非线性广义变延迟方程稳定性分析(英文)-豆柴文库

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非线性广义变延迟方程稳定性分析(英文)
NonlinearTime-DelayDifferentialEquationStabilityAnalysis
Introduction:
Time-delaydifferentialequations(DDEs)areaclassofmathematicalmodelsthatdescribesystemswherethecurrentstatedependsnotonlyonthecurrentinputsbutalsoonthepaststates.Manynaturalandengineeredsystemsexhibittime-delayeffects,suchaschemicalreactions,physiologicalprocesses,andcontrolsystems.Analyzingthestabilityoftime-delaysystemsisofgreatimportanceinvariousfieldsofscienceandengineering.Thispaperaimstodiscussthestabilityanalysisofnonlineartime-delaydifferentialequations,specificallyfocusingonaclassofequationsknownasnonlineargeneralizeddelayequations(NGDE).
NonlinearGeneralizedDelayEquations:
Nonlineargeneralizeddelayequations(NGDEs)areaspecialclassoftime-delaydifferentialequationsthatexhibitnonlinearityinboththefunctionalformandthedelayterm.ThegeneralformofanNGDEisgivenby:
x'(t)=f(x(t),x(t−τ))
wherex(t)isthestatevariable,f(·)isanonlinearfunction,andτisthetimedelay.NGDEscanmodelawiderangeofsystemswithvariousnonlinearityanddelaycharacteristics.
StabilityAnalysisTechniques:
Stabilityanalysisofnonlineartime-delaysystemsposessignificantchallengescomparedtolinearsystems.InthecaseofNGDEs,analyticalstabilitycriteriaareoftendifficulttoderiveduetothepresenceofnonlinearityanddelay.Therefore,numericalandcomputationalmethodsplayacrucialroleinthestabilityanalysisofNGDEs.SomewidelyusedtechniquesincludetheLyapunov-Krasovskiifunctionalapproach,themethodofaveraging,andthemultiple-delayapproach.
Lyapunov-KrasovskiiFunctionalApproach:
TheLyapunov-Krasovskiifunctionalapproachisapowerfultoolforstabilityanalysisoftime-delaysystems.ItinvolvesconstructingaLyapunovfunctionalthatcanguaranteethestabilityofthesystem.ThebasicideaistofindascalarfunctionV(x(t))andatimederivativeV'(x(t))thatsatisfycertainconditions.Byutilizingdelay-dependentinequalitiesorintegralinequalities,stabilitycriteriacanbeobtainedforNGDEs.However,theconstructionandanalysisofLyapunov-KrasovskiifunctionalsforNGDEscanbecomputationallyintensive.
Met