Study of Piecewise Multi-affine models for Genetic Regulatory Networks via a Lyapunov approach: an LMI framework

Abstract

In this work we study convergence properties of Piecewise Multi-Affine models of genetic regulatory networks, by means of a Lyapunov approach. These models, quantitatively more accurate than their Piecewise Affine counterpart, are obtained by a Piecewise Linear approximation of sigmoids regulation functions. In this work, using a linear matrix inequalities framework, we are able to find, if one exists subject to a box partitioning of the state space, a Piecewise Quadratic Lyapunov function, which is non-increasing along any system trajectory. In the first part of the paper we describe the considered model, defining and motivating the hyper-rectangular partition of the state space, while in the second part, using a result on the expression of multi-affine functions on an hyper-rectangle, we can define a set of linear matrix inequalities, whose solution gives the description of a piecewise quadratic Lyapunov function for the system. Convergence properties based on such functions are discussed and a numerical example will show the applicability of the results.