chỉnh hóa bài toán moment tổng quát chương 3

Chuang III: Chlnh h6a bai toaD moment t6ng quat

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CHU0NG III: CHINH HOA BAI TOAN MOlVIENT
TONG QUAT
3.1 Bai toan moment t6ng quat:
TIm mQt ham u tren mi~n D c Rdthoa di~u ki~n :

Iu(x)gn (x) dx =

~n voi

(3.1)

nEN

trong d6, (gn) la day ham cho truac, /.l=(/.ll, /.l2,...) 13.day so trong 12 hay
B3.i roan moment

r"

thuong khong chlnh rhea nghIa 13. chung khong c6

nghi~m va trong truong h<;:ip

ton t~i nghi~m thl chung kh6ng phlf thuQc lien tlfc
V3.0dli ki~n. Ta se ki~m chung l~i kh~ng dinh tren qua ket qua sau:
Cho (gn)n=I.2 la mQt hQ dQc l?p tuyen tinh trong khong gian Hilbert vai tfch vo
huang la (,,')H , chuin Wong ling la II.IIH l?p th3.nh co 56 cua H. Ta dn Urn u EO
va
H
sao cho : (u, gn)H= /.In

n = 1,2, ... vai /.l=(/.ll, /.l2,...) 13.day so trong 12hay 1':0

(3.1)
Truong h<;:ip lay khong gian dli ki~n 13.12tuc 13.khong gian cae day so
ta
oC

/.l =(/.In) saG cho II/.lJ
n=1

<

00

thl12 13. kh6ng gian Hilbert

vai rich trong (~l,rU

=

'"

I ~tlllln < 00 V3. kh6ng
n=1

mat tinh t6ng quat ta c6 th~ gia SLIrang ho (gn) tn(c

chuin. Khi d6 (xem [1]) : 12d~ng cau Hilbert vai H b6i anh x~:
U H ((u, gJH )n=I,2,...

Suy ra bai roan moment (u, gn) = /.In
13.chlnh (do ton tai duy nhat nghiem,
nghi~m phl,l thuQc lien tl,lc).

.:. Truong

h<;:ipta xet kh6ng gian dli ki~n 13. let), tilc 13.kh6ng gian Banach

cac day so /.l= (/.In)n = 1.2

thoa

II~III"

=

SuPI~n

n

I

(go) la trll m?t trong H nghla 13.: < gn InEO > = H
N

Xet A: H

--7

UH

let)

Au

= ((u, gJH t=!,2....

19

< 00.

Ta cling gia sLrhQ


Chuang III: Chlnh h6a bfli toan moment t6ng quat

Khi do nghi<$m cua bai tmln (3.1), neu co, thl duy nha't. Thc}tvc}y: gQi lit. uz
1ahai nghi~m ung vdi ~n,nghja 1a Au, = ~n, Auz = ~n' f)~t W = Uj- Uz => (w, gn)
= 0 "in => (w, g) = 0 vdi'\f g E < gll :>. Vc}yW E X.Lvdi X 1a kh6ng gian sinh
bdi cac gj, i

= 1,2,

..., n . Theo tinh chat trong gian Hilbert, ta co:

Lz =X + X.L ma X

=Lz

X.L ={a} => w =0 => UI

nen

=Uz .

Vc}yta co tinh duy nhat nghi~m (neu co nghi~m) cua bai roan (3.1) trong

truonghQpkh6ng gian da k.i~nla t' .
Vdi cach d~t roan ta A nhutren ta co m~nh d~ sail:

Menh d~:
Cho A: H ~ tJ
U H Au = ((u, gn)H )n=1.2,,'

1. A 1a anh X? tuyen tinh lien t\1c.
2. A 1a don anh.
3. Range A chua h~n trong 100
.
4. A-I: Range A ~

H kh6ng lien t\1c.

Chung minh:
1. A tuyen tinh do tinh tuyen tinh cua rich v6 hudng.
Ta co: IIAu!!t"' supl(u, O)H ::; supllgllllH =
=0
g
.lIullH
n
I

IluliH

=> A lien t\1C

2. Gia sa Au =«u, gn)H)n=1,2,..,= O. Do (gn) 1a co sd cua H nen UE H.L
=> u

=O. Vc}yA la don anh.

3. Ta chung rninh b~ng phan chung. Gia sa Range A = 100
.Theo h~ qua cua
dinh 1yanh X?md (xem [2] ) thl A la d6ng ph6i tit H vao 100
.
VI H phan X?, A la d6ng ph6i tuyen tinh, nen 100phan X? (xem [1] ). f)i~u
nay v6ly.

20


Chuang III: Chinh hoa bai roan moment t6ng quat
V~y Range A chua h:1n trong

r-c

nghla la bai roan 3.1 khong luau t6n t~i

nghi€$mtrong tntong h<;1p hong gian dG'ki€$nla t' .
k

4. Xet :" A: H -+ 1':0
u H Au = ((u, gn)H )n=I,2.."
Ta c:ln chung minh A-I: Range A -+ H khong lien n,lC.
Xet day (~P) trong Range A nhu sau: voi pEN,
1-

!J. -

(

I

.Ji

Ilk

1 ~
= (Ilk) = ( .fi'.fi'

!J.2

!J.

(

(Il~ )k=I,2,...

) = (~ ,0,0,...)

'

3

!J.P =

3

= Ilk

1

1

00,...)

,

1

) - (- -,-,0,
- .Jj'.Jj .Jj

0 ,.,. )

.................-....-....-..

Khi d6, II~pll"

=

~

p-+oo
->

0

khi

1

M~t khac tUAu =((u, gn)H)n= 1,2,... !J.n du'<;1Cf' =
=
ta
U

Khi do

II upll2H

=1

voi mQip

JP (gl

+ ... + g f')'

=> A-I khong lien tl,lc.

V~y bai toan (3.1) co khi la chlnh, co khi khong chlnh. Nhu'ng ph:ln IOnbai
roan moment la khong chlnh. Ta c:ln tlm cach chlnh hoa no.
- Phu'dng phap ch~t Cl,lt a'p Xlbai roan (3.1) bdi bai roan moment hii'uh~n
x
.bu(y)gj(y)dy

(3.2)

= ~j

du'<;1Ciai trong khong gian con <g I, g2,... , go> sinh bdi g I. g2,", , go. hill
g

Y

r~ng bai

roan (3.2) la 6n dtnh.
M~t khac voi n E N du'<;1C
chQn thich h<;1p
nghi€$m cua (3.2) xa'p Xl nghi€$m
cua bai roan g6c (3.1). Trang tntong hQp dG'ki€$n!J.= (~l ,..., !J.n) la khong chinh

xac ta c:ln u'oc lu'<;1ngai so' giii'a nghi€$mchinh hoa va nghi€$m chinh xac.
s
Phu'dng phap chinh hoa nay dl,l'atren nhG'ng tinh cha't cua phep chie'u trl,l'c
giao trong L2 (0 ).

21


Chuang III: Chinh h6a bili roan moment tong quat
3.2 Cach xCiy dung nghiem chinh hoc:

Ta gia sa gr. g~.. Ia dQcI~p tuyen tinh (theo nghla d<;li o) va kh6ng gian vec
s
to sinh bdi gr",gz...latrUm~t trong Lz (Q).
Kyhi~ull.11

,(,)lachuffnvatichv6huongcuaL2(Q).Cho

{e[,e2,'"

}la

h~ tn,tc chuffn dU<;1Cay d1,1'ngU (gl, g2 ...) bai phuong phap tr1,1'c iao h6a Gramx
t
g
Schmidt nhu sail:
e[

=gl /11 gill
-1

II-I

II-I

n
en =lIgn - ~(gn,e)ej

.(gn - ~(gll,e,)e)

11

= 2,3,...

Khi do {el, ez, ... } Ia co sa tr1,1'c huffn cua L2(Q) va hon nua t6n t<;liduy nh:1t
c
h~ng so Cij; Mij E R (i, j, E N) sac cho Cij
i

ej

= L Cjjgj;

= Mij = 0 neu

i < j va

I

=" M..e

gI
.

L...J
j=1

j=1

IJ J

Vi E N

Th~t v~y tu cach d~t cua enta co:
e[ =C11gl
e2 = C2lgr+Czz gz
e3 = C3Igl+C3Z gz + C33g3

..-..
Khi do tinh ngu<;1cl<;ligi thee ej , ta cling du<;1c
cach bi~u di~n tuong t1,1'.

Ta xem ma tr~n Cn = [Cij ], Mn =[Mij]

.
1:::; 1, J. < n
-

Khi do Mn = Cn-I
Bay giG ta lien ket voi m6i day so th1,1'C = (~l. ~2 ...)

~
day A = A (Jl )= (AI, A2,..' ) dU<;1Cac dinh bdi:
x
i

Ai

=Ai (Jl)=

LCijfJj

i = 1,2, ...

(3.3)

j=1

Neu ~ Ia mQt day hG'uh<;ln = (~I. ~2.'" ~) thl A = A (~ ) cling Ia day hG'u
Jl
h<;ln AiduQc cho bdi (3.3).
voi
- Ket qua sail cho ta nghi~m co chuffn c1,1'ci~u cua bfd roan moment hG'u
t
h<;ln
(3.2)
))


r- Chuang III: Chinh h6a bfli tmin moment

tong quat

,

3.2.1 Menh de 3.1:

= (Ill, Ilz,...

Cho 11

) la day so thvc , cho U E L z (0 ) va n E N. Ta e6 cae ke't

qua sau tuong duong.
1. U E <gl, gz,..., go> (kh6ng gian vectO sinh bdi { gl. gz

= Ilj

va (u, gj)

.go})

(1 :::;j :::; n)

2. u thoa (3.4) va

u II

II

=mill

{II v II: vEL

z (0),

(3.4)
(v, g)

= Ilj}

vdi 1 :::; :::;n
j
n

3. u

= LAi(JL).ei
i=\

. Ynghia
Ta xae dinh duQcnghit%m u cua bai roan moment (3.2) theo Ai , Ai du<;Je
eua
xae dinh tit Cij Ilj, Cij xac dinh tit ej va gj vdi di~u kit%nu E < gl .gz

.

go>

Chung minh

(1) Q (3)

Cho u E < gl, gz,

go>. Do {e" e2,

eo} la co sd trve ehmln nen:

11

U

= '"' ae}
L..

}

.

j=1

n

Khid6

(u,gJ = (Iajej,gi)
j=\
II

=Laj(ej,g,)

j

=I

n

=~a.M..
~

}

I}

j=1
,

(Do gi

I

= (LMijej
J=I

~ «u, gl),'"

~(gi'eJ)=(IM"e"e,)=Mij)
pI

T

,(u, go» = Mn(al,'"

Do d6 U thoa (3.4)

<=:>

Mn(a"

, an)

T

az,..., an)T

23

(3.5)

=(111.""

Iln )T


Chuang III: Chinh h6a bai toan moment t6ng quat
(do ta thay (u, gj) =~j VaG (3.5))
<=:> (ai,

T

a2, ..., an) = Cn (~l, ..., ~n)

T

(do Mn = Cn-I)
i

(do tli (3.3) Aj= 2:CijfL)

(a(, a2 ..., an)T = (}"1,A2,...,An)T

<=:>

)=1

<=:>

aj

=Aj
n

M?t khac u = IajeJ
j=1

(2)

=> u = ~ Ice J
~ J

}

j=1

<=:> (3)

Gia sa u thoa (2), Xet phan tich u = v + w voi v E <g(, gz, ..., gn>;
W E < gl, g2, ...,gn>1-, vIa hlnh chieu cua u xuong < g" g2, ..., gn > nen:
( v, gi,) = ( u, gj,) = ~i

(3.6)

(1 :::; :::; )
i n

(do tinh cha't phep chieu va thoa (3.4))
n

V~y v thoa man tinh cha't tli (1)

<=:> (3)

nen ta duQc v = LAiei
i=1

M?t khac rhea tinh cheithlnh chieu vuong goc ta co:
II vii:::;

II till

Mall ull=min{11

vII :vEL2(0),(v,gj)=~j}=>11

=II

Tli hai ke't qua tren suy ra : II u II

V~y II w 112 = II U 112-II

ull :::;11
vii'

v II

V 112 (dinh 1y Pythagore)

=0

=>w=o
n

Ta lai co u = v + w => u
.

=v =~

Xe
II

i=1

n

(do ta da chung minh duQc v = LAiei
i=1

)

V~y ta co (3)

. Dao lai:
Gia sa ta co (3). Cho v E L2 (D) sao cho <v, gj>
U

= 1,...,

= ~j

n). Cho Pyla hlnh chieu vuong goc cua v len < gJ, g2, ..., gn>'

2-1-


Chuang III: Chinh hoa bfli loan moment t6ng quat

Khi do rhea cach d~t tu (3.6) ta c6:
(Py, gj) = (v, gj) = Ilj

(Py Ia hinh chie'u cua v Ien < gl, g2,

gn> ~ Pythoa (1)).

n

Thea chung minh (1) Q (3) ta du<;cPy =
~

II u II

=II

Py"

:::;

v

II

II

I
i=l

Ajei

=u

(do pv Iii hlnh chie'u cua v)

voi 'v' v thoa (v, gj) = Ilj
~

II u II

=mill

{II v II ; v E L2 (0 ); (v, gj)

= Ilj}

V~y ta da chung minh xong dinh 1y.

.

Cho ilIa day so th1,1'c. oi voi moi n = 1, 2, ... ta ky hi~u pn = pn (J.l ) Ia
D
phan n1 duy nhat L2 (0 ) saG cho no thoa di€u ki~n tuong duang cua
m~nh d€ 3.1. pn co th€ xac dinh tu (3.3) va (3) (pn dong vai tra nhu u)

.

Do m<$nh d€ 3.1 pn(J.l) E <g" g2, ..., gn >

n
~

= I~igi
i=l

pn(ll)

(3.8)

M~t khac pn(ll) thoa man di€u ki~n (1) cua dinh 1y 3.1 nen
(3.9)

(pn(Il), gj) = Ilj
n

The'(3.8)vao(3.9)

:

(I

i=1

~ig/!gJ) = J-ij

n

Q

I~I(gl,g)=!lj

O=l,...,n)

i=1

Day Ia h<$
Crammer cua h~ phuong trlnh tuyen tinh. Do tinh dQc I?p tuye'n

tinh cua g\> gz, ..., gn nen ta co del ((gj, g)) :j;0 (1:::;i, j :::;n)
~ ~I, ~2, ... ,~n duQc xac dinh duy nhat qua d~ng thuc:
n

I

i=1

Tu ~i,

~i (gj gj )
J

= Il j

gi. ta tinh duQc pn(ll) nghla Iii ta xac dinh duQc nghi~m cua bai roan

moment (3.2).
Tu ke't qua cua m<$nhd€ (3.1) ta suy ra s1,1'
ph\,I thuQc lien t\,Ic cua pn(ll)
theo Il.

2S


Chuang III: Chlnh h6a bfli roan moment t6ng qmlt

3.2.2 Menh d~ 3.2:
Cho ~m = (~t); ~ =(~j) VOlm = 1,2,...la nhii'ng day so thl;l'csaG cho ~jm
--+ ~j khim --+ ex)V j

E

N

Khi d6 dol VOlm6i n E N co dinh pn(~m) --+pn(~) trang Lz (0 ) khi m --+ ex)

. Y nghia:
Sai so cua dii' k.i~n nho clan den sai so cua ket qua nho, nghla la bai roan
moment (3.2) la 6n dinh
3.2.3 Su' hoi tu cua nghiem chinh hoa va lJoc IlJdng sai s6:
Ta thila nh~n ket qua sau:
D6i voi m6i m E N thi ton tqi

C

=C

(m, 0 ) > 0 saG cha voi

V v E Hm (0 ) ta c6:
Ilv -

pYn vii

~ c.llvll~:(Q)n

VOl Yn = < gl, gz, ..., go>;

n

=1,2, ...

Hm(0)

(3.10)

la khong gian Sobolev thong thuang

(m=1,2,...); pYn la hinh chieu trl;l'cgiaa L2 (0) len Vn

- Ta

chung minh s1!hQi tl;lcua nghi~m xap Xl pn(~) trong truang hQp du k.i~n

chinh xac.

Dinh IV 3.3:
Cho ~

= (~j) la day

so th1!c. Khi d6:

1. Di~u k.i~nc~n va du d~ t6n t(,iinghi~m cua (3.1) la:
'"

i

I (I Cij~)2

(3.11 )

< co

;=1 j=1

2. Neu u la nghi~m duy nhat cua (3.1) thi pn(~) --+u trong L2 (0 )
khin--+ex)
Hon nua neu u E Hm(0) va (3.10) dung 'if m thi :
II pn(~)-ull

:::;cllull

~:(Q)n

2(')

Vn


Chuang III: Chinh hoa bai toan moment tc)ng quat

. YnghTa:
Truong help dli ki~n chinh xac thl nghi~m cua bai roan moment (3.2) se hQi t1,l
de'n nghi~m cua bai roan moment (3.1) trong L2 (Q ).

Chung minh:
1. Cho u la mQt nghi~m clh (3.1). Ta co vdi mQi i E N
,

,
(u, ej ) = (u,

I Cijg, )
J=1

(do e, =

I

)

Cijgj

,=1

I

= ICij(u,gj)
j=1
I

= IC,j~J
j=1

(VI u la nghi~m cua (3. ! ) nen (1I, gj)
00

Dodo:

,

= ~j )

lfJ

I(ICij~ti)2

=I(u,e,)2

i=1 J=I

=llul12
J=I

V~y (3.11) dung.

.

Dao lai:
Ne'u (3.11) dung, khi d() ta xet:
:c
U

,

I

= I(ICjj~j)ei

(Ai

;=1 j=1

=I

do

C'YJ

(3.3))

J=I

Ta chung mint U E L 2 (Q)

va la nghi~m cLla (3.1)

00

,

Ta co: (u, eJ = (LA,e ,ej) = A, = ICii~J
i=1

,(do (3.3))
n

11

Mi;H khac (u, ej)

= (u, I

(do

CiJgl)

ej

,= i

n

Tu(3.12)va(3.13)

=> (u,ICqg,)
T

gl)

,00'(u, go))

= C n (~!

=

I

J=I

C 'jg J)

n

-"C
L...

J=1
=> Cn ((u,

(3.12)

J=1

J=I

,

=> (u, gn ) = ~n

Vay u la nghi~m Cl'ta(3.1)

'7
-,

,un)

T

u J ) Vi=I,...,n

II'
.

(3.13)


Chuang III: Chinh h6a bai toan moment t6ng quat
2. Cho U E L2 (n) va la nghi~m cua (3.1).
Do t6n t<;li ghit$m nen U co d<;lng:
n
CX)

:f;

",;

= L (L Cij,uj)e;

U

=> u = IXi(,u)e;
;=1

;=\ j=1

I

ma thee

=
( 3 .3 ) "CII.IJI'"'"JA.
~
j=1

Do m~nh d~ 3.1 (3) va trong ph§n chung minh (2) Q (3) ta co:

u=v=P. u

}

Vn

Ma pn (p)= U

(pn(/l) dong vai tro nhu' u)

~ pn (p) = Pr-.(u)

Sl,l'hQi t1,lcua pn(~) d€n u trong L2 (Q ) theo tinh cha't cua khong gian Hilbert
:f;

Th~t v~y voi u E H ta co: U =
CX)

I

n=\

n

n=\

:f;

= Lakek

Ta co: U = Lanen

+ Iakek

k=1

k=,,+1

n

:f;

Iakek
k=1

= PYn(U)= pn(~)

V~y pn(~)

.

a"e" voi an = (u,en)

~

U

= Lakek
k=1

u trong L2 (Q )

~

Chung minh U E L2 en ):
CX)

Ta co: (u, ej)

i

= ( I (2:Cjjll)ej,ej
i=1

)

j=l

i

I Cijllj

j=1

=
ex)

=> IIuI12=

.

;

2: (2: Cijpj)2<
;=1

(vi (ej) la h~ trl,l'cchucfn)

00

( do gia thuy€t (3.11))

j=1

Trong (3.10) the' Pvn = pn (11) ta dliqc:
U

IIpn(Il)

-

till

~ cllull~:(Q)n

'v'n

28


Chuang III: Chlnh h6a bai toan moment tong quat

Dinh 1y sail chI ra r~ng trong tru'ong hc;5P kic$nkh6ng chinh xac, nghic$m
du
cua bai roan moment (3.4) 1a xap Xl 6n oinh nghic$m Clla bai roan (3.1)

Ky hi~u II

Il II

ro = sup {Illi I; i EN}

3.2.4 Dinh IV 3.4:
Cho Un E L2 (D ) 1a nghic$m Clla (3.1) tu'ang ling vdi:
0

I'

0

0

.1

= ( 1'1,1'2 ,... ) . vOl VOl 0 < £ < i I gl
I:'\~"

"

2
11

ham tang ng~t tu [ 1,00) 1en [II g&l, co) sao rho

rt-

.

ta ui;n: n(£ )
IICnl1

=[[

1

(£

-112

"

'

)] VOl f1 a

~ fen) \j n 2: 1.

Khi do t5n t~i ham s6'1 (£ ) sao rho lim17(E)= 0 va d6i vdi nhung day Il thoa:
i:~1)

111l-IlOlloo<£

taco:11

pn
<1'1(£)

Han nua ne'u (3.10) dung vdi Un E Hm (D) thl:

II pn(E )(fl)

-

Un II ~ £1I2 +

c.lluo (Q)
IIH'"
(n(£))"1

.

Y nghla

Tru'ong hQp du kic$nkh6ng chinh Xi.1C b~ng each chQn so n (£ ) thich hc;Jp
thl
nghic$m cua (3.2) se xap Xlnghic$mciia (3.1) va ta cach danh gia du'c;5c so:
sai

.

Chung minh

Theo tinh chat cua chua'n:
II pn(fl) -Un

II ~ II pn(ll)

- pn(fl()) II + II pn(~h- UnII

Ta ky hic$uII. 112va II . 1100
doi vdi chua'n dclit va chua'n sup tren Rn
II Cn II 1a chua'n cua Cn = [Cjj] vdi i, j = 1, 2

..,

, n xem nhu' 1a anh x~ tuye'n

tinh cua (Rn.11 .1100)de'n (Rn,II . 112),
II cnl! =sup{IICnvTI12;vERn:11

vilw ~l}

/1

Dodo:11 pn(Il)-pn(IlO)11 = IIIV-,(p)-Ai(,Ll°))e,
1=1

"

(Do mc$nh d~ 3.1 (3) pn(ll) = L>,,(,LL)e,)
1=1

29


Chuang III: Chinh h6a bai toan moment t6ng quat

= II

(AI (~) - AI(~O),

, An(/1) - An (/10 )11

=IICn(~I-~IO""'~n

-~()n)TII

(Do(3.3)Aj=

IC,;f'/)
):1

::;; Ilcnll.II(~I-~nO""

::;; Ilcnll

(/1n -~on)Tllw

.11(/1- ~olloo

Cho f: [1, 00 ) ~ [

II

(3.14 )

g 11-1 , 00)

la ham tang ng~t rhea

Voi 0 < 8 < II gj 112 ~ 8-1/2 ~ II gll-I
ChQn n (8 )

= [fl

~

II Cnll ::;;fen) voi 'v' n ~ 1.

fl (8-1/2) ~ 1

(3.15)

( 8-112
)]

Khi d6: II pn(e)(/1) - pn(e)(~o) II < 81/2
Th?t V?y rhea

(3.16)

(3.14)

Ilpn(e)(~)_pn(e)(~(~11

::;;11cn(e)II.IIIl-~lollw

::;;f(n(8)).8(dogiathuyetll/1-/1°llcn<8

vallcn(e)11

::;;f(n(8))

::;; [fl(8 -1/2)] 8
f

=8 -1/2 .8

(do d~t n (8 )

=fl(8-1/2)]

= 8112
M~t khac:
II pn(e)(~o)

-

~II = (

f(Aj(,u°))C)1

C

j:n(E)+1

Th?t V?y thee ph~n chung minh (2) (j dinh 1y 3.3 ta c6 pn(e)(/1°) 1a hlnh chieu
0
n(£ )
n(£ J 0
vuong goc cua Uoxuong V n(e) nen Uo = P ( /1 )+ ( Uo- P ( /1 ))
A

,?

K

A

Theo dinh 1y Pythagore:
II Uo 112

~

= II

p* )(~O) 112+ II p* )(~to)- Uo112

II pn(e )(~O) - Uo II

= ( II Uo 112 - II pn(£ )(~o) 112 ) 1/2

Ta l~i c6:
n

(UO, ej)

= (uo,ICijgj)
):1
n

= ICij(uo,g)
):1
30

(3.17)


Chu'ClngIII: Chinh hoa bfli toan moment t6ng quat
n

=I

Cij.,uJ (VI Uola nghi~m cua (3.1»

J=I

= Ai (JlO) ( do

(3.3»

V~y bai roan (3.1) tuang duang vdi bai roan (uo, eJ

=Ai (1.1.0) i = 1,2;

""

Ne'u bai roan tren co nghi~m Uothl :

I

A~(Jlo) =

i=1

I (f
;=1

(3.18)

= IIUoll" < Cf)

CijJln2

j=l

M~t khac pn(JlO)dong vai tra u d m~nh d€ 3.1 (3) nen
n

pn(JlO)=

L Ai (j.1°).ei
i=\

n(E)

(3.19)

=> II pn(E:)(JlO) 12 = LA~(j.1°)
1
j=l
00

Tu (3.18) va (3.19) II Uo112-II

L A~ (j.1°)

pn(E:
)(JlO)112 =

j=n(E)+1

The' vao (3.17) ta du<;jc:
00

II pn(E: )(JlO)

-

=(

Uo II

I

(Aj(JLO))2)h

(3.20)

j=n(E)+\
00

=

I

(

.

c:tCijJLiO)2)h

(do Ai(JlO)= (Uo, ei».

j=n(E)+1 i=l

Tu(3.16) va (3.20) ta du<;jc:
=> II pn(E:)(Jl)-uoll:::;

II p*

~ E

1/2

\Jl)-

pOlE)(JlO) II + II pO(E:
)(Ilo)-uoll

~ (L.. C
~

+ ( L..

1/

if j.1 ~

) 2 ) /2

j=n(E)+1 ;=1

-

f) ~t 11
(8)

=E 1/2

+(

~ (L..C JL
~
L..
if

iO

j=n(E)+\ i=1

1/

) 2 ) /2

=> II p* )(Jl)-uoll < 11(8) vdi di€u ki~n 0 < 8 <
Khi 8 -? 0 thl 8-112
-?

00

Man(8) = [f -I (8-112)]nen

n (8 ) -? 00

31

II gl 112


Chuang III: Chlnh h6a bili toan moment t6ng quat
w

M~t khac tU'dinh 1"93.3 ta co:

I (I Cijf-Lj)2 00
<
i=1 )=1

j

00

I (I Cij!-lnz

nen

(

j=n(e)+\

-70 khi n (8) -7

ex)

i=\

V~y 11(8) -7 0 khi 8 -7 0 nghla la sai so cua dfi' kit$n nho diin de'n sai so cua
ke't qua nho.

.

Bay giG ta gia sd 110E Hffi(0 ) va (3.10) dung. Do dfnh ly (3.3)
-/II

II

pn(~O)- uo II ~

Tli

duac:
.

11

c.

11

pn(~)- uo 11 ~

11

uo II

pn(~)- pn(~o) II +11 pn(~o)-

uo II ~ 8\/2 + C II

11 pn(e )(~)-

(3.21)

H"'(O)"

UO II

Uo

II va

(3.16) (3.21) ta

-/~:,
n
H (12)

Binh 1"9chung minh xong.

Voi nhfi'ng gia thuye't khac tren Uo, ~ota se du

danh gia sai so d~c bi~t bon.

3.3 £>onh qio sai so dung gia tri rien9 cua toon 111
Laplace:
- Gia sll'0 la mi~nbi ch~n vdi bien trail.Baitoan gia tri rieng cua roan ill'
Laplace:

- L1U= au
{ u

trang Q

=0 tren aQ

Co mQt day tang cua gia tri rieng 0 < at ~ a2 ~ ... ~ an saG cho
1. an -7 0 khi n -7 ex)
2. Day tu'dng ung (chucfn boa) ham gia tf! rieng bl. bz,... la ht$ tfllc chucfn
cua L2 (0).
Voi di~u kit$n kEN
0

Hk

ta d~t:
00

= {v EL2 (0):

Ia~

m=1

(v,bn,)" < oo}

0

Khi do

Hk

(0) 1ftkhong gian Hilbert vai rich vo huang:
0

00

(v,w)=
hay

Ia~(v,bm)(w,bm)
m=!

II I v 1112

vdiv,w

E Hk vachucfnlll

= I a~ (v,bin)"
m=!
""'J

j.;..

viII =((V,V»)1/2


Chuang III: Chlnh hoa bai toaD moment tong quat
.2

.::

3.3.1 Bo de 3.1:
Cho 0 la mi~n dong tren Ru.
0

"

1. Ta co:

Hk

= { V E Hk (0

):,0.j v = 0 tren 00 , j 0

va chuin III. III tu'ong ung voi chuin cua Hk (0) earn sinh tren

Hk

(0)

Hon nua:
0

Illulll
voi u
I

~dk/2Iulk.Q
I

k,Q

VU

(0)

Hk

(3.22)

= (IIIDauf)1!2
'al=k

. Chungminh:
Ph~n d~u cua b6 d~ la ket qua clla b6 d~ 1, chuang 3, [5], ta kh6ng chung
rninh 0 day. Ta dung ket qua do d6 chung minh (3.22).

.

Truong hop k chii'n:
0

k

= 21voi 1 ~ 1. Khi do V v E Hk (0

) ta co:

,0.J = 0 tren ao voi j ~ 1- 1 ( VItheo ph~n d~u cua b6 d~: j ~ 1-1 <1= k/2
v
=> j < k/2 => t:}v = 0)
Tli c6ng thuc Green thu hai : 1 (f6g)dx-1 (g6f)dx = !of ~CY

-

L g ~ dCY

trong do u la phap vec to don vi ngoai cua rn;rtta 0 ; f, g la hai ham du trail.
Ap dl:lllg c6ng thuc tren voi f

= u , g =bm ta duQc:

(,0.u , bm ) =( u , ,0.bm ) (do f = u = 0 tren ao )

(3.23)

M;rttkhac rhea gia thuyet - ,0.u= a u suy ra 6 bm = - ambm
V~y (u , ,0.bm ) = ( u, - am bm ) = - ameli, bm)
Tli (3.23) (3.24) ta co: (,0.u , bm ) =

(3.24)

(u,,0. bm) = - ameli, bm)

vdi u 1aham du trail va tri~t tieu tren bien a O.
Thay u = ,0.j voi j ~ 1- 1 vao ket qua tren ta duQC:
v
(,0.1 bm) =
v,

- am (6]-1 V, bm)=...

= (-1)1 a~(vJbm)

..,..,

-) _1

(3.25)


Chuang III: Chlnh h6a bai toan moment t6ng quat
un

Vt).y

ex)

IIIV 1112 =

I a~ (v, bm)2
m=!
ex)

"\' a21 ( V, b m )
L
m

=

2

(vi k=2l)

m=!
ex)

= I(a~(v,bm))2
m=!

=

=

(do 3.25)

II(~/v,bmr
m=!
II ~lv 112

2

a2

= 1 { (~+
ax!

a2

!

+~)
axel

V

}

dx

(vi d la s6 chi~u cua Q va dinh nghla cua ~ )

= 1(LDav)2dx
aeI,
(d~ohamca'p21=1

0.1)

~

1 dl

I(Dav)2

aeJI

dx

(co d1 s6 h~ng)

(ap d\lilg ba't d~ng thlic Cauchy - Schwartz:
(al.bl+...+anbn)2~(aI2+

I

= d1 1 ae!,

~

d21

.2:

...+an\(bI2

Ca (DaV)2 dx

J (DaV)2dx

lal=21=k[2

= d21

L f (DClV)2dx

aEl:Q

34

+...+ bn2) vdin=d1;aj=1)


Chuang III: Chlnb boa bfli roan moment t6ng qmit

LIIDav)r

=dk.

( chuffn trong L 2 (0) )

aEl2

?:

= dk

.Ivl

~,Q

L (DaV)2 1a t6ng

Trong d6:

co du'Qc khi khai tri~n mQt cach hint thli'c

aEII
.,

?

?

bi~u thli'c (~-2 +
I

+ :2)1 v ma khong nit gQn cac sO' h<;lngb~ng nhau (cac sO'
d

h<;lngnay chlkhac

nhau v~ thli'tv l:1y d<;loham rieng. Do d6 t6ng nay chua dI sO'

h<;lng
va a
I

=21=k 'If a

I

M~t khac:

Ell)

L Ca (DaV)2 1a t6ng noi tren nhu'ng da nit gQn cac sO' h<;lng

aEI2

b~ng nhau voi cac h~ s6tu'ong li'ng 1a Ca < dl.

.

Truong hop k Ie:

=21 +

k

0

1 voi 1 ~ O. Khi d6 'If Y E Hk(O) ta co:

!::J.Jv0 tren an
=

voi j ~ 1

(3.26)

(do j ~ 1 < 1 + Y2=k/2 ~ j ~ k/2 thoa man ph~n d~u cua b6 d~ (3.1))

Tu'ong tv nhu' tren ta c6:
(!::J.+IV , bm) = (_1)1+1. amI+l (v, bm)
I

(!::J. bm) = (_1)1 . amI (v, bm)
Iv,
Ap

dl,lng cong thli'c Green thli' nh:1t:

I(Mg) dx + I('vf\7g)dx = If ag dcr

Q

(!::J.l+IV

voi g = f = !::J.l ta du'<;1C:
V

nanD

, !::J.I =
v)

- ()
II

(do yoi 1 ~ 1 dung ~

do

(3:27)

Y'(~Jv) 12dx

!::J.l = 0 tren aQ)
Y

.

a>

V~ylll vll12 =Ia~I+I.(v,bm)2 (/0 k=21+1)
m=1

co

= I[a~+l.(v,bl1/)] [a'(v,bm)ll
m=l
3S


Chuang III: Chinh h6a hili roan moment tong quat
00

L

-

=

m=l

{(L~I+IV1
bll1) (.61v, bm)}

= - (.61+IV, .61V)
(do 3.27)

= SIV(~IV)12
dx
Q

..,..,J

d

0

= Q J~1 -(~+
S2: ax. j
{

J

aJ-

0-

I

~)

ax i

v

ax d

}

dx

,

it~?( ~j

s

D"v

r dx

(Khai tritin kh6ng nh6m cae so h9-ngb~ng nhau va ap dl,mg bat ding thuc
Cauchy - Schwartz)
J

2:

= d' SI

Ca

Q j~1 aeI2

~

- dx

DUV

( ax

)

.I

(do nh6m cae s6 h(;lngb~ng nhau)
J

:;;

d"

It

6:,[

:::;dklvl2

k.Q

~, D' v)r dx

(do C"


(dok=2f+!>21~k>21)

V~y ta da chung rninh xong dint 19.

3.3.2 f>anh gia sai s6:

.

Dinh If 3.5:

,
,
,.
()
Gla su Uo1a ng h l~m cua (3 . 1) tu'ang ling VOl~
.??

'~

'"

?

()

=( ~ I )

,()
1 ="1 2 ... va ~ th oa

I

'"

k

L,.a;
i~1

< 2
( "'C ijJl)0 ) 2 - £
L,.
j~1

E la h~ng s6 du'dngnaG d6 va kEN. Khi d6 ydi mQiday s6 ~ saGcho:
II

~

II P

- ~o II

n(E)

00

<E (E>O) va lay nee) nhu'd tren ta c6:

(~)- Uo II :::;E

-k/2

1/2

+

E .a n(c)+1

ho~c ta c6 danh gia khac:

36

?


Chuang III: Chlnh h6a bai toan moment t6ng quat

II pn
'

()

g

(J.l)- Uo

II ;S;E 1/2 + d kI2 . I~(uo) IbQ.

-k/2

an(e)+l
0

trong do:

.

~ (lIo) =

I (Uo,e;)bj va la ph<1ntli'cua Hk (D.).
j=l

Y nghTa:

Ta danh gia du<;1C so giua nghit$mchinh xac cua bai roan 3.1 va 3.4 vdi
sai
cae di~u kit$nuo, J.l° bai roan Laplace
cua

.

Chum! minh:

Uo
Ta phan rich Uo= PvnUo + w (Pvn la hlnh chi€u tn,I'cgiao cua Uoxuong Vn)
= Pn(J.l°)+ w
Khi do : Uo - Pn(J.l)= Pn (J.l O) - Pn (J.l) + w
Ta co W E ynl. nen rhea dinh 1;' Pythagore

ta dU<;1c:

IIpn(J.l)- uoll = (IIpn(J.l°)- pn(J.l)112 Ilwl12)h
+
;S; lpn
I

(do .Ja2 +b2 ;S;a+b

(3.28)

(J.lO) - pn (J.l)11+ Ilwll

vdi a,b~O)

ChQn n = neE) va ap dt,mg k€t qua trudc ta du<;1c:
lip n(e)(J.l0)

M~t khac do

- p n(e)

(J.l)11

;S;Eh

Y;,Pn(J.l°) E Yn, hQ (en) tn,I'c chuin nen:

WE

N€u i :::;n thi

(u ° - P n(J.l° ), e J = (u 0' e i ) - (p n (J.l° ), e i )

= ,-,(~')

-(t '-J(~')ej,e,)

= Ai (J.l ° ) - Ai (J.l ° )

=0
(do (eJ la ht$tn!c chu:1nva i;S;n)
n

- N€u i>n thl

I Aj(J.l°)ej,e;) = 0
j=l

Y~y (w,eJ=(uo-Pn(J.l°),ej)

nen ta chi con (uo, ej)

(do Uo =Pn(J.l°)+w)

o neu i :::;n

(3.29)

= { (uo,e;)neui>n
37


Chuang III: Chlnh h6a bfli roan moment t6ng quat
?

n

Tli do ta co: Ilwll- = L(w,eY
i=1


(do 3.29)

= L(uo,ei)2
;=n+\


n

=i~}ct>(uJ,bJ(Vi

(~(uJ,bi)=

( ~(uOJej)bi,bi ) =(uoJej))

-k ~
k

'
)
_all+1L.,a, (ct>(lIo).b,<

1=11+\

a
(do an la day tang => an+1 < an+2 < an+3 ...

=> IIwl12:::;a~~IIII<t>(ulJII12 (do d~nh nghla

=> ~
anTI

'

~

1 voi i ~ 1)

chu.1n 111.111)

(3.30)

M~t khac :
? un


(3.31)

'111ct>(u,,)f =La;(ct>(uJ,bIlJ
,=1

Ta l~i c6: «!>(uJ, bm) =

(t (u"e, )bi, b",]

= (Zlole,)

(3.32)

Tli (3.31) va (3.32) ta du<;:1c.
111=
1=1

2

= fa;

(

1=1

uo,tCijgI
J=I

J

= ta:[~C,f(U,.g,)r

~ta;(tc;

1':r

5: £2

(3.33)
0

V~y

q)(uo)eHk(Q)

Ta l~i co:
Ilpn(c) (I') -110115:Ilpn(c)(1'0)- pn(6')(p~, + IIpnlel ~o)-

3X

ZloII

(3.34)


Chu'dngIII: Chinh hoa bai loan moment t6ng quelt

Thea ket qua tntoc ta co :
(3.35)

IIpn(c) (Uo ) - pn(c) CLLI< £h
~

Thea (3.28) ta c6 :

.

IIpn (Uo )- UoII~ Ilpl/(Jlo )- pn (Jlo ~I + Ilwll
Suy ra :

(3.36)

IIpn(Jlo)-uoll~lIwll
M~t khac tli (3.30) ta c6 :
IIwl12 ~ a;:IIII~(uo
<

~

E2

-k

-

~112

al/(c)+1

(3.37)

Ilwll~ E.a,~(:/t,
'

I

k

(3.38)
I

Tli

(3.36)va (3.37)taaugc Ipo",~I')- U"I,;E.a,;(~).,

The (3.35) va (3.38) vfw (3.34) ta Guqc:
II

p"(t:)(JL)-u

~Eh

()

+E.a-kl1

",,>-I

II

Ta viet lc;ti I\wl12~ a~(:)+I.III)If
wll~ a:~+l

III


III

(3.39)

A.p dl;lngketqua bode (3.1)
1I1
Tli(3.38)va(3.39):

Ilwll:::;a~t£!):,d7j.1 (UO)lk

The VaG(3.36) ta duQc :
IIpn(JLo)-uoll:::;

:~+,.d7jlqj(uo)1

U2

Tli (3.40), (3.34), (3.35) ta duQc :
Ilpn(£) (J-l) - Uo 11<£~

+ d7j .lqj(Uo)1ul.a~2+,

V~y ta da chung minh xang djnh 19.

3Y

(3.40)