Global Exponential Stability Analysis of Coupled Cyclic Genetic Regulatory Networks With Constant Delays

For a class of coupled cyclic genetic regulatory networks (CGRNs) with constant delays, this article focuses on the problem of global exponential stability analysis. To this end, one technique lemma is first given, which provides computed formulations of several induced norms and the characteristic polynomial of a constant matrix. Then, by employing Brouwer's fixed-point theorem, it is shown that the considered CGRNs have at least one nonnegative equilibrium. Furthermore, a novel approach is presented to prove that the CGRNs under consideration have a unique nonnegative equilibrium, which is globally exponentially stable. These exponential stability criteria can be verified by solving several simple linear matrix inequalities or computing an induced norm of a constant matrix, which can be easily realized by usual software tools. Several numerical examples are given to illustrate the effectiveness of the proposed approach. It is worth emphasizing that after a simple modification, the approach proposed in this article can be applied to many models of nonlinear systems with bounded time-varying delays, including GRNs and neural networks.