥,𝑦∈ℝ. (1.4)(H2)𝜏_𝑖(𝑡)(𝑖=1,2,…,𝑛) are bounded differential functions of time 𝑡 and satisfy 0≤𝜏_𝑖(𝑡)≤𝜏,0≤𝜏̇_𝑖(𝑡)≤ℎ_𝑖<1.

Denote 𝐿=diag(𝑙₁,𝑙₂,…,𝑙_𝑛), 𝐿_𝑀=max_{1≤𝑖≤𝑛}{𝑙_𝑖}, ℎ=max_{1≤𝑖≤𝑛}{ℎ_𝑖}, and 𝛿=max{𝜏,𝜎}.

The organization of this paper is as follows. In Section 2, some preliminaries are given. In Section 3, sufficient conditions for the existence, uniqueness, and global robust exponential stability of the equilibrium point for system (1.1) are presented. In Section 4, some numerical examples are provided to illustrate the effectiveness of the obtained results and comparisons are made between our results and the previously published ones. A concluding remark is given in Section 5 to end this work.

2. Preliminaries

We give some preliminaries in this section. For a vector 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛), ‖𝑥‖₂=(∑^𝑛_𝑖=1𝑥²_𝑖)^1/2. For a matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐴^T denotes the transpose; 𝐴⁻¹ denotes the inverse; 𝐴>(≥)0 means that 𝐴 is a symmetric positive definite (semidefinite) matrix; 𝜆_max(𝐴) and 𝜆_min(𝐴) denote the largest and the smallest eigenvalues of 𝐴, respectively; ‖𝐴‖₂=𝜆_max(𝐴^T𝐴) denotes the spectral norm of 𝐴. 𝐼 denotes the identity matrix. ∗ denotes the symmetric block in a symmetric matrix.

Definition 2.1 (see [20]). The neural network (1.1) with the parameter ranges defined by (1.3) is globally robustly exponentially stable if for each 𝐷∈𝐷_𝐼, 𝐴∈𝐴_𝐼, 𝐵∈𝐵_𝐼, 𝐶∈𝐶_𝐼, and 𝐽, system (1.1) has a unique equilibrium point 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)^T, and there exist constants 𝑏>0 and 𝛼≥1 such that 𝑥(𝑡)−𝑥^∗≤𝛼𝜙(𝜃)−𝑥^∗e^−𝑏𝑡,∀𝑡>0, (2.1) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^T is a solution of system (1.1) with the initial value 𝜙(𝜃)=(𝜙₁(𝜃),𝜙₂(𝜃),…,𝜙_𝑛(𝜃)), 𝜃∈[−𝛿,0].

Definition 2.2 (see [31]). Let 𝑍_𝑛={𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛∈𝑀_𝑛(ℝ)∶𝑎_𝑖𝑗≤0 if 𝑖≠𝑗, 𝑖,𝑗=1,2,…,𝑛}, where 𝑀_𝑛(ℝ) denotes the set of all 𝑛×𝑛 matrices with entries from ℝ. Then a matrix 𝐴 is called an M-matrix if 𝐴∈𝑍_𝑛 and all successive principal minors of 𝐴 are positive.

Definition 2.3 (see [31]). An 𝑛×𝑛 matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 is said to be an H-matrix if its comparison matrix 𝑀(𝐴)=(𝑚_𝑖𝑗)_𝑛×𝑛 is an M-matrix, where 𝑚_𝑖𝑗=𝑎_𝑖𝑖−𝑎if𝑖=𝑗,_𝑖𝑗if𝑖≠𝑗. (2.2)

Lemma 2.4 (see [19]). For any vectors 𝑥,𝑦∈ℝ^𝑛 and positive definite matrix 𝐺∈ℝ^𝑛×𝑛, the following inequality holds: 2𝑥^T𝑦≤𝑥^T𝐺𝑥+𝑦^T𝐺⁻¹𝑦.

Lemma 2.5 (see [31]). Let 𝐴,𝐵∈𝑍_𝑛. If 𝐴 is an M -matrix and the elements of matrices 𝐴, 𝐵 satisfy the inequalities 𝑎_𝑖𝑗≤𝑏_𝑖𝑗, 𝑖,𝑗=1,2,…,𝑛, then 𝐵 is an M-matrix.

Lemma 2.6 (see [31]). The following LMI: _𝑆^{𝑄(𝑥)𝑆(𝑥)}_{T(𝑥)𝑅(𝑥)}>0, where 𝑄(𝑥)=𝑄^T(𝑥), 𝑅(𝑥)=𝑅^T(𝑥), is equivalent to 𝑅(𝑥)>0 and 𝑄(𝑥)−𝑆(𝑥)𝑅⁻¹(𝑥)𝑆^T(𝑥)>0 or 𝑄(𝑥)>0 and 𝑅(𝑥)−𝑆^T(𝑥)𝑄⁻¹(𝑥)𝑆(𝑥)>0.

Lemma 2.7 (see [1]). 𝐻(𝑥)∶ℝ^𝑛→ℝ^𝑛 is a homeomorphism if 𝐻(𝑥) satisfies the following conditions:(1)𝐻(𝑥) is injective, that is, 𝐻(𝑥)≠𝐻(𝑦), ∀𝑥≠𝑦; (2)𝐻(𝑥) is proper, that is, ‖𝐻(𝑥)‖→+∞ as 𝑥→+∞.

Lemma 2.8. Suppose that the neural network parameters are defined by (1.3), andΞ=Φ−𝑆−𝑃𝐵_∗−𝑃𝐶_∗∗(1−ℎ)𝑅0∗∗𝑄>0, (2.3) where 𝑃=diag(𝑝₁,𝑝₂,…,𝑝_𝑛), 𝑅=diag(𝑟₁,𝑟₂,…,𝑟_𝑛), and 𝑄=diag(𝑞₁,𝑞₂,…,𝑞_𝑛)are positive diagonal matrices, Φ=(Φ_𝑖𝑗)_𝑛×𝑛=2𝑃𝐷𝐿⁻¹−((2−ℎ)/(1−ℎ))‖𝑃𝐵^∗‖₂𝐼−2‖𝑃𝐶^∗‖₂𝐼−𝑅−𝑄, 𝑆=(𝑠_𝑖𝑗)_𝑛×𝑛with 𝑠_𝑖𝑗=2𝑝_𝑖𝑎_𝑖𝑖𝑝if𝑖=𝑗,max_𝑖𝑎_𝑖𝑗+𝑝_𝑗𝑎_𝑗𝑖,𝑝_𝑖𝑎_𝑖𝑗+𝑝_𝑗𝑎_𝑗𝑖if𝑖≠𝑗. (2.4) Then, for all 𝐴∈𝐴_𝐼, 𝐵∈𝐵_𝐼, and 𝐶∈𝐶_𝐼, we have Θ=Φ−𝑆^′∗()−𝑃Δ𝐵−𝑃Δ𝐶1−ℎ𝑅0∗∗𝑄>0, (2.5) where 𝑆^′=(𝑠^′_𝑖𝑗)_𝑛×𝑛=𝑃𝐴+𝐴^T𝑃.

Proof. Denote 𝑇=𝑡_{𝑖𝑗𝑛×𝑛}=11−ℎ𝑃𝐵_∗𝑅⁻¹𝐵^T_∗𝑃+𝑃𝐶_∗𝑄⁻¹𝐶^T_∗𝑇𝑃,_′=𝑡^′_{𝑖𝑗𝑛×𝑛}=11−ℎ𝑃Δ𝐵𝑅⁻¹Δ𝐵^T𝑃+𝑃Δ𝐶𝑄⁻¹Δ𝐶^T𝑃. (2.6) By Lemma 2.6, Ξ>0 is equivalent to Ω=Ω_{𝑖𝑗𝑛×𝑛}=Φ−𝑆−𝑇>0. (2.7) Obviously, Ω∈𝑍_𝑛, and it follows by Definition 2.2 that Ω is an M-matrix.
By Lemma 2.6, Θ>0 is equivalent to Λ=(Λ_𝑖𝑗)_𝑛×𝑛=Φ−𝑆^′−𝑇^′>0. Therefore, we need only to verify that Λ>0. Noting that 𝑡_𝑖𝑗≥𝑡^′_𝑖𝑗≥0(𝑖,𝑗=1,2,…,𝑛) and 𝑠^′_𝑖𝑖=2𝑝_𝑖𝑎_𝑖𝑖≤2𝑝_𝑖𝑎_𝑖𝑖=𝑠_𝑖𝑖, we have Λ_𝑖𝑖=Φ_𝑖𝑖−𝑠^′_𝑖𝑖−𝑡^′_𝑖𝑖≥Φ_𝑖𝑖−𝑠_𝑖𝑖−𝑡_𝑖𝑖=Ω_𝑖𝑖>0,𝑖=1,2,…,𝑛. (2.8)
Denote the comparison matrix of Λ by 𝑀(Λ)=(𝑚_𝑖𝑗)_𝑛×𝑛, where 𝑚_𝑖𝑗=Λ_𝑖𝑖−Λif𝑖=𝑗,_𝑖𝑗if𝑖≠𝑗. (2.9) Considering 𝑠_𝑖𝑗≥|𝑠^′_𝑖𝑗|(𝑖,𝑗=1,2,…,𝑛,𝑖≠𝑗), we can obtain that 𝑚_𝑖𝑗=−𝑠^′_𝑖𝑗+𝑡^′_𝑖𝑗≥−𝑠^′_𝑖𝑗−𝑡^′_𝑖𝑗≥−𝑠_𝑖𝑗−𝑡_𝑖𝑗=Ω_𝑖𝑗,𝑖,𝑗=1,2,…,𝑛,𝑖≠𝑗. (2.10) It follows from (2.8) and (2.10) that 𝑚_𝑖𝑗≥Ω_𝑖𝑗,𝑖,𝑗=1,2,…,𝑛. From Lemma 2.5, we deduce that 𝑀(Λ) is an M-matrix, that is, Λ is an H-matrix with positive diagonal elements. It is well known that a symmetric H-matrix with positive diagonal entries is positive definite, then Λ>0, which implies that Θ>0 for all 𝐴∈𝐴_𝐼, 𝐵∈𝐵_𝐼, and 𝐶∈𝐶_𝐼. The proof is complete.

3. Global Robust Exponential Stability

In this section, we will give a new sufficient condition for the existence and uniqueness of the equilibrium point for system (1.1) and analyze the global robust exponential stability of the equilibrium point.

Theorem 3.1. Under assumptions (H1) and (H2), if there exist positive diagonal matrices 𝑃=diag(𝑝₁,𝑝₂,…,𝑝_𝑛), 𝑅=diag(𝑟₁,𝑟₂,…,𝑟_𝑛), and 𝑄=diag(𝑞₁,𝑞₂,…,𝑞_𝑛)such that Ξ>0, or equivalently Ω>0, where Ξ and Ω are defined by (2.3) and (2.7), respectively, then system (1.1) is globally robustly exponentially stable.

Proof. We will prove the theorem in two steps.
Step  1. We will prove the existence and uniqueness of the equilibrium point of system (1.1).
Define a map: 𝐻(𝑥)=−𝐷𝑥+(𝐴+𝐵+𝐶)𝑓(𝑥)+𝐽. We will prove that 𝐻(𝑥) is a homeomorphism of ℝ^𝑛 into itself.
First, we prove that 𝐻(𝑥) is an injective map on ℝ^𝑛. For 𝑥,𝑦∈𝑅^𝑛, 𝑥≠𝑦, we have 𝐻(𝑥)−𝐻𝑦=−𝐷𝑥−𝑦+(𝐴+𝐵+𝐶)𝑓(𝑥)−𝑓𝑦. (3.1) If 𝑓(𝑥)=𝑓(𝑦), then 𝐻(𝑥)≠𝐻(𝑦) for 𝑥≠𝑦. If 𝑓(𝑥)≠𝑓(𝑦), multiplying both sides of (3.1) by 2(𝑓(𝑥)−𝑓(𝑦))^T𝑃, and utilizing Lemma 2.4, assumptions (H1), (H2) and the compatibility of vector 2-norm and matrix spectral norm, we deduce that 2𝑓(𝑥)−𝑓𝑦^T𝑃𝐻(𝑥)𝑦𝑓(𝑥)𝑦−𝐻=−2−𝑓^T𝑃𝐷𝑥−𝑦+2𝑓(𝑥)−𝑓𝑦^T𝑃()𝑓(𝑥)𝑦𝑓(𝑥)𝑦𝐴+𝐵+𝐶−𝑓≤−2−𝑓^T𝑃𝐷𝐿⁻¹𝑓(𝑥)−𝑓𝑦+2𝑓(𝑥)−𝑓𝑦^T𝑓(𝑥)𝑦𝑓(𝑥)𝑦𝑃𝐴−𝑓+2−𝑓^T𝑃𝐵^∗+Δ𝐵+𝐶^∗𝑓(𝑥)𝑦𝑓(𝑥)𝑦+Δ𝐶−𝑓≤−2−𝑓^T𝑃𝐷𝐿⁻¹𝑓(𝑥)−𝑓𝑦+2𝑓(𝑥)−𝑓𝑦^T𝑓(𝑥)𝑦+𝑃𝐴−𝑓2−ℎ1−ℎ𝑃𝐵^∗₂𝑓(𝑥)−𝑓𝑦²₂+1𝑓(𝑥)𝑦1−ℎ−𝑓^T𝑅𝑓(𝑥)𝑦+𝑓(𝑥)𝑦−𝑓−𝑓^T𝑃Δ𝐵𝑅⁻¹Δ𝐵^T𝑃𝑓(𝑥)−𝑓𝑦+2𝑃𝐶^∗₂𝑓(𝑥)−𝑓𝑦²₂+𝑓(𝑥)−𝑓𝑦^T𝑃Δ𝐶𝑄⁻¹Δ𝐶^T𝑃𝑓(𝑥)−𝑓𝑦+𝑓(𝑥)−𝑓𝑦^T𝑄𝑓(𝑥)𝑦𝑓(𝑥)𝑦−𝑓=−−𝑓^TΛ𝑓(𝑥)−𝑓𝑦. (3.2) By Lemma 2.8, we have shown that Λ>0 if Ξ>0, which leads to 𝑓(𝑥)−𝑓𝑦^T𝑃𝐻(𝑥)−𝐻𝑦<0,𝑓(𝑥)≠𝑓𝑦. (3.3) Therefore, 𝐻(𝑥)≠𝐻(𝑦) for all 𝑥≠𝑦, that is, 𝐻(𝑥) is injective.
Next, we prove that ‖𝐻(𝑥)‖₂→+∞ as ‖𝑥‖₂→+∞. Letting 𝑦=0 in (3.2), we get 2𝑓(𝑥)−𝑓(0)^T𝑃(𝐻(𝑥)(0𝑓(𝑥)(0)−𝐻))≤−−𝑓^TΛ𝑓(𝑥)−𝑓(0)≤−𝜆_min(Λ)𝑓(𝑥)−𝑓(0)²₂. (3.4) It follows that 2‖𝑃‖₂‖𝐻(𝑥)−𝐻(0)‖₂≥𝜆_min(Λ)𝑓(𝑥)−𝑓(0)₂, (3.5) which yields ‖𝐻(𝑥)‖₂+‖𝐻(0)‖₂≥𝜆_min(Λ)2‖𝑃‖₂𝑓(𝑥)₂−𝑓(0)₂. (3.6) Since ‖𝐻(0)‖₂ and ‖𝑓(0)‖₂ are finite, it is obvious that ‖𝐻(𝑥)‖₂→+∞ as ‖𝑓(𝑥)‖₂→+∞. On the other hand, for unbounded activation functions, by (H1), ‖𝑓(𝑥)‖₂→+∞ implies ‖𝑥‖₂→+∞. For bounded activation functions, it is not difficult to derive from (3.1) that ‖𝐻(𝑥)‖₂→+∞ as ‖𝑥‖₂→+∞.
By Lemma 2.7, we know that 𝐻(𝑥) is a homeomorphism on ℝ^𝑛. Thus, system (1.1) has a unique equilibrium point 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)^T.
Step  2. We prove that the unique equilibrium point 𝑥^∗ is globally robustly exponentially stable.
Let 𝑦(𝑡)=𝑥(𝑡)−𝑥^∗; one can transform system (1.1) into the following system: 𝑦̇_𝑖(𝑡)=−𝑑_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑔_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑡−𝜏_𝑗(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗^𝑡_𝑡−𝜎𝑘_𝑗(𝑡−𝑠)𝑔_𝑗𝑦_𝑗(𝑠)d𝑠,𝑖=1,2,…,𝑛, (3.7) or equivalently 𝑦̇(𝑡)=−𝐷𝑦(𝑡)+𝐴𝑔𝑦(𝑡)+𝐵𝑔𝑦(𝑡−𝜏(𝑡))+𝐶^𝑡_𝑡−𝜎𝐾(𝑡−𝑠)𝑔𝑦(𝑠)d𝑠, (3.8) where 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^T, 𝑔(𝑦(𝑡))=(𝑔₁(𝑦₁(𝑡)),𝑔₂(𝑦₂(𝑡)),…,𝑔_𝑛(𝑦_𝑛(𝑡)))^T, 𝑔(𝑦(𝑡−𝜏(𝑡)))=(𝑔₁(𝑦₁(𝑡−𝜏₁(𝑡))),𝑔₂(𝑦₂(𝑡−𝜏₂(𝑡))),…,𝑔_𝑛(𝑦_𝑛(𝑡−𝜏_𝑛(𝑡))))^T with 𝑔_𝑗(𝑦_𝑗(𝑡))=𝑓_𝑗(𝑦_𝑗(𝑡)+𝑥^∗_𝑗)−𝑓_𝑗(𝑥^∗_𝑗).
We define a Lyapunov functional: 𝑉(𝑡)=∑⁴_𝑖=1𝑉_𝑖(𝑡), where 𝑉₁(𝑡)=𝑠e^𝑏𝑡𝑦^T(𝑡)𝑦(𝑡),𝑉₂(𝑡)=2e^𝑛𝑏𝑡_𝑖=1𝑝_𝑖^𝑦𝑖₀^(𝑡)𝑔_𝑖(𝜉)𝑉d𝜉,₃(𝑡)=^𝑛_𝑖=1^𝑡_{𝑡−𝜏𝑖(𝑡)}𝑟_𝑖+𝑃𝐵^∗₂e^{𝑏(𝜉+𝜏)}𝑔²_𝑖𝑦_𝑖(𝜉)𝑉d𝜉,₄(𝑡)=^𝑛_𝑖=1𝑞_𝑖+𝑃𝐶^∗₂^𝜎₀𝑘_𝑖(𝑣)e^𝑏𝑣𝑡_𝑡−𝑣e^𝑏𝜉𝑔²_𝑖𝑦_𝑖(𝜉)d𝜉d𝑣. (3.9) Calculating the derivative of 𝑉(𝑡) along the trajectories of system (3.7), we obtain that 𝑉̇₁(𝑡)=𝑠𝑏e^𝑏𝑡𝑦^T(𝑡)𝑦(𝑡)+2𝑠e^𝑏𝑡𝑦^T(𝑡)𝑦̇(𝑡)=𝑠𝑏e^𝑏𝑡𝑦^T(𝑡)𝑦(𝑡)+2𝑠e^𝑏𝑡𝑦^T(𝑡)−𝐷𝑦(𝑡)+𝐴𝑔𝑦(𝑡)+𝐵𝑔𝑦(𝑡−𝜏(𝑡))+𝐶^𝑡_𝑡−𝜎𝐾()𝑔𝑦(𝑠),𝑉̇𝑡−𝑠d𝑠₂(𝑡)=2𝑏e^𝑛𝑏𝑡_𝑖=1𝑝_𝑖^𝑦𝑖₀^(𝑡)𝑔_𝑖(𝜉)d𝜉+2e^𝑛𝑏𝑡_𝑖=1𝑝_𝑖𝑔_𝑖𝑦_𝑖(𝑡)−𝑑_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑔_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑡−𝜏_𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗^𝑡_𝑡−𝜎𝑘_𝑗(𝑡−𝑠)𝑔_𝑗𝑦_𝑗(𝑠)d𝑠≤𝑏e^𝑏𝑡𝑦^T(𝑡)𝑃𝐿𝑦(𝑡)−2e^𝑏𝑡𝑔^T𝑦(𝑡)𝑃𝐷𝐿⁻¹𝑔𝑦(𝑡)+2e^𝑏𝑡𝑔^T𝑦(𝑡)𝑃𝐴𝑔𝑦(𝑡)+e^𝑏𝑡𝑃𝐵^∗₂1𝑔𝑦(𝑡)1−ℎ²²+(1−ℎ)𝑔𝑦(𝑡−𝜏(𝑡))²₂+2e^𝑏𝑡𝑔^T𝑦(𝑡)𝑃Δ𝐵𝑔𝑦(𝑡−𝜏(𝑡))+2e^𝑏𝑡𝑔^T𝑦(𝑡)𝑃Δ𝐶^𝜎₀𝐾(𝑠)𝑔𝑦(𝑡−𝑠)d𝑠+e^𝑏𝑡𝑃𝐶^∗₂𝑔𝑦(𝑡)²₂+^𝜎₀𝐾(𝑠)𝑔𝑦(𝑡−𝑠)d𝑠²₂,𝑉̇₃(𝑡)=^𝑛_𝑖=1𝑟_𝑖+𝑃𝐵^∗₂e^{𝑏(𝑡+𝜏)}𝑔²_𝑖𝑦_𝑖(𝑡)−1−𝜏̇_𝑖(𝑡)e^{𝑏(𝑡−𝜏𝑖(𝑡)+𝜏)}𝑔²_𝑖𝑦_𝑖𝑡−𝜏_𝑖(𝑡)≤e^{𝑏(𝑡+𝜏)}𝑔^T𝑦(𝑡)𝑅𝑔𝑦(𝑡)+e^{𝑏(𝑡+𝜏)}𝑃𝐵^∗₂𝑔𝑦(𝑡)²₂−(1−ℎ)e^𝑏𝑡𝑔^T𝑦(𝑡−𝜏(𝑡))𝑅𝑔𝑦(𝑡−𝜏(𝑡))−(1−ℎ)e^𝑏𝑡𝑃𝐵^∗₂𝑔𝑦(𝑡−𝜏(𝑡))²₂,𝑉̇₄(𝑡)=^𝑛_𝑖=1𝑞_𝑖+𝑃𝐶^∗₂^𝜎₀𝑘_𝑖(𝑣)e^𝑏𝑣e^𝑏𝑡𝑔²_𝑖𝑦_𝑖(𝑡)−e^{𝑏(𝑡−𝑣)}𝑔²_𝑖𝑦_𝑖(𝑡−𝑣)d𝑣≤e^{𝑏(𝑡+𝜎)}𝑔^T𝑦(𝑡)𝑄+𝑃𝐶^∗₂𝐼𝑔𝑦(𝑡)−e^𝑏𝑡𝜎₀𝐾(𝑠)𝑔𝑦(𝑡−𝑠)d𝑠^T×𝑄+𝑃𝐶^∗₂𝐼^𝜎₀𝐾(𝑠)𝑔𝑦(𝑡−𝑠)d𝑠Schwarz^′sinequality. (3.10) Therefore, one can deduce that 𝑉̇(𝑡)≤𝑠𝑏e^𝑏𝑡𝑦^T(𝑡)𝑦(𝑡)+𝑏e^𝑏𝑡𝑦^T(𝑡)𝑃𝐿𝑦(𝑡)−2𝑠e^𝑏𝑡𝑦^T(𝑡)𝐷𝑦(𝑡)+2𝑠e^𝑏𝑡𝑦^T(𝑡)𝐴𝑔𝑦(𝑡)+2𝑠e^𝑏𝑡𝑦^T(𝑡)𝐵𝑔𝑦(𝑡−𝜏(𝑡))+2𝑠e^𝑏𝑡𝑦^T(𝑡)𝐶^𝑡_𝑡−𝜎𝐾(𝑡−𝑠)𝑔𝑦(𝑠)d𝑠−2e^𝑏𝑡𝑔^T𝑦(𝑡)𝑃𝐷𝐿⁻¹𝑔𝑦(𝑡)+2e^𝑏𝑡𝑔^T𝑦(𝑡)𝑃𝐴𝑔𝑦(𝑡)+e^𝑏𝑡11−ℎ𝑃𝐵^∗₂𝑔𝑦(𝑡)²₂+2e^𝑏𝑡𝑔^T𝑦(𝑡)𝑃Δ𝐵𝑔𝑦(𝑡−𝜏(𝑡))+2e^𝑏𝑡𝑔^T𝑦(𝑡)𝑃Δ𝐶^𝜎₀𝐾(𝑠)𝑔𝑦(𝑡−𝑠)d𝑠+e^𝑏𝑡𝑃𝐶^∗₂𝑔𝑦(𝑡)²₂+e^{𝑏(𝑡+𝜏)}𝑔^T𝑦(𝑡)𝑅𝑔𝑦(𝑡)+e^{𝑏(𝑡+𝜏)}𝑃𝐵^∗₂𝑔𝑦(𝑡)²₂+e^𝑏𝑡𝑃𝐵^∗₂𝑔𝑦(𝑡)²₂−e^𝑏𝑡𝑃𝐵^∗₂𝑔𝑦(𝑡)²₂+e^𝑏𝑡𝑔^T𝑦(𝑡)𝑅𝑔𝑦(𝑡)−e^𝑏𝑡𝑔^T𝑦(𝑡)𝑦(𝑡)−()e𝑅𝑔1−ℎ^𝑏𝑡𝑔^T𝑦(𝑡−𝜏(𝑡))𝑅𝑔𝑦(𝑡−𝜏(𝑡))+e^{𝑏(𝑡+𝜎)}𝑔^T𝑦(𝑡)𝑄+𝑃𝐶^∗₂𝐼𝑔𝑦(𝑡)−e^𝑏𝑡𝜎₀𝐾(𝑠)𝑔𝑦(𝑡−𝑠)d𝑠^T𝑄^𝜎₀𝐾(𝑠)𝑔𝑦(𝑡−𝑠)d𝑠+e^𝑏𝑡𝑔^T𝑦(𝑡)𝑄+𝑃𝐶^∗₂𝐼𝑔𝑦(𝑡)−e^𝑏𝑡𝑔^T𝑦(𝑡)𝑄+𝑃𝐶^∗₂𝐼𝑔𝑦(𝑡)≤𝑏e^𝑏𝑡𝑦^T(𝑡)(𝑠𝐼+𝑃𝐿)𝑦(𝑡)+e^𝑏𝜏−1e^𝑏𝑡𝑔^T𝑦(𝑡)𝑅+𝑃𝐵^∗₂𝐼𝑔𝑦(𝑡)+e^𝑏𝜎−1e^𝑏𝑡𝑔^T𝑦(𝑡)𝑄+𝑃𝐶^∗₂𝐼𝑔𝑦(𝑡)−e^𝑏𝑡𝑧^T(𝑡)Ψ𝑧(𝑡), (3.11) where Ψ=2𝑠𝐷−𝑠𝐴−𝑠𝐵−𝑠𝐶∗Φ−𝑆^′(),𝑧(𝑡)=𝑦−𝑃Δ𝐵−𝑃Δ𝐶∗∗1−ℎ𝑅0∗∗∗𝑄_T(𝑡)𝑔^T𝑦(𝑡)𝑔^T𝑦(𝑡−𝜏(𝑡))^𝜎₀𝐾(𝑠)𝑔𝑦(𝑡−𝑠)d𝑠^TT. (3.12) Denote Υ=(𝐴𝐵𝐶), from Lemma 2.6, Ψ>0 is equivalent to Θ−(𝑠/2)Υ^T𝐷⁻¹Υ>0, where Θ is defined by (2.5). By Lemma 2.8, we have Θ>0. Letting 0<𝑠<2min_{1≤𝑖≤𝑛}{𝑑_𝑖}𝜆_min(Θ)/𝜆_max(Υ^TΥ), we can derive that 𝑠Θ−2Υ^T𝐷⁻¹𝑠Υ≥Θ−2min_{1≤𝑖≤𝑛}𝑑_𝑖Υ^TΥ>0, (3.13) which yields Ψ>0. Choosing 0<𝑏<min_1≤𝑖≤3{𝑏_𝑖} with 𝑏₁=𝜆_min(Ψ)𝑠+max_{1≤𝑖≤𝑛}𝑝_𝑖𝑙_𝑖,𝑏₂=1𝜏𝜆ln^min(Ψ)2max_{1≤𝑖≤𝑛}𝑟_𝑖+‖𝑃𝐵^∗‖₂,𝑏+1₃=1𝜎𝜆ln^min(Ψ)2max_{1≤𝑖≤𝑛}𝑞_𝑖+𝑃𝐶^∗₂+1, (3.14) we can get 𝑉̇(𝑡)≤𝑏₁e^𝑏𝑡𝑦^T(𝑡)(𝑠𝐼+𝑃𝐿)𝑦(𝑡)+e^𝑏2𝜏−1e^𝑏𝑡𝑔^T𝑦(𝑡)𝑅+𝑃𝐵^∗₂𝐼𝑔𝑦(𝑡)+e^𝑏3𝜎−1e^𝑏𝑡𝑔^T𝑦(𝑡)𝑄+𝑃𝐶^∗₂𝐼𝑔𝑦(𝑡)−e^𝑏𝑡𝑧^T(𝑡)Ψ𝑧(𝑡)<e^𝑏𝑡𝜆_min(Ψ)𝑦^T(𝑡)𝑦(𝑡)+𝑔^T𝑦(𝑡)𝑔𝑦(𝑡)−e^𝑏𝑡𝑧^T(𝑡)Ψ𝑧(𝑡)≤0. (3.15) Consequently, 𝑉(𝑡)≤𝑉(0) for all 𝑡≥0.
On the other hand, 𝑉(0)=𝑠𝑦^T(0)𝑦(0)+2^𝑛_𝑖=1𝑝_𝑖^𝑦𝑖₀⁽⁰⁾𝑔_𝑖(𝜉)d𝜉+^𝑛_𝑖=1⁰_{−𝜏𝑖(0)}𝑟_𝑖+𝑃𝐵^∗₂e^{𝑏(𝜉+𝜏)}𝑔²_𝑖𝑦_𝑖(𝜉+)d𝜉_𝑛𝑖=1𝑞_𝑖+𝑃𝐶^∗₂^𝜎₀𝑘_𝑖(𝑣)e^𝑏𝑣0_−𝑣e^𝑏𝜉𝑔²_𝑖𝑦_𝑖(𝜉)𝑦(0d𝜉d𝑣≤𝑠)²₂+max_{1≤𝑖≤𝑛}𝑝_𝑖𝑙_𝑖𝑦(0)²₂+1𝑏max_{1≤𝑖≤𝑛}𝑟_𝑖+𝑃𝐵^∗₂𝐿²_𝑀e^𝑏𝜏−1sup_{−𝜏≤𝜃≤0}𝑦(𝜃)²₂+1𝑏max_{1≤𝑖≤𝑛}𝑞_𝑖+𝑃𝐶^∗₂𝐿²_𝑀e^𝑏𝜎−1sup_{−𝜎≤𝜃≤0}𝑦(𝜃)²₂≤𝛼sup_{−𝛿≤𝜃≤0}𝑦(𝜃)²₂, (3.16) where 𝛼=𝑠+max_{1≤𝑖≤𝑛}{𝑝_𝑖𝑙_𝑖}+(1/𝑏)(max_{1≤𝑖≤𝑛}{𝑟_𝑖}+max_{1≤𝑖≤𝑛}{𝑞_𝑖}+‖𝑃𝐵^∗‖₂+‖𝑃𝐶^∗‖₂)𝐿²_𝑀(e^𝑏𝛿−1). Hence, 𝑠e^𝑏𝑡‖𝑦(𝑡)‖²₂≤𝑉(𝑡)≤𝑉(0)≤𝛼sup_{−𝛿≤𝜃≤0}‖𝑦(𝜃)‖²₂, that is, 𝑥(𝑡)−𝑥^∗≤𝛼𝑠𝜙(𝜃)−𝑥^∗e^{−𝑏𝑡/2},𝑡>0. (3.17) Thus, system (1.1) is globally robustly exponentially stable. The proof is complete.

Remark 3.2. For the case of infinite distributed delays, that is, letting 𝜎=∞ in (1.1), assume that the delay kernels 𝑘_𝑗(⋅)(𝑗=1,2,…,𝑛) satisfy (H3)∫^∞₀𝑘_𝑗(𝑠)d𝑠=1 and ∫^∞₀𝑘_𝑗(𝑠)e^𝜇𝑠d𝑠<∞for some positive constant 𝜇. A typical example of such delay kernels is given by 𝑘_𝑗(𝑠)=𝑠^𝑟/𝑟!𝛾_𝑗^𝑟+1e^{−𝛾𝑗𝑠} for 𝑠∈[0,∞), where 𝛾_𝑗∈[0,∞), 𝑟∈{0,1,…,𝑛}, which are called the Gamma Memory Filter in [32]. From assumption (H3), we can choose a constant 𝑏^′₃∶0<𝑏^′₃<𝜇 satisfying the following requirement: ^∞₀𝑘_𝑗(𝑠)e^𝑏′3𝑠𝜆d𝑠≤^min(Ψ)2max_{1≤𝑖≤𝑛}𝑞_𝑖+𝑃𝐶^∗₂+1,1≤𝑗≤𝑛. (3.18) In a similar argument as the proof of Theorem 3.1, Under the conditions of Theorem 3.1 and assumption (H3), we can derive that 𝑥(𝑡)−𝑥^∗≤𝛼^′𝑠𝜙(𝜃)−𝑥^∗e^{−𝑏𝑡/2},𝑡>0, (3.19) where 0<𝑏<min{𝑏₁,𝑏₂,𝑏^′₃}, 𝛼^′=𝑠+max_{1≤𝑖≤𝑛}{𝑝_𝑖𝑙_𝑖}+(1/𝑏)(max_{1≤𝑖≤𝑛}{𝑟_𝑖}+max_{1≤𝑖≤𝑛}{𝑞_𝑖}+‖𝑃𝐵^∗‖₂+‖𝑃𝐶^∗‖₂)𝐿²_𝑀max{e^𝑏𝜏−1,𝜆_min(Ψ)/2max_{1≤𝑖≤𝑛}{𝑞_𝑖+‖𝑃𝐶^∗‖₂}}. Hence, for the case of 𝜎=∞, system (1.1) is also globally robustly exponentially stable.

Remark 3.3. Letting 𝑃=𝑝𝐼 be a positive scalar matrix in Theorem 3.1, we can get a robust exponential stability criterion based on LMI.

Remark 3.4. In [8, 13, 15–18], the authors have dealt with the robust exponential stability of neural networks with time-varying delays. However, the distributed delays were not taken into account in models. Therefore, our results in this paper are more general than those reported in [8, 13, 15–18]. It should be noted that the main results in [15] are a special case of Theorem 3.1 when 𝐶=0. Also, our results generalize some previous ones in [2, 6, 7] as mentioned in [15].

Remark 3.5. In previous works such as [2, 6, 7, 17], ‖𝐵_∗‖₂ is often used as a part to estimate the bounds for ‖𝐵‖₂. Considering that 𝐵_∗ is a nonnegative matrix, we develop a new approach based on H-matrix theory. The obtained robust stability criterion is in terms of the matrices 𝐵_∗ and 𝐵^T_∗, which can reduce the conservativeness of the robust results to some extent.

4. Numerical Simulations and Comparisons

In what follows, we give some examples to illustrate the results above and make comparisons between our results and the previously published ones.

Example 4.1. Consider system (1.1) with the following parameters: 𝐴=,−0.3−0.2−0.5−0.6𝐴=0.30.20.20.1,𝐵=,−0.8−0.9−0.4−1𝐵=0.50.60.71,𝐶=,−0.5−0.5−0.3−0.8,𝑑𝐶=0.50.60.31₁=3.6,𝑑₂=5.8,𝐽₁=𝐽₂=0,𝑓₁(𝑥)=𝑓₂(𝑥)=12|+|,𝜏(|𝑥+1𝑥−1|)₁(𝑡)=𝜏₂(𝑡)e=1−^−𝑡2,𝜎=+∞,𝑘_𝑗(𝑡)=𝑡e^−𝑡. (4.1) It is clear that 𝑙₁=𝑙₂=1, 𝜏=1, ℎ=0.5, 𝜇=1. Using the optimization toolbox of Matlab and solving the optimization problem (2.3), we can obtain a feasible solution: 𝑃=diag(1.3970,1.0335),𝑅=diag(1.9323,2.0000),𝑄=diag(1.7742,1.9982). (4.2) In this case, Ω=1.7869−2.8928−2.89284.6886>0. (4.3) By Theorem 3.1, system (1.1) with above parameters is globally robustly exponentially stable. To illustrate the theoretical result, we present a simulation with the following parameters: 𝐴=0.20.1−0.1−0.4,𝐵=0.10.50.60.5,𝐶=0.20.360.20.8,𝐷=3.6005.8. (4.4) We can find that the neuron vector 𝑥=(𝑥₁(𝑡),𝑥₂(𝑡))^T converges to the unique equilibrium point 𝑥^∗=(0.4024,0.2723)^T (see Figure 1). Further, from (3.14) and (3.18), we can deduce that 𝑏₁=0.086, 𝑏₂=0.0445, and 𝑏^′₃=0.014. Thus the exponential convergence rate is 𝑏=0.014.

Figure 1 

Time responses of the state variables 𝑥(𝑡) with different initial values in Example 4.1.

Next, we will compare our results with the previously robust stability results derived in the literature. If 𝑐_𝑖𝑗=0, 𝜏_𝑗(𝑡)≡𝜏_𝑗, 𝜏_𝑗 is a constant, 𝑖,𝑗=1,2,…,𝑛, system (1.1) reduces to the following interval neural networks: 𝑥̇_𝑖(𝑡)=−𝑑_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑗+𝐽_𝑖,𝑖=1,2,…,𝑛, (4.5) which was studied in [3, 4, 8] and the main results are restated as follows.

Theorem 4.2 (see [3]). Let 𝑓∈ℒ. Then neural network model (4.5) is globally asymptotically robustly stable, if the following condition holds: 𝛿=𝑑_𝑚−𝐿_𝑀𝐴^∗2₂+𝐴_∗²₂+2𝐴^T_∗𝐴^∗₂−𝐿_𝑀𝐵₁𝐵_∞>0, (4.6) where 𝐿_𝑀=max(𝑙_𝑖), 𝑑_𝑚=min(𝑑_𝑖), 𝐴^∗=(1/2)(𝐴+𝐴), 𝐴_∗=(1/2)(𝐴−𝐴), and 𝐵=(𝑏̂_𝑖𝑗)_𝑛×𝑛 with 𝑏̂_𝑖𝑗=max{|𝑏_𝑖𝑗|,|𝑏_𝑖𝑗|}.

Theorem 4.3 (see [4]). For the neural network defined by (4.5), assume that 𝑓∈ℒ. Then, neural network model (4.5) is globally asymptotically stable if the following condition holds: 𝜀=𝑑_𝑚−𝐿_𝑀‖𝑄‖₂−𝐿_𝑀√𝑛1−𝑝‖𝑅‖_∞+𝑝‖𝑅‖₁>0, (4.7) where 𝑑_𝑚=min(𝑑_𝑖), 𝐿_𝑀=max(𝑙_𝑖), 0≤𝑝≤1, ‖𝑄‖₂=min𝐴^∗₂+𝐴_∗2,𝐴^∗2₂+𝐴_∗²₂+2𝐴^T_∗𝐴^∗₂,𝐴₂ (4.8) with 𝐴^∗=(1/2)(𝐴+𝐴), 𝐴_∗=(1/2)(𝐴−𝐴), 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 with 𝑎_𝑖𝑗=max{|𝑎_𝑖𝑗|,|𝑎_𝑖𝑗|}, 𝑅=(𝑟_𝑖𝑗)_𝑛×𝑛 with 𝑟_𝑖𝑗=𝑏̂²_𝑖𝑗 and 𝑏̂_𝑖𝑗=max{|𝑏_𝑖𝑗|,|𝑏_𝑖𝑗|}.

Theorem 4.4 (see [8]). Under assumption (H1), if there exists a positive definite diagonal matrix 𝑃=diag(𝑝₁,𝑝₂,…,𝑝_𝑛), such that Π=2𝑃𝐷𝐿⁻¹−𝑆−2‖𝑃‖₂𝐵^∗₂+𝐵_∗2𝐼>0, (4.9) where 𝑆 is defined in Lemma 2.5, then system (4.5) is globally robustly exponentially stable.

Example 4.5. In system (4.5), we choose 𝐴=−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1−1,𝐵=,−1−1−1−1000000000000𝐴=−𝐴,𝐵=−𝐵,𝐷=diag(8,3,7,7.5),𝐽_𝑖=0,𝑓_𝑖(𝑥)=12(|𝑥+1|+|𝑥−1|),𝜏_𝑖=1(𝑖=1,2,3,4). (4.10) It is clear that 𝑙_𝑖=1(𝑖=1,2,3,4). Solving the optimization problem (2.3), we obtain ()(),(),𝑃=diag0.4556,0.9154,0.5650,0.5271,𝑅=diag0.8122,0.5392,1.1254,1.1470𝑄=diag0.4983,0.1274,0.8477,0.8710Ω=3.0558−1.3709−1.0206−0.9826−1.37092.9948−1.4804−1.4424−1.0206−1.48044.8070−1.0921−0.9826−1.4424−1.09214.8338>0. (4.11) By Theorem 3.1, system (4.5) is globally robustly exponentially stable. To illustrate the theoretical result, we present a simulation with 𝐴=𝐴, 𝐵=𝐵, and 𝐷=𝐷. We can find that the neuron vector 𝑥(𝑡) converges to the unique equilibrium point 𝑥^∗=(1.2000,1.6000,0.6857,0.6400)^T (see Figure 2).

Figure 2 

Time responses of the state variables 𝑥(𝑡) with initial value 𝜙(𝑠)=(0.6,1.6,1,−0.5)T for 𝑠∈[−1,0] in Example 4.5.

Now, applying the result of Theorem 4.2 to this example yields 𝛿=𝑑_𝑚−𝐿_𝑀𝐴^∗2₂+𝐴_∗²₂+2𝐴^T_∗𝐴^∗₂−𝐿_𝑀𝐵₁𝐵_∞=𝑑_𝑚−6. (4.12) The choice 𝑑_𝑚>6 ensures the global robust stability of system (4.5).

If we apply the result of Theorem 4.3 to this example and choose 𝑝=1, which can guarantee 𝑑_𝑚 reaches the minimum value when 𝜀>0, then we have 𝜀=𝑑_𝑚−𝐿_𝑀‖𝑄‖₂−𝐿_𝑀√𝑛1−𝑝‖𝑅‖_∞+𝑝‖𝑅‖₁=𝑑_𝑚−6, (4.13) from which we can obtain that system (4.5) is globally robustly exponentially stable for 𝑑_𝑚>6.

In this example, noting that system (4.5) is globally robustly exponentially stable for 𝑑₂=3 in our result, hence, for the network parameters of this example, our result derived in Theorem 3.1 imposes a less restrictive condition on 𝑑_𝑚 than those imposed by Theorems 4.2 and 4.3.

Example 4.6. In system (4.5), we choose 𝐴=,0.80.60.81𝐴=1111,𝐵=,−0.9−0.8−0.2−0.6,𝐷𝐵=0.60.510.8=diag(3.8,3.8),𝐽_𝑖=0,𝑓_𝑖(𝑥)=12(|𝑥+1|+|𝑥−1|),𝜏_𝑖=1(𝑖=1,2). (4.14) Using Theorem 4.4, one can obtain Π=7.6𝑝₁007.6𝑝₂−2𝑝₁𝑝₁+𝑝₂𝑝₁+𝑝₂2𝑝₂−3.6077×max𝑝₁,𝑝₂≤×𝐼1.9923𝑝₁−𝑝₁+𝑝₂−𝑝₁+𝑝₂1.9923𝑝₂. (4.15) Clearly, there do not exist suitable positive constants 𝑝₁ and 𝑝₂ such that Π>0. As a result, Theorem 4.4 cannot be applied to this example.

Solving the optimization problem (2.3), we obtain that 𝑃=diag(1.7909,1.5070),𝑅=diag(2.0000,1.9893),𝑄=diag(0.0001,0.0001),Ω=5.0299−4.5224−4.52244.0661>0. (4.16) By Theorem 3.1, system (4.5) is globally robustly exponentially stable.

5. Conclusion

In this paper, we discussed a class of interval neural networks with time-varying delays and finite as well as infinite distributed delays. By employing H-matrix and M-matrix theory, homeomorphism techniques, Lyapunov functional method, and LMI approach, sufficient conditions for the existence, uniqueness, and global robust exponential stability of the equilibrium point for the neural networks were established. It was shown that the obtained results improve and generalize the previously published results. Therefore, our results extend the application domain of neural networks to a larger class of engineering problems. Numerical simulations demonstrated the main results. At last, in order to guide the readers to future works of robust stability of neural networks, we would like to point out that the key factor should be the determination of new upper bound norms for the intervalized connection matrices. Such new upper bound estimations for intervalized connection matrices might help us to derive new sufficient conditions for robust stability of delayed neural networks.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11071254) and the Science Foundation of Mechanical Engineering College (YJJXM11004).