Các hệ động lực tuyến tính bị động và đơn nguyên

z M(I - z AflB, KaCz)=(I - z AflB, \jf*a(z)=R +z C(I - z AflQ, HaCz)=C (I - zAfl, tacocacd1lngthucsau 1- 8(z)*8(z')=waCz)*waCz')+(1- zz')KaCz)*KaCz'), 1- 8(z)8(z')*=\jf*a (z) \jf*a(z') * +(1- zz')HaCz)Iix(z')*, (4.3) (4.4) trongdoz, z' thuQcdlatrimdonvi (jJJ.Trongtrlfdngh9Pa la h~donnguyen, cactoantVM, N, Q, R bfu1g0nentaco 1- 8(z)*8(z')= (1- zz')Ka(z)*Ka(z'), 1- 8(z)8(z')*=(1- zz')Ha(z)HaCz')*. (4.5) (4.6) 52 H~bi dQngex=( x, U, V, A, B, C, D ) dlf<;5CgQila t6ilfU (tlfongling d6i-t6ilfU(1»)n~uvdimQih~bi dQngex'=(X', U, V, A', B', C',D') cocling hamtruy~nvdih~ex, taco fA kBuk l1 ::; II fA,k B'ud ; \in EN, \iUkEU; k=O k=O ( Tlfongling, Ilk~AkBUkI121Ik~oA'kB'Ukll; \in EN, \iUkEU). H~bi dQngexdlf<;5CgQila *- t6ilfU (tlfongling, *-d6i t6i lfU) n~uvdi mQi h~bi dQnga' vdi8a,(z)=8a(z),taco fA *k c*vk ll ::; II fA'*kC'*vd ' \in EN, \iVkEV; k=O k=O (4.7) (tlfongling, II fA *k C *vk l1 2 11 fA'*kC'*vkll, \in EN, \iVkEV) . k=O k=O Hi€n OOientacon~uex=(X, U,V,A,B,C,D)la t6ilfU ( tlfongling,d6i- t6ilJU)thih~d6ing~ua*=( X, U,V,A*, C*,B*,D* ) la *- t6ilfU (tlfongling, *- d6i t6i lfU). Nglfdi tachlingminhdlf<;5Crbg ( [1] , [2])mQth~bi dQngt6i lfUva t6i thi€u (h~bi dQngd6i-t6i lfUvat6i thi€u) till dlf<;5CxacdiM duyoofitbbiham truy~nsai khacmQtphepbi~nd6i donnguyen;va vdi m6iham8(z) E $(U,V), d~ut6nt?imQth~bidQngt6ilfUvat6ithi€ucohamtruy~nla8(z). G.? ? 8() ( 8(Z) JU V F d' ()I ' h' :. oo:'?la s11 z = : ~ Ef>,trong 0 <pz a amnontot atcua <pCz) (1) Trangbeli[IJ. Arovdungthu(jtngil" *-t6ilIu"thayvi "d6it6ilIu" 53 hamI - 8(z)*8(z).Hi~nnhientacoham8(z) covi cp(z)*cp(z)::; 1- 8(z)*8(z).Giasu a =( X, U, V EBF,A, B, C, D) lah~dongian,don nguyencohamtruy~nla 8(z).D~t 0 _0_0 _0 - A =A, B=B, C=PyC, D=PyD, (4.8) trongdoPyla phepchi~uvuonggoct2fV EBF tenV. Khi dotaco([1])h~ 0 0 0 0 0 a =( X, U, V, A, B, C, D) lah~bidQngt6i11U,cohamtruy~nla 80(z)=8(z) ;va a 0 0 0 0 0 0 N=PFD, M=PFC, \jfo(z)=N+zM(I-zAflB =cp(z). (4.9) a 4.1.Di~ukifD t&itfuva d&i-t&itfucuahf hj dQDg. Giasu a =(X, U, V, A, B, C,D) lah~bidQng.Do(4.3),taco waCz)*waCz)::;I - 8(z)*8(z). Tli diM ng.l}Iacuahamcp(z),tasuyra waCz)*wa(z)::; cp(z)*cp(z). Trongphfu1sailtasechUngminhrfulgn~uh~a la t6i 11Uthi waCz)*Wa(z)se d(;ltc~ tren,nghlala taco waCz)*WaCz)=cp(z)*cp(z);van~uh~a la d6i- t6i 11Uthi waCz)*WaCz)sed(;ltc~ dudi,nghlala Wa(z)=O. Djnh Iy 4.1. H~bidQnga =(X,U,V,A,B,C,D)lad6i-t6i11Un~uvachIn~uwaCz)=O. 54 ChUngminh. Gia sua 1ah~d6i-t6iuu.GQia' 1ah~ddnnguyencocUnghamtruy~n1a 8(z).Vi a' cling1ah~bi dQngnentacovdimQiZE@ valiEU, 11(I-zA)-lBu 112~ 11(I-zA')-lB'u 112 <Ka(z)*Klz)u, u ) ~ <~, (z)*lea, (z)u, u )=? III (4.3)va(4.5),tasuyfa <(I - 8(z)*8(z)- \Viz)*\Va(z))u,u ) ~<(I - 8(z)*8(z))u,u ) ; di~unaydfuld~n<\Viz)*\Viz)u,u )=0,vadodo\Va(z)=O. NguQcl?i, n~u\Va(z)=0,tU(4.3)taco 1- 8(z)*8(z')=(1- zz')Kiz)*Kiz') ; z, Z'E @. Gis.sua' =( X', U, V, A', B', C',D' ) 1ah~bi dQngcocUnghamtruy~n1a 8(z).taco n n II L (l-zkA') - 1B' Uk 112=II L Ka, (Zk):Ik 112 k~ k~ n n =L L <Ka, (Zj)* lea,(Zj)Uj , Uj ) i=lj=l n n - =L L (1-zj Zj) - 1 <( I - 8(zj)*8(zj)- \Va'(Z)*\Va'(Zj))~, Uj) i=l j=l n n n n = L L <Ka(z)*Kizj)~ , Uj)- L L (1- Zjz) - l<\Va'(Zj)*\Va,(Zj)Uj,Uj) i=lj=l i=lj=l n n n = II L (I - zkA)-lBuk 112-L L (1-ZjZj)-l <\Va'(Zj)*\Va'(Z)~, Uj) (4.10) k=l i=lj=l vdimQi{Zk}~=lc @va {Uk}~=lcU. 55 nGia s11\Va'(z) : U ~ W'. D~t h (~)= L (~- Zk)-lUb UkEU. Ta CO k=l n h(~) E L2(U), P - ,'Va,h= L (~-Zk)-l'Va'(Zk)Uk Va L2(W ) k=l n II PL"2(W')'Valh 112L2(W')=II ~1 (~- Zk)-l'Va,(Zk)Uk 112L2(WI) n n -1 -1 =L L «(~- zJ 'Va,(Zi)Ui, (~- Zj) 'Va,(Zj)Uj >L2(W') i=lj=l n n --1 = L L «(1- ZjzJ 'Va,(ZJUi' \Va'(Zj)Uj>W' i=lj=l n n - 1 =L L «(1-ZjZJ - 'Va'(z j) *\Va'(Zi)Uj,Uj>U. i=lj=l Nhl1v~ytU (4.10)taSuyfa n n II L(I-zkA)-lBuk 112~II L(I-zkA,)-lB'uk 112;\fn=1,2,.., \fZkEqj), k=l k=l \fUk E U. Do doa lah~d6i-t6il1Uvadinhly dU<;5cchUngminh. Djnh Iy 4.2. Gia s11a =(X,U,V, A, B, C,DYlah~bidQngco<p(z)lahamnont6t nhfttUngvdihamtruy~n8(z).N~uh~a la t6il1Uthi \VaCz)*\Va(z)=<p(z)*<p(z). Khi dotfmt~imQthamtrongb(z)saocho\Va(z)=b(z)<p(z). ChUngminh. 0 0 Gia s11a lah~bidQngt6il1Udu<;5cmotadtren.D6ivdih~a, taco 56 I - e(z)*e(z')= q:>(z)*q:>(z')+ (1- zz')K 0 (z)*K 0 (z'); z, Z'E rz/). a a 0 Vi exvaa lacach~t6ilfUcoclinghamtruy~nlae(z),nentacovdimQi Z E rz/)va U E U : 1 II 0 0 112 II(I - ZA)-l Bull-=I(I - ZA)-l Bu 2 => IIKa(z)uI12=IIKo(z)u a => (KcJ z) * Ka (z)u,u)~(K~(z)*K~(z)u,u) ( 1- 1 Z 12)«( I - e(z)*e(z)- \VaCz)*\VaCZ))u,u ) = ( 1- 1 Z 12 )«(I - e(z)*e(z)- q:>(z)*q:>(z))u,u ) => => «(q:>(z)*q:>(z)- \Va(z)*\Va(z))U,U ) =0 Suyfa q:>(z)*q:>(z)= \VaCz)*\Va(z). Vi q:>(z)lahamngoainentheom~nhd~4.1( [31]),t6nta)mQthamtrong b(z):F ~ W saocho\lfa(z)=b(z)q:>(z).DiM ly du<;5cchUngminh. Gia sv 8*(z) =(e(z) q:>*(z)):U EDF' ~ V, trongdo q:>* (z) la*-hamnon 0 t6tnhfttcuahamI - e(z)e(z)*vaa * lah~bi dQngt6ilfUdu<;5cdiM nghianhu ~ nhumrthav8(z) bdi8*(z)= ( 8(z) ) : V ~ U ffiF'.B(mgcachap dung ~ ~ / <1'*(z) ~. ~ chUngminhcuadinhly 4.1va diM ly 4.2choh~a*=(X,V,U,A*,C*,B*,D*) 57 d6ingducuah~a =( X, U, V, A, B, C, D ), vabfulgcachS11d1JIlgh~mo 0 hiOObi dQngvat6iuu a * va d~g thuc(4.4); tadll<;1Ccack~tquad6ingdu cuadiM Iy 4.1vadiMIy 4.2. Binb Iy4.3. H~bidQnga Ia*- d6it6iuun~uvachin~u'If*a(Z)=0 . Binb Iy4.4. GiaS11a Iah~bi dQngco*(z)Ia *-hamnont61nMt tl1dngUngvdi hamtruy~n8(z) cua h~.D@h~ a Ia *-t6i uu, di~uki~nc§n Ia 'If*a(z) 'If*a(z) *=*(z)*(z)*. H~qua4.1. MQth~ddnnguyenIah~bidQngd6i-t6iuuva*- d6it6iuu. ChUngminh. Hi@nOOienmQth~ddnnguyena Ia h~bi dQng.Hdn mlan~ua Ia ddn nguyenthicactoootUM, N, Q vaR bfulg0;dodo 'lfa(Z)=0va 'If*a(z)=O. TheodiM Iy 4.1vadiM Iy 4.3,tacoh~ala d6i-t6iuuva*- d6i-t6iuu. H~qua4.2. N~uh~bi dQngdi~ukhi@ndll<;1C(1.11.quansatdllQC)a =(X, U, V, A, B, C, D) lit d6i-t6i lIu (1.11.*-d6i t6i lIu) till toanW T = (~ ~): X E9U ~ 58 (A* X EBV (t.u. T*= B* C: J : X EBV~ X EBU) 1adkg cu. D . ChUngminh. Th~tv~y,vi h~ex1ad6i-t6iuunentaco\VaCz)=O.Tli daydfuld~n 00 WaCO)=N=OvadodozM( I - zA) -IB = L Zk+IMAkB=O.SuyfaMAkB =0, k=O Vk E N. Do h~ex1adi~ukhi~nduQcnenM =O.V~y I - A*A - C*C =0 1- B*B - D*D=0 A*B +C*D=0, , , va T 1adkg c1;f. Phfu1cuah~qualien qUaild~ntinh qUailsatduQCva *- d6i t6i uu cling dUngvi tacohebi dQngex1aqUailsatduQCva *- d6i t6i uun~uvachin~uh~ (X*1adi~ukhi~nduQcva d6i t6i uu. Tli h~qua 4.1vah~qua...4.2,tathuduQcdi~uki~nd~mQth~bi dQng 1adonnguyen. H~qua 4.3. Gia suh~bi dQngex1at6i thi~u.Khi doh~ex1adonnguyenn~uva chi n~uh~ex13.d6i-t6iuuva*-d6it6iuu. 4.2.H~hi dQngva hamnont8tnhfttti'ngvrohamtruy~ncuah~. Cho cp(z)1ahamnont6tnhMcuahamI - 8(z)*8(z),taco 0 ~ cp(z)*cp(z)~I - 8(z)*8(z). 59 Trongphfu1nayta sexett£OOch~tcuah~bi d<)nga trongtnionghQpbi~u thuccp(z)*cp(z)cuahamnont6tOO~tlingvdihamtruy~n8(z)cuah~dq.tduQc c~ tren,nghla1acp(z)*cp(z)=I - 8(z)*8(z);vaxettrongtnionghQpbi~uthuc dodq.tc~ dudi,nghla1acp(z)=0. TuongUJ,taclingxetchotrlJonghQp* - hamnont6tnh~tcp.(z)lingvdiham truy~n8(z)cuah~a. 4.2.1Trudch~ttaphatbi~uk~tquachotnionghQpcp(z)*cp(z)=I - 8(z)*8(z) vacp.(z)cp.(z)*=I - 8(z)8(z)*. Djnh Iy 4.5. Cho a 1ah~bi d<)ngt6ilfU( t.u.*-t6ilfU)cohamtruy~n1a8(z).Khi do h~a hoantoankhongdi~ukhi~nduQc( t.u.hoantoankhongqUailsatduQc) n~uvacmn~ucp(z)*cp(z)=1- 8(z)*8(z)( t.u.cp.(z)cp,,(z)*=1- 8(z)8(z)*). (Bdi diM nghla,h~a duQcgQi1ahoantoankhongdi~ukhi~nduQcn~u X~={O},vaduQCgQi1ahoantoankhongqUailsatduQcn~uX~={0}). Chungminh. Gia sVh~a hoantoankho~gdi~ukhi~nduQc.Vdi mQiZE:llJva liEU, ta ,co (Ka(z)*Ka(z)u,u)= II Ka(Z)U112 = II(I-zA)-lBu 112=0. Tll (4.3),taco ( ( 1- 8(z)*8(z) - \jJaCz)*\jJaCz))u, u >=O. Suy ra \jJaCz)*\jJa(z)=I - 8(z)*8(z).Vi h~a 1at6ilfU,dodiM 1y4.2taco 60 wcxCz)*wa(Z) =cp(z)*cp(z)vadodocp(z)*cp(z)=I - 9(z)*9(z). NgllQCl'.li,gia sv cp(z)*cp(z)=I - 9(z)*9(z).Ap dlJl1gh~thuc(4.3)cho h~bid9nga, taco 1- 9(z)*9(z)= WcxCz)*Wa(z)+( 1- I zl2)KaCz)*Ka(z) (4.11) dUngvdimQiZEq}).Vi h~a 1at6iuunen cp(z)*cp(z)= WcxCz)*Wa(z) III (4.11)va(4.12), tasuyfa (4.12) lieI - zA r IBu 112==0 vdi mQiZEf!lJva liEU. Di~unaydfu:1d6n x~={a}vadodoa hoan.toan khongdi~ukhi~ndllQC. D~chUngminhphfmlienqUaild6nrinhhoantoankhongqUailsatdllQC, taapdlJl1gphfmchungminhnaychoh~d6ing~ua* cuah~a. Dinh 1ydllQC chUngminh. Tli diM 1ynay,tath~yn6uhamI-9(z)*9(z) nhantVhmidllQC( t.ll. 1-8(z)9(z)*)nhantVhoadllQC)nghia1aphlldngtrinh1- 9(z)*8(z)=y(z)*y(z) (t.ll. I - 9(z)9(z)*=y(z)y(z)*) co nghi~m,thi khongth~t6nt'.J.im9th~bi dQnga t6i uu ( *- t6i uu) co khonggiandi~ukhi~ndllQC( t.ll, khonggian qUailsatdll<;5C)khac {O}, 4"" T 11'" ta 't t i' h hI' b ' d" " b' tr " , h'...rangp a..l1nay, xe~ITl..dng QP y ! <;mgVd1 ...amruyenco am nont6tnhMbfulg0 61 M~nhd~4.1. Gis.sv0(z):U ~ V 1ahamtruy~ncuah~bid(>nga vacp(z)=01aham nont6tnh~tcuahamI - 0(z)*0(z) ( t.11.cp*(z)=0 1a*- hamnont6tnh~tcua ham1- 0(z)0(z)* ). Khi doh~a 1at6i11Uvad6i-t6i11U( t.11.*-t6i11Uva *-d6i-t6i 11U). ChUngminh. Do \VaCz)*\VaCz):::;I - 0(z)*0(z)vagis.thi~tcp(z)=0,taSurra\Va(z)=o. D6i vdi h~a, tUh~thuc(4.3)ta d11<;5c I - 0(z)*0(z')=(1- z z')Ka(z)*KaCz'); \j z,z'E9/). (4.13) Gis.sva' 1~h~bi d(>ngcohamtruy~n1a0(z),taclingco \Va'(z)*\Va'(z) :::; cp(z)*cp(z).Di~unaydfuld~n\Va'(z)=0 va 1- 0(z)*0(z') =( 1- zz' )Ka, (z)*Ka, (z') ; \j Z,Z'E9/). (4.14) Tli (4.13) va (4.14),tacovdimQin =1,2/..'{Zk}~=lC 9lJva {Uk}~=lc U, n n II L ( I - zkA rIB Uk 112= II L KaCZk )Uk 112 k~ k~ n n =L L i=lj=l n n = L L i=lj=l n = II L ( I - zkA'rIB 'Uk112. k=l T ' d.3 h " k':' 1 " 1'1-.",:,. ' d ':"':" 11 a..'1gtitlC nay,ta et U<?lla a BytO1u-ruva O1-tO1UU. Phfu11ienqUaild~ntnfongh<;5pcp*(z)= 0 d11<;5CchUngminhbfulgcachdUng d6ing~u. 62 H~qoil 4.4. Gia Slr cp(z)=0 lahamnont6tnh~tcuaham1- 8(z)*8(z) ( t.u.cp*(z)=0 la*-hamnont6tnh~tcuahamI - 8(z)8(z)*).Khi doh~bi d<)ng,di~ukhiSn du<;5c(t.u.qUailsatdu<;5c) du<;5cxacdinhduynhMb6ihamtruy~n8(z). ChUngminh. Tli m~nh4.1,h~exla t6i uu. Ta d~dangth~ym<)th~bi d<)ng,t6i uu, di~ukhiSndu<;5c,lat6ithiSu,nenh~exdu<;5cxacdinhduynh~tb6ihamtruy~n 8(z). Phfmcon l<:licuah~quadu<;5CchUngminh Wongtl;f. Mfnh d~4.2. Gia Slrexlah~bi d<)ngcocp(z)=0lahamnont6tnh~t( t.u.cp*(z)=Ola *-hamnont6tnh~t) Ungvdihamtruy~n8 (z).N~uh~exdi~ukhiSndu<;5c(t.u. quail sat dll<Jc ) !hi tmm h1 T = (~ ~): X $ U ~ X $ V ( t.1I. ( A * C * ] T*= :X EBV ~ X EBU ) ladiingcu. B* D* . ChUngminh. Theom~nhd~4.1,h~exla d6i-t6iuu.Do h~qua4.2,tak~tlu~ toantV T la diingclJ . B6i d6i ng~u,taco k~tquachotn.fdngh<;5pc *(z)=O. 63 Tli m~OOd~4.2,taco m<)tdi~uki~ndu d~m<)th~bi d<)nga hi m<)th~ donnguyen. Hf qua4.5. N~uh~bi d<)ngt6i thi~ua co caehamnont6tOO~tva *-hamnont6t ooMlingvdihamtruy~ne(z)d~ubfulg0,thia lah~donnguyen. 4.3. M8i lienhf giuacaehf hi dQngt8i tin ( *-t8i tin) . Tfong lu~ annay,ta gQim<)th~tuy~nt1OOa =( X,U,V,A,B,C,D ) la m<)tndi f<)ngbelltraicuah~tuy~nriOOa' =( X',U,V,A',B',C',D') n~ut6ntq.i m<)tkhonggianconG" saDeho x =G" EBX', A*G cG B*G ={O} A'=A I B'=B C'=C I D'=D" ¥, " , X" , x., . Tuongtlf,m<)th~tuy~nt1OOa =( X,U;V,A,B,C,D ) la m<)tndi f<)ngbell phaicuah~tuy~nriOOa' =( X',U,V,A',B',C',D' ) n~ut6ntq.im<)tkhonggian con G saDeho X =X' EBG , AG c G CG ={O} A'* =A* I B'*=B* I C'*=C* D'*=D*, , x" x" , . Ta d~dangki~mITaduQcn~uh~a lam<)tndif<)ngbelltraiho~cbellphaicua h~ a', thihamtruy~ncuacach~a vaa' bfulgOOautrongm<)tIanc~ naodo cuaO.Ngoaifa,n~ua lam<)tndir<)ngbelltraicuaa' thih~d6ingftua* la m<)tndir<)ngbellphaicuaa'* . 64 Djnh If 4.6. Di~uki~ndm va du d~haih~bi dQngt6i liu ( tv. haih~bi dQng*-t6i liu) aI, ~ co clinghamtruy~n,8cq(z) =8a;2(z), la t6ntc;timQth~bi dQngt6i thi~u,t6iliu(tv. h~bidQngt6ithi~u,*-t6iliu) a' 8aochoab ~ landirQng belltnii(tv. ndirQngbellphai)cuah~a'. ChUngminh. Gia 811cach~t6iliu ab a2coclinghamtruy~nla6(z).D~t X, - xc x, - xc x, , h" ,- ( x, DV A' B' C' D' ) d ' I - 0.1' 2 - (X2' et cac yak - k", k, k, b k trong 0 A'k= Akl X'k' B'k=Bb C'k=Ck IX'k ' D'k=Dk, tacoh~aklamQtndirQngbelltraicuah~a'kva AkBk =A'k Brbk=1,2. Gia 811 T: x\~ X'2 A'r B'l u1--7A'~B'2u, \fnEN, \fuED, n n ." Ark B' "A,kB'L 1 1Uk 1--7 L 2 2 Uk . k=O k=O Dogiathietcaeh~ab ~ lat6iliuvacoclinghamtruy~n,tad~dangki~m chUngdv<JcT dv<JcxacdiM valatoantVd6nnguyen.Tli diM nghTacuaT, ta , co TA' (A,nB' u)- A,n+lB' U- A' T(A,n B' U)III - 22- 2 11, nen TA'I=A'2T. (4.15) (4.16)H6nmlataclingco TB\=B'2' 65 Vi cach~aj, a2coclinghamtruyenlien 8(z)=Dj+zCj(I- zAjfjBj= D2+zCiI- zA2fjB2 tUdaydfu1d~n8(O)=Dj=D2, (4.17) 00 00 khi do IznC1AfB1 = IznC2A~B2, do do CjAfBj= C2A~B2.Tv dinh n=O n=O nghiacuacach~a\ vaa'2tasuyraC\A'fB\= C'2A'2B'2"V~y C'j=C'2T. (4.18) ToantVT 1adonnguyenvathoacach~thuc(4.15)- (4.18), lien a\ vaa'2 Wongdvongdonnguyen.Honmla,vi A~Bk =A'~B'kvaX' k=X~k' lien h~ a'k1at6ithi~uvat6iuu. Ngu<jcl?i, giltsut6nt?im<)th~bi d<)ngt6ithi~uvat6iuua' saocho aj va a2 1andi r<)nghentrai cuah~a', khi do 8al(Z) =8a,(z) =8a2(Z). Ngoaifa, dotacovdi k =1,2: Xk =G*kEBX', A \ C G*k , B\G*k ={a}, A' =Aklx' , B' =Bk, lientasuyraA ~Bk =AmB'.Vi h~a' 1at6iuu,liencach~al va a2clingt6i uu. BfulgeachapdlplgchUngminhnayehoeach~d6ing§.ual* va~*, ta co k~tquacho trlfongh<jplien quand~n*-t6i 1111. D~ndaytaduarakhaini~mveWongduongdonnguyenb<)ph~ va *- Wongduongdonnguyenb<)ph~. Caekhaini~mnaysedu<jcsudlplgd~n 66 (t.u.*-WongduongdonnguyenbQph~). ChUngminh. Giltsv aI, a2la cach~bi dQngt6i1lUcoclinghamtruyen.D~tX'I= X~I' X'2= X~2'khidotoantV T: X'I~ X'2trongphclnchUngminhcua diM ly 4.6thoadieuki~ndScach~al va~ laWongduongdonnguyenbQ ph~. Ngu<jcl(;li,giltsVhaih~al vaa2 laWongduongdonnguyenbQph~. Khi dotfmt(;licackh6nggianconX'I C Xl , X'2C X2vatoantVdonnguyen T: X'I ~ X'2thoa(4.19)va(4.20).Vdi k=1,2,d~t A'k=AklXk' B'k=Bk, C'k=Cklxk' D'k=Dk. (4.21) Tli (4.19)va(4.21)tacoaklamQtndifQnghentraicilaa'knenakvaa'kco clinghamtruyen.RonmJa,do(4.20)tasurfa a'l vaa'2laWongduongdon nguyen,nenchUngcoclinghamtruyen,v~dodocach~al va~ clingco clinghamtruyen. Tr1ldngh<jpcach~bi dQng*-t6i 1lUdu<jcchUngminhWongUJ. l 68 ._.