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On the stability of projected dynamical systems. (English) Zbl 0837.93063
Summary: A class of projected dynamical systems (PDS), whose stationary points solve the corresponding variational inequality problem (VIP), was recently studied by P. Dupuis and and the second author [Ann. Oper. Res. 44, 9-42 (1993; Zbl 0785.93044)]. 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. In a more direct way, we then establish a series of local and global stability results of the PDS, under various conditions of monotonicity.

MSC:
93D20 Asymptotic stability in control theory
93B11 System structure simplification
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