Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- 1 John Williams Calkin: a short biography
- 2 On Calkin's abstract symmetric boundary conditions
- 3 Boundary triplets and maximal accretive extensions of sectorial operators
- 4 Boundary control state/signal systems and boundary triplets
- 5 Passive state/signal systems and conservative boundary relations
- 6 Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples
- 7 Boundary triplets and Weyl functions. Recent developments
- 8 Extension theory for elliptic partial differential operators with pseudodifferential methods
- 9 Dirac structures and boundary relations
- 10 Naĭmark dilations and Naĭmark extensions in favour of moment problems
4 - Boundary control state/signal systems and boundary triplets
Published online by Cambridge University Press: 05 November 2012
- Frontmatter
- Contents
- List of contributors
- Preface
- 1 John Williams Calkin: a short biography
- 2 On Calkin's abstract symmetric boundary conditions
- 3 Boundary triplets and maximal accretive extensions of sectorial operators
- 4 Boundary control state/signal systems and boundary triplets
- 5 Passive state/signal systems and conservative boundary relations
- 6 Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples
- 7 Boundary triplets and Weyl functions. Recent developments
- 8 Extension theory for elliptic partial differential operators with pseudodifferential methods
- 9 Dirac structures and boundary relations
- 10 Naĭmark dilations and Naĭmark extensions in favour of moment problems
Summary
Abstract This chapter is an introduction to the basic theory of state/signal systems via boundary control theory. The ℒC-transmission line illustrates the new concepts. It is shown that every boundary triplet can be interpreted as an impedance representation of a conservative boundary control state/signal system.
Introduction
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 symmetric 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 4.2, where we also introduce the boundary control state/signal system. In Section 4.3 we briefly discuss the concept of conservativity in the state/signal framework and in Section 4.4 we illustrate the abstract concepts we have introduced using the example of a finite-length conservative ℒC-transmission line with distributed inductance and capacitance.
We conclude this chapter in Section 4.5, 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 5, where we treat more general passive state/signal systems, not only conservative systems or systems of boundary-control type.
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- Operator Methods for Boundary Value Problems , pp. 73 - 86Publisher: Cambridge University PressPrint publication year: 2012
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