Finite-Time Stability Analysis of Switched Genetic Regulatory Networks

This paper investigates the finite-time stability problem of switching genetic regulatory networks (GRNs) with interval time-varying delays and unbounded continuous distributed delays. Based on the piecewise Lyapunov-Krasovskii functional and the average dwell time method, some new finite-time stability criteria are obtained in the form of linear matrix inequalities (LMIs), which are easy to be confirmed by the Matlab toolbox. The finite-time stability is taken into account in switching genetic regulatory networks for the first time and the average dwell time of the switching signal is obtained. Two numerical examples are presented to illustrate the effectiveness of our results. 1. Introduction In the last decade or so, the genetic regulatory networks (GRNs) have become an important research area in molecular Biology. On the one hand, how to construct GRNs from gene expression data; on the other hand, what is the dynamic characteristics of gene regulatory networks. The stability is one of vital dynamic characteristics of GRNs and is researched in this paper. There are also many references on the stability of GRNs [1–6]. High throughput biological experiments have proved that time delays were ubiquitous in GRNs. The existence of time delays influences the stability of GRNs, which can give rise to oscillatory or unstable networks. Therefore, it is necessary to study the influences of time delays for the stability of GRNs. There are a few theoretical results on GRNs with time delays [7–15]. In [2], random time delays are taken into account, and some stability criteria for the uncertain delayed genetic networks with SUM regulatory logic where each transcription factor acts additively to regulate a gene were obtained. In [12], some stochastic asymptotic stability conditions were established for a class of uncertain stochastic genetic regulatory networks with mixed time-varying delays by constructing appropriate Lyapunov-Krasovskii functionals and employing stochastic analysis method. In [13, 14], authors studied GRNs with constant delay, and asymptotical stability criteria were proposed for GRNs with interval time-varying delays and nonlinear disturbance in [10]. Cell-cycle regulatory processes can be viewed as finite-state processes. So some complex GRNs were described by continuous time switched system. Usually, a finite state Markov chain was used to simulate this switch process [16–19]. In [17], authors investigate the global robust stability of uncertain stochastic GRNs with Markovian switching process. In [19], control theory and mathematical tools