On the robust stability of time‐varying uncertain genetic regulatory networks

This paper investigates robust stability of time‐varying uncertain genetic regulatory networks (GRNs). In particular, the considered model includes, as special cases, SUM and PROD regulatory functions typically considered in the literature. It is supposed that the coefficients of the GRN are affine linear functions of an uncertain vector constrained in a polytope, and that the activation functions are uncertain into sector‐type regions. As the first problem, we consider to establish whether the GRN is robustly globally stable for all admissible uncertainties. It is shown that this problem can be addressed by solving a linear matrix inequality (LMI) feasibility test built by exploiting homogeneous polynomial Lyapunov functions. As the second problem, we consider to determine the slowest speed with which the concentrations of mRNAs and proteins reach their equilibrium values. It is shown that a guaranteed underestimate of such a speed can be provided by solving a generalized eigenvalue problem built from the proposed stability condition. Some numerical examples illustrate the proposed approaches. It is worth remarking that this paper proposes for the first time in the literature the use of nonquadratic Lyapunov functions for studying robust stability of uncertain GRNs, whereas existing works have addressed the problem only via quadratic Lyapunov functions (either common or parameter‐dependent), which are known to be conservative for time‐varying uncertainty. Copyright © 2011 John Wiley & Sons, Ltd.