Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 767456, 16 pages
http://dx.doi.org/10.1155/2015/767456

Research Article

New Delay-Dependent Exponential Stability Criteria for Neural Networks with Mixed Time-Varying Delays

Wu Wen¹ and Kaibo Shi2,3

¹Department of Academic Affairs Office, Sichuan University of Arts and Science of China, Dazhou 635000, China
²School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
³Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1

Received 17 December 2014; Revised 8 April 2015; Accepted 15 April 2015

Academic Editor: Asier Ibeas

Copyright © 2015 Wu Wen and Kaibo Shi. 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 study is concerned with the problem of new delay-dependent exponential stability criteria for neural networks (NNs) with mixed time-varying delays via introducing a novel integral inequality approach. Specifically, first, by taking fully the relationship between the terms in the Leibniz-Newton formula into account, several improved delay-dependent exponential stability criteria are obtained in terms of linear matrix inequalities (LMIs). Second, together with some effective mathematical techniques and a convex optimization approach, less conservative conditions are derived by constructing an appropriate Lyapunov-Krasovskii functional (LKF). Third, the proposed methods include the least numbers of decision variables while keeping the validity of the obtained results. Finally, three numerical examples with simulations are presented to illustrate the validity and advantages of the theoretical results.

1. Introduction

Over the course of the past decade, neural networks have become an important area of research and attracted increasing attention due to their extensive applications in many practical systems, such as power systems [1], pattern recognition [2], signal detection [3], landmark recognition [4], and other scientific areas [5–7].

Moreover, it is inevitable to introduce time delay into the signals transmitted among neurons because the processes of transcription and translation are not instantaneous. However, it is a well-known fact that time delay as a source of instability and poor performance usually appears in many dynamical systems, for instance, Cohen-Grossberg neural networks, cellular neural networks, BAM neural network, chaotic neural networks, filtering, and nonlinear systems [8–53]. Therefore, stability analysis for neural networks with delays has been an attractive subject of research in recent years [8–42, 50–53].

Furthermore, neural networks (NNs) often have a spatial nature due to the presence of many parallel pathways of a variety of axon sizes and lengths. Thus, in order to have a more accurate model, a distributed delay over a certain time of duration needs to be included in NNs such that the distant past has less influence compared to the recent behavior of the state. Therefore, there has been a growing interest in the study of neural networks with discrete and distributed delays during the past two decades. To date, some results on delay-dependent exponential stability for neural networks with mixed time-varying delays have been reported in [18–23, 34–36]. In [18], the authors considered the global asymptotic stability for a class of delayed cellular neural networks with mixed time-varying delays by using LMIs approach, Lyapunov theory, and Leibniz-Newton formula. However, the activation functions in [18] were assumed to be monotonically nondecreasing. In [19], several delay-dependent sufficient conditions are obtained to guarantee the global asymptotic and exponential stability of the addressed neural networks by employing appropriate LKF and linear matrix inequality (LMI) technique. In [20], an exponential stability criterion is proposed by constructing an augmented LKF, where the discrete delay must be differentiable. In [22, 23], some improved delay-dependent stability criteria are derived in terms of linear matrix inequalities by dividing the discrete delay interval into multiple segments. Different from [35], the appropriate LKF not only divides the discrete delay interval into two ones and , but also divides the discrete delay interval into three ones , , and . Although this approach seems to be effective for achieving less conservative conditions, it can increase the larger numbers of computed variables. Hence, there exists great room for further improvement. To the best of our knowledge, it is of a great significance for the current research to find a more effective approach to get rid of the strict constraint and obtain less conservative conditions.

Motivated by the above discussion, combining effective mathematical techniques and a convex optimization approach, we choose a more general type of LKF to study the delay-dependent exponential stability criteria for neural networks (NNs) with mixed time-varying delays in the paper. Some improved delay-dependent stability conditions derived benefit mainly from using firstly a new integral inequality approach, which is proved to be less conservative than the celebrated Jensen’s inequality and showed having a great potential efficient in practice. Both theoretical and numerical comparisons have been provided to show the effectiveness and efficiency of the proposed method. Besides, the main merit of this method lies in containing the least numbers of decision variables while keeping the validity of the obtained results. Finally, the stability criteria obtained turn out to be less conservative than some recently reported ones via three numerical examples.

Notation. Notations used in this paper are fairly standard: denotes the -dimensional Euclidean space, and   is the set of all dimensional matrices;   is the identity matrix of appropriate dimensions, and   is the matrix transposition of the matrix . By (resp., ), for , we mean that the matrix is real symmetric positive definite (resp., positive semidefinite); denotes block diagonal matrix with diagonal elements , , and the symbol represents the elements below the main diagonal of a symmetric matrix; is defined as .

2. Preliminaries

Consider the following neural networks with mixed time-varying delays:

where is the neural state vector and is the neuron activation function; is an external constant input vector, , and , , and are the constant matrices of appropriate dimensions.

Assumption A. The time-varying delay is continuous and differential function satisfying

Assumption B. For the constants and the bounded activation function in (1) satisfies the following condition:

We denote , , , , and .

Under Assumption B, by using Brouwer’s fixed-point theorem [25], it can be easily proven that there exists one equilibrium point for system (1). Assuming that is an equilibrium point of system (1). For convenience, we firstly shift the equilibrium point to the origin by letting and , and then system (1) can be converted to

where . It is easy to check that the function satisfies , and

Due to the influence of external factors, cannot express the actual state of the accurate information. Therefore, by translating   to function   , we have

In the paper, we will attempt to formulate some practically computable criteria to check the global exponential stability of system (6). The following lemmas are useful in deriving the criteria.

Definition 1 (see [22]). The equilibrium point of system (6) is said to be globally exponentially stable, if there exist scalars and such that

Lemma 2 (see [22]). The following inequalities are true:

Lemma 3 (see [24]). For any positive definite matrix and , , the following inequalities hold:

Lemma 4 (see [54]). For any constant matrix , , a scalar function , and a vector-valued function such that the following integrations are well:

Lemma 5 (see [17]). For any positive semidefinite matrices,

the integral inequality holds as follows:

3. Main Results

In this section we will give sufficient conditions under which system (6) is globally exponentially stable.

Theorem 6. For given scalars , , and , the origin system (6) with the neuron activation function satisfying condition (5) and the time-varying delay satisfying (2) is globally exponentially stable with the exponential convergence rate index if there exist ,   ,   , diagonal matrices , , , , , , any matrices   , and

such that the following symmetric linear matrix inequality holds:

where

Proof. Consider an augmentation of LKF for system (6) as follows:

where

The time derivative of along with the trajectory of system (6) is given as

where

Using Lemma 4, we can have

Using Lemma 5, we may get

By Lemma 3, we can obtain

From (5), for any diagonal matrices   , the following inequality holds:

Furthermore, for arbitrary matrices , , , and with appropriate dimensions, we have

The combination of (19)–(24) gives

where .
From (14), we know that , which means the asymptotically stability of system (5). This completes the proof.
Furthermore, setting  , we can have

It is easy to have

According to with ,

Thus according to (26)–(28), there exists a positive constant such that

where

On the other hand, we have

Therefore

Then, from Definition 1, system (6) is exponentially stable with convergence rate , and the proof is completed.

Remark 7. In the paper, the reduced conservatism of Theorem 6 benefits primarily from a new integral inequality, which is proved to be less conservative than the celebrated Jensen’s inequality, and takes fully the relationship between the terms in the Leibniz-Newton formula within the framework of LMIs into account. In order to lower the conservatism of stability criteria, we further deal with the integral terms of and via Lemma 4. Different from that of [17], this kind of processing method can reduce ulteriorly the conservatism of stability criteria.

Remark 8. As a matter of fact, Theorem 6 gives a stability criterion for system (6) with satisfying , , where is given constant. In many cases, is unknown. Considering this situation, a rate-independent corollary for the delay satisfying is derived by setting , , and in the proof of Theorem 6.

Theorem 9. For given scalars and , the origin of system (6) with the neuron activation function satisfying condition (5) is globally exponentially stable with the exponential convergence rate index if there exist , ,   , diagonal matrices , , , , , , any matrices   , and

such that the following symmetric linear matrix inequality holds:

where

The other procedure is straight forward from the proof of Theorem 6, so we omit it.

Remark 10. In the paper, we make full use of the relationship between and , which can reduce the conservatism of stability criteria once again. However, these useful terms of and were always ignored in [18–20], which may lead to considerable conservatism to certain extent.

Remark 11. Due to constructing a simple type of Lyapunov-Krasovskii functional and taking full advantage of effective mathematical techniques, the conservatism of improved delay-dependent stability criteria obtained is reduced to a great degree in this study. Compared with those in previous articles [22, 23], we employ a few free variables and do not use a delay decomposition method and add some zero terms, that is, not only dividing the discrete delay interval into two ones and , but also dividing the discrete delay interval into three ones , , and and adding the following equalities:

By using this method, the conservatism of the obtained stability condition in [22, 23] is reduced to some degree. However, the computing complexity is also improved since more variables are involved. Besides, we provided a comparison of the numbers of the variables involved in [22, 23] and our paper in Table 1. From Table 1, it is clear to see that the number of decision variables in our paper is much less than those in [22, 23]. Thus, it also expounds validity and applicability of the proposed method.

Table 1: Comparison of the numbers of the involved variables.

Method/number of decision variables
[22]
[23]
This paper

Theorem 6

Theorem 9

Remark 12. In many actual applications, maximum allowable time-delay upper bounds   are of interest. In Theorems 6 and 9, with a fixed and , can be obtained through following optimization procedure:

Besides, maximum allowable time-delay upper bounds obtained are very valuable. In Theorems 6 and 9, with a fixed and , can be also acquired through following optimization procedure:

Inequalities (37) and (38) are a convex optimization problem and can be obtained efficiently by using the MATLABLMI Toolbox.

4. Numerical Examples

In this section, three examples are given to demonstrate the feasibility and effectiveness of the main results derived above.

Example 1. Consider a delayed neural network in (6) with parameters as follows:

Case A (set , ). For different and , the allowable upper bounds of the exponential convergence rate index calculated by Theorem 6 in this paper and Theorem in [20–24] are listed in Table 2. According to Table 2, this example is given to indicate significant improvements over some existing results.

Table 2: Allowable upper bounds of for for Case A in Example 1.

Method
[20]
[21]
[24]
[23]
[22]
Theorem 6

, ,
0.46
0.58
0.56
0.67
0.80
0.88

, ,
0.21
0.35
0.35
0.45
0.50
0.65

, ,
0.06
0.20
0.33
0.29
0.34
0.45

, ,
0.00
0.05
0.10
0.11
0.14
0.21

Besides, for the parameters listed above, let , , , and . Then we can obtain the following feasible parameters by Theorem 6 in our paper. Due to the limitation of the length of this paper, we only provide a part of the feasible solutions here as follows:

Case B (set , ). When and , the maximum allowable upper bounds obtained from Theorem 6 for different values are listed in Table 3. From Table 3, it can be seen clearly that our results have less conservatism over the results acquired by [20, 22, 34–36] and illustrate the effectiveness and the advantage of the proposed methods. The active functions are chosen as , , and . Moreover, the globally exponentially stable with the initial value is shown in Figures 1–3.

Table 3: Maximum allowable time delay upper bounds for Case B in Example 1.

Method
[20]
[34]
[35]
[36]
[22]
Theorem 6

2.2931
3.3294
5.8415
7.2820
9.6503
9.8356

2.1010
2.6363
4.0130
5.8958
8.0394
8.2175

1.7939
1.7200
3.6574
4.0847
5.8974
6.0561

Figure 1: State trajectories of in the plane for in Example 1.

Figure 2: Phase trajectory of model of in the plane for in Example 1.

Figure 3: Phase trajectory of model of in the space for in Example 1.

Example 2. Consider a delayed neural network in (6) with parameters as follows:

Let and let .
Case A. For the corresponding upper bounds of for unknown obtained by Theorem 9 and the results in [18–20] are listed in Table 4. According to Table 4, this example shows that the stability criterion in this paper gives much less conservative results than those in the previous literatures.
The active functions are set as , , and ; the globally exponentially stable with the initial value is shown in Figures 4–6.
Case B. For , the corresponding upper bounds of for unknown obtained by Theorem 9 are listed in Table 5. From Table 5, it is shown clearly that our results have significant improvement over the existing results.

Table 4: Maximum allowable time delay upper bounds in Example 2.

Method
[19]
[18]
[20]
Theorem 9

1.833
3.597
5.068
5.220

Table 5: Maximum allowable time delay upper bounds for different values in Example 2.

Method
0.1
0.3
0.5
0.8

[22]
9.082
4.774
3.313
2.217

Theorem 9
9.235
4.963
3.518
2.363

Figure 4: State trajectories of in the plane for in Example 2.

Figure 5: Phase trajectory of model of in the plane for in Example 2.

Figure 6: State trajectories of on the space for in Example 2.

Example 3. Consider a delayed neural network in (6) with parameters as follows:

Let and let .
When and , the corresponding upper bounds of for different obtained by Theorem 6 are listed in Table 6. The active functions are chosen as and ; the globally exponentially stable with the initial value is given in Figures 7–9. Therefore, we can say that the results in this paper are less conservative than those in [24].

Table 6: Comparison of the maximum delay between different methods for various in Example 3.

0
0.2
0.5
0.7

[24]
3.377
3.336
3.268
3.264

Theorem 6
3.518
3.427
3.358
3.386

Figure 7: State trajectories of in the plane for in Example 3.

Figure 8: Phase trajectory of model of in the plane for in Example 3.

Figure 9: State trajectories of on the space for in Example 3.

5. Conclusions

In this paper, the delay-dependent exponential stability problem for NNs with mixed time-varying delays has been investigated. By using a new integral inequality approach to express the relationship between the terms in the Leibniz-Newton formula within the framework of LMIs for the first time, several less conservative delay-dependent exponential stability criteria are obtained. Moreover, combining effective mathematical techniques and a convex optimization approach, new delay-dependent exponential stability conditions are derived by constructing a proper LKF. Finally, three numerical examples are given to illustrate the feasibility and effectiveness of the proposed methods. The foregoing methods have the potential to be useful for the further study of neural network. Meanwhile, it is expected that these approaches can be further applied to other delayed systems.

Disclaimer

The authors promise that this paper has not been published and will not be simultaneously submitted or published elsewhere.

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 National Basic Research Program of China (2010CB732501), National Natural Science Foundation of China (61273015), the National Defense Pre-Research Foundation of China (Grant no. 9140A27040213DZ02001), the Fundamental Research Funds for the Central Universities (ZYGX2014J070), and the Program for New Century Excellent Talents in University (NCET-10-0097).

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