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1、DifferentialGameswithImpulseControlArkadiiA.ChikriiGlushkovInstituteofCyberneticsNASUkraine40GlushkovPrsp.,Kiev03187,Ukrainedept165@insyg.kiev.uaIvanI.MatychynGlushkovInstituteofCyberneticsNASUkraine40GlushkovPrsp.,Kiev03187,UkraineKirillA.ChikriiGlushkovInstituteofCyberneticsNASUkraine4
2、0GlushkovPrsp.,Kiev03187,UkraineAbstractThischapterdealswiththepursuitgamesinwhichplayers(pursuer,evader,orboth)employimpulsecontrol.Weconsidercontinuous-timedynamicalsys-temsmodeledbyordinarydifferentialequationsthatareaffectedbyjumpsinstateatdiscreteinstants.Themomentsofjumpcomplywitht
3、heconditionforafinitenumberofjumpsinfinitetime.Insodoing,theDiracdeltafunctionisusedtodescribetheimpulsecontrol.Suchsystemsrepresentaspecialcaseofhybridsystems.Themethodofresolvingfunctionsprovidesageneralframe-workforanalysisoftheabove-mentionedproblems.Thismethodessentiallyusesthetechniq
4、ueofthetheoryofset-valuedmappings.Thefollowingcasesareexaminedinsuccession:impulsecontrolofthepursuer;impulsecontroloftheevader;impulsecontrolofbothplayers.Theproblemofapproachingacylindricalterminalsetisstudiedforeachcase,andthesufficientconditionsforitssolvabilityarederived.Obtainedresu
5、ltsaresupportedbyamodelexampleofthegamewithsimplemotiondynamics.1IntroductionInthischapteraclassoflinearpursuit-evasiondifferentialgamesisstudied,whereoneorbothoftheplayersemployimpulsecontrol.TheimpulsecharacterofcontrolissymbolicallyconveyedbyusingtheDiracdeltafunction.Quiteanextensive
6、literaturecoverstheimpulsedynamicgames.Apparentlythepapers[1,2]wereamongthefirst.Krasovskii’smethodsareusedintheworksofthereseachersfromtheUralschool(see,e.g.,[3]).Inthepapers[4,5]thenotionofbackwardprocedureisemployed.Therecentpapers[6,7]treatimpulsediffer-entialgamesintheframeworkofthev
7、iabilityofcontrolsystems.Inthesepapers38A.A.Chikrii,I.I.Matychyn,andK.A.Chikriithemethodsdevelopedforcontinuousdynamicsystemsareextendedtothecaseofimpulsesystems.Thischapterisnoexception.Theapproachpresentedhereisbasedonthemainideasofthemethodofresolvingfunctions[8].Itisa
