Mở rộng và ứng dụng bổ đề gronwall bellman

D~I HQC Quac GIA TP. HO CHi MINH TRUONG DAI HOC KHOA HOC TU NHIEN HoANG THANH LONG MO RaNGvA UNGDUNG . . ~ BODE GRONWALL-BELLMAN ~ Chuyen nganh : Toan Ghii Tich Ma s6 : 1.01.01 LUA.N . VAN THAC . 51 ToAN HOC . NGUOI HUaNG DAN KHOA HOC: PGS. TS. NGDYE N DINH PHU _ . I;Ht.~H.TlI' NHIEN ~.::(."; lTHtJ \t1~N 001103 I. TP. Hfi CHi MINH - 2005 MlJ ri)ng va ung d~ng Bd d~ Gronwall-Bellman Hoang Thanh Long MUCLUC . . Trang M1).C 11).c. Loi Carn do. Danh rn1).ccac ky hi~u. - Chu'dng T6ng quail. Chu'dng - B6 dS Gronwall- Bellman va mQt sf{ m(j rQng d(;lngtuy€n tinh. Chu'dng - MQt sf{m(j rQng d(;lngphi tuy€n. 13 Chu'dng - MQt sf{ m(j rQng d(;lng ham exponent. 34 Chu'dng - MQt sf{ling dl;!ng 38 §4.1. Sl;!'duy nha't nghi~m cua phuong trlnh vi 38 phan va tich phan. §4.2. Sl;!'lien tl;!Ccua nghi~m theo diSu ki~n 43 d~u va theo v€ phai. 45 §4. 3. £hinh gia tinh b~ ch~n cua nghi~m. §4.4. Sai l~ch nghi~m hai phuong trlnh vi phan. 48 §4.5. Sl;!'phl;! thuQc cua nghi~m theo thalli sf{. A' §4.6. On d~nh mil kh6ng gian Banach. Lufjn wIn th[Jc Sf loan h{JC 50 52 Mil nganh : 1.01.01 Mlf ri)ng va ung d¥ng Btf di Gronwall-Bellman Hoang Thanh Long §4. 7. On dinh cae h~ tlfa di~u khi€n. ~ §4.8. On dinh h~ kich dQngthu'ong xuyen. 56 59 Ke't lu~n. 61 Tfti li~u tham khao. 62 Mi'Jri)ng va ling d~tngBd di Gronwall-Bellman , Hoang Thanh Long ? L(J/ CAM (j N Ddu lien toi xin chan cam an sau sdc den Thiiy PGS.TS. Nguy~n Dinh Phu:dii t(in tlnh huang ddn toi tit d~ cuang den luc hoan lu(in van. Toi xin chan cam an haT Thiiy phiin bifn PGS.TS. Dinh Ngl)c Thanh va Thiiy TS. Nguy~n Thanh Long dii dQc lu(in van va dang gap nhi~u y kien quy bau. Toi xin chan cam an cac Thdy Co TruiJng Dc;zi HQc Khoa HQc T1;CNhien, TruiJng Dc;zi HQc Su Phc;zm, TruiJng Dc;zi HQc Bach Khoa dii t(in tlnh giang dC;Zyva truy~n dc;ztnhi~u kien thac mal, b6 ich giup tai lam quen ddn Val vi~c nghien cau khoa hQc. Tai ciing xin chan cam an gia dlnh, cam an cac bc;zndii luon luon df)ng vien, giup diJ va tc;zodi~u ki~n v~ mQi m(it di tai hoan to/t vi~c hQc. Luljn van tlUJC sl loan h{JC Mil nganh 1.01.01 MlJri)ng va u'ng d~tngBlf di Gronwall-Bellman Hoang Thanh Long DANH MUC . KY HIEU . Trang lu~n van co sa dl,mgcae ky hi~u va quy tide cgn thi€t. 1. IRll:Khong gian thl;icn chi~u. 2. IR+ = [0,00). 3. Q = [to,tr] c IR. 5. 1.1: Gia tfi tuy~t d6i . 6. 11.11 : Chu§'n. Tren IRllta lfiy chu§'n euclide. 7. exp(u) = ell. 8. Sup : C~n tren. 9. inf : C~n dtidi. 10. max : Gia tri IOn nhfit. 11. L : T6ng. 12. A = (aik), i,k = I, .,n, la ma tr~n vuong cfip n. n 13. IIAII = Supk Ii=l laik I, la chu§'n cua ma tr~n A. I 14. IIullL2= (1:,lu(xW dX)2 < 00. 15. Dam : Mi~n xac dinh. 16. Re : Phgn thl;ic. 17. (D) : K€t thlic chung minh. Lllqn van thl!c si loan h(JC Mil nganh : 1.01.01 , MlJri)ngva llng d~tngBd dil Gronwall-Bellman Hoang Thanh Long CHUaNG ~ TONG QUAN Trang loan hQc t6n t~i mOt sf{ phuong trlnh va ba-t phuong trlnh ra-t quail trQng. Chung mang nhi~u y nghla th1fc ti~n cho nhi~u ling dvng khac nhau. Ba-td~ng thuc Gronwall hay B6 d~ Gronwall-Bellman la mOt sf{do. B6 d~ co d~ng h€t suc ra-tdon gian, nhung duQc ling dvng ra-thi~u qua dS chung minh s1fduy nha-t nghi~m cua phuong trlnh vi phan, dung danh gia s1f sai l~ch nghi~m, dung dS tlm di~u ki~n du cho mOt sf{ bai loan 6n dinh nghi~m, VI v~y doi h6i phai hoan thi~n ba-t d~ng thuc nhu mOt di~u ta-t nhien cua quy lu~t phat triSn. Va co nhi~u lac gia ma rOng theo y tuang va mvc dich khac (xem [1, 2, 3, 4, 5, 6]). Tuy theo mvc dich giai quy€t bai loan ma cac lac gia ma rOng khac nhau. Theo chung Wi v~n can ra-t nhi~u d~ng, nhu d~ng lilY thua, d~ng ham exponent, cgn duQc ma rOng. Mf:lCdich cua lu4n van la t6ng k€t cac d~ng cua B6 d~ GronwallBellman, d6ng thai ti€p tvc ma rOng va trlnh bay mOt sf{ling dvng cua chung. Lu~n van duQc chi a lam nam chuang. Chuang - T6ng quail. Chuang - B6 d~ Gronwall-Bellman va mOt sf{ma rOng d~ng tuy€n tinh. Chuang g6m B6 d~ Gronwall-Bellman va dinh ly 1.1-1.9. Chuang - MOt sf{ma rOng d~ng phi tuy€n. Chuang g6m mOt b6 d~ b6 trQ va 23 dinh ly 2.1-2.23. Luljn van lh{lc si loan h(JC Mil nganh : 1.01.01 . Mli rf)ng va ling dl!-ngBd di Gronwall-Bellman Hoang Thanh Long Chuang - MQt 86 md rQng d~lllg ham exponent. Chuang g6m illQtb6 d€ va 04 dinh 19 3.1-3.4. Chuang - MQt 86 ling dvng. Chuang g6m lInh v1,1'cling dvng khac nhau. Nhling ling dvng 1a: Chang minh sf! nh{{t nghi~m cua phuong trinh vi phan va rich phan; Sf! lien tl:lCcua nghi~m rhea di~u ki~n ddu va rhea vi phdi; Danh gia tfnh hj chc;incua nghi~m; Sai l~ch nghi~m > thi nghi~m x - to)J, (4.6.9) = cua phurJng trinh (4.6.1) tin dinh mil. Chung minh dinh Iy 6.1. Ta co nghi~m cua (4.6.1) 1ft: x(t) = Wet, to)x(to) + rt W(t,s)R(x,s)ds. Jto (4.6.10) Suy fa II x(t) II ~ II W(t,to) ~ Bexp[ 1111Keto) -aCt II + II (W(t,S)R(x,s)ds II - to)] II Keto) II + Jto r BLexp[-a(t - s)] IIxes) IIds. (4.6.11) Ap dl!ng dinh 1y 1.8 chuang 1, ta dU 0, nen phuong trlnh (4.6.1) 6n dinh mil.(D) 4.6.4. H~ qua. Ne'u phu(jflg trinh x'(t) = A(t)x(t) + f(t)x(t), co hamf(t) thoa man IIf(t) II ~ L (to ~ t < 00), co ma trgn crJban thoa man (4.6.9), va A nghi~m x = a- BL > 0, A(t) roan tiituye'n tfnh, lien tl;lc,bi chi;inthi = cua no tin dinh mil. Vi d\l 2. X6t h~ phuong trlnh vi phan sail: Lllf)n van thfJc Sl loan hf)C Mil nganh : 1.01.01 55 Mi'Jri)ng va zing dljng Btl d~ Gronwall-Bellman Hoang Thanh Long \ =-Xl (t) X (t) (4.6.17) X'2(t) = -2X2(t) ( xJto) = 1;x2(to) = £)~t A= -1 [0 -2 ] , x(t) = XJt) [ X2(t)] . Khi (4.6.17) ducjc vi€t l~i X'(t) = Ax(t). Phuong trlnh co nghit%m 1a x(t) = exp[A(t (4.6.18) - to)]x(to). M~ t khac, ta l~i co: e -(Ho) exp[A(t - to)] = [ (4.6.19) e -2(Ho) ] , lien Ilx(t)1I~ 2I1x(to)11exp[ -(t -to)]. V?y nghit%m kh6ng cua ht%phuong trlnh (4.6.17) 6n dinh mil. Luljn van lhCJc si loan h(JC Mil nganh : 1.01.01 56 Hoang Thanh Long MlJ ri)ng va ung dl;mgBd dl Gronwall-Bellman ~ """ ;::; ,,? §4.7. ON DJNH CAC H~ TtjA DIED KHIEN Xet phuong trlnh: x'(t) = Ax(t)+R(x(t),t), (4.7.1 ) R(x,t) la ham di6u khi€n, lien Wc tren D'; Ala ma tr~n h~ng. N€u phuong trlnh (4.7.1) dua v6 d,;mgggn dung thti'nha't, nghla la R(x,t) thoa man di6u ki~n: 11111 . IIR(x,t)11 Ilxll~O I Ix II = 0, (4.7.2) va Ala ma tr~n 5n dinh thl nghi~m x =0 cua phuong trlnh (4.7.1) cling 5n dinh. Bay gio ta xet truong h 0, 38, h saD cho IIxoll< 8, ho < h thl IIx(t)1I< E. 4.8.2. Dfnh If. Gid sit cae ddu ki~n (4.6.2), (4.6.9) durjc thoa man, va /L = a - BL > O. Ntu V'E>O,llx(to)11 [...]...10 Hoang Thanh Long Md rqng va ung dljng Bd di Gronwall- Bellman c) Ke't qua a) va b) win dung neu thay r Jto biJi r' va Jt t Ss biJi r' Jt Chung minh dinh Iy 1.4 Xem [3].(0) 1.2.5 Dinh Iy 1.5 (Xem [4 D V6i cac gid thie/; nhu djnh ly 1.3 Va gid sit a( t)... (k(s)u(s)ds, ( 1.20) btEQ, thE u(t) ~ betHc(to)exp[ + Lll{jn van thlJc Sf loan hQc ( b(r )k(r )dr] r c'(s)exp[ fb(r)k(r)dr]ds), ~ s tltEQ (1.21) Mil nganh : 1.01.01 11 Mil ri)ng va ung d1!ngBd di Gronwall- Bellman trang do crt) Hoang Thanh Long = art) b(t) Chung minh djnh Iy 1.6 Chia hai v6 cua (1.20) cho bet), ap dlJngdinh ly 1.5.(0) 1.2.7 Djnh Iy 1.7 (Xem[10], tr.191-192) Cho u(t), art) la cac ham... to)]}' \?tEn (1.25) Chung minh djnh Iy 1.8 Tlnh loan tnjc ti6p tu dinh ly 1.5 ho~c chung ta Luqn van th{lc sf loan h(JC t)H.~H.TtfNH'EN THtr\lIEN Mil nganh : 1.01.01 12 MlJrQngva u'ngdlJng Bii d€ Gronwall- Bellman Hoang Thanh Long c6 thS chung minh nhl1 sail: B~t x(t) = u(t)exp( at) ( 1.26) Khi d6, tu (1.24), ta dl1Qc: ( 1.27) x(t) S Keto)+ i~ [axes) + bexp(as)]ds, Ap dl;lng dinh 1:91.5, ta dl1Qc: x(t)... Binh 1:91.9 t6ng qu:H h6a dinh 1:91.8 Binh 1:91.8 dl1Qcsuy ra tu dinh 1:91.9 trong trl1dng hQp a, b 1a cac ham h~ng, Lllqn van thfJc sl loan h{Jc Mii nganh : 1.01.01 13 MiJr{}ngva ung d1!ngBli d€ Gronwall- Bellman Hoang Thanh Long CHUaNG 2 " ,," " ? "" MOT SO MO RONGDANG PHI TUYEN Trang chuang 1 chung Wi da trlnh bay mQt 86 k€t qua md rQng d(;mg tuy€n tinh d6i voi ham u(t) Trang chuang nay chung Wi... b6 d~ b6 trQ Xem [5].(0) 2.2 Dinh Iy 2.1 Cho u(t) la ham lien tl:lC tren Q Gid si/:art), bet), cp(t)la cac ham lien Luijn van thlJc si loan h{Jc Mil nganh 1.01.01 14 Mil ri)ng va ring d(tng Bd d~ Gronwall- Bellman Hoang Thanh Long tf:lC,khong am tren £2 Ne'u rt 2 u (t)~a(t)+2b(t) Jtocp(s)u(s)ds, (2.4 ) l7tEQ, thi 1 r [cp2(s)+a(s)]exp[ lu(t)1 ~(a(t)+b(t) to fb(r)drJdsj2, s [7tE£2(2.5) Chung minh dfnh... trang do tp = SUp{tEQ I[~Jq b(to) Luqn van th{lc sl loan h(JC r 1 thz 1 u(t)::; c{exp[ q 1: a(s)ds] + c-'1q1: b(s)exp[ q r a(r )dr ]ds}q (2.23) 1 I rta(s)ds]/q{-q VtE[to,tp), tfJ =Sup{tEQlexp[q rt b(s)ds]}q >c} Jto Jto Chung minh... u(t)::;exp[ rta(s)ds] Jto 1 (Cl-'1 +(1-q) rt b(s)exp[(q-1) Jto r~a(r)dr]dsp-'1 Jto (2.26) b Ntu p < 1 thz Lu{jn van th{Jc si loan h(Jc Mii nganh : 1.01.01 18 Hoang Thanh Long MlJrl)ng va ung dljng Bd dff Gronwall- Bellman I-p I rt a(s)dst-p, u(t) So[ZI-q(t)+(1-p) trong do Z(t) = Sup{K( (2.27) Jto s) I s E[ta,t]} c.Ne'u p > 1 thi I p-l I rt a(s)K~(s)dsJ'-p, u(t) SoK1-q(t)[1+(1-p) (2.28) Jto p-l rt a(s)KI-q(s)ds . lIng d(tng Bd di Gronwall- Bellman Hoang Thanh Long CHUaNG 1 N ~ , BO DE GRONWALL- BELLMAN VA " K ? " K " MOT SO MO RONG DANG TUYEN TINH . Trang B6 d~ Gronwall- Bellman co nhi~u. cua B6 d~ Gronwall- Bellman, d6ng thai ti€p tvc ma rOng va trlnh bay mOt sf{ling dvng cua chung. Lu~n van duQc chia lam nam chuang. Chuang 0 - T6ng quail. Chuang 1 - B6 d~ Gronwall- Bellman va. va ung d~ng Bd d~ Gronwall- Bellman Hoang Thanh Long MUCLUC . . M1).C 11).c. Loi Carn do. Danh rn1).ccac ky hi~u. Chu'dng 0 -T6ng quail. Chu'dng 1 - B6 dS Gronwall- Bellman va mQt sf{ m(j