Control theory from the geometric viewpoint.
type: Item
instance of: scholarly article
title: Control theory from the geometric viewpoint.
zbMATH Open document ID: 1062.93001
author: Yu L. Sachkov
author: A. A. Agrachev
published in: Encyclopaedia of Mathematical Sciences
publication date: 2004-10-04T00:00:00Z
review text: This monograph on geometric control theory, is the new reference text for researchers and graduate students in mathematical control theory who are interested in differential-geometric methods. These methods are particularly useful for control problems formulated on manifolds or on Lie groups, as it happens very often, for instance, in robotics and in quantum control. The book, written by leader experts in the subject, is full of details that are not published elsewhere. It is translated also in Russian. The first chapter is devoted to standard material in differential geometry and control systems. The second chapter presents the chronological calculus, which is a non-standard language developed by Agrachev and Gamkrelidze to deal with flows of ODEs having an explicit dependence on the time. It permits to transform nonlinear objects living on a finite-dimensional manifold (e.g. points, vector fields, flows etc.) into linear operators acting on an infinite dimensional space, namely, the space of smooth functions on the manifold. Chronological calculus, albeit sometimes difficult at a first glance for neophytes, allows to simplify deeply many of the proofs appearing in the remaining of the book and is a very interesting subject in itself. Chapters 3 and 4 deal with linear systems and with state linearizability. Chapters 5,6,7,8 are devoted to the problem of controllability, having as cornerstone the orbit theorem by Hermann-Nagano (analytic case) and Sussmann (smooth case). An application to the rotation of the rigid body is presented in Chapter 6. Chapter 9 is about feedback and state equivalence of control systems. The core of the book lies in the remaining chapters, where Optimal Control is presented. The proof of Pontryagin Maximum Principle has a highly geometrical taste and is written directly for systems on manifolds. Some non-trivial applications (e.g., linear time-optimal problems, linear-quadratic problems, left invariant problems on Lie groups) are also give…
Mathematics Subject Classification ID: 93-02
Mathematics Subject Classification ID: 93B03
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93B17
Mathematics Subject Classification ID: 93B29
Mathematics Subject Classification ID: 49K15
Mathematics Subject Classification ID: 58A30
Mathematics Subject Classification ID: 70E18
Mathematics Subject Classification ID: 70G45
Mathematics Subject Classification ID: 70H20
Mathematics Subject Classification ID: 70F25
zbMATH DE Number: 2105696
zbMATH Keywords: chronological calculus
zbMATH Keywords: maximum principle
zbMATH Keywords: differential-geometric methods in control
zbMATH Keywords: chronological exponential
zbMATH Keywords: linearizability
zbMATH Keywords: Frobenius theorem
zbMATH Keywords: rigid body
zbMATH Keywords: attainable sets
zbMATH Keywords: feedback linearization
zbMATH Keywords: optimal control
zbMATH Keywords: time-optimal control
zbMATH Keywords: L-Q problem
zbMATH Keywords: second-order conditions
zbMATH Keywords: moving frame
zbMATH Keywords: curvature
zbMATH Keywords: rolling bodies
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2105696