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**Differential inclusions and optimal control.**
*(English)*
Zbl 0731.49001

Mathematics and Its Applications. East European Series 44. Dordrecht etc.: Kluwer Academic Publishers; Warszawa: PWN-Polish Scientific Publishers (ISBN 0-7923-0675-9). xix, 240 p. (1991).

This book is devoted to the analysis and optimal control of dynamical systems described by functional-differential equations. This type of mathematical models arises whenever the past exerts a significant influence upon the future behaviour of the system. The optimal control problem for this class of systems leads to functional-differential inclusions and, in particular, to neutral functional-differential inclusions. The main body of the book deals with the latter type of problems and is based on recently published monographs on the subject of functional-differential inclusions [see J. P. Aubin and A. Cellina, “Differential inclusions” (1984; Zbl 0538.34007); M. C. Castaing and M. Valadier, “Convex analysis and measurable multifunctions”, Lect. Notes Math. 580 (1977; Zbl 0346.46038); A. F. Filippov, “Differential equations with discontinuous right hand sides” (Russian) (1985; Zbl 0571.34001) (for the 1988 English ed. see Zbl 0664.34001); T. Parthasarathy, “Selection theorems and their applications” (1972; Zbl 0239.54011); A. A. Tolstogonov, “Differential inclusions in a Banach space” (Russian) (1986; Zbl 0689.34014)].

The book is divided into five chapters. Chapter I repeats basic notations and theorems of topology, functional analysis and vector measure. Chapter II covers the fundamental theory of set-valued functions. Chapter III presents the properties of subtrajectory and trajectory integrals of these functions depending on parameters. Chapters IV and V deal with neutral functional-differential inclusions and their applications in optimal control theory.

This book is written on a high mathematical level. It is intended for students and professionals in mathematics and specialists in optimal control theory. It is too theoretical for engineers.

The book is divided into five chapters. Chapter I repeats basic notations and theorems of topology, functional analysis and vector measure. Chapter II covers the fundamental theory of set-valued functions. Chapter III presents the properties of subtrajectory and trajectory integrals of these functions depending on parameters. Chapters IV and V deal with neutral functional-differential inclusions and their applications in optimal control theory.

This book is written on a high mathematical level. It is intended for students and professionals in mathematics and specialists in optimal control theory. It is too theoretical for engineers.

Reviewer: D.Franke (Hamburg)

### MSC:

49J24 | Optimal control problems with differential inclusions (existence) (MSC2000) |

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