Robust Stability Of Gene Regulatory Networks Analysis

Along with the development of Bioinformatics, it has been hot research problem to understand underlying mechanism in biological system by modeling genetic regulatory networks (GRNs) and studying interaction between genetics and proteins. GRNs are biochemical dynamical systems, and their basic feature is stability, which is of great research significance.A continuous-time GRNs model is proposed by considering nonlinear, time delay and stochastic perturbation, which has a form of Lurie system. Then a semi-discretization technique is introduced to obtain a discrete-time GRNs model formulated by difference equations.For the continuous-time GRNs with both interval time-varying delays and stochastic perturbation, by using novel augmented Lyapunov-Krasovskii functional, considering the relationship among the time-varying delays, its upper bound and their difference, and introducing an improved-free-weighting matrix approach and the basic inequality, a less conservative delay-dependent/delay-rate-dependent stability condition is proposed without ignoring any useful terms in the weak infinitesimal operator of the Lyapunov-Krasovskii functional. On this basis, other criteria are obtained, which have less decision variables. The delay-dependent/delay-rate-independent and delay-independent stability criteria are presented respectively. The criteria are then extended to analyze the robust stability for GRNs with time-varying structured uncertainties. Two numerical examples are given to illustrate the effectiveness and the reduced conservativeness of the obtained results.For the discrete-time GRNs with both interval time-varying delays and stochastic perturbation, a global asymptotical stability condition is presented, which is delay-dependent/delay-range-dependent, by using augmented Lyapunov-Krasovskii functional combined with improved-free-weighting matrix approach. Meanwhile, a delay-independent stability condition is proposed. Based on these conclusions, the robust stability is analyzed for discrete-time GRNs with time-varying structured uncertainties. Two numerical examples are given to illustrate the effectiveness of the obtained results.Finally, a summary of study on stability of GRNs is presented, and prospects for futher studies are mentioned.