© Lizi Yin and Xilin Fu. 2011
Received: 13 November 2010
Accepted: 3 March 2011
Published: 14 March 2011
This paper is concerned with the problem of -stability of impulsive neural systems with unbounded time-varying delays and continuously distributed delays. Some -stability criteria are derived by using the Lyapunov-Krasovskii functional method. Those criteria are expressed in the form of linear matrix inequalities (LMIs), and they can easily be checked. A numerical example is provided to demonstrate the effectiveness of the obtained results.
In recent years, the dynamics of neural networks have been extensively studied because of their application in many areas, such as associative memory, pattern recognition, and optimization [1–4]. Many researchers have a lot of contributions to these subjects. Stability is a basic knowledge for dynamical systems and is useful to the real-life systems. The time delays happen frequently in various engineering, biological, and economical systems, and they may cause instability and poor performance of practical systems. Therefore, the stability analysis for neural networks with time-delay has attracted a large amount of research interest, and many sufficient conditions have been proposed to guarantee the stability of neural networks with various type of time delays, see for example [5–20] and the references therein. However, most of the results are obtained based on the assumption that the time delay is bounded. As we know, time delays occur and vary frequently and irregularly in many engineering systems, and sometimes they depend on the histories heavily and may be unbounded [21, 22]. In such case, those existing results in [5–20] are all invalid.
How to guarantee the desirable stability if the time delays are unbounded? Recently, Chen et al. [23, 24] proposed a new concept of -stability and established some sufficient conditions to guarantee the global -stability of delayed neural networks with or without uncertainties via different approaches. Those results can be applied to neural networks with unbounded time-varying delays. Moreover, few results have been reported in the literature concerning the problem of -stability of impulsive neural networks with unbounded time-varying delays and continuously distributed delays. As we know, the impulse phenomenon as well as time delays are ubiquitous in the real world [25–27]. The systems with impulses and time delays can describe the real world well and truly. This inspire our interests.
In this paper, we investigate the problem of -stability for a class of impulsive neural networks with unbounded time-varying delays and continuously distributed delays. Based on Lyapunov-Krasovskii functional and some analysis techniques, several sufficient conditions that ensure the -stability of the addressed systems are derived in terms of LMIs, which can easily be checked by resorting to available software packages. The organization of this paper is as follows. The problems investigated in the paper are formulated, and some preliminaries are presented, in Section 2. In Section 3, we state and prove our main results. Then, a numerical example is given to demonstrate the effectiveness of the obtained results in Section 4. Finally, concluding remarks are made in Section 5.
Let denote the set of real numbers, denote the set of positive integers, and denote the -dimensional real spaces equipped with the Euclidean norm . Let or denote that the matrix is a symmetric and positive semidefinite or negative semidefinite matrix. The notations and mean the transpose of and the inverse of a square matrix. or denote the maximum eigenvalue or the minimum eigenvalue of matrix denotes the identity matrix with appropriate dimensions and . In addition, the notation always denotes the symmetric block in one symmetric matrix.
where the impulse times satisfy is the neuron state vector of the neural network; is a diagonal matrix with are the connection weight matrix, the delayed weight matrix, and the distributively delayed connection weight matrix, respectively; is an input constant vector; is the transmission delay of the neural networks; represents the neuron activation function; is the delay kernel function and is the impulsive function.
Throughout this paper, the following assumptions are needed.
Obviously, the -stability analysis of the equilibrium point of system (2.1) can be transformed to the -stability analysis of the trivial solution of system (2.5). For completeness, we first give the following definition and lemmas.
Definition 2.1 (see ).
Lemma 2.2 (see ).
3. Main Results
This completes the proof of Theorem 3.1.
Theorem 3.1 provides a -stability criterion for an impulsive differential system (2.5). It should be noted that the conditions in the theorem are dependent on the upper bound of the derivative of time-varying delay and the delay kernels , and independent of the range of time-varying delay. Thus, it can be applied to impulsive neural networks with unbounded time-varying and continuously distributed delays.
In [23, 24], the authors have studied -stability for neural networks with unbounded time-varying delays and continuously distributed delays via different approaches. However, the impulsive effect is not taken into account. Hence, our developed result in this paper complements and improves those reported in [23, 24]. In particular, if we take , , , then the following result can be obtained.
4. A Numerical Example
In the following, we give an example to illustrate the validity of our method.
In this paper, some sufficient conditions for -stability of impulsive neural networks with unbounded time-varying delays and continuously distributed delays are derived. The results are described in terms of LMIs, which can be easily checked by resorting to available software packages. A numerical example has been given to demonstrate the effectiveness of the results obtained.
This paper is supported by the National Natural Science Foundation of China (11071276), the Natural Science Foundation of Shandong Province (Y2008A29, ZR2010AL016), and the Science and Technology Programs of Shandong Province (2008GG30009008).
- Chua LO, Yang L: Cellular neural networks: theory. IEEE Transactions on Circuits and Systems 1988,35(10):1257-1272. 10.1109/31.7600MathSciNetView ArticleMATHGoogle Scholar
- Cohen MA, Grossberg S: Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Transactions on Systems, Man, and Cybernetics 1983,13(5):815-826.MathSciNetView ArticleMATHGoogle Scholar
- Hopfield JJ: Neurons with graded response have collective computational properties like those of two-state neurons. Proceedings of the National Academy of Sciences of the United States of America 1984,81(10 I):3088-3092.View ArticleGoogle Scholar
- Kosko B: Bidirectional associative memories. IEEE Transactions on Systems, Man, and Cybernetics 1988,18(1):49-60. 10.1109/21.87054MathSciNetView ArticleGoogle Scholar
- Zhang Q, Wei X, Xu J: Delay-dependent global stability results for delayed Hopfield neural networks. Chaos, Solitons & Fractals 2007,34(2):662-668. 10.1016/j.chaos.2006.03.073MathSciNetView ArticleMATHGoogle Scholar
- Mohamad S, Gopalsamy K, Akça H: Exponential stability of artificial neural networks with distributed delays and large impulses. Nonlinear Analysis: Real World Applications 2008,9(3):872-888. 10.1016/j.nonrwa.2007.01.011MathSciNetView ArticleMATHGoogle Scholar
- Wang Q, Liu X: Exponential stability of impulsive cellular neural networks with time delay via Lyapunov functionals. Applied Mathematics and Computation 2007,194(1):186-198. 10.1016/j.amc.2007.04.112MathSciNetView ArticleMATHGoogle Scholar
- Huang Z-T, Yang Q-G, Luo X: Exponential stability of impulsive neural networks with time-varying delays. Chaos, Solitons & Fractals 2008,35(4):770-780. 10.1016/j.chaos.2006.05.089View ArticleMATHGoogle Scholar
- Lou XY, Cui B: New LMI conditions for delay-dependent asymptotic stability of delayed Hopfield neural networks. Neurocomputing 2006,69(16–18):2374-2378.View ArticleGoogle Scholar
- Singh V: On global robust stability of interval Hopfield neural networks with delay. Chaos, Solitons & Fractals 2007,33(4):1183-1188. 10.1016/j.chaos.2006.01.121MathSciNetView ArticleMATHGoogle Scholar
- Arik S: Global asymptotic stability of hybrid bidirectional associative memory neural networks with time delays. Physics Letters, Section A 2006,351(1-2):85-91. 10.1016/j.physleta.2005.10.059View ArticleMATHGoogle Scholar
- Zhang Y, Sun J: Stability of impulsive neural networks with time delays. Physics Letters, Section A 2005,348(1-2):44-50. 10.1016/j.physleta.2005.08.030View ArticleMATHGoogle Scholar
- Liao X, Li C: An LMI approach to asymptotical stability of multi-delayed neural networks. Physica D 2005,200(1-2):139-155. 10.1016/j.physd.2004.10.009MathSciNetView ArticleMATHGoogle Scholar
- Mohamad S: Exponential stability in Hopfield-type neural networks with impulses. Chaos, Solitons & Fractals 2007,32(2):456-467. 10.1016/j.chaos.2006.06.035MathSciNetView ArticleMATHGoogle Scholar
- Ou O: Global robust exponential stability of delayed neural networks: an LMI approach. Chaos, Solitons & Fractals 2007,32(5):1742-1748. 10.1016/j.chaos.2005.12.026MathSciNetView ArticleMATHGoogle Scholar
- Rakkiyappan R, Balasubramaniam P, Cao J: Global exponential stability results for neutral-type impulsive neural networks. Nonlinear Analysis: Real World Applications 2010,11(1):122-130. 10.1016/j.nonrwa.2008.10.050MathSciNetView ArticleMATHGoogle Scholar
- Rakkiyappan R, Balasubramaniam P: On exponential stability results for fuzzy impulsive neural networks. Fuzzy Sets and Systems 2010,161(13):1823-1835. 10.1016/j.fss.2009.12.016MathSciNetView ArticleMATHGoogle Scholar
- Raja R, Sakthivel R, Anthoni SM: Stability analysis for discrete-time stochastic neural networks with mixed time delays and impulsive effects. Canadian Journal of Physics 2010,88(12):885-898. 10.1139/P10-086View ArticleGoogle Scholar
- Sakthivel R, Samidurai R, Anthoni SM: New exponential stability criteria for stochastic BAM neural networks with impulses. Physica Scripta 2010.,82(4):Google Scholar
- Sakthivel R, Samidurai R, Anthoni SM: Asymptotic stability of stochastic delayed recurrent neural networks with impulsive effects. Journal of Optimization Theory and Applications 2010,147(3):583-596. 10.1007/s10957-010-9728-8MathSciNetView ArticleMATHGoogle Scholar
- Niculescu S-I: Delay Effects on Stability: A RobustControl Approach, Lecture Notes in Control and Information Sciences. Volume 269. Springer, London, UK; 2001:xvi+383.Google Scholar
- Kolmanovskiĭ VB, Nosov VR: Stability of Functional Differential Equations, Mathematics in Science and Engineering. Volume 180. Academic Press, London, UK; 1986:xiv+217.Google Scholar
- Chen T, Wang L: Global μ -stability of delayed neural networks with unbounded time-varying delays. IEEE Transactions on Neural Networks 2007,18(6):705-709.Google Scholar
- Liu X, Chen T: Robust μ -stability for uncertain stochastic neural networks with unbounded time-varying delays. Physica A 2008,387(12):2952-2962.MathSciNetView ArticleGoogle Scholar
- Lakshmikantham V, Baĭnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.View ArticleGoogle Scholar
- Baĭnov DD, Simeonov PS: Systems with Impulse Effect: Stability Theory and Applications, Ellis Horwood Series: Mathematics and Its Applications. Ellis Horwood, Chichester, UK; 1989:255.Google Scholar
- Li X: Uniform asymptotic stability and global stabiliy of impulsive infinite delay differential equations. Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):1975-1983. 10.1016/j.na.2008.02.096MathSciNetView ArticleMATHGoogle Scholar
- Li X, Fu X, Balasubramaniam P, Rakkiyappan R: Existence, uniqueness and stability analysis of recurrent neural networks with time delay in the leakage term under impulsive perturbations. Nonlinear Analysis: Real World Applications 2010,11(5):4092-4108. 10.1016/j.nonrwa.2010.03.014MathSciNetView ArticleMATHGoogle Scholar
- Boyd S, El Ghaoui L, Feron E, Balakrishnan V: Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics. Volume 15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA; 1994:xii+193.View ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.