Stability analysis for continuous-time and discrete-time genetic regulatory networks with delays

This paper investigates the global exponential stability problem for the genetic regulatory networks (GRNs) with delays in both the continuous-time case and the discrete-time case. First, Dini derivative method is applied for a new Lyapunov functional to obtain a sufficient condition of the global exponential stability for a class of the continuous-time GRNs, which is given in the form of several elementary inequalities. This sufficient condition is simple to be easily implemented on the computer. Second, by using the semi-discretization technique and the IMEX-θ method, two discrete-time GRN systems have been derived, which can preserve the dynamical characteristics of the continuous-time systems. Furthermore, it is shown that under the same sufficient conditions obtained earlier, these two discrete-time GRN systems are globally exponentially stable. Finally, a pair of examples are given to show the validity of the obtained stability analysis results.