Wang G., Wang L., Xu Y., Zhang Y. Time Optimal Control of Evolution Equations

Springer, 2018. — 344 p. — (Progress in Nonlinear Differential Equations and Their Applications: Subseries in Control 92). — ISBN: 978-3-319-95362-5.This monograph develops a framework for time-optimal control problems, focusing on minimal and maximal time-optimal controls for linear-controlled evolution equations. Its use in optimal control provides a welcome update to Fattorini’s work on time-optimal and norm-optimal control problems. By discussing the best way of representing various control problems and equivalence among them, this systematic study gives readers the tools they need to solve practical problems in control.
After introducing preliminaries in functional analysis, evolution equations, and controllability and observability estimates, the authors present their time-optimal control framework, which consists of four elements: a controlled system, a control constraint set, a starting set, and an ending set. From there, they use their framework to address areas of recent development in time-optimal control, including the existence of admissible controls and optimal controls, Pontryagin’s maximum principle for optimal controls, the equivalence of different optimal control problems, and bang-bang properties.
This monograph will appeal to researchers and graduate students in time-optimal control theory, as well as related areas of controllability and dynamic programming. For ease of reference, the text itself is self-contained on the topic of time-optimal control. Frequent examples throughout clarify the applications of theorems and definitions, although experience with functional analysis and differential equations will be useful.Mathematical Preliminaries
Time Optimal Control Problems
Existence of Admissible Controls and Optimal Controls
Maximum Principle of Optimal Controls
Equivalence of Several Kinds of Optimal Controls
Bang-Bang Property of Optimal Controls