Stability and stabilization of infinite dimensional systems with applications.

*(English)*Zbl 0922.93001
Communications and Control Engineering Series. London: Springer. xiii, 403 p. (1999).

The time evolution of many physical phenomena in nature can be described by partial differential equations. In this very important book, the authors concentrate on the analysis and control of the dynamic behavior of such systems. The book reports upon some recent achievements in stability and feedback stabilization of infinite dimensional systems, with the emphasis on the second order partial differential equations which arise from the control of numerous mechanical systems such as flexible robot arms and large space structures. A number of new features obtained by the authors include: The integrated semigroup theory, the theorems and characterizations on weak strong stabilities of \(C_0\)-semigroups and a new characterization of the growth rate of a \(C_0\)-semigroup, the \(A\)-dependent operator concept, which has proven to be a powerful tool for establishing the well-posedness of some non-standard abstract second order equations, the application of the energy multiplier method to the proof of the closed-loop stability of beam equations with dynamic boundary control, the exponential stability analysis for a wide class of systems with boundary stabilizers, based on the verification of the spectrum-determined growth condition.

It is the authors’ aim to introduce the reader to this very important research field and to some of its applications. Detailed proofs for most lemmas and theorems are given. The book is thus adequate as a textbook for students in applied mathematics or as a reference book for control engineers and applied mathematicians interested in the analysis and control of infinite dimensional systems. The book presents not only some new theorems on semigroups and their stability but also some useful techniques for solving practical engineering problems.

It is the authors’ aim to introduce the reader to this very important research field and to some of its applications. Detailed proofs for most lemmas and theorems are given. The book is thus adequate as a textbook for students in applied mathematics or as a reference book for control engineers and applied mathematicians interested in the analysis and control of infinite dimensional systems. The book presents not only some new theorems on semigroups and their stability but also some useful techniques for solving practical engineering problems.

Reviewer: J.Lovíšek (Bratislava)

##### MSC:

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

93D15 | Stabilization of systems by feedback |

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

93C25 | Control/observation systems in abstract spaces |

74M05 | Control, switches and devices (“smart materials”) in solid mechanics |

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