The concept of Genetic Regulatory Network (GRN) is ubiquitous in systems biology, as it describes the transcriptional interactions between genes and proteins, illustrating an intricate set of relations that ultimately guide the genetic expression in a cell. In this dissertation we mainly consider a Piecewise Affine model for the dynamics of Genetic Regulatory Networks, where steep Hill regulatory functions are substituted by step approximations. After an introduction on the model, we define an LMI framework, whose solution describes a generally discontinuous Piecewise Quadratic Lyapunov function for the system. After a discussion on systems
solutions when considering isolated Zeno behaviours, the Lyapunov function is proved to be eventually
non-increasing along any system trajectory, including any possible sliding motion, and is showed that a Lasalle-like result holds, a result that will allow to characterise convergence properties of the system.
After the analysis in the nominal case, acknowledging the fact that biological systems are inherently uncertain, because of environmental changes or imperfect parameters knowledge, polytopic uncertainties are introduced in the model, and the framework is extended to find a Lyapunov function which depends affinely on the uncertain parameters, and that allows to conclude convergence results robust to such uncertainties.
In the last Chapter, a more accurate Piecewise Multi-Affine model of the dynamics of Genetic Regulatory Networks is considered, resulting from a Piecewise Linear approximation of Hill functions. It is shown how, using a result on convexity of Multi-Affine functions on hyper-rectangles, a similar LMI framework can be defined, to obtain a Piecewise Quadratic Lyapunov function which is non-increasing along any system trajectory.
Throughout the work, many examples show how the LMI Feasibility Problems can be setup, how the theoretical results hold and how these can be used to estimate the convergence sets for the analysed systems.