M-matrix-based stability conditions for genetic regulatory networks with time-varying delays and noise perturbations.

Stability is essential for designing and controlling any dynamic systems. Recently, the stability of genetic regulatory networks has been widely studied by employing linear matrix inequality (LMI) approach, which results in checking the existence of feasible solutions to high-dimensional LMIs. In the previous study, the authors present several stability conditions for genetic regulatory networks with time-varying delays, based on M-matrix theory and using the non-smooth Lyapunov function, which results in determining whether a low-dimensional matrix is a non-singular M-matrix. However, the previous approach cannot be applied to analyse the stability of genetic regulatory networks with noise perturbations. Here, the authors design a smooth Lyapunov function quadratic in state variables and employ M-matrix theory to derive new stability conditions for genetic regulatory networks with time-varying delays. Theoretically, these conditions are less conservative than existing ones in some genetic regulatory networks. Then the results are extended to genetic regulatory networks with time-varying delays and noise perturbations. For genetic regulatory networks with n genes and n proteins, the derived conditions are to check if an n × n matrix is a non-singular M-matrix. To further present the new theories proposed in this study, three example regulatory networks are analysed.