Luận văn Các hệ động lực tuyến tính bị động và đơn nguyên

CÁC HỆ ĐỘNG LỰC TUYẾN TÍNH BỊ ĐỘNG VÀ ĐƠN NGUYÊN NGUYỄN MINH HẰNG Trang nhan đề Mục lục Phần mở đầu Chương1: Tổng quan về các vấn đề đặt ra trong luận án. Chương2: Khai triển tường minh các hệ mô hình đơn nguyên. Chương3: Hàm non tốt nhất của tích các hàng toán tử co giải tích trên dĩa tròn đơn vị. Chương4: Hệ bị động và tính tối ưu. Chương5: Hệ nối và hàm non tốt nhất. Kết luận Tài liệu tham khảo

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CHUaNG 1 1 , - - , TONG QUAN vB CAC VAN DE BAT RA TRONG LUAN AN Giai tichhamnoi chung,va d~cbi~t1a1ythuy~tmohiOOtoantlf co r~t OOi~uling d\illg trongcac 1InhVL;fckhacOOaucua toanhQCva v~t1y.Ly thuy~tcactoantlf k~th<;5pvdi mo hiOOcach~d9ngllJC tuy~ntiOOda dfu1d~n cack~tquathll vi trongvi~cnghiencoo cactiOOch~tcach~tuy~ntiOOvo h?ll chi~u.N9i dungcua1u~an1anghiencoocactiOOch~tcuacach~d9ngllJC tuy~ntioovo h?ll chi~ubfulgcongClfmohiOOtoantlf va mohiOOh~tuy~n tiOO. 1.Ly tbuy~tfiG binb tmintn. Nho vaoOOilngk~tquan6i ti~ngcuaHilbertv~ph6toantlf, ta daco du<;5CmohiOOcuacactoantlf tlJ lienh<;5pva toantlf ddnnguyen,chUngdu<;5c bi€u di~nd d?llgtichphan T = fAdP). a(T) dm9thudngkhac,tadabi~tcack~tquacuaShillv~duam9tmatr~ tlJ lienh<;5pv~d?llgduongcheo000phepbi~nd6iddnnguyen. Phattri€n haihudngtren,vaodftuOOilngnam50,cacOOatoanhQcXo Vi~tb~tdftuxay d1..fng1ythuy~tmohiOOchocactoantlf k.hongtlJ lienh<;5p , ho~ckhong ddn nguyenva nguoi di tien phongtrong 1InhVL;fcnay 1a M.S.Livsis.Nam1946,Livsis dacongb6congtriooqUailtrQng[38],trongdo l§.nd~utienhamtoantitd~ctnfngcuatoantitdu<jcduafa.Khaini~mham toantitd~ctnfngsaDnaytrdthanhmQtcongclfquantrQngtrongnghiencUu cuar~tnhi~unhatoanhQC.Quatrinhti~nboacualy thuy~thamtoantitd~c trlfngdi~nrakh<idaivakhokhan.NgaytUd~unam1946,Livsis[38]dalieU congthuchamtoantit d~ctnfngcuatoantitcod?Ilgkh<iphuct;tpnhusaD eA(Z)=-A sign(I-AA*)+zI I-AA * 11/2(I-zA *f1 II-A *A 1112 (1.1) D?Ilghamd~ctnfngnhutrenkhati~ndlfngchotoantit"g§.n"donnguyen. D6i vdi toantit"g§.n"tVlienh<jp,nam1954,Livsis [39JduadinhnghTa saD W(z)=I+2i(SignAl)1 AI I 112(A*-zIfl I AI 1112 * A = A-A I 2i vdi Dinhnghlatrendadu<jcM.S.Brodskii[36JmdrQngvaonam1956d d?IlgsaD W(z)=I-2iK*(A-zI) -lKJ trongdo J=J*, J2=I,KJK*=Ar vdiJ, K lacactoantitbi ch~ lienh~,vdi toantitA bdicaccongthuctren. Sl}caiti~nhamd~ctnfngd?Ilg(1.1)chotoantit "g§.n"donnguyenmai d~nnam1972mdidu<jckhkg dinh,dolahamtoantit eA(z)=D+zC(I-zAflB trongdoB, C, D lacactoantitb6tr<j,thoadi~uki~n I-A *A=C*C I-AA *=BB*I-D*D=B*B I-DD*=CC* -A *B=C*D, , , , . 4 Cilia khoad€ sud\lIlgcachamtoantVd~ctningdtrenduejcth€ hi~nqua dinhly cobansauday. Dinh ly Livsis [37].N~uhai toantVbi ch~, dongianco clInghamtoantV d~ctningthiWongduongd6nnguyen. Ti~psau,nhonhi1ngnghiencdusaus~cvehamtoantVcuaPotapov [32],Ginzburg[18];Brodskiva Livsis [37]dakhaitri€n hamtoantVd~c tningvaod~g "J 1 1 de:(t) 00 "J eA(Z)=7eel-aCt) IT(I - 1 q~qk) 0 k=l Ak - Z trong docactoantVaCt),set),~ thoamanmQts6h~thuc. Nho vaobi€u di~nnay,Livsis daxaydvngmohinhcuatoantV"gfu1"tl! lienhejp,daychinhlabudcd~utienqUailtrQngtrongly thuy~tmohinhtoan tV khongdonnguyenho~ckhongtl!li~nhejp.Mo hinhcu~Livsisdadu<Jc cacd6ngnghi~pva cachQctrocuaongcM.ti~n, mdrQngV3.0nhUngnam60 va70[41].Mo hinhnaytrongtniongh<Jph6roi f?Ccod~g : (Af)k=AJk+i I, fjqjJq~; j=k+l trongtniongh<Jph6lientvccod~g : 1 (Af)(x)=a(x)f(x)+if f(t)q(t)Jq*(x)dt. x D~unhvngnam60,dmQthudngkhac,cacnhatoanhQcDongAu, Nagy va Foiasdati~nhanhnhUngnghiencilil r~tsaus~cve cactoantVco trong khonggianHilbertmamQtv~ detrQngtamlaphepgian(dilation)toantV. 5 TrongquatrinhxtlydlJngphepgiffilcactoantv,cacnhatoanhQcnaydathi~t l~pm9td<:lih1Qngd~ctnfngcuatoantv,va th~thuvi la d<:lilv(jngd~ctnfng nayl<:litrimgvdikhaini~mv~hamd~ctnfngcuaLivsis.DlJavaokhaini~m hamtoancld~ctnfngdo,NagyvaFoiasdaxtlydlJngm9tmahinhrfttti~n d1Jngmasannaylienh;1cxuftthi~ntrencaccangtrinhcuacacnhatoanhQc trenth~gidi. Ly thuy~tmahinhtoancl cuaNagyvaFoiascoth~tomt~tnhvsan. ChoA latoanclcotrenkhanggianHilbert,hamtoancld~ctnfngcuaA dvQcdinhnghlab6i eA(z)=-A +z(I - AA *)1/2(I - zA*)-1(I - A *A)1/2 Ngv(jcl<:li,chotrlfdchame(Z)E$ (U,V), Nagy-FoiasxtlydlJngmahinhtoan clconhvsan. x =[L~(V)Ef)&2(U) ]e{(eO)Ef)ilO))/ 0)E L~(U)}, A( <pEf) \jf) =e-it «p(eit) - <p(O))Ef)e-it\jf( eit), trongdo il( eit)=(I - e(eit)*e(eit))1/2. ToanclA nayddngianvacohamtoancl d~ctnfngtrimgvdie(z).Sando, vaonam1972BrodskidaxtlydlJngth~mcactoancl Bu =e-it (e(eit) - e(o))uEf)e-itil( eit)u , C( <pEf)\jf) = <p(O), Du =e(O)u, d~dVav~mahinhcuah~.H~tuy~ntinhdv(jcxtlydlJngnhvtrenla ddngian, ddnnguyenva co hamtruy~nla e(z).Ta co k~tqua(diM ly Livsis-Brodski) 6 L1haih~dongian,donnguyenco cUnghamtruy~nthi Wongduongdon nguyen.Nhuv~ymohinhcuaNagy-Foiasdacom9tvaitroqUailtr<;mgd~ nghiencUucach~donnguyen,mohinhnaycoth~chotanhi~uthu~1Qivi cactoantUd~uduQcxaydlJIlgtlfdngminh. Hudngthllbatrong1ythuy~tmohinhtoantUduQcphattri~nb6icac nhatoanhQcMy, DeBranges,Rovnyak[13].SaDday1amohinhcuaDe BrangesvaRovnyakchotoantU coduQcxaydlJIlgtheoham8(Z)E$ (U,V) chotrudc.GQiBe1akhonggiang6mcacphfu1tU (f(z),g(z))vdi f(z)EH2(V), g(z)EH2(U)saocho «f(z),g(z)), Kew,X,/Z»Be=y+u vdi Kew,x,/z)1ahamcactoantUduQcdinhnghiab6i K e (z)=w,x,y ( I - 8(z)8(w)* 8(z)- 8(w) 8(z)- 8(w) 1- 8(z)8(w)* J x+ y, . x + y 1-zw z-w z-w 1-zw trongdo 8(z) =8("2)*,WEq}),XEV, YEU. ToantUmohinhtrenBeduQcdiM nghiab6i A : (f(z),g(z))H (zf(z)-8(z)g(O),g(z)- g(O)) z cohamtoantUd~ctrlfng1a8(z).Vi cactoantU A trongcacmohiOOcua Nagy-FoiasvacuaDe Branges-Rovnyakd~udongianlienchUngtlfong duc5ngdonnguyen.TrongmorJlli~cuaDeBranges-Rovnyak,tuykhonggian Bekhongcobi~udi~ntlfdngminhnhungcouu di~m1acacphfu1tUfez),g(z) d~u1acachamgildtich. 7 Ngoai cacGongtriOOchuy~utren,ly thuy~tmohiOOtoantVconduQc mdr<)ngchocacloptoantVkhac,kScatoantVkhongbi ch~ [6],[8]. Tronglu?nannay,chUngWiS11dlJIlgchuy~umohiOOcuaNagy-Foias. LUll Y lacacmohiOOtrenlamohiOOham.Ngoairaconcohuangxfty dlJIlgmohiOOd~g tichphant:.heocackhonggianconbfttbi~nduQcxftyd\fng bdiBrodski[14],Gohberg,Krein [19]. 2.Ly thuy~th~dQnghfetuy~ntinh. BM d~utUnam1960,sailcacGongtrioocuaKalman[23],m<)ts6huang nghiencoocactiOOchfttdiOOtiOOcuah~d<)ngh.Jctuy~ntiOOphattriSnm~. KalmandadUafa cackhaini~mrfttqUailtr<;mg: tiOOdieDkhiSnduQc,qUail satduQc,xftydlJngmohiOOcach~(ly thuy~thShi~n),sqd6ngd~g cuacac h~tuy~ntiOO[23],... Xet h~d<)nglqc tuy~ntiOOa=(X,U,V,A,B,C,D) duQcmohiOOhoabdi h~phuongtrioosail dx =Ax(t)+Bu(t), dt vet)=Cx(t)+Du(t); x(t),u(t),vet),la cachamvectovoigiatri la cacvectdl~ luQthu<)ccac khonggianHilbertkhatachX, U, V. Hamx(t)duQcgQila hamtr~gthai, u(t)duQcgQiIahamdieDkhiSnvavet)duQcgQiIahamqUailsat. H~a duQcgQiIa dieDkhiSnduQctUtr~g thaiXod~ntr~g thaiXl trong khoangthdigian[to,tl]n~ut6nt(;lim<)thamdieDkhiSnu(t)xacdiOOtren[to,t1] saochon~uh~b~td~utUtr~g thaiXo(tUcla x(to)=Xo)thi t(;lithdidiSmtl no 8 cotr?llgthaiXl' tUGlax(tl)=Xl'Di~udod6ivdih~tuy~nt1nhxetd trenco nghiala X(tl) =eA(tl-tO)Xo+ ftt~eA(tl-S)Bu(s)ds. H~a duQcgqi la di~ukhi€n duQchoantoann~ua di~ukhi€n duQctUtr?llg thaibM10'Xov~tr?llgthaib~t10'Xl trongkhO<lngthaigianb~t10'[to,tl]. Trongdi~uki~nX,U,V la cackhonggianhituh?ll chi~uthi h~di~u khi€nduQckhivacmkhi rang(B,AB, ...,An-lB)=n=dimX. D6i vdi h~vo h?ll chi~u,khaini~mdi~ukhi€n duQcthuangduQchi€u d d~lgdi~u~~i€nduQcx~pxi, nghia1.1vdi ill<)tIanc~ chotrVdccuaXb luon t6nt?i m9thamdi~ukhi€n u(t)di~ukhi€n quyd?ocuah~tUtr?llgthaiXod~n Ianc~ cuatr?llgthaiXl trongm9tthaigianhituh?ll,Khi ~ydi~uki~ncfu1va du d€ h~di~ukhi€n duQcla :AkBU=X 0 D6i ng~uvdi khaini~mdi~ukhi€n duQc,Kalmanduarakhaini~mqUail satduQc.V~ d~d~tralakhibi~thamqUailsatvet)(t2 to)thi tr?llgthaiban d~uXo=x(to)coduQcxacdinhduynh~tkh6ng?N~uh~a cotr?llgthaix(to) =Xo"*0,hamdi~ukhi€n u(t)=0(t2 to)l'itico hamqUailsatvet)=0 (t2 to)thi tr?llg thai Xogqi la kt~ongqUailsatduQCt'itithai di€m to.H~duQCgqi la qUail satduQchoantoann~ut'itimqithaidi€m , khongcovectonaokhongqUailsat duQc.Khi dotacok~tquad6ing~uchotinhqUailsatduQc.H~hituh?ll chi~u , ,,', l' qUailsatduQchoantoanneuvachineu 9 rang(C*, A *C*,.. .,A*n-1C*)=n=dimX; \trongtru6nghpvo h~ chi~uthidi~uki~nc§nvadud~h~qUailsatduQc hoantoanla 00 vA *kc*v=x. 0 MQtkhaini~mqUailtr(;mgtrongh~tuyentioodUnglakhaini~mham truy~n,hamnayducxa dinhb6icongthuc eaCz)=D+zC(I-zAr1B:U ~ V. Ly thuyeth~dQngl\fCtuyentinh d\fatrenhamtruy~nva ly thuyetmohiOO toantUtronggiaitichphattri~ndQcl~psongsongOOungcoOOi~udi~mWong d6ngth1.ivi. D6i vdimQts6ldpcach~thihamtruy~ntrUngvdihamd~ctrung ? , ? Acuatoanill . Hamtruy~nmangynghiaOOusail:giasith~a covectotr~g thaixCi)= XoeZ\vectovaou(t)=uoezt,vectoravet)=voezt,hivet)=e(l(z)u(t).Nhu v~y haih~coclinghamtruy~ncoth~coilaWongduongvi tr~gthaibentrong cuahaih~coth~khacOOaunhungkhi choclingtinhi~uvaou(t),tad~c clingtin hi~ura vet).TiOOqUailtrQngcuahamtruy~ncon duQcth~hi~n6 dinhly Kalman[23]: neu"haih~hituh~ chi~uai, ~ di~ukhi~nduQc,qUail satduQCcoclinghamtruy~nthichUngd6ngd~g,nghialakhidot6nt?imQt toantUkhanghichlien1:\1cW:xi~ X2saocho A2 =WA1W-1 , B2 =WE1 , C2 =C1W-1, 10 D2=Dl ; vara ranghaih~d6ngd~g till chUngcoclIngmQts6cactinhch~tqUailtn;mg nhl1tinhdi~ukhi€n dl1<JC,qUailsatdl1<JC,dndinh,phd... N~uhon1l11alOantli W ladonnguyentill ng116itanoihaih~laWongdl1ongdonnguyen. Trenco s6dinhly d6ngd~g, KalmandaxaydljIlgcacmohinhcuah~ tuy~ntinhIDeomQthamtruy~n8(z) chotrUocmaongtagQila cacth€ hi~n (realization)cuaham8(z) [23].Ly thuy~th€ hi~ndapilattri€n kham~, khongnhltngchoh~tuy~ntinhdung,h~khongdUngmaCelh~phituy~n. Di Sailhonnltad6i voi h~tuy~ntinh,cacnhalOanhQcMy (Brockett, Barass...[10],Israel(Gohberg[11]),... danghienCUllSlJlienk~tcaeh~.Cac h~tuy~ntinh khi lien k~tn6~ti~pnhau IDeonghia : Cho hai h~ a k= (Xk,Uk,VbAbBbCbDk),k = 1,2,saochoU2=Vl . H~a =(X,U,V,A,B,C,D ) dl1<JCgQila lienk~tn6i ti~p(tichn6iti~p)cuahaih~al , a2va dl1<JCkY hi~u laa =a2aln~u: U =Vl ;V =V2 ;X =Xl EBX2, A =A1P1+A2P2+B2ClP1, B =Bl +B2Dl, C =C2P2+D2C1P1, D =D2D1, trongdo Pk la phepchi~uvuonggoctUkhonggianX lenkhonggianXk, k=12., , thicactiOOdi~ukhi€n dUdC,qUail satdUdC,t6i thi€u, ddn gian, t6i ULl...coth€ ~. . khong du\Jc baa toan.Cac taGgia trenda co mQts6 k@tqua v~di~uki~nd€ baa toancac tiOOch~tdo khi lien k@tcach~.Cac k@tquanay du\Jcphatbi€u trenligonngvb~cMacMilancilahammatr~.Trongt~tcacacth€ hi~ncila ham8(z),th€ hi~ncos6chi~ucilakhonggiantr~gthaila006OO~tdu\Jcgqi la th€ hi~nt6i thi€u. S6chi~ucilakhonggiantr~g thaitrongth€ hi~nt6i thi€ucila8(z)du\Jcgqilab~cMacMilancila8(z)vadu\JCkYhi~uladeg8(z). MQtk@tquav~di~uki~nd€ baatoantiOOch~t6i thi€u du\Jcphatbi€u trong diOOly Gohberg: Lien k@tn6i ti@pa cilahaih~t6i thi€u aj va a2la t6i thi€u . n@uvachin@udeg8a(z)=degeaj(z)+degea2(z). CongClfcilahudngnghiencoonayla GongClfd?is6matr~, r~tkho phatri€nchotr1.fdngh\Jpvoh~ chi~u. 3.Ly thuy~th~dQngI1fctuy~ntlnh trenkhonggianHilbert. Livsis la ngudid~utiennghiencooly thuy@th~tuy@ntiOOtrongkhong gianvo h?il chi~u[40].Ongdakhaosatcach~dQngh.;tctuy@ntiOOdissipative d~g x =Ax+Bu, v =Cx+Du; trong do C=B* D=I, , A - A * , BB* ,. h' ,,' d ~= , VOl am truyen co ~g S(z)=I+2iB*(zI-AflB;vanghiencoonhi~ulingd\lilgcilachUngtrongv~tly. 12 ? ~, ,- cachQctracuaangclingd3:tienhanhcacnghiencUuvecach~ngaunhien (Iancevich[30]),v~ th€ hi~ncua cac hamphanhinh (meromorphic- D.C.Khanh [42]).D~cbi~tArov d3:nghiencUusailv~cach~bi dQng (passive).Do lah~rair~cd~g ~+l=~+B~, vn =C~+D~; vcii(~ ~):XEBU~XEj)V1aloanttJcovamUlltruy~n8(z)=D+zC(I-zArIB. Ongd3:xaydl,fngcach~mahinhcualopcach~bi dQngt6iliu,xaydl,fngcac th€ hi~nbi dQngkhacnhaucuacaclophamtoanhf trongkhanggianHilbert voinhifngynghlav~tly hfongling,d6ngthailienh~voiphepgiancach~. Arovd3:xaydlJIlgphepgiancuahamtoanhfcogi::iitichS(z),hIcla timma tr~kh6i ~ ( Sll (z) S(z) ) " S(z) = S21(z) S22(z) don nguyentren yang trOll don vi fj}Jjva thoa di~uki~nt6i thi€u KerSll(z)={O}h~ukh~ptrenfj}Jj.sv dl)11gcack~tquacuaArov, D.C.Khanh d3:khaosatcacbaitoanv~lienk~tcach~,slJbaatoancactinhchMdiOOtiOO cuah~trongquatriOOlienk~t[24],[44],[45],[46].PhuongphapnghienCUlld daylaOOanhfhoacachamtoanhf covaly thuy~tmahiOOtoanhf.Duavao cackhaini~mmoi(:t) nhanhfhoachiOOquycuahamtoanhf,D.C.Khanhd3: thi~tl~pcacdi~uki~nc~ va du d€ baatoantioodi~ukhi€n du<;jc,qUailsat du<;jc,t6ithi€u khi lienk~tcach~donnguyenho~ccach~bi dQng. 13 Cho 8(z) E ,%\U,V-j,8k(z)E $(UbVk), k=l, 2,U)=U, V)=U2,V?=V. Nhan111hoahamtoan1118(z)=8iz)8)(z)dllQCgQila (+)chiOOquyn~utoan 111 Z+: Llli ~ .1.28)hEB.1.)h,Vh E H2(U) sail iliac tri€n tuy~nt£OOlien h;1c1atoan111ddnnguyentUkhonggian .1.H2(U) 1enkhonggian .1.2H2(U2) EB .1.1H2(U1); trongtrlidnghQptoan111 * Z-: .1.*h~.1.2*hEB.1.1*82h,hE 1"2(V) sail iliac tri€n tuy~ntiOOlien h;1c1atoan111ddn nguyentU khong gian .1.*1"2(V) 1en.1.2*1"2(V 2) EBL11*L2(VI) ; vdi .1.*(eit)=(I - 8(eit)8(eit)*)112, .1. (eit) =( 1-8 (eit)8 (eit) *)112k=12'k* k k " , thi OOan111hoahamtoan111dtrendllQCgQila (-) chiOOquy. . SaildaylamQtvaik~tquadllQCdUngtrong1u~an. DiM 1y1.Choh~a 1alienk~tn6iti~pcuahaih~ddngian,ddnnguyen,di~u khi€n dllQCa) va 0.2'Khi doh~a la di~ukhi€n dllQCn~uva chi n~uOOan111 hoahamtruy~n8a(z)= 8aj(Z)8a2(z)1a(-) chiOOquy. DiM 1y2. Choa) vaa21acach~bi dQngt6ithi€u.N~uOOan111hoaham truy~n8a(z)=8aj(Z)8a2(z)1a(:1:)chiOOquythih~a=~aj1ah~t6ithi€u. 4.Caeviin d~nghienetfutrong lu~nan. Ti~ph;1chlldngnghiencootren,d6ivdi cach~dQngh;fctuy~ntioordi f?Cbi dongvdihamtruy~n1ahamcactoan111cogiaitichtrendratrimddnvi, 14 m9t10<;1tcacbailoanmaiduqcd~traho~cdj thi~ncack~tquacuacaclac gianeutren.Nguqcl<;1ivaivfu1d~lienk~tcach~,chUngtoixetbailoantach m9th~thanhn6icuahaih~ddngiand~nghiencootUngh~rieng.Trongbai loannay,chungtoidatimduqcd?llgWongminhcuacach~thanhphftnvada tachh~theohaihl1dng: hudngthllOO~tla tachh~theorinhchiOOquycua hamtruy~n,huangthllhailatachh~theokhonggianconb~tbi~ncualoantV chiOOA. ChUngtoi clingdatimduqcm6ilienh~gi11ahaicachkhaitri~nnoi tren.Cac k~tquanayduqctriOObaytrongchudng2 cualu~ anva daduqc congb6 trong[25].K~ d~n,chUngWi clingxet d~ncacriOOch~tdinhriOO cuacach~vo h<;1nchi~uvavfu1d~lienk~tcach~Dhungphudngphapnghien cood daychuy~ula dUngkhaini~mhamnont6tOO~tcuahamtruy~n.Cho 8(z): U ~ V lahamcacloantVcogiairichtrendlatrimddnvi qj).Nagy- FoiasdachUngmiOOduqct6nt<;1im9thamngoai<p(z)trenqj),oo~ giatri la cacloantVcotU khonggianU vaokhonggianF saocho , <p(eit)*<p(eit) < 1- 8(eit)*8 (eit) a.e. tren ff!)) , va n~u~(z)lahamgiairichcacloantVcosaocho ~(eit)*~(eit)< 1- 8(eit)*8(eit) a.e.thi ~(eit)*~(eit) < <p(eit)*<p(eit) a.e. Ham<p(z)duqcxacdiOOduyOO~tsaikhacm9tloantVh~g ddnnguyenOOan v~belltraivaduqcgQilahamnont6tOO~tcuahamI - 8(z)*8(z). 15 Theodinhnghla,hamcactoantUco giai richtrendla trimddnvi cp(z):U~Fdl1<;5cgqi la hamngoain~u<pH2(U)=H2(F),trongdo cp: H2(U) ~H2(F), (cpu)(z)=cp(z)u(z),uEH2(u);va cp(z)dl1<;5cgQilahamtrongn~u cpla mQtd&ngclJ. Ham non t6tnhfttcp(z)nay da co nhi~uvai tra trongvi~ckhao satcac h~ddn nguyen: khaosatcackhonggiancon b:ltbi~n,thanhphfu1khongqUailsat dl1<;5c,khongdi~ukhiSndl1<;5ccuah~...[9J,[dinhly 3.4,chl1dng3J; xflydt;fng mohinhcach~bi dQngt6i00, t6ithiSu[35J... Do k~tqua: n~uh~a larich n6iti~pcuahaih~exIvaa2thihamtruy~ncuah~exlaeiz) =eal(Z)8a2(z). NenmQtbaitoandl1<;5cd~trachochUngWi laxflydt;fnghamnont6tnhfttcua ham8(z)=8iz)8I(z)tUcachamnont6tnhfttcua8I(z) va 8iz). Cac k~tqua thudl1<;5cdl1<;5ctrinhbaytrongmQtphfu1nQidungcuachl1dng3 va da dl1<;5c congb6trong[26J.K~tquanaydadl1<;5csitdlJIlgdSchUngminhdinhly baa toantinht6i00 cua h~n6i.DlJatrenkhaini~mhamnont6tnh:lt,chUngWi clingtimdl1<;5cdi~uki~ndSmQth~bi dQnglat6i00, d6it6i00;di~uki~ndS mQth~bi dQngla ddnnguyen...Cack~tquanaydl1<;5ctrinhbaytrongchl1dng 4va dadl1<;5Ccongb6trong[28J.Clingtrencdsdhamnont6tnh:lt,chUngtoi dathi~tl~pcacdi~uki~ncfu1vadudSbaatoantinht6i00, tinht6ithiSu,tinh hoantoankhongqUailsatdl1<;5c... trongquatrinhlienk~tcach~bi dQng.Cac k~tquathu dl1<;5Cdl1<;5ctrinh bay trongchl1dng5 cua lu~ an va da dl1<;5ccong b6 mQtphfu1trong[27J va mQtphfu1sedl1<;5Ccongb6 trong[20J. 16 Gia sV CP1(z), CP2(z) va cp(z)Ifm lu<jtla cachamnont6tnhfttling vdi , , « P2(Z)8I(Z) ) , , 81(z),8iz) va8(z),8(z)=82(z)81(z).Ta luonco la hamnon <PI(z) , ,. 8( ) v :. d~ dXt 1 , kh . , « P2(Z)8I (Z» ) - 1 , h' :. nh :. VngVOl z. an e <:;tra a 1nao se a amnontot at, <PI(z) nghlalakhinaotrongbfttd~g thUG « P2(Z)8I (Z) J '" « P2(Z)8I (Z» ) ::;;cp(z)*cp(z) <PI(Z) <PI(Z) codftubfu1gxayfa.Bfu1gcacphuongphapkhacMati, chUngtoithudu<JcmQt s6k~tquav~vftnd~nayxettrencach~mohinhkhacMati.ChUngdu<Jctrinh baytrongchuang3va5cilalu~ anvadadu<JCcongb6trong[26]va [27]. Ngoaifa,chUngWiclingtimdu<JCdi~uki~nd€ haih~t6il1Uco cUngham truy~n;k~tquaco du<JCdu<Jctrinhbaytrongchuang4 va sedu<jcGongb6 trong[20]. 17

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