Exponential stability of Markovian jumping stochastic Cohen–Grossberg neural networks with mode-dependent probabilistic time-varying delays and impulses☆
Introduction
The Cohen–Grossberg neural network (CGNN) model, proposed by Cohen and Grossberg [1] in 1983, has attracted considerable attentions due to their extensive applications in classification of patterns, associative memories, image processing, quadratic optimization and other areas. Over a decade, many scientific and technical workers have been joining the study fields with great interest, and various interesting results for CGNNs with/without delays have been reported [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Because time delays are often encountered in very large scale integration (VLSI) implementations of artificial neural networks due to delay transmission line and partial element equivalent circuit (PEEC), delayed neural networks (DNNs) have become a focus of research and a great number of results have been reported in the literature. As is well known, in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes [12]. On the other hand, it has been pointed out in [13] that a neural network could be stabilized or destabilized by certain stochastic inputs. Besides time delays, in the applications and designs of networks, some unavoidable uncertainties, which result from using an approximate system model for simplicity, parameter fluctuations, data errors, and so on, must be integrated into the system model. Such time delays, parametric uncertainties and stochastic disturbances may significantly influence the overall properties of a dynamic system. Therefore, it is of practical importance to study the stochastic effects on the stability property of delayed CGNNs [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26].
Meanwhile, in a real system, time-delay often exists in a random form, i.e., some values of time-delay are very large but the probability taking such large values is very small, which will lead to some conservatism if only the information of variation range of time-delay is considered. Thus, recently, some researchers have considered the stability for various neural networks with probability-distribution delays [27], [28], [29], [30], [31], [32], [33], [34]. Markovian jump systems, introduced by Krasovskii and Lidskii [35] in 1961, have received increasing attentions in the past years [36], [37]. It should be noted that Markovian jump systems can be considered a special class of hybrid systems, which can be described by a set of linear systems with the transitions between models determined by a Markovian chain in a finite mode set. This kind of systems has applications in economic systems, modeling production systems and other practical systems. In this regard, a great number of results on stability analysis for neural networks with Markovian jumping parameters have been reported in the literature and various approaches have been proposed [38], [39] and references therein.
On the other hand, impulsive effects exist widely in many evolution processes in which states are changed abruptly at certain moments of time, involving such fields as medicine and biology, economics, mechanics, electronics and telecommunications, see for example [40] and references therein. Thus, the study of impulsive neural networks with delays is a very good research topic in recent years and many researchers have investigated the problem of stability analysis of impulsive neural networks with delays [41], [42]. Neural networks are often subject to impulsive perturbations that in turn affect dynamical behaviors of the systems [43]. Therefore, it is necessary to take impulsive effects into account on dynamical behaviors of neural networks [44], [45], [46], [47]. To the best of the author's knowledge, very few results on the problem of exponential stability analysis for Markovian jumping stochastic Cohen–Grossberg neural networks with mode-dependent probabilistic time-varying delays and impulses have been studied in the literature. This motivates our present research.
Inspired by the above discussions, in this paper, the robust exponential stability results for Markovian jumping stochastic Cohen–Grossberg neural networks (MJSCGNNs) with mode-dependent probabilistic time-varying delays, continuously distributed delays and impulsive perturbations are considered. By constructing of novel Lyapunov–Krasovskii functional having the triple integral terms and introducing of free-weighting matrices, several new criteria for global exponential stability of MJSCGNNs are derived, which are expressed in terms of LMIs. Finally, the results are illustrated through some numerical simulation examples.
Notation: Let denote the n-dimensional Euclidean space and the superscript “T” denote the transpose of a matrix or vector. I denotes the identity matrix with compatible dimensions. denotes a block diagonal matrix. For square matrices, M1 and M2, the notation denotes is a positive-definite (positive-semi-definite, negative, negative-semi-definite) matrix. and stand for the minimum and maximum eigenvalues of a given matrix. Let be a complete probability space with a natural filtration and stand for the correspondent expectation operator with respect to the given probability measure . Also, let and denote the family of continuously differentiable function ϕ from to with the norm , where is the Euclidean norm in and . Denote by the family of bounded , random variables such that .
Section snippets
Problem description and preliminaries
In this paper, the Markovian jump stochastic CGNNs with both impulsive perturbations and mixed time delays are described as follows:for and , where is the state vector associated with the n neurons, ,
Main results
In this section, we will investigate the problems of exponential stability for MJSCGNNs with delays and impulsive perturbations. Firstly, the following result is given for system (10). Theorem 3.1 Under Assumption 2.1, Assumption 2.2, Assumption 2.3, Assumption 2.4, Assumption 2.5, Assumption 2.6 the trivial solution of (10) is exponentially stable in the mean square, if there exist positive scalars , positive diagonal matrices , , , , ,
Numerical examples
In this section, two numerical examples are provided to demonstrate the effectiveness and applicability of the proposed method. Example 4.1 Consider the two dimensional Markovian jump impulsive stochastic CGNNs (10) with mode dependent probabilistic time varying delays as follows:where
Conclusion
In this paper, we have dealt with robust exponential stability problem for MJSCGNNs with mode-dependent probabilistic time-varying delays, continuously distributed delays and impulsive perturbations. By novel Lyapunov–Krasovskii functional and free-weighting matrices, several new stability criteria have been derived to ensure global exponential stability for MJSCGNNs, which are expressed in terms of LMIs. Finally, two numerical examples are provided to show the effectiveness of the proposed
R. Rakkiyappan undergraduated in the field of Mathematics during 1999–2002 from Sri Ramakrishna Mission Vidyalaya College of Arts and Science. He postgraduated in Mathematics from PSG College of Arts and Science affiliated to Bharathiar University, Coimbatore, Tamil Nadu, India, during 2002–2004. He was awarded Doctor of Philosophy in 2011 from the Department of Mathematics, Gandhigram Rural University, Gandhigram, Tamil Nadu, India. His research interests are in the field of qualitative theory
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R. Rakkiyappan undergraduated in the field of Mathematics during 1999–2002 from Sri Ramakrishna Mission Vidyalaya College of Arts and Science. He postgraduated in Mathematics from PSG College of Arts and Science affiliated to Bharathiar University, Coimbatore, Tamil Nadu, India, during 2002–2004. He was awarded Doctor of Philosophy in 2011 from the Department of Mathematics, Gandhigram Rural University, Gandhigram, Tamil Nadu, India. His research interests are in the field of qualitative theory of stochastic and impulsive systems, neural networks and delay differential systems. He has published more than papers in international journals. Now he is working as an Assistant Professor in Department of Mathematics, Bharathiyar University, Coimbatore.
A. Chandrasekar was born in 1989. He received the B.Sc. degree in Mathematics from Thiruvalluvar Government Arts college affiliated to Periyar University, Salem, in 2009 and the M.Sc. degree in Mathematics from Bharathiar University, Coimbatore, Tamil Nadu, India, in 2011. He is pursuing M.Phil. Mathematics in Bharathiar University, Coimbatore, Tamil Nadu, India. His research interests include neural networks, stochastic and impulsive systems.
S. Lakshmanan undergraduated in the field of Mathematics during 2002–2005 from Government Arts College, Salem-7. He postgraduated in Mathematics from Sri Ramakrishna Mission Vidyalaya College of Arts and Science affiliated to Bharathiar University, Coimbatore, Tamil Nadu, India, during 2005–2007. He received the Master of Philosophy and Doctor of Philosophy from Department of Mathematics, Gandhigram Rural University, Gandhigram, Tamil Nadu, India, in 2008 and 2012, respectively. His research interests are in the field of qualitative theory of stochastic systems and neural networks. Now he is working as a Post-Doctoral Research Associate in Department of Electrical Engineering/Information and Communication Engineering, Yeungnam University, Kyongsan, Republic of Korea.
Ju H. Park received the Ph.D. degree in Electronics and Electrical Engineering from POSTECH, Pohang, Republic of Korea, in 1997. From May 1997 to February 2000, he was a Research Associate in ERC-ARC, POSTECH. In March 2000, he joined Yeungnam University, Kyongsan, Republic of Korea, where he is currently the Chuma Chair Professor. From December 2006 to December 2007, he was a Visiting Professor in the Department of Mechanical Engineering, Georgia Institute of Technology. Prof. Park's research interests include robust control and filtering, neural networks, complex networks, and chaotic systems. He has published a number of papers in these areas. Prof. Park severs as a Editor of International Journal of Control, Automation and Systems. He is also an Associate Editor/Editorial Board member for several international journals, including Applied Mathematics and Computation, Journal of The Franklin Institute, Journal of Applied Mathematics and Computing, etc.
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This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2A10005201).