Geometric Optimal Control: Theory, Methods and Examples PDF
Preview Geometric Optimal Control: Theory, Methods and Examples
Geometric Optimal Control Interdisciplinary Applied Mathematics Editors S.S.Antman P.Holmes K.Sreenivasan SeriesAdvisors L.Glass P.S.Krishnaprasad R.V.Kohn J.D.Murray S.S.Sastry Problems in engineering, computational science, and the physical and biological sciences are using increasingly sophisticated mathematical techniques. Thus, the bridgebetweenthemathematicalsciencesandotherdisciplinesisheavilytraveled. The correspondingly increased dialog between the disciplines has led to the establishmentoftheseries:InterdisciplinaryAppliedMathematics. Thepurposeofthisseriesistomeetthecurrentandfutureneedsfortheinteraction betweenvariousscienceandtechnologyareasontheonehandandmathematicson the other. This is done, firstly, by encouragingthe ways that mathematicsmay be applied in traditional areas, as well as point towards new and innovative areas of applications;and,secondly,byencouragingotherscientificdisciplinestoengagein adialogwithmathematiciansoutliningtheirproblemstobothaccessnewmethods andsuggestinnovativedevelopmentswithinmathematicsitself. Theserieswillconsistofmonographsandhigh-leveltextsfromresearchersworking ontheinterplaybetweenmathematicsandotherfieldsofscienceandtechnology. Forfurthervolumes: http://www.springer.com/series/1390 Heinz Scha¨ttler • Urszula Ledzewicz Geometric Optimal Control Theory, Methods and Examples 123 HeinzScha¨ttler UrszulaLedzewicz WashingtonUniversity SouthernIllinoisUniversity St.Louis,MO, 63130 EdwardsvilleIL,62026 USA USA SeriesEditors: S.S.Antman P.Holmes DepartmentofMathematics DepartmentofMechanicalandAerospace and Engineering InstituteforPhysicalScience PrincetonUniversity andTechnology 215FineHall UniversityofMaryland Princeton,NJ,08544 CollegePark,MD,20742 USA USA [email protected] [email protected] K.Sreenivasan DepartmentofPhysics NewYorkUniversity 70WashingtonSquareSouth NewYorkCity,NY,10012 USA [email protected] ISSN0939-6047 ISBN978-1-4614-3833-5 ISBN978-1-4614-3834-2(eBook) DOI10.1007/978-1-4614-3834-2 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2012939056 MathematicsSubjectClassification(2010):49K15,49L20,34H05,93C15 ©SpringerScience+BusinessMedia,LLC2012 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To ourparents, Elfriedeand Oswald,Teresa andJerzy, andoursons, FilipandMax Preface Optimal control theory is a discipline that has its historical origin in the calculus ofvariations,datingbacktotheformulationofJohannBernoulli’sbrachistochrone problemmorethan300yearsago.Itdevelopedintoafieldofitsowninthe1960sin connectionwiththedevelopmentofspaceexploration.Infact,itwastheengineering problemoflaunchingasatelliteintoasustainedorbit—Sputnik—thatgeneratedthe activitiesintheSovietUnionthatledtotheearlydevelopmentsofthetheory.This theory provides techniques to analyze far-reaching problems in various fields of science,engineering,economics,andmorerecentlyalsobiomedicine.Generally,in these problems the underlying task is to transfer the state of a dynamical system fromagiveninitialpositionintoadesiredterminalcondition,e.g.,deployasatellite into a prescribed orbit; guide a spacecraft to some remote planet, possibly even making a soft landing (Mars rover); perform tasks using robotic manipulators; achieve positions of wealth in economic endeavors through investment decisions; eliminate,ifpossible,cancercellsfromourbodyorinfectedcellsfromabiological host;andsoon,withthenumberofrealisticexampleslimitless.Alltheseproblems havein commonthatthe dynamicsof thesystem can be influenced(“controlled”) bymeansofsomeexternalvariable,e.g.,thefuelburnedinsidearockettogenerate thrust,theallocationofeconomicresourcesbetweenconsumptionandinvestment, the amounts of therapeutic agents given to treat cancer. Naturally, there always exist practical constraints that are imposed by a particular situation—the amount of fuel that a rocket can carry is limited, drugs cannot be given without careful considerationoftheirsideeffects,andsoon.Still,andwithinthephysicalandother limitsimposedbyaparticularsituation,generallythereexiststremendousfreedom in the choice of the controls over time to achieve a desired objective. This leads to optimization problems. Sometimes, problems are naturally associated with an objective function to be minimized or maximized; in other instances, there is no suchchoice,andimposingacriterionmaysimplybeameanstogenerateprocedures (i.e.,throughnecessaryconditionsforoptimality)thatallowonetocomeupwitha reasonablesolutiontotheunderlyingproblem.Theproblemoftransferringthestate ofa dynamicalsystemfroma giveninitialconditionintoa setofdesiredterminal conditions,while atthe same time minimizingsome objectiveassociated with the vii viii Preface motion, and possibly a penalty on the terminal state, thus is a most natural one. These belongto the generaltypeof problemsthatare analyzedwith the toolsand techniquesofoptimalcontroltheory. More precisely, the kinds of problems that will be considered in this text are finite-dimensionaldeterministicoptimalcontrolproblems.Thesearecharacterized bythefactthatthetimeevolutionoftheunderlyingdynamicalsystemisdescribed bysolutionstoordinarydifferentialequations(“finite-dimensional”)asopposedto partialdifferentialequations(“infinite-dimensional”),andstochasticeffectsrelated to noise and other modeling perturbationsassociated with random effects are not includedin themodeling(“deterministic”).Clearly,these are importantaspectsas well.However,methodsthatdealwiththese structuresareofa verydifferenttype andarewell representedin the literature.The problemswe areanalyzinghereare among the most classical ones in mathematics and physics and have their origin in the calculus of variations. In fact, a calculus of variation problem simply is a specialoptimalcontrolproblemwithatrivialdynamics(givenbythederivativeof the curve) and no constraints imposed on the control. While such constraints are importantinpracticalproblems,thesedonot,fromourpointofview,constitutethe maindifferencebetweenproblemsinthecalculusofvariationsandoptimalcontrol problems.Rather,itisthepresenceofanontrivialandtypicallynonlineardynamics that connects the controls and states. For this reason, optimal control problems becomemuchmoredifficultthanmereextensionsofoptimizationproblemsfroma finite-toaninfinite-dimensionalsetting.Whiletherehasbeentremendousprogress in numericalmethodsin optimalcontroloverthe pastfifteen yearsthat has led to the solutions of some specific and very difficult problems—the design of optimal controlsbyNASAforthepositioningoftheinternationalspacestationusinggyros with pseudospectral techniques or the experimental design of highly complicated pulse sequences in nuclear magnetic resonance (NMR) spectroscopy, to mention justtwooftheoutstandingachievements—therestilldonotexistreliablenumerical proceduresthatcouldsimplybeappliedtoanyoptimalcontrolproblemandgivethe solution. Specific methods, such as pseudospectral techniques, shooting methods, andarcparameterizationtechniques,havetheirstrengthsandshortcomings,simply becausethereexistsfartoogreatavarietyinthedynamics.Nonlinearsystemsdefy simpleclassifications,andfromthepracticalside,problemsoftenhavetobesolved onacase-by-casebasis. Yet, there does exist a common framework that can be used to tackle these problems, and it is this framework that we describe in our text. We give a comprehensivetreatmentofthefundamentalnecessaryandsufficientconditionsfor optimalityandillustratehowthesecanbeusedto solveoptimalcontrolproblems. Ouremphasisisonthegeometricaspectsofthetheory,andinthiscontext,wealso provide tools and techniques that go well beyond standard conditions (including a comprehensivetreatmentof envelopesand singularitiesin the flow of extremals as well as a Lie-algebra-based framework for explicit computations in canonical coordinates)andcanbeusedtoobtaina fullunderstandingoftheglobalstructure ofsolutionsfortheunderlyingproblem,notjustanisolatednumericalcomputation for specific parameter values. We include a palette of examples that are worked Preface ix out in detail and range from classical to novel and from elementary to the highly nontrivial. All these examples, in one way or another, illustrate the power of geometrictechniquesandmethods. The text is quite versatile and contains material on different levels ranging from the introductory and elementary to the quite advanced. In this sense, some parts of our text can be viewed as a comprehensive textbook for both advanced undergraduate and all levels of graduate courses on optimal control in both mathematics and engineering departments. In fact, this text grew out of lecture notesoftheauthorsforcoursestaughtattheDepartmentsofSystemsScienceand MathematicsandElectricalandSystemsEngineeringintheSchoolofEngineering andAppliedScienceatWashingtonUniversityinSt.LouisandtheDepartmentof MathematicsandStatisticsandvariousengineeringdepartmentsatSouthernIllinois University Edwardsville. The variety of fully solved examples that illustrate the theory,rarelypresenttothisextentinmoreadvancedtextbooksandmonographsin thisfield,makesthistextastrongeducationalasset.Thetextmovessmoothlyfrom themoreintroductorytopicstothosepartsthatareinamonographstylewheremore advancedtopicsarepresented.Whilethispresentationismathematicallyrigorous, itiscarriedoutinatutorialstylethatmakesthetextaccessibletoawideaudience ofresearchersandstudentsfromvariousfields, notjustthe mathematicalsciences and engineering. In a sequel, in which applications of geometric optimal control to biomedical problems will be analyzed, the tools and techniques developed in thistextwillbeusedtosolvevariousoptimalcontrolproblemsthatariseincancer treatmentsthatrangefromclassicalproceduressuchaschemo-andradiotherapyto novelapproachesthatincludeantiangiogenicagentsandimmunotherapy. Wearegreatlyindebtedtonotonlyourteacherswhohaveinfluencedourviews onthesubject,especiallytoourdoctoraladvisors,HectorSussmannandStanislaw Walczak, but also to H.W. Knobloch,who introducedthe first author to the fields ofdifferentialequationsandoptimalcontrol.Thisbookwouldnothavecomeinto existencewithouttheguidinginfluenceandpassionforthesubjectinstilledinusby ourmentorandgoodfriendHectorSussmann,whointroducedustothebeautyof thegeometricapproachtooptimalcontrol.Manyprofessionalcolleagueshavebeen instrumental in our academic careers, and we would like to take the opportunity to thank some of them, in particular E. Roxin, A. Nowakowski, A. Krener, V. Lakshmikantham,andT.J.Tarn.Especiallywewouldliketoacknowledgethelate J. Zaborszky, a true engineer who appreciated mathematics, but always insisted on a practical connection. Thanks are also due to all our students who at one stage or another have contributed to the writing of this text. We also would like tothankouruniversities,WashingtonUniversityinSt.LouisandSouthernIllinois UniversityEdwardsville,andtheNationalScienceFoundation,whichhassupported our research at various stages for by now over 20 years. Finally, we would like to thank David Kramer, who so carefully read our text, and all the editors at Springer,especiallyAchiDosanjhandDonnaChernyk,whohavebeenveryhelpful throughouttheentireproductionprocess. Edwardsville,Illinois,USA HeinzScha¨ttler UrszulaLedzewicz Outline of the Chapters of the Text Below we givea briefoutlineof thechaptersthatcan serve asa roadmap forthe scientificjourneythroughourtext. Chapter 1 introduces the fundamental results of the calculus of variations organized around complete solutions of two cornerstone classical examples: the brachistochrone problem and the problem of surfaces of revolution of minimum area.Theideasandconceptspresentedinthischapterservebothasanintroduction toandasamotivationforthecorrespondingnotionsinoptimalcontroltheorytobe discussedinsubsequentchapters. The Pontryagin maximum principle, which gives the fundamental necessary conditionsforoptimalityinoptimalcontrolproblems,willbeintroducedinChap.2 withthefocusonillustratinghowthisresultcanbeusedtosolveproblems.Tothis end, we introduce important Lie-derivative-based techniques that form the basis for geometric optimal control and use them to give a detailed derivation of H. Sussmann’sresultsonthestructureoftime-optimalcontrolsfornonlinearcontrol- affine systems in the plane. These results serve as a first illustration of the power ofgeometricmethodsthatgowellbeyondtheconditionsofthemaximumprinciple andleadtodeepresultsaboutthestructureofoptimalsolutions. Whiletheemphasisofourtextisonmethodsfornonlinearsystems,inChaps.2 and3wealsogivesomeoftheclassicalresultsaboutlineartime-invariantsystems. Theyincludeaproofoftheconvexityofthereachablesetsandtwoformulationsof thecelebratedbang-bangtheorem. In Chap.4 we then prove the Pontryagin maximum principle. Necessary con- ditions for optimality follow from separation results about convex cones that approximatethe reachableset andthe set of pointswherethe objectivedecreases, respectively. These constructions equally apply to the classical needle variations used by Pontryagin et al. and to high-order variations. Specific variations will be madetoprovetheLegendre–Clebschcondition,theKelleycondition,andtheGoh condition for optimality of singular controls. For this, an adequate computational framework is needed that is provided by exponential representations of solutions to differentialequations and the associated Lie-algebraic formalism related to the Baker–Campbell–Hausdorffformula. xi xii OutlineoftheChaptersoftheText Chapters 5 and 6 then deal with sufficient conditionsfor optimality, both local andglobal.InChap.5weintroduceparameterizedfamiliesofextremals,i.e.,collec- tionsofcontrolledtrajectoriesthatsatisfytheconditionsofthemaximumprinciple. Throughoutthetext,weemphasizetheroletheyplayintheconstructionofsolutions totheHamilton–Jacobi–Bellmanequation,afirst-orderpartialdifferentialequation coupledwiththesolutionofaminimizationproblemforthecontrolsthatdescribe theminimumvalueoftheoptimalcontrolproblemasafunctionoftheinitialdata. Weadaptthemethodofcharacteristics,aclassicalsolutionprocedureforfirst-order partialdifferentialequations,totheoptimalcontrolsettinganduseittoconstructthe value function associated with a parameterized family of extremals. For example, in this way we give an elementaryproofof the optimality of the synthesisfor the Fullerproblemforwhichoptimalsolutionsconsistofchatteringarcswhosecontrols switchinfinitelyoftenonfiniteintervals.Thesegeometricconstructionsprovidethe generalizationoftheconceptofafieldofextremalsfromthecalculusofvariationsto optimalcontroltheoryandclearlybringouttherelationshipsbetweenthenecessary conditionsof the maximumprincipleandthe sufficientconditionsof the dynamic programmingprinciple. WhiletheresultsinChap.5haveamostlylocalcharacterandarealldeveloped in the context of continuous controls, in Chap.6 we extend the constructions to broken extremals that are finite concatenations of bang and singular controls. Geometric transversality and matching conditions will be developed that allow us to investigate the optimality of the flow of extremals as various patches are combined. The main result of this chapter is a verification theorem due to H. Sussmann that implies the optimality of a synthesis of controlled trajectories if the associated value function satisfies some weak continuity properties and is a continuously differentiable solution of the Hamilton–Jacobi–Bellman equation awayfromalocallyfiniteunionofembeddedsubmanifoldsofpositivecodimension. TheresultsthatwillbedevelopedinChaps.5and6preciselyleadtothesepiecewise differentiabilityproperties.Itisnotrequiredthatthevaluefunctionbecontinuous. Chapter7concludesourtextwithillustratinghowthesetechniquescanalsobe used in low dimensions to determine small-time reachable sets exactly. This also providesan alternativegeometric viewpointto the results on time-optimalcontrol fornonlinearsystemsintheplanethatwerederivedinChap.2.Thematerialinthis chapterhasneverbeenpresentedbeforeinbookform.Bitsandpiecesareavailable intheresearchliterature,andheretheseapproachesareunified,andforthefirsttime anaccessibleaccountofthissubjectisgiven. Throughoutourpresentation,thetextisasmuchself-containedaspossible,and wedoincludemoretechnicalanddifficultcomputationsiftheyarerequiredinthe proofsortogivecompletesolutionsforsomeoftheexamples.Atvariousstages,we revisitthesametopicfromdifferentangles,andbelowisashortroadmaptosome ofthesetopics: • Linear-QuadraticRegulatorandPerturbationFeedbackControl:Sects.2.1,2.4, and5.3 • Time-OptimalControlforLinearSystems:Sects.2.5and2.6andChap.3
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