#### Abstract

This paper researches global asymptotic stability of impulsive cellular neural networks with proportional delays and partially Lipschitz activation functions. Firstly, by means of the transformation , the impulsive cellular neural networks with proportional delays are transformed into impulsive cellular neural networks with the variable coefficients and constant delays. Secondly, we provide novel criteria for the uniqueness and exponential stability of the equilibrium point of the latter by relative nonlinear measure and prove that the exponential stability of equilibrium point of the latter implies the asymptotic stability of one of the former. We furthermore obtain a sufficient condition to the uniqueness and global asymptotic stability of the equilibrium point of the former. Our method does not require conventional assumptions on global Lipschitz continuity, boundedness, and monotonicity of activation functions. Our results are generalizations and improvements of some existing ones. Finally, an example and its simulations are provided to illustrate the correctness of our analysis.

#### 1. Introduction

Cellular neural networks (CNNs) introduced by Chua and Yang [1, 2] have found many important applications in biology, the solving of optimization problem, image processing, and pattern recognition [3]. In fact, CNNs can be characterized by an array of identical nonlinear dynamical systems (called cells) locally interconnected in the paper [4] which presented a set of sufficient conditions ensuring the existence of at least one stable equilibrium point in terms of the template elements. As we know, time delays are inevitable in electronic implementation of CNNs [5]. However, time delays may destroy stability of the networks and even lead to the oscillation behaviors. Hence, it is necessary to study the stability of CNNs with different types of delays. Time delays may be proportional delays; that is to say, the delay function is a monotonically increasing function with respect to , where is a constant and satisfies . The type of proportional delays is usually required in Web quality of service routing decision and one may be convenient to control the network’s running time according to the network allowed delays. Moreover, one can refer to the paper [6] about more information on the proportional delay engineering. Proportional delays [7–10] are unbounded time-varying ones different from constant delays [11], bounded time-varying delays [12–18], and unbounded distributed delay [19–23]. It is relatively difficult to deal with this class of the unbounded time-varying delays because none of any other assumptions are imposed on it compared with other unbounded time-varying delays, such as, unbounded distributed delays often require that the delay kernel functions satisfy , or there exists a positive number such that [20–23]. Several stability criteria of CNNs with proportional delays have been obtained [7]. Moreover, the abrupt changes in the voltages produced by faulty circuit elements are exemplary of impulse phenomena which can affect the transient behavior of the network [24]. Hence, it is significant to discuss the stability of the CNNs with impulses and proportional delays. However, to the best of the authors’ knowledge, few authors have handled the stability of CNNs with impulses and proportional delays.

Among the existing research results about neural networks, some activation functions are assumed to be globally Lipschitz continuous [25–30], bounded and monotonic [31], and bounded [24, 32]. However, these assumptions make these existing results unapplicable to some important engineering problems. For example, when the neural networks are used to solve optimization problems with the presence of constraints (linear, quadratic, or more general programming problems), unbounded (or nonmonotonic, non-globally Lipspchitz continuous) activations modeled by diode-like exponential-type functions are needed such that constraints are satisfied [33]. Motivated by this, we attempt to abandon these assumptions and only require activation functions to be partially Lipschitz continuous. Moreover, the relative nonlinear measure is more efficient than the nonlinear measure for exponential stability analysis of different classes of neural networks without delays where the equilibrium points are given [20, 34].

According to the foregoing analysis, this paper is devoted to analyzing stability of impulsive CNNs with proportional delays and Lipschitz continuous activation functions by relative nonlinear measure. The remainder of this paper is arranged as follows. Section 2 describes the model of proportion-delayed impulsive CNNs with partial Lipschitz continuous activation functions and provides its equivalent form by some transformation. Being preliminaries, Section 3 is devoted to uniqueness and exponential stability of equilibrium point of a nonlinear impulsive functional differential equation with variable coefficients and constant delays by means of relative nonlinear measure. In Section 4, a sufficient condition is obtained for global asymptotic stability of equilibrium point of impulsive proportion-delayed CNNs with partially Lipschitz continuous activation functions by results derived in Section 3. Furthermore, an example and its simulations are presented to illustrate that our method is valid and that our derived results are new and correct. Conclusions are given in Section 5.

#### 2. Model Description and Its Equivalent Form

We consider the following CNNs with impulses and multiproportional delays: for , where is the number of cells in the networks; denotes the potential of the th cell at time ; represents the rate with which the th cell resets its potential to the resting state when isolated from other cells and inputs at time ; , and denote the strengths of connectivity between the th and the th cells at time , , and , respectively; and are proportional delay factors and satisfy , , and , , in which correspond to the time delays required in processing and transmitting a signal from the th cell to the th cell, and , as ; is the impulse at moments and is a strictly increasing sequences such that ; , , and are the nonlinear activation functions; denotes the th component of an external input source introduced from outside the network to the th cell at time .

To discuss stability of the networks (1), we only assume the following.(H)Activation functions , , and are partially Lipschitz continuous on for .

In what follows, we plan to transform model (1) into a model what we can directly deal with. Motivated by this paper [7], we define the transformation by

(I) When and , then , and ; that is, Taking and then , then the transformation (2) is written as From (1) and (4), we derive that is, From transformation (2), we obtain where , .

By (2), (6), and (7), we enjoy

(II) When and , then and . By transformation (2), we have

(III) When , from (1) we have where . Hence, the initial functions associated with (8) are given by

Conversely, let , in (8); by transformation (2), then (8) can be written as (1) for and , for and , and for , from (10) and (11), the initial function associated with (1) is given by , .

In conclusion, in the sense of solutions, the CNNs with impulses and multiproportional delays (1) is equivalent to the following CNNs with constant delays and variable coefficients for , where , , , denoting the space of all continuous functions from to for and .

#### 3. Preliminaries

Let -dimensional real vector space be endowed with -norm defined by where the superscript denotes the transpose. Let denote the inner product in and the sign vector of , where represents the sign function of . Obviously, the relations hold for all .

In order to discuss the stability of the neural networks (1), we firstly consider exponential stability of the following differential equation with variable coefficients, delays, and impulses where , denotes the space of all continuous functions from into the open subset of ; is defined by for all and ; and are nonlinear operators; is a strictly increasing sequence such that ; is defined as follows: The nonlinear operators and are defined, respectively, by

*Definition 1 (see [20]). * A nonlinear operator is called be Lipschitz continuous on if there exists a nonnegative constant such that
where is called the Lipschitz constant of on . The constant
is called the minimal Lipschitz constant (MLC) of on . Furthermore, the operator is called globally Lipschitz continuous if .

A nonlinear operator is said to be partially Lipschitz continuous on if, for any , there exists a constant such that
The constant
is called minimal partial Lipschitz constant (MPLC) of on with respect to . Furthermore, the operator is called partially Lipschitz continuous if .

From the paper [20] we conclude that every Lipschitz continuous operator on is partially Lipschitz continuous on and for any Lipschitz continuous operator and .

*Definition 2 (see [34]). *Assume that is an open subset of , is a nonlinear operator from into , and is any vector. The constant
is called relative nonlinear measure of at .

*Definition 3. * is said to be an equilibrium point of (15) if and for all .

*Definition 4. *Let be an equilibrium point of (15) and an open neighborhood of . is exponentially stable on if there exist two positive constants and such that
holds for , where is the unique solution of (15) initiated from the function .

Particularly, if holds, then is the unique equilibrium point and (15) is said to be globally exponentially stable.

Lemma 5 (see [35]). *If , for every nonnegative real number , the equation
**
has a unique positive solution.*

Lemma 6 (see [36]). *Let for and . Suppose that
**
If , there exist constants and such that
**
holds for .*

Theorem 7. *Let be an open neighborhood of the equilibrium point of (15). Equation (15) has no other equilibrium point in different from if .*

*Proof. *Assume that is any equilibrium point of (15) different from ; that is,
Then, we derive
which contradicts .

Theorem 8. *Let be a neighborhood of the equilibrium of (15). Assume and to be partially Lipschitz continuous on with respect to and
**
If there exists some diagonal matrix with such that the inequality
**
holds, then is exponentially stable on . Particularly, the solution of (15) initiated from decays by
**
where is the unique positive solution of the equation
*

*Proof. *Let for all . From the relations (14) we derive that
holds for all . Consequently, the function is absolutely continuous in , which implies that derivatives of exist almost everywhere in . Furthermore, from (15) we conclude that derivatives of satisfy
The combination of condition (30) and Lemmas 6 and 5 implies that
holds for all , where is the unique positive solution of the equation
It needs to point out that the positive solution of (36) is strictly monotonically increasing with respect to . In fact, let be the positive solution of the equation
By subtracting (37) from (36), we derive
Furthermore, we have
Since both of and are positive, we have
It is obvious that is not equal to , that is, or . If , then and from inequality (40), that is,
Since , contradict the assumption . This means that holds for ; that is, the positive solution of (36) is strictly monotonically increasing with respect to . Hence, , where is the unique positive solution of (37) at ; that is,
Since for all , , inequality (35) means that
holds for all , , where is the unique positive solution of (32). Inequality (43) is accordingly changed into the following form:
which holds for all , . According to condition (29), we enjoy
This implies
From (35) and (46) we derive
In conclusion, we obtain inequality (31).

*Remark 9. *Our proof idea mainly comes from Theorem 2 of [20] investigating the exponential stability of the special case of (15) (i.e., (15) with constant coefficients). However, they are essentially different because Theorem 8 in this paper has to deal with time-varying coefficients. Consequently, Theorem 8 in this paper is a generalization of Theorem 2 in [20]. Moreover, it needs to point out that the exponential stability criterion (30) and exponential decay index in (31) are independent of time although the abstract equation (15) enjoys time-varying coefficients, which means that our method is essential to qualitatively and quantitatively characterize exponential stability of (15). Moreover, Theorem 8 is not only generalization and improvement of Theorem 1 in [35] because there indeed exists a nonlinear Lipschitz continuous map on such that is strictly less than for any and (15) enjoys time-varying coefficients.

It is obvious that the CNNs model (12) can be changed into the form of (15). By Theorem 8, we can obtain the exponential stable criterion of equilibrium point of the CNNs model (12). Since the model (1) is equivalent to the model (12) in the sense of solution, models (12) and (1) enjoy the same equilibrium point , where and are the equilibrium point of models (12) and (1), respectively. What qualitative property of the model (1) can be derived from the global exponential stability of the model (12)? The next theorem can answer this problem.

Theorem 10. *Suppose that the equilibrium point of the model (12) is globally exponentially stable, that is, that exist two positive constants and such that
**
holds for , where is the unique solution of the model (12) initiated from . Then of the model (1) is globally asymptotic stable. Particularly, the inequality
**
holds for , where is the unique solution of the model (1) initiated from , .*

*Proof. *By the transformation (2) and the inequality (48), we derive
Let , then and . Let , then . The inequality (50) implies
Taking , we furthermore derive
This implies that the equilibrium point of the model (1) is globally asymptotic stable.

*Remark 11. *It need point out that the paper [7] has obtained not exponential stability, but asymptotic stable criteria of CNNs with multi-proportional delays because it mistakes asymptotic stability as exponential stability, which can be easily seen from the Remark 3.2 in [7] and Theorem 10 in this paper.

#### 4. Uniqueness and Global Asymptotic Stability of Equilibrium Point of Model (1)

In this subsection, we firstly prove that model (1) has a unique equilibrium point in . It is enough to prove that model (12) has a unique equilibrium point in because models (12) and (1) enjoy the same equilibrium point. For this, we define that and are defined, respectively, by

Theorem 12. *Suppose that the assumption holds and is an equilibrium point of the model (1). For each set of external inputs, , model (1) has no other equilibrium point in different from if there exist positive real numbers () such that
**
holds, where , , and denote MPLC of , , and on with respect to , respectively.*

*Proof. *Obviously, it is enough to prove that model (12) has no other equilibrium point in different from if the inequality (54) holds. Define and we need only prove according to Theorem 7. In detail, for , we enjoy
The combination of (55) and (54) implies that , which implies that model (12) enjoys no other equilibrium point in different from . That is to say, is the unique equilibrium point in of model (1).

Secondly, we prove that condition (54) also guarantees global asymptotic stability of equilibrium point of model (1) by Theorems 8 and 10.

Theorem 13. *Assume that assumption holds, is the equilibrium point of the model (1), and , , for and . If there exist a set of positive real numbers () such that condition (54) holds, then for each set of external input, , model (1) is globally asymptotic stable. Particularly, if is the solution of the model (1) initiated from , then the inequality
**
holds for , where is the unique positive solution of the equation
**
with
*

*Proof. *Obviously, is also the equilibrium point of model (12) because model (1) is equivalent to the model (12) in the sense of solution. Firstly, we prove that model (12) is globally exponentially stable by Theorem 8. For this, let and . It immediately follows from the condition (54) that
For all ,
which implies that . For all , we have
thus,
Consequently, from (54) we conclude that
By Theorem 8, the solution of the functional differential equation
satisfies
where is the unique positive solution of (57). It is obvious that is the solution of (64) if is a solution of model (12). Consequently, the equilibrium point of model (12) is globally exponentially stable; that is,
holds for , where is unique positive solution of (57), , , , and . By Theorem 10, we derive that the solution of model (1) initiated from satisfies inequality (56) for . That is to say, the solution of model (1) is globally asymptotically stable.

*Remark 14. *Theorems 12 and 13 provide a sufficient condition (54) to the uniqueness and global asymptotic stability of the equilibrium point of impulsive CNNs (1) with multiproportional delays and partially Lipschitz continuous activation functions. On the one hand, the proportional delay is time varying, unbounded, and monotonic and the model (1) does not require the proportional delays to meet any other condition. Hence, compared with the results in papers [11–23] with the constant, bounded time varying, or unbounded distributed delays, our results are new. Moreover, the stability of CNNs with general unbounded time varying delays in [37, 38] and proportional delays in [7–10] has been investigated. Compared with these results, our results are their generalizations because model (1) has impulsive perturbations. On the other hand, model (1) only requires activation functions to be partially Lipschitz continuous. In fact, partial Lipschitz continuity is less conservative; that is, it does not meet conventional assumptions, such as, boundedness, global Lipschitz continuity, or monotonicity. Compared with these excellent results on neural networks with globally Lipschitz continuous [25–30], bounded and monotonic [31], or bounded [24, 32] activation functions, our results are new. Furthermore, our results are also generalizations of the papers [24, 29, 35] and even improvement of the paper [35] with globally Lipschitz continuous activation functions and there indeed exists a nonlinear globally Lipschitz continuous map on such that is strictly less than for any .

#### 5. Illustrative Example

In this section, we present an illustrative example to verify effectiveness of our method.

*Example 1. *Consider impulsive CNNs with proportional delays and partially Lipschitz continuous activation functions
where
for and , , , and , , is a strictly increasing sequence such that and , and , . and for .

From the definition of we can conclude
holds for , which means that is partially Lipschitz continuous on . It is easily verified that is the equilibrium point of the model (67) and ; that is, for . Consequently, the criteria of the papers [24, 29] are not applied to this the model (67) because they require activation functions to be globally Lipschitz continuous. Moreover, none of the stability criteria in