Luận văn Phương trình sai phân và phương trình vi phân Riccati: Hội tụ, đơn điệu và ổn định

Clntd'ngI , ~ A;.t , BAI TOitN DIEU KHIEN TOI U'U TOAN PHU'dNG ,..! ~" , TUYEN TINH Rdl R~C VA PHU'dNG TRINH RICCATI 1.1Cac djllh nghia Xetp]llfo'llgtr1nhviphan: x(t) = Ax (t) (1.1) va plllfcJng1.1'1n11sa.ip11an: Xk+l =AXk (1.2) trongdo A la ma tr~nvuongC2lpn. GQi J(A) la 1.~pcactri riengcuaA . Ma 1.r~nA dlfQ'CgQila 5ndinhungv6iphucJng1.dnh(1.1)n~uRe()..)< 0 vo'i lllQi A thuOc0"(A) ho;?,cA duQ'cgQi la Ondinh ling vo'i phuo'ngtrlnh (1.2) Il~U IAI < 1 vo'in19iA 1.huOcO"(A). . C~p(A,B) (11I'9'cgQila 6ndinhhoadu9'cn~ucomatr~nf( saocho matr~llA - B f( 6n dinh. . D~\,t°A,n = ](eTB n !(eTBA n... n !(eTBAn--l 1.rongdon la C2lpcllamatr~nA. C~p(A, B) dU'9'cgQila quaIlsat dU'9'Cn~u0 A,E=0 1.2 Bai toan di~u khi~n t6i tttl toan phu'o'ngtuy(fn Hnh rdi r{tc hii'u h<;tn 1.2.1 Phat bit;u lJ~titocln Xet h~phu'o"ngtrlnh sai pilau: Xk+l=FXk +GUt (1.3) trongdo xk lan-vector tr(;1Ilgthai Utla Tn-vectordi~ukhi~n F la matr:;lnvuongcapn khanghich G la n x Tnmatr~n Haytlmda,ydi~ukhi~nt6iu'u{udt=0,1,..., N -1} d~qrcti~uhoa phi~mham 2 N-1 ( ) T. ~--, { TQ T }J N, x 0, 1.L =:1;N Po XN +- L.; :c/,:. N - k--1XA:+-1LA:RN - A:-171 A: k=O (1.4) VcJ'j Xola tr<;Lngthai band~LUclla h~(1.3) Q/,: la,rnatr~nXc1.cdjnh khongam;k =0,1".., N - 1 Rk la,matr~nxac clinhdlwng; k =0,1,. . ., N - 1 Po 1~1ma tr8.11kholJgam ling V(5'itr~UlgthciiXN 1.2.2 Lo'i gifti D~Hm cla,ycli~ukhi&nt6i nu {1.Lt}C U, vcJ'iU ={1L =(1.Ldf-=-/(Uk E IRm},qI'C ti~tlhoa phiem ham J(N, Xo,1L)ta dung plnr(jng phap nhan tu Lagrange.Qua,trlnh tlm dihl khi~nt6i nu gamnhungbl1<1Csail da,y: Bu"o-c1 Xet b9 nh2t1lt I'lagrange/\1,A2"'" AN' Ta clinhnghia: N-1 L =X~POXN +- L {X[QN-k-1Xk +- 1.L[RN-k-11.Lk}+- k=O N-l Y' { \ T (1 -1 ( "( ) (p. I )1' \ }+-L-t /\k+1 (Xk +- I'Uk - Xk+1 +- I Xk +- (I1.Lk- Xk+1 /\k+1 k::::O Di~u ki~nc5,ncl~J (N, Xo,1L)d<;Ltql'C ti~ula: oL ---=0 OXk -~~=O i~f' =0 OAk k =1,2,...,N k =0,1,...,N-1 (1.5) k=1,2,...,N '1'rl1cJ'Ckhi tlllfC hi~ncacbl1'cJ'ctiep theota din caem~nhcl~: M~nh ere1.1 Cho A la,matr~,nca:pn. 'fa co: (i) a OXx*Ay (J ~:)-x*Axu:r - Ay (ii) - Ax l' vo'i ~ ( !!- -(~--... --~ )ox o:fl' 0:i:2' , ()£n 3 Cluing lninh (i) 'fa co: 0 -x* Ay - OX ( 0- x*Ay, 0- x*Ay,. . ., 0- x*Ay ) T OXI OX2 oxn , 0 -x* Ay OXI 0 ;r*Ay 0;i;-2 0 -x*A OX yn 0 'fa huh =-x*Ay; i = 1,2,. . . ,n. ox'z x*Ay = (xi, X2,. . . ,xri) all a12 ... aln a21 a22 ... a2n Yl Y2 anI an2 ... ann Yn Yl Y2 ( n n n ):L xjajl, :L xjaj2"'" :L xjajnj=1 j=1 j=1 Yn ( f X;ajl ) Yl+ (f X;aj2 ) Y2+... +(f x;ajn ) YnJ=1 J=1 J=1 Do do 0 n ~x* Ay =ailYl +ai2Y2+ .. .+ainYn= L aijYj OXi j=1 NhU' v~,y 0 ox,x*Ay =z n L aljYi j=1 n :L a2jYi j=l n L anjYi j;:::::l 4 all a12 ... aln ) (a21 a22 ... a2n I = Ay nl an2 ... 'ann) l n I (i) ciadU9'Cchlhlgminh. (ii) 'fhay y bc)'ixt1'ong(i) ta cltrqc(ii). M~nh de 1.2 1171+ YZ! = 11m+ ZYI v6i I1l'1m121cAcmat1'~ndo'nvi co c~phaml~IllU'qt121n va rn va Y, Z 121 CClCma tr~nlam clIoca,cdinh tlllfC co,ynghia. Cllltng rninh 'I' 1 / . 1. a,c nrng rum 1: x y 0 T x 0 Z T v (Ji I){ IIl' I =-J () Th~tv~,y,ta co: (1.6) ( X Y ) = ( _X 0 ) ( In 0 ) ( InK -1y )0 T 0 1m 0 T 0 1m Sur 1'a: "x Y 1= IJYIITI() T M<'Jttkha.c ( X () ) ( JY 0 ) ( In 0 ) ( In 0 )Z T = 0 1m () T T ~ 1 1m Do do x 0 Z 1'1=IXII1'1 V~y )( y 0 T JY 0 Z l' . Titp theota ch(rngminh: x Y I -- { IXII . T--ZX lYlncuIXI=-J° Z T -- ITIIX - YT--1Z! I1~U ITI =-J 0 (1.7) rl' Ia co: ( X Y \ c- ( X 0 0 ) ( In X --1 Y )Z T} - Z 1m 0 T - ZJY--IY 5 vo'iIXI#O theo(1.6): ( X Y ) = ( In Z T 0 x y Z TI Y ) ( X - YT-IZ 0 )T T-IZ 1m X y Z fl--'-) .. = IXIIT-ZX-IYI; IXI#O = ITII-"Y - YT-IZI; 1'1'1#0 V~y(1.7) c1U'q'cclllfngminh. Bay giG'ta cllll'ngminh m~nhc1~1.2. X I I In - Y et rna tr~n Z I rn A.p d\lIlg (1.7)vo'iX = In' Y = -Y; T = 1mta dU'q'c In -Y { 1171111171- ZI;l( -Y)I = 11m+ ZYI; /1711#0 Z 1m - IImllIn - (-Y)11:;lZI = 1171+ YZ/; 11ml#0 VlIIml #0 va IInl #0 Hen: 11m +ZYI = JIn + Y ZI M~nhd~1.2cltrq'cllll'ngminh. M~nh d~ 1.3 (-"Y+ YTZ)-l = -"Y-l - X-IY(T-l + Z-"Y-IY)-lZX-l Chltng rninh Ta co: eX + YTZ)[X-l - -"Y-.IY(T-l +ZX-IY)-lZX-l] - )(-"Y-l -XX-IY(T-l + Z-"Y-IY)-lZX-l + YTZX-IYTZX--IY(T--l +ZX-IY)-lZX-l = 1+ YTZ-"Y-l - Y[(T-l +ZX-IY)-l +TZX-IY(T-l +ZX-IY)-l]ZX'-:l = 1+ YTZX.-l - Y[(I +TZ-"Y-IY)(T-l +Z~Y-IY)-l]ZX-l = 1+ YTZ-"Y-l - Y[(T(T-l +TZX-IY)(T-l +TZX-IY)-l]ZX-l = 1+YTZ)(-l - YTZ-"y-l= I 6 SHY ra (X +YTZ)-1 =X-I - X-IY(T-l + ZX-IY)-IZX-l Bli'o'C2 D~J(N, xu,1(,)c10,tqL'Cti~uthl tir (1.5)va,m~nhde1.1ta din: o£ =-=QN-k-lXk +FI' /\k+l- /\k= 0 ox~: aD ---=- - POXN - AN =0 ~y ~ =RN-k-lUk +GTAk-I-I =0 (Ju l'at; ---=- =FXk-l +GUk-l - xk =0 OAk /\k = QN-k-IXk +FT Ak+l /\N = PoxN - - R-1 . GT\uk - N-k-l /\k+l Xk+l = FXk +GUk { Ak =QN-k-lXk +FT Ak+l {::? /\N = PoXN " -1 I' Xk+l =l'Xk - GRN-k-IG Ak+l {::? (1.8a) (1.8b) (1.8c) Bli'dc 3 Ta chungminh ding n~uPk la nghi~mnlraxac dinh dU'O'ngcua plnI'o'ngtrlnh sai pilau Riccati RD E: Pl.>+-!= FTPkF - FT PkG(GTPkG +lid-leT PkF +Qk thl (1.8a),(1.8b)va(1.8c) c1lI'Q'cthoa man. Th~t v~y,chQnAI.~= PN-kXk ta co: AN = POXN (1.9) hie do (1.8b)dlI'Q'cthoaman. Ta l;;ticl1QnUl.~= -RN-k-IGTAk+1 thl (l.8e) clingxa.yfa. Bay giG'ta chLrngminh (1.8a)chrQ'cthoa. Ta co: /\k = PN-kXf" = {QN-,~-l +FI' PN-k-IF- -FI' PN-k-lG(GT PN-k-IG +RN-k-l)-IGl'PN-k-lF}Xk - QN-k-lxk +FTPN-k-l[I - . -G(GT PN-k-IG +RN-k~-d-lGT PN-k-l] (1.10) 7 Tir (1.8c)ta co: Xk+l=FXk - GRN~k-lGTPN-k-lXk+l Suy fa: (I + GRN~k-lGTPN-k-lXk+l =FXk Ap d\lllg m~nh c1~1.2 ta dU'Q'c: II +GRN~I.:-lGTPN-k-ll = II +RN~k-lRN-k-l+RN~k-lGTPN-k-IGI = IRN~k-lRN-k-l+RN~k-lGTPN-k-IGI = InN~k-lllnN-k-l +G'l'PN-k-lG! I- 0 (vI RN-k-l >0; PN-k-l >0) Nlnrv~ymatr~nI +GRN~k-lGTPN-k-l khanghich. Theam~llhd~1.3taco: (1.11) (I +GRN~k-lGTPN-k-d-l =I - G(RN-k-l +GT PN-k-lG)GT PN-k-l (1.12) Da dotir (1.10),(1.11)va(1.12)8UYfa: Ak = QN-k-lXk +FT PN-k-lXk+l Ak = QN-k-lXk +FTAk+l V~y(1.8a)c1U'Q'cthoaman. i Bade 4 Ham di~ukhi~nt6i U'uUk,chi 86t6i U'u!(k vagia tri t6i U'uJmin. . Ham di~ukhi~ll t6i Uu Uk, chi 86t6i Uu !{k Thea bU'o'c3 ta co: Uk - - RN~k-l GT Ak+l - R-1 GTP- - N-k-lN-k-lxk+l - -RN~k-lGT PN-k-l(Fxk + Gud Suy fa: (I +RN~k-lGT PN-k-lG)Uk =-RN~k-lGT PN-k-lFxk hay Uk = -(1 + RN~k-lGl'PN-k-lG)-lRN~k-lGT PN-k-lFxk = -[RN-k-l(1 + RN~k-lGT PN-k-lG)]-lGT PN-k-lFxk = -(RN-k-l +GT PN-k-lG)-lGTPN-k-lFxk - -!(N-k-lXk 8 ',' 1" =- (1J G'l' }C> G)--lt"~'L' }C> ]i 'VOl \.k tk k \.J k . . Tinh gici,tri t6i UU Jmin' Ta c1nl'ngmillh T T - T T Xk QN-'~-IXk +u,~RN-k-lUk - Xk PN-kXk - Xk+lPN-k-lXk+l Xu 11ve phaiclla (1.13)xI'QN-k-1Xk+ u[RN-k-1Uk (1.13) - xl'[QN-k-l + FTPN-k-IF - FT PN-k-1G(GT PN-k-IG + t] ") )-l GTP T(']'l ' X l' fJ 'e- LN-/~-l 1 N-k-.Jl' "k - 'k-Il N--k-l' Idol = X[QN-k-1Xk +x[FT PN-k~I[I - G(GT ~V-k-IG + +RN -k-l )-IGl' PN -k-dFxk - X[+1PN -'~-1Xk+l = XrQN-k-1Xk +xl'FT PN-k-lXk+l - Xr+1PN-k-1Xk+l(xembuck 3) = X[QN-k-1Xk +(FXk - Xk+l)TPN-k-1Xk+l = XrQN-k-1Xk - (GudTPN-k-lXk+l (do (1.3)) - XrQN-k-lJ.:k + u[RN-k-l (-RN~k-1GT PN-k-1Xk+l VI Uk = -RN-_/,:-lGT>"',>llva,/\kll = PN--k--l:J:k-llDcn: Ve phai clla (1.13)b~ngX[QN-k-1Xk +u[RN-k-1Uk V~y (1.15) clu'Q'c1nl'ngminh. N-l J(N, Xo,u) = XJ:rPOXN + L (X[QN-k-lXk +u[RN-k-IUd k=O N-l = xTrPoxN+ L (x[ PN-kXk - xI'-I-IPN-k-IXd k=O T ( T T )= xNPOXN + XoPNxo - Xl PN-IXI +... + +(Xh-1P1XN-l - xTrPOXN) - x$PNxo Nay !.acllll'ng !.C)Jlllin = J(N, :1:0,u) = :v'J'PN:rO'Th\LLv~.y,gi<itill"co<.la,y di~ukhi~n{vdf=,/saGclIoJ(N, Xu,v) < x$PNxo. Khi ay,theablrac1, bU'<fc2, btf(5'c3 ta co: >"1= PNx1vo'ixLI-l= Fx1+GVkvaxfi=Xu_ I") (i 1'\ 1vk - LN-k-l . /\k+l Thea ph5,nd~tUclla buo'c4 till Vk =-/{N-k--l::d vo-i /(k =-(Rk +GTPkG)-IGT PkF va xlTQN-k-lXk +vIRN-k-1Vk=xlTPN-kXl - Xl~IPN-k-lXl+l 9 trong do {Pdf=1 la.clayI1ghi~1l1clla plllfO'ngtrlnh RD E (1.9) Do do: J(N, Xu,v) N-l IT - ~ ( IT 1 l' )- XN POXN+ 0 Xk QN-k-IXk + vkRN-k-IVk k=O - IT? ( ,IT p 1 IT p 1)+- XN OXN+ Xo NXO-XI N-IXI ( IT ]'J IT p 1)+ +( .11' P I IT f) ,,1 )Xl N-IXI-X2 N-2X2 .,. XN-IIXN-I-XN1OXN IT }) 1 T p T p \T~ 1 , - Xo -NXO =Xo NXO <Xo NXO' vO Y Nluf v~yJ( N, Xu,v) ~ X6Pxo vo'iffiQiv E U Hay JlIlin= 11" !(N, Xu,u) = xifPNxo.uEU 1,' 'L' t ~ j '- I' ll '11cac ulfo'c,rcll La.co (.pI 1 y: Dinh If 1.4Xit h~phu'CfngtTinhsaiphan Xk+I=FXk +GUk (1.14) tTOngdo ;X:k:17,-vectortn}ngthii tln!c Uk: m- vectort1'(;,ngthai th7,tc F: nw trtj,ncapn khdnghich G: 11x Tnma trtj,n J(hi a'yne'u{Pdf=l la day n.Qhi~1T"Ctlaph71.CfngtTinhRDE(1.9) vdi ditu ki~ndau Po >0 thiphie'1Ttham, N-I J(N, XU,u}=XhPOXN +L (X[QN-k-lXk +u[ RN-k-iUd k=O vdi Qk >0;k =0,1,. , , ,N - 1 Rk>O)'k=O,l,.,.,N-l Xolatr(1,ngthaibandliu cua(1.14)thi co ditu khiln ta'i'I1u u~~= -!(N-k--I:rkvdi!(k= (Rk+GTPkG)GTPkF (1.1Ga) vecto1't1'(1,ngthc£i:rk-i-l=(F - !(N-k-l)Xk (1.15b) va gid tTi ta'i U'U(nho Ttha't)Jlllin =X6PNxo (1.15c) Den daybaj Loanc1i~ukhi~Ilt6i uu to~lIlph11o'ngtuyent1nhrCiir;;1chuu h;;1nc1aduQ'cgiaiquyet, Tir lo'igiai cuabaiLoan,ta coth~dlia fa m<)tgiai tllU~ttIm lCiigiai bai to<:1,11b~l1gcongql mayHnll 111msail: G11-\I THU ~'l' 10 1. Nh~1,p::ro,Fo 2. Tinh Pdk = 1,. . ., N) theo phu'O'ngtrlnh (1.9) Tinh J(dk =0,..., N - 1) theo(1.146) 3. Tinh X~~,Ukva Jmin Tinh xdk =0,. .., N) theo(1.14c) Tinh udk =O,...,N -1) theo(1.14a) Tinh Jmintheo(1.14d) 4, Xui1t: Xk, Uk, Jmin 1.3 J3~tito<.lllui~nkhi~ll t6i un toall phudllg vo h~nva plnto"ngtrlnh Riccati 1.3.1Phat bien bai toan Xet h~p1nfO'ngtr'tnhsaiphan tuy(fn Huh ro"i r<;tc Xk+l =FXk +GUk (1.16) vo'i Xk: n- vector tr9-ngthai, Xk Uk: rIL- vector di~ukhi~n,llk thuOc(lm F: matr~nvuongcapn, khanghichvacaeph~nttrcuamatr~n1as6 plni'c . G: n x Tnma tr~n,cacphhnttrcuaIDa.tr~n1as6plllfc. Hay tl111dieukhi~nt6i lfU {udo d~qfC ti~uhoa.phi~mham CX) ( Q S ) ( Xk )J(XO,ll) =Eo(x!;u!:) s* R Uk trongdo . Q ]a,11130tr~,I1vuong cap 17"Hermite 111'1'30xac din h chro'ng . R ]amatr~Ilvuongcap171"Hermitexacdinhd(fO'ng ( Q S ) , I l ' 1 1. S* R lllfa. xac C!Il 1 Clwng, Ta k] hi~u:11 =(~,~),dogiathi~t11 >0 11enta djnh nghia . ( Xk ) 2 = (x'ku!:)R( Xk )Uk R Uk J (X ) =illf J (x 'u)0 uEU 0,r trongdoU = {{Uk}~O:UkE(Cm} 1.3.2 Lo'i gi:li . Tnfac hetta codinhnghia: M~nh d~1.4Neuc~p(F,G) 5nc1jnhc5ac1tfQ'cthi l(xo) <00 vai Xo tuy ythu(>c(Cn. Chang minh Vi c<;tP(F,G) 6ndinhhoadtfQ'cHencomatr~nK saoclIo ma tr~nF - GI( 5n djnh. ChQnUk=-I(xk ta co: Xk+l =FXk - GUk =FXk - GI{xk =(F - GK)Xk Suy ra Xk =(F - G!<)kxO(xo Iii tr(;tngthai band~ucuah~(1.3)) Dodo ( Xk ) ( Xk ) ( ! )Uk = -KXk = !< Xk = (1){F ~ GI()kxo Nhtf the ) "2 Xk ( Uk 1111= [en(F - GK)kxor R[(:) (F - GK)kxO) < IIRIIII(:)rIIXol1211F- GKI12k <Mak vai M = IIRIIII( I~) II' IlxolLa =IIF - GKI12 vi F - GI( 5ndjnhlienIIF - GKII <1)(xemb5d~5.6.10[10])dodocy< Tlf do 00 ( Xo ) 2 00 M l(xo,u)=I: k <M I: exk= <00 k=O Uk k=O 1 - ex l(xo) =inf l(xo,u) < l(xo,u) <00uEU M~nhd~c1u'Q'cclllfngminh. . Cac m~nhc1~1.5, 1.6. 1.7 phat bi~uc1i~uki~nd~l(xo) >0 M~nh d~ 1.5 Vo'imQiXoE (Cnco m(>tv E U d~khi l(xo, u) d(;ttgia trj 12 t6i Uti J(xo) thl J(xo) = J(xo, v). HO'nnira v6i mQi day {xk,oh hQi t\l v~ Xo ta co J(xo) < J~;~J(x,,~,o)n~u gio'i h~n (j v~ phai t6n t~i, ChU'llg luillh Theo dinh ngliia cua J(xo) tOll t~i day {Uk}C U sao clio 1 J(xo) < J(xo,ud <J(xo)+k Cho k -t 00 ta UtfQ'c k = 1,2,.., lim J( Xo,ud = J( xo) k--+oo Khidotheob6de16.2.3p353[10]t6nt~iv E U d~ J(xo, v) < lim(xo,ud = J(xo)k--+oo Th~nhungJ( xo,v) > J( xo),dodo J( xo,v) = J( xo) Theo (1.16)vo'imQik =1,2,... t6n t~iUk E U saoclIo J(Xk,O' Uk) = J(Xk,O)' V~ntheob6d~16.2.3p 353[10]t6nt~iv E U d~: (1.16) J(xo, v) < lim J(Xk 0,ud = lim J(Xk 0)k--+oo' k--+oo' V~yJ(xo) < J(xo,v)<kl~~J(Xk,O) M~nh d1;1.6 mu R=(~, ~) >0,R >0vaQ >0tillQ>0 vai A Q =Q - SR-lS*. Cllltng Ininh rr 'a co: ( Q - S~(-I S' ~) - ( I -SR-l ) A ( I 0 )0 I R -R-IS I ( I O ) *A ( I 0 )-R-lS* I -R-lS* I V6i mQi XoE (f}n,ta co: XoQXo = (xo) ( Q - SR-lS* o ) ( I 0 )0 R -R-lS* I - [( I 0 ) ] * A [( I 0 ) ] A R-lS* I Xo R R-lS* I Xo >o(vIR >0) 13 V~yQ ~o. M~nh de 1.7GiaSt?!': . RankR=Rank R +Rank Q . (!" Q) quaIlsat . R >0,Q >0vaR >0 Khi ay J(xo) >0vai illQiXoi- O. Chu'ng luinh Hi~IlnhienJ(xo,u) >0 do it >0 D~t Vk=R-lS*l;k +uk,taco ( * *' ) ( Q S ) ( Xk ) *Q *s xkuk S* R 'iLk = Xk Xk +Uk *Xk +XZSUk+ UZRUk = x*Qx* - x*SP-l S*x +X *Sl J-l S*xk k k L k k L k +UkS*Xk+ XkSUk+ ukRuk = xk(Q - SR-lS*)Xk +vkRVk A - XkQXk +VZRVk Do do: 00 ( Q S ) ( xk )J(xo,u) = Eo(XkUk) s* R Uk 00 A - L (XkQXk+VZRVk) k=O Nay gia S11co Xo i- 0 saoclIo J( xu) = 0,khiaytheoill~nhd~1.5,co 'itE U thoaJ(xo,'it)=O.Lucdo 00 A J(xo, u) =L (xkQXk+v'kRvd=0 k=O A A VI R >0vaQ >0 HenVk=0 vai illQi k =0,1,2,... vaQXk=0vai ffiQik=O,1,2,... Do Vk= 0vo"ill19ik =0,1,2,. .. nenta co: Xk'il = FXk +GUk =FXk - GR--18*Xk = FXk vai F = F - GR-lS* Suy fa 14 Xk = pk:/:O (:rOl~,tr;:1I1gth;l,j hall dh,l1C\'I;1,h(~(1.:1)). va 0 = QX1~= QPlkxOvo'imQik = 0,1,2, Do do XoE °fr,Qnghia 121(F, Q) khongquaIlsiltdLf<;fC.Nl11fngtheogiathittthl (F,Q) quaIlsat (11£Q'c,tu deSclanc1~Ilc1i~uvo Ii (xembe)c1~16.2.7p 355 [10]). V~y J( xo) >0 vo'imQi Xo i- O. . M6i lien 11~giua phi~mham J(N, xo,u) va phLfo'ngtrlnh Riccati ro'i r<;1cDTARE'. M~nh d~ 1.8 N~uX 121nghi~mHermiteIl11axac c1inhdLfo'ngb:1tky cua plwang trlnh DTARE': x = P*X F --- (8*-I- G*./Y F)*(R -I- G* ./YG)--J (8* -I- G* X F) -I- Q thl vo'imOi:roE(])nva u b:1tky thu9CU ta co: R -I-G*./YG >0va N ( ) "2 J(N,xo,u) L\ L Xk k=O Uk IIA N-I - xlv./Y:r:N -I- xoX Xo -I- L IIRxI(S* -I- G*./Y F)Xk -I- k=O (1.17) -I-v'klI~x trong do Rx =R -I- G*XG xtt-IX Xk+l - xZ./YXk = (FXk -I- Gud* X(FXk -I- Gud - xZXXk = xZF*./YFxk -I- xZF*XGu/,~ -I- 'u,ZG*XFXk -I-uZG*XGuk - xZ./YXk = xk(F*X F - ./Y)Xk -I- xZF* ./YGuk -I-uZG*XFXk -I- uZG*XGuk ' - xk(S*-I- G*./YF)* Rxl(S* -I- G* X F)Xk -I- -I-xZ(S -I- F* XG)Uk -I- uk(S* -I- G* X F)Xk -I- -I-uZRxuk - {XZQXk -I- XZSU1;-I- UZS*Xk +-uZRud - [RXl(8* -I- G* X F)Xk -I- ud* Rx[R-I(S* -I- ( ) 112 -I-G*XF)Xk-l-Uk]-11 Xk V'k IIA Suy fa: ( Xk ) 2 . =-xZ+IXXk -I- XZXXk +IIRxl(S*+G*XF)Xk+Uk"~x V'k A 15 (1.18) Do do: 2 ) 11 N Xk J(N,xo,u) = k~O (Uk 1111 N-l - -XN.X XN +;r~/Y-Xu+L IIRxl(S*+G*/YF)Xk +ukl11h k=O M~nhde(hfQ'Cclnfngminh. . Dieu khi~lltoi lru ILkvagia tri toi U'ul(xo). M~nh ct~1.9Gia Slr(F,G) 6ndinhhoaclU'Q'c,(F,Q) quailsatcltfQ'Cva X la nghi~mHermitecuaphU'O'ngtrlnh (1.17)khido 00 l(xo,u) J(xo,u)=xo/Yxo+L IIR-l(S*+G*XF)Xk+UkI11x ko:-.:::O (1.19) Cluing lllillh . VI J(xo, u) < 00va (F,Q) quaIlsatdU'Q'cHentheom~nhde16.2.9p 356[10]co {xN(xu,u)}N hQit1,1ve0 khi N ti~nra vo C1!C.Theo m~nhde 1.8thl: l(xo, u) = lim l(N, Xo,u)N -tOO { N-l } = lim -xjyXxN +xo/Yxo+ L IIRxl(S*+G*XF)Xk+UkI11x N-+oo k=O N-l - xo.Xxo+ L IIRxl(S*+G*XF)Xk+Ukll~x k=O Dinh If 1.10 Ntu (F, G) dn ainh hoa au'Q'c}rankR =rankR +rank Q va(F,Q) quansatdu'(lc}R > 0 vaQ > 0 thiphuangtrinh(1.17)co nghi?mduynht{t.LY>O.Ho'nniia vdi xobeltky thuQc(]}nbaitaan1.3.1 co duynha'tdieukhie"nt6'iuu duQ'cxacdinhbJi uk=-!{ Xk trong ao !{ =(R+G*x G)--l(S*+G*x G)vagiatri t6'iuulal( xu)=x(jXXo Chli'ng Ininh VI R > 0, Q > 0 varankR =rankR +rankQ nentheom~nhde 1.6,Q >0 vaiR=(~, ~) vaQ=Q - SR-1S'. Han nila (F,G) 511dinh hoa dU'Q'c,R > 0 vaCd > 0 nentheodinhly 16.6.1p 367[10] phU'o'ngtrlnh (1.17)conghi~mHermitecluynha:tX >0. Nayta chll'ng 16 minh l(xo) =xoXXo. Th~.tv~yvo'iXobat ky thuQc(Enva u tllY Y thuQc U ta co: J(xo,u) >xoXxo(rn~nhd~1.9) Sur ra J( xo)>xoXXo Xet h~llnUk=-](Xk vo'i J( =(R+G*XG)-l(S*+G*XF) TheoIn~nhde1.8:J(N, xo,uc)= -xivX XN+xuJ\'"Xo<xoXXo Sur ra l(xo,uc) <xoXxo Do do l(xo) <xoXXo V~yl(xo) =xoXXo Ti~ptheota clllrngminhdi~ukhi~nuk=-}( Xk t5n t;;ticluynhat. Di;itI/7LI/Rx=(v,*Rxu)l/2 VI Rx =R+G*XG vaR >0,x >0nenRx >O.M~.tkha.cR+G*XG 13.ma tr~nHermiteHenta co matr~nB saGclIo B*B =Rx. Nhu'v~yu*Rxv,= (Bu)*(Bu)v6iBu E(]}n,theo(1)trang13[21]11.IIRx dung1alll(>tchuElntrongkhonggian(En. Gia sit'co V E U saGclIo J(xo,v) =xoXxo 00 J(xo,v). xoXxo+ L IIRi1(S*+G*XF)Xk+Vkll~x=xoXxo k=O Di~unayxityfa khi IIRi1(s* +G*.XF)Xk+VkllRx= 0, Do 1I.IIRx la mot clmantrong khonggianG;'nnen Ri1(S* + G*~XF)Xk + Vk =0 hay Vk =-J(Xk vo-i ]( = RiI(S* +G*XF). S1)'cluynhatdIa dieukhi~nt6i uu duQ'chtfngminh. 1.4 Bai toctUcti~ukhi~n t6i l1'utoau plutd'ngtuy~nHnh ro'i rc,tc tiuh ti~n 1.4,1Pilat bi~u Xet h~p111t'o'ngtrlnhsaiphan(1.3)vci'icacgiathi~tnInt'baitoan1.2.1. TIm daydi~ukhi~lltoi Uti{uHsls=0,1,...}d~qt'cti~uhoaphi~mham. N-l J(N, Xl' u) =XZ+NPOXt+N+L (xt+QN-k-lXH +uZ+kRN-k-lUHk) (1.21) k=O . trong do Xt latr~ngthaicuah~(1.3)t~ith(jidi~mt Qk 13. matr~nnLt'axacdtnhduo'ng(k =0,1,..., N - 1) Rk 1amatr~nxac dtnhdlt'O'Ilg(k=0,1,.., N - 1) 1.4,2 Lo'i gicii 17 D~t Yj =Xt-Ij, Vj = ut+j Luc do (1.21)co th~phat vi~t: J(N,XtJ1i) - J(N,yo,v) N-l T ~ ( T T )- YNPOYN+ ~ YkQN-k-lYk + vkRN-k-lVk k=O Nillf v~ybai toan 1.5.1trff thanhbai toan 1.2.1.Luc do ta co: . H~Hnclibukhi&ll t6i IfLL Vk= -!(N-~~-lYkvo'i!(k = -(Rk +GTPkG)-lGTPkF' . Gia tri t6i IfLL . Jmin =Y5PNYo trongdo{Pd langhi~mcuaphuo'ngtrlnhRDE (1.9) Do do: 'Ut= -]{N-IXt vo'i ]{N-l= -(RN--l -/- GTPN-IG)-lGTPN-IF Jmin =xtPNXt 18