Stability criteria for Markovian jump neural networks with mode-dependent additive time-varying delays via quadratic convex combination
Introduction
During the past decades, Neural Networks (NNs) have been extensively studied and have also found many applications in various fields, such as image processing, pattern recognition, signal processing, combinatorial optimization, power systems, associative memory, and so on (for example [1], [2], [3], [4], [5]). All of these applications tediously depend on the dynamical characteristics. So the stability is an important property to many systems [6], [7], [8], [9], [10], [11]; much effort has been done to study the stability problem of NNs with time delays because the existence of time delays may cause the system like instability and oscillation of NNs. So there exist several results on stability of NNs with either constant or time-varying delays [6], [7], [8], [10], [11], [13], [14], [15], [16], [17], [21], [22], [23], [24], [25], [26].
Meanwhile, a new type of time-varying delay with two additive components in the state of NNs are introduced in [12]. Such a system may be encountered in many practical situations such as remote control and networked control system. For example, in networked controlled systems, signals transmitted from one point to another may experience a few segments of networks, which can possibly induce successive delays, one from the sensor to the controller and the other from the controller to the actuator, having different properties due to the variable network transmission conditions. This implies that the system with additive time-varying delays become more complicated and very interesting. Therefore, a great number of researchers investigated the system with additive time-varying delays (for example [13], [14], [15], [16], [17]). In [13], the authors investigated the synchronization of singular Markovian jumping complex dynamical networks with two additive time-varying delay components using the pinning control. In [14], the authors studied the problem of exponential synchronization of complex dynamical networks with two additive time-varying delay components and control packet loss using the stochastic sampled-data control. The authors in [17] analyzed the stability criteria for continuous time systems with additive time-varying delays.
In the real world, the NNs may exhibit the network mode jumping characteristic. Such jumping can be determined by the Markov chain. Recently, NNs with Markovian jump parameters have received much interest among researchers. This class of NNs is recognized as the best system to model the phenomenon of information latching and the abrupt phenomena, such as random failures or repairs of the components, sudden environmental changes, changing subsystem interconnections, and so forth. To deal with this situation, the authors in [18], [19], [20] considered the model of NNs with Markovian jumping parameters, which are also called Markovian jump neural networks (MJNNs). Also these papers give the extensive applications of such models in manufacturing systems, power systems, actuator saturation, and communication systems and network-based control systems. Thus MJNNs is a hybrid system with two components x(t) and r(t). Here x(t) is referred as the state, which is described by a differential equation and the r(t) is referred as the mode. In its operation, this class of systems will switch from one mode to another mode in a random way and it is also governed by a continuous time Markov chain with a finite state space . Therefore, it is important to study the dynamic behaviors of neural networks with Markovian jumping parameters and mode-dependent time-varying delays (for example [21], [22], [23], [24], [25], [26]).
In [21], the authors discussed about the robust stochastic convergence for an uncertain Markovian jumping Cohen–Grossberg NNs with mode-dependent time-varying delays. The delay dependent stochastic stability criteria are studied in [23] for MJNNs with mode-dependent time-varying delays and partially known transition rates. In [25], the problem of asymptotic stability of MJNNs with randomly occurring nonlinearities is investigated in the mean square sense. In [26], the robust exponential stability of Markovian jumping stochastic Cohen–Grossberg NNs with mode-dependent probabilistic time-varying delays and continuously distributed delays are studied by using the impulsive perturbations.
Motivated by the above discussion, in this paper, we investigate the global asymptotic stability for MJNNs with mode-dependent two additive time-varying delays. At first, we construct a new augmented LKF terms like as , , . Here, the quadratic terms are multiplied by the scalar function and respectively. By using Jensen׳s inequality and some new integral inequalities to solve these LKFs. Also, we will claim that the function is a quadratic convex combination on . Then the sufficient conditions are employed in terms of LMIs which ensuring the globally asymptotically stable of the proposed NNs. Finally, the effectiveness of theoretical results is validated by the numerical examples. However, to the best of our knowledge, until now there are no results on the stability problem of MJNNs with mode-dependent two additive time-varying delays based on the quadratic convex combination approach.
The outline of this paper is organized as follows: the NNs model is introduced and some necessary lemmas are given in Section 2. Section 3 includes the stability problem of MJNNs with mode-dependent two additive time-varying delays based on quadratic convex combination approach. Section 4 provides numerical examples to illustrate the effectiveness of the theoretical results. Finally, the conclusion is given in Section 5. To ease the analysis, let us provide the following notations.
Notations: Throughout this paper, denotes the n-dimensional Euclidean space. Sym(M) is defined as . The superscript T denotes the transposition. The notation (similarly, ) denotes that M is a positive semi-definite matrix (similarly, negative semi-definite matrix). The identity and zero matrices of appropriate dimensions are denoted by I and 0, respectively. . The notation ⁎ in a block matrix always represents the symmetric terms.
Section snippets
Problem formulation
Let is a right-continuous Markov chain on a complete probability space taking values in a finite space with operator given by where and is the transition rate from i to j, if while .
In this paper, we consider the following MJNNs with mode-dependent additive time-varying delays:In
Main results
In this section, we investigate the global asymptotic stability of MJNNs with mode-dependent two additive time-varying delays described by the system (2). By introducing the main theorem, for presentation convenience, in the following, we denote
Numerical example
In this section, we provide a numerical examples to demonstrate the effectiveness of our delay dependent stability criteria. Example 4.1 Consider the two dimensional MJNNs (2) with mode-dependent additive time-varying delays with the following parameters:In this example, the nonlinear activation functions are assumed to be with and . Thus,
Conclusion
In this paper, the stability problem of MJNNs with mode-dependent two additive time-varying delays has been investigated. A new augmented LKF has been constructed to derive the sufficient conditions in the form of LMIs, based on the Jensen׳s inequality and quadratic convex combination approach. It is also guaranteed that the MJNNs are globally asymptotically stable. Finally, the numerical examples are predicted to illustrate the effectiveness of our proposed theoretical results. Here, we would
Acknowledgments
The authors wish to thank the editor and reviewers for a number of constructive comments and suggestions that have improved the quality of this manuscript. This work was supported by Science and Engineering Research Board (SERB), New Delhi, India under the File no. YSS/2014/000447 Dated 20-November-2015.
P. Muthukumar received his M.Sc. degree from the Department of Mathematics of Gobi Arts and Science College affiliated to Bharathiar University, Coimbatore, Tamilnadu, India, in 2002. He completed his M.Phil. degree in Mathematics in 2006 from Bharathiar Univeristy, Coimbatore, Tamilnadu, India. He obtained his Ph.D. degree in 2009 from the Department of Mathematics, Gandhigram Rural Institute-Deemed University, Gandhigram, Tamilnadu, India. Since 2010, he is working as an Assistant Professor
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P. Muthukumar received his M.Sc. degree from the Department of Mathematics of Gobi Arts and Science College affiliated to Bharathiar University, Coimbatore, Tamilnadu, India, in 2002. He completed his M.Phil. degree in Mathematics in 2006 from Bharathiar Univeristy, Coimbatore, Tamilnadu, India. He obtained his Ph.D. degree in 2009 from the Department of Mathematics, Gandhigram Rural Institute-Deemed University, Gandhigram, Tamilnadu, India. Since 2010, he is working as an Assistant Professor at the Department of Mathematics, Gandhigram Rural Institute-Deemed University, Gandhigram, Tamilnadu, India. He was awarded IUSSTF Research Fellow 2012 award from Department of Science and Technology (DST), India and worked as a visiting faculty with Frank L. Lewis, UTA Research Institute, University of Texas at Arlington (UTA), Texas, U.S.A. for a year. He received UGC-SAP Project Fellow 2005 and CSIR-SRF 2009 awards from the Indian government. His research interests include control theory, stochastic differential systems, nonlinear control and its applications.
K. Subramanian undergraduated and postgraduated in the field of Mathematics during 2008–2011 and 2011–2013 respectively, from Sri Ramakrishna Mission Vidyalaya College of Arts and Science affiliated to Bharathiar University, Coimbatore, Tamilnadu, India. He received the Master of Philosophy from Department of Mathematics, Bharathiar University during 2013–2014. He is pursuing Ph.D. Mathematics in Gandhigram Rural Institute - Deemed University, Gandhigram, Tamil Nadu, India. His research interests include neural networks, optimal control, stochastic and impulsive systems.