Asymptotic Stability and Exponential Stability of Impulsive Delayed Hopfield Neural Networks

Abstract

A criterion for the uniform asymptotic stability of the equilibrium point of impulsive delayed Hopfield neural networks is presented by using Lyapunov functions and linear matrix inequality approach. The criterion is a less restrictive version of a recent result. By means of constructing the extended impulsive Halanay inequality, we also analyze the exponential stability of impulsive delayed Hopfield neural networks. Some new sufficient conditions ensuring exponential stability of the equilibrium point of impulsive delayed Hopfield neural networks are obtained. An example showing the effectiveness of the present criterion is given.

1. Introduction

In the last several years, Hopfield neural networks (HNN) have received especially considerable attention due to their extensive applications in solving optimization problem, traveling salesman problem, and many other subjects in recent years [1–9]. In hardware implementation of neural networks, time delays are inevitably present due to the finite switching speeds of the amplifiers. Hence, it is vital to investigate the stability of delayed HNN. Recently, various results for the stability of delayed HNN are obtained via different approaches. In [3], Rakkiyappan and Balasubramaniam studied the exponential stability for fuzzy impulsive neural networks by utilizing the Lyapunov-Krasovskii functional and the linear matrix inequality approach. In [8], Li studied the global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type based on the similar methods. In [9], Xia et al. derived some sufficient conditions for the synchronization problem of coupled identical Yang-Yang type fuzzy cellular neural networks with time-varying delays based on using the invariance principle of functional differential equations.

On the other hand, impulsive differential equations have attracted a great deal of attention due to its potential applications in biological systems, chemical reactions, and various results are obtained; for instance, see [10–14]. Impulses can make unstable systems stable, and stable systems can become unstable after impulse effects. Hence, the stability properties of impulsive HNN with time delays have become an important topic of theoretical studies and have been investigated by many researchers; see [5, 6, 15–22]. In [5], Zhang and Sun obtained a result for the uniform stability of the equilibrium point of the impulsive HNN systems with time delays by using Lyapunov functions and analysis technique. In [6], global exponential stability of impulsive delay HNN is investigated by applying the piecewise continuous vector Lyapunov function.

The purpose of this paper is to present some sufficient conditions for uniform asymptotic stability and global exponential stability of impulsive HNN with time delays by means of constructing the extended impulsive Halanay inequality which is different from that given in [23], Lyapunov functional methods, and linear matrix inequality approach. The results here are also discussed from the point of view of thier comparison with the earlier results. Our results improve and generalize the earlier results. At last, we discuss an example to illustrate the advantage of the results we obtained.

2. Systems Description and Preliminaries

Let ℝ denote the set of real numbers, let ℝ₊ denote the set of nonnegative real numbers, and let ℝ^𝑛 denote the 𝑛-dimensional real space equipped with the Euclidean norm ‖⋅‖.

Consider the following impulsive and delayed HNN model: 𝑥^′_𝑖(𝑡)=−𝑐_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑏_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡)+𝐼_𝑖,𝑡≠𝑡_𝑘,𝑡≥𝑡₀,Δ𝑥_𝑖∣_{𝑡=𝑡𝑘}=𝑥_𝑖𝑡_𝑘−𝑥_𝑖𝑡⁻_𝑘,𝑖∈Λ,𝑘=1,2,…, (1) where Λ={1,2,…,𝑛}. 𝑛≥2 corresponds to the number of units in a neural network; the impulse times 𝑡_𝑘 satisfy 0≤𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯,lim_𝑘→+∞𝑡_𝑘=+∞; 𝑥_𝑖 corresponds to the membrane potential of the unit 𝑖 at time 𝑡; 𝑐_𝑖 is positive constant; 𝑓_𝑗, 𝑔_𝑗 denote, respectively, the measures of response or activation to its incoming potentials of the unit 𝑗 at time 𝑡 and 𝑡−𝜏_𝑗(𝑡); constant 𝑎_𝑖𝑗 denotes the synaptic connection weight of the unit 𝑗 on the unit 𝑖 at time 𝑡; constant 𝑏_𝑖𝑗 denotes the synaptic connection weight of the unit 𝑗 on the unit 𝑖 at time 𝑡−𝜏_𝑗(𝑡); 𝐼_𝑖 is the input of the unit 𝑖;𝜏_𝑗(𝑡) is the transmission delay of the 𝑗th neuron such that 0<𝜏_𝑗(𝑡)≤𝜏,𝑡≥𝑡₀, and 𝑗∈Λ.

Assume that system (1) is supplemented with initial conditions of the form 𝑥(𝑠)=𝜙(𝑠),𝑠∈𝑡₀−𝜏,𝑡₀, (2) where 𝑥(𝑠)=(𝑥₁(𝑠),𝑥₂(𝑠),…,𝑥_𝑛(𝑠))^𝑇,  𝜙(𝑠)=(𝜙₁(𝑠),𝜙₂(𝑠),…,𝜙_𝑛(𝑠))^𝑇∈𝑃𝐶([𝑡₀−𝜏,𝑡₀],ℝ^𝑛), and 𝑃𝐶(𝐼,ℝ^𝑛)={𝜓∶𝐼→ℝ^𝑛, which is continuous everywhere except at finite number of points 𝑡_𝑘, at which 𝜓(𝑡⁺_𝑘) and 𝜓(𝑡⁻_𝑘) exist and 𝜓(𝑡⁺_𝑘)=𝜓(𝑡_𝑘)}. For any given 𝑡≥𝑡₀, 𝜓∈𝑃𝐶([𝑡−𝜏,𝑡],ℝ^𝑛), the norm of 𝜓 is defined by ‖𝜓‖_𝜏=sup_{𝑡−𝜏≤𝜃≤𝑡}|𝜓(𝜃)|. For any𝜎≥0, let𝑃𝐶_𝛿(𝜎)={𝜓∈𝑃𝐶([𝜎−𝜏,𝜎],ℝ^𝑛)∶‖𝜓‖<𝛿}.

In this paper, we assume that some conditions are satisfied, so that the equilibrium point of (1) without impulse does exist denoted by 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)^𝑇; see [2, 5]. Impulsive operator is viewed as perturbation of the equilibrium point of system (1) without impulsive effects. We assume that Δ𝑥_𝑖∣_{𝑡=𝑡𝑘}=𝑥_𝑖(𝑡_𝑘)−𝑥_𝑖(𝑡⁻_𝑘)=𝑑_𝑘^(𝑖)(𝑥_𝑖(𝑡⁻_𝑘)−𝑥^∗), 𝑑_𝑘^(𝑖)∈ℝ, and 𝑖∈Λ,  𝑘=1,2,….

Since 𝑥^∗ is an equilibrium point of (1), one can derive from (1) that the transformation 𝑦_𝑖=𝑥_𝑖−𝑥^∗_𝑖, 𝑖∈Λ, transforms system (1) into the following system: 𝑦^′_𝑖(𝑡)=−𝑐_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗Ω_𝑗𝑦_𝑗(𝑡+_𝑛𝑗=1)𝑏_𝑖𝑗Γ_𝑗𝑦_𝑗𝑡−𝜏_𝑗(𝑡),𝑡≠𝑡_𝑘,𝑡≥𝑡₀,𝑦_𝑖𝑡_𝑘=1+𝑑_𝑘^(𝑖)𝑦_𝑖𝑡⁻_𝑘,𝑖∈Λ,𝑘=1,2,…, (3) where Ω_𝑗(𝑦_𝑗(𝑡))=𝑓_𝑗(𝑥^∗_𝑗+𝑦_𝑗(𝑡))−𝑓_𝑗(𝑥^∗_𝑗), Γ_𝑗(𝑦_𝑗(𝑡))=𝑔_𝑗(𝑥^∗_𝑗+𝑦_𝑗(𝑡−𝜏_𝑗(𝑡)))−𝑔_𝑗(𝑥^∗_𝑗).

Clearly, 𝑥^∗ is uniformly asymptotically stable for system (1) if and only if the trivial solution of system (3) is uniformly asymptotically stable. Hence, we only need to prove the stability of the trivial solution of system (3).

Remark 1. If 𝑥_𝑖(𝑡_𝑘)=𝑑_𝑘^(𝑖)𝑥_𝑖(𝑡⁻_𝑘), then we cannot get 𝑦_𝑖(𝑡_𝑘)=𝑑_𝑘^(𝑖)𝑦_𝑖(𝑡⁻_𝑘) through the transformation 𝑦_𝑖=𝑥_𝑖−𝑥^∗_𝑖. So some of the results [5] are incorrect.

The following notations will be used throughout the paper. The notation 𝐴^𝑇 and 𝐴⁻¹ means the transpose of and the inverse of a square matrix 𝐴. Let 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇, 𝑦_𝜏=(𝑦₁(𝑡−𝜏₁(𝑡)), 𝑦₂(𝑡−𝜏₂(𝑡)),…,𝑦_𝑛(𝑡−𝜏_𝑛(𝑡)))^𝑇; 𝐶=diag[−𝑐₁,−𝑐₂,…,−𝑐_𝑛]^𝑇, 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛, 𝐷_𝑘=diag[1+𝑑_𝑘⁽¹⁾,1+𝑑_𝑘⁽²⁾,…,1+𝑑_𝑘^(𝑛)]^𝑇; Ω(𝑦)=[Ω₁(𝑦₁),Ω₂(𝑦₂),…,Ω_𝑛(𝑦_𝑛)]^𝑇, Γ(𝑦_𝜏)=[Γ₁(𝑦₁(𝑡−𝜏₁(𝑡))), Γ₂(𝑦₂(𝑡−𝜏₂(𝑡))),…,Γ_𝑛(𝑦_𝑛(𝑡−𝜏_𝑛(𝑡)))]^𝑇. Then system (3) with initial condition becomes 𝑦^′(𝑡)=𝐶𝑦(𝑡)+𝐴Ω𝑦(𝑡)+𝐵Γ𝑦_𝜏,𝑡≠𝑡_𝑘,𝑡≥𝑡₀,𝑦𝑡_𝑘=𝐷_𝑘𝑦𝑡⁻_𝑘𝑦𝑡,𝑘=1,2,…,₀+𝜃=𝜑(𝜃),𝜃∈[−𝜏,0], (4) where 𝜑(𝜃)=𝑥(𝑡₀+𝜃)−𝑥^∗.

We introduce some definitions as follows.

Definition 2 (see [10]). The function 𝑉(𝑡,𝑠)∶[0,∞)×ℝ^𝑛→ℝ₊ belongs to class 𝑣₀ if(𝐴₁) 𝑉 is continuous on each of the sets [𝑡_𝑘−1,𝑡_𝑘)×ℝ^𝑛 and lim_{(𝑡,𝑧)→(𝑡}⁻_𝑘,𝑠)𝑉(𝑡,𝑧)=𝑉(𝑡⁻_𝑘,𝑠) exists;(𝐴₂)𝑉(𝑡,𝑠) is locally Lipschitzian in 𝑠 and 𝑉(𝑡,0)≡0.

Definition 3 (see [10]). Let 𝑉∈𝑣₀, for any (𝑡,𝑠)∈[𝑡_𝑘−1,𝑡_𝑘)×ℝ^𝑛; the upper right-hand Dini derivative of 𝑉(𝑡,𝑠) along the solution of (4) is defined by 𝐷⁺𝑉(0=𝑡,𝜓)limsup_ℎ→0+𝑉𝑡+ℎ,𝑦𝑡,𝜓(𝑡+ℎ)−𝑉𝑡,𝜓(0)ℎ. (5)

Definition 4 (see [11]). Assume that 𝑦(𝑡)=𝑦(𝜎,𝜑)(𝑡) is the solution of (4) through (𝜎,𝜑). Then the zero solution of (4) is said to be(1)uniformly stable, if for any𝜎≥𝑡₀ and𝜀>0, there exists a 𝛿=𝛿(𝜀)>0 such that𝜑∈𝑃𝐶_𝛿(𝜎) implies that ‖𝑦(𝜎,𝜑)(𝑡)‖<𝜀,𝑡≥𝜎;(2)uniformly asymptotically stable, if it is uniformly stable, and there exists a 𝛿>0 such that for any𝜀>0, 𝜎≥𝑡₀, there is a 𝑇=𝑇(𝜀)>0 such that 𝜑∈𝑃𝐶_𝛿(𝜎) implies that ‖𝑦(𝜎,𝜑)(𝑡)‖<𝜀,𝑡≥𝜎+𝑇;(3)globally exponentially stable, if for any 𝜑∈𝑃𝐶([−𝜏,0],ℝ^𝑛), there exist constants 𝑀, 𝜇>0 such that 𝑦(𝑡)<𝑀𝜑_𝜏𝑒^{−𝜇(𝑡−𝜎)},𝑡≥𝜎. (6)

In this paper, we always assume that the following assumption holds:

(𝐻₀) there exist constants 𝑀, 𝑁>0 such that Ω^𝑇𝑦Ω𝑦≤𝑀𝑦^𝑇𝑦,Γ^𝑇𝑦_𝜏Γ𝑦_𝜏≤𝑁𝑦^𝑇_𝜏𝑦_𝜏. (7)

In addition, we have the following basic lemmas.

Lemma 5 (see [24]). For any vectors 𝑎, 𝑏∈ℝ^𝑛, the inequality ±2𝑎^𝑇𝑏≤𝑎^𝑇𝑋𝑎+𝑏^𝑇𝑋⁻¹𝑏 (8)

holds, in which 𝑋 is any 𝑛×𝑛 matrix with 𝑋>0.

Lemma 6 (see [25]). Assume that there exist constants 𝑃, 𝑄>0 and 𝑚(𝑡)∈𝑃𝐶([𝑡₀−𝜏,∞),ℝ₊) such that(i)for 𝑡=𝑡_𝑘, 𝑚(𝑡_𝑘)≤𝛾_𝑘𝑚(𝑡⁻_𝑘), 𝛾_𝑘>0 are constants and satisfy max_𝑘∈ℤ+{1/𝛾_𝑘,1}<𝑃/𝑄;(ii)for 𝑡≥𝑡₀, 𝑡≠𝑡_𝑘, 𝐷⁺𝑚(𝑡)≤−𝑃𝑚(𝑡)+𝑄𝑚(𝑡), (9)where 𝑚(𝑡)=sup_{𝑡−𝜏≤𝑠≤𝑡}𝑚(𝑠).

Then for 𝑡≥𝑡₀, 𝑚(𝑡)≤𝑚𝑡_{0𝑡0<𝑡𝑘≤𝑡}𝛾_𝑘𝑒^{−𝜆(𝑡−𝑡0)}, (10) where 𝜆 satisfies the following inequality: 0<𝜆≤𝑃−𝑄max_𝑘∈ℤ+1𝛾_𝑘,1⋅𝑒^𝜆𝜏. (11)

3. Main Results

In this section, we will establish some theorems which provide sufficient conditions for uniformly asymptotically stable and global exponential stability of system (1).

Theorem 7. The equilibrium point the system (1) is uniformly asymptotically stable, if there exists 𝑛×𝑛 symmetric, and positive definite matrix 𝑃 satisfies the following conditions:(𝐻₁)𝜂≐∏^∞_𝑘=1max{𝜂_𝑘,1}<∞, where 𝜂_𝑘 is the largest eigenvalue of 𝑃⁻¹𝐷_𝑘𝑃𝐷_𝑘;(𝐻₂)𝜆₃<(−𝑀−𝑁)/𝜆₁, where 𝜆₁ is the smallest eigenvalue of 𝑃 and 𝜆₃ is the largest eigenvalue of 𝑃⁻¹(𝐶𝑃+𝑃𝐶+𝑃𝐴𝐴^𝑇𝑃+𝑃𝐵𝐵^𝑇𝑃).

Proof. First, we will prove that the zero solution of system (4) is uniformly stable. For any 𝜀>0, we may choose a 𝛿>0 such that 𝛿≤(𝜆₁/𝜂𝜆₂)𝜀, where 𝜆₂ is the largest eigenvalue of 𝑃. For any 𝜎≥𝑡₀,𝜑∈𝑃𝐶_𝛿(𝜎), let 𝑦(𝑡)=𝑦(𝜎,𝜑)(𝑡) be a solution of (4) through (𝜎,𝜑),𝜎≥𝑡₀ (for convenience, that we assume 𝜎=𝑡₀); then we can prove that ‖𝑦(𝑡)‖<𝜀,𝑡≥𝑡₀.
Consider the following Lyapunov function: 𝑉(𝑡,𝑦(𝑡))=𝑦^𝑇(𝑡)𝑃𝑦(𝑡); then we have 𝜆₁𝑦(𝑡)²≤𝑉𝑡,𝑦(𝑡)≤𝜆₂𝑦(𝑡)². (12)
By virtue of Lemma 5, we obtain for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝑘=1,2,…, 𝐷⁺𝑉𝑡,𝑦(𝑡)₍₄₎=𝑦^𝑇(𝑡)^′𝑃𝑦(𝑡)+𝑦^𝑇(𝑡)𝑃𝑦^′(𝑡)=𝐶𝑦(𝑡)+𝐴Ω𝑦(𝑡)+𝐵Γ𝑦_𝜏^𝑇𝑃𝑦(𝑡)+𝑦^𝑇(𝑡)𝑃𝐶𝑦(𝑡)+𝐴Ω𝑦(𝑡)+𝐵Γ𝑦_𝜏=𝑦^𝑇(𝑡)𝐶𝑃𝑦(𝑡)+Ω^𝑇𝑦(𝑡)𝐴^𝑇𝑃𝑦(𝑡)+Γ^𝑇𝑦_𝜏𝐵^𝑇𝑃𝑦(𝑡)+𝑦^𝑇(𝑡)𝑃𝐶𝑦(𝑡)+𝑦^𝑇(𝑡)𝑃𝐴Ω𝑦(𝑡)+𝑦^𝑇(𝑡)𝑃𝐵Γ𝑦_𝜏=𝑦^𝑇(𝑡)(𝐶𝑃+𝑃𝐶)𝑦(𝑡)+2Ω^𝑇𝑦(𝑡)𝐴^𝑇𝑃𝑦(𝑡)+2Γ^𝑇𝑦_𝜏𝐵^𝑇𝑃𝑦(𝑡)≤𝑦^𝑇(𝑡)(𝐶𝑃+𝑃𝐶)𝑦(𝑡)+Ω^𝑇𝑦(𝑡)Ω𝑦(𝑡)+𝑦^𝑇(𝑡)𝑃𝐴𝐴^𝑇𝑃𝑦(𝑡)+Γ^𝑇𝑦_𝜏Γ𝑦_𝜏+𝑦^𝑇(𝑡)𝑃𝐵𝐵^𝑇𝑃𝑦(𝑡)≤𝑦^𝑇(𝑡)𝐶𝑃+𝑃𝐶+𝑃𝐴𝐴^𝑇𝑃+𝑃𝐵𝐵^𝑇𝑃𝑦(𝑡)+Ω^𝑇𝑦(𝑡)Ω𝑦(𝑡)+Γ^𝑇𝑦_𝜏Γ𝑦_𝜏≤𝜆₃𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+𝑀𝑦^𝑇(𝑡)𝑦(𝑡)+𝑁𝑦^𝑇_𝜏𝑦_𝜏≤𝜆₃𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+𝑀𝜆₁⁻¹𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+𝑁𝜆₁⁻¹𝑦^𝑇_𝜏𝑃𝑦_𝜏≤𝜆₃+𝑀𝜆₁⁻¹𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+𝑁𝜆₁⁻¹𝑦^𝑇_𝜏𝑃𝑦_𝜏. (13)
First, it is obvious that for 𝑡₀−𝜏≤𝑡≤𝑡₀, 𝜆₁𝑦(𝑡)²≤𝑉𝑡,𝑦(𝑡)≤𝜆₂𝛿²≤𝜂⁻¹𝜆₁𝜀². (14)
Then we can prove that for 𝑡∈[𝑡₀,𝑡₁), 𝑉𝑡,𝑦(𝑡)≤𝜂⁻¹𝜆₁𝜀². (15)
Suppose that this is not true; then there exists 𝑡̂∈[𝑡₀,𝑡₁) such that𝑉(𝑡,̂𝑦(𝑡̂))>𝜂⁻¹𝜆₁𝜀².
Set 𝑡̌=sup𝑡∣𝑠∈𝑡₀,𝑡,𝑉𝑠,𝑦(𝑠)≤𝜂⁻¹𝜆₁𝜀². (16)
It is obvious that 𝑡̌<𝑡̂. Then it follows that(𝐼_𝑎)𝑉(𝑡,𝑦(𝑡))≤𝜂⁻¹𝜆₁𝜀²,𝑡∈[𝑡₀,𝑡)̌; (𝐼𝐼_𝑎)𝑉(𝑡,̌𝑦(𝑡)̌)=𝜂⁻¹𝜆₁𝜀²; (𝐼𝐼𝐼_𝑎) for any 𝛿>0, there exists 𝑡_𝛿∈(𝑡,̌𝑡̌+𝛿) such that 𝑉(𝑡_𝛿,𝑦(𝑡_𝛿))>𝜂⁻¹𝜆₁𝜀².
So 𝑉𝑡,̌𝑦𝑡̌=𝜂⁻¹𝜆₁𝜀²≥𝑉𝑡,𝑦_𝜏,𝑡̌−𝜏≤𝑡≤𝑡.̌ (17)
In view of condition (𝐻₂), from (13), we obtain 𝐷⁺𝑉𝑡,̌𝑦𝑡̌≤𝜆₃+𝑀𝜆₁⁻¹𝑦^𝑇𝑡̌𝑃𝑦𝑡̌+𝑁𝜆₁⁻¹𝑦^𝑇_𝜏𝑃𝑦_𝜏≤𝜆₃+𝑀𝜆₁⁻¹+𝑁𝜆₁⁻¹𝑉𝑡,̌𝑦𝑡̌<0, (18)
which is a contradiction with (𝐼𝐼𝐼_𝑎). Hence, (15) holds.
Considering 𝑉𝑡₁,𝑦𝑡₁=𝑦^𝑇𝑡₁𝑃𝑦𝑡₁=𝑦^𝑇𝑡⁻₁𝐷₁𝑃𝐷₁𝑦𝑡⁻₁≤𝜂₁𝑦^𝑇𝑡⁻₁𝑃𝑦𝑡⁻₁=𝜂₁𝑉𝑡⁻₁,𝑦𝑡⁻₁≤𝜂₁𝜂⁻¹𝜆₁𝜀²≤max𝜂₁,1𝜂⁻¹𝜆₁𝜀², (19)
we will prove that for 𝑡∈[𝑡₁,𝑡₂), 𝑉𝑡,𝑦(𝑡)≤max𝜂₁,1𝜂⁻¹𝜆₁𝜀². (20)
Suppose that this is false; then we can define 𝑡⃗=sup𝑡∣𝑠∈𝑡₁,𝑡,𝑉𝑠,𝑦(𝑠)≤max𝜂₁,1𝜂⁻¹𝜆₁𝜀². (21)
Similarly, we can obtain(𝐼_𝑏)𝑉(𝑡,𝑦(𝑡))≤max{𝜂₁,1}𝜂⁻¹𝜆₁𝜀²,𝑡∈[𝑡₁,𝑡)⃗; (𝐼𝐼_𝑏)𝑉(𝑡,⃗𝑦(𝑡)⃗)=max{𝜂₁,1}𝜂⁻¹𝜆₁𝜀²; (𝐼𝐼𝐼_𝑏) for any 𝛿>0, there exists 𝑡_𝛿∈(𝑡,⃗𝑡⃗+𝛿) such that 𝑉(𝑡_𝛿,𝑦(𝑡_𝛿))>max{𝜂₁,1}𝜂⁻¹𝜆₁𝜀².
So 𝑉𝑡,⃗𝑦𝑡⃗=max𝜂₁,1𝜂⁻¹𝜆₁𝜀²≥𝑉𝑡,𝑦_𝜏,𝑡⃗−𝜏≤𝑡≤𝑡.⃗ (22)
In fact, if 𝑡⃗−𝜏≥𝑡₁, then it is obvious that inequality (22) holds. If 𝑡⃗−𝜏<𝑡₁, then 𝑉(𝑡,𝑦_𝜏)≤𝜂⁻¹𝜆₁𝜀²≤max{𝜂₁,1}𝜂⁻¹𝜆₁𝜀²=𝑉(𝑡,⃗𝑦(𝑡)⃗). So, inequality (22) still holds.
Considering condition (𝐻₂), from (13), we obtain 𝐷⁺𝑉𝑡,⃗𝑦𝑡⃗≤𝜆₃+𝑀𝜆₁⁻¹𝑦^𝑇𝑡⃗𝑃𝑦𝑡⃗+𝑁𝜆₁⁻¹𝑦^𝑇_𝜏𝑃𝑦_𝜏≤𝜆₃+𝑀𝜆₁⁻¹+𝑁𝜆₁⁻¹𝑉𝑡,⃗𝑦𝑡⃗<0, (23) which contradicts (𝐼𝐼𝐼_𝑏). Hence, (20) holds.
By induction hypothesis, we may prove, in general, that for 𝑡∈[𝑡_𝑚,𝑡_𝑚+1), 𝑉𝑡,𝑦(𝑡)≤^𝑚_𝑘=1max𝜂_𝑘,1𝜂⁻¹𝜆₁𝜀²; (24)
that is, 𝑉𝑡,𝑦(𝑡)≤_{𝑡0<𝑡𝑘≤𝑡}max𝜂_𝑘,1𝜂⁻¹𝜆₁𝜀². (25)
Finally, we arrive at 𝜆₁𝑦(𝑡)²≤𝑉𝑡,𝑦(𝑡)≤𝜆₁𝜀²,𝑡≥𝑡₀. (26)
Therefore, we obtain ‖𝑦(𝑡)‖<𝜀,𝑡≥𝑡₀. In view of the choice of 𝛿, the zero solution of (4) is uniformly stable; that is, the equilibrium point of (1) is uniformly stable.
Next we show the uniformly asymptotical stability. For any given 𝐺>0, we find a corresponding 𝛿>0 such that for any 𝜑∈𝑃𝐶_𝛿(𝜎) implies that ‖𝑦(𝑡)‖=‖𝑦(𝑡₀,𝜑)(𝑡)‖≤𝐺, 𝑡≥𝜎=𝑡₀; that is, 𝑉(𝑡,𝑦(𝑡))≤𝜆₂𝐺².
For any small 𝜀∈(0,𝐺), we choose 𝑁=𝑁(𝜀)∈ℤ₊ such that 𝑁𝜆₁2𝜆₂𝜀<𝐺≤𝑁+1𝜆₁2𝜆₂𝜀,𝑁>−𝑁𝜆₃𝜆₁.+𝑀+𝑁 (27)(28)
In fact, it is feasible to choose small enough 𝜀∈(0,𝐺) such that 𝑁 in (27) is large enough to satisfy (28).
Since 𝜂≐∏^∞_𝑘=1max{𝜂_𝑘,1}<∞ implies that ∑^∞_𝑘=1(max{𝜂_𝑘,1}−1)<∞, there exists sufficient large 𝑁^⋆∈ℤ₊ such that max𝜂^∞_{𝑖=𝑁⋆𝑖}≤𝜆,1−1¹𝜀²6𝜆₂𝐺²,max𝜂_𝑘<1,16𝑁+1,𝑘≥𝑁^⋆. (29)
Let 𝑤₁=1+∑^∞_𝑘=1max𝜂_𝑘,1−1𝜆₂𝐺²𝜆₁𝜆₃⋅2+𝑀+𝑁𝑁+𝑁𝜀²+1. (30)
Next we show that there exists 𝑇₁∈[𝑁^⋆,𝑁^⋆+𝑤₁] such that 𝑉𝑇₁,𝑦𝑇₁<𝜆₁𝜀²2𝑁. (31)
Or else, for all 𝑡∈[𝑁^⋆,𝑁^⋆+𝑤₁], 𝑉(𝑡)≥𝜆𝑡,𝑦¹𝜀²2𝑁. (32)
Thus, we get 𝑉(𝑡)+𝜆𝑡,𝑦¹𝜀²2≥𝜆₁𝜀²2𝑁+1≥𝜆₂𝐺²>𝑉𝜉,𝑦_𝜏,𝑡−𝜏≤𝜉≤𝑡. (33)
From (13), we have 𝐷⁺𝑉𝑡,𝑦(𝑡)₍₄₎≤𝜆₃+𝑀𝜆₁⁻¹𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+𝑁𝜆₁⁻¹𝑦^𝑇_𝜏𝑃𝑦_𝜏≤𝜆₃+𝑀𝜆₁⁻¹𝑉𝑡,𝑦(𝑡)+𝑁𝜆₁⁻¹𝑉(𝑡)+𝜆𝑡,𝑦¹𝜀²2≤𝜆₃+𝑀𝜆₁⁻¹+𝑁𝜆₁⁻¹𝑉(𝑡)+𝜀𝑡,𝑦²2𝑁≤𝜆₃+𝑀𝜆₁⁻¹+𝑁𝜆₁⁻¹𝜆₁𝜀²2𝜀𝑁+²2𝑁≤𝜆₃𝜆₁𝜀+𝑀+𝑁𝑁+𝑁²2. (34)
Integrating the above inequality from 𝑁^⋆ to 𝑁^⋆+𝑤₁, we have 𝑉𝑁^⋆+𝑤₁,𝑦𝑁^⋆+𝑤₁≤𝑉𝑁^⋆,𝑦𝑁^⋆+𝜆₃𝜆₁𝜀+𝑀+𝑁𝑁+𝑁²2𝑤₁+_{𝑁⋆<𝑡𝑘<𝑁⋆+𝑤1}𝑉𝑡_𝑘−𝑉𝑡⁻_𝑘≤𝑉𝑁^⋆,𝑦𝑁^⋆+𝜆₃𝜆₁𝜀+𝑀+𝑁𝑁+𝑁²2𝑤₁+_{𝑁⋆<𝑡𝑘<𝑁⋆+𝑤1}max𝜂_𝑘,1−1𝑉𝑡⁻_𝑘≤𝜆₂𝐺²+𝜆₃𝜆₁𝜀+𝑀+𝑁𝑁+𝑁²2𝑤₁+_{𝑁⋆<𝑡𝑘<𝑁⋆+𝑤1}max𝜂_𝑘,1−1𝜆₂𝐺²≤𝜆₃𝜆₁𝜀+𝑀+𝑁𝑁+𝑁²2𝑤₁+1+_{𝑁⋆<𝑡𝑘<𝑁⋆+𝑤1}max𝜂_𝑘,1−1𝜆₂𝐺²<0, (35) which is a contradiction. So (31) holds. We may choose 𝑇₁=𝑁^⋆+𝑤₁.
We next claim that for all 𝑡>𝑇₁, 𝑉(𝑡<𝑁𝑡,𝑦)2+14𝜆₁𝜀². (36)
Suppose that this is not true; then there exists a 𝜏₂>𝑇₁ such that 𝑉𝜏₂,𝑦𝜏₂≥𝑁2+14𝜆₁𝜀², (37) and for 𝑇₁<𝑡<𝜏₂, 𝑉(𝑡<𝑁𝑡,𝑦)2+14𝜆₁𝜀². (38)
Suppose that 𝑇₁∈[𝑡_𝑚,𝑡_𝑚+1), 𝑚∈ℤ₊. We claim that 𝜏₂≥𝑡_𝑚+1. Otherwise, 𝜏₂∈[𝑇₁,𝑡_𝑚+1). Since (31) holds, it is clear that there exists a 𝜏₁∈[𝑇₁,𝜏₂) such that 𝑉𝜏₁,𝑦𝜏₁=𝑁2𝜆₁𝜀²,𝑉𝜏₁,𝑦𝜏₁≤𝑉(𝑡,𝑥(𝑡))≤𝑉𝜏₂,𝑥𝜏₂,𝜏₁≤𝑡≤𝜏₂. (39)
Furthermore, we note that 𝑉(𝑡)+𝜆𝑡,𝑦¹𝜀²2≥𝜆₁𝜀²2𝑁+1≥𝜆₂𝐺²>𝑉𝜉,𝑦_𝜏,𝑡−𝜏≤𝜉≤𝑡. (40)
From (13), we have 𝐷⁺𝑉𝑡,𝑦(𝑡)≤𝜆₃𝜆₁𝜀+𝑀+𝑁𝑁+𝑁²2𝜏<0,₁≤𝑡≤𝜏₂, (41) which implies that 𝑉𝜏₂,𝑦𝜏₂≤𝑉𝜏₁,𝑦𝜏₁. (42)
This is a contradiction.
Hence, we obtain 𝜏₂≥𝑡_𝑚+1; without loss of generality, we may suppose that 𝜏₂∈[𝑡_𝑚+𝑞,𝑡_𝑚+𝑞+1),𝑞≥1. Next we first claim that there exists 𝜏₁>0 satisfying 𝜏₂>𝜏₁>𝑇₁ such that 𝑉𝜏₁,𝑦𝜏₁≥𝑁2𝜆₁𝜀². (43)
Suppose that this is false; then for all 𝑡∈(𝑇₁,𝜏₂), 𝑉(𝑡<𝑁𝑡,𝑦)2𝜆₁𝜀², (44) which implies that 𝜏₂=𝑡_𝑚+𝑞 in view of (37). Consequently, we have 𝑉𝑡_𝑚+𝑞,𝑦𝑡_𝑚+𝑞≥𝑁2+14𝜆₁𝜀²,𝑉𝑡⁻_𝑚+𝑞,𝑦𝑡⁻_𝑚+𝑞<𝑁2𝜆₁𝜀², (45) which implies that max𝜂_𝑚+𝑞,1−1𝜆₂𝐺²≥max𝜂_𝑚+𝑞,1−1𝑉𝑡⁻_𝑚+𝑞,𝑦𝑡⁻_𝑚+𝑞>14𝜆₁𝜀². (46)
Hence, we get max{𝜂_𝑚+𝑞,1}>1+(𝜆₁𝜀²/4𝜆₂𝐺²), which contradicts (29). So (43) holds.
Therefore, there are two situations 𝜏₂>𝑡_𝑚+𝑞 and 𝜏₂=𝑡_𝑚+𝑞. Next we discuss them, respectively.(𝑏₁) If 𝜏₂>𝑡_𝑚+𝑞, let 𝜏₁=inf_{𝜏1≤𝑡<𝜏2}𝑡∣𝑠∈𝑡,𝜏₂,𝑁2𝜆₁𝜀²(𝑠𝜏≤𝑉𝑠,𝑦)<𝑉₂,𝑦𝜏₂. (47)
We first show that 𝜏₁<𝑡_𝑚+𝑞. Suppose on the contrary that 𝜏₁∈[𝑡_𝑚+𝑞,𝜏₂); then 𝑉(𝑡)+𝜆𝑡,𝑦¹𝜀²2≥𝜆₁𝜀²2𝑁+1≥𝜆₂𝐺²>𝑉𝜉,𝑦_𝜏,𝑡−𝜏≤𝜉≤𝑡,𝑡∈𝜏₁,𝜏₂. (48)
From (13), we have 𝐷⁺𝑉𝑡,𝑦(𝑡)≤𝜆₃𝜆₁𝜀+𝑀+𝑁𝑁+𝑁²2𝜏<0,₁≤𝑡≤𝜏₂, (49) which implies that 𝑉𝜏₂,𝑦𝜏₂≤𝑉𝜏₁,𝑦𝜏₁; (50) which is a contradiction with the definition of 𝜏₁. Thus we obtain that 𝜏₁<𝑡_𝑚+𝑞. Suppose that 𝜏₁∈[𝑡_𝑚+𝑘,𝑡_𝑚+𝑘+1),1≤𝑘<𝑞.
We also have two cases.(𝑏_1𝑎) If 𝜏₁ is not impulsive point, that is to say 𝜏₁>𝑡_𝑚+𝑘, then considering the definition of 𝜏₁, we have 𝑉𝜏₁,𝑦𝜏₁=𝑁2𝜆₁𝜀². (51)
By the same argument as the above mentioned, we obtain that (48) still holds.
Hence, from (13), we get 𝑁2+14𝜆₁𝜀²≤𝑉𝜏₂,𝑦𝜏₂≤𝑉𝜏₁,𝑦𝜏₁+^𝑚+𝑞_{𝑖=𝑚+𝑘+1}𝑉𝑡_𝑖−𝑉𝑡⁻_𝑖≤𝑁2𝜆₁𝜀²+^𝑚+𝑞_{𝑖=𝑚+𝑘+1}max𝜂_𝑖,1−1𝑉𝑡⁻_𝑖≤𝑁2𝜆₁𝜀²+^𝑚+𝑞_{𝑖=𝑚+𝑘+1}max𝜂_𝑖,1−1𝜆₂𝐺², (52) which implies that 14𝜆₁𝜀²≤^𝑚+𝑞_{𝑖=𝑚+𝑘+1}max𝜂_𝑖,1−1𝜆₂𝐺², (53) which is a contradiction with (29). So 𝜏₁ is some impulsive point. (𝑏_1𝑏) If 𝜏₁ is some impulsive point, that is to see 𝜏₁=𝑡_𝑚+𝑘, then from the definition of 𝜏₁, it is clear that 𝑉𝜏₁⁻,𝑦𝜏₁⁻=𝑉𝑡⁻_𝑚+𝑘,𝑦𝑡⁻_𝑚+𝑘<𝑁2𝜆₁𝜀², (54)which implies that 𝑉𝜏₁,𝑦𝜏₁=𝑉𝑡_𝑚+𝑘,𝑦𝑡_𝑚+𝑘≤𝜂_𝑚+𝑘𝑉𝑡⁻_𝑚+𝑘,𝑦𝑡⁻_𝑚+𝑘<𝜂_𝑚+𝑘𝑁2𝜆₁𝜀²≤max𝜂_𝑚+𝑘𝑁,12𝜆₁𝜀². (55)
On the other hand, note that inequality (48) still holds; from (13) and (29), we have 𝑁2+14𝜆₁𝜀²≤𝑉𝜏₂,𝑦𝜏₂≤𝑉𝜏₁,𝑦𝜏₁+^𝑚+𝑞_{𝑖=𝑚+𝑘+1}𝑉𝑡_𝑖−𝑉𝑡⁻_𝑖<max𝜂_𝑚+𝑘𝑁,12𝜆₁𝜀²+^𝑚+𝑞_{𝑖=𝑚+𝑘+1}max𝜂_𝑖,1−1𝑉𝑡⁻_𝑖, (56) which implies that 14𝜆₁𝜀²≤max𝜂_𝑚+𝑘𝑁,1−12𝜆₁𝜀²+^𝑚+𝑞_{𝑖=𝑚+𝑘+1}max𝜂_𝑖,1−1𝜆₂𝐺²≤16𝑁𝑁2𝜆₁𝜀²+^𝑚+𝑞_{𝑖=𝑚+𝑘+1}max𝜂_𝑖,1−1𝜆₂𝐺². (57)
That means 16𝜆₁𝜀²<^𝑚+𝑞_{𝑖=𝑚+𝑘+1}max𝜂_𝑖,1−1𝜆₂𝐺², (58) which contradicts (29).
Hence, the first situation 𝜏₂>𝑡_𝑚+𝑞 is impossible. (𝑏₂) If 𝜏₂=𝑡_𝑚+𝑞, then by the same arguments as in the proof in (𝑏₁) and (43), we have 𝑉𝜏⁻₂,𝑦𝜏⁻₂>𝑁2𝜆₁𝜀². (59)
Then let 𝜏₁=inf_{𝜏1≤𝑡<𝜏2}𝑡∣𝑠∈𝑡,𝜏₂,𝑁2𝜆₁𝜀²≤𝑉𝑠,𝑦(𝑠)<𝑉𝜏₂,𝑦𝜏₂. (60)
The rest of the arguments are omitted. Finally we can find our desirable contradiction. Hence, (36) holds.
With above mentioned, the same arguments as before, if we replace 𝑇₁ with 𝑁^⋆, then there exists a 𝑇₂=𝑇₁+𝑤₁=𝑁^⋆+2𝑤₁ such that for 𝑡>𝑇₂, 𝑉(𝑡<𝑁𝑡,𝑦)2𝜆₁𝜀². (61)
Let 𝑇₂ replace 𝑇₁; then there exists a 𝑇₃=𝑁^⋆+3𝑤₁ such that for 𝑡>𝑇₃, 𝑉(𝑡<𝑁𝑡,𝑦)2−14𝜆₁𝜀². (62)
By induction hypothesis, we may prove, in general, that there exists a 𝑇_2𝑁=𝑁^⋆+2𝑁𝑤₁ such that 𝑡>𝑇_2𝑁, 𝑉(𝑡<𝑁𝑡,𝑦)2−𝑁−22𝜆₁𝜀²=𝜆₁𝜀². (63)
Therefore, we obtain that ‖𝑦(𝑡)‖<𝜀,𝑡>𝑁^⋆+2𝑁𝑤₁. In view of the choice of 𝑁^⋆, 𝑁, and 𝑤₁, the zero solution of (4) is uniformly asymptotically stable; that is, the equilibrium point of (1) is uniformly asymptotically stable. The proof of Theorem 7 is therefore complete.

Let |1+𝑑_𝑘^(𝑖)|≠|1+𝑑_𝑘^(𝑗)|, 𝑐_𝑖≠𝑐_𝑗, 𝑖≠𝑗, 𝑖,𝑗∈Λ, 𝑘=1,2,…, in Theorem 7; then we can have the following result.

Corollary 8. The equilibrium point of system (1) is uniformly asymptotically stable, if there exists 𝑛×𝑛 symmetric and positive definite matrix 𝑃 satisfying(𝐼₁) 𝜂≐∏^∞_𝑘=1max{max_𝑖∈Λ(1+𝑑_𝑘^(𝑖))²,1}<∞; (𝐼₂) 𝜆₃<(−𝑀−𝑁)/𝜆₁, where 𝜆₁ is the smallest eigenvalue of 𝑃 and 𝜆₃ is the largest eigenvalue of 2𝐶+𝐴𝐴^𝑇𝑃+𝐵𝐵^𝑇𝑃.

Remark 9. For using the less conservative conditions in Theorem 7, our results obviously improve some results established in the earlier references. In [5], condition 𝑦(𝑡_𝑘)=𝐷𝑦(𝑡⁻_𝑘) holds for all 𝑘∈ℤ₊; here note in our Theorem 7 that we only require that the solutions satisfy the hypothesis (𝐻₁) at impulsive points. In addition, our conditions are without requirement of the range of the largest eigenvalues of 𝑃⁻¹𝐷^𝑇𝑃𝐷 on (0,1), which are milder than the restrictions in [5].

By utilizing Lemma 6, we will give some sufficient conditions for globally exponential stability of the equilibrium point of system (1).

Theorem 10. Assume that there exists 𝑛×𝑛 symmetric and positive definite matrix 𝑃 such that 𝜆₁>0 is the smallest eigenvalue of 𝑃, 𝜆₃ is the largest eigenvalue of 𝑃⁻¹(𝐶𝑃+𝑃𝐶+𝑃𝐴𝐴^𝑇𝑃+𝑃𝐵𝐵^𝑇𝑃), 𝜂_𝑘 is the largest eigenvalue of 𝑃⁻¹𝐷_𝑘𝑃𝐷_𝑘, and 𝜆₁, 𝜆₃ and 𝜂_𝑘 satisfy the following conditions: (𝐻₃) one has 𝜆₃+𝑀𝜆₁+𝑁𝜆₁⋅max_𝑘∈ℤ+1𝜂_𝑘,1<0, (64)(𝐻₄) there exist constants 𝕌(>0), 𝛿(≥0) such that 𝛿<𝜇 and the following inequality ^𝑚_𝑘=1ln𝜂_𝑘−𝛿𝑡_𝑚−𝑡₀<𝕌∀𝑚∈ℤ₊ℎ𝑜𝑙𝑑𝑠, (65)

where 𝜇 satisfies the following inequality:0<𝜇≤−𝜆₃−𝑀𝜆₁−𝑁𝜆₁max_𝑘∈ℤ+1𝜂_𝑘,1⋅𝑒^𝜇𝜏. (66)

Then the equilibrium point of the system (1) is globally exponentially stable, and the exponential convergence rate is equal to (𝜇−𝛿)/2.

Proof. Let 𝑦(𝑡)=𝑦(𝜎,𝜑)(𝑡) be a solution of (4) through (𝜎,𝜑), 𝜎≥𝑡₀ (for convenience, we assume that 𝜎=𝑡₀). We next will prove that the zero solution of (4) is globally exponentially stable. Construct the Lyapunov functional as in Theorem 7; that is, 𝑉(𝑡,𝑦(𝑡))=𝑦^𝑇(𝑡)𝑃𝑦(𝑡); then we obtain the following:(1)𝜆₁‖𝑦(𝑡)‖²≤𝑉(𝑡,𝑦(𝑡))≤𝜆₂‖𝑦(𝑡)‖², where 𝜆₂ is the largest eigenvalue of 𝑃;(2)for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝑘=1,2,…, 𝐷⁺𝑉𝑡,𝑦(𝑡)₍₄₎≤𝜆₃+𝑀𝜆₁⁻¹𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+𝑁𝜆₁⁻¹𝑦^𝑇_𝜏𝑃𝑦_𝜏; (67)(3)𝑉(𝑡_𝑘,𝑦(𝑡_𝑘))≤𝜂_𝑘𝑉(𝑡⁻_𝑘,𝑦(𝑡⁻_𝑘)).
From (2), we have for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝑘=1,2,…, 𝐷⁺𝑉𝑡,𝑦(𝑡)₍₄₎≤−𝑃𝑉𝑡,𝑦(𝑡)+𝑄𝑉𝑡,𝑦(𝑡), (68) where 𝑉(𝑡)=sup_{𝑡−𝜏≤𝑠≤𝑡}𝑉(𝑠),   𝑃=−𝜆₃−𝑀𝜆₁⁻¹, and 𝑄=𝑁𝜆₁⁻¹.
For any 𝑡≥𝑡₀, suppose that 𝑡∈[𝑡_𝑚,𝑡_𝑚+1), 𝑚≥0. By Lemma 6 and condition (𝐻₄), we obtain 𝑉(𝑡)≤𝑉𝑡₀^𝑚_𝑘=1𝜂_𝑘𝑒^{−𝜇(𝑡−𝑡0)}≤𝑉𝑡₀𝑒^𝕌⋅𝑒^{𝛿(𝑡𝑚−𝑡0)}⋅𝑒^{−𝜇(𝑡−𝑡0)}≤𝑉𝑡₀𝑒^𝕌⋅𝑒^{𝛿(𝑡−𝑡0)}⋅𝑒^{−𝜇(𝑡−𝑡0)}≤𝜆₂𝑒^𝕌𝜑²_𝜏𝑒^{−(𝜇−𝛿)(𝑡−𝑡0)}, (69) where 𝜇 satisfies inequality (66).
Hence, we obtain for any 𝑡≥𝑡₀, 𝑦(𝑡)≤𝑒^(1/2)𝕌𝜆₂𝜆₁𝜑_𝜏𝑒^{−((𝜇−𝛿)/2)(𝑡−𝑡0)},𝑡≥𝑡₀, (70) where 𝜇 satisfies inequality (66).
Therefore, the zero solution of system (4) is globally exponentially stable; that is, the equilibrium point of system (1) is globally exponentially stable. The proof of Theorem 10 is complete.

Remark 11. In Theorem 10, if sup_𝑛∈ℤ+(∏^𝑛_𝑘=1𝜂_𝑘)<∞, then we can choose 𝛿=0 in condition (𝐻₃).
Let 𝜂_𝑘∈(0,1],𝑘=1,2,…, in Theorem 7; then we can have the following result.

Corollary 12. Assume that there exists 𝑛×𝑛 symmetric and positive definite matrix 𝑃 such that 𝜆₁ is the smallest eigenvalue of 𝑃, 𝜆₃ is the largest eigenvalue of 𝑃⁻¹(𝐶𝑃+𝑃𝐶+𝑃𝐴𝐴^𝑇𝑃+𝑃𝐵𝐵^𝑇𝑃), 𝜂_𝑘 is the largest eigenvalue of 𝑃⁻¹𝐷_𝑘𝑃𝐷_𝑘, and condition 𝜆₁𝜆₃+𝑀+𝑁<0ℎ𝑜𝑙𝑑𝑠. (71)

Then the equilibrium point of the system (1) is globally exponentially stable, and the exponential convergence rate is equal to 𝜇/2, where 𝜇 satisfies the following inequality:0<𝜇≤−𝜆₃−𝑀𝜆₁−𝑁𝜆₁max_𝑘∈ℤ+𝜂_𝑘⋅𝑒^𝜇𝜏. (72)

4. Example

In this section, we present a numerical example to illustrate our results.

Example 1. We consider Example  1 in [2] as follows: 𝑢^′₁(𝑡)=−2.5𝑢₁(𝑡)−0.5𝑓𝑢₁(𝑡)+0.1𝑓𝑢₂(𝑡)−0.1𝑓𝑢₁()𝑢𝑡−𝜏+0.2𝑓₂()𝑢𝑡−𝜏−1,_′2(𝑡)=−2𝑢₁(𝑡)+0.2𝑓𝑢₁(𝑡)−0.1𝑓𝑢₂(𝑡)+0.2𝑓𝑢₁()𝑢𝑡−𝜏+0.1𝑓₂(𝑡−𝜏)+4, (73) with impulses 𝑑_𝑘⁽¹⁾=11+2𝑘²𝑑−1,_𝑘(2)=11+𝑘²−1,𝑡_𝑘=𝑘,𝑘∈ℤ₊. (74)
The delayed feedback matrixes 𝐴, 𝐵, and 𝐶 are ,.𝐶=−2.500−2,𝐴=−0.50.10.2−0.1𝐵=−0.10.20.20.1 (75)
Since the activation function in [2] is described by 𝑓=𝑓_𝑖=0.5(|𝑥+1|−|𝑥−1|), 𝑖=1,2, then we have 𝑀=𝑁=1. We may choose 𝑃=𝐸 (unit matrix); then 𝜆₁=𝜆₂=1. Note that 𝑃⁻¹𝐷_𝑘𝑃𝐷_𝑘=𝐷²_𝑘=11+2𝑘²0101+𝑘²,𝑃⁻¹𝐶𝑃+𝑃𝐶+𝑃𝐴𝐴^𝑇𝑃+𝑃𝐵𝐵^𝑇𝑃=2𝐶+𝐴𝐴^𝑇+𝐵𝐵^𝑇=−4.69−0.11−0.11−3.9=Δ. (76) then we get 𝜂_𝑘=1+1/𝑘², 𝜂=∏^∞_𝑘=1(1+1/𝑘²)<∞. Matrix Δ^,s characteristic equation is 𝜆²+8.59𝜆+18.2789=0. (77)

By a straightforward calculation, we obtain that the largest characteristic root 𝜆₃≈−3.885<−2. By Theorem 7, the equilibrium point of system (73) is uniformly asymptotically stable with impulses (74) for any 𝜏>0, which is shown in Figure 1(a).

(a)

(b)

Figure 1 

(a) State trajectories of system (73) with impulses (74) and 𝜏=3. (b) State trajectories of system (73) with impulses (79) and 𝜏=0.17.

However, the criteria in [5] are invalid here. In fact, condition 𝑦(𝑡_𝑘)=𝐷𝑦(𝑡⁻_𝑘) is not satisfied here. Moreover, because of the impulsive effect, the criteria in [2] are also invalid here. Therefore, our results are less conservative than those given result in [2, 5].

Furthermore, let 𝜏=0.17. Note that ∏^∞_𝑘=1(1+1/𝑘²)<∞, 𝜆₃=−3.885<−2; then one can choose 𝛿=0 in Theorem 10. So all conditions of Theorem 10 are satisfied. Therefore, the equilibrium point of system (73) is globally exponentially stable, and the exponential convergence rate is equal to 𝜇, where 𝜇>0 satisfies 𝜇+𝑒^0.17𝜇−2.885<0. (78)

If 𝑑_𝑘⁽¹⁾, 𝑑_𝑘⁽²⁾ in above example are given as follows: 𝑑_𝑘⁽¹⁾=√√2.2−1,𝑘=2𝑛−1,0.23−1,𝑘=2𝑛,𝑛∈ℤ₊,𝑑_𝑘⁽²⁾√√1.7−1,𝑘=2𝑛−1,0.4−1,𝑘=2𝑛,𝑛∈ℤ₊, (79) then we finally get 𝜆₃=−3.885<−3.500, 𝛿=0. So all conditions of Theorem 10 are still satisfied. Therefore, the equilibrium point of system (73) is globally exponentially stable with impulses (79), and the exponential convergence rate is equal to 𝜇, where 𝜇>0 satisfies 𝜇+2.5𝑒^0.17𝜇−2.885<0. (80)

The simulation is shown in Figure 1(b). However, it is easy to check that the impulsive delayed Halanay inequality in [23, 26] is not feasible here. Thus our results can be applied to the case not covered in [23, 26].

5. Conclusion

The uniform asymptotic stability and global exponential stability of impulsive HNN with time delays are considered in this paper. Some new stability conditions are obtained by means of constructing the extended impulsive Halanay inequality, Lyapunov functional methods, and linear matrix inequality approach. Moreover, our results can be applied to the case not covered in some other existing criteria. Hence, the results extend and improve the earlier publications. An example is given to illustrate the feasibility of the results and the effects of impulses.

Acknowledgments

This work was jointly supported by the Project of Shandong Province Higher Educational Science and Technology Program (J12LI04), Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (BS2012DX039), and National Natural Science Foundation of China (11226136, 11301308).