© 2013

Invariance Entropy for Deterministic Control Systems

An Introduction


  • The first book to present results on the relatively new notions of feedback and invariance entropy

  • Provides detailed proofs of all key results, together with a comprehensive exposition of the mathematical background material used therein

  • Will be particularly interesting for readers with a background in dynamical systems theory

  • The basic knowledge of control theory needed for its understanding is presented at great length in the first chapter


Part of the Lecture Notes in Mathematics book series (LNM, volume 2089)

Table of contents

  1. Front Matter
    Pages i-xxii
  2. Christoph Kawan
    Pages 1-42
  3. Christoph Kawan
    Pages 43-87
  4. Christoph Kawan
    Pages 89-105
  5. Christoph Kawan
    Pages 107-120
  6. Christoph Kawan
    Pages 151-175
  7. Christoph Kawan
    Pages 177-220
  8. Back Matter
    Pages 221-272

About this book


This monograph provides an introduction to the concept of invariance entropy, the central motivation of which lies in the need to deal with communication constraints in networked control systems. For the simplest possible network topology, consisting of one controller and one dynamical system connected by a digital channel, invariance entropy provides a measure for the smallest data rate above which it is possible to render a given subset of the state space invariant by means of a symbolic coder-controller pair. This concept is essentially equivalent to the notion of topological feedback entropy introduced by Nair, Evans, Mareels and Moran (Topological feedback entropy and nonlinear stabilization. IEEE Trans. Automat. Control 49 (2004), 1585–1597). The book presents the foundations of a theory which aims at finding expressions for invariance entropy in terms of dynamical quantities such as Lyapunov exponents. While both discrete-time and continuous-time systems are treated, the emphasis lies on systems given by differential equations.


Controlled Invariance Feedback Entropy Invariance Entropy Minimal Data Rates

Authors and affiliations

  1. 1.Institute of MathematicsUniversity of AugsburgAugsburgGermany

Bibliographic information

  • Book Title Invariance Entropy for Deterministic Control Systems
  • Book Subtitle An Introduction
  • Authors Christoph Kawan
  • Series Title Lecture Notes in Mathematics
  • Series Abbreviated Title Lect.Notes Mathematics
  • DOI
  • Copyright Information Springer International Publishing Switzerland 2013
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-319-01287-2
  • eBook ISBN 978-3-319-01288-9
  • Series ISSN 0075-8434
  • Series E-ISSN 1617-9692
  • Edition Number 1
  • Number of Pages XXII, 270
  • Number of Illustrations 1 b/w illustrations, 1 illustrations in colour
  • Topics Dynamical Systems and Ergodic Theory
    Systems Theory, Control
  • Buy this book on publisher's site
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From the book reviews:

“The topics are well organized, the exposition is clear and there are examples to illustrate the theoretical results. This book is highly recommended for both researchers and graduate students in dynamical systems, control theory and related areas. This book is particularly useful because it collects in one place a large number of seminal results that are essential for an understanding of invariance entropy.” (Alexandre José Santana, Mathematical Reviews, August, 2014)