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

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