A criterion for stability of matrices

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1、JOURNALOFMATHEMATICALANALYSISANDAPPLICATIONS225,249]2641998.ARTICLENO.AY986020ACriterionforStabilityofMatricesMichaelY.Li*andLianchengWangDepartmentofMathematicsandStatistics,MississippiStateUni¨ersity,MississippiState,Mississippi39762SubmittedbyHalL.SmithReceivedOctober23,1997Anecessaryandsu

2、fficientconditionforthestabilityofn=nmatriceswithrealentriesisproved.Applicationstoasymptoticstabilityofequilibriaforvectorfieldsareconsidered.Theresultsofferanalternativetothewell-knownRouth]Hurwitzconditions.Q1998AcademicPressKeyWords:Matrix;stability;Routh]Hurwitzconditions;Lozinskiõmeasure

3、s;Ïcompoundmatrices1.INTRODUCTIONLetAbeann=nmatrixandlets.Abeitsspectrum.ThestabilitymodulusofAissA.smaxRel:lgs.A4,andAissaidtobestableifsA.-0.ThestabilityofamatrixisrelatedtotheRouth]Hurwitzproblemonthenumberofzerosofapolynomialthathavenegativerealparts.Muchresearchhasbeendevotedtothelat

4、ter.ThefirstsolutiondatesbacktoSturm21,p.304.UsingSturm'smethod,Routhdevelopedawxsimplealgorithmtosolvetheproblem.Hurwitzindependentlydiscoverednecessaryandsufficientconditionsforallofthezerostohavenegativerealparts,whichareknowntodayastheRouth]Hurwitzconditions.AgoodandconciseaccountoftheRout

5、h]Hurwitzproblemcanbefoundinwx5.AccordingtotheRouth]Hurwitzconditions,a2=2realmatrixAisstableifandonlyiftr.A-0anddet.A)0;a3=3realmatrixAisstableifandonlyiftr.A-0,det.A-0,andtr.A?a-det.A,where2a2isthesumofall2=2principalminorsofA.*ResearchsupportedinpartbyNationalScienceFoundationgrantDMS

6、9626128.E-mail:mli@math.msstate.edu.2490022-247Xr98$25.00CopyrightQ1998byAcademicPressAllrightsofreproductioninanyformreserved.250LIANDWANGStabilityofmatricesisintimatelyrelatedtothestabilityofstationarysolutionsofvariouskindsinthetheoryandapplicationsofdynamicalnsystems.Letfbeavectorfielddefi

7、nedinanopensetofR.Anequilibriumpointxoffissuchthatfx.s0.Itisasymptoticallystableif,foreachneighborhoodUofx,thereexistsaneighborhoodVsuchthatxgV;U,andx.0gVimpliesthatthesolutionxt.satisfiesxt.gUforallt)0,andthatxt.ªxastª`see11.Ifwx