Time optimal control of evolution equations
The book under review is a welcome addition to the increasing number of monographs in the theory of linear-controlled dynamical systems in Hilbert spaces and provides an inspired presentation of the tools of optimal control theory in solving time-optimal problems. The main objective of this monograph is to set up a general mathematical framework for minimal and maximal time control problems governed by some linear evolution equations and is a nice update of the book [\textit{H.O. Fattorini}, Infinite Dimensional Linear Control Systems, the Time Optimal and Norm Optimal Problems. North-Holland Mathematics Studies, 201. Amsterdam: Elsevier (2005; Zbl 1135.93001)]. It consists of six chapters each of which is followed by miscellaneous notes and references. Chapter 1 offers a short recall of fundamental elements of functional analysis, theory of set-valued maps, and controllability and observability, which are necessary later in presentation of other chapters. An overview on time optimal control problems is given in Chapter 2. It includes the definitions and main ingredients of such problems, connections between minimal and maximal time controls, and several examples. Chapter 3 is dedicated to study the existence of admissible controls and optimal controls from different perspectives for certain classes of minimal time problems. In Chapter 4 various (Pontryagin's) Maximum Principles for time optimal controls associated to some evolution systems are derived using a geometric approach. Chapter 5 is devoted to the study of equivalence between different kinds of optimal control problems. More specifically, it develops the equivalence of minimal time optimal control problem and minimal norm optimal control problem in the first part and the equivalence of maximal time optimal control problem and minimal norm optimal control problem in the second part. Finally, Chapter 6 presents the bang-bang property for the time optimal control problems. To sum up, this book contains many ideas, methods and recent advances in the field, is quite well-written and well-organized and suits best to researchers and graduate-level treatment. Despite the self-consistent content of the book, some experience in functional analysis and differential equations will be useful.
scientific article; zbMATH DE number 6909997
scientific article