Global Robust Attractive and Invariant Sets of Fuzzy Neural Networks with Delays and Impulses

Abstract

A class of fuzzy neural networks (FNNs) with time-varying delays and impulses is investigated. With removing some restrictions on the amplification functions, a new differential inequality is established, which improves previouse criteria. Applying this differential inequality, a series of new and useful criteria are obtained to ensure the existence of global robust attracting and invariant sets for FNNs with time-varying delays and impulses. Our main results allow much broader application for fuzzy and impulsive neural networks with or without delays. An example is given to illustrate the effectiveness of our results.

1. Introduction

The theoretical and applied studies of the current neural networks (CNNs) have been a new focus of studies worldwide because CNNs are widely applied in signal processing, image processing, pattern recognition, psychophysics, speech, perception, robotics, and so on. The scholars have introduced many classes of CNNs models such as Hopfield-type networks [1], bidirectional associative memory networks [2], cellular neural networks [3], recurrent back-propagation networks [4–6], optimization-type networks [7–9], brain-state-in-a-box-(BSB-) type networks [10, 11], and Cohen-Grossberg recurrent neural networks (CGCNNs) [12]. According to the choice of the variable for CNNs [13], two basic mathematical models of CNNs are commonly adopted: either local field neural network models or static neural network models. The basic model of local field neural network is described as 𝑥̇_𝑖(𝑡)=−𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝜔_𝑖𝑗𝑓_𝑖𝑥_𝑗(𝑡)+𝐼_𝑖,𝑖=1,2,…,𝑛, (1) where 𝑓_𝑗 denotes the activation function of the 𝑗th neuron; 𝑥_𝑖 is the state of the 𝑖th neuron; 𝐼_𝑖 is the external input imposed on the 𝑖th neuron; 𝜔_𝑖𝑗 denotes the synaptic connectivity value between the 𝑖th neuron and the 𝑗th neuron; 𝑛 is the number of neurons in the network.

It is well known that local field neural network not only models Hopfield-type networks but also models bidirectional associative memory networks and cellular neural networks. In the past few years, there has been increasing interest in studying dynamical characteristics such as stability, persistence, periodicity, robust stability of equilibrium points, and domains of attraction of local field neural network. Many deep theoretical results have been obtained for local field neural network. We can refer to [14–32] and the references cited therein.

However, in mathematical modeling of real world problems, we will encounter some other inconveniences, for example, the complexity and the uncertainty or vagueness. Fuzzy theory is considered as a more suitable setting for the sake of taking vagueness into consideration. Based on traditional cellular neural networks (CNNs),T. Yang and L.-B. Yangproposed the fuzzy CNNs (FCNNs) [33], which integrate fuzzy logic into the structure of traditional CNNs and maintain local connectedness among cells. Unlike previous CNNs structures, FCNNs have fuzzy logic between its template input and/or output besides the sum of product operation. FCNNs are very useful paradigm for image processing problems, which is a cornerstone in image processing and pattern recognition. In addition, many evolutionary processes in nature are characterized by the fact that their states are subject to sudden changes at certain moments and therefore can be described by impulsive system. Therefore, it is necessary to consider both the fuzzy logic and delay effect on dynamical behaviors of neural networks with impulses. Nevertheless, to the best of our knowledge, there are few published papers considering the global robust domain of attraction for the fuzzy neural network (FNNs). Therefore, in this paper, we will study the global robust attracting set and invariant set of the following fuzzy neural networks (FNNs) with time-varying delays and impulses: 𝑢̇_𝑖(𝑡)=−𝑐_𝑖(𝜆)𝑢_𝑖(𝑡)+^𝑛_𝑗=1𝜔_𝑖𝑗(𝜆)𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑖𝑗(𝑡+_𝑛𝑗=1)𝛾_𝑖𝑗(𝜆)𝜈_𝑗+𝐼_𝑖+^𝑛_𝑗=1𝑎_𝑖𝑗(𝜆)𝑓_𝑗𝑢_𝑗(𝑡)+^𝑛_𝑗=1𝛼_𝑖𝑗(𝜆)𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗(𝜆)𝜈_𝑗+^𝑛_𝑗=1𝑏_𝑖𝑗(𝜆)𝑓_𝑗𝑢_𝑗(𝑡)+^𝑛_𝑗=1𝛽_𝑖𝑗(𝜆)𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑖𝑗(𝑡+_𝑛𝑗=1)𝛿_𝑖𝑗(𝜆)𝜈_𝑗,𝑡≠𝑡_𝑘,𝑡≥0,Δ𝑢_𝑖(𝑡)=𝑢_𝑖𝑡⁺−𝑢_𝑖(𝑡⁻)=𝜇_𝑖𝑘(𝜆)𝑢_𝑖(𝑡),𝑡=𝑡_𝑘𝑢,𝑡≥0,_𝑖(𝑡)=𝜙_𝑖(𝑡),𝑡∈[−𝜏(𝜆),0], (2) where 𝛾_𝑖𝑗(𝜆) are elements of fuzzy feed-forward template, 𝑎_𝑖𝑗(𝜆) and 𝛼_𝑖𝑗(𝜆) are elements of fuzzy feedback MIN template, 𝑏_𝑖𝑗(𝜆) and 𝛽_𝑖𝑗(𝜆) are elements of fuzzy feedback MAX template, and 𝑐_𝑖𝑗(𝜆) and 𝛿_𝑖𝑗(𝜆) are elements of fuzzy feed-forward MIN template and fuzzy feed-forward MAX template, respectively. 𝜔_𝑖𝑗(𝜆) is the weight of connection between the 𝑖th neurons and the 𝑗th neurons. 𝑢_𝑖(𝑡),𝐼_𝑖, and 𝜈_𝑖 stand for state, input, and bias of the 𝑖th neurons, respectively. 𝜏_𝑖𝑗(𝑡) is the transmission delay and 𝑓_𝑗 is the activation function. ∧ and ∨ denote the fuzzy AND and fuzzy OR operation, respectively. Δ𝑢_𝑖(𝑡_𝑘) is the impulses at moments 𝑡_𝑘, and 0≤𝑡₁<𝑡₂<⋯ is a strictly increasing sequence such that lim_𝑘→∞𝑡_𝑘=+∞. 𝑔_𝑖=𝜇_𝑖𝑘(𝜆)𝑢_𝑖(𝑡) is the impulsive function. Function 𝜙_𝑖 is the initial function. 𝜏(𝜆)>0 is a constant. 𝜆∈Ξ⊂𝑅 is the parameter.

The main purpose of this paper is to investigate the global robust attracting and invariant sets of FNNs (2). Different from [34, 35], in this paper, we will introduce a new nonlinear differential inequality, which is more effective than the linear differential inequalities for studying the asymptotic behavior of some nonlinear differential equations. Applying this new nonlinear delay differential inequality, sufficient conditions are gained for global robust attracting and invariant sets.

The rest of this paper is organized as follows. In Section 2, we will give some basic definitions and basic results about the attracting domains of FNNs (2). In Section 3, we will obtain the proof of the usefully nonlinear delay differential inequality. In Section 4, our main results will be proved by this delay differential inequality. Finally, an example is given to illustrate the effectiveness of our results in Section 5.

2. Preliminaries

As usual, 𝐶(𝑋,𝑌) denotes the space of continuous mappings from the topological space 𝑋 to the topological space 𝑌. In particular, let 𝐶([−𝜏(𝜆),0],ℝ^𝑛) denote the set of all real-valued continuous mappings from [−𝜏(𝜆),0] to ℝ^𝑛 equipped with supremum norm ‖⋅‖_∞ defined by 𝜙=max_{1≤𝑖≤𝑛}sup_{(𝜆)−𝜏<𝑡≤0}𝜙_𝑖(𝑡), (3) where 𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛)^𝑇∈𝐶([−𝜏(𝜆),0],ℝ^𝑛). Denote by 𝑢(𝑡,𝜙,𝜆) the solution of FCNNs (2) with initial condition 𝜙∈𝐶([−𝜏(𝜆),0],ℝ^𝑛).

Let 𝐸 denote the 𝑛-dimensional unit matrix. For 𝐴,𝐵∈ℝ^𝑚×𝑛 or 𝐴,𝐵∈ℝ^𝑛,𝐴≥𝐵(𝐴>𝐵) means that each pair of the corresponding elements of 𝐴 and 𝐵 satisfies the inequality “≥(>)”. For any 𝑥∈ℝ^𝑛,𝐴∈ℝ^𝑛×𝑛,𝜑∈𝐶([−𝜏,0],ℝ^𝑛), we define [𝑥]⁺=(|𝑥₁|,|𝑥₂|,…,|𝑥_𝑛|)^𝑇=col{|𝑥_𝑖|}, [𝐴]⁺=(|𝑎_𝑖𝑗|)_𝑛×𝑛, [𝜑(𝑡)]_𝜏=([𝜑₁(𝑡)]_𝜏,[𝜑₂(𝑡)]_𝜏,…,[𝜑_𝑛(𝑡)]_𝜏)^𝑇, [𝜑(𝑡)]⁺_𝜏=[[𝜑(𝑡)]⁺]_𝜏, [𝜑_𝑖(𝑡)]_𝜏=sup_{−𝜏≤𝜃≤0}{𝜑_𝑖(𝑡+𝜃)}. For an 𝑀-matrix 𝐷 [36], we denote 𝐷∈ℳ and Ω_𝑀(𝐷)={𝑧∈𝑅^𝑛∶𝐷𝑧>0,𝑧>0}. For the sake of simplicity, we denote that 𝑔=sup_𝜆∈Ξ𝑔(𝜆),𝑔=inf_𝜆∈Ξ𝑔(𝜆), where 𝑔(𝜆) is bounded in Ξ.

As usual, in the theory of impulsive differential equations, at the points of discontinuity 𝑡_𝑘,𝑘=1,2,…, we assume that 𝑢_𝑖(𝑡_𝑘)≡𝑢_𝑖(𝑡⁻_𝑘) and 𝑢^′_𝑖(𝑡_𝑘)≡𝑢^′_𝑖(𝑡⁻_𝑘).

Inspired by [37], we construct an equivalent theorem between (2) and (4). Then we establish some lemmas which are necessary in the proof of the main results.

Throughout this paper, we always assume the following.(𝐴₁) For all 𝜆∈Ξ,0<|𝜇_𝑖𝑘(𝜆)|<1 and ∑^∞_𝑘=1𝜇_𝑖𝑘(𝜆) is uniformly absolute convergence.(𝐴₂)𝑐_𝑖(𝜆)>0, 𝜔_𝑖𝑗(𝜆), 𝛾_𝑖𝑗(𝜆), 𝑎_𝑖𝑗(𝜆), 𝑏_𝑖𝑗(𝜆), 𝑐_𝑖𝑗(𝜆), 𝛼_𝑖𝑗(𝜆), 𝛽_𝑖𝑗(𝜆), and 𝛿_𝑖𝑗(𝜆) are bounded in Ξ(𝑖,𝑗=1,2,…,𝑛).(𝐴₃) The activation function 𝑓_𝑖(⋅) with 𝑓_𝑖(0)=0,𝑖=1,2,…,𝑛 is second-order differentiable and Lipschitz continuous; that is, there exist positive constants 𝐿_𝑖 such that |𝑓_𝑖(𝑥)|≤𝐿_𝑖|𝑥| for any 𝑥∈ℝ.(𝐴₄) Functions 𝜏_𝑖𝑗(𝑡),𝑖,𝑗=1,2,…,𝑛 are nonnegative, bounded, and continuous defined on ℝ⁺ and 0≤𝜏_𝑖𝑗(𝑡)≤𝜏(𝜆).(𝐴₅) Let 𝐷=−(𝑃+𝑄)∈ℳ, where 𝑃=(𝑝_𝑖𝑗)_𝑛×𝑛,𝑝_𝑖𝑖=−𝑐_𝑖+(|𝑎_𝑖𝑖|+|𝑏_𝑖𝑖|)𝐿_𝑖, 𝑝_𝑖𝑗=𝐿_𝑖∏^∞_𝑘=1(1−|𝜇_𝑖𝑘|)⁻¹∑^𝑛_{𝑗=1,𝑗≠𝑖}∏^∞_𝑘=1(1+ |𝜇_𝑗𝑘|)(|𝑎_𝑖𝑗|+|𝑏_𝑖𝑗|), 𝑄=(𝑞_𝑖𝑗)_𝑛×𝑛,𝑞_𝑖𝑗=𝐿_𝑖∏^∞_𝑘=1(1−|𝜇_𝑖𝑘|)⁻¹∑^𝑛_𝑗=1∏^∞_𝑘=1(1+|𝜇_𝑗𝑘|)(|𝜔_𝑖𝑗|+|𝛼_𝑖𝑗|+|𝛽_𝑖𝑗|), and 𝐼̂=col{|𝐼_𝑖|+∑^𝑛_𝑗=1(|𝛾_𝑖𝑗|+|𝑐_𝑖𝑗|+|𝛿_𝑖𝑗|)|𝜈_𝑗|}.Consider the following non-impulsive system (4): 𝑥̇_𝑖(𝑡)=−𝑐_𝑖(𝜆)𝑥_𝑖(𝑡)+_{0≤𝑡𝑘<𝑡}1+𝜇_𝑖𝑘⁻¹×^𝑛_𝑗=1𝜔_𝑖𝑗(𝜆)𝑓_{𝑗0≤𝑡𝑘<𝑡}1+𝜇_𝑗𝑘𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝜆)𝑓_{𝑗0≤𝑡𝑘<𝑡}1+𝜇_𝑗𝑘𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑏_𝑖𝑗(𝜆)𝑓_{𝑗0≤𝑡𝑘<𝑡}1+𝜇_𝑗𝑘𝑥_𝑗(𝑡+_𝑛𝑗=1)𝛼_𝑖𝑗(𝜆)𝑓_{𝑗0≤𝑡𝑘<𝑡}1+𝜇_𝑗𝑘𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝛾_𝑖𝑗(𝜆)𝜈_𝑗+^𝑛_𝑗=1𝑐_𝑖𝑗(𝜆)𝜈_𝑗+^𝑛_𝑗=1𝛽_𝑖𝑗(𝜆)𝑓_{𝑗0≤𝑡𝑘<𝑡}1+𝜇_𝑗𝑘𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝛿_𝑖𝑗(𝜆)𝜈_𝑗+𝐼_𝑖𝑥,𝑡≥0,_𝑖(𝑡)=𝜙_𝑖(𝑡),𝑡∈[−𝜏(𝜆),0]. (4) We have the following lemma, which shows that system (2) and (4) is equivalent.

Lemma 1. Assume (𝐴₁) holds, thenwe have the following.(i)If 𝑥_𝑖(𝑡) is a solution of (4), then 𝑢_𝑖(𝑡)=∏_{0≤𝑡𝑘<𝑡}(1+𝜇_𝑖𝑘)𝑥_𝑖(𝑡) is a solution of (2). (ii)If 𝑢_𝑖(𝑡) is a solution of (2), then 𝑥_𝑖(𝑡)=∏_{0≤𝑡𝑘<𝑡}(1+𝜇_𝑖𝑘)⁻¹𝑢_𝑖(𝑡) is a solution of (4).

Proof. Firstly, let us prove (i). For a given 𝜆∈Ξ, it is easy to see that 𝑢_𝑖(𝑡)=∏_{0≤𝑡𝑘<𝑡}(1+𝜇_𝑖𝑘(𝜆))𝑥_𝑖(𝑡) is absolutely continuous on the interval (𝑡_𝑘,𝑡_𝑘+1] and for any 𝑡≠𝑡_𝑘, 𝑘=1,2,…, 𝑢(𝑡)=_{0≤𝑡𝑘<𝑡}1+𝜇_1𝑘(𝜆)𝑥₁(𝑡),…,_{0≤𝑡𝑘<𝑡}1+𝜇_𝑛𝑘(𝜆)𝑥_𝑛(𝑡) (5) satisfies system (2). In addition, for every 𝑡_𝑘∈{𝑡_𝑘}^∞_𝑘=1, 𝑢_𝑖𝑡⁺_𝑘=lim_{𝑡→𝑡+𝑘0≤𝑡𝑗<𝑡}1+𝜇_𝑖𝑗(𝜆)𝑥_𝑖(𝑡)=_{0≤𝑡𝑗≤𝑡𝑘}1+𝜇_𝑖𝑗(𝜆)𝑥_𝑖(𝑡),𝑢_𝑖𝑡_𝑘=_{0≤𝑡𝑗<𝑡𝑘}1+𝜇_𝑖𝑗(𝜆)𝑥_𝑖(𝑡). (6) Thus, for every 𝑘=1,2,…, 𝑢_𝑖𝑡⁺_𝑘=1+𝜇_𝑖𝑘(𝜆)𝑢_𝑖𝑡_𝑘. (7) The proof is complete.
Next, we prove (ii). Since 𝑢_𝑖(𝑡)=∏_{0≤𝑡𝑘<𝑡}(1+𝜇_𝑖𝑘(𝜆))𝑥_𝑖(𝑡) is absolutely continuous on the interval (𝑡_𝑘,𝑡_𝑘+1] and, in view of (7), it follows that, for any 𝑘=1,2,…, 𝑥_𝑖𝑡⁺_𝑘=_{0≤𝑡𝑗≤𝑡𝑘}1+𝜇_𝑖𝑗(𝜆)⁻¹𝑢_𝑖𝑡⁺_𝑘=_{0≤𝑡𝑗<𝑡𝑘}1+𝜇_𝑖𝑗(𝜆)⁻¹𝑢_𝑖𝑡_𝑘=𝑥_𝑖𝑡_𝑘,𝑥_𝑖𝑡⁻_𝑘=_{0≤𝑡𝑗≤𝑡𝑘−1}1+𝜇_𝑖𝑗(𝜆)⁻¹𝑢_𝑖𝑡⁻_𝑘=𝑥_𝑖𝑡_𝑘,𝑖=1,2,…, (8) which implies that 𝑥_𝑖(𝑡) is continuous on [0,∞). It is easy to prove that 𝑥_𝑖(𝑡) is absolutely continuous on [0,∞). Now, one can easily check that 𝑥(𝑡)=_{0≤𝑡𝑘<𝑡}1+𝜇_1𝑘(𝜆)⁻¹𝑢₁(𝑡),…,_{0≤𝑡𝑘<𝑡}1+𝜇_𝑛𝑘(𝜆)⁻¹𝑢_𝑛(𝑡) (9) is the solution of (4). The proof is complete.

Definition 2. Let 𝑆 be subsets of 𝐶([−𝜏(𝜆),0],ℝ^𝑛)≜𝐶 which is independent of the parameter 𝜆∈Ξ and let 𝑢(𝑡,𝜙,𝜆) be a solution of FNNs (2) with 𝜙∈𝐶.(i)For any given 𝜆∈Ξ, if for any initial value 𝜙∈𝑆 implies that 𝑢(𝑡,𝜙,𝜆)∈𝑆 for all 𝑡≥0, then 𝑆 is said to be a robust positive invariant set of system of FNNs (2).(ii)For any given 𝜆∈Ξ, if for any initial value 𝜙∈𝑆, the solution 𝑢(𝑡,𝜙,𝜆)∈𝑆 converges to 𝑆 as 𝑡→∞, that is, dist(𝑢(𝑡,𝜙,𝜆),𝑆)→0 as 𝑡→∞, then 𝑆 is said to be a global robust attracting set of system of FNNs (2), where dist(𝜑,𝑆)=inf_𝜓∈𝑆dist(𝜑,𝜓), and dist(𝜑,𝜓)=sup_{𝑠∈[−𝜏,0]}|𝜑(𝑠)−𝜓(𝑠)| for 𝜑∈𝐶.
For a class of differential equations with the term of fuzzy AND and fuzzy OR operation, there is the following useful inequality.

Lemma 3 (see [33]). Let 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇 and 𝑣=(𝑣₁,𝑣₂,…,𝑣_𝑛)^𝑇 be two states of (2), then we have ^𝑛_𝑗=1𝛼_𝑖𝑗𝑓_𝑗𝑢_𝑗−^𝑛_𝑗=1𝛼_𝑖𝑗𝑓_𝑗𝑣_𝑗≤^𝑛_𝑗=1𝛼_𝑖𝑗𝑓_𝑗𝑢_𝑗−𝑓_𝑗𝑣_𝑗,_𝑛𝑗=1𝛼_𝑖𝑗𝑓_𝑗𝑢_𝑗−^𝑛_𝑗=1𝛼_𝑖𝑗𝑓_𝑗𝑣_𝑗≤^𝑛_𝑗=1𝛼_𝑖𝑗𝑓_𝑗𝑢_𝑗−𝑓_𝑗𝑣_𝑗. (10)

3. Nonlinear Delay Differential Inequality

In this section, we will establish a new nonlinear delay differential inequality which will play the important role to prove our main results.

Lemma 4. Assume that 𝑢(𝑡)∈𝐶([𝑡₀,∞),ℝ^𝑛) satisfies 𝑑⁺[𝑢(𝑡𝑑𝑡)]⁺≤𝑃[𝑢(𝑡)]⁺+𝑄[𝑢(𝑡)]⁺_𝜏+𝐼,𝑡≥𝑡₀,𝑢𝑡₀+𝜃=𝜑(𝜃)∈𝐶[−𝜏,0],ℝ^𝑛,𝜃∈[−𝜏,0], (11) where 𝑃=(𝑝_𝑖𝑗)_𝑛×𝑛 and 𝑝_𝑖𝑗≥0 for 𝑖≠𝑗, 𝑄=(𝑞_𝑖𝑗)_𝑛×𝑛≥0, 𝐼=(𝐼₁,𝐼₂,…,𝐼_𝑛)^𝑇≥0. If 𝐷=−(𝑃+𝑄)∈ℳ and 𝐿=𝐷⁻¹𝐼, then we have the following.(i) For any constant 𝑑≥1, the solution 𝑢(𝑡) of (11) satisfies [𝑢(𝑡)]⁺≤𝑑𝐿,𝑡≥𝑡₀, (12) provided that [𝜑]⁺_𝜏≤𝑑𝐿.(ii)  Consider that[𝑢(𝑡)]⁺≤𝑧𝑒^{−𝜅(𝑡−𝑡0)}+𝐿,𝑡≥𝑡₀, (13) provided that [𝑢(𝑡)]⁺≤𝑧𝑒^{−𝜅(𝑡−𝑡0)}+𝐿,𝑡∈𝑡₀−𝜏,𝑡₀, (14) where 𝑧=col{|𝑧_𝑖|}∈Ω_𝑀(𝐷) and the positive constant 𝜅 is determined by the following inequality: 𝜅𝐸+𝑃+𝑄𝑒^𝜅𝜏<0. (15)

Proof. Since 𝐷=−(𝑃+𝑄)∈ℳ, we have 𝐷⁻¹≥0. Let 𝜀=𝐷⁻¹col{1}𝜖 (𝜖>0 small enough), then 𝜀>0. In order to prove (12), we will first prove that [𝑢(𝑡)]⁺≤𝑑𝐿+𝜀=col𝑥_𝑖=𝑥,∀𝑡≥𝑡₀, (16) for any given initial function 𝜑∈𝐶([𝑡₀−𝜏,𝑡₀],ℝ^𝑛) with [𝜑]⁺_𝜏≤𝑑𝐿.
If (16) does not hold, then there exist 𝑖∈{1,2,…,𝑛} and 𝑡₁>𝑡₀ such that 𝑢_𝑖𝑡₁=𝑥_𝑖,[𝑢(𝑡)]⁺≤𝑥,for𝑡≤𝑡₁,𝑑⁺𝑢𝑑𝑡^𝑖𝑡₁≥0. (17)(18) It follows from (11) and (17) that 𝑑⁺𝑢𝑡𝑑𝑡¹⁺≤𝑃𝑢𝑡₁⁺+𝑄𝑢𝑡₁⁺_𝜏+𝐼≤(𝑃+𝑄)[{1}]{1}𝑥+𝐼=−𝑑𝐼+col𝜖−𝐼≤−col𝜖<0, (19) which contradicts the inequality (18). So (16) holds for all 𝑡≥𝑡₀. Letting 𝜖→0 in (16), we have [𝑢(𝑡)]⁺≤𝑑𝐿,𝑡≥𝑡₀. (20) The proof of part (i) is complete.
Since 𝐿=𝐷⁻¹𝐼, we have (𝑃+𝑄)𝐿+𝐼=0. Then ^𝑛_𝑗=1𝑝_𝑖𝑗+𝑞_𝑖𝑗𝐿_𝑗+𝐼_𝑖=0,𝑖=1,2,…,𝑛. (21) From (15), we can get ^𝑛_𝑗=1𝑝_𝑖𝑗+𝑞_𝑖𝑗𝑒^𝜅𝜏𝑧_𝑗<−𝜅𝑧_𝑖,𝑖=1,2,…,𝑛. (22) In the following, we at first will prove that for any positive constant 𝜖, 𝑢_𝑖(𝑡)≤(1+𝜖)𝑧_𝑖𝑒^{−𝜅(𝑡−𝑡0)}+𝐿_𝑖=𝑤_𝑖(𝑡),𝑡≥𝑡₀,𝑖=1,2,…,𝑛. (23) We let ℘=𝑖∈{1,2,…,𝑛}∶𝑢_𝑖(𝑡)>𝑤_𝑖(𝑡)forsome𝑡∈𝑡₀,𝜗,+∞_𝑖=inf𝑡∈𝑡₀,+∞∶𝑢_𝑖(𝑡)>𝑤_𝑖(𝑡),𝑖∈℘. (24) If inequality (23) is not true, then ℘ is nonempty set and there must exist some integer 𝑚∈℘ such that 𝜗_𝑚=min_𝑖∈℘𝜗_𝑖∈[𝑡₀,+∞).
By 𝑢_𝑚(𝑡)∈([𝑡−0,+∞),ℝ) and the inequality (23), we can get 𝜗_𝑚>𝑡₀,𝑢_𝑚𝜗_𝑚=𝑤_𝑚𝜗_𝑚,𝑑⁺𝑢𝑑𝑡^𝑚𝜗_𝑚≥𝑤̇_𝑚𝜗_𝑚,𝑢_𝑖(𝑡)≤𝑤_𝑖(𝑡),𝑡∈𝑡₀−𝜏,𝜗_𝑚,𝑖=1,2,…,𝑛. (25)(26) By applying (11) and (21)–(26), we obtain 𝑑⁺𝑢𝑑𝑡^𝑚𝜗_𝑚≤^𝑛_𝑗=1(1+𝜖)𝑧_𝑗𝑒^{−𝜅(𝜗𝑚−𝑡0)}𝑝_𝑚𝑚𝑗+𝑞_𝑚𝑗𝑒^𝜅𝜏−𝜖𝐼_𝑚≤^𝑛_𝑗=1𝑝_𝑚𝑗+𝑞_𝑚𝑗𝑒^𝜅𝜏(1+𝜖)𝑧_𝑗𝑒^{−𝜅(𝜗𝑚−𝑡0)}<−𝜅𝑧_𝑚(1+𝜖)𝑒^{−𝜅(𝜗𝑚−𝑡0)}=𝑤̇_𝑚𝜗_𝑚, (27) which contradicts the inequality in (25). Thus the inequality (23) holds. Therefore, letting 𝜖→0, we have (13). The proof is complete.

By the process of proof of Lemma 4, we easily derive the following theorem

Theorem 5. Under the conditions of Lemma 4, then lim_𝑡→+∞[𝑢(𝑡)]⁺≤𝐿. (28)

4. Main Results

In this section, we will state and prove our main results. The following lemma is very useful to prove Theorem 7.

Lemma 6. Assume that 0<|𝑎_𝑘|<1,(𝑘=1,2,…) and the series of number ∑^∞_𝑘=1𝑎_𝑘 is absolute convergence, then the infinite products ∏^∞_𝑘=1(1−|𝑎_𝑘|)⁻¹, ∏^∞_𝑘=1(1+𝑎_𝑘) and ∏^∞_𝑘=1(1+|𝑎_𝑘|) are convergent and ∏^∞_𝑘=1(1−|𝑎_𝑘|)⁻¹≥∏^∞_𝑘=1(1+𝑎_𝑘)⁻¹,∏^∞_𝑘=1(1+|𝑎_𝑘|)≥∏^∞_𝑘=1(1+𝑎_𝑘).

Proof. In fact, by the assumption 0<|𝑎_𝑘|<1,(𝑘=1,2,…), we have 11−𝑎_𝑘1>1,1−𝑎_𝑘≥11+𝑎_𝑘𝑎>0,1+_𝑘>1,1+𝑎_𝑘≥1+𝑎_𝑘≥1−𝑎_𝑘>0 (29) which imply that ln1−𝑎_𝑘≤ln1+𝑎_𝑘≤ln1+𝑎_𝑘,ln1−𝑎_𝑘<0,ln1+𝑎_𝑘>0. (30) On the other hand, since ∑^∞_𝑘=1𝑎_𝑘 is absolute convergence, we derive that lim_𝑘→∞𝑎_𝑘=lim_𝑘→∞𝑎_𝑘=0. (31) Equations (30) and (31) give that lim_𝑘→∞−ln1−𝑎_𝑘𝑎_𝑘=1,lim_𝑘→∞ln1+𝑎_𝑘𝑎_𝑘=1,lim_𝑘→∞ln1+𝑎_𝑘𝑎_𝑘=1. (32) According to (32) and considering that ∑^∞_𝑘=1𝑎_𝑘 is absolute convergence, we get that the series of positive number −∑^∞_𝑘=1ln(1−|𝑎_𝑘|)=∑^∞_𝑘=1ln(1−|𝑎_𝑘|)⁻¹, ∑^∞_𝑘=1ln(1+𝑎_𝑘), and ∑^∞_𝑘=1ln(1+|𝑎_𝑘|) is convergent. Thus the infinite products ∏^∞_𝑘=1(1−|𝑎_𝑘|)⁻¹, ∏^∞_𝑘=1(1+𝑎_𝑘), and ∏^∞_𝑘=1(1+|𝑎_𝑘|) are convergent. At the same time, combined with (29), we conclude that ∏^∞_𝑘=1(1−|𝑎_𝑘|)⁻¹≥∏^∞_𝑘=1(1+𝑎_𝑘)⁻¹, ∏^∞_𝑘=1(1+|𝑎_𝑘|)≥∏^∞_𝑘=1(1+𝑎_𝑘). The proof of Lemma 6 is complete.

Theorem 7. Assume that (𝐴₁)–(𝐴₅) hold, then 𝑆=𝜙∈𝐶∶𝜙⁺_𝜏(𝜆)≤Π𝐷⁻¹𝐼̂ (33) is a robust positive invariant and global robust attracting set of FNNs (2),where Π=diag^∞_𝑘=11+𝜇_1𝑘,^∞_𝑘=11+𝜇_2𝑘,…,^∞_𝑘=11+𝜇_𝑛𝑘. (34)

Proof. Calculating the upper right derivative (𝑑⁺/𝑑𝑡)[𝑥(𝑡)]⁺ along system (4) and by using Lemma 6, we have 𝑑⁺𝑥𝑑𝑡^𝑖(𝑡𝑥)=sgn_𝑖(𝑡)𝑥̇_𝑖(𝑡)≤−𝑐_𝑖+𝐿_𝑖𝑎_𝑖𝑖+𝐿_𝑖𝑏_𝑖𝑖𝑥_𝑖(𝑡)+𝐿_{𝑖𝑡𝑘<𝑡}1+𝜇_𝑖𝑘^𝑛−1_{𝑗=1,𝑗≠𝑖𝑡𝑘<𝑡}1+𝜇_𝑗𝑘×𝑎_𝑖𝑗+𝑏_𝑖𝑗𝑥_𝑗(𝑡)+𝐿_{𝑖𝑡𝑘<𝑡}1+𝜇_𝑖𝑘^𝑛−1_{𝑗=1𝑡𝑘<𝑡}1+𝜇_𝑗𝑘×𝜔_𝑖𝑗+𝑎_𝑖𝑗+𝑏_𝑖𝑗×𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡+𝐼_𝑖)+^𝑛_𝑗=1𝛾_𝑖𝑗+𝑐_𝑖𝑗+𝛿_𝑖𝑗𝜈_𝑗≤−𝑐_𝑖+𝐿_𝑖𝑎_𝑖𝑖+𝐿_𝑖𝑏_𝑖𝑖𝑥_𝑖(𝑡)+𝐿_{𝑖𝑡𝑘<𝑡}1−𝜇_𝑖𝑘^𝑛−1_{𝑗=1,𝑗≠𝑖𝑡𝑘<𝑡}1+𝜇_𝑗𝑘×𝑎_𝑖𝑗+𝑏_𝑖𝑗𝑥_𝑗(𝑡)+𝐿_{𝑖𝑡𝑘<𝑡}1−𝜇_𝑖𝑘^𝑛−1_{𝑗=1𝑡𝑘<𝑡}1+𝜇_𝑗𝑘×𝜔_𝑖𝑗+𝑎_𝑖𝑗+𝑏_𝑖𝑗×𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡+𝐼_𝑖)+^𝑛_𝑗=1𝛾_𝑖𝑗+𝑐_𝑖𝑗+𝛿_𝑖𝑗𝜈_𝑗≤−𝑐_𝑖+𝐿_𝑖𝑎_𝑖𝑖+𝐿_𝑖𝑏_𝑖𝑖𝑥_𝑖(𝑡)+𝐿_𝑖^∞_𝑘=11−𝜇_𝑖𝑘^𝑛−1∞_{𝑗=1,𝑗≠𝑖𝑘=1}1+𝜇_𝑗𝑘×𝑎_𝑖𝑗+𝑏_𝑖𝑗𝑥_𝑗(𝑡)+𝐿_𝑖^∞_𝑘=11−𝜇_𝑖𝑘^𝑛−1∞_𝑗=1𝑘=11+𝜇_𝑗𝑘×𝜔_𝑖𝑗+𝑎_𝑖𝑗+𝑏_𝑖𝑗×𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖+^𝑛_𝑗=1𝛾_𝑖𝑗+𝑐_𝑖𝑗+𝛿_𝑖𝑗𝜈_𝑗. (35) From (𝐴₅), (35) can be rewritten as follows:𝑑⁺[𝑥(𝑡𝑑𝑡)]⁺≤𝑃[𝑥(𝑡)]⁺+𝑄[𝑥(𝑡)]⁺_𝜏(𝜆)+𝐼,̂𝑡≥0. (36) Then from the conclusion (i) of Lemma 4, we can obtain [𝑥(𝑡)]⁺≤𝐿₁,𝑡≥0, (37) provided that [𝜙]⁺_𝜏(𝜆)≤𝐿₁, where 𝐿₁=𝐷⁻¹𝐼̂.
According to Lemma 1 and (37), one has [𝑢(𝑡)]⁺=diag_{0≤𝑡𝑘<𝑡}1+𝜇_1𝑘,…,_{0≤𝑡𝑘<𝑡}1+𝜇_𝑛𝑘[𝑥(𝑡)]⁺≤diag^∞_𝑘=11+𝜇_1𝑘,…,^∞_𝑘=11+𝜇_𝑛𝑘[𝑥(𝑡)]⁺=Π[𝑥(𝑡)]⁺≤Π𝐿₁=𝐿, (38) provided that [𝜙]⁺_𝜏(𝜆)≤𝐿, where 𝐿=Π𝐷⁻¹𝐼̂. In view of Definition 2, we get that 𝑆 denoted by (33) is a robust positive invariant set of FNNs (2).
On the other hand, since 𝐷∈ℳ, there exists a positive vector 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑛)^𝑇 such that 𝐷𝑧>0,thatis,𝑃+𝑄𝑧<0. (39) By using continuity, we know that there must exist a positive scalar 𝜅(𝜆) such that 𝜅(𝜆)𝐸+𝑃+𝑄𝑒^{𝜅(𝜆)𝜏(𝜆)}𝑧<0, (40) where 𝑑≥1 is a constant such that [𝜙]⁺_𝜏(𝜆)≤𝑑𝐿.
Then by (36), (40), and (𝐴₄), all the conditions of Theorem 5 are satisfied, and we have lim_𝑡→+∞[𝑥(𝑡)]⁺≤𝐿. (41) According to Lemma 1 and Definition 2, we yield that 𝑆 denoted by (33) is also a global robust attracting set of FNNs (2). The proof is complete.

Theorem 8. In addition to (𝐴₁)–(𝐴₅), further assume 𝐼̂=0. Then FNNs (2) has a zero solution and the zero solution is global robust exponential stability and the exponential convergent rate equals 𝜅 which is determined by (40).

5. Illustrative Example

The following illustrative example will demonstrate the effectiveness of our results. Consider the following FNNs with time-varying delays and impulses: 𝑢̇_𝑖(𝑡)=−𝑐_𝑖(𝜆)𝑢_𝑖(𝑡)+²_𝑗=1𝜔_𝑖𝑗(𝜆)𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑖𝑗(𝑡+_𝑛𝑗=1)𝛾_𝑖𝑗(𝜆)𝜈_𝑗+𝐼_𝑖+²_𝑗=1𝑎_𝑖𝑗(𝜆)𝑓_𝑗𝑢_𝑗(𝑡)+²_𝑗=1𝛼_𝑖𝑗(𝜆)𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗(𝜆)𝜈_𝑗+²_𝑗=1𝑏_𝑖𝑗(𝜆)𝑓_𝑗𝑢_𝑗(𝑡)+²_𝑗=1𝛽_𝑖𝑗(𝜆)𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝛿_𝑖𝑗(𝜆)𝜈_𝑗,𝑡≠𝑡_𝑘,𝑡≥0,Δ𝑢_𝑖(𝑡)=𝑢_𝑖𝑡⁺−𝑢_𝑖(𝑡⁻)=𝜇_𝑖𝑘(𝜆)𝑢_𝑖(𝑡),𝑡=𝑡_𝑘𝑢,𝑡≥0,_𝑖(𝑡)=𝜙_𝑖(𝑡),𝑡∈[−𝜏(𝜆),0], (42) where 𝑖=1,2,𝑡_𝑘∈{1,2,3,…}, Ξ=[𝜋/4,𝜋/2], 𝜏(𝜆)=1, 𝑐₁(𝜆)=ℎ(8+sin(2𝜆)), 𝑐₂(𝜆)=ℎ(9+cos(2𝜆)), 𝑓₁(𝑥)=𝑓₂(𝑥)=(1/2)(|𝑥+1|−|𝑥−1|), 𝜏_𝑖𝑗(𝑡)=|sin(𝑖−𝑗)𝑡|, 𝜇_1𝑘(𝜆)=𝜇_2𝑘(𝜆)=(−1)^𝑘sin𝜆/2^𝑘, 𝑎_𝑖𝑖(𝜆)=ℎ(1+sin2𝑖𝜆)/4, 𝑎_𝑖𝑗(𝜆)=((1+sin(𝑖+𝑗)𝜆)/4)(𝑖≠𝑗), 𝑏_𝑖𝑖(𝜆)=ℎ(1+2sin(𝑖+𝑗)𝜆)/6, 𝑏_𝑖𝑗(𝜆)=((1+2sin(𝑖+𝑗)𝜆)/6)(𝑖≠𝑗), 𝛼_𝑖𝑗(𝜆)=(2+sin(𝑖+𝑗)𝜆)/9, 𝛽_𝑖𝑗(𝜆)=(1+2sin(𝑖+𝑗)𝜆)/9, 𝜔_𝑖𝑗(𝜆)=(1+sin(𝑖+𝑗)𝜆)/6, 𝑐_𝑖𝑗(𝜆)=sin(𝑖+𝑗)𝜆/4,𝛾_𝑖𝑗(𝜆)= −sin(𝑖+𝑗)𝜆/4, and 𝛿_𝑖𝑗(𝜆)=cos(𝑖+𝑗)𝜆/4, 𝐼_𝑖=−((−1)^𝑖/4), 𝜈_𝑖=(−1)^𝑖. By the simple calculation, we obtain,,̂11𝑃=ℎ−711−7𝑄=ℎ1111𝐼=, (43) where ℎ=∏^∞_𝑘=1(2^𝑘+1)/(2^𝑘−1). Then 𝐷=−(𝑃+𝑄) is a nonsingular 𝑀-matrix, and taking 𝜅=0.1, 𝑧=(1,1)^𝑇, we get 𝜅𝐸+𝑃+𝑄𝑒^𝜅𝜏𝑧<0. (44) Therefore, by Theorem 7, we obtain that 𝑆=𝜙∈𝐶∶𝜙⁺_𝜏(𝜆)≤Π𝐷⁻¹̂1𝐼=4,14^𝑇 (45) is a robust positive invariant and global robust attracting set of FNNs (42).

Acknowledgments

The author would like to thank the anonymous referees for their useful and valuable suggestions. This work is supported by the National Natural Sciences Foundation of China under Grant no. 11161025, Yunnan Province Natural Scientific Research Fund Project (no. 2011FZ058), and Yunnan Province Education Department Scientific Research Fund Project (no. 2011Z001).