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**An introduction to infinite-dimensional linear systems theory.**
*(English)*
Zbl 0839.93001

Texts in Applied Mathematics. 21. New York, NY: Springer-Verlag. xviii, 698 p. (1995).

The book is an introductory text that treats both state-space and frequency domain aspects of infinite-dimensional control systems in an integrated fashion. The theory covered includes material on strongly continuous semigroups of linear operators, the abstract Cauchy problem, controllability, observability, stabilizability, detectability, linear quadratic optimal control, the Callier-Desoer class of transfer functions, Hankel operators, the Nehari problem, and robust control. Whilst some of the state-space material is standard and can already be found in the lecture notes by the first author and A. Pritchard [Infinite-dimensional linear systems theory, Lect. Notes Control Inf. Sci. 8 (Springer-Verlag, 1978; Zbl 0389.93001)], many results appear for the first time in book form. Throughout the book, it is assumed that the state-space is a Hilbert space and that the control and observation operators are bounded. The really novel aspect of the book is the inclusion of a frequency-domain theory for irrational transfer functions that treats coprime factorizations, input-output stability, and robust control design (in particular \(H^\infty\)-robustness optimization).

Moreover, the relationship between the state-space and frequency-domain descriptions is treated carefully. All system and control theoretic concepts introduced in the text are illustrated by the same three physical examples: namely, a diffusion equation, a wave equation (both one-dimensional in space) and a delay equation. The book contains many worked examples, and every chapter contains up to 40 exercises. Each chapter ends with a section entitled “Notes and References” which gives a short discussion of the literature relevant to that chapter (the list of references contains 276 items). Finally, the book contains an extensive appendix (104 pages out of 698) on mathematical background material such as complex analysis, normed spaces, spectral theory, vector-valued integration etc., which should make it self-contained even for readers with a minimal background in functional analysis.

Moreover, the relationship between the state-space and frequency-domain descriptions is treated carefully. All system and control theoretic concepts introduced in the text are illustrated by the same three physical examples: namely, a diffusion equation, a wave equation (both one-dimensional in space) and a delay equation. The book contains many worked examples, and every chapter contains up to 40 exercises. Each chapter ends with a section entitled “Notes and References” which gives a short discussion of the literature relevant to that chapter (the list of references contains 276 items). Finally, the book contains an extensive appendix (104 pages out of 698) on mathematical background material such as complex analysis, normed spaces, spectral theory, vector-valued integration etc., which should make it self-contained even for readers with a minimal background in functional analysis.

Reviewer: H.Logemann (Bath)

### MSC:

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |

93C25 | Control/observation systems in abstract spaces |

93C20 | Control/observation systems governed by partial differential equations |

93D15 | Stabilization of systems by feedback |

93B36 | \(H^\infty\)-control |

93B07 | Observability |

93D25 | Input-output approaches in control theory |

93B05 | Controllability |