Delay-Probability-Distribution-Dependent Stability of Uncertain Stochastic Genetic Regulatory Networks with Time-Varying
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Delay-Probability-Distribution-Dependent Stability of Uncertain Stochastic Genetic Regulatory Networks with Time-Varying Delays R. Rakkiyappan · S. Lakshmanan · P. Balasubramaniam
Received: 11 February 2011 / Revised: 8 March 2012 © Springer Science+Business Media New York 2013
Abstract This paper investigates the delay-probability-distribution-dependent stability problem of uncertain stochastic genetic regulatory networks (SGRNs) with time-varying delays. The information of the probability distribution of the time-delay is considered and transformed into parameter matrices of the transferred SGRNs model. Based on the Lyapunov–Krasovskii functional and a stochastic analysis approach, a delay-probability-distribution-dependent sufficient condition is obtained in the linear matrix inequality (LMI) form such that delayed SGRNs are robustly globally asymptotically stable in the mean square for all admissible uncertainties. Three numerical examples are given to illustrate the effectiveness of our theoretical results. Keywords Asymptotic stability · Delay-probability-distribution dependent · Stochastic genetic regulatory networks · Linear matrix inequality · Lyapunov–Krasovskii functional
P. Balasubramaniam also working as a Visiting Professor, Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia for six months since 12th September 2011. S. Lakshmanan · P. Balasubramaniam () Department of Mathematics, Gandhigram Rural Institute-Deemed University, Gandhigram 624 302, Tamilnadu, India e-mail: [email protected] S. Lakshmanan e-mail: [email protected] R. Rakkiyappan Department of Mathematics, Bharathiar University, Coimbatore 641 046, Tamilnadu, India e-mail: [email protected] P. Balasubramaniam Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia
Circuits Syst Signal Process
1 Introduction The past few years have witnessed significant progress in the research area of gene engineering and other biological sciences. The mechanisms that have evolved to regulate the gene expression are known as genetic regulatory networks (GRNs). With the study of GRNs, scientists would be able to explain the interactions between genes and protein that form complex biological systems, see for example [28, 33]. A great number of genes and proteins either directly or indirectly interact with one another in living cells. Such interactions make up a dynamic GRN, which acts as a complex dynamic system for controlling cellular functions, see for example [20]. The modeling of GRNs is largely dependent on powerful tools of mathematics theory. In general, the GRNs can be described by two types of models, that is, the discrete model (such as Boolean networks) and the continuous model (such as the differential equation model), for details see [4, 8, 12, 15, 26, 29]. Boolean network models represent the state of a gene as a Boolean variable (on/off) and interactions between genes as Boolean functions, which determine the state of a gene on t
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