On converse Lyapunov theorems for fluid network models

TL;DR

This paper develops state-dependent Lyapunov functions for fluid network models, establishing a converse Lyapunov theorem for a subclass called strict GFN models, and demonstrates its applicability to common fluid network models.

Contribution

It introduces strict GFN models with conditions enabling a converse Lyapunov theorem and constructs corresponding state-dependent Lyapunov functions.

Findings

Counterexamples show closed GFN models lack sufficient info for converse theorems.

Strict GFN models satisfy concatenation and semicontinuity conditions.

Common fluid network models are examples of strict GFN models.

Abstract

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.