New Approach to the Stability of Chemical Reaction Networks: Piecewise Linear in Rates Lyapunov Functions

TL;DR

This paper introduces Piecewise-Linear in Rates Lyapunov functions for chemical reaction networks, providing a robust method to ensure stability and convergence of trajectories towards equilibria, applicable beyond mass-action kinetics.

Contribution

The paper defines, proves properties of, and offers algorithms for constructing PWLR Lyapunov functions, extending stability analysis tools for CRNs beyond traditional approaches.

Findings

PWLR Lyapunov functions guarantee asymptotic stability of CRNs.

Construction algorithms for PWLR functions are provided.

PWLR functions are robust to various monotone reaction rates.

Abstract

Piecewise-Linear in Rates (PWLR) Lyapunov functions are introduced for a class of Chemical Reaction Networks (CRNs). In addition to their simple structure, these functions are robust with respect to arbitrary monotone reaction rates, of which mass-action is a special case. The existence of such functions ensures the convergence of trajectories towards equilibria, and guarantee their asymptotic stability with respect to the corresponding stoichiometric compatibility class. We give the definition of these Lyapunov functions, prove their basic properties, and provide algorithms for constructing them. Examples are provided, relationship with consensus dynamics are discussed, and future directions are elaborated.