On certain infinite-dimensional lie algebras and related system-theoretic problems

Abstract

In some recent work on nonlinear filtering, current algebras play a significant role. In the study of certain control problems arising in spacecraft attitude control, it has also become apparent that Lie-Poisson algebras arise naturally. It is our purpose in this paper to draw attention to some of the mathematical and system-theoretic aspects of these two types of infinite-dimensional Lie algebras. To this end, this paper contain brief background material on the structure of these Lie algebras. Part I of this paper deals mainly with current algebras and Part II of this paper deals with Lie-Poisson algebras. In both parts of the paper we indicate the specific applications that we are most familiar with. We then proceed to suggest other types of application areas where the structure of these infinite-dimensional Lie algebras may be exploited. Specifically we have in mind the control of infinite dimensional bilinear systems and complex multi-body mechanical systems. Details of the existing applications may be found in the references.