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

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.