On converse Lyapunov theorems for fluid network models

Michael Schönlein; Fabian Wirth

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arXiv · 1111.1990

On converse Lyapunov theorems for fluid network models

Michael Schönlein·Fabian Wirth

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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.

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BibTeX RIS

Michael Schönlein, Fabian Wirth. 2011-11-08. On converse Lyapunov theorems for fluid network models. https://arxiv.org/abs/1111.1990

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