  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.
