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1、Chapter1WhatisOptimalControlTheory?Manymanagementscienceapplicationsinvolvethecontrolofdynamicsystems,i.e.,systemsthatevolveovertime.Theyarecalledcontinuous-timesystemsordiscrete-timesystemsdependingonwhethertimevariescontinuouslyordiscretely.Weshalldealwithbothkindsofsystemsinthisbook,altho
2、ughthemainemphasiswillbeoncontinuous-timesystems.Optimalcontroltheoryisabranchofmathematicsdevelopedtofindoptimalwaystocontroladynamicsystem.Thepurposeofthisbookistogiveanelementaryintroductiontomathematicaltheoryandthentoapplyittoawidevarietyofdifferentsituationsarisinginmanagementscience.W
3、ehavedeliberatelykeptthelevelofmathematicsassimpleaspossibleinordertomakethebookaccessibletoalargeaudience.Theonlymathematicalrequirementsforthisbookareelementarycalculus,includingpartialdifferentiation,someknowledgeofvectorsandmatrices,andelementaryordinaryandpartialdifferentialequations.Mo
4、reover,thelasttopicisbrieflycoveredinAppendixA.AnexceptionisChapter13onstochasticoptimalcontrol,whichalsorequiressomeconceptsinstochasticcalculus.Thoseconceptswillbeintroducedatthebeginningofthatchapter.Theprinciplemanagementscienceapplicationsdiscussedinthisbookareinthefollowingareas:financ
5、e,economics,productionandinventory,marketing,maintenanceandreplacement,andtheconsumptionofnatu?ralresources.Ineachmajorareawehaveformulatedoneormoresimplemodelsfollowedbyamorecomplicatedmodel.Thereadermaywishat21.WhatisOptimalControlTheory?firstreadingtocoveronlythesimplermodelsineachareatog
6、etanover?allideaofwhatispossibletodowithoptimalcontroltheory.Laterthereadermaywishtogointomoredepthintooneormoreoftheappliedareas.Worked-outexamplesareprovidedinmostchapterstofacilitatetheexposition.Attheendofeachchapter,wehavelistedexercisesthatthereadershouldsolvefordeepermiderstandingofth
7、ematerialpresentedinthechapter.Hintsaresuppliedwithsomeoftheexercises.Furthermore,difficultexercisesareindicatedwithanasterisk(*).1.1BasicConceptsandDefinitionsWeshallusethewordsystemasaprimitiveterminthisbook.Theonlypropertythatwerequireofasystemi
