Abstract and Applied Analysis

Volume 2014 (2014), Article ID 832892, 11 pages

http://dx.doi.org/10.1155/2014/832892

Research Article

Global Asymptotic Stability of Impulsive CNNs with Proportional Delays and Partially Lipschitz Activation Functions

Xueli Song¹ and Jigen Peng²

¹Department of Mathematics and Information Science, Chang’an University, Xi’an 710064, China

²School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Received 15 April 2014; Accepted 26 June 2014; Published 23 July 2014

Academic Editor: Ademir F. Pazoto

Copyright © 2014 Xueli Song and Jigen Peng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [11–23] is applied to model (67) since the delay factors and are unbounded and monotonic and not required to meet any other additional conditions. Although the papers [7–10] have investigated the stability of CNNs with proportional delays, these methods are not able to apply to model (67) because model (67) enjoys the impulsive perturbations.

However, taking , and , we obtain

that is, the condition (54) holds for , , and . Moreover, for . According to Theorems 10 and 12, we conclude that the equilibrium point of the model (67) is globally asymptotically stable and the solution satisfies the following inequality (Figure 1):

where is the solution of (67) initiated from and is the unique positive solution of the equation .

Figure 1: The simulation for the solution to the neural networks (67) at the impulsive moments .

Remark 2. It needs to be pointed out that the impulsive instants are only selected as in the simulation of this example to simplify the simulation, which is obviously not enough to illustrate the impulsive effect. In order to accurately characterize wider of impulses, the papers [39, 40] proposed the concepts of average dwell time and average impulsive interval. Moreover, the papers [41, 42] presented single impulsive controller and pinning impulsive stabilization criterion, respectively. Consequently, their methods are recommended to simulations with more general impulses.

6. Conclusions and Further Work

By means of relative nonlinear measure and transformation, this paper has discussed global asymptotic stability of impulsive cellular neural networks with proportional delays and partially Lipschitz activation functions. We have obtained the novel criterion of uniqueness and global asymptotic stability of the equilibrium point of this CNNs model. Our method does not require conventional assumptions on global Lipschitz continuity, boundedness, and monotonicity of activation functions and proportional delays to meet other requirements, which demonstrates that our criteria derived are less restrictive than some existing ones and that they are generalizations and improvements of some existing ones. Finally, the example with three cells has illustrated that our method is effective and that our results are correction.

Our method only requires the activation functions of the cellular networks to be partially Lipschitz continuous. The relative weak assumption makes our results applicable to more general engineering problems. In the future, we attempt to design a cellular neural networks model to solve optimization problems with some constraints, where unbounded (or nonmonotonic, nonglobally Lipspchitz continuous) activations are required such that these constraints are satisfied.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation of China under the Contact (nos. 11201038, 11171043), Youth Science and Technology Nova Program of Shaanxi Province under the Contact no. 2014KJXX-55, the Scientific Research Program Funded by Shaanxi Provincial Education Department under the Contact no. 2013JK0591, the Special Fund for Basic Scientific Research of Central Colleges in Chang’an University under the Contact (nos. 2013G2121017, CHD2012TD015), and Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China the Natural Science.

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