  We consider the class of closed generic fluid networks (GFN) models, which
provides an abstract framework containing a wide variety of fluid networks.
Within this framework a Lyapunov method for stability of GFN models was
proposed by Ye and Chen. They proved that stability of a GFN model is
equivalent to the existence of a functional on the set of paths that is
decaying along paths. This result falls short of a converse Lyapunov theorem in
that no state dependent Lyapunov function is constructed. In this paper we
construct state-dependent Lyapunov functions in contrast to path-wise
functionals. We first show by counterexamples that closed GFN models do not
provide sufficient information that allow for a converse Lyapunov theorem. To
resolve this problem we introduce the class of strict GFN models by forcing the
closed GFN model to satisfy a concatenation and a semicontinuity condition of
the set of paths in dependence of initial condition. For the class of strict
GFN models we define a state-dependent Lyapunov and show that a converse
Lyapunov theorem holds. Finally, it is shown that common fluid network models,
like general work-conserving and priority fluid network models as well as
certain linear Skorokhod problems define strict GFN models.
