Book contents
- Frontmatter
- Contents
- List of figures
- Preface
- List of notation
- 1 Introduction and overview
- 2 Basic properties of well-posed linear systems
- 3 Strongly continuous semigroups
- 4 The generators of a well-posed linear system
- 5 Compatible and regular systems
- 6 Anti-causal, dual, and inverted systems
- 7 Feedback
- 8 Stabilization and detection
- 9 Realizations
- 10 Admissibility
- 11 Passive and conservative scattering systems
- 12 Discrete time systems
- Appendix
- Bibliography
- Index
Appendix
Published online by Cambridge University Press: 13 October 2009
- Frontmatter
- Contents
- List of figures
- Preface
- List of notation
- 1 Introduction and overview
- 2 Basic properties of well-posed linear systems
- 3 Strongly continuous semigroups
- 4 The generators of a well-posed linear system
- 5 Compatible and regular systems
- 6 Anti-causal, dual, and inverted systems
- 7 Feedback
- 8 Stabilization and detection
- 9 Realizations
- 10 Admissibility
- 11 Passive and conservative scattering systems
- 12 Discrete time systems
- Appendix
- Bibliography
- Index
Summary
In this appendix we present a number of auxiliary results about regulated functions, the positive square root and polar decomposition of a closed unbounded operator convolutions, and inversion of block matrices.
Regulated functions
Definition A.1.1 Let X be a Banach space, and let I = [a, b] ⊂ ℝ be a closed bounded interval.
function f: I → X is a step function if f is constant on successive intervals Ik = [ak, ak+1), 0 ≤ k < n, where a0 = a and an = b.
function f : I → X is regulated if f is right-continuous and has a left-hand limit at every point of [a, b]. We denote the class of all regulated X-valued functions on I = [a, b] by Reg(I; X).
an X-valued step function on a closed unbounded subinterval J of ℝ we mean a function whose restriction to any closed bounded interval is a step function in the sense of (i).
an X-valued regulated function on a closed unbounded subinterval J of ℝ we mean a function whose restriction to any closed bounded interval is regulated in the sense of (ii). We denote this class of functions by Regloc(J; X), and let Reg(J; X) be the space of all functions which are both regulated and bounded.
Proposition A.1.2Let X be a Banach space, and let I = [a, b] be a closed bounded interval.
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- Well-Posed Linear Systems , pp. 730 - 749Publisher: Cambridge University PressPrint publication year: 2005