Stability of fluid network models and Lyapunov functions

We consider the class of closed generic fluid networks (GFN) models. This class contains for example fluid networks under general work-conserving and priority disciplines. Within this abstract 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 property that for every path of the model a Lyapunov like function is decaying. In this paper we construct state-dependent Lyapunov functions in contrast to pathwise functionals. We first show by counterexamples that closed GFN models do not provide sufficient information that allow for a converse Lyapunov theorem with state-dependent Lyapunov functions. To resolve this problem we introduce the class of strict closed GFN models by forcing the closed GFN model to satisfy a concatenation and a perfectness condition and define a statedependent Lyapunov function. We show that for the class of strict closed GFN models 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 closed GFN models.