Stability of genetic networks with SUM regulatory logic: Lur'e system and LMI approach

In this paper, we present a nonlinear model for genetic regulatory networks with SUM regulatory functions. We show that the genetic network can be transformed into Lur'e form. Based on the Lyapunov method and the Lur'e system approach, sufficient conditions for the stability of the genetic networks are derived, in particular for the cases with time delays owing to the slow processes of transcription, translation, and translocation, and for the cases with stochastic perturbations due to natural random intra- and inter-cellular fluctuations. All the stability conditions are given in terms of linear matrix inequalities (LMIs), which are very easy to be verified. To test the effectiveness of our theoretical results, several examples of genetic networks are also presented in this paper.