##
**Systems with hysteresis. Transl. from the Russian by Marek Niezgódka.**
*(English)*
Zbl 0665.47038

Berlin etc.: Springer-Verlag. xviii, 410 p. DM 148.00 (1989).

Hysteresis phenomena have become a well-established part of applied sciences covered by quite a vast literature in physics, mechanics, engineering, and technology. Loosely speaking, whenever one is concerned with irreversible phenomena “with memory”, both material (e.g. in elasticity) and immaterial (e.g. in magnetism), hysteresis phenomena may arise quite naturally. A rigorous mathematical treatment, however, was long in coming for many years. It is the authors’ incontestable merit to have made the first serious attempt in this direction in the book under review. Usually, when Krasnosel’skij writes a new book on some mathematical topics, you can find everything therein. This is in part true, in part false for the present work. In fact, one can find a wealth of interesting new (and sometimes even surprising) results on problems involving hysteresis-type nonlinearities in the book. On the other hand, some of these problems have become clearer in the meantime (the Russian original appeared about 6 years ago), and certainly some parts of such a book would look different nowadays (even if written by the same authors).

It is interesting to note that, according to the physical background of the topics covered in the book (and perhaps also to the objective of the authors’ financiers), purely mathematical facts are expressed throughout in the colloquial language of engineers: thus, the authors write of “vibro-correct transducers W with domain of feasible states D, inputs u(t), and outputs x(t)”, rather than of “well-posed operators W with domain of definition D, arguments u(t), and values x(t)”, which would sound more familiar to mathematicians. Of course, the mathematics treated in the book are, with a very few exceptions, high-standard and challenging.

The book consists of 7 chapters of nearly equal size. The first chapter introduces the fundamental concept of “hysteron” which is nothing else but an operator associated with a given hysteresis loop with certain “natural” properties. Mathematically, the main problem consists in extending operators from narrow classes of inputs (e.g. smooth) to larger ones (e.g. continuous or monotone). The somewhat shorter Chapter 2 deals with various types of identification problems (which refers, roughly speaking, to the mathematical modelling of a “black box”). While the first two chapters mainly deal with the stationary theory, Chapter 3 is concerned with evolutionary (i.e. time-dependent) hysterons. Mathematically, this amounts to a thorough study of a class of “vibro- correct” differential equations with and without constraints. Finally, in Chapter 4 a parallel theory is developed in the vector-valued case, covering, for instance, the classical models of Mises and Saint-Venant.

As hysteresis nonlinearities may be viewed as multivalued input-output- relations with specific features, it seems natural to treat hysteresis transducers within the very advanced theory of multivalued maps. This is the purpose of the fifth chapter which may be considered as a self- contained survey on several topological and analytical properties of multivalued superposition operators. Surprisingly, in this part the exposition is more “narrative”, rather than mathematically rigorous, compared with the other chapters.

The final Chapters 6 and 7 are concerned with mathematical notions which are modelled on self-magnetization phenomena and complex nondeterministic nonlinearities related, in their physical roots, with the work of Madelung, Ishlinskij, and Preisach. The book concludes with three pages of very brief bibliographical comments, a list of 170 references (covering mostly books and papers by Soviet authors), and a subject index.

The Russian original is very clear and understandable; unfortunately, this is no longer true for the English translation. The translator was obviously stricking very closely to the Russian text, sometimes translating even word-for-word; as a consequence, the English is rather poor and clumsy. Quite the opposite may be stated about the editorial part: both the text and the formulas are very readable throughout, and whoever has typed the English manuscript has done an excellent job. All in all, this is another masterpiece of one of the champions in nonlinear analysis which will become, without any doubt, a standard reference. Maybe the rather high price will prevent the book from the overall distribution it would deserve.

It is interesting to note that, according to the physical background of the topics covered in the book (and perhaps also to the objective of the authors’ financiers), purely mathematical facts are expressed throughout in the colloquial language of engineers: thus, the authors write of “vibro-correct transducers W with domain of feasible states D, inputs u(t), and outputs x(t)”, rather than of “well-posed operators W with domain of definition D, arguments u(t), and values x(t)”, which would sound more familiar to mathematicians. Of course, the mathematics treated in the book are, with a very few exceptions, high-standard and challenging.

The book consists of 7 chapters of nearly equal size. The first chapter introduces the fundamental concept of “hysteron” which is nothing else but an operator associated with a given hysteresis loop with certain “natural” properties. Mathematically, the main problem consists in extending operators from narrow classes of inputs (e.g. smooth) to larger ones (e.g. continuous or monotone). The somewhat shorter Chapter 2 deals with various types of identification problems (which refers, roughly speaking, to the mathematical modelling of a “black box”). While the first two chapters mainly deal with the stationary theory, Chapter 3 is concerned with evolutionary (i.e. time-dependent) hysterons. Mathematically, this amounts to a thorough study of a class of “vibro- correct” differential equations with and without constraints. Finally, in Chapter 4 a parallel theory is developed in the vector-valued case, covering, for instance, the classical models of Mises and Saint-Venant.

As hysteresis nonlinearities may be viewed as multivalued input-output- relations with specific features, it seems natural to treat hysteresis transducers within the very advanced theory of multivalued maps. This is the purpose of the fifth chapter which may be considered as a self- contained survey on several topological and analytical properties of multivalued superposition operators. Surprisingly, in this part the exposition is more “narrative”, rather than mathematically rigorous, compared with the other chapters.

The final Chapters 6 and 7 are concerned with mathematical notions which are modelled on self-magnetization phenomena and complex nondeterministic nonlinearities related, in their physical roots, with the work of Madelung, Ishlinskij, and Preisach. The book concludes with three pages of very brief bibliographical comments, a list of 170 references (covering mostly books and papers by Soviet authors), and a subject index.

The Russian original is very clear and understandable; unfortunately, this is no longer true for the English translation. The translator was obviously stricking very closely to the Russian text, sometimes translating even word-for-word; as a consequence, the English is rather poor and clumsy. Quite the opposite may be stated about the editorial part: both the text and the formulas are very readable throughout, and whoever has typed the English manuscript has done an excellent job. All in all, this is another masterpiece of one of the champions in nonlinear analysis which will become, without any doubt, a standard reference. Maybe the rather high price will prevent the book from the overall distribution it would deserve.

Reviewer: J.Appell

### MSC:

47J05 | Equations involving nonlinear operators (general) |

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

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

74B99 | Elastic materials |

74C99 | Plastic materials, materials of stress-rate and internal-variable type |

78A99 | General topics in optics and electromagnetic theory |

54C60 | Set-valued maps in general topology |