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Mathematical models of hysteresis. (English) Zbl 0723.73003
New York etc.: Springer-Verlag. xx, 207 p. DM 114.00 (1991).
This book is concerned with mathematical models of hysteresis phenomena with “nonlocal memory”; this means that a given input function $$u=u(t)$$ causes an output function $$w=w(t)$$ in such a way that $$w(t_ 0)$$ depends on all values u(t) for $$t\leq t_ 0$$ (more precisely, on all extrema of u(t) for $$t\leq t_ 0).$$
Important progress in the mathematical theory of nonlinear problems involving hysteresis operators is due to, among others, M. A. Krasnosel’skij and A. V. Prokrovskij, M. Brokate, P. Krejci, A. Visintin [see the preceding review (Zbl 0723.73002)], and the author of the present book. In particular, the already classical monograph by M. A. Krasnosel’skij and A. V. Pokrovskij [Systems with hysteresis (1989; Zbl 0665.47038)] is mainly concerned with the problem of extending hysteresis operators (which are, by their very nature, primarily given on spaces of piecewise monotone functions) by continuity to larger (Banach) spaces. In contrast to this, the present monograph is written “by an engineer for engineers”, and thus favours a more phenomenological approach above mathematical abstraction.
The book consists of three chapters of nearly equal size, whose contents is described by the author as follows. The book is primarily concerned with Preisach-type models of hysteresis. All these models have a common generic feature; they are constructed as superpositions of simplest hysteresis nonlinearities - rectangular loops. The discussion is by and large centered around the following topics: various generalizations and extensions of the classical Preisach model (with special emphasis on vector generalizations), finding of necessary and sufficient conditions for the representation of actual hysteresis nonlinearities by various Preisach-type models, solution of identification problems for these models, and numerical implementation and experimental testing of Preisach-type models. Although the study of Preisach-type models constitutes the main subject of the book, some effort is also made to establish some interesting connections between these models and such topics as the critical state model for superconducting hysteresis, the classical Stoner-Wolfarth model for vector magnetic hysteresis, thermal activation-type models for viscosity, magnetostrictive hysteresis and neural networks.
The book is clearly written and well-organized. It fills a gap in an important new field of nonlinear analysis of rapidly growing interest. The only serious drawback concerns the list of references, where the author only quoted authors’ names and abbreviations of journals, but not the titles of papers. This may be excusable at the end of a short note (for lack of space), but certainly not for a book of 200 pages.

##### MSC:
 74-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 74S99 Numerical and other methods in solid mechanics