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Book review

Author(s) Kolmanovskii, V.
Myshkis, A.
Title Applied theory of functional differential equations
Publisher Kluwer Academic Publishers
Year of publication 1992
Reviewed by Radu Precup

After a period of abstractization and formalization the mathematicians return to their major source of problems: the life. They give up the mathematics of empty concepts that are to fill for the mathematics of concepts already filled. The activity of making aspects of experience accessible to the mathematical instrument is the modeling. Its results are the mathematical models, condensed and simplified representations of available information. The reason to model something and to study the corresponding model is to enhance understanding and to predict, influence and regulate the future developments of the real process. This is a book on mathematical models described by functional differential equations which are applicable to phenomena of quite different natures including: immunology, nuclear power generation, heat transfer, track signal processing, regulation systems, medicine, economy, etc. The common trait of the phenomena under consideration is the memory that makes the state of a system at any time depend on the past history of the process and not only on its initial state. The rough contents are as follows, Chapter 1: Models (about 34pp of models in mechanics, physics, techniques, biology, medicine, economy and other sciences); Chapter 2: General theory (a broad introduction to the basic principles of functional differential equations. Some important special classes of such equations are described and analyzed in some detail); Chapters 3-5: Stability of retarding differential equations, Stability of neutral type functional differential equations, Stability of stochastic functional differential equations; Chapters 6-8: Problems of control for deterministic functional differential equations, Optimal control of stochastic delay systems, State estimates of stochastic systems with delay. A general Index and a good list of references including more than 500 authors, complete the book.

The book adresses primarily to applied mathematicians, engineers and physicists whose work involves mathematical modeling of hereditary systems. It can also be useful for mathematicians who want to find applications of their theories. This volume can also be recommended for undergraduate students in applied mathematics and differential equations.

This is a new valuable book in the prestigious series Mathematics and Its Applications edited by Professor M. flazewinkel.