##
**Observation and control for operator semigroups.**
*(English)*
Zbl 1188.93002

Birkhäuser Advanced Texts. Basler Lehrbücher. Basel: Birkhäuser (ISBN 978-3-7643-8993-2/hbk; 978-3-7643-8994-9/ebook). xi, 483 p. (2009).

This book, written at the level of a graduate textbook, is devoted to observation and control operators for operator semigroups. Many systems modeled by linear partial differential equations or linear delay equations can be described by operator semigroups. Observation and control operators are needed to model the interaction of a system with the outside world. This book focuses mainly on topics like admissibility, observability and controllability of these operators.

The main aim of the authors is to provide a self-contained introduction into the topics admissibility, observability and controllability. In particular, no prior knowledge of finite-dimensional control theory, operator semigroups or unbounded operators is assumed, and only results are included for which they can provide a complete proof. More advanced results which use deeper tools from functional analysis or PDEs are mentioned in the bibliographic comments. In addition, a large number of PDE examples is included in the book in order to illustrate the abstract results. The authors combine in an ideal manner the abstract functional analytic approach and the PDE approach to observability and controllability.

Let us describe the content of the monograph in more detail. Chapter 1 is devoted to finite-dimensional systems theory. In particular, the concepts of controllability and observability are introduced and studied.

For the study of infinite-dimensional systems a large quantity of background material on the theory of strongly continuous semigroups of operators on Hilbert spaces is required, and this is developed in Chapter 2 and 3. Here the authors derive the basic properties of the spectrum and the resolvent of the semigroup generator, and of invariant subspaces for semigroups. Further, special classes of semigroups, such as diagonalizable semigroups, strongly continuous groups, adjoint semigroups, unitary groups and contraction semigroups are introduced and studied. Results concerning bounded perturbations of a generator and duality of two Hilbert spaces with respect to a pivot space are included as well.

Admissible control and observation operators are the subject of Chapter 4. Basic properties of these operators are given and the duality between these two concepts is shown.

The theme of admissibility is continued in Chapter 5, where sufficient and necessary conditions for admissibility are developed. Gramians and Lyapunov inequalities, the Carleson measure criterion and admissible control operators for perturbed semigroups are discussed. Further tests for admissibility are included without proofs in the notes.

In Chapters 6 to 9 the notion of observability is covered. Chapter 6 starts with a discussion of several observability concepts such as exact observability, approximate observability and final state observability. Note that in finite dimensions all these concepts are equivalent. Further, robustness of exact observability with respect to admissible perturbations of the generator and simultaneous exact observability are discussed. In finite dimensions the Hautus test is a powerful test for observability. Various extensions to the infinite-dimensional setting are shown. In particular, a Hautus test is presented for skew-adjoint generators with an admissible observation operator. Furthermore, the derivation of exact observability results for abstract Schrödinger-type equations from properties of abstract wave-type equations is included.

This theme is continued in Chapter 7, where observability for the wave equation is studied in more detail, and consequences for the Schrödinger and plate equation are discussed. The main idea in this chapter is to split the system governed by PDEs into a low- and a high-frequency part. The high-frequency part can be tackled by various methods. The authors choose multiplier and perturbation techniques. The low-frequency part is tackled by the finite-dimensional Hautus test in combination with unique continuation for elliptic operators.

Chapter 8 is devoted to classical results on non-harmonic Fourier series and the implications to exact observability for some systems governed by PDEs. This method is limited to rectangular domains in \(\mathbb R^n\), but for some examples this method provides sharp estimates on the observability time and on the observation region. The observability of parabolic equations is studied in Chapter 9. Observability results for systems governed by the wave equation are used to show final state observability for systems governed by the heat or related parabolic equations. Moreover, the global Carleman estimate for the heat equation is given. This estimate is needed to prove final state observability for arbitrary observation regions.

In Chapter 10 the subject of boundary control systems is covered. Basic properties are summarized and several examples such as Euler-Bernoulli beam, heat and Schrödinger equations with boundary control are shown to fit into this framework.

Finally, in Chapter 11 controllability questions are studied. Several controllability concepts are discussed, the duality between controllability and observability is addressed, and the controllability of parabolic as well as hyperbolic PDEs are investigated.

Each chapter concludes with a section containing bibliographic notes and further results. In the appendix much of the needed background, such as Fourier and Laplace transformations, distributions and Sobolev spaces, is summarized.

In summary, the book is a welcome addition to infinite-dimensional systems theory. The book is suitable for researchers in other areas who wish to enter the field and also for specialists in the field as a general reference.

The main aim of the authors is to provide a self-contained introduction into the topics admissibility, observability and controllability. In particular, no prior knowledge of finite-dimensional control theory, operator semigroups or unbounded operators is assumed, and only results are included for which they can provide a complete proof. More advanced results which use deeper tools from functional analysis or PDEs are mentioned in the bibliographic comments. In addition, a large number of PDE examples is included in the book in order to illustrate the abstract results. The authors combine in an ideal manner the abstract functional analytic approach and the PDE approach to observability and controllability.

Let us describe the content of the monograph in more detail. Chapter 1 is devoted to finite-dimensional systems theory. In particular, the concepts of controllability and observability are introduced and studied.

For the study of infinite-dimensional systems a large quantity of background material on the theory of strongly continuous semigroups of operators on Hilbert spaces is required, and this is developed in Chapter 2 and 3. Here the authors derive the basic properties of the spectrum and the resolvent of the semigroup generator, and of invariant subspaces for semigroups. Further, special classes of semigroups, such as diagonalizable semigroups, strongly continuous groups, adjoint semigroups, unitary groups and contraction semigroups are introduced and studied. Results concerning bounded perturbations of a generator and duality of two Hilbert spaces with respect to a pivot space are included as well.

Admissible control and observation operators are the subject of Chapter 4. Basic properties of these operators are given and the duality between these two concepts is shown.

The theme of admissibility is continued in Chapter 5, where sufficient and necessary conditions for admissibility are developed. Gramians and Lyapunov inequalities, the Carleson measure criterion and admissible control operators for perturbed semigroups are discussed. Further tests for admissibility are included without proofs in the notes.

In Chapters 6 to 9 the notion of observability is covered. Chapter 6 starts with a discussion of several observability concepts such as exact observability, approximate observability and final state observability. Note that in finite dimensions all these concepts are equivalent. Further, robustness of exact observability with respect to admissible perturbations of the generator and simultaneous exact observability are discussed. In finite dimensions the Hautus test is a powerful test for observability. Various extensions to the infinite-dimensional setting are shown. In particular, a Hautus test is presented for skew-adjoint generators with an admissible observation operator. Furthermore, the derivation of exact observability results for abstract Schrödinger-type equations from properties of abstract wave-type equations is included.

This theme is continued in Chapter 7, where observability for the wave equation is studied in more detail, and consequences for the Schrödinger and plate equation are discussed. The main idea in this chapter is to split the system governed by PDEs into a low- and a high-frequency part. The high-frequency part can be tackled by various methods. The authors choose multiplier and perturbation techniques. The low-frequency part is tackled by the finite-dimensional Hautus test in combination with unique continuation for elliptic operators.

Chapter 8 is devoted to classical results on non-harmonic Fourier series and the implications to exact observability for some systems governed by PDEs. This method is limited to rectangular domains in \(\mathbb R^n\), but for some examples this method provides sharp estimates on the observability time and on the observation region. The observability of parabolic equations is studied in Chapter 9. Observability results for systems governed by the wave equation are used to show final state observability for systems governed by the heat or related parabolic equations. Moreover, the global Carleman estimate for the heat equation is given. This estimate is needed to prove final state observability for arbitrary observation regions.

In Chapter 10 the subject of boundary control systems is covered. Basic properties are summarized and several examples such as Euler-Bernoulli beam, heat and Schrödinger equations with boundary control are shown to fit into this framework.

Finally, in Chapter 11 controllability questions are studied. Several controllability concepts are discussed, the duality between controllability and observability is addressed, and the controllability of parabolic as well as hyperbolic PDEs are investigated.

Each chapter concludes with a section containing bibliographic notes and further results. In the appendix much of the needed background, such as Fourier and Laplace transformations, distributions and Sobolev spaces, is summarized.

In summary, the book is a welcome addition to infinite-dimensional systems theory. The book is suitable for researchers in other areas who wish to enter the field and also for specialists in the field as a general reference.

Reviewer: Birgit Jacob (Paderborn)

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

47D03 | Groups and semigroups of linear operators |

93B05 | Controllability |

93B07 | Observability |

93B28 | Operator-theoretic methods |

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

93C25 | Control/observation systems in abstract spaces |

93C05 | Linear systems in control theory |