Mở rộng và ứng dụng bổ đề Gronwall -BellMan 9

Luận văn thạc sĩ toán học -ngành Toán Giải Tích -Chuyên đề :Mở rộng và ứng dụng bổ đề Gronwall -BellMan

MlJ r(mg va llng dlJng Bd dl Gronwall-Bellman Hoang Thanh Long

CHUONG 4

MOT s6 UNG DT)NG

Nhuda gidi thi~u a ph~n d~u tien, B6 d~ Gronwall-Bellman va cac

ma rQng cua no dong mQt vai tro ra't quail tn;mg trong 1:9thuy€t dinh tinhphuong trlnh vi phan No nhu mQt cong Cl;lhUll fch d~ chung minh mQt s6k€t qu~ quail trQng nhu stf duy nha't, tinh 6n dinh, danh gia tinh bi ch~ncua nghi~m, Bay giC5chung toi xin trlnh bay mQt s6 ung dl;lng cua no

§4.1 s1j DUY NHAT NGHI~M CUA PHUONG

TRINH VI PHAN vA TicH PHAN

Cho E ={u: I =[to,to+T]~IRll I u lien tl;lC}.Tren E ta trang bi mQtchu§'n Ilull= Supllu(t)III' 11.111 la chu§'n Euclide tren IRll.DS dang nh~n tha'y

trong do x'(t) la d~o ham theo t cua ham xCi).

X: IRn x I ~I Rnla ham lien tl;lctren mi~n D.

D = {(x,t)EIRllxIIIIx- xolls r, It- tols T}.

Gia sa X la mQtham thoa man di~u ki~n Lipschitz theo bi€n x,

MlJ rl)ng va ung dl.mg B6 dl Gronwall-Bellman Hoang Thanh Long

nghla la ::JK> 0 sao cho:

IIX(x,t) - X(y,t)1I ~ KIIx - yll, V(x,t), (y,t) ED (4.1.2)

Do X(x,t) la ham lien wc tren D lien ham X(xo,t) la bi ch~n tren

It- tal ~ T E>~tMo = Sup IIX(x(»t) II va M =Mo + r K Tli tlnh Lipschitz

11-lol$T

cua ham X, ta co duQc:

IIX(x,t)1I~ M, V(x,t) ED

4.1.1 Dinh Iy 1.1

Ne'u X thoa adu kifn (4.1.2) thi hili loan Cauchy cho phuang trlnh

(4.1.1) co duy nhli't nghifm x(t) tren It - tol < rM-1.

Chung minh dinh Iy 1.1.

Gia sa x(t), yet) la hai nghi<%m cua phuong trlnh (4.1.1) thoa di~u

ki<%ndftu x(to) =y(to) =Xo.Ta co

IIyet) - x(t) II ~ II( X(y(s),s) - X(x(s),s)ds II.

~ fl K IIyes) - xes) IIds.

E>~tu(t) =Ily(t) - x(t) II2 0, ap dlJng dinh ly 1.1 chuang 1, ta duQCu(t)

=0, vdi t thoa man It - tal < rM-1

Vi d\l Xet bai loan sail:

40 MiJ rfjng va ling dlJng Bfl dff Gronwall-Bellman Hoang Thanh Long

Ta chung minh phuong trlnh (4.1.6) voi di~u ki~n d~u (4.1.7) co nghi~m duy nh~t tren [xo, Xo+ T], voi T > 0 n~lOdo.

Th~t v~y, gicl sli'phuong trlnh tren co hai nghi~m Yl, Y2khcl vi lien

Wc Wi b~c hai tren [xo, Xo+ T]

D:)t w =Yl - Y2 Khi do, w thoa man:

w"(x) + aw'(x) + pw(x) = O.

w(xo) = w' (xo) = O

(4.1.8)

(4.1.9)

Ta chung minh: w(x) =O,'v'XE[Xo, Xo+T]

Nhan hai vii cua (4.1.8) voi w' va rut gQn, ta duQc:

Iw'\x) I ~ {IpiT + 21 a I}foIw'\s) Ids (4.1.14)

Ap d\lng b6 d~ 1 chuang 1 (c =21al + IpIT, k =0, u(xo) =0 ), ta thuduQc w'\x) ~ O Suy ra

Mi'Jri)ng va ling d(tng B6 dl Gronwall-Bellman Hoang Thanh Long

w(x) = c, 'v'XE[Xo,xo+T], C la m9t h~ng sf{nao do

Do (4.1.10) nen ta duQc:

w(x) =0, 'v'XE[Xo,xo+T]

4.1.2 Dinh Iy 1.2.

Ne'u ~~ tbn t[Ji, lien tl:lctren D thz hai roan Cauchy cho phuang trenh

(4.1.1) tren It - tol < rM-1co duy nhat nghi~m.

Ti€p theo la m9t sf{k€t qua v~ st! duy nha't nghi~m cua phuong trlnhtkh phan.

co duy nhat nghi~m.

Chung minh dinh Iy 1.3.

St! t6n t~i nghi~m cua (4.1.15) co thS chung minh b~ng nguyen 1:9anh X~ co ho~c co thS thalli khao trong [10]

Gia sa x(t), yet) la hai nghi~m cua phuong trlnh (4.1.15) Ta co:

Mlf ri)ng va ung dl;lngBtl d~ Gronwall-Bellman Hoang Thanh Long

Suy ra

I yet) - x(t) 1~ I f.t 1 (I K(t,s) IIyes) - xes) 1ds. (4.1.18)

Do K lien tlJc tren ~ la mQt t~p compact lien t6n t~i M > 0 sao cho

IK(t,s)1 ~ M, '\i(t,S)E~ Tli (4.1.18), ta suy fa:

1yet) - x(t) I ~ 1f.tIM (I yes) - xes) Ids. (4.1.19)Tli ba't d~ng thuc nay ta suy ra yet) =x(t), '\itEQ bang cach ap dlJng

h<$qua trong chuang 1 (mlJc 1.1.3) va chli yrang Iy(t) - x(t)l?: 0.(0)

4.1.4 Dinh Iy 1.4.

Cia sit KEL2(tJ.;IR), a(t)EF, j.1la hling s(5'thiphuang trinh (4.1.15) tren F co duy nhllt nghi~m.

Chung minh dinh Iy 1.4 Tu'ang tv nhu dinh ly 1.3.

Ap dlJng ba't d~ng thuc Holder, ta duqc:

Iy(t) -x(t)12 ~ 1f.t1211 K 112 2 ft Iyes) - xes) 12ds

D~t u(t) =Iy(t) - x(t)12?: O Ta co

Ap dlJng b6 de 2 trong chuang 1 (mlJc 1.2.1), ta duqc u(t) =O,'\itEQ,

nghIa la phuong trlnh (4.1.15) tren F co duy nha't nghi<$m.(O)

MlJ rQng va ung d~tngBd d€ Gronwall-Bellman Hoang Thanh Long

II x(t) - yet) II ~ IIx(to) - y(to) II+( K IIxes)- yes) IIds.

Ap dlJng dinh 1y 1.1 chuang 1, ta du(,1c(4.2.3).(0)

Gid sit X la ham thoa man diiu ki~n (4.1.2) va

IIX(x,t) - Y(x,t)11 ::::&, V(x,t) ED. (4.2.7)

Niu x(t), yet) lein IU(ftla hai nghi~m cua (4.2.1) va (4.2.2) tren /, thz ta co ddnh gid:

Ilx(t) - y(t) II ::::llx(to)-y(to)llexp[K(t - to)J

[;

+ -{exp[K(t- to)J -1}.

Chung minh dinh Iy 2.2 Tuang tlj nhu chung minh dinh 1y 2.1.

Th?t V?y, ta co danh gia:

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 dlJng dinh 1y 1.1 chuang 1, ta du(,1c(4.2.9).(0)

MlJ rfjng va ung d(tng Bd di Gronwall-Bellman Hoang Thanh Long

§4.3 DANH GIA TINH BJ CH~N CUA NGHIEM

Danh gia nghi~m cua mQt phuong trlnh vi phan co th~ cho bie't tinh

6n dinh, bi ch~n,

Xet phudng trlnh vi phan:

trong do x: 1 + IRllva f: IRHxl + IRH

Gia sa f la ham lien t\lc tren mi~n xac dinh cua no Ta xet mQt sf)d~ng cua f tho a man:

a IIf(x(t),t)1I ~ g(t)lIx(t)11 + h(t), (4.3.2)

g, h la cac ham duong, kha tich tren 1

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'u hamf(x(t),t) cila phztclng trinh (4.3.1) thoa man diiu ki~n (4.3.2) thEta co:

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:

Suy fa

II xCi) II~ IIKeto)II+ (II f(x(s),s) IIds

MlJ ri)ng va lIng dl,mg Bd d€ Gronwall-Bellman Hoang Thanh Long

II x(t) II ::;II x(to) II +1~ {g(s) IIxes) II+h(t) }ds (4.3.7)

Ap dl;1ngdinh 19 1.5 chuang 1, ta duQC(4.3.5).(0)

4.3.2 Dinh Iy 3.2.

Ne'u hamf(x(t),t) cuaphliang trinh (4.3.1) thoa man di~u ki~n (4.3.3) thi ta co:

I

Ilx(t)ll:::{exp[C(t-to)J{[11 x(to) III-a + IJ -l}l-a. (4.3.8)

Chung minh dinh Iy 3.2.

Tli (4.3.3) va (4.3.7), ta Suyfa:

Ap dl;1ngdinh 19 2.6 chuang 2, ta duQc:

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'u hamf(x(t),t) cuaphliang trinh (4.3.1) thoa man ddu ki~n (4.3.4)

va g la ham tang, lien tl;lctren [0,00) thi ta co:

IIx(t) II::; \jf-I [\!f(11x(to) II)+ K(t - to)]' 'ritE/, (4.3.10)

trang do

ex 1

\!f(x) = JI -ds E g(s) (£>0, x>O). (4.3.11)

Chung minh dinh Iy 3.3.

Tli (4.3.1) va (4.3.4), ta suy fa:

4.3.4 Dinh Iy 3.4.

Nghi~m cua phuong tdnh Riccati sau

y'(t) =a(t)/(t) + b(t)y(t) + kef) (4.3.13)

trang do a( t), b( t), k(t) la cac ham lien tl;lctren Q, y ECl (.0), va nh~n gia trj th1!Cse thou man danh gia

'rItE[to,tp),tp =SUp(tEQ I exp[- Jtortb(s)dsJ( rta(s)dsJ}-i >Mj,Jto

M=Sup(ly(to)+ fk(s)dslj.

(4.3.15)

Chung minh dinh Iy 3.4.

Tli phuong trlnh (4,3.13), ta suy fa:

Iyet) I ~ Iy(to) + L k(s)ds I+ L Ia(s) IIy\s) Ids + L Ib(s) IIyes) Ids

~ M + rl Ia(s) IIyes) 12ds + rl Ib(s) IIyes) Ids

Ap dl;lng dinh ly 2.4 chuang 2, ta duQC(4.3.14).(0)

Lllqn van th{lc si loan h(Jc Mil nganh 1.01.01

M/J rf)ng va ll'ng dl;lngBd di Gronwall-Bellman Hoang Thanh Long

§4.4 SAI LtCH NGHItM HAl PHUONG TRINH

VI PHAN

MQt phuong trlnh vi phan khi bi thay d6i vri phai, VI dl;! bC'1icaenhi~u, di~u khi€n se diin drin slf sai khae nghi~m Chung ta se sa dl;!ngcae mC'1rQng eua B6 d~ Gronwall-Bellman d€ danh gia slf sai khae do,

Cho g: IR+~ (0,00) thoa man cae tfnh eha't:

a, g lien tl;!eva tang tren [0,00)

b g(x) s; X,VXE[O,oo)

Xet hai phuong trlnh vi phan sail:

x' =X(x,t)y' = X(y,t) + R(y,t)

(4.4.1)(4.4.2)Gia sa x, R la cae ham lien tue tren D va thoa man cae di~u kien:~ .

vdi 8(t) la mQtham kha tfch tren I va

IIX(x,t) - X(y,t)1I s; Kg(lIx - yll), V(x,t), (y,t)ED

Mi'JrQng va ling dl,mg Bd di Gronwall-Bellman Hoang Thanh Long

M= Sup{15+ fte(s)dsl tEl},

Chung minh djnh If.

Tli (4.4.1) va (4.4.2), ta Suy fa:

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

Ne'u g(u) =u thi ta co:

Ily(t) - x(t) II ::;15exp[K(t - to)] + (exp[ K(t - s)]e( s)ds (4.4.12)

MlJ rQllg va Ullgd(l1lgBiJ dl Grollwall-Bellman Hoang Thanh Long

§4.5 SV PHT}THUQC CUA NGHItM THEO

Chung minh dinh Iy

Ta celn chung torAng:

VE> 0, 38(E,~o) > 0: I~- ~ol< 8 => IIcp(t,~)- cp(t,~o)1I< E

Th~t v~y, tu (4.5.1), ta co:

cp(t,~o) = cp(to'~o) + rl X(cp(s'~o),s,Jlo)ds

Til (4.5.3) va (4.5.4), ta thu duQc:

II <p(t,f.l) - <p(t,f.lo) II < II <p(to'f.l) - <p(to'f.lo) II

Mi'irfjng va ll'ng d~tngBli di Gronwall-Bellman Hoang Thanh Long

§4.2 Stj LIEN T{)C CUA NGHItM THEO

Tinh lien t\lC cua mQt ham s6 la ra'"tquail trQng vi dt!a vao tinh lient\lC ta co th€ xa'"pXl gia tri cua ham s6 ling vdi st! thay d6i nho ban dgu.Tinh lien wc cua nghi<%m cua mQtphudng trlnh vi phan cling khong phiii

la ngo(;li 1<%.

Xet hai bai loan Cauchy sau:

(4.2.2)y'= Y(y,t); y(to) = Yo,

x, Y la cac ham lien t\lCtren D.

Giii sa x(t), yet) la hai nghi<%mcua (4.2,1) Ta co:

x(t) =x(to) + rt X(x(s),s)ds,Jto \itEr, (4.2.4 )yet) = y(to) + rt X(y(s),s)ds, \itEI.Jto (4.2.5)Tli (4.2.4) va (4.2.5), ta thu ducjc:

IIx(t) - yet) II :::; II x(to) - y(to) II+(II X(x(s),s) - X(y(s),s) IIds

Mli r(mg va ling dljng BiJ dl Gronwall-Bellman Hoang Thanh Long

Tli (4.5.5), ta thu duqc:

II <pC t, f.l) - <pC t, f.lo) II~ II <pCtIp f.l) - <pCtIp J.lo) II

Ap dl;lng dinh 19 1.1 chudng 1, ta duqc:

E:

2exp[L(ti - to)]

(4.5.8)Khi do ta chQn y(to) va ta duqc di~u cffn chung minh.(D)

Khi xet de'n Hnh 6n dinh nghi~m cua mQt phuong trlnh vi phan,

chung ta thudng xet slf 6n dinh cua nghi~m t~m thudng, tuc la nghi~m x

= o Ne'u x = XI"*0, ta co thS d?t y = x - Xl va xet Hnh 6n dinh nghi~m y.

Trong ph~n nay chung ta xet Hnh 6n dinh mil cua nghi~m Gia sa mQi

nghi~m d~u co thS keo dai de'n 00

D'={(x,t) IlIxll ~ H, to ~ t <oo}, 0 < H la h~ng s6.

Xet phuong trlnh vi phan:

voi A(t) la loan ta tuye'n Hnh, bi ch?n, lien Wctheo t, R(x,t) la ham lien

t\lCtrong D' va thoa man di~u ki~n:

trang do W(t,s) lil taan tit Cauchy ( ma trqn ca ban), W(t,s) =X(t)X./ (s)

Mi'Jri)ng va u'ng d1;lngBi} dl Gronwall-Bellman Hoang Thanh Long

vdi x( t) lit ma trcJ-nnghi~m cua phuclng trinh

D~ tha'y X(t) thoa man phuong trlnh (4.6.8) va detX(t) = et2"*0,

\ftE [to,oo) Tli day ta d~ dang tinh dU<;5cWet,s) =X(t)X-1(s).

Mi'Jri)ng va ling dl!ng Bd dl Gronwall-Bellman Hoang Thanh Long

4.6.3 Dinh Iy

Ne'uphu(jflg trlnh (4.6.1) co ham R(x,t) thoa man di~u ki~n (4.6.2),

A =a - BL > 0 thi nghi~m x =0 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) 1111Keto) II+ II (W(t,S)R(x,s)ds II

+ r BLexp[-a(t - s)] IIxes) IIds

Ap dl!ng dinh 1y 1.8 chuang 1, ta dU<;5c:

IIx(t) II~ B IIKeto) IIexp[ -( a - BL)(t - to)] (4.6.12)

VI A = a - BL > 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 =a- BL > 0, A(t) roan tiituye'n tfnh, lien tl;lc,bi chi;inthi nghi~m x =0cua no tin dinh mil.

Vi d\l 2.

X6t h~ phuong trlnh vi phan sail:

Mi'Jri)ng va zing dljng Btl d~ Gronwall-Bellman Hoang Thanh Long

(

X\(t) =-Xl (t)X'2(t) = -2X2(t)xJto) = 1;x2(to) = 2

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

trong do 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:

Ma tr~n A du<;jcgQi la 5n dinh n€u Re(Ai) < 0, i =1, ,n, trong do Ai, i

=1, ,n, la cac gia tri rieng cua ma tr~n A

4.7.2 Djnh Iy 7.1.

Gid sit R(x,t) thoa man di~u ki?n:

IIR(x,t)ll::;y(t)llxll & J~y(s)ds<oo, (4.7.3)

va ma trcJ-nA an dinh thi nghi?m x =0 cua phuong trrnh (4.7.1) an dinh.

Chung minh djnh Iy 7.1.

Nghi~m cua (4.7.1)dudi d~ng c6ng thuc Cauchy:

x(t) =exp[A(t - to)]x(to) + rt exp[A(t - s)]R(x(s),s)ds (4.7.4)

JtD

MlJ ri)ng va zing d~tngBIl di Gronwall-Bellman Hoang Thanh Long

Ta suy ra duqc:

II x(t) IIs II exp[A(t - to)]1111 Keto) II

+ (II exp[A(t - s)] 1111R(x(s),s) lids (4.7.5)M~t khac do nghi~m X= 0 cua phuong trinh x'(t)=Ax(t) 6n dinh nen::3K> 0 sao cho IIexp(At) lis K Tli (4.7.5), ta suy fa:

Do (4.7.3) nen tli (4.7.7), ta duqc:

(4.7.7)

Ta co th~ chQn Keto)d~ cho nghi~m x = 0 cua (4.7.1) 6n dinh.(D)

Ta co mQts6 di6u ki~n khac d~ danh gia s116n dinh cua (4.7.1).

Giii sa ma tr~n A co cac gia tri rieng Ajva Re(Aj) < O 'v'j=1" ,n.D~t A=maxReA.iCA), 'v'j =1, ,n

Chung minh djnh Iy 7.2 Ta co:

II x(t) IIs II exp[A(t - to)]1111 Keto) II

+ rtIIexp[A(t - s)]1111 R(x(s),s) lids.

Do A la ma tr~n 6n dinh nen ::3B > 0, la mQt h~ng s6 sao cho:

MiJ rf)ng va ung dl!ng B6 di Gronwall-Bellman Hoang Thanh Long

II exp[A(t - s)] II~ Bexp[A(t - s)] ,lit 2: s 2: to

Tli (4.7.9), ta duQc:

II x(t) II ~ Bexp[A(t - to)] II Keto) II

+BAo rt exp[A(t- s)] IIxes)II ds.

Ap dlJng dinh ly 1.8 chuang 1, ta duQC:

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 dinh ly 7.1 va dinh ly 7.2 suy fa

4.7.4 H~ qua.

Gid SU:A la m(Jt ma tr(m an djnh Ne'u (4.7.1) thoa man m(Jt trang hai ddu ki~n sail:

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)

trang do h( u) la ham duclng, lien tl;lc(u > 0), va thi phuclng trlnh (4.7.1)

co nghi~m an djnh.

Mi'Jr(Jng va ung d~tngBd d~ Gronwall-Bellman Hoang Thanh Long

trong d6 A(t) la loan tu tuy6n tinh, bi ch?n, lien Wc theo t u(x,t) la ham

kich dQng,lien tl;lctren D' va thoa man Ilu(x,t)1I::;;r(t) voi r(t) la ham kha

tich trong khoang thai gian hUll h£;lnba't ky; R(x,t) la ham lien tl;lctren D'.D?t ho=Sup{r(t)1 t ~ to}

4.8.1 Dfnh nghia.

(4.8.2)

Nghi~m x =0 cua phuong trlnh (4.8.1) 6n dinh duoi lac dQng thuang

xuyen cua kich dQng u(x,t), n6u '\IE> 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 <~,ho <~A, thEnghiem x =0 cuaphuang trinh

(4.8.1) an d;nh duai lac d{)ng thuiJng xuyen.

D~ chung minh dinh ly chung ta sa dl;lng b6 d~ sail:

60 Mil ri)ng va dng dljng Bfl d€ Gronwall-Bellman Hoang Thanh Long

rPj(t) =exp[ -A(t - to)] IIx(to) II,

II x(t) II ::; Bexp[ -ex(t - to)] II x(to) II

Ap dt,mg dinh ly 1.9 chudng 1, ta duQc:

IIx(t) II::; Bexp[-(a - BL)(t - to)] IIx(to) II

+B ft exp[-(ex - BL)(t - s)]r(s)ds

Jto

::;Bexp[ -(ex - BL)(t - to)]

::;Bexp[ -A(t - to)] {IIx(to) II+ ( exp[A(s - to)]r(s)ds}