# On the robust existence of piecewise quadratic Lyapunov functions for hybrid models of gene regulatory networks

## TL;DR

This paper develops conditions for the existence of piecewise quadratic Lyapunov functions to analyze the stability of uncertain hybrid gene regulatory network models, accounting for parameter uncertainties and sliding modes.

## Contribution

It provides a novel theoretical framework for stability analysis of uncertain hybrid gene regulatory networks using piecewise quadratic Lyapunov functions.

## Key findings

- Conditions for Lyapunov function existence are derived for all parameter uncertainties.
- The approach accounts for sliding modes in hybrid systems.
- An example demonstrates the applicability of the theoretical results.

## Abstract

In this work we addressed the problem of stability analysis for an uncertain piecewise affine model of a genetic regulatory network. In particular we considered polytopic parameter uncertainties on the proteins production rate functions, giving conditions for the existence of a piecewise quadratic Lyapunov function for any possible realization of the system. In the spirit of other works in literature, the resulting conditions will be given on the vertices of the parameter polytope, while still taking into consideration the piecewise nature of the Lyapunov function and the presence, in general, of sliding modes solutions. An example is shown to prove the validity and applicability of the theoretical results.