Optimal control theory with applications in economics

373 Pages·2011·2.187 MB·English

by  Thomas A. Weber, A.V. Kryazhimskiy

OptimalControlTheorywithApplicationsinEconomics

OptimalControlTheorywithApplicationsinEconomics
ThomasA.Weber
ForewordbyA.V.Kryazhimskiy
TheMITPress
Cambridge,Massachusetts
London,England
©2011MassachusettsInstituteofTechnology
Allrightsreserved. Nopartofthisbookmaybereproducedinanyformbyanyelec-
tronicormechanicalmeans(includingphotocopying,recording,orinformationstorage
andretrieval)withoutpermissioninwritingfromthepublisher.
Forinformationaboutspecialquantitydiscounts,pleaseemail
[email protected].
ThisbookwassetinPalatinobyWestchesterBookComposition.Printedandboundin
theUnitedStatesofAmerica.
LibraryofCongressCataloging-in-PublicationData
Weber,ThomasA.,1969–
Optimalcontroltheorywithapplicationsineconomics/ThomasA.Weber;forewordby
A.V.Kryazhimskiy.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-262-01573-8(hardcover:alk.paper)
1.Economics—Mathematicalmodels. 2.Controltheory. 3.Mathematicaloptimization.
4.Gametheory. I.Title.
HB135.W433 2011
(cid:2)
330.01515642—dc22 2010046482
10 9 8 7 6 5 4 3 2 1
ForWim

Contents
ForewordbyA.V.Kryazhimskiy ix
Acknowledgments xi
1 Introduction 1
1.1 Outline 3
1.2 Prerequisites 5
1.3 ABriefHistoryofOptimalControl 5
1.4 Notes 15
2 OrdinaryDifferentialEquations 17
2.1 Overview 17
2.2 First-OrderODEs 20
2.3 Higher-OrderODEsandSolutionTechniques 67
2.4 Notes 75
2.5 Exercises 76
3 OptimalControlTheory 81
3.1 Overview 81
3.2 ControlSystems 83
3.3 OptimalControl—AMotivatingExample 88
3.4 Finite-HorizonOptimalControl 103
3.5 Infinite-HorizonOptimalControl 113
3.6 Supplement1:AProofofthePontryagin
MaximumPrinciple 119
3.7 Supplement2:TheFilippovExistenceTheorem 135
3.8 Notes 140
3.9 Exercises 141
viii Contents
4 GameTheory 149
4.1 Overview 149
4.2 FundamentalConcepts 155
4.3 DifferentialGames 188
4.4 Notes 202
4.5 Exercises 203
5 MechanismDesign 207
5.1 Motivation 207
5.2 AModelwithTwoTypes 208
5.3 TheScreeningProblem 215
5.4 NonlinearPricing 220
5.5 Notes 226
5.6 Exercises 227
AppendixA:MathematicalReview 231
A.1 Algebra 231
A.2 NormedVectorSpaces 233
A.3 Analysis 240
A.4 Optimization 246
A.5 Notes 251
AppendixB:SolutionstoExercises 253
B.1 NumericalMethods 253
B.2 OrdinaryDifferentialEquations 258
B.3 OptimalControlTheory 271
B.4 GameTheory 302
B.5 MechanismDesign 324
AppendixC:IntellectualHeritage 333
References 335
Index 349
Foreword
Sincethediscovery,byL.S.Pontryagin,ofthenecessaryoptimalitycon-
ditions for the control of dynamic systems in the 1950s, mathematical
controltheoryhasfoundnumerousapplicationsinengineeringandin
thesocialsciences.T.A.Weberhasdedicatedhisbooktooptimalcon-
trol theory and its applications in economics. Readers can find here a
succinct introduction to the basic control-theoretic methods, and also
clearandmeaningfulexamplesillustratingthetheory.
Remarkable features of this text are rigor, scope, and brevity, com-
bined with a well-structured hierarchical approach. The author starts
withageneralviewondynamicalsystemsfromtheperspectiveofthe
theoryofordinarydifferentialequations; onthisbasis, heproceedsto
the classical optimal control theory, and he concludes the book with
more recent views of game theory and mechanism design, in which
optimalcontrolplaysaninstrumentalrole.
The treatment is largely self-contained and compact; it amounts to
a lucid overview, featuring much of the author’s own research. The
characteroftheproblemsdiscussedinthebookpromisestomakethe
theoryaccessibletoawideaudience.Theexercisesplacedatthechapter
endingsarelargelyoriginal.
I am confident that readers will appreciate the author’s style and
studentswillfindthisbookahelpfulguideontheirpathofdiscovery.
A.V.Kryazhimskiy
SteklovInstituteofMathematics,RussianAcademyofSciences
InternationalInstituteforAppliedSystemsAnalysis