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Research Article | Open Access
New Delay-Dependent Exponential Stability Criteria for Neural Networks with Mixed Time-Varying Delays
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.
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 , pattern recognition , signal detection , landmark recognition , 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 , 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  were assumed to be monotonically nondecreasing. In , 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 , 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 , 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 .
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 , 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 towhere . It is easy to check that the function satisfies , andDue to the influence of external factors, cannot express the actual state of the accurate information. Therefore, by translating to function , we haveIn 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.
Lemma 2 (see ). The following inequalities are true:
Lemma 3 (see ). For any positive definite matrix and , , the following inequalities hold:
Lemma 4 (see ). For any constant matrix , , a scalar function , and a vector-valued function such that the following integrations are well:
Lemma 5 (see ). 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 , andsuch that the following symmetric linear matrix inequality holds:where
Proof. Consider an augmentation of LKF for system (6) as follows:whereThe time derivative of along with the trajectory of system (6) is given as where
Using Lemma 4, we can haveUsing Lemma 5, we may getBy Lemma 3, we can obtainFrom (5), for any diagonal matrices , the following inequality holds:Furthermore, for arbitrary matrices , , , and with appropriate dimensions, we haveThe combination of (19)–(24) giveswhere .
From (14), we know that , which means the asymptotically stability of system (5). This completes the proof.
Furthermore, setting , we can haveIt is easy to haveAccording to with , Thus according to (26)–(28), there exists a positive constant such thatwhereOn the other hand, we haveThereforeThen, 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 , 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 , andsuch that the following symmetric linear matrix inequality holds:whereThe 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.
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: