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

~nvuongcapn IanltrQ'tfngvo'icaephu'o'ngtdnh (3.1)va(3,2)vaXJ >0,X;r >o. B6 d~3.1 [23]Xit haiphuo'ngtrinh RDE (3.1)va (3.2)vdi caegid thitt ve caernatrtJnF, G,R, QbQ2,XJ, .X;rvitaneu a tren. GQixl, ,Xf tan l1cQ'tta nghi~1neuaeaephuO'ngtrinh (3.1)va (3.2)va ky hi?u- - Xk =Xk - Xk lie (t6Xk thdamJin phuo'ngtrinh sau day: Xk+l =FTXkF - FTXkG(GTXkG +R)-lGTXkF+QI ./Yk+I =PITXkPI - PIT ./YkG(GT./YfG+R)-IGT XkP1 +Q hay - - '"?1T . - ~1I ~1T - T - - -1 T - - 1 - Xk+I - 1 "'(kJl - F ./YkG(G XkG +Rd G ./YkFk+Q trongd6 PI =F - G(GT./YlG +R)-IGl'XlF Q =Q2- QI 31 Itk=GTXIG+R B6d~3.1dU'(!cC.B ueSouzath\fchi~n[14],nodU'\"cS11dvngd~chung minht{nhdO'ndi~ucuanghi~mphU'O'ngtrlnhRDE (3.1)vaSlfhQitveua nghi~meuaplllI'o'llgtrlnh(3.1).B6d~naysosaHlIcaenghi~meuaphuO'ng tdnh RDE (3.1)va (3.2)nola m&rQngcaenghi~meuacaek~tquama Nishimura[15]vaPoubeJle[16],[17]v~SI,1'sosanhgiuacaellghi~mC1Jacae phuO'ngtrlnh tuO'ngt\I'nlllI'so saHlInghi~meuacaephuo'ngtrlnh RD E GQi_xl, _Xl1~n1U'Q't1anghi~meua(3.1)va (3.2) M~nh ct~3.2 Vo'ieacgic\thi~tnhub6d~3.1vagiaslI'themdingQ1 > Q2 va co Nosan clIo vI v2 -ANa >-"'\No L I 11 I' . k 0 vI > v2 I' X l X 2 I'" 1 I' Iue C0 VO"l Il1Ql n > , -"'\No-J-k - -"'\No+k VO'l -k , k an ll'Q't a cae nghi~Il1cuaplnrejngtrlnh (3.1)va(3~2) Cluing Ininh - - 2 1Bat X 7\T =)( 7\T - X 7\T. - lvO lvO lvO Q =Q2 - Q1 1 2' A - - Vi XNo ~XNo va Q1 >Q2 Hen XNo <0,Q <0 Ta vi~tl<;l.i(3.1)va(3.2)nhusan: -"'X-l+1=(F +G}(1)xl(F+GI(l) +I{lRI{l+Q1 -"'X-f+1=(F +Gl{f)-"'X-f(F+GI(f)+I(fRI(f +Q2 Vo'i}{L=-(GT-",X-LG+R)-lGTXLF (i =1,2) DoXJ >0,-"'X-J> 0,Q1 >0,Q2 >0vaR >0nenXf >0,X~>0 (k=0,1,2,...) Theo biS d~ 3.1: -:r __-11'- -=1) -='1T-r -2 -' - -1 - ~\No+1- FNoXNollNo - fiNo~\NoG(GTXNoG + R) 1GT-"'X-NoFNo+ Q v&i }~o=F - C(CP'X.lvoG +R)-lGTXlvoF - - 2 - - Vi -"'X-No 0, Q < 0 HenXNo+1< O. SllY ra -Xlvo+1~~X-Jro+1 Nlnr th~ta c1a,dnrng minh m~nhc1~dung v&i k =1. Gia Sll'm~nhd~dungvo'ik, tuc1aj(No+h<O.Ta chungminh XNo+h+l<0 Th~tv~y,apchlllgb6d~3.1rnQtDinnuata co: -- - - -,IT -- ~'1 -;'11' - - l' 2 -1 l' - ~1 XNo+h+l - FNo+hXNo+hl No+h-FNo+k-",X-NoHG(G XNo+hG+R) G XNo+kFNo+k- 32 "" Do '/YNo+k 0vaQ <0DentasuyraXNol-k-ll<O. Theo ly tllllyeLqui Il(;1PLakeLlui;lllXNo+k O. D d' vI v2 ,. . k 00 .o./\. No+k 2': ./\.No+k VO'l mQl n > . M~nh d~3.3 Neu./Ykla nghi~mxacc1inhkhongamcuaphlwngtrlnh RDE ./Yk+l=FT./YkF- FTXtG(GTXkG +R)-lGTXkF +Q (3.3) khongtangt9.imi?tthCiic1i~mNo XNo+k< XNo Khi ay./Y k c1o'ndi~ukhong tang ti?-inwi thCiidi~msail do ./YNo+k+1 O. Clutng ruinh Ta d6ngnhatXk =Xl, Xk+l=./Yl(doQl =Q2=Q) VI X No+1 <./Y No Hell X~r <.xLno - no Do m~nhd~3.1ta suyfa: v2 vI ,. . k 0./\.No+k NllU' the XNo+k+1 O. M~nh d~3.4 NeuXk la nghi~mxacdinhkhongamcuapillfangtrlnh (3.3)khonggiamti?-im9tthCiidi~mNonaodo,tuc la X.No+1>XNo Khi ayX' k do'ndi~ukhonggiamti?-imQitho'idi~msail do nghiala XNo+k+l>XNo+kV(J'imQi k >0 Cluing lllinh Ta dongnhatXNo+1= Xlvo' XNo =X'lvo(do Ql =Q2 =Q) Do XNo+1>XNo ncnXlv-a>./YFvo Theo m~nhd~3.2sur fa .X-lvo+k >./Y'lvo+kvo'imQik > 0 Do do Xlvo+k+l > X No+kvo'imQik > 0 M~nh d~3.5 Neuplnfo'ngtrlnh (3.3)cosaiphancaphaixac dinhkhong dl1o'ngt<;tim9t tho'idi~mNo naGdo, tu.'cla XNo+2- 2XNo+l+ XNo <0 th!vo'imQik >0 ta co XNo+k+2- 2XNo+k+l+ XNo+k< 0 33 Cllltng rninh DM LtXk=.Xk+l - Xk Theo b6 d~3.1L1Xt th6amallpl~U'o'ngtrlnh RD E: L1Xk+1= ifT L1.XkF'l- F'l'l'L1XkG(GTL1XkG+Ad-IGT L1XkFl vo'i Fl =F - G(GTXkG +R)GTXkF Rk=GTXkG +R Q=Q2 - QI =0 (vIQI =Q2) Do Xk >0,R >0neniLk=GTXkG +R >0 Ta co .XNo+2- 2XNo+1+ XNo<0 nen .XNo+2- XNo+1< XNo+1- XNo hay L1XNo+1<L1XNo Theo m~nhd~3,3 ta co: L1X No+k+l 0 Do do .XNo+k+2- 2XNo+k+1+XNo+k O. 3.1.2Sl! do'ndi~uciianghi~nlcuaplnto"ngtrlnh vi phanRiccati M~nhd~3.6CoiX(t) langhi~md6ixlrngphU'o'ngtrlnh /Y(t)=AT/Y(t)+X(t)A - X(t)BR-IBT/Y(t) +Q (3.4) vo'idi~uki~nd~uX(O) >O.D~tA(t) =A - BR-IBTX(t). Lucdota co: (i) X (t)=X(t)A(t)+AT/Y(t) 0 . (ii)X (t)=X(t)}l+ATX (t)- 2X(t)BR-IBT X(t) ChUng minh (i) D~ohamvecua(3.4)ta dU'Q'c: X (t) = ATX(t)+/Y(t)A- [~Y(t)BR-IBT/Y(t) +/Y(t)BR-IBTX(t)] = (AT - /Y(t)BR-IBT)X(t) +X(t)[A - BR-IBTX(t)] = ~Y(t)A(t)+AT(t)X(t) (ii) D~oham2vecua(i) ta dU'Q'C: 0" .. - 0 --1' --1'. - o' X (t) = X (t)A(t)+X(t)A (t)+A (t)X(t)+ATX = X (t)A(t)+ATX (t)+X(t)(-BR-IBTX(t) +(-X(t)BR-IB1')X(( = X (t)A(t)+ATX (t)- 2X(t)BR-I B1'X(t) 34 M~nh d~ 3.7 N~u./Y(t)la nghi~mxaec1inhkhOngameuaphlfO'ngtdnh (3.4)khongtangt<;tim9t tho'ic1i~mt naodo, t1jela ./Y(t)<0 tIll ./Y(t)clancli~ukhOngtangt<;timQithCiidi~msaltdo, nghiala X(t+5)0 Clni'ng nlillh D~ehil'ngminhm~nhd~3,7ta din b6d~: Bo ue 3.8 [8]Xit phuo'ngtdnh Lya]J1lnovthayctditheothcligian ./Y (t) = X (t)M (t) + AfT (t)./Y(t) + vV(t), X (0) = ./Y 0 (3.5) J(Jj hifU if>(t:,T) lr},ma h'iJn Ch1/,y/n tn}.ng thAi 11719vdi J\;[(t). [(hi ({y nghi~mczlaphuo'ngtrinh (3.5)rho bd'i X(t) =pT(t;,O)Xow(t,0)+lotq5T(t,T)TV(T)P(t, T)dT Bay gill'fa ch'ling1ninhm~nhae 3.7 Theom~nhae 3.6)X(t) to'nghi~mcuaphuO'ngtTinh(3.4)nen X (t)=XA(t)+AT(t)X(t) (3.6) vdiA(t)=A - BR-IBT./Y(t) Nhu the'(3.6) Ia phuo'ngtTinhLyapunovvdi TiV(t) =0) do ao theobd ae 3.8 X(t +5) =cjJT(t +5,t)X(t)w(t+5,t) v6i mQi5 >0 trangao W(t,T) Ia 17Wtn7,nchuye'ntrg,ngthai 'lingvdi A(t) , Theogid thie't-,Y(t)0 M~nh de 3.9N~u./Y(t) la nghi~mxaedinhkh6ngameuaphlfO'ngtrlnh (3.4)kh6nggiamt<;tim9ttho'idi~mt naodo,tlfela X(t) >0 lile do ./Y(t) la do'ndi~ukhOnggiamt<;timQitho'idi~msail do,nghiala 35 x(t+ s) >0 vo'imQis >0 ChUng Ininh tlfo'ngt1!nlnf chungminhm~nhd~3.6. 3.2 8\1'hQi t\l Ilghi~ln cua nghi~m cua pl111cjngtdnh Riccati 3.2.1 Sl).'h<)it\l cila nghi~nl cila plUI'd'ngtdnh RDE M~nhde 3.10Xernphu'o'ngtdnhRDE ungVQ'ibaitoandi~ukhi~nt6i l1uLoan pllU'o'ngtuy~ntlnh ro'ir9'cVOh~n1.3.1 x =FTXF - FXG(GTXG +R)-lGTXF +Q (3.6) trongdo . (F, G) 6n d~nhhoadlfQ'C. . (F, Q) quan sat c111<;ic. . Q >0vaR >O. Luc do plnro'ngtrlnh(3.5)coduynhatnghi~mX C1!Cd~i, d6i xung, xcicdinh d110'ng. ChUng Ininh Vi fl >0,Q >0,(F,G)611 djnhboaduQ'cvaR=(~~)tacorank ii = rankR+ rankQ nentheodinhIf 1.10phU'O'ngtdnh(3.6)co nghi~m cluynha:tX qrc d0-i,c16ixlfng,xacdinhkhongam. Nay ta chil'ngminh/Y >O.Th~tv~y,theodinhly 1.10ta cogiatri t6i U'ucuabaiLoan1.3.1la J( xu)=X6/Y Xo M~tkhactheodillhly 16.5.3[10]cou E U saoclIo J(xo,u) =J(xo)=X6XXo Gic\ Slr co Xo ::I 0 eMX6X Xo lilc do <X) J(xo, u) L (X[QXk +u[Ru,J=0 k=O Sur ra uk = 0v&iillQik =0,1,2,.. . (vI R >0) QXk =0 vo'i Il1Qik =0,1,2, . . . VI xk+l =F:rk+G1lknen xk+l =FXk vaxk=Fkxo.Dodo QXk =QFkxO =0 v&i Il1Qik =0,1,2,... NhU'v~yXoE °F,Q = }(er Qn }(erQF n... n ](erQFn-l 36 Dieunaychungto (F,Q) khongquansatdU'Q'c.Vo ly V~y -"Y>o. Dinh Iy 3.11 Coiph7lo'ngtdnh (3.7) -"Yk+l=FTXkF - FI'-"YkG(GTXkG +R)-IGI' XkF +Q (3.3) trongdo . (F,G) o"ndinh hoa duQ'c. . (F, Q) quansat duQ'C. . R >0) Q >o. . Xo >0 Luc doXk -t )( khik -t 00vdiX Langhifmcuaph'l1cJngtrinh (3.5). Clnfng Ininh Vooicaedieuki~ncuadinhlyvatheom~nhde3.10,phU'o'ng trlnh (3.6)collghi~lllcluynhat-"y- qrc cli;ti,cloixung,xac dinh dU'O'ng.Ta chu:ngminhphlfO'ngtrlnh(3.7)luonconghi~mvooimQik =0,1,..° Theogiathi~tta co: R > 0;Xo >0 HenGT-"YoG+R > O. Do etoma tr~nGTXoG+R kh.1nghich. D9-t1(0=_(GTXOG+R)-lGI'-"YoF Lucdo (3.8) Xl = FT -"yoP- FT-"YOG(GT-"YoG+R)-lGT XoF +Q = FTXoF +FTXoG1(o+Q = (FI' +1(6'GT)Xo(F+G1{o)- 1<6'GTXoF - 1<6'GI'XoGJ(o +Q Tir (3.8)suyfa: GTXoF=_(GTXoG+R)1(o Nen -"Y1 = (F +G1(o)1'-"Yo(P+G1(o)+ 1(6'(GT-"YoG+R)1(o- K6'GXoGJ(o+Q - (F +G1(o)TXo(F+G1(o)+J(6'RKo+Q Do -"Yo>0,R >0,Q >0 Hen-"YI>O. Gi.1SlrXk > O. 'fa cllLl'ngminh )(k+1>O. Th~t v~y: D~tKk =_(GTXkG+R)-lGI' XkF Khi do, ly lu~ntlfO'ngt\!OnhU'trenta co: Xk+1= (F +G1(dT-"YdF+GJ(d +1<[R1<[+Q VI. Xk >0, R >0,Q >0 HenXk+l>0 37 Theo nguyenly qui n~pthl plnfO'ngtrlnh (3.7)luan co nghi~mXk vOii m9ik=1,2,... Bay giG'ta clnl'ngto {Xd h<)it~l. 'fa dabi~toX- la nghi~mcuaphU'O'ngtrlnh(3.6)Hen /Y = pT X F --FT XG(GX G + R) GT /Y F + Q vacoiplnfo'ngtrlnh(3.7): /Yk+l=FT/YkF - pT/YkG(GTXkG+R)GTXkF+Q Khi aytheob6d~3,1ta co /Yk-I-l=pT/YkP - PTXkG(GTXkG+R)-lGT/YkPl trongdo:- - Xk =Xk -/Y P =F - G(GT/YG +R)-lGT}(F VIGT/YkG+R >0Den: - - - Xk+l <FXkF forallk E N Suy ra /Yk <(FT)k/Yo(F)k,dodo (3,8) VI (P, G) 611 dinh hOa dlf<!C, (A, Q) qua11 sat dlfQ'C, R=(~ ~) do e16rank R=rankR+ rankQ nentheodtnhly 16.6.4[10]F 121matr~n 6ndinhnenIIFII < 1. Dodo IIXkl1 <IIXoIIIIPI12k lim IIXkl1 =0k---+00 hay lim IIXk - XII =0 k-~oo V~yX" --+X khi k -+00. Dtnh ly duQ'cchungminh, 3.2.2 81,1'hQi t~lcua nghi~In cua phl1'o'ngtdnh vi phan Riccati M~nh c1~3.12Xemphu'O'ngtrlnhRiccatid~i86ARB (3.9)lien k~tvOii bai toan di~ukhi~nt6iU'u to21nphuo'ngtuy~nHuh tren khoangtho'igian va h<;tIl2.2.1 AT/y +/YA - XBR-IBTX + Q = 0 (3.9) 38 Lrongdo . (A,B) 511djnh hoadLfQ'c. . (A,Q) quaIlsa,tchrQ'c. . Q >0va,R >O. Khi ayplllwngtrlnh(:3.9)tOnt;;ticluynhatnghi~mX qrc d;;ti,d6ixu-ng, xac c1!nhdll'o'ng. CIllfng ruinh Ta co (A, B) 5n c1!nhhoadHQ'C,Q > 0, R > 0, (A, Q) quansat duQ'C nenR = (~ ~?)corankII =rankR+rank (2, thea dinh 1]116.3.3 [10],plnl'O'ngtrIllh p.9) co nghi~mcluynhat ./Y ql'C d;;ti,d6i xu'ng,xac d!nhkhong21,r11.Ta chu'ngto ./Y >O. Th~tv~y,theobaj toan 2.2.1ta co J(xo) =x3'./Y:rotheo cljnhIy 16.2.4co 1£E U saocho J(xo, u) = J(xo). Gia str co Xo =f0 sao clIo X6XXo =0tuc la J(xo, lL)=10=(xT(t)QX(t)+117(t)Ru(t))dt=J(xo)=0 Suyrau(t)- 0 (vIR >0) Qx(t) 0 (vIQ >0) Dox(t)=Ax(t)+Bu(t)nenx(t)=Ax(t) I<hiayx(t)=eAtxo ClIonen Q1:(t;)=QeAtxo 0v6iXo=f 0 Dieunaychungto (Q,A) khongquansatduQ'c(clef1 trang30[14]). Vo If V~y./Y>o. Dinh If 3.13Xemplnto'ngtTinhvi phdnRiccati ./Y(t)=J'P'X(t) +X(t)A- )((t)BR-IBT./Y(t) +Q (3.10). tTOngao . (A,B) dn afnhhoaauQ'c. . (A,Q) quansata'l1Q'c. . R >O! Q 2:: ()va X (())>(). J(hi d/y./Y(t) -t X khit -t 00 vdi X la nghi~mduynluitJ qtc ar;riJa6/i xungJ xac dinh duO'ngcuaphuo'ngtTinh(3.9). Chang lllinh Ta chungminhd!uhIi chotnl'cmghQ'p X(O)=o. 39 Theogiathietta co (A,B) 5ndinhhoadlfQ'C,(A,Q) quailsatduQ'C, If > 0, Q 2::0 Hell plnwng trlnh (3.9) co cluy nhat nghi~mX qfc d<;ti,d6i xli'ng,xac dinh duO'ng. VI R >0,Q >0va.X(0)=0 nen theo dinh ly 16.4.3[10]pl1l1'O'ngtdl1h (3.10)t6n t9,jcluynhat nghi~mX(t) va vo'jmQit thu<?c[0,00). 81/ton t<;tinghi~mclla plnfO'ngtrlnh 3.10lam cho bai loan 2,3.1d<;ttgia tri t6i uu J(xo) =x6X(T)xo,trongdo x(O)=Xo la tr<;tngthaiband~ucuah~ phuongtdnhx(t) =Ax(t)+Bu(t). Theom~nhde2.8t5nt9,i1LE U d~clIo -']' J(xo) =10(xT(t)Qx(t)+1L1'(t)R1L(t))dt=X6/Y(T)xo vo'i1L(t)=_R-IB1'/Y(t)x(t). Nayta clnfngminhx6X.(T)xola hamtang.Th~tv~y,v6imQiTl >0, T2>0 saoclIoTl <T2ta tImctieukhi~nt6i uuclIohaiphiemham(xem bai loan2.3.1) r1'l J*(Tb xu,1L) = Jo (x1'(t)Qx(t) +1L1'(t)R1L(t))dtva 1*(72,l:O,u) = 101'2(xl'(t)Qx(t)+1LT(t)R1L(t))dt (3.11) Ta co J*(T1,xu,u) 0,R >0) Dodoplnwngtrlnh(3.10)conghi~mvo'imQit E [0,00)HenJ(T1,xu,u), J(T2, xu,u) l~nluQ.td<;ttgiatri t6i uula xl X(Tl)XOvaxl X(T2)XO'Tir do theo(3.11)taco: xl' /Y ('l'})XO<xl' /Y (T2)XO V~yxo/Y(T)xolahamtang. Ngoai fa v6i lllQi Xo ERn, = fo1'(x1'(t)Qx(t)+1L1'(t)Ru(t))dt < fooo(x1'(t)Qx(t)+uT(t)Ru(t))dt= J(x,O) (xembailoan2.2.1) d~Y ding(]dayta xetbailoan2.2.1v6i S =O. Tif do sur fa J*(xo) < J(xo) hayxoX(T):ro < :ro~Y:ro' I-lamx3'x(T)xotangvabich~nen7FIll xo/Y(T)xo ton t~ivo'imQixo-+00 thuQcRn J*(T, xu,u) 40 D~thay1'YT(t)clingla nghi~mcua(3.10)nhU'ngVI tlnhcluynhatcua nghi~mo'la phu'o'ngtdnh (3.10)nenX(t) =XT(t). VI X(t) la matr~n d6ixungva Iim X (T), ta chungminhX =1'Y. T-hX) Th~t v~yta co: AT 1'Y+ 1'YA = 101(A]'X~+X A)dt = rI lim (ATX(t +T) +1'Y(t+T)A)dtJo 1'-+00 = rI lim[X(t+T)BR-I BTX(t +T) - Q+X(t +T)]dt(do3.1C./0 1'-+00 = 1'YBR-I BT X - Q + lim r1X(t +T)dt1'-+00Jo - X BR-I BTX - Q+ Iim(1Y(t+T) - X(T))1'-+00 VI Iim (X(T + 1)- X(T)) =0 nen 1'-+00 ATX +X A =)( BR-I BTX -Q hayATX +1'YA-X BR-IBT X +Q =O. . 81,1'ki~nnaydni'ngto 1'YIa nghi~mcua(3.9).NhU'ng(3.9)conghi~m cluynha:tnenX =X. NhU'v~y IimX (t)=X.Too M~nhd~dlfQ'Cchu'ngminh. 41 ._.