Convergence Assessment of the Trajectories of a Bioreaction System by Using Asymmetric Truncated Vertex Functions

• Published in: Symmetry (Basel) 2020-04, Vol.12 (4), p.513
• Main Authors: Rincón, Alejandro, Florez, Gloria Yaneth, Olivar, Gerard
• Format: Article
• Language: English
• Subjects: asymptotic convergence; global stability; invariant set; Lyapunov-like function; vertex truncation
• ISSN: 2073-8994; 2073-8994
• DOI: 10.3390/sym12040513

In several open and closed-loop systems, the trajectories converge to a region instead of an equilibrium point. Identifying the convergence region and proving the asymptotic convergence upon arbitrarily large initial values of the state variables are regarded as important issues. In this work, the convergence of the trajectories of a biological process is determined and proved via truncated functions and Barbalat’s Lemma, while a simple and systematic procedure is provided. The state variables of the process asymptotically converge to a compact set instead of an equilibrium point, with asymmetrical bounds of the compact sets. This convergence is rigorously proved by using asymmetric forms with vertex truncation for each state variable and the Barbalat’s lemma. This includes the definition of the truncated V i functions and the arrangement of its time derivative in terms of truncated functions. The proposed truncated function is different from the common one as it accounts for the model nonlinearities and the asymmetry of the vanishment region. The convergence analysis is valid for arbitrarily large initial values of the state variables, and arbitrarily large size of the convergence regions. The positive invariant nature of the convergence regions is proved. Simulations confirm the findings.