流体网络模型的逆李雅普诺夫定理
On converse Lyapunov theorems for fluid network models
摘要：
我们考虑一类封闭的通用流体网络（GFN）模型，它提供了一个抽象的框架，包含各种各样的流体网络。在这个框架内，叶和陈提出了一个稳定的GFN模型的李亚普诺夫方法。他们证明了GFN模型的稳定性等价于路径集上存在一个沿着路径衰减的泛函。这一结果不符合李雅普诺夫逆定理，因为没有构造状态相关的李雅普诺夫函数。在本文中，我们构造状态依赖的李雅普诺夫函数相反，路径泛函。我们首先通过反例表明，封闭的GFN模型不提供足够的信息，允许逆李雅普诺夫定理。为了解决这个问题，我们引入严格GFN模型类，迫使封闭的GFN模型，以满足一个连接和连续条件的依赖于初始条件的路径集。对于一类严格的GFN模型，我们定义了一个状态依赖的李雅普诺夫，并证明了逆李雅普诺夫定理成立。最后，它表明，常见的流体网络模型，如一般的工作守恒和优先流体网络模型，以及某些线性Skorokhod问题定义严格的GFN模型。
英文摘要：
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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