Abstract This chapter is a continuation and deepening of Chapter 4. In the present chapter the state/signal theory is extended beyond boundary control and beyond conservative systems. The main aim is to clarify the basic connections between the state/signal theory and that of (conservative) boundary relations. It is described how one can represent a state/signal system using input/state/output systems in different ways by making different choices of input signal and output signal. There is an “almost one-to-one” relationship between conservative state/signal systems and boundary relations, and this connection is used in order to introduce dynamics to a boundary relation. Consequently, a boundary relation is such a general object that it mathematically has rather little to do with boundary control. TheWeyl family and γ-field of a boundary relation are connected to the frequency-domain characteristics of a state/signal system.

Introduction

The theory of boundary relations has been developed by a number of authors in the framework of the theory of self-adjoint extensions of symmetric operators and relations in Hilbert spaces; see e.g. the recent articles [Derkach et al., 2006; Derkach, 2009; Derkach et al., 2009; Behrndt et al., 2009]. Boundary relations are described in detail in Chapter 7.

One way of introducing the notion of a state/signal (s/s) system is to start from an input/state/output (i/s/o) system. By a standard i/s/o system we mean a system of equations of the type

where ẋ stands for the time derivative of x.