Observation and inverse problems in coupled cell networks

Romain Joly

Description

Observation and inverse problems in coupled cell networks Romain Joly March 2011 Abstract A coupled cell network is a model for many situations such as food webs in ecosys- tems, cellular metabolism, economical networks... It consists in a directed graph G, each node (or cell) representing an agent of the network and each directed arrow rep- resenting which agent acts on which one. It yields a system of differential equations x˙(t) = f(x(t)), where the component i of f depends only on the cells xj(t) for which the arrow j ? i exists in G. In this paper, we investigate the observation problems in coupled cell networks: can one deduce the behaviour of the whole network (os- cillations, stabilisation etc.) by observing only one of the cells? We show that the natural observation properties holds for almost all the interactions f . Key words: coupled cell networks, observability, inverse problems, genericity, tran- versality theorems. AMS subject classification: 93B07, 34C25, 34H15, 92B25. 1 Introduction The coupled cell networks. In the recent years, the mathematical study of coupled cell networks has been quickly developing. It combines several interests: it is strongly related with applications and real phenomena, the setting is very simple and it leads to a rich class of mathematically inter- esting problems.

• can also


• also stop


• x1 ?


• interaction


• admissible vector fields


• many concrete


• networks


• coupled cell

Sujets

• Interaction

Informations

• Publié par: pefav
• Nombre de lectures: 14
• Langue: English

Extrait

BDhIgkCiPdc CcH PckIghI egdDaIbh GIaa cIiwdgTh

(

Eˆ“NV] 9ˆ[i

?NbPU ”`´´

Pc

GdjeaIH

AAbtacta AagbI1.-1II˙1fiadAie)Wa.1I:adW)˙yeifg)fia˙eeg-?)e:aa.i1,ei˙1-aeye( - f1Weff-1IIgI)dW1f),aIieWff1-a˙aWi-)I˙1fiadAe.di-1˙i.)eiefa-e˙d)b?f1.=IfGff 1)-?˙a.1(ad-1II)d1bd1e1˙fi˙=)˙)=1˙fa:f?1˙1fiadA)˙.1)-?.id1-f1.)ddaid1b( d1e1˙fi˙=i?i-?)=1˙f)-fea˙i?i-?a˙1.Ifiy1I.e)eyef1Wa:.i1d1˙fi)I1cg)fia˙e x_ (t) =f(x(t)ff i?1d1 f?1 -aWba˙1˙fia:f.1b1˙.e a˙Iy a˙ f?1 -1IIex J (t?i-i?:)da f?1 )ddaij!i1kiefe i˙Ga,?11f)fi=efh1i˙1iffd1b)bei?f˙I.eI,W1b˙adf)ai1ehd i˙-agbI1.-1II˙1fiadAe:-)˙a˙1.1.g-1f?1,1?)hiagda:f?1i?aI1˙1fiadA(ae( -iII)fia˙eff ef),iIie)fia˙ 1f-.) ,y a,e1dhi˙= a˙Iy a˙1 a: f?1 -1IIe? W1 e?ai f?)f f?1 ˙)fgd)Ia,e1dh)fia˙bdab1dfi1e?aI.e:ad)IWaef)IIf?1i˙f1d)-fia˙ef.

KAywoH:K:˙(d)b1edI,adffyfi1h˙i-idiffffyeff1W˙1=1-.I1˙If1a-bg1Ie1dh),iIiadAeffa, h1de)Iifyf?1ad1We. 3MSKu7jA9M9l5KKBfB95MBoC:204,7001C.5,01H,5,2.4..5

IcigdHjGiPdc

TOIGdjeaIHGIaacIiwdgTh. 8] dUR bRPR]d iRNbcffi dUR “NdUR“NdVPN[ cdeQi ˆS Pˆe˙[RQ PR[[ ]RdgˆbYc UNc ORR] aeVPY[i QRfR[ˆ˙V]T. 8d Pˆ“OV]Rc cRfRbN[ V]dRbRcdc: Vd Vc cdbˆ]T[i bR[NdRQ gVdU N˙˙[VPNdVˆ]c N]Q bRN[ ˙UR]ˆ“R]Nffi dUR cRddV]T Vc fRbi cV“˙[R N]Q Vd [RNQc dˆ N bVPU P[Ncc ˆS “NdUR“NdVPN[[i V]dRbı RcdV]T ˙bˆO[R“c. 0 Pˆe˙[RQ PR[[ ]RdgˆbY “ˆQR[c N Tbˆe˙ ˆS NTR]dcffi RNPU ˆ]R V]dRbNPdV]T

´

gVdUNTVfR]˙NbdˆSdURˆdURbcdUbˆeTUQVRbR]dVN[RaeNdVˆ]c.8dVcbR˙bRcR]dRQOiN]QVı bRPdRQ TbN˙Uffi RNPU ]ˆQR ORV]T ˆ]R ˆS dUR NTR]dc N]Q RNPU QVbRPdRQ Nbbˆg bR˙bRcR]dV]T gUVPU NTR]d NPdc ˆ] gUVPU ˆ]R. GUVc “ˆQR[[V]T N˙˙RNbc V] “N]i Pˆ]PbRdR cVdeNdVˆ]c: ]RdgˆbYc ˆS ]Rebˆ]cffi PR[[e[Nb “RdNOˆ[VP ]RdgˆbYcffi SˆˆQ gROc V] RPˆcicdR“cffi RPˆ]ˆ“VP ]RdgˆbYc RdP. ?N]i TˆˆQ NbTe“R]dc Sˆb cdeQiV]T Pˆe˙[RQ PR[[ ]RdgˆbYc NbR TVfR] V] J´,L. ?N]i RhN“˙[Rc ˆS Pˆe˙[RQ PR[[ ]RdgˆbYc PN] N[cˆ OR Sˆe]Q V] J,L N]Q J´-L N]Q dUR bRSRbR]PRc dURbRV]. GUR “NdUR“NdVPN[ cRddV]T Vc dUR Sˆ[[ˆgV]T. =RdGOR N QVbRPdRQ TbN˙U gVdUNPR[[c [V]YRQ gVdU Nbbˆgc. Gˆ dUR PR[[Jffi gR NccˆPVNdR N ˙UNcR c˙NPR< I ˆS Γ]VdR QV“R]cVˆ]D I ffi gUVPU Vc Ncce“RQ V] dUVc ˙N˙Rb dˆ OR dUR dˆbec fiRµZ) , i =T , i fidUVc Ncce“˙dVˆ] Vc “NQR Sˆb cNYR ˆS cV“˙[VPVdi V] ˆeb ˙bˆˆScffi Oed ˆeb bRce[dc N[cˆ Uˆ[Q VS< I Vc N “ˆbR TR]RbN[ “N]VSˆ[Qffi V]P[eQV]T< I =R , i RRcffi+]ˆVdPRF.)RWRcd<=< ) ζ ζ ζ< N N]QD=D ) flζ ζ ζflD N . 5ˆb N]iP2<N]Q N]i cRd ˆS V]QVPRcξ=⊂J ) θ ζ ζ ζ θ J N →ffiP γ QR]ˆdRc dUR V“NTR ˆS dUR PN]ˆ]VPN[ ˙bˆWRPdVˆ] ˆSPˆ]dˆ< γ =< I 0 ζ ζ ζ< I k fiVSξ=⊂J→ffi gR cV“˙[i gbVdRP I ).WRN[cˆQR]ˆdR OiD γ =D I 0 flζ ζ ζflD I k dUR QV“R]cVˆ] ˆS< γ . 5ˆb RNPU PR[[Jffi dURMSdNLf S“bgfeˆSJVc dUR cRd ˆS PR[[cKcePU dUNd dUR NbbˆgK!JOR[ˆ]Tc dˆG ]ˆQR. 0KVc N]S“MSdNLf S“bgfˆSJVS dURbR RhVcdc N cRaeR]PR ˆS NbbˆgcK!k ) !k + !ζ ζ ζ!JOR[ˆ]TV]T dˆG. =RdX ) fi<) OR dUR c˙NPR ˆS dURC )  fRPdˆb ΓR[Qc ˆ]<R]QˆgRQ gVdU dUR eceN[ 1N]NPU dˆ˙ˆ[ˆTi ficRR Sˆb RhN“˙[R J´”L ˆb J´L). Gˆ cV“˙[VSi dUR ]ˆdNdVˆ]cffi Sˆb N]iP2<ffi gR VQR]dVSi , h <gVdUR , N]QX ) fi<) gVdUC ) fi<θR , RdU[[SNˆRPN˙cOecRUdRPeQ]dbˆ.WRV)HMWSeeSIVN hNLfad ffNVMe

C G =⊂I2X ) fi<)θ I I QR˙R]Qc ˆ][i ˆ] dUR QVbRPd V]˙edc ˆS dUR PR[[J→ WVdURNPUI2 C G ffi gR NccˆPVNdR dUR Qi]N“VPN[ cicdR“.fit)P ( Pfit)ffi PN[[RQLagbVNM LNVV “NfiadURbR]dVN[RaeNdVˆ]ffiR]RTRdNbdiOQVQRU

 _P P fi fi t=) =)`P Ifi ( Pfit)θ t2R fi´´) . 2< AˆdVPR dUNdffi cV]PR<Vc Pˆ“˙NPd N]QIVc ˆS P[NccC ) ffi dUR cˆ[edVˆ]c ˆS fi´.´) RhVcd Sˆb N[[ dV“Rc. GUR ˙bˆ˙RbdVRc ˆS.fitˆ] dUR cdbePdebR ˆS dUR QVbRPdRQ TbN˙U) “Ni cdbˆ]T[i QR˙R]Q G WRV]dbˆQePRURbRcˆ“RQRΓ]VdVˆ]c.

DIcPiPdch (.(. WN eHk fRHf H LNVVJSe H“ˆOcRbfNdVˆ] PR[[SO H“k LNVVKaOGiSfRJ−=KSe H“ S“MSdNLf S“bgf aOJ. WN eHk fRHf H eNf aO LNVVeξSe H“V]QR˙R]QR]d ceOı]RdgˆbYSOξLa“fHS“e HVV Sfe S“MSdNLf S“bgfe´ S.N. SO fRN ekefNWfi´.´)dNefdSLfNM fa fRN eNf aO LNVVeξSe efSVV H“ Hgfa“aWage A-.. WNeHkfRHffRNPdHbRGSecdbˆ]T[i Pˆ]]RPdRQSO H“k LNVV Se H“ aIeNdhHfSa“ LNVV ad NcgShHVN“fVk SO H“k LNVV Se H“ S“MSdNLf S“bgf aO H“k afRNd a“N. WNeHkfRHffRNPdHbRGSecR[SıQR˙R]QR]dSO H“k LNVV Se H MSdNLf S“bgf aO SfeNVO.

”