On the stability of projected dynamical systems
On the stability of projected dynamical systems
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DOI：
10.1007/bf02192301
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发表时间：
1995-04
影响因子：
1.9
通讯作者：
Ding Zhang;A. Nagurney
中科院分区：
文献类型：
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作者：
Ding Zhang;A. Nagurney
Dupuis和Nagurney （Ref. 1）最近研究了一类投影动力系统（PDS），其平稳点求解相应的变分不等式问题（VIP）。本文首先研究了这类PDS在平稳点周围的稳定性，从而引起了VIP解的动态稳定性研究。实例表明，这种研究与经典的动力系统稳定性研究有很大的不同。给出了VIP的正则解的定义，并引入了由PDS引起的最小面流的概念，PDS是一种低维的标准DS。然后，我们证明，在VIP的正则解下，PDS的局部稳定性与其最小面流的局部稳定性基本相同。因此，在这种情况下，我们将问题简化为DS的经典稳定性研究之一，这是一个更发达的学科。然后，我们更直接地建立了PDS在各种单调性条件下的一系列局部和全局稳定性结果。
A class of projected dynamical systems (PDS), whose stationary points solve the corresponding variational inequality problem (VIP), was recently studied by Dupuis and Nagurney (Ref. 1). This paper initiates the study of the stability of such PDS around their stationary points and thus gives rise to the study of the dynamical stability of VIP solutions. Examples are constructed showing that such a study can be quite distinct from the classical stability study for dynamical systems (DS). We give the definition of a regular solution to a VIP and introduce the concept of a minimal face flow induced by a PDS, which is a standard DS of a lower dimension. We then show that, at the regular solutions of the VIP, the local stability of the PDS is essentially the same as that of its minimal face flow. Hence, we reduce the problem, in this case, to one of the classical stability study of DS, a more developed discipline. In a more direct way, we then establish a series of local and global stability results of the PDS, under various conditions of monotonicity.