Boundary control state/signal systems and boundary triplets

Abstract

We give an introduction to the basic theory of state/signal systems via boundary control. More precisely, we discuss the connection between some basic notions of boundary control state/signal systems on one hand, and classical boundary triplets on the other hand. Boundary triplets and their generalizations have been extensively utilized in the theory of self-adjoint extensions of symmetrical operators in Hilbert spaces, see e.g. [Gorbachuk and Gorbachuk, 1991], [Derkach and Malamud, 1995], [Behrndt and Langer, 2007], and the references therein. The notions related to standard input/state/output boundary control systems are discussed in Section 2, where we also introduce the boundary control state/signal system. In Section 3 we briefly discuss the concept of conservativity in the state/signal framework and in Section 4 we illustrate the abstract concepts using the example of a finitelength conservative LC-transmission line with distributed inductance and capacitance. We conclude this chapter in Section 5, where we recall the definition of a boundary triplet for a symmetric operator and compare this object to a boundary control state/signal system. In particular, we show that every boundary triplet can be transformed into a conservative boundary control state/signal system in impedance form, but that the converse is not true. We make a few final remarks about common generalizations of boundary triplets, which leads over to Chapter ??,