Globally Exponential Stability of Impulsive Neural Networks with Given Convergence Rate

Abstract

This paper deals with the stability problem for a class of impulsive neural networks. Some sufficient conditions which can guarantee the globally exponential stability of the addressed models with given convergence rate are derived by using Lyapunov function and impulsive analysis techniques. Finally, an example is given to show the effectiveness of the obtained results.

1. Introduction

Recently, special interest has been devoted to the dynamics analysis of neural networks due to their potential applications in different areas of science. Particularly, there has been a significant development in the theory of neural networks with impulsive effects [1–9], since such neural networks with impulsive effect can be used as an appropriate description of the phenomena of abrupt qualitative dynamical changes of essential continuous time systems. Based on the theory of impulsive differential equations [10–17], some sufficient conditions guaranteeing the exponential stability are derived [18–24]. For example, in [8], the author has obtained a criterion of exponential stability for a Hopfield neural network with periodic coefficients; in [18], by constructing the extended impulsive delayed Halanay inequality and Lyapunov functional methods, authors have got some sufficient conditions ensuring exponential stability of the unique equilibrium point of impulsive Hopfield neural networks with time delays. They all have obtained exponential stability for some kinds of neural networks through different methods. However, most of the existing results about the exponential stability of impulsive neural networks have a common feature that the exponential convergence rate cannot be derived, or derived but not the given one [8, 18, 23, 24]. The purpose of this paper is to establish some criteria which can guarantee the globally exponential stability of impulsive neural networks with the given convergence rate by using Lyapunov function and impulsive analysis techniques. This work is organized as follows. In Section 2, we introduce some basic definitions and notations. In Section 3, the main results are presented. In Section 4, an example is discussed to illustrate the results.

2. Preliminaries

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

Consider the following impulsive neural networks: 𝑥̇_𝑖(𝑡)=−𝑎_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+𝐼_𝑖(𝑡),𝑡≥𝑡₀,𝑡≠𝑡_𝑘,Δ𝑥_𝑖|_{𝑡=𝑡𝑘}=𝑥_𝑖𝑡_𝑘−𝑥_𝑖𝑡⁻_𝑘,𝑘∈ℤ₊,𝑖∈Λ, (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 state of the neurons, 𝑓_𝑗 denotes the measures of response to its incoming potentials of the unit 𝑗 at time 𝑡; 𝐼_𝑖(𝑡) is the input of the unit 𝑖 at time 𝑡. 𝑃𝐶[𝐼,ℝ]≜{𝜑∶𝐼→ℝ∣𝜑(𝑡⁺)=𝜑(𝑡) for 𝑡∈𝐼, 𝜑(𝑡⁻) exists for 𝑡∈𝐼, 𝜑(𝑡⁻)=𝜑(𝑡) for all but points 𝑡_𝑘∈𝐼}, where 𝐼⊂ℝ is an interval, 𝜑(𝑡⁺) and 𝜑(𝑡⁻) denote the left limit and right limit of function 𝜑(𝑡), respectively. 𝑎_𝑖(𝑡)> 0, 𝑏_𝑖𝑗(𝑡),𝐼_𝑖(𝑡)∈𝑃𝐶[[𝑡₀,+∞),ℝ]. For given 𝑡₀,𝑥(𝑡₀)=𝑥₀=(𝑥⁰₁,𝑥⁰₂,…,𝑥⁰_𝑛)∈ℝ^𝑛, we denote by 𝑥(𝑡) the solution of system (1) with initial value (𝑡₀,𝑥(𝑡₀)).

In this paper, we assume that some conditions are satisfied so that the equilibrium point of system (1) does exist, see [16, 17]. Assume that 𝑥^⋆=(𝑥^⋆₁,𝑥^⋆₂,…,𝑥^⋆_𝑛)^𝑇 is an equilibrium point of system (1). Impulsive operator is viewed as perturbation of the point 𝑥^⋆ of such system without impulsive effects. We assume that the following impulsive condition holds.?(H₀) Δ𝑥_𝑖|_{𝑡=𝑡𝑘}=𝑥_𝑖(𝑡_𝑘)−𝑥_𝑖(𝑡⁻_𝑘)=𝜎_𝑖𝑘(𝑥_𝑖(𝑡⁻_𝑘)−𝑥^⋆_𝑖), 𝜎_𝑖𝑘∈ℝ, 𝑖∈Λ, 𝑘∈ℤ₊.

Furthermore, we will assume that the response function 𝑓_𝑖 satisfies the following condition.?(H₁) 𝑓_𝑖 is globally Lipschizian with Lipschitz constant 𝑙_𝑖>0, that is, |𝑓_𝑖(𝑠₁)−𝑓_𝑖(𝑠₂)|≤𝑙_𝑖|𝑠₁−𝑠₂|, for all 𝑠₁,𝑠₂∈ℝ, 𝑖∈Λ.

Note that 𝑥^⋆ is an equilibrium point of system (1), one can derive from system (1) that the transformation 𝑧_𝑖=𝑥_𝑖−𝑥^⋆_𝑖, 𝑖∈Λ transforms such system into the following system: 𝑧̇_𝑖(𝑡)=−𝑎_i(𝑡)𝑧_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝐹_𝑗𝑧_𝑗(𝑡),𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑧_𝑖𝑡_𝑘=1+𝜎_𝑖𝑘𝑧_𝑖𝑡⁻_𝑘,𝑖∈Λ,𝑘∈ℤ₊, (2) where 𝐹_𝑗(𝑧_𝑗(𝑡))=𝑓_𝑗(𝑥^⋆_𝑗+𝑧_𝑗(𝑡))−𝑓_𝑗(𝑥^⋆_𝑗), and from the condition (H₁), it holds that ‖𝐹_𝑗(𝑧_𝑗(𝑡))‖≤𝑙_𝑗‖𝑧_𝑗(𝑡)‖, 𝑗∈Λ.

Furthermore, let 𝑦_𝑖(𝑡)=𝑧_𝑖(𝑡)𝑒^{𝛼(𝑡−𝑡0)}, 𝑖∈Λ, then system (2) becomes as follows: 𝑦̇_𝑖(𝑡)=𝛼−𝑎_𝑖(𝑡)𝑦_𝑖(𝑡)+𝑒^{𝛼(𝑡−𝑡0)𝑛}_𝑗=1𝑏_𝑖𝑗(𝑡)𝐹_𝑗𝑦_𝑗(𝑡)𝑒^{−𝛼(𝑡−𝑡0)},𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑦_𝑖𝑡_𝑘=1+𝜎_𝑖𝑘𝑦_𝑖𝑡⁻_𝑘,𝑖∈Λ,𝑘∈ℤ₊. (3)

To prove the stability of 𝑥^⋆ of system (1), it is equal to prove the stability of the zero solution of system (2), and also equal to the boundedness of system (3).

In the following, the notion 𝐴^𝑇 means the transpose of a square matrix 𝐴. We will use the notation 𝐴>0 (or 𝐴<0, 𝐴≥0, 𝐴≤0) to denote that the matrix 𝐴 is a positive definite (negtive definite, positive semidefinite, an negative semidefinite) marix.

Let 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇, 𝐴(𝑡)=diag[𝑎₁(𝑡),𝑎₂(𝑡),…,𝑎_𝑛(𝑡)], 𝐵(𝑡)=(𝑏_𝑖𝑗(𝑡))_𝑛×𝑛, 𝐿=diag[𝑙₁,𝑙₂,…,𝑙_𝑛], 𝐼=diag[1,1,…,1], 𝐷_𝑘=diag[1+𝜎_1𝑘,1+𝜎_2𝑘,…,1+𝜎_𝑛𝑘], 𝐹(𝑦)=(𝐹₁(𝑦₁),𝐹₂(𝑦₂),…,and𝐹_𝑛(𝑦_𝑛))^𝑇 then system (3) with initial condition becomes as follows: 𝑦̇(𝑡)=(𝛼𝐼−𝐴(𝑡))𝑦(𝑡)+𝑒^{−𝛼(𝑡−𝑡0)}𝐵(𝑡)𝐹𝑦(𝑡)𝑒^{−𝛼(𝑡−𝑡0)},𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑦𝑡_𝑘=𝐷_𝑘𝑦𝑡⁻_𝑘,𝑖∈Λ,𝑘∈ℤ₊. (4)

We introduce a definition as follows.

Definition 1. Assume 𝑥^⋆=(𝑥^⋆₁,𝑥^⋆₂,…,𝑥^⋆_𝑛)^𝑇∈ℝ^𝑛 is the equilibrium point of system (1), then the equilibrium point 𝑥^⋆ of system (1) is said to be globally exponential stable with given convergence rate 𝛼>0. If for any initial data 𝑥(𝑡₀)=𝑥₀∈ℝ, there exists a constant 𝑀⩾1, such that 𝑥𝑡,𝑡₀,𝑥₀−𝑥^⋆⩽𝑥₀−𝑥^⋆𝑀𝑒^{−𝛼(𝑡−𝑡0)},𝑡⩾𝑡₀. (5)
From the transformation 𝑧_𝑖=𝑥_𝑖−𝑥^⋆_𝑖, 𝑖∈Λ, and 𝑧(𝑡₀)=𝑧₀=𝑥₀−𝑥^⋆, the globally exponential stability of the equilibrium point 𝑥^⋆ of system (1) can be transformed into the globally exponential stability of trivial solution of system (2), so (5) can be rewritten as follows: 𝑧𝑡,𝑡₀,𝑧₀≤𝑧𝑡₀𝑀𝑒^{−𝛼(𝑡−𝑡0)},𝑡⩾𝑡₀. (6)
Furthermore, form the transformation 𝑦(𝑡)=𝑧(𝑡,𝑡₀,𝑧₀)𝑒^{𝛼(𝑡−𝑡0)}, 𝑖∈Λ, the globally exponential stability of trivial solution of system (2) can be transformed into the boundedness of the solution of system (4) and it can be rewritten as follows: 𝑦(𝑡)≤𝑦𝑡₀𝑀,𝑡⩾𝑡₀. (7)

3. Main Results

Theorem 2. Given constant 𝛼>0. The equilibrium point of the system (1) is globally exponentially stable with the given convergence rate 𝛼, if the conditions (H₀) and (H₁) are fulfilled; moreover, suppose that (i)𝛼𝐼−𝐴(𝑡)+𝐵(𝑡)𝐿≤0, for all 𝑡>𝑡₀, (ii)∏^∞_𝑘=1max_𝑖∈Λ(1+𝜎_𝑖𝑘)<∞, and 𝜎_𝑖𝑘≥0, 𝑖∈Λ, 𝑘∈ℤ₊.

Proof. We only need to prove 𝑦(𝑡)=𝑧(𝑡)𝑒^{𝛼(𝑡−𝑡0)} is bounded when 𝑡≥𝑡₀, where 𝑧(𝑡)=𝑧(𝑡,𝑡₀,𝑧₀) is a solution of (3) through (𝑡₀,𝑧₀).
Consider the Lyapunov function as follows: 𝑉(𝑡)=𝑦(𝑡)^𝑇𝑦(𝑡)=^𝑛_𝑖=1𝑦²_𝑖(𝑡). (8) Particularly, 𝑉(𝑡₀)=∑^𝑛_𝑖=1𝑦²_𝑖(𝑡₀).
Then from conditions (H₀)-(H₁) and (i), we get the upper right-hand derivative of 𝑉(𝑡) along the solutions of system (3), for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝑘∈ℤ₊𝐷⁺𝑉(𝑡)=2^𝑛_𝑖=1𝑦_𝑖(𝑡)𝑦̇_𝑖(𝑡)=2^𝑛_𝑖=1𝑧_𝑖(𝑡)𝑒^{2𝛼(𝑡−𝑡0)}𝑧̇_𝑖(𝑡)+𝛼𝑧_𝑖(𝑡)=2^𝑛_𝑖=1𝑧_𝑖(𝑡)𝑒^{2𝛼(𝑡−𝑡0)}×−𝑎_𝑖(𝑡)𝑧_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝐹_𝑗𝑧_𝑗(𝑡)+𝛼𝑧_𝑖(𝑡)=2^𝑛_𝑖=1𝑒^{2𝛼(𝑡−𝑡0)}×𝛼−𝑎_𝑖(𝑡)𝑧²_𝑖(𝑡)+𝑧_𝑖(𝑡)^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝐹_𝑗𝑧_𝑗(𝑡)≤2^𝑛_𝑖=1𝑒^{2𝛼(𝑡−𝑡0)}×𝛼−𝑎_𝑖(𝑡)𝑧_𝑖(𝑡)²+𝑧_𝑖(𝑡)^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑙_𝑗𝑧_𝑗(𝑡)=2^𝑛_𝑖=1𝑒^{2𝛼(𝑡−𝑡0)}𝑧²_𝑖(𝑡)𝛼−𝑎_𝑖(𝑡)+2^𝑛_𝑖=1𝑒^{2𝛼(𝑡−𝑡0)}𝑧_𝑖(𝑡)^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑙_𝑗𝑧_𝑗(𝑡(𝑡))=2𝑦^𝑇[𝛼𝐼−𝐴(𝑡)+𝐵(𝑡)𝐿]𝑦(𝑡)≤0, (9) which implies the functional 𝑉(𝑡) is nonincreasing for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝑘∈ℤ₊. By condition (ii), it holds that 𝑉𝑡_𝑘=^𝑛_𝑖=1𝑦²_𝑖𝑡_𝑘=^𝑛_𝑖=1𝑧²_𝑖𝑡_𝑘𝑒^{2𝛼(𝑡𝑘−𝑡0)}=^𝑛_𝑖=1𝑒^{2𝛼(𝑡−𝑡0)}𝑧_𝑖𝑡⁻_𝑘+𝐽_𝑖𝑧_𝑖𝑡⁻_𝑘²=^𝑛_𝑖=1𝑒^{2𝛼(𝑡−𝑡0)}1+𝜎_𝑖𝑘²𝑧²_𝑖𝑡⁻_𝑘≤max_𝑖∈Λ1+𝜎_𝑖𝑘²𝑉𝑡⁻_𝑘. (10) For any 𝑡∈[𝑡₀,𝑡₁), since 𝑉(𝑡) is nonincreasing, it holds that 𝑉(𝑡)≤𝑉(𝑡₀); moreover, 𝑉𝑡₁≤max_𝑖∈Λ1+𝜎_𝑖1²𝑉𝑡⁻₁≤max_𝑖∈Λ1+𝜎_𝑖1²𝑉𝑡₀. (11) Similarly, for any 𝑡∈[𝑡₁,𝑡₂), it holds that 𝑉(𝑡)≤𝑉(𝑡₁)≤max_𝑖∈Λ(1+𝜎_𝑖1)²𝑉(𝑡₀), and 𝑉𝑡₂≤max_𝑖∈Λ1+𝜎_𝑖2²𝑉𝑡⁻₂≤max_𝑖∈Λ1+𝜎_𝑖2²max_𝑖∈Λ1+𝜎_𝑖1²𝑉𝑡₀. (12) Thus, it can be deduced that for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), ?𝑘∈ℤ₊, 𝑉(𝑡)≤^𝑘_𝑗=1max_𝑖∈Λ1+𝜎_𝑖𝑗²𝑉𝑡₀. (13) Hence, we obtain that for any 𝑡≥𝑡₀, 𝑉(𝑡)≤^∞_𝑗=1max_𝑖∈Λ1+𝜎_𝑖𝑗²𝑉𝑡₀, (14) which implies that 𝑦(𝑡)≤𝑀𝑦𝑡₀,𝑡≥𝑡₀, (15) where 𝑀=∏^∞_𝑘=1max_𝑖∈Λ(1+𝜎_𝑖𝑘)<∞.
The proof of Theorem 2 is complete.

Remark 3. Most of the existing results about the exponential stability of impulsive neural networks cannot effectively control the convergence rate. It is interesting to see that Theorem 2 can guarantee the globally exponential stability of impulsive neural networks with the given convergence rate.

Remark 4. In particular, if 𝐴(𝑡)≡𝐴,𝐵(𝑡)≡𝐵 in Theorem 2, where 𝐴, 𝐵 are constant matrices, then condition 𝛼𝐼−𝐴+𝐵𝐿<0 can be easily checked via Matlab.

Theorem 5. Given constant 𝛼>0. The equilibrium point of the system (1) is globally exponentially stable with the given convergence rate 𝛼, if the conditions (H₀)-(H₁) are fulfilled; moreover, suppose that (i)there exists a constant 𝜆>0, such that (𝛼+𝜆)𝐼−𝐴(𝑡)+𝐵(𝑡)𝐿<0, for all 𝑡>𝑡₀, (ii)𝜏≜min_𝑘∈ℤ+{𝑡_𝑘+1−𝑡_𝑘}, max_𝑖∈Λ(1+𝜎_𝑖𝑘)≤𝑀_𝑘𝑒^𝜆𝜏, where 1≤𝑀_𝑘<∞ and ∏^∞_𝑘=1𝑀_𝑘<∞, 𝑖∈Λ, 𝑘∈ℤ₊.

Proof. We only need to prove that 𝑦(𝑡)=𝑧(𝑡)𝑒^{𝛼(𝑡−𝑡0)} is bounded when 𝑡≥𝑡₀, where 𝑧(𝑡)=𝑧(𝑡,𝑡₀,𝑧₀) is a solution of (3) through (𝑡₀,𝑧₀).
Consider the Lyapunov function as follows: 𝑉(𝑡)=𝑦(𝑡)^𝑇𝑦(𝑡)=^𝑛_𝑖=1𝑦²_𝑖(𝑡). (16) In particular, 𝑉(𝑡₀)=∑^𝑛_𝑖=1𝑦²_𝑖(𝑡₀).
Then from conditions (H₀)-(H₁) and (i), we get the upper right-hand derivative of 𝑉(𝑡) along the solutions of system (3), for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝑘∈ℤ₊𝐷⁺𝑉(𝑡)=2^𝑛_𝑖=1𝑦_𝑖(𝑡)𝑦̇_𝑖(𝑡)≤2𝑦(𝑡)^𝑇(𝛼𝐼−𝐴(𝑡)+𝐵(𝑡)𝐿)𝑦(𝑡)<−2𝜆𝑉(𝑡). (17) Thus, 𝑉(𝑡)<𝑉𝑡_𝑘𝑒^{−2𝜆(𝑡−𝑡𝑘)}<𝑉𝑡_𝑘𝑒^−2𝜆𝜏,𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘∈ℤ₊. (18) By condition (ii), it holds that 𝑉𝑡_𝑘=^𝑛_𝑖=1𝑦²_𝑖𝑡_𝑘=^𝑛_𝑖=1𝑒^{2𝛼(𝑡−𝑡0)}1+𝜎_𝑖𝑘²𝑧²_𝑖𝑡⁻_𝑘≤𝑀²_𝑘𝑒^{𝑛2𝜆𝜏}_𝑖=1𝑒^{2𝛼(𝑡−𝑡0)}𝑧²_𝑖𝑡⁻_𝑘≤𝑀²_𝑘𝑒^2𝜆𝜏𝑉𝑡⁻_𝑘. (19) For any 𝑡∈[𝑡₀,𝑡₁), it holds that 𝑉(𝑡)≤𝑉(𝑡₀)𝑒^−2𝜆𝜏, moreover 𝑉𝑡₁≤𝑀²₁𝑒^2𝜆𝜏𝑉𝑡⁻₁≤𝑀²₁𝑉𝑡₀. (20) Similarly, for any 𝑡∈[𝑡₁,𝑡₂), it holds that 𝑉(𝑡)≤𝑉(𝑡₁)𝑒^−2𝜆𝜏≤𝑀²₁𝑉(𝑡₀)𝑒^−2𝜆𝜏, and 𝑉𝑡₂≤𝑀₂𝑒^2𝜆𝜏𝑉𝑡⁻₂≤𝑀²₂𝑀²₁𝑉𝑡₀. (21) Without loss of generality, when 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝑘∈ℤ₊, it can be deduced that 𝑉(𝑡)≤^𝑘_𝑗=1𝑀²_𝑗𝑒^−2𝜆𝜏𝑉𝑡₀. (22) Hence, we obtain that for any 𝑡≥𝑡₀, 𝑉(𝑡)≤^∞_𝑗=1𝑀²_𝑗𝑉𝑡₀, (23) which implies that 𝑦(𝑡)≤𝑀𝑦𝑡₀,𝑡≥𝑡₀, (24) where 𝑀=(∏^∞_𝑗=1𝑀_𝑗)<∞.
The proof of Theorem 5 is complete.

Remark 6. Although Theorem 5 enhances the restriction on condition (i), the impulsive restriction in (ii) is weaker; that is, 𝜎_𝑖𝑘 is not necessary to converge to 0 as 𝑘 is large enough, provided that the impulsive intervals are not too small.

Theorem 7. Given constant 𝛼>0. The equilibrium point of the system (1) is globally exponentially stable with the given exponential convergence rate 𝛼, if the conditions (H₀)-(H₁) are fulfilled; moreover, suppose that (i)there exists a constant 𝜆>0, such that (𝛼−𝜆)𝐼−𝐴(𝑡)+𝐵(𝑡)𝐿<0, for all 𝑡>𝑡₀, (ii)𝜏≜max_𝑘∈ℤ+{𝑡_𝑘+1−𝑡_𝑘} and 𝛽_𝑘𝑒^𝜆𝜏≤1, where 𝛽_𝑘=max_𝑖∈Λ(1+𝜎_𝑖𝑘), −1≤𝜎_𝑖𝑘≤0, 𝑖∈Λ, 𝑘∈ℤ₊.

Proof. We only need to prove that 𝑦(𝑡)=𝑧(𝑡)𝑒^{𝛼(𝑡−𝑡0)} is bounded when 𝑡≥𝑡₀, where 𝑧(𝑡)=𝑧(𝑡,𝑡₀,𝑧₀) is a solution of (3) through (𝑡₀,𝑧₀).
Consider the Lyapunov functional as follows: 𝑉(𝑡)=𝑦(𝑡)^𝑇𝑦(𝑡)=^𝑛_𝑖=1𝑦²_𝑖(𝑡). (25) Particularly, 𝑉(𝑡₀)=∑^𝑛_𝑖=1𝑦²_𝑖(𝑡₀).
Then from conditions (H₀)-(H₁) and (i), we get the upper right-hand derivative of 𝑉(𝑡) along the solutions of system (1), for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝑘∈ℤ₊𝐷⁺𝑉(𝑡)=2^𝑛_𝑖=1𝑦_𝑖(𝑡)𝑦̇_𝑖(𝑡)=2𝑦(𝑡)^𝑇(𝛼𝐼−𝐴(𝑡)+𝐵(𝑡)𝐿)𝑦(𝑡)<2𝜆𝑉(𝑡). (26) Thus, 𝑉(𝑡)<𝑉𝑡_𝑘𝑒^{2𝜆(𝑡−𝑡𝑘)}<𝑉𝑡_𝑘𝑒^2𝜆𝜏,𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘∈ℤ₊. (27) By condition (ii), it holds that 𝑉t_𝑘=^𝑛_𝑖=1𝑦²_𝑖𝑡_𝑘≤max_𝑖∈Λ1+𝜎_𝑖𝑘²𝑉𝑡⁻_𝑘=𝛽²_𝑘𝑉𝑡⁻_𝑘,𝑘∈ℤ₊. (28) By simple induction, we can prove that for any 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝑘∈ℤ₊, 𝑉(𝑡)≤^𝑘_𝑗=1𝛽²_𝑗𝑒^{2(𝑘+1)𝜆𝜏}𝑉𝑡₀≤𝑒^2𝜆𝜏𝑉𝑡₀, (29) which implies that 𝑦(𝑡)≤𝑀𝑦𝑡₀,𝑡≥𝑡₀, (30) where 𝑀=𝑒^𝜆𝜏.
The proof of Theorem 7 is complete.

4. Applications

The following illustrative example will demonstrate the effectiveness of our results.

Example 8. Consider the following impulsive neural networks: 𝑥̇₁(𝑡)=−𝑎₁(𝑡)𝑥₁(𝑡)+𝑏₁₁(𝑡)𝑓₁𝑥₁(𝑡)+𝑏₁₂(𝑡)𝑓₂𝑥₂(𝑡)+𝐼₁(𝑡),𝑥̇₂(𝑡)=−𝑎₂(𝑡)𝑥₂(𝑡)+𝑏₂₁(𝑡)𝑓₁𝑥₁(𝑡)+𝑏₂₂(𝑡)𝑓₂𝑥₂(𝑡)+𝐼₂(𝑡),𝑥₁𝑡_𝑘=1+𝜎_1𝑘𝑥₁𝑡⁻_𝑘−𝜎_1𝑘,𝑘∈ℤ₊,𝑥₂𝑡_𝑘=1+𝜎_2𝑘𝑥₂𝑡⁻_𝑘−𝜎_2𝑘,𝑘∈ℤ₊, (31) where 𝑓₁(𝑢)=𝑓₂(𝑢)=(|𝑢+1|−|𝑢−1|)/2.
It is easy to see that 𝑓_𝑗, 𝑗=1,2 satisfying hypothesis (H₁) with 𝑙_𝑖=𝑙₂=1. We have 𝑎₁(𝑡)=3, 𝑎₂(𝑡)=2−(3/2)cos(2𝑡), 𝑏₁₁(𝑡)=|sin𝑡|/2, 𝑏₁₂(𝑡)=1/2, 𝑏₂₁(𝑡)=2, 𝑏₂₂(𝑡)=−1−(cos(2𝑡)/2), 𝐼₁=(5/2)−(|sin𝑡|/2), 𝐼₂=1−cos(2𝑡), 𝜎_1𝑘=1+(1/5𝑘²)−1, 𝜎_2𝑘=1+(1/6𝑘²)−1.
Let 𝛼=3. It can be deduced that 𝛼𝐼−𝐴(𝑡)+𝐵(𝑡)𝐿≤0 and ∏^∞_𝑘=1max_𝑖=1,2|1+𝜎_𝑖𝑘|<∞. Hence, all the conditions of Theorem 2 are satisfied; then the equilibrium point 𝑥^⋆=(1,1) of the above system (31) is globally exponentially stable with the given convergence rate 𝛼=3.

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, 11171192).