Existence and Global Exponential Stability of Equilibrium for Impulsive Cellular Neural Network Models with Piecewise Alternately Advanced and Retarded Argument

Abstract

We introduce impulsive cellular neural network models with piecewise alternately advanced and retarded argument (in short IDEPCA). The model with the advanced argument is system with strong anticipation. Some sufficient conditions are established for the existence and global exponential stability of a unique equilibrium. The approaches are based on employing Banach’s fixed point theorem and a new IDEPCA integral inequality of Gronwall type. The criteria given are easily verifiable, possess many adjustable parameters, and depend on impulses and piecewise constant argument deviations, which provides exibility for the design and analysis of cellular neural network models. Several numerical examples and simulations are also given to show the feasibility and effectiveness of our results.

1. Introduction

Chua and Yang [1] proposed a novel class of information-processing systems called cellular neural networks (CNNs) in 1988. Like neural networks, it is a large-scale nonlinear analog circuit which processes signals in real time. Like cellular automata [2] it is made of a massive aggregate of regularly spaced circuit clones, called cells, which communicate with each other directly only through its nearest neighbors. Each cell is made of a linear capacitor, a nonlinear voltage-controlled current source, and a few resistive linear circuit elements. The key features of neural networks are asynchronous parallel processing and global interaction of network elements. Impressive applications of neural networks have been proposed for various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision. For the circuit diagram and connection pattern implementing the CNN, one can refer to [1]. The CNN can be applied in signal processing and can also be used to solve some image processing and pattern recognition problems [3]. However, it is necessary to solve some dynamic image processing and pattern recognition problems by using delayed cellular neural networks (DCNN) [4–6]. The study of the stability of CNN and DCNN is known to be an important problem in theory and applications.

On the other hand, in real world, many evolutionary processes are characterized by abrupt changes at certain time. These changes are known to be impulsive phenomena, which are included in many fields such as physics, chemistry, population dynamics, and optimal control. Fundamental theory of impulsive differential equations has been developed in [7]. Furthermore, researches of impulsive differential equations have been received much interesting in recent years [8–18]. Meanwhile, several kinds of neural networks with impulse have been investigated. In particular, Xu and Yang established the delay differential inequalities with impulsive initial conditions; some new sufficient conditions for global exponential stability of impulsive delay model were obtained [15, 16].

Most neural networks can be classified into two types, continuous or discrete. However, many real-world systems and natural processes cannot be categorized into one of them. They display characteristics both continuous and discrete styles. For instance, some biological neural networks in biology, bursting rhythm models in pathology, and optimal control models in economics are characterized by abrupt changes of state. These are the familiar impulsive phenomena.

It is well known that applications of CNN depend crucially on the dynamical behavior of the networks. In these applications, stability and convergence of neural networks are prerequisites. However, in the design of neural networks one is interested not only in the uniform asymptotic stability but also in the global exponential stability, which guarantees a neural network to converge fast enough in order to achieve fast response. In addition, in the analysis of dynamical neural networks for parallel computation and optimization, to increase the rate of convergence to the equilibrium point of the networks and reduce the neural computing time, it is necessary to ensure a desired exponential convergence rate of the networks' trajectories, starting from arbitrary initial states to the equilibrium point which corresponds to the optimal solution. Thus, from the mathematical and engineering points of view, it is required that the neural networks have a unique equilibrium point which is globally exponentially stable. Therefore, the problem of stability analysis has received great attention and many results on this topic have been reported in the literature. See, for instance, [4, 9, 13, 19–27] and references cited therein.

1.1. Piecewise Constant Impulsive Systems

Differential equations with piecewise constant argument (in short DEPCA) are first considered by Shah and Wiener [28] and Cooke and Wiener [29] in the 80s and have been developed by many authors. Applications of DEPCAs are discussed in [30]. Theory and practice of DEPCA of general type, have been discussed extensively in [31–37]. Piecewise constant systems exist in widely expanded areas such as biomedicine, chemistry, mechanical engineering, and physics. The systematical studies with mathematical models involving piecewise constant arguments were initiated for solving some biomedical problems. These kinds of equations are similar in structure to those found in certain sequential-continuous models of disease dynamics. In [38], the following system of equations describing the dynamics of the disease for generation 𝑛=1,2,… is investigated: 𝑑𝐼^(𝑛)(𝑡)(𝑡)𝐼𝑑𝑡=−𝑐^(𝑛)(𝑡)+𝑘(𝑡)𝑆^(𝑛)(𝑡)𝐼^(𝑛)(𝑡),𝑛<𝑡≤𝑛+1,𝑑𝑆^(𝑛)(𝑡)(𝑡)𝑆𝑑𝑡=−𝑐^(𝑛)(𝑡)−𝑘(𝑡)𝑆^(𝑛)(𝑡)𝐼^(𝑛)(𝑡),𝑛<𝑡≤𝑛+1, (1) while 𝐼⁽¹⁾(1)=𝐼₀,𝑆⁽¹⁾(1)=𝑆₀, (2) where 𝑐 is the death rate and 𝑘 is the horizontal transmission factor. These types of models are special cases of the general form 𝑑𝑥(𝑡)𝑑𝑡=𝐹𝑡,𝑥_𝑡,[𝑡]<𝑡≤[𝑡]+1,𝑥_[𝑡]=𝜙_[𝑡],𝜙_[𝑡]=𝐺[𝑡],𝑥_[𝑡],[𝑡]≥2,𝜙₁=𝐻, (3) which arise naturally in a number of models of epidemic. DEPCAs usually describe hybrid dynamical systems (a combination of continuous and discrete) and so combine properties of both differential and difference equations.

Impulsive differential equations with discontinuous argument are proposed as an open problem by Wiener [30] in 1994, namely, the impulsive differential equations with piecewise constant argument: IDEPCA. As we know, impulsive differential equations with piecewise constant arguments (in short IDEPCA) are studied in a few papers [8, 39, 40].

1.2. Model Description

First, let us give a general description of the mathematical model of ICNNs with piecewise alternately advanced and retarded argument:𝑑𝑥_𝑖(𝑡)𝑑𝑡=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑚𝑡+𝑙𝑚+𝑑_𝑖,𝑡≠𝑚𝑘−𝑙,Δ𝑥_{𝑖𝑡=𝑚𝑘−𝑙}=𝐽_𝑖𝑘𝑥_𝑖(𝑚𝑘−𝑙⁻),𝑖=1,2,…,𝑛,𝑘∈ℕ, (4a)(4b)where [⋅] signifies the greatest integer function, 𝑙 and 𝑚 are positive real numbers such that 𝑙<𝑚, 𝑡, 𝑥_𝑖∈ℝ⁺, 𝑖=1,2,…,𝑛, Δ𝑥_𝑖(𝑚𝑘−𝑙)=𝑥_𝑖(𝑚𝑘−𝑙)−𝑥_𝑖(𝑚𝑘−𝑙⁻), and 𝑥_𝑖(𝑚𝑘−𝑙⁻) = lim_ℎ→0⁻𝑥_𝑖(𝑚𝑘−𝑙+ℎ). Moreover, 𝑛 denotes the number of neurons in the network, 𝑥_𝑖(𝑡) corresponds to the state of the 𝑖th unit at time 𝑡, 𝑓_𝑗(𝑥_𝑗(𝑡)) and 𝑔_𝑗(𝑥_𝑗(𝑚[(𝑡+𝑙)/𝑚])) denote, respectively, the measures of activation to its incoming potentials of the unit 𝑗 at time 𝑡 and discrete-time 𝑚[(𝑡+𝑙)/𝑚], 𝑎_𝑖 denotes the rate with which the unit 𝑖 resets its potential to the resting state when isolated from other units and inputs, 𝑏_𝑖𝑗 denotes the synaptic connection weight of the unit 𝑗 on the unit 𝑖 at time 𝑡, 𝑐_𝑖𝑗 denotes the synaptic connection weight of the unit 𝑗 on the unit 𝑖 at discrete-time 𝑚[(𝑡+𝑙)/𝑚], and 𝑑_𝑖 is the input from outside the network to the unit 𝑖. The numbers 𝑥_𝑖(𝑚𝑘−𝑙⁻) and 𝑥_𝑖(𝑚𝑘−𝑙) are, respectively, the states of the 𝑖th unit before and after impulse perturbation at the moment 𝑚𝑘−𝑙, 𝑘∈ℕ, and represent the abrupt change of the state 𝐽_𝑖𝑘(𝑥_𝑖(𝑚𝑘−𝑙⁻)) at the impulsive moment 𝑚𝑘−𝑙.

Let us clarify why the IDEPCA (4a)-(4b) is of alternately advanced and retarded type; that is, the argument can change its deviation character during the motion. The argument is deviated if it is advanced or retarded. Fix 𝑘∈ℕ, and consider the IDEPCA on the interval 𝐼_𝑘=[𝑚𝑘−𝑙,𝑚(𝑘+1)−𝑙). Then, the identification function 𝑚[(𝑡+𝑙)/𝑚] is equal to 𝑚𝑘. If 𝑡∈𝐼⁺_𝑘=[𝑚𝑘−𝑙,𝑚𝑘), then 𝑚[(𝑡+𝑙)/𝑚]≥𝑡 and IDEPCA (4a)-(4b) is an equation with advanced argument. Similarly, if 𝑡∈𝐼⁻_𝑘=(𝑚𝑘,𝑚(𝑘+1)−𝑙) then 𝑚[(𝑡+𝑙)/𝑚]<𝑡 and IDEPCA (4a)-(4b) is an equation with retarded argument. Consequently, IDEPCA (4a)-(4b) changes the type of deviation of the argument during the process. In other words, the IDEPCA (4a)-(4b) is of alternately advanced and retarded type.

For any solution 𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^𝑇 of IDEPCA (4a)-(4b), the model can be summarized as follows:𝑑𝑥(𝑡)(𝑡)(𝑥(𝑡𝑥𝑚𝑑𝑡=−𝐴𝑥+𝐵𝑓))+𝐶𝑔𝑡+𝑙𝑚|+𝐷,𝑡≠𝑚𝑘−𝑙,Δ𝑥_{𝑡=𝑚𝑘−𝑙}=𝐽_𝑘(𝑥(𝑚𝑘−𝑙⁻)),𝑘∈ℕ, (5a)(5b)where 𝐴=diag(𝑎₁,…,𝑎_𝑛), 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛, and  𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛 are constant matrices and 𝐷=(𝑑₁,…,𝑑_𝑛) is a constant vector. Moreover, the functions 𝑓∶ℝ^𝑛↦ℝ^𝑛, 𝑔∶ℝ^𝑛↦ℝ^𝑛 satisfy (𝜕𝑓_𝑖/𝜕𝑥_𝑗)=(𝜕𝑔_𝑖/𝜕𝑥_𝑗)=0 when 𝑖≠𝑗.

To the best of our knowledge, cellular neural network with piecewise constant argument has been developed by few authors, for example, Huang et al. Reference [41] considered first the following cellular neural network with piecewise constant delay: 𝑑𝑥_𝑖(𝑡)𝑑𝑡=−𝑎_𝑖([𝑡])𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗([𝑡])𝑔_𝑗𝑥_𝑗([𝑡])+𝑑_𝑖([𝑡]), (6) where [⋅] signifies the greatest integer function. Some sufficient conditions of existence and attractivity of almost periodic sequence solution were given for the corresponding discrete-time analogue: 𝑥_𝑖(𝑘+1)=𝑥_𝑖(𝑘)𝑒^{−𝑎𝑖(𝑘)}+1−𝑒^{−𝑎𝑖(𝑘)}𝑎_𝑖(𝑘)×^𝑛_𝑗=1𝑐_𝑖𝑗(𝑘)𝑔_𝑗𝑥_𝑗(𝑘)+𝑑_𝑖(𝑘). (7) In 2010, Akhmet and Yılmaz [8] considered first the following impulsive neural network with only piecewise constant retarded argument: 𝑑𝑥_𝑖(𝑡)𝑑𝑡=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑚_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝛽(𝑡)+𝑑_𝑖,𝑡≠𝜃_𝑘,Δ𝑥_{𝑖𝑡=𝜃𝑘}=𝐼_𝑘𝑥_𝑖𝜃⁻_𝑘,𝑖=1,2,…,𝑚,𝑘∈ℕ, (8) where 𝛽(𝑡)=𝜃_𝑘 if 𝜃_𝑘<𝑡<𝜃_𝑘+1, 𝑘∈ℕ, 𝑡∈ℝ⁺, is an identification function and 𝜃_𝑘>0, 𝑘∈ℕ, is a sequence of real numbers. Several sufficient conditions are obtained for the existence and stability of a unique 𝜔-periodic solution.

In this paper, we for the first time study the dynamic behavior of impulsive cellular neural network models that combine the properties of impulsive differential equations and discrete-time difference equations, that is, the following impulsive cellular neural network with piecewise alternately advanced and retarded argument: 𝑑𝑥_𝑖(𝑡)𝑑𝑡=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑚𝑡+𝑙𝑚+𝑑_𝑖,𝑡≠𝑚𝑘−𝑙,Δ𝑥_{𝑖𝑡=𝑚𝑘−𝑙}=𝐽_𝑖𝑘𝑥_𝑖(𝑚𝑘−𝑙⁻),𝑖=1,2,…,𝑛,𝑘∈ℕ. (9) The purpose of this paper is to derive some new and simple sufficient conditions for the existence and uniqueness of solutions of the ICNNs with IDEPCA system (5a)-(5b), which is globally exponentially stable. This paper is organized as follows. In Section 2, we establish several criteria for the existence and uniqueness of a unique equilibrium of the ICNNs with IDEPCA system and the equivalence lemma for (5a)-(5b). Here, a new IDEPCA Gronwall-type inequality is very useful. In Section 3, we derive some sufficient conditions which ensure that a unique equilibrium of the ICNNs with IDEPCA system (5a)-(5b) is globally exponentially stable. In Section 4, two illustrative examples and the numerical simulations are given to demonstrate the effectiveness of our results. The conclusions are drawn in Section 5.

2. Existence and Uniqueness Theorems

In this section, sufficient conditions that govern the network parameters and the activation functions are established for the existence of a unique equilibrium state of the impulsive cellular neural network models (5a)-(5b).

2.1. Preliminaries and Definition

In this section, we will focus our attention on some preliminary results which will be used in the existence and uniqueness of solutions of the ICNNs with IDEPCA system (5a)-(5b).

For every 𝑡∈ℝ, let 𝑖=𝑖(𝑡)∈ℕ be the unique integer such that 𝑡∈𝐼_𝑖=[𝑚𝑖−𝑙,𝑚(𝑖+1)−𝑙).

For the sake of convenience, two of the standing assumptions are formulated below.

Lipschitz Condition(L)The activation functions 𝑓_𝑗 and 𝑔_𝑗 with 𝑓_𝑗(0)=0, 𝑔_𝑗(0)=0 (𝑗=1,2,…,𝑛) satisfy the Lipschitz condition; that is, there are constants ℒ^𝑓_𝑗, ℒ^𝑔_𝑗>0 such that 𝑓_𝑗(𝑢)−𝑓_𝑗(𝑣)≤ℒ^𝑓_𝑗||,𝑔𝑢−𝑣_𝑗(𝑢)−𝑔_𝑗(𝑣)≤ℒ^𝑔_𝑗|𝑢−𝑣| (10) for all 𝑢, 𝑣∈ℝ⁺.

The impulsive operator 𝐽_𝑘 satisfies 𝐽_𝑗𝑘(𝑢)−𝐽_𝑗𝑘(𝑣)≤ℒ^𝐽_𝑘|𝑢−𝑣|, (11) for all 𝑢, 𝑣∈ℝ⁺, 𝑗=1,…,𝑛,𝑘∈ℕ, where ℒ^𝐽_𝑘 is a positive Lipschitz constant.

Existence condition(E)Consider max_{[]𝑖∈1,…,𝑛}1−𝑒^{−𝑙⋅𝑎∗}𝑎_∗^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_𝑖𝑗+ℒ^𝑔_𝑗𝑐_𝑖𝑗<1, (12) where min_{𝑖∈[1,…,𝑛]}𝑎_𝑖=𝑎_∗.

First, we prove the existence and uniqueness of solutions of IDEPCA system (5a)-(5b). A natural extension of the original definition of a solution of DEPCA [28–30, 42] allows us to define a solution of IDEPCA system.

Definition 1. A function 𝑥 is a solution of IDEPCA system (5a)-(5b) in ℝ⁺=[0,∞) if (i)𝑥(𝑡) is continuous for 𝑡∈ℝ⁺ with the possible exception of the points 𝑡=𝑚𝑘−𝑙, 𝑘∈ℕ,(ii)𝑥(𝑡) is right continuous and has left-hand limits at the points 𝑡=𝑚𝑘−𝑙, 𝑘∈ℕ,(iii)𝑥(𝑡) is differentiable and satisfies (5a) for any 𝑡∈ℝ⁺, with the possible exception of the points 𝑡=𝑚𝑘−𝑙, 𝑘∈ℕ, where one-sided derivatives exist,(iv)𝑥(𝑛) satisfies (5b) for 𝑛=𝑘𝑚−𝑙, 𝑘∈ℕ.

To study nonlinear IDEPCA system, we will use the approach based on the construction of an equivalent integral equation. Let us give the following proposition.

Proposition 2. Let (𝜏,𝑥₀)∈ℝ⁺×ℝ^𝑛. The function 𝑥(𝑡)=𝑥(𝑡,𝜏,𝑥₀) is a solution on ℝ⁺ of the IDEPCA system (5a)-(5b) in the sense of Definition 1 if and only if it is a solution of the integral equation 𝑥(𝑡)=𝑒^{−𝐴(𝑡−𝜏)}𝑥₀+^𝑡_𝜏𝑒^{−𝐴(𝑡−𝑠)}(𝑥(𝑠𝑥𝑚𝐵𝑓))+𝐶𝑔𝑠+𝑙𝑚++𝐷𝑑𝑠_{𝑖(𝑡)(𝜏)𝑘=𝑖+1}𝑒^{−𝐴(𝑡−(𝑚𝑘−𝑙))}𝐽_𝑘(𝑥(𝑚𝑘−𝑙⁻)),𝑡∈ℝ⁺. (13) In particular, one has the following integral equations: for 𝑖=1,…,𝑛,  𝑡∈ℝ⁺, 𝑥_𝑖(𝑡)=𝑒^{−𝑎𝑖(𝑡−𝜏)}𝑥_𝑖(𝜏)+^𝑡_𝜏𝑒^{−𝑎𝑖(𝑡−𝑠)𝑛}_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑚𝑠+𝑙𝑚+𝑑_𝑖+𝑑𝑠_{𝑖(𝑡)(𝜏)𝑘=𝑖+1}𝑒^{−𝑎(𝑡−(𝑚𝑘−𝑙))}𝐽_𝑖𝑘𝑥_𝑖(𝑚𝑘−1⁻). (14)

The proof of Proposition 2 is almost identical to the verification in [7] with slight changes which are caused by the piecewise constant argument.

In the next, we give the following lemma about IDEPCA integral inequality of Gronwall type, which is one of the most important auxiliary results of the present paper.

Lemma 3. Let 𝑢∶ℝ→[0,∞) be a function such that 𝑢 is continuous with possible points of discontinuity of the first kind at 𝑡=𝑚𝑘−𝑙,𝑘∈ℕ, and 𝜂₁, 𝜂₂ are nonnegative real constants satisfying 𝜐∶=𝜂₁+𝜂₂𝑙<1. (15) Suppose that for 𝑡≥𝜏 the inequality 𝑢(𝑡)≤𝑢(𝜏)+^𝑡_𝜏𝜂₁𝑢(𝑠)+𝜂₂𝑢𝑚𝑠+𝑙𝑚+𝑑𝑠_{𝑖(𝑡)(𝜏)𝑘=𝑖+1}𝛽_𝑘𝑢(𝑚𝑘−𝑙⁻) (16) holds. Then for 𝑡≥𝜏, 𝑢(𝑡)≤𝑢(𝜏)^𝑖(𝑡)_{(𝜏)𝑘=𝑖+1}1+𝛽_𝑘exp𝜂₁+𝜂₂(,𝑢𝑚1−𝜐𝑡−𝜏)𝑡+𝑙𝑚≤𝑢(𝜏)1−𝜐^𝑖(𝑡)_{(𝜏)𝑘=𝑖+1}1+𝛽_𝑘×exp𝜂₁+𝜂₂(,𝑢()≤()1−𝜐𝑡−𝜏)𝑚𝑖1−𝜐⁻¹𝑢(𝑚𝑖−𝑙),𝑖∈ℕ. (17)(18)(19)

Proof. Call 𝑣(𝑡) the right member of (16). So 𝑣(𝜏)=𝑢(𝜏), 𝑢≤𝑣, and 𝑣 is a piecewise differentiable and nondecreasing function and, by (16), it satisfies 𝑣^′(𝑡)≤𝜂₁𝑣(𝑡)+𝜂₂𝑣𝑚𝑡+𝑙𝑚,𝑣(𝑚𝑘−𝑙)≤1+𝛽_𝑘𝑣(𝑚𝑘−𝑙⁻), (20)𝑘∈ℕ and for any 𝑡≥𝑟 with 𝑡,𝑟∈𝐼_𝑖𝑣(𝑡)−𝑣(𝑟)≤^𝑡_𝑟𝜂₁𝑣(𝑠)+𝜂₂𝑣𝑚𝑠+𝑙𝑚𝑑𝑠. (21) With 𝑡=𝑚𝑖 and 𝑟=𝑚𝑖−𝑙 in (21) for 𝑡∈𝐼_𝑖, since 𝑣 is a nondecreasing function, we get 𝑣(𝑚𝑖)≤𝑣(𝑚𝑖−𝑙)+^𝑚𝑖_{𝑚𝑖−𝑙}𝜂₁𝑣(𝑠)+𝜂₂𝑣()()+𝜂𝑚𝑖𝑑𝑠≤𝑣𝑚𝑖−𝑙₁+𝜂₂𝑙⋅𝑣(𝑚𝑖). (22) Considering the particular case 𝜏=𝑡_𝑖 and taking 𝑣(𝑡_𝑖)=𝑢(𝑡_𝑖) and 𝑢≤𝑣, by (15) and (22), estimate (19) follows. Take now in (21) 𝑡∈𝐼_𝑖 and 𝑟=𝑚𝑖−𝑙 to obtain 𝑣(𝑡)≤𝑣(𝑚𝑖−𝑙)+^𝑡_{𝑚𝑖−𝑙}𝜂₁𝑣(𝑠)+𝜂₂𝑣()()+𝑚𝑖𝑑𝑠≤𝑣𝑚𝑖−𝑙^𝑡_{𝑚𝑖−𝑙}𝜂₁𝑣(𝑠)+𝜂₂𝑣()()+1−𝜐𝑚𝑖−𝑙𝑑𝑠≤𝑣𝑚𝑖−𝑙^𝑡_{𝑚𝑖−𝑙}𝜂₁+𝜂₂𝑣(𝑠)1−𝜐𝑑𝑠 (23) because 𝑣 is a nondecreasing function. Now, we can apply the classical Gronwall’s Lemma to get 𝑣(𝑡)≤𝑣(𝑚𝑖−𝑙)exp𝜂₁+𝜂₂((1−𝜐𝑡−𝑚𝑖−𝑙))for𝑡∈𝐼_𝑖. (24) By the impulsive effect (20), we have 𝑣(𝑚(𝑖+1)−𝑙)≤1+𝛽_𝑖+1𝑣()𝜂𝑚𝑖−𝑙×exp𝑚⋅₁+𝜂₂.1−𝜐 (25) From (25), recursively we obtain 𝑢(𝑡)≤𝑣(𝑡)≤𝑣(𝜏)^𝑖(𝑡)_{(𝜏)𝑘=𝑖+1}1+𝛽_𝑘×exp𝜂₁+𝜂₂(,1−𝜐𝑡−𝜏) (26) using 𝑣(𝜏)=𝑢(𝜏) and (19); then we give (17) and (18). The proof is complete. This IDEPCA inequality of Gronwall type seems to be new.

We need to have the global unique existence of solutions 𝑥(𝑡)=𝑥(𝑡,𝜏,𝑥₀) on ℝ⁺ of the nonlinear IDEPCA system (5a)-(5b).

One can easily see that IDEPCA system (5a)-(5b) has the form of DEPCA system without impulsive effect within the intervals [𝑚𝑖−𝑙,𝑚(𝑖+1)−𝑙),𝑖∈ℕ; then using the same technique of [34, 35, 37] we have the following results.

Proposition 4. Suppose that conditions (L) and (E) hold. For any (𝜏,𝑥₀)∈ℝ⁺×ℝ^𝑛 there exists a unique solution 𝑥(𝑡)=𝑥(𝑡,𝜏,𝑥₀) of the IDEPCA system (5a)-(5b) on [𝑚𝑖(𝜏)−𝑙,𝑚(𝑖(𝜏)+1)−𝑙).

Theorem 5. Under conditions (L) and (E), for every (𝜏,𝑥₀)∈ℝ⁺×ℝ^𝑛, there exists a unique solution 𝑥(𝑡)=𝑥(𝑡,𝜏,𝑥₀) of the IDEPCA system (5a)-(5b) with 𝑥(𝜏)=𝑥₀ for 𝑡∈[𝜏,∞) in the sense of Definition 1.

Proof. Fix 𝜏∈ℝ⁺; then 𝜏∈𝐼_𝑖(𝜏)=[𝑚𝑖(𝜏)−𝑙,𝑚𝑖(𝜏)+𝑚−𝑙). Use Proposition 4 with 𝑥(𝜏)=𝑥₀ to obtain the unique solution 𝑥(𝑡)=𝑥(𝑡,𝜏,𝑥₀) on 𝐼_𝑖(𝜏). Then apply the impulse condition to evaluate uniquely 𝑥𝑚𝑖(𝜏)+𝑚−𝑙,𝜏,𝑥₀=𝑥𝑚𝑖(𝜏)+𝑚−𝑙⁻,𝜏,𝑥₀+𝐽_𝑖(𝜏)+1𝑥𝑚𝑖(𝜏)+𝑚−𝑙⁻,𝜏,𝑥₀. (27) Next, on the interval 𝐼_𝑖(𝜏)+1=[𝑚𝑖(𝜏)+𝑚−𝑙,𝑚𝑖(𝜏)+2𝑚−𝑙) the solution satisfies the DEPCA: 𝑑𝑦_𝑖(𝑡)𝑑𝑡=−𝑎_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡)+𝑐_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑚𝑡+𝑙𝑚+𝑑_𝑖,𝑖=1,2,…,𝑛. (28) The IDEPCA system has a unique solution 𝑦(𝑡,𝑚𝑖(𝜏)+𝑚−𝑙,𝑥(𝑚𝑖(𝜏)+𝑚−𝑙,𝜏,𝑥₀)). By definition of the solution of IDEPCA system (5a)-(5b), 𝑥(𝑡,𝜏,𝑥₀)=𝑦(𝑡,𝑚𝑖(𝜏)+𝑚−𝑙,𝑥(𝑚𝑖(𝜏)+𝑚−𝑙,𝜏,𝑥₀)) on 𝐼_𝑖(𝜏)+1=[𝑚𝑖(𝜏)+𝑚−𝑙,𝑚𝑖(𝜏)+2𝑚−𝑙). The mathematical induction completes the proof.

2.2. Existence and Uniqueness of Equilibrium

When impulsive cellular neural network models are used for the solution of optimization problems, one of the fundamental issues in the design of a network is concerned with the existence of a unique globally exponentially stable equilibrium state of network (5a)-(5b). Without requiring the boundedness, differentiability, or monotonicity, we establish easily verifiable sufficient conditions for the existence of a unique equilibrium state in this section.

Let us denote an equilibrium state of the impulsive cellular neural network models (5a)-(5b) by the constant vector 𝑥^∗=(𝑥^∗₁,𝑥^∗₂…,𝑥^∗_𝑛)^𝑇∈ℝ^𝑛, where each 𝑥^∗_𝑖 is governed by the algebraic system 0=−𝑎_𝑖𝑥^∗_𝑖+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥^∗_𝑗+𝑐_𝑖𝑗𝑔_𝑗𝑥^∗_𝑗+𝑑_𝑖,𝑖=1,…,𝑛. (29) Here, it is assumed that the impulse functions 𝐽_𝑖𝑘(⋅) satisfy 𝐽_𝑖𝑘(𝑥^∗_𝑖)=0 for all 𝑖=1,…,𝑛, 𝑘∈ℕ.

In the following theorem, we obtain sufficient conditions for the existence of a unique equilibrium, 𝑥^∗=(𝑥^∗₁,𝑥^∗₂…,𝑥^∗_𝑛)^𝑇∈ℝ^𝑛, of the impulsive cellular neural network models (5a)-(5b).

Theorem 6. Suppose that conditions (L) and (E) hold and the neural parameters 𝑎_𝑖, 𝑏_𝑖𝑗,  and  𝑐_𝑖𝑗 and Lipschitz constants ℒ^𝑓_𝑖, ℒ^𝑔_𝑖 satisfy 𝑎_𝑖>ℒ^𝑓_𝑖^𝑛_𝑗=1𝑏_𝑖𝑗+ℒ^𝑔_𝑖^𝑛_𝑗=1𝑐_𝑖𝑗,𝑖,𝑗=1,…,𝑛. (30) Then there exists a unique equilibrium state 𝑥^∗ of the ICNNs with IDEPCA system (5a)-(5b).

Proof. Let us consider a mapping 𝐺(𝑢)=(𝐺₁(𝑢),𝐺₂(𝑢),…,𝐺_𝑛(𝑢))^𝑇∈ℝ^𝑛, where 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇∈ℝ^𝑛 and 𝐺_𝑖(𝑢)=1𝑎_𝑖^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑢_𝑗+𝑐_𝑖𝑗𝑔_𝑗𝑢_𝑗+𝑑_𝑖,𝑖=1,…,𝑛. (31) By applying the hypotheses, max_{1≤𝑖≤𝑛}𝐺_𝑖(𝑢)−𝐺_𝑖(𝑣)=max_{1≤𝑖≤𝑛}1𝑎_𝑖^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑢_𝑗+𝑐_𝑖𝑗𝑔_𝑗𝑢_𝑗+𝑑_𝑖−1𝑎_𝑖^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑣_𝑗+𝑐_𝑖𝑗𝑔_𝑗𝑣_𝑗+𝑑_𝑖≤max_{1≤𝑖≤𝑛}1𝑎_𝑖^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑢_𝑗−𝑓_𝑗𝑣_𝑗+1𝑎_𝑖^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑢_𝑗−𝑔_𝑗𝑣_𝑗≤max_{1≤𝑖≤𝑛}1𝑎_𝑖^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_𝑖𝑗𝑢_𝑗−𝑣_𝑗+1𝑎_𝑖^𝑛_𝑗=1ℒ^𝑔_𝑗𝑐_𝑖𝑗𝑢_𝑗−𝑣_𝑗≤max_{1≤𝑖≤𝑛}1𝑎_𝑖^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_𝑖𝑗+ℒ^𝑔_𝑗𝑐_𝑖𝑗⋅max_{1≤𝑗≤𝑛}𝑢_𝑗−𝑣_𝑗, (32) where the number 1𝜌=𝑎_𝑖^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_𝑖𝑗+ℒ^𝑔_𝑗𝑐_𝑖𝑗 (33) satisfies 0<𝜌<1 by virtue of condition (30). Thus, max_{1≤𝑖≤𝑛}𝐺_𝑖(𝑢)−𝐺_𝑖(𝑣)≤𝜌max_{1≤𝑗≤𝑛}𝑢_𝑗−𝑣_𝑗 (34) for any two vectors 𝑢,𝑣∈ℝ^𝑛 implying that the mapping 𝐺∶ℝ^𝑛→ℝ^𝑛 is a global contraction on ℝ^𝑛 endowed with the supremum norm. Hence, there is a unique fixed point 𝑥^∗∈ℝ^𝑛 that satisfies 𝐺(𝑥^∗)=𝑥^∗ (i.e., 𝐺_𝑖(𝑥^∗)=𝑥^∗_𝑖 for 𝑖=1,…,𝑛). This point defines the unique equilibrium state of the impulsive cellular neural network models (5a)-(5b). The proof is now complete.

3. Global Exponential Stability of Equilibrium

The existence and stability of a unique equilibrium state are usually a requirement in the design of cellular neural network models for various applications, particularly when there are destabilizing agents such as retarded arguments and impulses. However, even if the unique stable state exists, these agents may affect the convergence speed of the network, which in turn can downgrade the performance of the network in applications that demand fast computation in real-time mode. Thus, exponential stability is usually desirable for an impulsive network, and sufficient conditions for the global exponential stability of the unique equilibrium state 𝑥^∗ of the ICNNs with IDEPCA system (5a)-(5b) are obtained in this section.

For analytical convenience, the ICNNs with IDEPCA system (5a)-(5b) can be simplified as follows. Let 𝑧_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥^∗_𝑖,𝑓𝑧_𝑖(𝑡)=𝑓𝑥_𝑖(𝑡)+𝑥^∗_𝑖−𝑓𝑥^∗_𝑖,𝑔𝑧_𝑖𝑚𝑡+𝑙𝑚=𝑔𝑥_𝑖𝑚𝑡+𝑙𝑚+𝑥^∗_𝑖−𝑔𝑥^∗_𝑖,𝐽̂_𝑖𝑘𝑧_𝑖(𝑚𝑘−𝑙⁻)=𝐽_𝑖𝑘𝑥_𝑖(𝑚𝑘−𝑙⁻)+𝑥^∗_𝑖, (35) so that the ICNNs with IDEPCA system (5a)-(5b) can be written as 𝑑𝑧(𝑡)(𝑡)𝑓(𝑧(𝑡𝑔𝑧𝑚𝑑𝑡=−𝐴𝑧+𝐵))+𝐶𝑡+𝑙𝑚|,𝑡≠𝑚𝑘−𝑙,Δ𝑥_{𝑡=𝑚𝑘−𝑙}=𝐽_𝑘̂(𝑧(𝑚𝑘−𝑙⁻)),𝑘∈ℕ, (36) where 𝑓(𝑧(𝑡))=[𝑓₁(𝑧₁(𝑡)),…,𝑓_𝑛(𝑧_𝑛(𝑡))]^𝑇, 𝑔(𝑧(𝑚[(𝑡+𝑙)/𝑚]))=[𝑔₁(𝑧₁(𝑚[(𝑡+𝑙)/𝑚])),…, 𝑔_𝑛(𝑧_𝑛(𝑚[(𝑡+𝑙)/𝑚]))]^𝑇and 𝐽_𝑘̂(𝑧(𝑚𝑘−𝑙⁻))=[𝐽_1𝑘(𝑥₁(𝑚𝑘−𝑙⁻)+𝑥^∗₁),…,𝐽_𝑛𝑘(𝑥_𝑛(𝑚𝑘−𝑙⁻)+𝑥^∗_𝑛)]^𝑇.

The activation functions 𝑓_𝑖(⋅), inheriting the properties of 𝑓_𝑖(⋅), satisfy 𝑓_𝑖(0)=0,𝑓_𝑖(𝑢)−𝑓_𝑖(𝑣)≤ℒ^𝑓_𝑖|𝑢−𝑣|, (37) the functions 𝑔_𝑖(⋅) inherit the properties of 𝑔_𝑖(⋅), namely, 𝑔_𝑖(0)=0,𝑔_𝑖(𝑢)−𝑔_𝑖(𝑣)≤ℒ^𝑔_𝑖|𝑢−𝑣|, (38) and the impulsive operator 𝐽̂_𝑖𝑘 satisfies 𝐽̂_𝑖𝑘(0)=0,𝐽̂_𝑖𝑘(𝑢)−𝐽̂_𝑖𝑘(𝑣)≤ℒ^𝐽_𝑘𝑢_𝑖−𝑣_𝑖, (39) for all 𝑢, 𝑣∈ℝ⁺, 𝑖=1,…,𝑛,𝑘∈ℕ.

It is clear that the stability of the zero solution of (36) is equivalent to that of the equilibrium 𝑥^∗ of the ICNNs with IDEPCA system (4a)-(4b). Therefore, we restrict our discussion to the stability of the zero solution of (36).

First of all, we give the following definition and lemma, which will be used in the proof of the stability of the zero solution for the ICNNs with IDEPCA system.

Definition 7. The equilibrium 𝑥^∗ of the ICNNs with IDEPCA system (5a)-(5b) is said to be globally exponentially stable if there exist positive constants 𝛼 and 𝜆 such that the estimation 𝑥(𝑡)−𝑥^∗≤𝛼𝑥(𝜏)−𝑥^∗𝑒^{−𝜆(𝑡−𝜏)} (40) is valid for all 𝑡≥𝜏.

Lemma 8. If (L) and (E) are satisfied, then the solutions 𝜑 and 𝜓 of the IDEPCA system (5a)-(5b) satisfy for all 𝑡≥𝜏 the inequality 𝜑(𝑡)−𝜓(𝑡)≤𝜑(𝜏)−𝜓(𝜏)exp−𝜆_𝑖(𝑡)⋅(𝑡−𝜏), (41) where 𝜆_𝑖(𝑡)=𝑎_∗−𝛽^∗−ℒ_𝑖(𝑡), 𝑎_∗=min_{𝑖∈[1,…,𝑛]}𝑎_𝑖, ℒ_𝑖(𝑡)=max_{𝑖(𝜏)+1≤𝑘≤𝑖(𝑡)}(ln(1+ℒ^𝐽_𝑘)/𝑚), 𝛽^∗=max_{𝑖∈[1,…,𝑛]}𝛽_𝑖=max_[]^𝑛_{𝑖∈1,…,𝑛𝑗=1}ℒ^𝑓_𝑗𝑏_𝑖𝑗+ℒ^𝑔_𝑗𝑐_𝑖𝑗(1−𝑣)⁻¹×exp𝑎_∗⋅(𝑚−𝑙),𝑣=max_{𝑖∈[1,…,𝑛]}^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_𝑖𝑗+ℒ^𝑔_𝑗𝑐_𝑖𝑗exp𝑎_∗⋅(𝑚−𝑙)⋅𝑙<1. (42)

Proof. Suppose that 𝜑(𝑡)=(𝜑₁,…,𝜑_𝑛)^𝑇 and 𝜓(𝑡)=(𝜓₁,…,𝜓_𝑛)^𝑇 are arbitrary solutions of the IDEPCA system (5a)-(5b). Let 𝑦(𝑡)=𝜑(𝑡)−𝜓(𝑡) and by (5a) and (5b) it follows that 𝑦(⋅) satisfies 𝑦(̇𝑡)(𝑡)𝑓𝑦(𝑡)(𝑡)𝜓(𝑡)𝑔𝑦𝑚=−𝐴𝑦+𝐵+𝜓−𝑓+𝐶𝑡+𝑙𝑚+𝜓𝑚𝑡+𝑙𝑚−𝑔𝜓𝑚𝑡+𝑙𝑚,Δ𝑦_{𝑡=𝑚𝑘−𝑙}=𝐽_𝑘𝑦(𝑚𝑘−𝑙⁻)+𝜓(𝑚𝑘−𝑙⁻)−𝐽_𝑘𝜓(𝑚𝑘−𝑙⁻),𝑘∈ℕ. (43) By the variation of parameters formula, it can be proved that 𝑦(𝑡)=𝑒^{−𝐴(𝑡−𝜏)}𝑦(𝜏)+^𝑡_𝜏𝑒^{−𝐴(𝑡−𝑠)}ℛ(𝑠)+𝑠,𝑦𝑑𝑠_{𝑖(𝑡)(𝜏)𝑘=𝑖+1}𝑒^{−𝐴(𝑡−(𝑚𝑘−𝑙))}𝒥_𝑘𝑦(𝑚𝑘−𝑙⁻), (44) where ℛ(𝑠)⋅𝑓𝑦(𝑠)(𝑠)𝜓(𝑠)𝑔𝑦𝑚𝑠,𝑦∶=𝐵+𝜓−𝑓+𝐶⋅𝑠+𝑙𝑚+𝜓𝑚𝑠+𝑙𝑚−𝑔𝜓𝑚𝑠+𝑙𝑚,𝒥_𝑘𝑦(𝑚𝑘−𝑙⁻)∶=𝐽_𝑘𝑦(𝑚𝑘−𝑙⁻)+𝜓(𝑚𝑘−𝑙⁻)−𝐽_𝑘𝜓(𝑚𝑘−𝑙⁻). (45) Notice that (L) implies that ℛ_𝑖𝑠,𝑦(𝑠)≤^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_𝑖𝑗(𝑠+𝑦𝑖)_𝑛𝑗=1ℒ^𝑔_𝑗𝑐_𝑖𝑗𝑦𝑖𝑚𝑠+𝑙𝑚,ℛ_𝑖𝑠,𝑦(𝑠)≤max_{[]𝑖∈1,…,𝑛}^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_𝑖𝑗𝑦(𝑠+_𝑛𝑗=1)ℒ^𝑔_𝑗𝑐_𝑖𝑗𝑦𝑚𝑠+𝑙𝑚,𝒥_𝑘𝑦(𝑚𝑘−𝑙⁻)≤ℒ^𝐽_𝑘𝑦(𝑚𝑘−𝑙⁻). (46) By (44), we can deduce that 𝑣_𝑖(𝑡)=exp(𝑎_∗⋅(𝑡−𝜏))|𝑦_𝑖(𝑡)| satisfies 𝑣_𝑖(𝑡)≤𝜑_𝑖(𝜏)−𝜓_𝑖(𝜏+_{𝑡𝜏𝑛𝑗=1})ℒ^𝑓_𝑗𝑏_𝑖𝑗𝑣_𝑗(𝑠+_𝑛𝑗=1)ℒ^𝑔_𝑗𝑐_𝑖𝑗𝑣_𝑗𝑚𝑠+𝑙𝑚×exp𝑎_∗⋅𝑠−𝑚𝑠+𝑙𝑚+𝑑𝑠_{𝑖(𝑡)(𝜏)𝑘=𝑖+1}ℒ^𝐽_𝑘𝑣_𝑖(𝑚𝑘−𝑙⁻), (47) or |𝑣(𝑡)|≤𝜑(𝜏)−𝜓(𝜏)+max_{[]𝑖∈1,…,𝑛}^𝑡_𝜏^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_𝑖𝑗𝑣(𝑠+|)|_𝑛𝑗=1ℒ^𝑔_𝑗𝑐_𝑖𝑗𝑣𝑚𝑠+𝑙𝑚×exp𝑎_∗⋅𝑠−𝑚𝑠+𝑙𝑚+𝑑𝑠_{𝑖(𝑡)(𝜏)𝑘=𝑖+1}ℒ^𝐽_𝑘|𝑣(𝑚𝑘−𝑙⁻≤𝜑(𝜏)(𝜏)|−𝜓)+max_{[]𝑖∈1,…,𝑛}×^𝑡_𝜏^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_𝑖𝑗|𝑣(𝑠)|+^𝑛_𝑗=1ℒ^𝑔_𝑗𝑐_𝑖𝑗×exp𝑎_∗⋅(𝑚−𝑙)𝑣𝑚𝑠+𝑙𝑚+𝑑𝑠_{𝑖(𝑡)(𝜏)𝑘=𝑖+1}ℒ^𝐽_𝑘|𝑣(𝑚𝑘−𝑙⁻)|, (48) for any finite 𝑡∈[𝜏,∞).
Hence, by Lemma 3 of IDEPCA Gronwall's inequality implies |𝑣(𝑡)|≤𝜑(𝜏)−𝜓(𝜏)^𝑖(𝑡)_{(𝜏)𝑘=𝑖+1}1+ℒ^𝐽_𝑘×expmax_{[]𝑖∈1,…,𝑛}^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_𝑖𝑗+11−𝑣^𝑛_𝑗=1ℒ^𝑔_𝑗𝑐_𝑖𝑗×exp𝑎_∗⋅(𝑚−𝑙)(𝑡−𝜏). (49) Then, we have 𝜑(𝑡)−𝜓(𝑡)≤𝜑(𝜏)−𝜓(𝜏)^𝑖(𝑡)_{(𝜏)𝑘=𝑖+1}1+ℒ^𝐽_𝑘×exp−𝑎_∗−max_{[]𝑖∈1,…,𝑛}𝛽_𝑖(𝑡−𝜏), (50) or 𝜑(𝑡)(𝑡≤𝜑(𝜏)(𝜏−𝑎−𝜓)−𝜓)×exp_∗−max_{[]𝑖∈1,…,𝑛}𝛽_𝑖−max_{𝑖(𝜏)(𝑡)+1≤𝑘≤𝑖}ln1+ℒ^𝐽_𝑘𝑚(𝑡−𝜏), (51) and the statement (41) follows.

The following result will show sufficient conditions for the global exponential stability of the unique equilibrium of the ICNNs with IDEPCA system (5a)-(5b).

Theorem 9. If the assumptions of Theorem 6, (42) and 𝑎_∗−𝛽^∗−ℒ_𝑖(𝑡)>0,𝑡∈ℝ⁺, (52) are satisfied, then the unique equilibrium 𝑥^∗ of the ICNNs with IDEPCA system (5a)-(5b) is globally exponentially stable.

Proof. By Theorem 6, we know that the IDEPCA system (5a)-(5b) has a unique equilibrium 𝑥^∗. Let 𝑥(𝑡,𝑥₀) be an arbitrary solution of (5a)-(5b) with initial condition 𝑥₀ and define 𝑧(𝑡)=𝑥(𝑡,𝑥₀)−𝑥^∗. By Lemma 8 and (42), we obtain |𝑧(𝑡)|≤|𝑧(𝜏)|exp−𝜆_𝑖(𝑡)⋅(𝑡−𝜏), (53) where 𝜆_𝑖(𝑡)=𝑎_∗−𝛽^∗−ℒ_𝑖(𝑡). So, using (52), we see that |𝑧(𝑡)|→0 as 𝑡→∞. That is, the zero solution of ICNNs with IDEPCA system (36) is globally exponentially stable. Therefore, the unique equilibrium 𝑥^∗ of the ICNNs with IDEPCA system (5a)-(5b) is globally exponentially stable.

Remark 10. To the best of the author's knowledge, this is the first time we investigate impulsive cellular neural network models with piecewise alternately advanced and retarded argument in equilibrium case. Sufficient conditions are gained for the existence and exponential stability of a unique equilibrium of the ICNNs with IDEPCA system. And our results can be extended to a unique equilibrium of the CNNs with DEPCA system. See Corollaries 11–12. Our results about exponential stability of a unique equilibrium of the ICNNs with IDEPCA system may give some insight into the application of neural networks.

As immediate corollaries of Lemma 8 and Theorem 9, the following results without impulsive effects are true.

Corollary 11. If (L) and (E) are satisfied, then the solutions 𝜑 and 𝜓 of the DEPCA system (5a) satisfy for all 𝑡≥𝜏 the inequality 𝜑(𝑡)−𝜓(𝑡)≤𝜑(𝜏)−𝜓(𝜏)exp(−𝜆⋅(𝑡−𝜏)), (54) where 𝜆=𝑎_∗−𝛽^∗, 𝑎_∗=min_{𝑖∈[1,…,𝑛]}𝑎_𝑖, 𝛽^∗=max_{[]𝑖∈1,…,𝑛}𝛽_𝑖=max_[]^𝑛_{𝑖∈1,…,𝑛𝑗=1}ℒ^𝑓_𝑗𝑏_𝑖𝑗+ℒ^𝑔_𝑗𝑐_𝑖𝑗(1−𝑣)⁻¹×exp𝑎_∗⋅(𝑚−𝑙),𝑣=max_{[]𝑖∈1,…,𝑛}^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_𝑖𝑗+ℒ^𝑔_𝑗𝑐_𝑖𝑗exp𝑎_∗⋅(𝑚−𝑙)⋅𝑙<1. (55)

Corollary 12. If the assumptions of Corollary 11 and 𝑎_∗−𝛽^∗>0 (56) are satisfied, then the unique equilibrium 𝑥^∗ of the CNNs with DEPCA system (5a) is globally exponentially stable.

Remark 13. In [41], authors investigated discrete-time cellular neural network without impulsive effects in almost periodic case. Simple sufficient conditions are gained for a unique almost periodic sequence solution which is globally attractive. When 𝑚=1, 𝑙=0, this conclusion of Corollary 12 cannot be derived by applying the corresponding stability result for cellular neural networks given in the literature [41] with 𝑎_𝑖, 𝑏_𝑖𝑗, 𝑐_𝑖𝑗, and 𝑑_𝑖 being constant coefficients.

4. Examples and Simulations

In this section, we give two examples with numerical simulations to illustrate the effectiveness of the proposed method and results.

Example 1. Consider the following impulsive cellular neural networks with piecewise alternately advanced and retarded argument: 𝑑𝑥(𝑡)𝑥𝑑𝑡=−1.2000.9¹(𝑡)𝑥₂(𝑡)+𝑥0.150.250.250.15tanh¹(𝑡)2𝑥tanh²(𝑡)8+𝑥0.150.250.250.15tanh¹(3[(𝑡+1)/3])8𝑥tanh²(3[(𝑡+1)/3])2+,|0.20.1Δ𝑥_{𝑡=3𝑘−1}=𝐽_1𝑘𝑥₁(3𝑘−1⁻)𝐽_2𝑘𝑥₂(3𝑘−1⁻)=𝑥₁(3𝑘−1⁻)−𝑥^∗₁2𝑥₂(3𝑘−1⁻)−𝑥^∗₂3,𝑘∈ℕ, (57) where 𝑥^∗₁=1.943, 𝑥^∗₂=1.57. One can check that the point 𝑥^∗=(𝑥^∗₁,𝑥^∗₂)^𝑇 satisfies the algebraic system −𝑎_𝑖𝑥^∗_𝑖+²_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥^∗_𝑗+𝑐_𝑖𝑗𝑔_𝑗𝑥^∗_𝑗+𝑑_𝑖=0,𝑖=1,2, (58) approximately. And it is clear that 𝐽_𝑖𝑘(𝑥^∗_𝑖)=0 for 𝑖=1,2. By simple calculation, we can see that 𝑎_∗=0.9, ℒ^𝑓₁=ℒ^𝑔₂=ℒ^𝐽₁=1/2, ℒ^𝑓₂=ℒ^𝑔₁=1/8, ℒ^𝐽₂=1/3, and sup_𝑡∈ℝ+ℒ_𝑖(𝑡)=ln(1+ℒ^𝐽₁)/3≈0.1351. Then 1−𝑒^{−𝑙⋅𝑎∗}𝑎_∗^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_1𝑗+ℒ^𝑔_𝑗𝑐_1𝑗≈0.1813<1,1−𝑒^{−𝑙⋅𝑎∗}𝑎_∗^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_2𝑗+ℒ^𝑔_𝑗𝑐_2𝑗𝑎≈0.1648<1,₁=1.2>0.275=ℒ^𝑓₁²_𝑗=1𝑏_1𝑗+ℒ^𝑔₁²_𝑗=1𝑐_1𝑗,𝑎₂=0.9>0.25=ℒ^𝑓₂²_𝑗=1𝑏_2𝑗+ℒ^𝑔₂²_𝑗=1𝑐_2𝑗. (59) By Theorem 6, we know that the ICNNs with IDEPCA system (57) have a unique equilibrium state 𝑥^∗, approximately with the error, which is less than 10⁻¹¹ (evaluated by MATLAB).

Moreover, we have 𝑣=max_{[]𝑖∈1,…,𝑛}^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_𝑖𝑗+ℒ^𝑔_𝑗𝑐_𝑖𝑗exp𝑎_∗⋅(𝑚−𝑙)⋅𝑙≈0.7522<1,_𝑛𝑗=1ℒ^𝑓_𝑗𝑏_1𝑗+ℒ^𝑔_𝑗𝑐_1𝑗(1−𝑣)⁻¹exp𝑎_∗⋅(𝑚−𝑙)≈0.7607<0.7648≈𝑎_∗−sup_𝑡∈ℝ+ℒ_𝑖(𝑡),_𝑛𝑗=1ℒ^𝑓_𝑗𝑏_2𝑗+ℒ^𝑔_𝑗𝑐_2𝑗(1−𝑣)⁻¹exp𝑎_∗⋅(𝑚−𝑙)≈0.58302<0.7648≈𝑎_∗−sup_𝑡∈ℝ+ℒ_𝑖(𝑡). (60) Thus, according to Theorem 9, the ICNNs with IDEPCA system (57) have a unique globally exponentially stable equilibrium. The numerical simulations, showing the convergence of the unique equilibrium 𝑥^∗ of the ICNNs with and without impulses (57), are given in Figures 1(a) and 1(b).

(a)

(b)

Figure 1 

(a) The simulation, where the initial value is chosen as (1,3)𝑇, illustrates that all trajectories uniformly converge to the unique equilibrium 𝑥∗=(1.943,1.57)𝑇 for the ICNNs (57) with impulsive effects. (b) The simulation, where the initial value is chosen as (1,3)𝑇, illustrates that all trajectories uniformly converge to the unique equilibrium 𝑥∗=(1.943,1.57)𝑇 for the ICNNs (57) without impulsive effects.

Example 2. Consider the following impulsive cellular neural networks model with piecewise constant argument: 𝑑𝑥(𝑡)𝑥𝑑𝑡=−1.3000.8¹(𝑡)𝑥₂(𝑡)+×𝑥0.10.250.250.35tanh¹(𝑡)3𝑥tanh²(𝑡)4+×𝑥0.160.260.250.15₁4𝑡+14+1−𝑥₁4𝑡+14−1𝑥10₂4𝑡+14+1−𝑥₂4𝑡+14−18+34,Δ𝑥|_{𝑡=4𝑘−1}=𝐽_1𝑘𝑥₁(4𝑘−1⁻𝐽_2𝑘)𝑥₂(4𝑘−1⁻)=𝑥₁(4𝑘−1⁻)−𝑥^∗₁4𝑥₂(4𝑘−1⁻)−𝑥^∗₂10,𝑘∈ℕ, (61) where 𝑥^∗₁=2.608, 𝑥^∗₂=5.719. One can check that the point 𝑥^∗=(𝑥^∗₁,𝑥^∗₂)^𝑇 satisfies the algebraic system (29) approximately and it is clear that 𝐽_𝑖𝑘(𝑥^∗_𝑖)=0 for 𝑖=1,2. The output functions are 𝑓₁(𝑥₁)=tanh(𝑥₁/3), 𝑓₂(𝑥₂)=tanh(𝑥₂/4), 𝑔₁(𝑥₁)=(|𝑥₁+1|−|𝑥₁−1|)/10, and 𝑔₂(𝑥₂) = (|𝑥₂+1|−|𝑥₂−1|)/8.

We can easily obtain that 𝑎_∗=0.8, ℒ^𝑓₁=1/3, ℒ^𝑓₂=ℒ^𝑔₂=ℒ^𝐽₁=0.25, ℒ^𝑔₁=0.2, ℒ^𝐽₂=0.1, and sup_𝑡∈ℝ+ℒ_𝑖(𝑡)=ln(1+ℒ^𝐽₁)/4≈0.0557.

Then we give 1−𝑒^{−𝑙⋅𝑎∗}𝑎_∗^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_1𝑗+ℒ^𝑔_𝑗𝑐_1𝑗≈0.1904<1,1−𝑒^{−𝑙⋅𝑎∗}𝑎_∗^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_2𝑗+ℒ^𝑔_𝑗𝑐_2𝑗𝑎≈0.1996<1,₁=1.3>0.2766≈ℒ^𝑓₁²_𝑗=1𝑏_1𝑗+ℒ^𝑔₁²_𝑗=1𝑐_1𝑗,𝑎₂=0.8>0.29=ℒ^𝑓₂²_𝑗=1𝑏_2𝑗+ℒ^𝑔₂²_𝑗=1𝑐_2𝑗. (62) By Theorem 6, we know that the ICNNs with IDEPCA system (61) have a unique equilibrium.

In addition, we have 𝑣=max_{[]𝑖∈1,…,𝑛}^𝑛_𝑗=1ℒ^𝑓_𝑗𝑏_𝑖𝑗+ℒ^𝑔_𝑗𝑐_𝑖𝑗exp𝑎_∗⋅(𝑚−𝑙)⋅𝑙≈0.4615<1,_𝑛𝑗=1ℒ^𝑓_𝑗𝑏_1𝑗+ℒ^𝑔_𝑗𝑐_1𝑗(1−𝑣)⁻¹exp𝑎_∗⋅(𝑚−𝑙)≈0.5826<0.7442≈𝑎_∗−sup_𝑡∈ℝ+ℒ_𝑖(𝑡),_𝑛𝑗=1ℒ^𝑓_𝑗𝑏_2𝑗+ℒ^𝑔_𝑗𝑐_2𝑗(1−𝑣)⁻¹exp𝑎_∗⋅(𝑚−𝑙)≈0.7286<0.7442≈𝑎_∗−sup_𝑡∈ℝ+ℒ_𝑖(𝑡). (63) From Theorem 9, the ICNNs with IDEPCA system (61) have the unique equilibrium 𝑥^∗ which is globally asymptotically stable and all other solutions of the IDEPCA system (61) converge exponentially to it as 𝑡→∞. The numerical simulations, showing the convergence of the unique equilibrium 𝑥^∗ of the ICNNs with and without impulses (61), are given in Figures 2(a) and 2(b).

(a)

(b)

Figure 2 

(a) The simulation, where the initial value is chosen as (3,3)𝑇, illustrates that all trajectories uniformly converge to the unique equilibrium 𝑥∗=(2.608,5.719)𝑇 for the ICNNs (61) with impulsive effects. (b) The simulation, where the initial value is chosen as (3,3)𝑇, illustrates that all trajectories uniformly converge to the unique equilibrium 𝑥∗=(2.608,5.719)𝑇 for the ICNNs (61) without impulsive effects.

5. Conclusions

This is the first time that impulsive differential equations with alternately advanced and retarded argument have been applied to the model of cellular neural network models, and this paper has provided sufficient conditions guaranteeing the existence, uniqueness, and global exponential stability of the unique equilibrium of the impulsive cellular neural network models for the considered system based on a new IDEPCA integral inequality of Gronwall type and fixed point theorem. In addition, our method gives new ideas not only from the modeling point of view but also from that of theoretical opportunities since the impulsive cellular neural network model equation involves piecewise constant arguments of both advanced and delayed types. The obtained results could be useful in the design and applications of impulsive cellular neural network models. Furthermore, the examples with numerical simulations are given to show the effectiveness of the proposed method and results.

Acknowledgments

The author thanks the referees and editor very much for their valuable suggestions which made this paper much improved. This research was in part supported by FIBE 01-12 DIUMCE.