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On converse Lyapunov theorems for fluid network models

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Abstract

We consider the class of closed generic fluid network (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 closed GFN models to satisfy a concatenation and a semicontinuity condition. For the class of strict GFN models we define a state-dependent Lyapunov function 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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Authors and Affiliations

  1. Institute for Mathematics, University of Würzburg, Emil-Fischer Straße 40, 97074, Würzburg, Germany

    Michael Schönlein & Fabian Wirth

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  1. Michael Schönlein
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  2. Fabian Wirth
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Correspondence to Michael Schönlein.

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This research was funded by the Volkswagen Foundation under grant I/83 087.

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Schönlein, M., Wirth, F. On converse Lyapunov theorems for fluid network models. Queueing Syst 70, 339–367 (2012). https://doi.org/10.1007/s11134-012-9279-9

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  • DOI: https://doi.org/10.1007/s11134-012-9279-9

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