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1、oBol.Soc.Esp.Mat.Apl.n0(0000),1{62ControlTheory:History,MathematicalAchievementsandPerspectives*E.Fernandez-Cara?1andE.Zuazua21DepartamentodeEcuacionesDiferencialesyAn?alisisNum?erico,UniversidaddeSevilla2DepartamentodeMatem?aticas,UniversidadAut?onomadeMadridcara@
2、us.es,enrique.zuazua@uam.esAbstractThesenotesaredevotedtopresentsomeofthemathematicalmilestonesofControlTheory.Todothat,weˉrstoverviewitsoriginsandsomeofthemainmathematicalachievements.Then,wediscussthemaindomainsofSciencesandTechnologieswhereControlTheoryarisesand
3、applies.Thisforcesustoaddressmodellingissuesandtodistinguishbetweenthetwomaincontroltheoreticalapproaches,controllabilityandoptimalcontrol,discussingtheadvantagesanddrawbacksofeachofthem.Inordertogiveadequateformulationsofthemainquestions,wehaveintroducedsomeofthem
4、ostelementarymathematicalmaterial,avoidingunnecessarytechnicaldi±cultiesandtryingtomakethepaperaccessibletoalargeclassofreaders.Thesubjectsweaddressrangefromthebasicconceptsrelatedtothedynamicalsystemsapproachto(linearandnonlinear)MathematicalProgrammingandCalculus
5、ofVariations.WealsopresentasimpliˉedversionoftheoutstandingresultsbyKalmanonthecontrollabilityoflinearˉnitedimensionaldynamicalsystems,Pontryaguin'smaximumprincipleandtheprincipleofdynamicalprogramming.Someaspectsrelatedtothecomplexityofmoderncontrolsystems,thedisc
6、reteversuscontinuousmodelling,thenumericalapproximationofcontrolproblemsanditscontroltheoreticalconsequencesarealsodiscussed.Finally,wedescribesomeofthemajorchallengingapplicationsinControlTheoryfortheXXICentury.Theywillprobablyin°uencestronglythedevelopmentofthisd
7、isciplineinthenearfuture.*TheauthorshavebeenpartiallyˉnancedbyD.G.E.S.-Espa~na,respectivelyGrantsBFM2000-1317andBFM2002-03345.12E.Fernandez-CaraandE.Zuazua?Keywords:ControlTheory,optimalcontrol,controllability,di?erentialequations,feedback,Optimization,CalculusofVa
8、riations.AMSsubjectclassiˉcations:49J15,49J20,70Q05,93B,93C1.IntroductionThisarticleisdevotedtopresentsomeofthemathematicalmilestonesofControlThe
