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Chapter1.Introduction
delays,areformulatedintheformofLMIs.Q.L.Han[61]studiedtheproblemof
stabilityoflineartimedelaysystems,bothretardedandneutraltypes,usingthe
discretedelayN-decompositionapproachtoderivesomemoregeneralnewdiscrete
delaydependentstabilitycriteria.Q.L.Han[62]employedthedelaydecomposition
approachtoderivesomeimprovedstabilitycriteriaforlinearneutralsystemsandto
deducesomesufcientconditionsfortheexistenceoftheLyapunovfunctionalfor
asystemwithk-non-commensurateneutraltimedelaysofadelayedstatefeedback
controller,whichensureasymptoticstabilityandaprescribedH1performancelevelof
thecorrespondingclosed-loopsystem.K.GuandY.Liu[54]investigatedthestability
ofcoupleddiferentialfunctionalequationsusingthediscretizedLyapunovfunctional
methodandsetforththestabilityconditionintheformofLMIs,suitablefornumerical
computation.
E.F.InfanteandW.B.Castelan[63]basedtheconstructionoftheLyapunovfunc-
tionalonsolutionofamatrixdiferential-diferenceequationonafnitetimein-
terval.V.L.KharitonovandA.P
.Zhabko[82]extendedtheresultsofE.F.Infante
andW.B.Castelanandproposedaprocedureofconstructionofthequadraticfunc-
tionalforlinearretardedtypedelaysystemswhichcouldbeusedforrobuststability
analysisoftimedelaysystems.ThisfunctionalwasexpressedbymeansofaLya-
punovmatrix,whichdependedonthefundamentalmatrixofatimedelaysystem.
V.L.Kharitonov[66]extendedsomebasicresultsobtainedforthecaseofretardedtype
timedelaysystemstothecaseofneutraltypetimedelaysystems,andtoneutraltype
timedelaysystemswithadiscreteanddistributeddelay[68].V.L.Kharitonovand
D.Hinrichsen[75]usedtheLyapunovmatrixtoderiveexponentialestimatesforsolu-
tionsofexponentiallystabletimedelaysystems.V.L.KharitonovandE.Plischke[81]
formulatednecessaryandsufcientconditionsfortheexistenceanduniquenessof
thedelayLyapunovmatrixforthecaseofaretardedsystemwithonedelay.
TheLyapunovquadraticfunctionalsarealsousedtocalculationofavalueof
aquadraticperformanceindexofqualityintheprocessofparametricoptimization
fortimedelaysystems.Thevalueofthatfunctionalattheinitialstateofthetime
delaysystemisequaltothevalueofaquadraticperformanceindexofquality.To
calculatethevalueofaperformanceindexofqualityoneneedstheformulasforthe
Lyapunovfunctionalcoefcients.ForthefrsttimeaLyapunovfunctionalfortime
delaysystemwasintroducedbyYu.M.Repin[98]forthecaseofalinearsystemwith
oneretarded-typedelay.Yu.M.Repindeliveredalsotheprocedurefordetermination
ofthefunctionalcoefcients.Theprocedureisasfollows.Atfrst,itisassumedthe
formofthefunctional,afterwordsitstimederivativeonthetrajectoryofsystem
withatimedelayiscomputedandequatedwiththenegativelydefnitequadratic
formofasystemstate.Inthiswayweobtainthesetofdiferentialandalgebraic
equations,whichenablesustodeterminetheformulasofthefunctionalcoefcients.
ThepresentedmethodgivesanalyticalformulasforthecoefcientsoftheLyapunov
functional.
InthismonographthemethodproposedbyRepin[98]isappliedtoobtainthe
Lyapunovfunctionals,withcoefcientsgivenbyanalyticalformulas.Inchapter2.
areconsideredsystemswiththeretardedtypetimedelay.Thismethodisappliedto
thesystemwithonedelay(chapter2.2),tothesystemwithtwodelays(chapter2.3),
