- •
4^'
'.
BO
GIAO
DUG
VA
DAO
TAO
DAI HOC QUOC
GIA
HA
NOI
n^l/CJNG
DAI
HOC KHOA HOC
Tl/NHIEN
&
NGUYEN
VAN
HUNG
MOT
SO
PHUONG
PHAP
HIEU
CHJNR
GIAI BAI TO
AN
DAT
KHONG CIliMH
'•J-v-'^'j'
•. N
ll^.
4
•;
'. j
>
i^ir,
Ha
nOi
-
1996
MUC
LUC
L51
noi
dau 1
CHUONG
I
PHUONG PHAP COMPACT
THU
HEP
CAI BIEN
§1.
Ma dau 5
§2.
Cac gia
thi^'t
cua bai toan
5
§3.
Danh gia
tinh
6n dinh cua
nghiem trSn
compact
y6u
27
§4.
Phuang
phap khai
tridn
ky di chat cut 29
§5.
Phuang trinh tich phan
dang
tich chAp
31
§6.
Danh gia diam
Vs
trong
phaang
phap compact thu hep cua Gaponenko 36
CHUONG III
MOT
SO
PHUONG PHAP LAP - HIEU CHINH
§1.
Phirang
dia
phuomg cua phuomg phap RSN 54
III -
Bdi
todn Men tudn hodn cho phuomg trinh Duffing - Van
derpol
57
nil
- Bdi todn bien tudn hodn doi
v&iphuomg
trinh Van
derpol
65
Phan ket
luan
66
Phu
luc
67
1-
Bai toan -
Lcfi
giai 67
2-
Thuat
giai va
chuong
trinh 72
Tai lieu
tham
toan.
Trong
thirc te
ta
thirofng
gap
nhiJng
bai toan ma
dfr ki6n
ban dau chi
dugc
biet gan
dung,
nhCmg
thay
ddi
nho cua
du*
kien ban
d'au
c6
thd dSn dfi'n
thay
do!
liiy
y cua
nghiem, do do
vi^c
tun
nghidm
Nhimg
den
cuo'i nhung nam 50
nguofi
ta phat hien ra rang, nhicu bai toan
ly
thuyet va
hau
het
Ccic
bai toan trong
ki
thuat va
ihirc
16'
deu dan
de'n
bai loan dat khong chinh.
Tikhonov A
.N
la
ngirai
c6 c6ng dau trong
nghi^n cuti vah de
nay[49,50].
6ng
da
d~e
xuat khai niem bai toan dat khong chinh cho
Icfp
Bakushinski A.B, Gaponenko Yu
.L,
Bertero M.P, Nashed M.Z, Groetsch CM.,
Khai niem dat chinh theo Hadamard la:
Xet phirang trinh
Ax
=
y (0,1)
Trong do A
la
toan tu dua khong gian
T5p6
X vao khong gian Topo Y
1 - Vai
m6i
y e
Ytbn
tai x
e
X
2 - Nghiem x la duy
nhat.
3 - Nghiem phu
thu5c
lien
liic
vao cac du kien cua bai loan.
Ne'u vi pham
it nhSit m6t
trong ba
khOng
phai
vdi
moi y
e
Y).
ii - Ne'u
A'
:
Y->
X c6 ton tai
thi
cung khong lien tuc.
Di^eu
nay chung to nghiem
cua bai toan (0,1) khong phu thuoc lien tuc cac
du"
kien ban dau
Tu* dinh nghia
v'e
tinh
chinh thay rang:
Tinh
chinh cua bai toan phu
thu6c
bo ba
{A,X,Y).
Ngucri
ta
thucmg
kh5ng dap
img
dugc yeu cau cua thirc te.
4 - Phuang phap hieu chinh : La phuang phap thay bai toan dat kh5ng chinh bang
m6t
ho bai toan dat chinh phu
thu5c
tham
s6'
ma nghiem cua bai toan dat chinh sc
d'an
de'n nghiem bai toan dat
kh6ng
chinh khi tham so
d'an
tai
khOng.
Phuang phap
Laverenliev, phuang phap Tikhonov [54], phuang phap tua nghich dao Lattes -
Lions,
phuang phap lap, su dung khai
tridn
ky di va khai
tri^n
ky di chat cut la
nhung phuang phap hieu chinh quen biet.
Phuang phap hieu chinh ciia Tikhonov dua tren b6 de sau:
Bo de Tikhonov A.N [54 ]
Gia su A: M
->
e Xo
: fi[x] < r} la tap compact tuang doi trong X
Phuang phap hieu chinh Tikhonov la lay nghiem
g^an
dung cua (0,1) la
didm
cue tieu
cii
phie'm ham tran:
M
"^ [x,y5
]
=
p^
(Ax,y5
) + a
0[x
]
->
min
x
e Xo
Vai
in6t
s6'
gia thie't nhat dinh c6
th^
chiing
minh dugc rang.
i/ 3 !
dugc quan tam trong luan an la:
1-
Giai bai toan dat kh6ng chinh tren tap compact.
2-
Danh gia tfnh 6n dinh ye'u cua bai loan dat
khOng
chinh tren compact va compact yeu
3 - Cac phuang phap lap hieu chinh giai bai toan dat kh6ng chinh.
Ban luan an
gbm
ba
chirang,
tai lieu tham khao va phan phu luc
Chuomg
I: Phuomg phap compact thu hep cat Men.
Chuang nay trinh bay phuang phap compact thu hep cai bien va danh gia dugc
toe
do
h6i
tu cua phuang phap.
Chuang II:
Bdi
todn tuyen tinh khong chinh tren compact yeu.
Trong bai toan (0,1) xet
truong
hgp X,Y la cac
khOng
gian Hilbert, A la toan
tur
tuye'n
nam thanh lap khoa Toan - Co - Tin DHTHHN hoc nam 1991, Hoi nghi khoa hoc khoa
toan DHSPHN 2 1992,
H5i
nghi khoa hoc khoa Toan - Co - Tin hoc DHTHHN
1994,
Hoi nghi quoc
te'
vt
bai toan ngugc 1995 ( Tai thanh pho Ho Chf Minh).
CHUONGI
PHUONG PHAP COMPACT THU HEP CAI
BICN
$1 - Ma dau:
Xet phuang trinh
Ax
=
y
• (LI)
6 day A
la
toan
tir
phi tuye'n, X,, Y la cac khong gian vector
t6p6.
Gaponenko
Yu.L [24,25] nam 1982 da de xua't phuang phap c6 ten la
"compact
thu
hep".
Ong da
mOt
nua
y'5
thoa man dieu kien.
II /5-yd
||c<5
(l<ji<m;i=l-r ,r>m/2)
(1.2)
Bang
each sir
dung ham Robust, Arsenin da dua ra mot phuang phap hieu chinh.
V6i ham
M"
[
x,y^5
,.•• ,y'"8 ] = ^^ (Ax)
+
aC)(x),
Arsenin V.
la
da churng minh dugc cac ke't qua sau.
l/3!Xa-ArgminM"[x,y^8
y"\]
2/
3 a
=
a(5) :
Xa(6) -> x*
(5
->
xac dinh cua phie'm ham on dinh
D.[x].
OO
-6
Xi
c:
X va ta c6
Xi ==uK(n)
(2.1)
trong do K(n)
=
(v e
X] ,D.{v)
<n )
Chung ta da bie't
m6i
tap K(n) la tap compact trong X
Gia
suf
phuang trinh
(L1) vdfi ve'phai
dung
yd
c6 nghiem duy nhat
Xd
Axd-
yd
Trong thue te'ta kh6ng bie't yd(t), ma chi bie't cac ihi hien cua no
y'5(t)
,
c
trong do r
=
max
{n[vi
],n[v2
]) ;
4^(l,r)
> 0 la ham lien tuc, dan dieu tang theo cac
bie'n, vai 0 < t,r <
+00,
4^(t,r)
->
0 khi t
^
0
Goi S(n,h)
e
K(n)
(Vn
> 1) la
(p(h) luai
huu ban cua tap compact K(n), c6
nghia la:
Vxe
K(n),
3 Xh e
S (n,h) :
||xh
- x|| < (p(h)
V
(2
k)
I
s
I
< k
I
s
I
>k
Trong d6 0 < K < C.6 , 0 < C - Const
-7-
Ham
(jjg
(y) c6
tinh
cha't sau (xem [5])
i/ T6n tai
ygCt) e
C[a,b] :
(t)s(y5) = Inf {(^sCy) : y e
C[a,b] )
ii/
3V>0
:
llyc-ysllc ^
V.5
iii/
3G>0
, t
e
[a,b],
5>0
S3
- Thuat toan Compact thu hep dang Robust.
Gia
sir
ta
c6
he
thiic.
b
m
(t)6(yd) =
J
Z P5(y'5(t)-yd(t))dt<©(8)
a
i^l
(3.1)
a day
a)(5)
la ham lien tuc, khong am hoi tu de'n 0 khi
5-^0.
Nhan xet 3.1:
Dfeu
kien (3.1)
hiin
nhien duac thue hien trong
tru&ng
c6 the
Ian
tuy y.
De dang chiing minh dugc cac danh gia sau:
(l)5(yd)<I||y'8-yd||
i=
1
Li
8-
||y'5-y||
<(b-a)||y^8-y||,||ys-y||
<
(b-a)^^^M|y5-y||
Li
C
Ll
Lp
d
dayqj>
1
:
l/qj+
1/pj
=
1,
Tir
he thue (2.2 ) va (3.2). suy ra he
thii-c
(3.1).
Gia su 5 > 0 la
Ne'u
Vi
9^
([) la'y T2
=
ri
sau do
ihirc
hien cac buac tie'p theo.
Birdc
n < N,
Vdi r„
dugc xac dinh tu
budre
truac.
Vn = i
ve S
(rn,
hn)
:
^^
(Av) <
GT((p
(hn),
rn)
+
CO
(5)}
Ne'u
Vn = (t)
c6
dinh thoa man he
thu"e
(j)5(Av) ^
0 (5
->
0 )
thi
I (t)8(Av)
-
Uyd
1^0
(5
->
0 )
va
V
=
Xd
Chung
minh day
dii
bd de nay xem trong [5 ].
Bo de
3,2:
T6n tai s6'
No = No (Xd)
sao cho
VN>No=> Q[Xd] <rN
Chvcng
vdi
gia thie't.
Tuang tu, vdi
ni = Ni +
1,
tbn
tai
s6'
N2
sao cho
v6i mi
<
m2 <
N2,
Vm2 ^
(j) .v.v
Tie'p tuc tie'n hanh nhu tren ta dugc day tap hgp
{Vmk
)"°
khac r6ng. Chon day
{Vn
)
k= 1
ba't ky trong
V m
,n=l,2
taco:
n[Vn] <r,T,< n[Xd]
,
n
Khong ma't
tinh
tong quat ta
c6
the coi
n
11
Vn —> Vo
( n
-^ 00
), va do do:
Q[Vo]
<
Q[Xd]
-
£
^
(3.3)
Mat khac :
(l)5(AVn
) <
GT
((p(h^),
Tn,
)
+
C0(5) <
GT((p(hn,
),n[Xd])
+
tu nhien N >
No
tap
VN
khong
rdng
va
chira
trong tap
compact K
Chirng
minh:
Tur bd
d'e
(3.2) suy ra
vbi
moi N >
No,
fi[Xd] <
TN
va do do
Xd
e
K{T^).
gia
surxh
e
S(rn,hn)
sao cho
||xh
d'e
dugc
chi^ng
minh [].
Nhan xet 3,2:
VN
^
i?,
doi
vdi
moi N >
No
ta
luOn
c6
rw < rNo ^ No
va
VN
<=
K (No)
==
K (K
-
Tap
compact)
Bd de 3.4: Day tap compact
{V5)
co
Ve didm Xd
khi 5
-^
v" khi k
-> o)
Khi do ta cung c6
Avk"
-^
Av"*".
Mat khac
(|)5
(Avk")
<
GT((p(hNk
),rNk)
+
co(5).
Tir
do suy ra.
0 <
(1)5 (Av*)
<
co(5) ->
0 ( 5
->
0)
Theo bd de (3.1) ta c6:
v*
= Xd
e X . Dieu nay mau thuan vox (3.4 ).
Nhu vay ta c6 diam
V5 ->
Nhu
vay
ta da chung minh dugc dinh ly sau:
Djnh
!y 3.1: Phuang phap compact thu
ht^p
dang Robuts
h(3i
tu va ta c6 danh gia.
IIX5
-
Xd 11
< diam
Va +
5
Nhan
xet
3.3:
Ne'u phie'm ham dn djnh
Q[x]
thoa man
di'eu
kien
Q[x]
> C
'
||x|
Vx e X
(C-const
> 0
(Av) <
G^
(cp
(hn),C.r ) +
co(5)
)
D^
y rang phuang phap compact thu hep [24] la truong hgp rieng cua
phircng
phap nay khi n
=
1,
Q[x ]=
||x||
1
va
||y5
- yd|| < 5, a day ||. ||
1
la
chu^n
nao do
irong
kh6ng gian con
Xo,
trii
mat va compact trong X
Nhan xet 3.4: Ne'u thay ddi each
xay
dung tap
diam
Vi
+ 8
Thuat loan
dimg
a day
Ne'u
Vi ^
(|), ta la'y
r2 = ri
+ 1 va
thue
hien buac 2
Budc
( n < N) dugc thirc hien ne'u
Vi,
V„-1 = ([>
Khi do
rn =
n, ta thanh lap tap
Vn
Vn-{v:veS(hn,rn):(l)6(Av)
<
GT(5,rn
) +
co(5))
Ne'u
Vn ^
(t>,
ta la'y
nay chung to: Ne'u
Qxd
la mot so nguyen
thi
so
bubc
cua thuat loan la fifxd
].
Ne'u
il[Xd
] khong phai la so nguyen
thi
so' bu6e cua thuat toan la
[Q[Xd
]]+ 1. Vay so
bu(5fc ciia
thuat toan
khOng
phu
thu()c
vao
5
khi 5 du be.
Nhan xet 3.6: Ne'u tien nghiem bie't rang
Q[Xd] <
R thi ta thanh lap tap
V5={v:veS(hN,R):(|)5(Av)
<
G4^(6,R)+co(5)}
va thuat toan
van
con dung trong truong hgp nay
§4 -
Truotig h(ifp
khong duy nhat nghiem:
Trong muc nay, ta't ca cac gia thie't cua muc
52
trtr gia thie't (1.1) c6 nghiem
duy nhat dugc
giu"
nguyen.
Goi
U=
{x e
Xi
: Ax
= Axd = yd)
la tap hgp nghiem cua (1.1) Gia su trong
Utbn
tai duy nhat
x*
sao cho
Q[x*] = minf^[x]
xe U
TrU(Je
he't ta xet thuat toan cai bien sau:
Vain
> l,dat:
V„ = {v:v
e
1
LtfyTn+i =
r,^
'^\~
")^^^^y<^v^gtap
Vn^
+ i = {
v:v
eS(hN,rn^+i):(t)6(Av)<GT(5,rn^+ i + co(5)
}
Ne'u
Vn^+1 =
(j),
thi
thuat toan
dimg
lai va ta la'y
vg
e
Vn
tuy y. Ngoai ra
||x*-V6||
<dimVn
+ 8
-T(8)
N
1
Ne'u
Vn^+,
^^ftaia'y
'
Khi do
thuat toan dimg
lai va ta la'y
vg e
V,,
+
,„-1
tuy y,
ngoai
ra
\^
T(5)
=diamV,,
.n,-i
+5
N
V<5i
each
xay
dung thuat toan
nhu
tren
ta c6 ke't qua sau.
Djnh
ly
4.1:
Phlln tir V5
dugc chon
a
, VN >
No => rN
>
0[x*]
Chiing minh
bd de nay
hoan toan
lap lai
chung minh
bd de
(3.2).
Bd
de 4.2 :
Vn +m-1 chiia
trong
tap
compact
Ko,
N
eon
ke't
luan
cua bd de 4.2 suy ra
tmc
tiep
tu
each
xay
dung
tap
>
e>0
vai k
=:
1,2,
Vi
Vn e
K
(nk)
- la
compact,
k - 1,2, nen
khong giam tdng quat
ta gia
sir
rang:
Vk*
~>
Vo"
khi
k
->
00
Tac6(i)5(Avk*)
<
G^((p(hnj^+
.iyi),rn^
+n,^
i)
+
= nN +
—•
•
N
"N^"' "N*""'
N N
Do nhan xet (3.5 ) thi
nN = ^[x*]
ne'u
n[x*]
la
sd
nguyen hoae
nN = n[x*]
+
1
trong
tnrbng hcDfp
ngugc lai.
a - Truong
hcrp Cllx*]
la sb nguyen thi m
=
0 va
Q[x*] - [Q[x*]] =
0
b - Truong
hcirp Q[x*]
khong la sb nguyen
thi.
cz {
) sao cho
N "
Ns
N
ms ms
lim =
a Do dinh nghia cua
{
) ta cung eo.
ms nris
1
1
<n[x*]-
[D.[x*]]<
1-
+
—
Ns Ns Ns
(4.2)
Cho s
-> OO
thi dugc he
thiJc
Theo each
1 -a<
Q[x*]-
[Q[x*]]
xay dung day (
Vk"
tinh
tdng quat coi k
=
s. Khi k
->
oo ta duac
15
n[vo^
]
-
[n[x']] -
1
- a
Tijr
(4.2
) suy ra
a[vo*
]
=
a [
X*
]
(4.3)
Til
(4.1)
. (4.3) va
tinh
duy
nha't
cua
nay
chung
ta gia
thie't rang: Trong phuang trinh
(1.1) ta
khong bie't
A
ma
chi bie't
A ^
A^
:
X
->
C[a,b] La toan tur lien
tuc
thoa man.
a
-
II
A^v
-
Av||
<
u(|a,Q[v])
V
v
e
Xi,
u(ja,s)
],Q[v2]),
ham
T(t,r)
thoa man
gia
thie't trong
S2
c
- Ta
van
gia
thie't
(|)5(
yj) = (t)5(AxT)
<
co(8).
Cac
gia
thie't khac
giiJ
nguyen nhu trong
52.
Chon
day (hn)
kh6ng tang;
0 <
h„
<
1,
hn —>
V G S(h,,ri)
:
U^^'v)
<
G[T((p (hi),ri) + u(vi,ri)
]+
co(8))
Ne'u
W\^^ia.
la'y
r2 = ri
16-
Ne'u
Vi
=
(|)
ta la'y
r2 = ri
+ 1
Sau do thue hien
budc
tie'p theo
Bir6c
n ( n < N )
Vn =
{v:
V e S(hn,rn)
:
UA^'v)
<
=^
^
l^
V
e Vg
tuy y ne'u
Vg^
^ ^
Bo de 5.1 : T6n tai sb
No = No (Xd)
sao cho:
V N >
No n[xd]
<
rN
Bd de nay dugc chung minh tuang tu bd de (3.2)
Bd de 5.2 : Vbi moi sb tu nhien N >
No
tap
VN
khong
rdng
va
chu-a
trong tap
compact
Ko
nao do.
Chtrng
minh: Tur bd
GlJA'^Xh
-
A^Xdll+Gu(vi,rN)+
oj(S)
<
G^(cp(hN),r)
+
Gu(p,rN)
+ 0(8)
<
GT ((p(hN),rN)
+
Gu(p,rN)
+
co(8)
Trong do r
= max(n[Xh],n[Xd])
<
rN
Dieu nay
chihig
to
xi,
e
VN
hay
VN
^
(j).
Bd
Ko,
kh6ng
m&i tinh
tdng quat chiing ta cb ihi coi rang:
Nk
Vk*
~>
v"^
khi K
-^ 00
Taco:
Tur
day suy ra:
U^x^-)
^ G[^((p( h
)
, r)
+
u(P,
r
) ] +
«(5)
k k k
lim lim
(|)5^(A^
Vk") = (l)6(Avo-)
<
co(8)
H-*
0
-
Xdll
< diam
V^g
+
cp(hN)
Hay
||x^g - Xdll
<
T(8,p) =
diam
V^g
+ 8 +
u
(5.2)
Ta phat
bidu
ke't qua vua nhan dugc
dudi
dang dinh ly sau:
Dinh ly 5.1: Phuang phap compact thu hep dang robust v6i ve' phai la toan
tir
A
bie't
g"an
dung
hOi
tu va co danh gia (5.2) .
Nhdn xet 5.1: Ne'u chi bie't
g"an
r2
U
=
U(0)
=
(pL
Trong do
D:d^^[0,l]
->
C[0,1] la toan tu vi phan phi tuye'n bac 2 thoa man
di^eu
kien
||Dvi-Dv2||
<M;(||
V,
-V2Lr),V vi,V2
e
^'^[0,1]
d dayr =
max{|lvi|U|v2||
) , 0 <
vKt,r)
la ham lien tuc, khong giam theo t
e
var,
0 < t,r < +
00 ,
va
M^(0,r)
=
(t»8(Du) -
<|)g
(/)
<
co(6)
6 day
co(6) ->
khi 6
->
0. Dual day ta se
sur
dung cac
chudn
||
. ||
2,
|| J
^^1.
dugc xac
dinh nhu sau 2
Vbix e
C'iOA]:
\\
x||2
= max {
||
x||^|| x||^|| x||^
},aday
1|
.||^aehuSn
1
S
=
S (h,x,p)
yk = kT ,k =
0,±l,±2, ,±px-'
Goi M
=
M (h,x,p) la tap ta't ca cac ham
luoi
tren S,
tiJc
la cac ham
lien
luc tren
[0,1],
tuye'n tfnh tren m6i khoang
(xi.
1,
Xi)
i
=
1,2,
, N, con tai cac didm
Xi
(i = 0,1
,N)
chung chi nhan cac gia tri trong tap {0, ±
x
,± 2x , ±
r6i rac cua ham
u"(x)eWj[0,l]
2,
Ham x t
Uh,
{x)= \ i
u"aTi)dTidt
+ u'(0)x + u(0), 0 < x < 1
0 0
•J
dugc
goi la tuang tu
rbi
rac cua ham u (x)
e
W
[0,1]
Bd de 6.1 [20]: Ne'u u(x)
e
W
^
[0,1],
eon
Uhx(x)
la tuang tu rai rac
ctia
u(x), thi tbn
tai hang sb
ho =
ho(u) > 0
(h,i,T,i,r,i)
trong do
r,i
se xac dinh sau. Goi
M„ = M(hn,x,i,r„)
la tap
ta'l
ca cac ham
ludi
tren
S,v
Kn(r) =
{a(x):a(x)eM„:
|| a ||
i
<r }
w
2
^
Uhg
vbi
mbi
ham a(x)
e Kn
(r)
a
C[ 0,1] ta xet ham
Va
(x)
e
tap
Vi = {ve
r'(v)
e
Ki(ri) :^^
(Dv) <
G\\f{
V"h7
+ xi +
25, r, + |
q),M
+
I q)2M
) +
w
(6)}
Ne'u
Vi =
(j),
ta la'y
r2 = ri
+ 1
Ne'u
Vi
1^
(j),
ta la'y
r2
=
ri
Ne'u
Vn ^ ^,
ta la'y
rn -n = rn
va thue hien buac n + 1 < N.
Birdc
N:
Vn = V5 =
{
V
:
r Vv) e Kn (rn
+
Co)
;
(t)6
(Dv) <
GvK (36,rN
+
i
cpi^
I +
i
ipi""
I)
+
co
(6))
La'y
u^
II u" II 1
] +
1
ta
tim
dugc
Nj
>
ni
va
mi
<
Ni
sao cho
W ^ ^.
That vay, ne'u
w
2
khOng,
theo
each
xay
dimg
tren ta eo
rNi =
Ni^
ni
> ||u"
||
1
<
||u"||
iViFn,
W^
n
w
n
la sb nguyen
nen
tbn tai sb s > 0 sao cho.
II^'IIV
-ll""llw'
-^
=
'•'',11=1,2,
Vi
hinh
cau dong trong
W2'[
0,1] la tap hgp compact trong
C[0,1]
nen tir day
{v"n} = (an) CZ
C[0,1], tbn tai day
{ak
}
e {an)sao cho
n
ak -> ao
khi n
Vi
Vne VtTin
nen
(t)6(DVn) <
G\jy(V
hn
+
Xn
+ 25,
Vn + I
(pi^
I
+
I
(P2^
I) + co(6)
Tu*
tinh
lien tuc cua
^^
va D suy ra
(t)6(Dvo)
<
Gy
(36,rN +
I (piM
+
I
^Pi"
I) +
khong
rOng
va
chiia
trong tap compact
Ko
nao db.
Chieng
minh: Tu bd de
(6.1)
suy ra:
l|u"L
^ II"11 1 ^rN va||u",,|l
<
rN
Lfo.i] "w Lro.i]
2 2
2
Theo each xay dung, tren
mOi
doan
[Xi,Xi +1]
u"ht(x)
CO
dang
ajX
+
bi,
tiif
da'y suy ra
Xet ham.
22
X t
uV =
J J
u"hx(r|)
dridt + cpi^x
+
(p2^
= Uhx
(x)
+((pi^
-(pi
)x
+
((p2^
-
(p2)
0 0
d
day
Uh t(x)
la tuang tu
rbi
rac
ciia
u(x)
Ta cb danh gia:
II u\ ,
-
= max{||u\,||2,||u||2)
<
rN
+
I cpiH
+
I (P2H
+ 25 (6.4)
Tijf
(6.2), (6.3) va (6.4) la cb he
thu:c.
^8(Du\,)
<
Gvj;
(35,
rN
+
I
cpi^
I
+
I (P2M
+ 25 ) +
co(5)
Suy ra
u\
^ eVn ,
hay
VN
^ ^.
trong C[ 0,1] khi n
-> 00
Khi do: x t
v'^n
=
J J
a'^n(ri)dridt
+
(pi ^x
+
(p2^->
v^
(n-^00)
0 0
trong
C^[
0,1],
b
day
v-„
(x)
G
VN
Mat khac
(|)5(Dv*n)
^
Gvj/
(
VTiI7
+
Tu bd rfe (3.1) suy ra
v*n
= Vo
mau
thuSn vdfi
(6.5). Mau
ihuan
nay suy ra bd de
dugc chung minh.[]
Tijr
cac bd de tren suy ra dinh ly sau:
Djnh ly 6.1: Vai moi 5 > 0 tbn lai sb N(5) sao
choVN =
V^
chiia Uhx
,vai moi
u^ e
V^
ta cb danh gia:
||u^
-
uh^llu"^
-
u\x
II2 + II
u\,
- u
II2
< diam
V^
qua n
( n < N
),
hoae bie't
||u"
|
1
< R ta cung eo
thd xay
dung thuat loan mot budfc nhu
^2
nhung nhan xet a cac muc
tru6c.
CHUONG II
BAI TOAN
TUYfiN
TINH KHONG CHINH TREN COMPACT YEU
|1
- Mo dau
Gaponenko Yu.L (1989[27])
d"e
xua't phuang phap xa'p xi tuofng thich giai bai
toan (0,1) trong truong hgp X
= Y = L2[
0,1] vdfi gia thie't lien nghiem |
Xd(l) I ^
R V
t
e
[0,1]
mCt
thuat loan
kidu
khai
tridn
ky di chat cut
dd
giai
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