Robust stochastic stability of genetic regulatory networks with time delays and parametric uncertainties

Abstract

This paper investigates robust stochastic stability of uncertain genetic regulatory networks. It is assumed that the networks have SUM regulatory functions affected by Wiener processes, time delays and parametric uncertainties. Contrary to existing works, the uncertainty is not restricted to belong to a polytope, but more generally it is assumed to belong to a multi-dimensional set described by polynomial inequalities. First, it is shown that a condition for robust stochastic stability with guaranteed disturbance attenuation for all admissible uncertainties in the absence of time delays can be obtained by solving a convex optimization problem built by exploiting the square matrix representation (SMR) of matrix polynomials and by introducing polynomially parameter-dependent Lyapunov functions. Then, it is shown how this condition can be extended to deal with the presence of time delays with bounded variation rates by introducing polynomially parameter-dependent Lyapunov-Krasovskii functionals (LKFs). Examples with fictitious and real biological models illustrate the use of the proposed methodology. © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society.