Mathematical Control Theory - An Introduction

347 Pages·2020·4.797 MB·English

by  Jerzy Zabczyk

Systems & Control: Foundations & Applications
Jerzy Zabczyk
Mathematical
Control
Theory
An Introduction
Second Edition

Systems & Control: Foundations & Applications
Series Editor
Tamer Başar, University of Illinois at Urbana-Champaign, Urbana, IL, USA
Editorial Board
Karl Johan Åström, Lund Institute of Technology, Lund, Sweden
Han-Fu Chen, Academia Sinica, Beijing, China
Bill Helton, University of California, San Diego, CA, USA
Alberto Isidori, Sapienza University of Rome, Rome, Italy
Miroslav Krstic, University of California, San Diego, La Jolla, CA, USA
H. Vincent Poor, Princeton University, Princeton, NJ, USA
Mete Soner, ETH Zürich, Zürich, Switzerland;
Swiss Finance Institute, Zürich, Switzerland
Former Editorial Board Member
Roberto Tempo, (1956–2017), CNR-IEIIT, Politecnico di Torino, Italy
More information about this series at http://www.springer.com/series/4895
Jerzy Zabczyk
Mathematical Control Theory
An Introduction
Second Edition
JerzyZabczyk
Institute of Mathematics
Polish Academy ofSciences
Warsaw,Poland
ISSN 2324-9749 ISSN 2324-9757 (electronic)
Systems &Control: Foundations& Applications
ISBN978-3-030-44776-2 ISBN978-3-030-44778-6 (eBook)
https://doi.org/10.1007/978-3-030-44778-6
MathematicsSubjectClassification(2010): 49N05,49J15,49J20
1stedition:©BirkhäuserBoston2008
2ndedition:©SpringerNatureSwitzerlandAG2020
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To Zofia, Veronika, Johannes, and Maximilian
my grandchildren
PREFACE
TO THE FIRST EDITION
Control theory originated around 150 years ago when the performance of
mechanicalgovernorsstartedtobeanalysedinamathematicalway.Suchgov-
ernors act in a stable way if all the roots of some associated polynomials are
contained in the left half of the complex plane. One of the most outstand-
ing results of the early period of control theory was the Routh algorithm,
which allowed one to check whether a given polynomial had this property.
Questions of stability are present in control theory today, and, in addition,
to technical applications, new ones of economical and biological nature have
beenadded. Control theory has beenstrongly linked with mathematics since
WorldWarII.Ithashadconsiderableinfluenceonthecalculusofvariations,
the theory of differential equations and the theory of stochastic processes.
TheaimofMathematicalControlTheory istogiveaself-containedoutline
ofmathematicalcontroltheory.Theworkconsciouslyconcentratesontypical
and characteristic results, presented in four parts preceded by an introduc-
tion. The introduction surveys basic concepts and questions of the theory
and describes typical, motivating examples.
PartIisdevotedtostructuralpropertiesoflinearsystems.Itcontainsbasic
results on controllability, observability, stability and stabilizability. A sepa-
rate chapter covers realization theory. Toward the end more special topics
are treated: linear systems with bounded sets of control parameters and the
so-called positive systems.
StructuralpropertiesofnonlinearsystemsarethecontentofPartII,which
issimilar insettingtoPartI.Itstarts fromananalysis ofcontrollability and
observability and then discusses in great detail stability and stabilizability.
It also presents typical theorems on nonlinear realizations.
Part III concentrates on the question of how to find optimal controls. It
discusses Bellman’s optimality principle and its typical applications to the
linear regulator problem and to impulse control. It gives a proof of Pontrya-
gin’smaximumprincipleforclassicalproblemswithfixedcontrolintervalsas
well as for time-optimal and impulse control problems. Existence problems
vii
viii Prefacetothefirstedition
are considered in the final chapters, which also contain the basic Filippov
theorem.
Part IV is devoted to infinite dimensional systems. The course is limited
to linear systems and to the so-called semigroup approach. The first chapter
treatslinearsystemswithoutcontrolandis,inasense,aconcisepresentation
of the theory of semigroups of linear operators. The following two chapters
concentrate on controllability, stability and stabilizability of linear systems
and the final one on the linear regulator problem in Hilbert spaces.
Besidesclassicaltopicsthebookalsodiscusseslesstraditionalones.Inpar-
ticular great attention is paid to realization theory and to geometrical meth-
ods of analysis of controllability, observability and stabilizability of linear
and nonlinear systems. One can find here recent results on positive, impul-
sive and infinite dimensional systems. To preserve some uniformity of style
discrete systems as well as stochastic ones have not been included. This was
a conscious compromise. Each would be worthy of a separate book.
Controltheoryistodayaseparatebranchofmathematics,andeachofthe
topics covered in this book has an extensive literature. Therefore the book is
only an introduction to control theory.
Knowledge of basic facts from linear algebra, differential equations and
calculus is required. Only the final part of the book assumes familiarity with
more advanced mathematics.
Several unclear passages and mistakes have been removed due to remarks
of Professor L. Mikołajczyk and Professor W. Szlenk. The presentation of
therealizationtheoryowesmuchtodiscussionswithProfessorB.Jakubczyk.
I thank them very much for their help.
Finallysomecommentsaboutthearrangementofthematerial.Successive
numberofparagraph,theorem,lemma,formula,example,exerciseispreceded
by the number of the chapter. When referring to a paragraph from some
other parts of the book a Latin number of the part is added. Numbers of
paragraphs,formulaeandexamplesfromIntroductionareprecededby0and
those from Appendix by letter A.
Warsaw and Coventry Jerzy Zabczyk
1992
PREFACE
TO THE SECOND EDITION
Asthefirstedition,thesecondoneisdividedintofourparts:PartI,Elements
of the classical control theory; Part II, Nonlinear control systems; Part III,
Optimal control; Part IV, Infinite-dimensional linear systems.
The new elements of the book can be described as follows.
IntoPartIanewchapteroncontrollabilitywithvanishingenergyisadded.
TheconceptwasintroducedbyE.Priolaandtheauthor.Applicationstothe
orbital transfer problem are elaborated. A short proof of the Routh stability
criteria is presented. Part II is essentially as before. New chapter Viscos-
ity solutions of Bellman’s equation is added to Part III. It includes detailed
proofs of the existence and uniqueness theorems. Boundary control systems
systemsarediscussedinanewchapterinPartIV.Approximatecontrollabil-
ity of one-dimensional heating system is established and null controllability
ofmultidimensionalheatingsystemispresented.Inadditiondelayedsystems
areinvestigated insomedetail. Inparticular theirstability, semigroup repre-
sentation and the linear regulator are covered. Improved stability results for
general hyperbolic systems are presented.
Theindexwassubstantiallyenlarged.Severalmisprintsandmistakeswere
corrected.SomeofthemwerenoticedbyDr.J.Kowalski,whoalsotypedthe
new material. I thank him for his help.
I thank Professors B. Gołdys, A. Święch and R. Triggiani for comments
on some parts of the book. I thank the reviewers of the second edition for
constructive suggestions. Writing of the second edition took more time than
expected and I thank the editors for their patience.
Warsaw Jerzy Zabczyk
2020
ix
CONTENTS
Preface to the first edition.................................... vii
Preface to the second edition ................................. ix
Introduction.................................................. xvii
§0.1. Problems of mathematical control theory ...... xvii
§0.2. Specific models............................. xix
Bibliographical notes ............................... xxvi
Part I. Elements of Classical Control Theory
Chapter 1. Controllability and observability................. 3
§1.1. Linear differential equations ................. 3
§1.2. The controllability matrix ................... 6
§1.3. Rank condition ............................ 10
§1.4. A classification of control systems ............ 14
§1.5. Kalman decomposition...................... 16
§1.6. Observability .............................. 18
Bibliographical notes ............................... 20
Chapter 2. Stability and stabilizability ...................... 21
§2.1. Stable linear systems ....................... 21
§2.2. Stable polynomials ......................... 25
§2.3. The Routh theorem......................... 27
§2.4. Stability, observability and the Lyapunov
equation .................................. 31
§2.5. Stabilizability and controllability ............. 34
§2.6. Hautus lemma ............................. 37
§2.7. Detectability and dynamical observers ........ 39
Bibliographical notes ............................... 41
xi