Time optimal control of evolution equations
type: Item
instance of: scholarly article
title: Time optimal control of evolution equations
zbMATH Open document ID: 1406.49002
DOI: 10.1007/978-3-319-95363-2
author: Gengsheng Wang
author: Yashan Xu
author: Yubiao Zhang
author: Lijuan Wang
published in: Progress in Nonlinear Differential Equations and Their Applications
publication date: 2018-07-24T00:00:00Z
review text: 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 i…
Mathematics Subject Classification ID: 49-02
Mathematics Subject Classification ID: 93-02
Mathematics Subject Classification ID: 37L05
Mathematics Subject Classification ID: 49J15
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49J30
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93B27
zbMATH DE Number: 6909997
zbMATH Keywords: controllability
zbMATH Keywords: semigroup
zbMATH Keywords: time-optimal control problems
zbMATH Keywords: source and target sets
zbMATH Keywords: constraint set
zbMATH Keywords: bang-bang theory
zbMATH Keywords: admissible control
zbMATH Keywords: maximum principle
reviewed by: Shokhrukh Yusufovich Kholmatov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-95363-2
OpenAlex ID: W2888080725
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description: scientific article; zbMATH DE number 6909997