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1、Chapter13StochasticOptimalControlInpreviouschaptersweassumedthatthestatevariablesofthesystemwereknownwithcertainty.Ifthiswerenotthecase,thestateofthesystemovertimewouldbeastochasticprocess.Inaddition,itmightnotbepossibletomeasurethevalueofthestatevariablesattimet.Inthiscase
2、,onewouldhavetomeasurefunctionsofthestatevariables.Moreover,themeasurementsareusuallynoisy,i.e.,theyaresubjecttoerrors.Thus,adecisionmakerisfacedwiththeproblemofmakinggoodestimatesofthesestatevariablesfromnoisymeasurementsonfunctionsofthem.Theprocessofestimatingthevaluesoft
3、hestatevariablesiscalledop•timalfiltering,InSection13.1,wewilldiscussoneparticularfilter,calledtheKalmanfilteranditscontinuous-timeanaloguecaUedtheKalman-Bucyfilter.Itshouldbenotedthatwhileoptimalfilteringprovidesoptimalestimatesofthevalueofthestatevariablesfromnoisymeasure
4、•mentsofrelatedquantities,nocontrolisinvolved.Whenacontrolisinvolved,wearefacedwithastochasticoptimalcontrolproblem.Here,thestateofthesystemisrepresentedbyacon•trolledstochasticprocess.InSection13.2,weshallformulateastochas•ticoptimalcontrolproblemwhichisgovernedbystochasti
5、cdifferentialequations.WeshallonlyconsiderstochasticdifferentialequationsofatypeknownasItoequations.Theseequationsarisewhenthestateequartions,suchasthosewehaveseeninthepreviouschapters,areperturbedbyMarkovdiffusionprocesses.OurgoalinSection13.2willbetosyn•thesizeoptimalfeed
6、backcontrolsforsystemssubjecttoItoequationsinawaythatmaximizestheexpectedvalueofagivenobjectivefunction.InSection13.3,weshallextendtheproductionplanningmodelof34013.StochasticOptimalControlChapter6toallowforsomeuncertaindisturbances.Weshallobtainanoptimalproductionpolicyfor
7、thestochasticproductionplanningprob•lemthusformulated.InSection13.4,wesolveanoptimalstochasticadvertisingproblemexplicitly.TheproblemisamodificationaswellasastochasticextensionoftheoptimalcontrolproblemoftheVidale-WolfeadvertisingmodeltreatedinSection7.2.4.InSection13.5,wew
8、iUintroduceinvestmentdecisionsinthecon•sumptionmodelofExample1.3.Wewillconsiderbot
