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1、Availableonlineatwww.sciencedirect.comScienceDirectFuzzySetsandSystems265(2015)121126www.elsevier.com/locate/fssShortcommunicationAnoteonAclassoflineardifferentialdynamicalsystemswithfuzzyinitialconditionMehranMazandarania,∗,bMarziehNajariyanaDepartmentofElectricalEngineering,FerdowsiUniversityofMas
2、hhad,Mashhad,IranbDepartmentofAppliedMathematics,FerdowsiUniversityofMashhad,Mashhad,IranReceived27February2014;accepted27May2014Availableonline2June2014AbstractInthisnote,weshowthattheproposedapproachinJ.Xuetal.(2007)[1]failstoobtainthestablesolutionstothestablelineardynamicalsystemswithfuzzyinitia
3、lconditions.Asitwillbeexplained,thisshortcomingiscausedbyneglectingtomovethestabilitypropertyofthefuzzysystemtothequasi-level-wisesystem.Moreover,tohandlethis,anotherapproachisproposedsimilartothepreviousoneforthestableandunstablesystems.©2014ElsevierB.V.Allrightsreserved.Keywords:Fuzzydifferentiale
4、quations;Stabilityproperty;Quasi-level-wisesystem1.IntroductionandbasicconceptsThroughoutthisnote,thesetofallrealnumbersisdenotedbyR,andthesetofalltype-1fuzzynumbersonRbyE1.Theleftandrightend-pointsoftheα-levelsetsofA˜,[A˜]α,aredenotedbyAαandA¯αrespectively.J.Xuetal.in[1]proposedanapproachtosolveacl
5、assoffuzzydifferentialequations[FDEs]system.Theapproachisbasedonacomplexnumberrepresentationoftheα-levelsetsofthefuzzysystem.Letthiskindofrepresentationbecalledquasi-level-wisesystem.ThefollowingclassofFDEssystemhasbeenconsideredin[1]˙˜X(t)=AX(t)˜+f(t),˜X(˜0)=X˜0(1.1)˙˜dX(t)˜nwhereX(t)=,X˜:[0,tf]→Ei
6、safuzzyfunctionontherealinterval[0,tf],A=[aij]n×nisamatrix,aij∈R,dt1f˜:[0,tf]→Enisafuzzyfunction,andX˜0istheinitialvalueofthesystem.Usingtheapproachasin[1],theauthors1haveobtainedanunstablesolutiontoastablesystem,seeExample4.6in[1].Inthisnote,wearegoingtoshowhowthisshortcomingstemsfromneglectingtomo
7、vethestabilitypropertyofthefuzzysystemtothequasi-level-wise*Correspondingauthor.E-mailaddresses:me.mazandarani@gmail.com,me.mazandarani@ieee.org(M.Mazandarani).http://dx.doi.org/10.1016/j.fss.2014.05.
