On convergence for hybrid models of gene regulatory networks under polytopic uncertainties: a Lyapunov approach

• Published in: Journal of mathematical biology Vol. 83; no. 6-7; p. 64
• Main Authors: Pasquini, Mirko, Angeli, David
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2021; Springer Nature B.V
• Subjects: Applications of Mathematics; Approximation; Biology; Chemical reactions; Control theory; Convergence; Convergence analysis; Engineering; Gene Regulatory Networks; Genes; Liapunov functions; Linear matrix inequalities; Lyapunov methods; Mathematical and Computational Biology; Mathematical models; Mathematics; Mathematics and Statistics; Parameters; Polytopic uncertainties; Systems biology; Systems stability; Transcription factors; Uncertainty; Gene regulatory networks; 92C42; Systems biology; Linear matrix inequalities; 93D30; Lyapunov methods; Convergence analysis; Polytopic uncertainties
• ISSN: 0303-6812; 1432-1416; 1432-1416
• DOI: 10.1007/s00285-021-01690-3

Hybrid models of genetic regulatory networks allow for a simpler analysis with respect to fully detailed quantitative models, still maintaining the main dynamical features of interest. In this paper we consider a piecewise affine model of a genetic regulatory network, in which the parameters describing the production function are affected by polytopic uncertainties. In the first part of the paper, after recalling how the problem of finding a Lyapunov function is solved in the nominal case, we present the considered polytopic uncertain system and then, after describing how to deal with sliding mode solutions, we prove a result of existence of a parameter dependent Lyapunov function subject to the solution of a feasibility linear matrix inequalities problem. In the second part of the paper, based on the previously described Lyapunov function, we are able to determine a set of domains where the system is guaranteed to converge, with the exception of a zero measure set of times, independently from the uncertainty realization. Finally a three nodes network example shows the validity of the results.