Fourier series in control theory.

*(English)*Zbl 1094.49002
Springer Monographs in Mathematics. New York, NY: Springer (ISBN 0-387-22383-5/hbk). ix, 226 p. (2005).

The multiplier method is widely employed in the area of control theory. This outstanding book discusses new theoretical approaches to the study of control problems based on harmonic analysis. The main purpose of the authors is to unify, as much as possible, the so-called harmonic (or nonharmonic) analysis method and also to make the subject as simple as possible. The book includes only evolutionary problems with time-reversible dynamics. The duality between the notions of observability and controllability is presented. Two important generalizations of Parseval’s inequality are introduced. They enable us to solve several simple but already nontrivial problems concerning the observalibity of strings. The propagation speed is infinite in beams (in strings it is finite). This fact is reflected in the results in which the critical optimisability time is determined. A general theorem for the operator (the infinitesimal generator of the semigroup – diagonizable in some simple sense) is proved and a large number of examples where this theorem applies is given. Some new results concerning the zeros of Bessel-type functions are established and new simpler proofs of some classical results are presented. A classical theorem of Kahane is improved. The authors’ results show an interesting connection between Ingham-type theorems and the spectral theory of the Lapacian operator. The related optimal results by using other norms of the Euclidean space are proved. The authors then apply these results in order to extend some surprising internal observability theorems of Haraux and Jaffard concerning rectangular plates to arbitrary spatial dimensions. The usefulness of the general theorem by proving optimal simultaneous observability results for string and beam systems is demonstrated. The authors’ results hold only under some number-theoretic hypotheses concerning the lengths and strings or beams. An optimal observability theorem for spherical shells with a central hole is established.

The applications of the methods developed in this monograph are not limited to control theory. A new simple proof of a celebrated generalization of Bernstein of Pólya’s theorem on the singularities of Dirichlet series is presented.

This volume is primarily addressed to applied mathematicians working in the field of control theory and harmonic analysis. However, the book will be also useful for scientists from the application areas, in particular, applied scientists from engineering and physics.

The applications of the methods developed in this monograph are not limited to control theory. A new simple proof of a celebrated generalization of Bernstein of Pólya’s theorem on the singularities of Dirichlet series is presented.

This volume is primarily addressed to applied mathematicians working in the field of control theory and harmonic analysis. However, the book will be also useful for scientists from the application areas, in particular, applied scientists from engineering and physics.

Reviewer: Jan Lovíšek (Bratislava)

##### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

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

42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |

74K05 | Strings |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74K15 | Membranes |

74K20 | Plates |