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4 - Boundary control state/signal systems and boundary triplets
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- By D.Z. Arov, Institute of Physics and Mathematics, M. Kurula, University of Twente, O.J. Staffans, Åbo Akademi University
- Edited by Seppo Hassi, University of Vaasa, Finland, Hendrik S. V. de Snoo, Rijksuniversiteit Groningen, The Netherlands, Franciszek Hugon Szafraniec, Jagiellonian University, Krakow
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- Book:
- Operator Methods for Boundary Value Problems
- Published online:
- 05 November 2012
- Print publication:
- 11 October 2012, pp 73-86
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- Chapter
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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.
5 - Passive state/signal systems and conservative boundary relations
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- By D.Z. Arov, South-Ukrainian National Pedagogical University, M. Kurula, University of Twente, O.J. Staffans, Åbo Akademi University,
- Edited by Seppo Hassi, University of Vaasa, Finland, Hendrik S. V. de Snoo, Rijksuniversiteit Groningen, The Netherlands, Franciszek Hugon Szafraniec, Jagiellonian University, Krakow
-
- Book:
- Operator Methods for Boundary Value Problems
- Published online:
- 05 November 2012
- Print publication:
- 11 October 2012, pp 87-120
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- Chapter
- Export citation
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Summary
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.