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Optimal control theory with applications in economics PDF

373 Pages·2011·2.187 MB·English
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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

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