
Author(s)  Kisielewicz, Michal 

Title  DIFFERENTIAL INCLUSIONS and OPTIMAL CONTROL 
Publisher  Kluwer Academic Publishers 
Year of publication  1991 
Reviewed by  S. Rogosin 
The main subject of this book are the functionaldifferential inclusions of the following type <formula> (1) where F is a given functions, xt = x(t+s). These inclusions are called neutral functionaldifferential inclusions (NFDI). They are quite analogous to the corresponding class of functionaldifferential equations the study of which began very intensive in the last twothree decades. Some motivations for the inclusions (1) follow from the consideration of some typical situations for oscillatory systems with some interconnection between them. The most typical property of these inclusions is a dependence of the present dynamics of the systems describing from their past behavior.
The NFDI need to use extremely accurate integrability technics. This branch of mathematical knowledge is highly developed in Poland. So the author follows the brilliant polish mathematical traditions in this sense. Besides one of the basic constructions is so called Aumann type of integration for setvalued functions for which one has to know also the deep measurability properties of these functions.
The NFDI are investigated by many authors, but main results for them are obtained by M. Kisielewicz. The most of them are presented in this book. Here is not only developing the method of the functionaldifferential inclusions of neutral type, but also constructing the ground for this branch of mathematics. So in spite the fact that this method is far away from completeness this monograph can be considered as a deep basis for further investigations and applications. All materials which are on need even for the beginners are gathered and deeply discussed here. From this point of view the book is a good collection of lectures on the neutral functionaldifferential inclusions.
The book consists of five chapters, the short list of references and subject index. Every chapters is ended by a number of notes and remarks (mainly of historical character).
The first three chapters has more or less auxiliary and even preparatory character. First of all are presented set and topological preliminaries and some basic facts of modern functional analysis which will be used further in the main constructions. Then is developing the theory of integration for setvalued functions (chapters 2 and 3). As it was mentioned above the main for this book is the construction of integrals given by R.J. Aumann in the middle sixties as well as trajectory and subtrajectory integrals of set valued functions.
The neutral functionaldifferential inclusions themselves are introduced in the chapter 4. Besides of their description and some motivations here are given some basic properties of these objects. At the beginning these properties are connected with the corresponding properties of functionaldifferential equations of the neutral type. But then it is appeared some differences between them. Here are presented the results of the both two types. Among the properties of the second type we can note controllability theorems obtained with the using of Bouligand's contingent cone construction.
At last in the chapter 5 some properties of the NFDI are used for obtaining some existence theorems for solution of some optimal control problems. On the consideration are the known Mayer and Lagrange optimal control problems for systems described by neutral functionaldifferential inclusions. The basic for these theorems are some compactness properties of the set of attainable trajectories and semicontinuities of given functionals, which were investigated in previous chapter.
The book is written in semiclassical style  here it is presented all necessary information for applying the proposed method. It makes possible to consider this book from one side as a textbook for study some methods of optimal control theory. So it can be useful even graduate and postgraduate students in Mathematics and Engineering which are interesting in the theory of functionaldifferential inclusions. From the other side it is undoubtedly the deep monograph containing some modern results on its subject. Therefore it is interesting for the specialists in functionaldifferential equations and inclusions, integration theory, control theory and their applications.