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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 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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 introduction. 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 separate 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