Polynomial relaxation-based conditions for global asymptotic stability of equilibrium points of genetic regulatory networks

An important problem in systems biology consists of establishing whether an equilibrium point of a genetic regulatory network (GRN) is stable or not. This article investigates this problem for GRNs with SUM or PROD regulatory functions. It is shown that sufficient conditions for global asymptotical stability of an equilibrium point of these networks can be derived in terms of convex optimisations with linear matrix inequality constraints. These conditions are obtained by looking for a Lyapunov function through the use of suitable polynomial relaxations, and do not introduce approximations of the nonlinearities present in the GRNs. The benefit of these conditions is that their conservatism can be decreased by increasing the degree of the introduced polynomial relaxations. Numerical examples illustrate the usefulness of the proposed conditions.