Introduction to the theory and applications of functional differential equations.

*(English)*Zbl 0917.34001
Mathematics and its Applications (Dordrecht). 463. Dordrecht: Kluwer Academic Publishers. xvi, 648 p. (1999).

When I received this book to review it, I thought: just a few mathematicians have had the great opportunity to be actors and witnesses of what happened in the last five decades in the theory and applications of functional-differential equations(FDEs); and V. Kolmanovskii and A. Myshkis certainly belong to this exclusive group. It is worth to remind that A. Myshkis’s book [Linear differential equations with retarded arguments (Russian (1951; Zbl 0043.30904); German translation (1955; Zbl 0063.31802)] laid the foundations for a general theory of linear systems. This new book written by the authors is devoted to a systematic exposition of the general theory of functional-differential equations of retarded and neutral type and applications to control theory, mathematical biology models, boundary value problems, and stochastics equations. The book is organized as follows.

The first chapter is devoted to basic concepts in FDEs.

In Chapter 2, the authors introduce the reader to a large number of models with delay in viscoelasticity, aftereffects in mechanics, biology, medicine, economy; hereditary phenomena in physics, models with delay in technical problems. In this way, the reader becomes familiar with numerous and important problems that arise in FDEs.

Chapter 3 contains the general theory of FDEs of retarded and neutral type, including applications to differential inclusions of retarded type.

Chapters 4-10 are devoted to thorough investigations of the stability of linear and nonlinear FDEs of retarded, neutral and stochastic type. The Lyapunov functional and the method of Razumikhin are considered.

Chapters 11-13 are an extensive study of boundary value problems and periodic solutions of FDEs.

Chapter 14-16 deal with optimal control problems described by FDEs of retarded, neutral and stochastic type; and estimation in hereditary systems.

The first chapter is devoted to basic concepts in FDEs.

In Chapter 2, the authors introduce the reader to a large number of models with delay in viscoelasticity, aftereffects in mechanics, biology, medicine, economy; hereditary phenomena in physics, models with delay in technical problems. In this way, the reader becomes familiar with numerous and important problems that arise in FDEs.

Chapter 3 contains the general theory of FDEs of retarded and neutral type, including applications to differential inclusions of retarded type.

Chapters 4-10 are devoted to thorough investigations of the stability of linear and nonlinear FDEs of retarded, neutral and stochastic type. The Lyapunov functional and the method of Razumikhin are considered.

Chapters 11-13 are an extensive study of boundary value problems and periodic solutions of FDEs.

Chapter 14-16 deal with optimal control problems described by FDEs of retarded, neutral and stochastic type; and estimation in hereditary systems.

Reviewer: Marcos Lizana (Tempe)

##### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

49J15 | Existence theories for optimal control problems involving ordinary differential equations |

34Kxx | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

34H05 | Control problems involving ordinary differential equations |

49J25 | Optimal control problems with equations with ret. arguments (exist.) (MSC2000) |

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