State space theory for resolution space operators

Abstract

The notion of state has played a fundamental role in modern systems theory both from a theoretical point of view and for the solution of problems in filtering, estimation, and optimal control. The first study of state for abstract inputoutput operators on Resolution Space was made by Saeks [7]. There most of the basic ideas for the construction of a state representation for Hilbert space operators were introduced. However, despite the fact that a Hilbert space structure was used, most of the notions were algebraic and had no topological meaning. Also, the relationship between the state space and the space of state trajectories was not clear. These problems were noticed by two authors, Schnure [9] and Steinberger [IO]. Both attempted to rigorize the theory of Saeks by introducing topological content into Saeks’ ideas. The work presented here, while owing a debt to both these authors, leans much more heavily on that of Schnurc. One basic difference between the notions of state of Schnure on the one hand and of Saeks and Steinberger on the other hand is that for time-varying systems, for Schnure the state space varies with time while for Saeks and Steinberger it is fixed. We adopt the approach of Schnure, This allows us to construct the space of state trajectories using the notion of a direct integral of Hilbert spaces due to von Neumann. As is expected, the space of state trajectories for an infinite time system will turn out to be an extended space [13, I l] of some Hilbert space. The Hilbert space will be precisely the direct integral integral of the state spaces. A fairly complete state space theory is presented here including a state space isomorphism theorem and a realization theorem and the existence of a family of transition operators. In the last part of the paper we consider the time-invar-iant case and show the connection between the state space theory considered here and the theory of C, semigroups.