Nghiệm một số bài toán uốn tấm nhiều lớp 6

Luận văn Thạc sĩ toán học-ngành Giải Tích-Chuyên đề :Nghiệm một số bài toán uốn tấm nhiều lớp

ChtidDg 2

Nghi~m ye'u cua bai toaD II

Trong chu'ang nay chung Wi nghien cUusv t6n t~i va duy nha't nghic$my€u cua bai tmln II l~n lu'cjttrong cae tru'ong hcjp

. f = f(x, i),

. f - f(x, t, w, w').

Nhan 2 v€ cua (6) cho v E V, la'y tkh phan tren [2, dung cong thuc

Divergence va cae diSu kic$nbien (7) ta du'cjc

([J2w(t) 82V)

(w"(t), v) + ,(Vw"(t), Vv) + Dn 8x2' 8x2

(82W(t) 82V ) (82w(t) 82V)

+ 2D12 8x8y' 8x8y + D22 8y2' oy2 = (f(t), v),

w(x,O) = wo(x), w'(x, 0) = Wl(X).

Bai tmin

Ch 2 Nghi?m ytu cua bili roan II 41 TIm ham w E L2(0, T, V) sao cho

82w(t) 82V

)

(w"(t), v) + '"'((Vw"(t),Vv) + Dll\ 8x2 ' 8X2

/ 82w(t) 82V

) / 82w(t) 82V)

+2DI2\ 8x8y , 8x8y + D22\ 8y2 ' 8y2 = (f(t), v), Vv E V,(41)

2.2 St! fon t~i va duy nb!t nghi~m

2.2.1 Tr1i<1nghqp f - f(x, t)

Gia thie't

f E L2(Q x [0,TD,

Wo E HJ nH2,WI E HI.

(43) (44)

Dfnh Iy 3 Cho T > 0 ed djnh, dlldi eae gid thilt (43)-(44), hai toan (41) - (42) t6n tgi duy nht{t nghi?m w saD eho w E LOO(O, T, V), va w' E

LOO(O,T, HI).

Chang minh Chung minh dinh 19 g6m nhi6u buoc

A Xffp xi Garlerkin

Trong m~nh d6 4 chudng 1 ta dfi chung minh duQc V 1a kh6ng gian Hibert tach duQc, do do t6n t(;limQt cd sa de'm duQC{VI, V2, } trong V,

nghia Ia:

if {VI,V2, ,Vm} 1a t~p dQc l~p tuye'n tinh Vm E N

iif T~p cac t6 hQp tuye'n tinh hfi'u h(;ln~j=I ajvj, (aj E R), mEN tru

m~t trong V.

Cho {Vj} Ia cd sa de'm duQCcua V Nghi~m xffp Xl cua bai tOaDbie'n phan

(41) duQc Hm duoi d(;lng

m

wm(t) = L Cmj(t)Vj,

j=I

Ch 2 Nghifm ye'u cua hili toan II 42

trong doCmj (t) thai! h~ phuong trlnh vi phan tuy6n tinh sau

(w~(t), Vj) + ,('lw~(t), 'lVj) + Dn (82Wm(t) 8X2 ' 8x2 82vj \ /

(82Wm(t) 82vj \ (82Wm(t) 82vj\

+2D12 8x8y' 8x8y/ + D22 8y2 ' 8y2/ = (f(t), Vj), (45)

1< j < m,

wm(O) = WOrn = L amiVi, w~(O) = WIm = L fJmivi,

(46)

voi

WOrn -+Wo m~nh trong HJ n H2,

WIm -+WI m~nh trong HI

(47) (48)

Cho T > 0 c6 dinh, h~ phuong trlnh (45)- (46) du<jcvi6t l~i

m

L C~i(t) ((Vi, Vj) + ,('lVi, 'lVj))

i=1

m

( (82vi 82Vj \

(82vi 82Vj \ + L Cmi(t) DII 8X2' 8X2/ + 2DI2 8x8y' 8x8y/

)

+D22( 8 Vi, 8 Vj\ = (f(t),Vj), 1 <j< m.

8y2 8y2 /

Cmi(O) = ami, C~i(O) = fJmi,

(49)

Theo 19 thuy6t phuong trlnh vi phan thuong h~ (49) co nghi~m duy nha't

Cmi(t),i = 1, , m trong [0,Tm] noi each khac h~ phuong trlnh (45) -(46)

co nghi~m duy nha't wm(t) tren do~n f) < t < Tm Hon nii'a ta co th~ thac tri~n nghi~m cua h~ (45)-( 46) d~ cho Tm = T, \/m theo each danh gia duoi

day

B Danh gia tien nghi~m

Ch 2 Nghi~m ytu cua bili loan II 43

Nhan 2 v€ cua (45) cho Gmj(t), Ia'y t6ng theo j, tich phan theo t ta

duQc

l' [(W;;'(T), W;"(T)) +')'(IlW;;,(T), IlW;"(T))

D /82Wm(T) 82W~(T)

) 2D /82wm(T) 82W~(T) )

/82Wm(T) 82W~(T)

)] dT = rt(!(T),W:n(T))dT

B6i thli tlf Ia'y tich phan ta thu duQc k€t qua

Sm(t) = Sm(O) + 21' (f(r), w'm(r))dr, (50)

trong do

Sm(t) = IIw;,,(t)lIi, + I'll Vw;"(t) IIi, + Dl1I182~~(t) IIL2

2D

II

82Wm(t)

11

2 D

II

82Wm(t)

11

2

+ 12 8 8X Y L2+ 22 8 2Y L2

Danh giG vt phdi

II

82WOm

II

Sm(O) = Ilw1mllL2 + ,llvW1mllL2+ D11 8x2 L2

+2D1211~;;'11:2 +D2211a;,:~mll:2 < Oi>- (51)

trong do ba't d~ng thlic (51) co duQc Ia nhd (47)- (48) Do do tu

Ch 2 Nghi?m ytu Guahili roan II 44

(50)-(51)

Sm(t) < C1 + 1'"f(T)II'i2dT + 1'"w'm(T)II'i2dT

< MTm+ l' IIw;"(r)II1"dr

< MTm+ 1t Sm(r)dr

trong do MTm = CI +Tmllflli2. Ap dlJng b6 d€ Gronwall ta nh~n duQc

vdi ffiQi0 < t < Tm < T Tu (52) ta suy ra

{wm} bi chi;introng L 00(0,T, V), {W~TJ bi chi;introng LOO(O, T, HI).

(53) (54)

C Qua gilii h~n

Tu (53),(54), ta co th~ Iffyra ffiQtday con cua day {wm} ffia ta vftn ky hi~u Ia {wm} sao cho

Wm -+ W trong LOO(O, T, V) ye'u * , (55)

w~ -+w' trong LOO(O, T, HI) ye'u * (56) Qua gidi h(;lntrong (45), b~ng cach dung (51)- (56) vdi chu Y

(w':n(t), Vj) = ~(w~(t), Vj) -+ :t(w' (t), Vj) = (w"(t), Vj)

(Vw':n(t), Vj) = :t (Vw~(t), Vj) -+ ~(Vw'(t), Vj) = (Vw"(t), Vj)

ta thu duQc

j82W(t) 82vo

)

(w"(t), Vj) + ,(Vw"(t), VVj) + Du \ 8x2 ' 8xg

j82W(t) 82Vj

) j82w(t) 82Vj )

+ 2DI2\ 8x8y , 8x8y + D22\ 8y2 ' 8y2 = (f(t), Vj),

Ch 2 Nghi?m ytu cila hili loan II 45

voi ffiQij E N, t E [0,T], Vj E V Do do

/82w(t) 82V

)

(w"(t) , v) + ,(Vw"(t), Vv) + Dll \ 8X2 '8x2

/82W(t) 82V

) /82w(t) 82V)

+ 2D12\ 8x8y '8x8y + D22\ 8y2 '8y2 = (f(t), v),

voi ffiQit E [0,T], v E v.

Diiu ki~n ddu

Vi w, Wm E 0°(0, T, £2) Den Wm(O)~ w(O) ffi~nh trong £2 guy fa

w(O) = woo M~t khac (w~(t), Vj) va (w'(t), Vj) thuQClop 0°(0, T) Den

(w~(O) - w'(O), Vj) ~ 0 khi m ~oo Vi the" w'(O) = WI.

D 811duy nha't nghif:ID

Giii sa w la nghi~ffi ye"ucua bai loan II tu'dng ung voi f = 0, Wo=

0,WI = 0, nhu' gall

" " ()4w

D ()4w D ()4w - 0

x E [2,t E [0,T],

8w_8w=0

w(x,O) = w'(x,O) = 0 x E [2,

w E £00(0,T, V), w' E £00(0,T, HI).

(59)

Ta se chung minh ding w = o Nhfin2 ve"-cua (57) cho w'(t), Iffy tfch phfin

Ch 2 Nghi?m ytu Gilahili loan II

tren r2, Iffy rich phan rhea t, bie'n d6i rich phan ta thu du'Qc

~llw'(t)lli2 + ; Ilvw'(t)lli2 + DU ll

02w(t)

11

2

+ D12

11

02w(t)

11

2 + D22

11

02w(t)

11

2 =0

oxoy £2 2 oy2 L2

TIT(59),(60) va daub gia (28) ta du'QCwet) =0 voi mQit E [0,T].

2.2.2 Truong h(jp f = lex, t, w,w')

Gia thie't

f E CO(r2x [0,+00) x R2),

of of of of E Co(r2 x [0, +00) X R2),

ox' By' ow' ow'

wo E HJ n H2, wi.E HI.

Cho tru'oc s6 M, T > 0 d~t

K = K(M, T, f) = suplf(x, t, w, w')!,

KI = KI(M, T, f) = sup(lfxl + Ifyl + Ifwi+ Ifw'I),

46

(60)

D

(61)

(62)

(63)

(64) (65)

trong m6i tru'ong hQp sup du'QcIffy tren mi~n (x, y) E r2, 0 < t < T, Iwl,Iw'l < M Ta du'a vao mQtkhong gian moi du'Qcdinh nghia nhu'sau

V(M, T) = {v E £00(0, T, V) : v' E £00(0, T, HI),

IlvIILOO(O,T,V) < M, IIv'IILOO(O,T,Hl) < M} (66)

Tren khOng gian V(M, T) ta xay dvng thu~t tmln

Bu'oc 1 ChQn Wo E V(M, T).

Bu'oc 2 Gia sa Wm-I E V(M, T). Chung ta lien ke't bai loan II voi bai loan bie'n phan sau

Ch 2 Nghi?m ye'u cila hili tolin II 47

Tim ham wm E V (M, T) saD cho

( wm -1/ ( ) ) t , v +, ( \lwm t , \lv + Dn \-1/ ( ) ) j82Wm(t)8x2 ' 8x282V)

j 82wm(t) 82V

) j 82wm(t) 82V)

+ 2D12\ 8x8y , 8x8y + D22\ 8y2 ' 8y2 = (Fm(t), v),(67)

Wm(O) = Wo,w~(O) = WI.

trong do Fm(t) = j(x,t,wm-l(t),w'm-l(t)). Svt6n t~i cua M,T > 0, va

wm(t) cho bai dinh 1y sau

Dinh Iy 4 DU:dieae giG thiet (61)-(65), ton tgi eae hang so' M, T > 0 va

day quy ngp {wm} C V(M, T) dU:f/edinh nghzQ biJi (67) - (68).

Chang minh Chung minh dinh 1y g6m nhi6u bu'oc

A Xa'p xi Garlerkin

Tu'dng tv nhu' tru'ong hQp j = j (x, t) La'y {vj} Ia day de'm du'Qccac ham cd sa cua V Nghi~m cua bai loan bie'n phan (67) - (68) du'Qctim du'oi

d~ng

(68)

k

w};:)(t) = L c~;(t)Vj

j=1

trong do c~; (t) thai! h~ phu'dng trinh tuye'n tlnh sau

(W'Jk) (t), Vj) + ,(\lW'Jk) (t), \lVj)

j82W};:)(t) 82Vj

) j82w};:)(t) 82Vj ) + D11\ 8x2 ' 8x2 + 2D12\ 8x8y , 8x8y

82W(k)(t) 82v.

+ D22\ 8;2 ' 8yg) = (Fm(t),Vj), 1 < j < k, (69)

-(k) ( ) - - ~ -'(k) ( ) - - ~

Wm 0 - WOk - 6 GmiVi, wm 0 - Wlk - 6 (3miVi.

(70)

Ch 2 Nghi~m ytu cila hili roan II 48

vdi

WOk -+Wo m~nh trong HJ n H2,

Wlk -+WI m~nh trong HI.

(71) (72)

Cho T > 0 c6 dinh, Gia sii' Wm-I E V(M, T) Ht$ phuong trlnh (69) -(70)

viSt l~i

k

"" lI(k)

~ cmi ((Vi, Vj) + ,,( \7Vi, \7Vj) )

i=I

k

(

!::\2!::\2 !::\2!::\2

(k) U Vi U Vj U Vi U Vj

+ 8Cmi Dll (ox2' ox2 ) + 2D12 (oxOy' oxOy)

(82Vi 82Vj)) ( ( ) )

.

+D22 8y2' 8y2 = Fm t ,Vj, 1 < J < k,

( ) Cmi (0) = ami, Cmi 0 = (3mi

(73)

Theo 1:9thuySt phuong trlnh phan thuong ht$ (73) c6 nghit$m duy nhtt

c~l(t),i = 1 k trong [O,T~)]. N6i each khac ht$ phuong trlnh (69) -(70) c6

nghit$m duy nha"t w~)(t) tren do~n 0::; t::; T~) ::;T Hon nua ta c6 th~ thac

tri~n nghit$mcua ht$ (69) d~ cho T~) = T,\/m,k theo each danh gia dudi day

B Danh gia tieD nghi~m

D~t

82-(k)( )

S~)(t) = Ilw~)(t)lli2 +,,1I\7w~k)(t)lli2+ Dnll ~~2 t 11£2

11

82W~)(t)

11

2

11

82W~)(t)

11

2

+ 2DI2 8 8X Y £2+ D22 8 2Y £2'

Ch 2 Nghi?m ytu cila bili loan II 49 Nhan 2 vt cua (69) cha c~~)(t), la'y t6ng rhea j ta du'<,1c

(W'Jk) (t), w~k)(t)) + 'Y(vw'Jk)(t), vw~k)(t))

82w~)(t) 82w~k)(t) 82w~)(t) 82w~k)(t) + D11( 8x2 ' 8x2 ) + 2D12( 8x8y , 8x8y )

/82w~)(t) 82w~k)(t)

) = (Frn(t),w~)(t)) + D22\ 8y2 ' 8y2

Tich phan rhea t, d6i thli tl! la'y rich phan ta thu du'<,1cktt qua

sf::)(t) = sf::)(O) + 21' (Fm(T), w;!.kJ(T))dT (74)

Danh giG ve' phdi cua (74)

S6 h~ng thli hai d v~ phai cua (74) du'<,1cdanh gia bdi

2l (Fm(-r),ii/,!,k) (T))dT < 2K l"w',!,k)( T)lIpdT

< TK2 + l S!::)(r)dr

S6 h~ng thli nha't d v~ phai cua (74) du'<,1cvi~t l~i

82

11

2

S~)(O) = IIWlklli2 + 'YllvWlkll£2*Dull 8x2 £2

II

82wOk

11

2

II

82wo

11

2

+ 2D12 8x8y £2 + D22 8y2 m £2

Ch 2 Nghi~m ylu cila bili roan II 50

TIT (71), (72) ta suy ra t6n t~i M > 0 sao cho

Vlthe'

Sm(t) < 6 +TK2 + Jo S~)(T)dT

TITgia thie't (64)- (65) ta co

lirn TK(M,T,f) =0, Hrn TK1(M,T,f) =0

do do ta Iuan chQn du'QcT > 0 sao cho

2KIT

(2- + 2C2 + 1C1 C2 ) < 1 (77) trong do 6 = rnax {I, " D1l, 2D12,D22, (2C~+1)}, ChAng s6 Poincare ph\l

thuQcvao [2, C1,O2 Ia cac hAngs6 dQc I~p T Ap d\lng b6 d~ Gronwall ta

co danh gia

S",(t} < 6M2e-T + l' sf:) (r}dr < 6M2e-TeT < 6M2

vdi ffiQit E [0,T] nghla Ia ta co th~ Iffy T = T;::) vdi ffiQim, k Do do

IIw~)(t)1112-+,II vw~k)(t)1112 < 6M2

II

a2w~)(t)

II II

a2w~)(t)

11

2

II

a2w~)(t)

11

Ch 2 Nghi?m ytu cila hili toan II 51

hay

Ilw~)(t)lI~l < M2,

IIw~)(t)ll: < M2

bfft d~ng thuc cu6i co du'Qcnha danh gia (28) V~y

U~) E V(M, T), Vm,k (78)

TIT(78) ta co th€ Iffy 1 day con cua day {u~)} ma ta vftn ky hi~u Ia {~)}

saDcho

U~) + Urntrong Loo(O,T, V) ye'u *

u~k) + u'm trong Loo(O,T, HI) ye'u *

Urn E V(M,T)

Qua gioi h(~llltrong (69) -(70) ta nh~n du'QcUrn thoa (67) -(68) trong L2(0, T)

D

Dinh Iy 5 DllOicac gid thilt (61)-(65) ta luon tim dllf/Cso M > 0, T > 0 thoa (75),(76) va (77) saD cho bai loan II co nghi?m ylu duy nhdt w E V(M, T), h(Jnnila

wI! E Loo(O, T, H2)

Noi cach khac day {wrn} cho bOi (67)-(68) hQi t1:lm{lnh vi nghi?m w trong

khong gian

~(T) = {vE Loo(O,T,V): v' E Loo(O,T,HI)}

Chang minh Chung minh dinh Iy g6m

A 811t'on t~i nghi~m

Theo mQt b6 dS trong [10] chu'dng 1 cr

Banach voi chuffn

57 thl ~(T) la khong gian

IlvIIVl(T) = IlvIlLOO(O,T,V)+ Ilv'IILOO(O,T,Hl) (79)

Ch 2 Nghifm ytu cua hili loan II 52

Ta chung minh day {wm} la day Cauchy trong VI(T) La'y vm = Wm+1 -Wm.

Luc do Vm thai!

(v~(t), v) + ,(Vv~(t), Vv) + Du j 82vm(t) 02V\ 8x2 'ox2 )

jO2Vm(t) 82V

) j82Vm(t) 82V) + 2DI2\ 8x8y 'oxoy + D22\ oy2 '8y2

= (Fm+l(t) - Fm(t), v), Vv E V

vm(O)= v;"(O) = 0

(80) (81)

La'y lftn Iu'Qtla v = Vj,j = 1 k Nhan 2 vt cua (81) cho c~~, Iffyt6ng

theo j, tich phan theo t ta thu du'Qc

Sm(t) = Sm(O)+ 21' (Fm+l(r) - Fm(r), v;")dr (82)

trong do

Sm(t) = Ilv;"(t)Ill, +'I'll V'v;"{t) Ill, + Dull &;;~t) t,

+ 2DI2

JJ

82Vm(t)

11 2 + D22

11

82Vm(t)

11

2

TIT(81) ta co Sm(O) = 0, va

IF m+ I (t) - F m (t ) I < K I (I Vm-I ( t) I + IV;"-I (t ) I)

Ch 2 Nghi?m ylu Gilabai toan II 53

ta co danh gia

Sm(t) < 2Kll' 1 (IVm-l(T) I+ IV;"-l(T) I) IV;"(T)ldxdT

< 2Kll' (1IVm-l(T)llv + Ilv;"-l(T)IIL') Ilv;,,(T)lIdT

< 2Kll'lIvm-tllv,(T)lIvmllv,(T)dT

< 2K1Tllvm-11IVl(T)IIVmIIVi(T) (83)

Danh gia vt trdi Gua (8]))

M~t khac

(1Iv:n(t) IIHl + Ilvm(t) Ilv)2

(

1 2C2 + 1

2 C2

11

2

)

< C1 + C2 C11Ivm(t)IIHl+ 2C2 + 11Ivm(t) V

<

) S ( )

(

1 2C2 + 1

)

<2K1T C1 + C2 Ilvm-11IVi(T)lIvmIlVl(T),\I O<t<T

trong do C1= min{1,,},C2 = min{Dl1,2D12,D22},C HihAngs6Poincare

hay

IIVm(t)ll~l(T)< kTllvm-1I1vl(T)IIVmllvl(T)

suy ra

IIvmllViT < kTllvm-11IVi(T) \1m

Ch 2 Nghi~m ytu Gilahili roan II 54

trong do kT = 2KIT(J1 + 2C~:I) < 1 Cu6i cung ta co danh gia sau

Ilvm+p- vmllV)(T) < Ilvm+p- vm+p-I/lV1(T)+ + IIVm+1 - Vmll~(T)

< k;+P-IlivI - VOIlV)(T)+ + kFlivI- VOIl~(T)

k:;(l - kfr) IlvI - VOIl~(T)

< 1 - kT

km

< T IlvI - VoIIVI(T)

TIT(77),(84) ta suy ra {wm} Ia day Cauchy trong ~(T). Do do t6n t(;li

W E ~(T) sac cho Wm + W m(;lnh trong VI(T). Ta cling chli y ding

Wm E V(M, T) Den tITday {Wm} ta Iffy ra du<JcmQt day con ma ta v~n ky

hi~u Ia {wm} sac cho

Wm + W trong £00(0, T, V) ye'u *

w~ + w' trong £00(0, T, HI) ye'u *

wE V(M,T)

Tuong tl! nhu (83) ta co danh gia sau

(85) (86)

IIFm- f(x, t, w, W')IILOO(O,T,£2) < KIllwm-1 - wllvl(T)

TIT(87) va sl! hQi t1;1ffi(;lnhcua Wm v~ W trong ~ (T) ta nh~n du<Jc

Qua gioi h(;lntrong (67), (68), dung cae ke't qua trong (85), (86), (88) suy

ra sl! t6n t(;linghi~m W E V (M, T) rhea phuong trlnh

(w"(t),v) + ,(Vw"(t), Vv)

(87)

j fPw(t) 82V

+ Dn \ 8x2 ' 8x2 + 2DI2\ 8x8y , 8x8y

j82W(t) 82V

) = (f(t),v),

Ch 2 Nghi?m ye'u Gua bili toan II

vdi di€u ki~n d~u

55

w(x,O) = WO(X),w' (X, 0) = WI (X).

TIT(88) va (89) ta duQc

w" - "Y~w" = H(x, t)

(90)

vdi

(

)

H(x, t) = - Dll 8x4 + 2D128x28y2 + D228y4 + f(x, t, w, w')

Vi

w" - "Y~w" = H E L2(0, T; L2)

Den

w" E LOO(O,T; L2).

M~t khac

"Y~w" = w"- H E LOO(O, T; L2),

do do ta nh~n duQc w" E LOO(O,T; H2).

B 811 duy nba't ngbi~m

GQi iili, i = 1,2 la hai nghi~m cua bai toaD (89), (90) thoa Wi E V(Mi, Ii) Luc do w = wi- w~, 0 < t < T = min{TI, T2} thoa bai

toaD

82w(t) 82v\ / 82w(t) 82v \

(w"(t),v) + "Y(Vw"(t),Vv) + Dll \ 8x2 ' 8X2/ + 2D12\ 8x8y , 8x8y /

/ 82w(t) 82v\

+ D22\ 8y2 ' 8y2/ = (FI(t) - F2(t),v)(91)

Lffy v = w' trong (91), tich phan theo t ta duQc

8(t) = 8(0) + 21' (F,( T) - F2(T), V)dT (93)

Ch 2 Nghifm ylu Gilahili roan II 56

trong d6

2

II

82w(t)

II

8(t) = Ilw'(t)lli2 + ,IIVw'(t)llv + Dn 8x2 £2

2D

II

82w(t)

11

II

82w(t)

11

2

+ 12 8x8y £2 + 22 8y2 £2'

Cht1 Y cac diSu ki~n (92) nh~n duQc 8(0) = 0, tuang tv nhu (83) va dung

ba't dAng thuc (xem chuang 0)

Ilw(s)lli2 < 2C2

(1I82w~s) 112 +

II

82w(s)

11

2

)

ta nh~n duQc

2l'(Fl(r) - F2(r),v)dr < 2Kll' (lIw(r) IlL' + IIw'(r)IIL') Ilw'(r)IIL,dr

< K, l' (IIW(T)lIi,+ 3I1w'(T)IIi,) dT

< K, l' (lIW(T)IIl"+ 3CIIVw'(T)IIl,,)dT

< MK11' S(T)dT,

trong d6 M = max{3C", Dn, 2D12,D22} A.p d\lng b6 dS Gronwall ta duQc 8(t) = 0 vdi mQi t E [0,T] , mQt cach tuang tv nhu trong truong hQp

f = f(x, t) ta phiii c6 W1(t) = W2(t), vdi mQi t E [0,T] Do d6 dinh ly da