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DidierAurouxUniversityofNiceSophiaAntipolisauroux@unice.frWorkincollaborationwith:J.Blum(U.Nice)+workofE.Cosme(U.Grenoble)INSU-CNRSLEFE-ASSIMBFN/BFN2,Villefranche-sur-Mer,France.BackandForthNudgingorSEEK(BFN,BF-SEEK)September29,2009
td=ForwardnudgingF(X)+K(XbosXdX(0)=0whereKisthenudging(orgain)matrix.X,H(X)),0<t<T,Inthelinearcase(whereFisamatrix),theforwardnudgingiscalledLuenbergerorasymptoticobserver.–Meteorology:Hoke-Anthes(1976)–Oceanography(QGmodel):Verron-Holland(1989)–Atmosphere(meso-scale):Stauffer-Seaman(1990)–Optimaldeterminationofthenudgingcoeffcients:Zou-Navon-LeDimet(1992),Stauffer-Bao(1993),Vidard-LeDimet-Piacentini(2003)LEFE/ASSIM2009,Villefranche-sur-Mer,September29200931/1
Forwardnudging:linearcaseLuenbergerobserver,orasymptoticobserver(Luenberger,1966)ddXt=FX+K(XobsHX),dXˆ=FXˆ,Xobs=HXˆ.tdddt(XXˆ)=(FKH)(XXˆ)IfFKHisaHurwitzmatrix,i.e.itsspectrumisstrictlyincludedinthehalf-plane{λC;Re(λ)<0},thenXXˆwhent+.LEFE/ASSIM2009,Villefranche-sur-Mer,September29200931/2
BackwardnudgingHowtorecovertheinitialstatefromthefinalsolution?Backwardmodel:dX˜=F(X˜),T>t>0,tdX˜(T)=X˜T.Ifweapplynudgingtothisbackwardmodel:td=F(˜X)dX˜X˜(T)=X˜T.K(XbosLEFE/ASSIM2009,Villefranche-sur-Mer,September292009H˜X),T>t>0,/331