Lyapunov Function Partial Different Equations for Stability Analysis of a C of Chemical Reaction Networks

TL;DR

This paper develops a Lyapunov function PDE approach to analyze the stability of complex-balanced-produced chemical reaction networks, extending the applicability of Lyapunov functions beyond traditional methods.

Contribution

It introduces a generalized pseudoHelmholtz free energy function as a Lyapunov function for CBP-CRNs, supporting the Lyapunov PDE approach for broader MAS stability analysis.

Findings

The generalized pseudoHelmholtz free energy function acts as a Lyapunov function for CBP-CRNs.

The Lyapunov PDE approach can potentially be applied to any mass action system.

The method ensures asymptotic stability of the studied chemical reaction networks.

Abstract

We investigate a broad family of chemical reaction networks (CRNs) assigned with mass action kinetics, called complex-balanced-produced-CRNs (CBP-CRNs), which are generated by any given complex balanced mass action system (MAS) and whose structures depend on the selection of producing matrices. Unluckily, the generally applied pseudoHelmholtz free energy function may fail to act as a Lyapunov function for the CBP-CRNs. Inspired by the method of Lyapunov function partial differential equations (PDEs), we construct one solution of their corresponding Lyapunov function PDEs, termed as the generalized pseudoHelmholtz free energy function, and further show that solution can behave as a Lyapunov function to render the asymptotic stability for the CBP-CRNs. This work can be taken as an argument of the conjecture that Lyapunov function PDEs approach can serve for any MAS.