Luận văn Mở rộng và ứng dụng bổ đề Gronwall-Bellman

38 MlJ r(mgvallngdlJngBddl Gronwall-Bellman HoangThanhLong CHUONG 4 MOT s6 UNG DT)NG Nhuda gidi thi~uaph~nd~utien,B6 d~Gronwall-Bellmanvacac marQngcuano dongmQtvai trora'tquailtn;mgtrong1:9thuy€t dinhtinh phuongtrlnhvi phan.No nhumQtcongCl;lhUllfchd~chungminhmQts6 k€t qu~quailtrQngnhustfduynha't,tinh 6n dinh,danhgia tinhbi ch~n cuanghi~m,...BaygiC5chungtoixin trlnhbaymQts6ungdl;lngcuano. ,..!' ,,? §4.1.s1jDUY NHAT NGHI~M CUA PHUONG TRINH VI PHAN vA TicH PHAN ChoE ={u:I =[to,to+T]~IRll I u lien tl;lC}.Tren E ta trangbi mQt chu§'nIlull=Supllu(t)III'11.111la chu§'nEuclidetrenIRll.DSdangnh~ntha'y 11-1,,1,;1' E la mQtkhonggianvecWva voi chu§'nnhutren,no trathanhmQtkhong gianBanach. X6t phuongtrlnhvi phan: x'=X(x,t), (4.1.1) trongdox'(t)lad~ohamtheotcuahamxCi). X: IRnxI ~I Rnlahamlientl;lctrenmi~nD. D ={(x,t)EIRllxIII x - xollsr, It- tolsT}. GiasaX lamQthamthoamandi~uki~nLipschitztheobi€n x, Lllqn vanth{lcsi loanh()c Mil nganh: 1.01.01 39 MlJ rl)ngvaungdl.mgB6dl Gronwall-Bellman HoangThanhLong nghlala ::JK>0 saocho: IIX(x,t)- X(y,t)1I~KII x - yll,V(x,t),(y,t)ED. (4.1.2) Do X(x,t) la hamlien wc trenD lienhamX(xo,t)la bi ch~ntren It- tal~T. E>~tMo =SupIIX(x(»t)II vaM =Mo+r K. Tli tlnhLipschitz 11-lol$T cuahamX, tacoduQc: IIX(x,t)1I~M, V(x,t)ED. 4.1.1.Dinh Iy 1.1. Ne'uX thoaadu kifn (4.1.2)thihili loanCauchychophuangtrlnh (4.1.1)coduynhli'tnghifmx(t)trenIt- tol<rM-1. ChungminhdinhIy 1.1. Giasax(t),yet)lahainghi<%mcuaphuongtrlnh(4.1.1)thoadi~u ki<%ndftux(to)=y(to) =Xo.Ta co x(t)=Xo + fl X(x(s),s)ds,VtEI. Jlo (4.1.3) yet)=Xo+ fl X(y(s),s)ds,VtEI. Jlo (4.1.4) Tli (4.1.3)va (4.1.4),taduQC: IIyet) - x(t) II ~II ( X(y(s),s)- X(x(s),s)ds II. ~ fl K IIyes)- xes)IIds. Jlo (4.1.5) E>~tu(t) =Ily(t)- x(t)II2 0,apdlJngdinhly 1.1chuang1,taduQCu(t) =0,vdi t thoamanIt- tal<rM-1. Vi d\l. Xet bai loansail: y"(x)+ay'(x)+~y(x)=f(x), (4.1.6) Lllqn vanth(lcsf loan hf)C Mil nganh: 1.01.01 40 MiJ rfjngvalingdlJngBfldffGronwall-Bellman HoangThanhLong y(xo) =a; y' (xo)=b, (4.1.7) Ta chungminhphuongtrlnh(4.1.6)voidi~uki~nd~u(4.1.7)co nghi~mduynh~tren[xo,Xo+T],voiT >0n~lOdo. Th~tv~y,giclsli'phuongtrlnhtrencohainghi~mYl, Y2khclvi lien WcWi b~chaitren[xo,Xo+T]. D:)tw=Yl - Y2. Khi do, wthoaman: w"(x)+aw'(x)+pw(x)=O. w(xo)=w' (xo)=O. (4.1.8) (4.1.9) Ta chungminh:w(x)=O,'v'XE[Xo,Xo+T]. Nhan hai vii cua (4.1.8)voi w' va rut gQn,ta duQc: 1 d --{w'\x) +pw\x)}+aw'\x) =O. 2 dx (4.1.10) L~ytichphanhaivii (4.1.10)tuxodiinx, taduQc: W'2(X)=_pW2(X)- 2aIx w'2(s)ds.Xo (4.1.11) M:)tkhac,tal?i co: W2(X)~T Ixw'\s)ds.Xo (4.1.12) Tuphuongtrlnh(4.1.11),tasuyfa: Iw'\x) I ~ IPII w\x) I+21a I foIW'2(S) Ids. (4.1.13) Thay (4.1.12)vao(4.1.13),taduQc: Iw'\x) I~{IpiT +21a I}foIw'\s) Ids. (4.1.14) Ap d\lngb6d~1chuang1(c=21al+IpIT,k =0,u(xo)=0 ), tathu duQcw'\x) ~O.Suyra Luqnvanthgcsl loanhl)c Mil nganh: 1.01.01 41 Mi'Jri)ngvalingd(tngB6dl Gronwall-Bellman HoangThanhLong w(x) =c, 'v'XE[Xo,xo+T],C la m9th~ngsf{naodo. Do (4.1.10)nentaduQc: w(x)=0, 'v'XE[Xo,xo+T]. 4.1.2.Dinh Iy 1.2. Ne'u~~ tbnt[Ji,lientl:lctrenD thzhai roanCauchychophuangtrenh (4.1.1)trenIt- tol< rM-1coduynhatnghi~m. Ti€p theola m9tsf{k€t quav~st!duynha'tnghi~mcuaphuongtrlnh tkh phan. £)~tLl={(t,S)EQxQlto~s~t~td. 4.1.3.Dinh Iy 1.3. ChoK: L1-f IR la mQthamlientl:lc.fJi,it F ={f If Q -f IR lientl:lc}. Gid sit art)EF vaJ-lE IR la mQthdngso:Khi dophuangtrenhrich phanVolterratuye'ntinhtrenF x(t)=art)+f.1 ft K(t,s)x(s)ds,tirED, Jto (4.1.15) co duynhatnghi~m. Chung minh dinh Iy 1.3. St!t6nt~inghi~mcua(4.1.15)cothSchungminhb~ngnguyen1:9 anhX~coho~ccothSthallikhaotrong[10]. Gia sax(t),yet)la hainghi~mcuaphuongtrlnh(4.1.15).Ta co: X(t)=aCt)+f.1 ft K(t,s)x(s)ds, Jto (4.1.16) yet)=aCt)+f.1ftK(t,s)y(s)ds. Jto (4.1.17) Lllqn vanth{lcsf loan h(JC Mil nganh: 1.01.01 42 Mlf ri)ngvaungdl;lngBtld~Gronwall-Bellman HoangThanhLong Suyra I yet) - x(t) 1~ I f.t1(I K(t,s) IIyes)- xes)1ds. (4.1.18) Do K lien tlJctren~la mQtt~pcompactlient6nt~iM >0 saocho IK(t,s)1~M, '\i(t,S)E~.Tli (4.1.18),tasuyfa: 1yet) - x(t)I~1f.tIM (I yes)- xes)Ids. (4.1.19) Tli ba'td~ngthucnaytasuyrayet)=x(t),'\itEQ bangcachapdlJng h<$quatrongchuang1(mlJc1.1.3)va chliyrangIy(t)- x(t)l?:0.(0) 4.1.4.Dinh Iy 1.4. Cia sitKEL2(tJ.;IR),a(t)EF, j.1lahlings(5'thiphuangtrinh(4.1.15) trenF coduynhlltnghi~m. ChungminhdinhIy 1.4.Tu'angtvnhudinhly 1.3. 1yet) - x(t) I ~I f.tI (I K(t,s) IIyes)- xes)1ds. (4.1.20) Ap dlJngba'td~ngthucHolder,taduqc: Iy(t)- x(t)12~ 1f.t1211K 112 ft Iyes)- xes)12ds. L Jto (4.1.21) D~tu(t) =Iy(t) - x(t)12?:O.Ta co u(t) ~ 1f.t1211K 12L ft u(s)ds.Jto (4.1.22) / ? , Ap dlJngb6de2 trongchuang1(mlJc1.2.1),taduqcu(t)=O,'\itEQ, nghIala phuongtrlnh(4.1.15)trenF coduynha'tnghi<$m.(O) Lllijn vanth{lcsi loanh{Jc Mil nganh..1.01.01 44 MlJ rQngvaungd~tngBd d€ Gronwall-Bellman HoangThanhLong II x(t) - yet) II ~ IIx(to) - y(to) II+( K IIxes)- yes) IIds. Ap dlJngdinh1y1.1chuang1,tadu(,1c(4.2.3).(0) (4.2.6) ChQn8= E . Khi do,taco lIy(t)- x(t)1I~E,co nghla13khi exp[KT] di~ukit%nd~uthayd6inhothinghit%mcua(4.2.1)clingthayd6inho. 4.2.2.Dinh Iy 2.2. Gid sitX la hamthoamandiiu ki~n(4.1.2)va IIX(x,t)- Y(x,t)11::::&,V(x,t)ED. (4.2.7) Niu x(t),yet)leinIU(ftla hainghi~mcua(4.2.1)va (4.2.2)tren/, thztaco ddnhgid: Ilx(t)- y(t)II ::::llx(to)-y(to)llexp[K(t- to)J [; + -{exp[K(t- to)J-1}.K (4.2.8) Chung minh dinh Iy 2.2.Tuangtlj nhuchungminhdinh1y2.1. Th?tV?y,tacodanhgia: II x(t) - yet) II ~II x(to) - y(to) II+(II X(x(s),s)- X(y(s),s)lids ~ IIx(to) - y(to) II+rtII X(x(s),s) - X(y(s),s) II ds Jto +rtIIX(y(s),s) - Y(y(s),s) lids Jto ~ II x(to) - y(to) II+([K IIxes)- yes) II+E]ds (4.2.9) Ap dlJngdinh1y1.1chuang1,tadu(,1c(4.2.9).(0) Lllqn vanth{lcsl loanh(Jc Mil nganh: 1.01.01 45 MlJ rfjngvaungd(tngBddi Gronwall-Bellman HoangThanhLong " "" ...,? ,... §4.3.DANH GIA TINH BJ CH~N CUA NGHIEM Danh gia nghi~mcua mQtphuongtrlnh vi phan co th~cho bie'ttinh 6n dinh, bi ch~n,.... Xetphudngtrlnhviphan: x'(t)=f(x(t),t), (4.3.1) trongdox: 1--+IRllva f: IRHxl --+IRH. Gia saf la hamlien t\lctrenmi~nxacdinhcuano.Ta xetmQtsf) d~ngcuaf thoa man: a. IIf(x(t),t)1I~g(t)lIx(t)11+h(t), (4.3.2) g,h la cachamduong,khatichtren1. b. Ilf(x(t),t)11~C(llx(t)1I+ IIx(t)IICX),0 ~a < 1. c. Ilf(x(t),t)1I~Kg(lIx(t)II). (4.3.3) (4.3.4) 4.3.1.Djnh If 3.1. Ne'uhamf(x(t),t)cilaphztclngtrinh(4.3.1)thoamandiiu ki~n(4.3.2) thEtaco: IIx(t)IIs IIx(to)IIexp[(g(s)dsJ + (h(s)exp[ fg(r)drJds.(4.3.5) Chung minh djnh If 3.1.Ta co: xCi)=Keto) + rtf(x(s),s)ds.Jto (4.3.6) Suy fa II xCi) II ~ II Keto)II +(II f(x(s),s) IIds. Luijn vanthfJcsl loan hflC Mii nganh: 1.01.01 46 MlJ ri)ngvalIngdl,mgBdd€ Gronwall-Bellman HoangThanhLong II x(t) II ::;II x(to) II +1~{g(s)IIxes) II+h(t) }ds. (4.3.7) Ap dl;1ngdinh191.5chuang1,taduQC(4.3.5).(0) 4.3.2.Dinh Iy 3.2. Ne'uhamf(x(t),t)cuaphliangtrinh(4.3.1)thoamandi~uki~n(4.3.3) thi taco: I Ilx(t)ll:::{exp[C(t-to)J{[11x(to)III-a+IJ -l}l-a. (4.3.8) ChungminhdinhIy3.2. Tli (4.3.3)va(4.3.7),taSuyfa: II x(t) II ::;IIx(to) II +C ( (lIx(s)II+llx(s)lIa)ds (4.3.9) Ap dl;1ngdinh192.6chuang2, taduQc: I IIx(t)1I::;exp[C(t- to)]{[IIx(to)III-a+1]-I}1-a.(0) 4.3.3.Dinh Iy 3.3. Ne'uhamf(x(t),t)cuaphliangtrinh(4.3.1)thoamanddu ki~n(4.3.4) vag la hamtang,lientl;lctren[0,00)thi taco: IIx(t)II::; \jf-I [\!f(11x(to)II)+K(t - to)]''ritE/, (4.3.10) trangdo ex 1 \!f(x)= JI -ds (£>0, x>O).E g(s) (4.3.11) ChungminhdinhIy 3.3. Tli (4.3.1)va(4.3.4),tasuyfa: II x(t) II ::;IIx(to) II +( Kg(1Ixes)lI)ds. (4.3.12) Lllqn vanth(lcsfloanh()c Mil nganh: 1.01.01 47 MlJ ri)ngvau'ngd(tngBdd€ Gronwall-Bellman HoangThanhLong Ap d\mgb6d~Bihari,taduQc(4.3.11).(0) Nhu chungta dffbie"tphuongtrlnhRiccatinIt kh6 tlm nghi~mgiai tich tudngminhtrongtrudnghQpt6ngquat,chi'c6 thStlm duQcnghi~m trongmQts6 trudnghQpd~cbi~t.Vi v~yvi~cdanhgia nghi~mcua n6 d6ngmQtvai trohe"tsucquailtrQng. 4.3.4.Dinh Iy 3.4. Nghi~mcuaphuongtdnhRiccatisau y'(t) =a(t)/(t) +b(t)y(t)+kef) (4.3.13) trangdoa(t),b(t),k(t) la cachamlientl;lctrenQ, yECl(.0),vanh~ngia trj th1!Cse thoumandanhgia !y(t)1~M(exp[-(lb(s)ldsJ-M (la(s)lexp[- f' b(r)IdrJds;-1(4.3.14) 'rItE[to,tp),tp=SUp(tEQ I exp[- rtb(s)dsJ(rta(s)dsJ}-i>Mj,Jto Jto M=Sup(ly(to)+ fk(s)dslj.tED. to (4.3.15) Chung minhdinh Iy 3.4. Tli phuongtrlnh(4,3.13),tasuyfa: Iyet)I~Iy(to)+L k(s)dsI+L Ia(s)IIy\s) Ids+L Ib(s)IIyes)Ids ~ M + rl Ia(s)IIyes)12ds+ rl Ib(s) IIyes)Ids . Jlo Jlo (4.3.16) Ap dl;lngdinhly 2.4chuang2, taduQC(4.3.14).(0) Lllqn vanth{lcsi loanh(Jc Mil nganh.. 1.01.01 48 M/J rf)ngvall'ngdl;lngBddi Gronwall-Bellman HoangThanhLong §4.4.SAI LtCH NGHItM HAl PHUONG TRINH VI PHAN MQt phuongtrlnh vi phankhi bi thay d6i vri phai, VI dl;!bC'1icae nhi~u,di~ukhi€n se diin drinslf sai khaenghi~m.Chungta se sa dl;!ng caemC'1rQngeuaB6 d~Gronwall-Bellmand€ danhgia slfsaikhaedo, Cho g: IR+~ (0,00)thoamancaetfnheha't: a,g lien tl;!evatangtren[0,00). b. g(x)s;X,VXE[O,oo). Xet haiphuongtrlnhvi phansail: x' =X(x,t) y' =X(y,t)+R(y,t) (4.4.1) (4.4.2) Giasax, R la caehamlientuetrenD vathoamancaedi~ukien:~. . IIR(x,t)1Is;8(t), (4.4.3) vdi8(t)lamQthamkhatfchtrenI va IIX(x,t)- X(y,t)1Is;Kg(lIx- yll),V(x,t),(y,t)ED. 4.4.1.Dinh IS'. (4.4.4) Niu caehamX(x,t),R(y,t)cuaphuangtrlnh(4.4.1),(4.4.2)thoaman caedi~uki~n(4.4.3)va(4.4.4)thEtaco.. Ily(t)- x(t)II s G-1[G(M)+K(t-to)],ME/, (4.4.5) trongdo 6= Ily(to)-x(to)ll, Luqnvanth{lcsi loanh{Jc Mil nganh..1.01.01 49 Mi'JrQngvalingdl,mgBddi Gronwall-Bellman HoangThanhLong M= Sup{15+fte(s)dsltEl}, Jto (4.4.6) fX 1 G(x)= J, -ds (c:>O,x>O). E g(s) (4.4.7) Chung minh djnhIf. Tli (4.4.1)va (4.4.2),taSuyfa: II yet) - x(t) II ::;II y(to) - x(to)II + (II X(y(s),s) - X(x(s),s) IIds +(II R(y(s),s) IIds ::;IIy(to)- x(to)II+ (Kg(1I yes)- xes)lI)ds +( e(s)ds (4.4.8) f)~tu(t)=lIy(t)- x(t)11~0,VtEI, va 8=lIy(to)- x(to)l!. Tli (4.4.8),taduQC: u(t)::;M +(Kg(U(S»dS. (4.4.9) Ap dvngb6d~Bihafi, taduQc: u(t)::;G-1[G(M)+K(t-to)], haylIy(t)- x(t)1I::;G-1[G(M)+K(t-to)].(D) (4.4.10) 4.4.2.H~qua4.1. Ne'uR(y,t)=0,MEl, thi taco: Ily(t)- x(t)II ::;G-1[G( 15)+ K(t-to)]. 4.4.3.H~ qua 4.2. (4.4.11) Ne'ug(u) =uthi taco: Ily(t)- x(t)II ::;15exp[K(t- to)]+ (exp[ K(t - s)]e(s)ds. (4.4.12) Luljn vanth[Jcsl loan h(Jc Mil nganh: 1.01.01 50 MlJ rQllgvaUllgd(l1lgBiJ dl Grollwall-Bellman HoangThanhLong §4.5.SV PHT}THUQC CUA NGHItM THEO THAM SO Ta danghiencUuslf lien tl;lccuanghi~mtheodi~uki~ndeluva theo v~phai.Bay giotanghienCUuslflien tl;lccuanghi~mtheothams6. X6t phudngtrlnhvi phan: x'(t) =X(x(t),t,~), (4.5.1) x: I ~ IRll; X: IRllxlxlR~ IRll,lahamlientl;lctheocacbi~nvatheo thams6Jl, vathoamandi~uki~nLipschitztheobi~nx,nghlala 3L >0 : IIX(x,t,Jl)- X(y,t,Jl)II::;Lllx - yll,V(x,t),(y,t)ED,VJlEIR. (4.5.2) Dinh Iy. Ne'uphu{fflgtrinh(4.5.1)cohamX(x,t,Jl)thoamanddu ki~n(4.5.2) thinghi~mxl/t) =rp(t,p)cilano lient1;lCrheathamsf;'J-l.. Chung minh dinh Iy. Ta celnchungtorAng: VE> 0, 38(E,~o)>0: I~- ~olIIcp(t,~)- cp(t,~o)1I<E. Th~tv~y,tu(4.5.1),taco: cp(t,~o)=cp(to'~o)+ rl X(cp(s'~o),s,Jlo)ds. Jlo (4.5.3) cp(t,~)=cp(to'~)+ rl X(cp(s,Jl),s,Jl)ds.Jlo (4.5.4) Til (4.5.3)va (4.5.4),tathuduQc: II <p(t,f.l)- <p(t,f.lo)II < II <p(to'f.l)- <p(to'f.lo)II Lllljn viill (h{lcsi (oanh(Jc Mil ngil1lh..1.01.01 43 Mi'irfjngvall'ngd~tngBli di Gronwall-Bellman HoangThanhLong §4.2.Stj LIEN T{)CCUA NGHItM THEO ;:: " ;:: '- "'? DIEU KIENDAU VA THEOVE PHAI. Tinh lien t\lCcuamQthams6la ra'"tquailtrQngvi dt!avao tinh lien t\lCta co th€ xa'"pXl gia tri cuahams6ling vdi st!thayd6i nhobandgu. Tinhlienwc cuanghi<%mcuamQtphudngtrlnhvi phanclingkhongphiii la ngo(;li 1<%. Xet haibai loanCauchysau: X'=X(x,t); x(to)=Xo. (4.2.1) (4.2.2)y'=Y(y,t); y(to)=Yo, x, Y lacachamlient\lCtrenD. 4.2.1.Djnh Iy 2.1. GiGsaX thoaman(4.1,2).Ne'ux(t),yet)fahai nghi~mcua(4.2.1)thi Ilx(t)- y(t)11.$'11x(to)- y(to)llexp[K(t- to)J, Chung minh djnh Iy 2.1. (4.2.3) Giii sax(t),yet)la hainghi<%mcua(4.2,1).Ta co: x(t) =x(to)+ rtX(x(s),s)ds,\itEr,Jto (4.2.4) yet)=y(to)+ rtX(y(s),s)ds,\itEI.Jto (4.2.5) Tli (4.2.4)va (4.2.5),tathuducjc: II x(t) - yet) II :::;II x(to) - y(to) II+(II X(x(s),s)- X(y(s),s)IIds Luqn van th[Jcsi loan h(JC Mil nganh: 1.01.01 51 Mli r(mgvalingdljngBiJ dl Gronwall-Bellman HoangThanhLong + rtII X«p(s,f.1),S,f.1) - X«p(s,f.1o),S,f.1o)IIds Jto II <p(t,/.-l)-<p(t,/.-lo) II < II <p(to,/.-l)-<p(to'/.-lo) II + rt II X«p(S,f.1),S,f.1) - X«p(s,f.1o)'S,f.1) II ds Jto + rtII X«p(s,f.1o)'S,f.1) - X«p(s,f.1o),S,f.1o)IIds.(4.5.5) Jto Do X la ham lien tl;lctheo f.1nen 381> 0, 1f.1-f.11<81, keo theo E:L II X( <pCs,f.lo), s, J.l) - X( <pCs, J.lo), s, f.lo) II < 2exp[L(t, - to)] Tli (4.5.5),tathuduqc: II <pCt, f.l)- <pCt, f.lo)II ~II <pCtIp f.l) - <pCtIp J.lo) II E:L }ds. + 1:.(L II <p(s,~)-<p(s'~o» II +2exp[L(t,-to)] (4.5.6) f)~tyet)=II <p(t,f.l) - <p(t,f.lo)II. Khi do,tli (4.5.6),taduqc: t 8L }ds. yet)~y(to)+II{Ly(s)+2exp[L(t]- to)] (4.5.7) Ap dl;lngdinh191.1chudng1,taduqc: E: yet)~y(to)exp[L(t - to)]+ {exp[L(t - to)]-I} 2exp[L(ti - to)] (4.5.8) Khi dotachQny(to)vataduqcdi~ucffnchungminh.(D) Lll{jnvanlh(lcsf loan h(JC Mii nganh ..1.01.01 52 Mli rf)ngvazIngdlJngB{Jde'Gronwall-Bellman HoangThanhLong N - A §4.6. ON DJNH MU TRONG KHONG GIAN BANACH Khi xet de'nHnh6n dinhnghi~mcuamQtphuongtrlnhvi phan, chungtathudngxetslf6ndinhcuanghi~mt~mthudng,tuclanghi~m x =o.Ne'ux =XI"*0,tacothSd?ty =x - XlvaxetHnh6ndinhnghi~my. Trongph~nnaychungta xetHnh6ndinhmil cuanghi~m.Gia samQi nghi~md~ucothSkeodaide'n00. D'={(x,t)IlIxll~H, to~t <oo},0<H lah~ngs6. Xet phuongtrlnhvi phan: X'(t)=A(t)x(t)+R(x(t),t), (4.6.1) voiA(t)la loantatuye'nHnh,bi ch?n,lienWctheot,R(x,t)lahamlien t\lCtrongD' vathoamandi~uki~n: IIR(x,t)1I~Lllxll,L >0, (x,t)ED'. 4.6.1.Dinh nghla. (4.6.2) Nghi~mX=0 cuaphuongtrlnh(4.6.1)duQcgQila 6ndinhmilne'u t6n t~ia> 0, B >0 saocho: Ilx(t)11~Bexp[-a(t - to)]lIx(to)11. 4.6.2.B6 d~. (4.6.3) Nghi~mcuaphuangtrrnh(2.6.1)cod(lng: x(t)=W(t,to)x(to)+ rlW(t,s)R(x(s),s)ds,JID (4.6.4) trangdo W(t,s)lil taantitCauchy( matrqncaban), W(t,s)=X(t)X./ (s) Lllqn vanthl!csf loanh(Jc Mil nganh : 1.01.01 53 Mi'Jri)ngvau'ngd1;lngBi}dl Gronwall-Bellman HoangThanhLong vdix( t) lit matrcJ-nghi~mcuaphuclngtrinh X'(t) =A(t)X(t). Chungminhbfld~. (4.6.5) Tinh toantrl;1'cti€p x(t)tli (4.6.4)r6i thayvaophuongtrlnh(4.6.1). x'et)=X'(t)X-I(to)x(to) +X'(t) rtX-I(s)R(x(s),s)ds+R(x(t),t). Jto (4.6.6) ThayX'(t) =A(t)X(t) vao(4.6.6)vanit gQn,tadU<;5c: xlet)=A(t)x(t)+R(x(t),t). (4.6.7) V~yx(t)langhi~mcua(4.6.1).(0) Vi d\l 1. X6t h~phuongtrlnhvi phan: { X\(t) =x2(t) , X'2(t)=2tx2(t) (4.6.8) d 2 Ta co:-X2(t) =2tX2(t),tasuyfa,x2(t)=Cetdt N€u C =0 thl x2(t)=0 vachQnXl(t)=1. 2 it 2N€u C =1thl X2(t)=et va chQnXl(t)= esds.to Bi:!t X(t)= [ 1 (eS2ds ]0 t2e D~tha'yX(t)thoamanphuongtrlnh(4.6.8)vadetX(t)= et2"* 0, \ftE [to,oo).Tli day ta d~dangtinh dU<;5cWet,s)=X(t)X-1(s). Lllijn vanlh{lcSl loanhQC Mil nganh ..1.01.01 54 Mi'Jri)ngvalingdl!ngBddl Gronwall-Bellman HoangThanhLong 4.6.3.DinhIy . Ne'uphu(jflgtrlnh(4.6.1)cohamR(x,t)thoamandi~uki~n(4.6.2), IIW(t,to)I1.5'Bexp[-a(t - to)J, (4.6.9) A =a - BL >0 thinghi~mx =0 cuaphurJngtrinh(4.6.1)tindinhmil. ChungminhdinhIy6.1. Ta conghi~mcua(4.6.1)1ft: x(t) =Wet,to)x(to)+ rtW(t,s)R(x,s)ds.Jto (4.6.10) Suy fa II x(t) II ~II W(t,to) 1111Keto) II +II (W(t,S)R(x,s)ds II ~ Bexp[ -aCt - to)] II Keto) II + r BLexp[-a(t - s)]IIxes)IIds. Jto (4.6.11) Ap dl!ngdinh 1y1.8chuang 1,tadU<;5c: IIx(t) II~B IIKeto)IIexp[-( a - BL)(t - to)]. (4.6.12) VI A =a - BL >0,nenphuongtrlnh(4.6.1)6ndinhmil.(D) 4.6.4.H~qua. Ne'uphu(jflgtrinh x'(t) =A(t)x(t)+f(t)x(t), co hamf(t)thoamanIIf(t) II ~L (to~t <00),comatrgncrJbanthoaman (4.6.9),vaA =a- BL >0,A(t) roantiituye'ntfnh,lientl;lc,bi chi;inthi nghi~mx =0 cuanotindinhmil. Vi d\l2. X6th~phuongtrlnhvi phansail: Lllf)n vanthfJcSl loan hf)C Mil nganh : 1.01.01 55 Mi'Jri)ngvazingdljngBtld~Gronwall-Bellman HoangThanhLong ( X\(t)=-Xl (t) X'2(t)=-2X2(t) xJto) =1;x2(to)=2 (4.6.17) £)~t [ -1 0 ] [ XJt) ] A = , x(t)= . 0 -2 X2(t) Khi do (4.6.17)ducjcvi€t l~ithanhX'(t)=Ax(t).Phuongtrlnhnay co nghit%m1a x(t)=exp[A(t- to)]x(to). (4.6.18) M~t khac,tal~ico: [ e-(Ho) 0 ]exp[A(t- to)]= 0 e-2(Ho) , (4.6.19) lien Ilx(t)1I~2I1x(to)11exp[-(t -to)]. V?y nghit%mkh6ng cua ht%phuongtrlnh (4.6.17)6n dinh mil. Luljn van lhCJcsi loan h(JC Mil nganh : 1.01.01 56 MlJ ri)ngvaungdl;mgBd dl Gronwall-Bellman HoangThanhLong ~ """ ;::; ,,? §4.7.ON DJNH CAC H~ TtjA DIED KHIEN Xet phuongtrlnh: x'(t)=Ax(t)+R(x(t),t), (4.7.1) trongdoR(x,t)lahamdi6ukhi€n, lienWctrenD'; Ala matr~nh~ng. N€u phuongtrlnh(4.7.1)duav6d,;mggndungthti'nha't,nghlala R(x,t) thoamandi6uki~n: . IIR(x,t)11=0,11111 I IIIlxll~O I x (4.7.2) vaAla matr~n5ndinhthlnghi~mx=0cuaphuongtrlnh(4.7.1)cling5n dinh. Bay giotaxettruongh<;jpR(x,t)khongthoamandi6uki~n(4.7.2). 4.7.1.Djnh nghia. Ma tr~nA du<;jcgQila 5ndinhn€u Re(Ai)<0, i =1,...,n,trongdoAi, i =1,...,n,la cacgiatri riengcuamatr~nA. 4.7.2.Djnh Iy 7.1. Gid sitR(x,t) thoamandi~uki?n: IIR(x,t)ll::;y(t)llxll & J~y(s)ds<oo, (4.7.3) va matrcJ-nA an dinh thi nghi?mx =0 cuaphuongtrrnh(4.7.1)andinh. Chung minhdjnh Iy 7.1. Nghi~mcua(4.7.1)dudid~ngc6ngthucCauchy: x(t) =exp[A(t- to)]x(to)+ rtexp[A(t- s)]R(x(s),s)ds. (4.7.4) JtD Luqn vanthgcsl loan h{JC Mii ngimh ..1.01.01 57 MlJ ri)ngvazingd~tngBIl di Gronwall-Bellman HoangThanhLong Ta suyraduqc: II x(t) II s II exp[A(t - to)]1111Keto)II +(II exp[A(t - s)]1111R(x(s),s) lids (4.7.5) M~tkhacdonghi~mX=0cuaphuongtrinhx'(t)=Ax(t)6ndinhnen ::3K>0 saocho IIexp(At)lis K. Tli (4.7.5),tasuyfa: II x(t) II s K II Keto) II +K ( yes) II xes) II ds (4.7.6) sK II Keto) II exp[K ( y(s)ds] Do (4.7.3)nentli (4.7.7),taduqc: (4.7.7) II x(t) II S k, II Keto) II vdi kj =Kexp[Kr y(s)ds]. Jto Ta coth~chQnKeto)d~chonghi~mx =0cua(4.7.1)6ndinh.(D) TacomQts6di6uki~nkhacd~danhgias116ndinhcua(4.7.1). Giii samatr~nA cocacgiatri riengAjvaRe(Aj)<O.'v'j=1"..,n. D~tA =maxReA.iCA),'v'j=1,...,n. 4.7.3.DinhIy 7.2. NeuphurJngtrlnh(4.7.1)comatr~nA andjnhvaR(x,t) thoaman IIR(x,t) II '1 ' '1 '1 h' h'" O ? (4 71) A' d ' hs I\.,()' va 1\.,0 <-I\." t 1ng lemx = cua .. on In . Ilxll ' , Chung minh djnh Iy 7.2.Ta co: II x(t) II s II exp[A(t - to)]1111Keto)II +rtIIexp[A(t - s)]1111R(x(s),s) lids. Jto (4.7.9) Do A la matr~n6ndinhnen::3B >0, la mQth~ngs6saocho: Lufjnvanth{lcsfloanh(Jc Mil nganh: 1.01.01 58 MiJ rf)ngvaungdl!ngB6di Gronwall-Bellman HoangThanhLong II exp[A(t - s)] II ~Bexp[A(t- s)],lit 2:s2:to. Tli (4.7.9),taduQc: II x(t) II ~ Bexp[A(t - to)] II Keto) II +BAortexp[A(t- s)] II xes)II ds. Jto (4.7.10) Ap dlJngdinhly 1.8chuang1,taduQC: IIx(t) II :s;B IIKeto)IIexp[(A+Ao)(t- to)]. (4.7.11) DOA+Ao <O,neil limllx(t)1I =0.(0)t~oo Tli dinhly 7.1vadinhly 7.2suyfa 4.7.4.H~qua. Gid SU:A la m(Jtmatr(mandjnh.Ne'u(4.7.1)thoamanm(Jttranghai ddu ki~nsail: 1.IIR(x,t)II~llxllay(t), a<O, IIXW-1~Ao<-A. 2.IIR(x,t)1I~h(llxll)y(t), h(U)~AoU<-AU, (4.7.12) (4.7.13) trangdoh(u) la hamduclng,lientl;lc(u >0),va thiphuclngtrlnh(4.7.1) co nghi~mandjnh. Lui)n vanthlJcsf loan hf)c Mil nganh : 1.01.01 59 Mi'Jr(Jngvaungd~tngBdd~Gronwall-Bellman HoangThanhLong ~ """ §4.8.ON DJNH H~ KICH DONG THU ONG XUYE N X6t phuongtrlnhvi phan: x'(t)=A(t)x(t)+R(x,t)+u(x,t), (4.8.1) trongd6A(t)la loantutuy6ntinh,bi ch?n,lienWctheot.u(x,t)laham kichdQng,lientl;lctrenD' vathoamanIlu(x,t)1I::;;r(t)voir(t)lahamkha tichtrongkhoangthaigianhUllh£;lnba'tky; R(x,t)la hamlien tl;lctrenD'. D?t ho=Sup{r(t)1t ~to}. 4.8.1.Dfnh nghia. (4.8.2) Nghi~mx =0 cuaphuongtrlnh(4.8.1)6ndinhduoilac dQngthuang xuyencuakichdQngu(x,t),n6u '\IE>0,38,h saDchoIIxoll<8,ho<h thl IIx(t)1I<E. 4.8.2.Dfnh If. Gidsitcaedduki~n(4.6.2),(4.6.9)durjcthoaman,va/L=a - BL > O.Ntu V'E>O,llx(to)11<~,ho <~A, thEnghiemx =0 cuaphuangtrinh2B 2B . (4.8.1)and;nhduailacd{)ngthuiJngxuyen. D~chungminhdinhly chungtasadl;lngb6d~sail: 4.8.3.B6 d~. MQinghi~mcuaphuongtrinh(4.8.1)sethoamandanhgia: Ilx(t)11::;;B(~Jt)+~2(t)), (4.8.3) trangdo Luijn vanth{lcsf loan h{JC Mjj nganh ..1.01.01 60 Mil ri)ngvadngdljngBfld€ Gronwall-Bellman HoangThanhLong rPj(t) =exp[-A(t - to)]IIx(to)II, rP2(t)=exp[-A(t-to)]r exp[A(s-to)]r(s)ds. Jlo (4.8.4) (4.8.5) Chungminhb6d~. Tit (4.8.1),tavie'tnghi~mdudid~ngcongthucCauchy: x(t)=W(t,to)xo+ ftW(t,s)[R(x(s),s)+u(x(s),s)]ds Jto (4.8.6) II x(t) II ::;Bexp[ -ex(t - to)] II x(to) II +B( exp[-a(t - s)][L IIx(s) II+r(s)]ds (4.8.7) Ap dt,mgdinhly 1.9chudng1,taduQc: IIx(t) II::;Bexp[-(a - BL)(t - to)] IIx(to) II +B ftexp[-(ex- BL)(t - s)]r(s)ds Jto ::;Bexp[-(ex- BL)(t - to)] {II x(to) II+( exp[(a- BL)(s - to)]r(s)ds} ::;Bexp[-A(t - to)]{IIx(to)II+(exp[A(s- to)]r(s)ds} ::;B(~l(t)+~2(t)).(0) ChungminhdinhIf. Tit (4.8.2)va(4.8.5),taduQc: it h E~2(t)::;hoexp[-A(t- to)] exp[A(S- to)]ds::;~::;-to A 2B (4.8.8) Ap dt,mgb6 d~tren,tasuyfa: II x(t) II <E .(0) Lllljn vanthq.,cs'iloanhfJc Mii nganh: 1.01.01