Mở rộng và ứng dụng bổ đề Gronwall -BellMan 6

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

MlJ rfJng va lIng d(tng Bd di Gronwall-Bellman Hoang Thanh Long

CHUaNG 1

BO DE GRONWALL-BELLMAN VA

Trang B6 d~ Gronwall-Bellman co nhi~u d?i luQng tham gia Ne'u chung ta 19n luQt thay d6i cac d?i luQng nay chung ta se co nhi~u ma fQng Cac ma rQng nay co ra't nhi~u ling d\lng va duQc phat biSu dudi d?ng cac dinh ly Cac ma rQng chli ye'u la cac d?ng ba't d~ng thlic tkh phan.

Trudc lien chung ta xem l?i B6 d~ Gronwall-Bellman dudi d?ng ba't

phudng trlnh vi phan va ba't d~ng thlic tkh phan

I B6 d~ Gronwall-Bellman.

1.1.1 B6 d~ 1.

Gid sa u(t) la ham sf;'khd vi tren n N e'u t6n tc;zicac hang so' k, c 7: 0 saD cho:

thi ta co:

k u(t) S u(to)exp[c(t - to)] +-(exp[c(t - to)] - l}, 't;1'tEn.

1.1.2 B6 d~ 2.

Gid sa u(t), art) la cac ham so'lien tl;lc,khong am tren n Ne'u t6n tc;zi hang so' k ;::0 saD cho:

MlJrf)ng va lIng d~mg Btl d€ Gronwall-Bellman Hoang Thanh Long

u(t)~a(t)+k f' u(s)ds, VtEQ,

the ta co:

u(t)~a(t)+k rta(s)exp[k(t-s)Jds, VtEo.

1.1.3 H~ qua.

N€u aCt)=a =constant,\ftEQ, thl ta co:

II MQt sf) md rQng d~ng tuye-n Hnh.

1.2.1 Djnh Iy 1.1 (Xem[3 D.

Gid sa u( t) la ham so'lien tl;lc, khong am tren 0 N e'u tbn tC:licac hang so' a ~O, k ~O, c > 0 saD cho:

u(t) ~ a + rt[cu(s) + kids, VtEQ,

the ta co:

k u(t) ~ aexp[ crt - to)J + -{exp[ crt - to)J -I}, Vt Eo.

Chung minh djnh Iy 1.1.

Ta co th6 chung minh bang cach ap dl;lng b6 d~ 1 nhusau

Di:it

v(t)=a+ rt[cu(s)+k]ds,\ftEQ

TO'(1.6) va (1.8), ta suy fa u(t) ~ vet), veto)=a

K€t hQp Iffy d~o ham hai v€ (1.8), ta duQc:

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

MlJri)ng va ung dlJng Bd di Gronwall-Bellman Hoang Thanh Long

Ap dlJng b6 a~ 1, ta au'qc (1.7).(0)

1.2.2 Djnh Iy 1.2 (Xem [2]).

Gid sit u( t), k( t) la cac ham so'lien tl;lc,khong am tren Q Niu tbn tqzi hang so' a ;? 0 saD cha

u(t)~a+ rt k(s)u(s)ds, b1EQ,

thE

u(t)~aexp[ rtk(s)ds], \1tEQ.

1.2.3 Djnh Iy 1.3 (Xem [6]).

Gid sit u(t), a(t), k(t) la cac ham so'lien tl;lc,khong am tren Q Niu

thE

u(t)~a(t)+ 1:a(s)k(s)exp[ fk(r)dr]ds, b1EQ.

Chung minh djnh Iy 1.3 Xem [6].(0)

(1.13)

1.2.4 Djnh Iy 1.4 (Beesack, Xem [3]).

Gid sit u(t), a(t), b(t), k(t) la cac ham lien tl;lc,khong am tren Q. a) Niu

thE

u(t) ~a(t)+b(t) 1: a(s)k(s)exp[ fb(r)k(r)dr]ds, \1tEQ. (1.15)

b) Kef qua a) vdn dung ntu thay dau H~' bJi dau H;?"trang (1.14)

va (1.15).

Md rqng va ung dljng Bd di Gronwall-Bellman Hoang Thanh Long

c) Ke't qua a) va b) win dung neu thay r biJi r' va S t biJi r'.

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) la ham khd vi tren Q Neu

u(t)~a(t)+ rt k(s)u(s)ds, btEQ,

thE

u(t)~a(to)exp[ (k(s)ds] + (a/(s)exp[ fk(r)dr]ds, tltEQ.(1.17)

Chung minh dinh Iy 1.5 f)~t vet) la vii phai cua (1.16), ta co:

v'et) = a let) + k(t)u(t)

Suy ra

vet) ~a(to)exp[ ( k(s)ds] + ( a '(s)exp[ fk(r)dr ]ds (1.19) Tli b~t d~ng thlic nay, ta du<;5c(1.17).(0)

1.2.6 Dinh Iy 1.6 (Xem [4D

Gid sit u(t), kef) la cac ham lien tf:lC,kh6ng am tren Q; art), bet) la cac

ham duCfng,khd vi tren Q Neu

u(t)~a(t)+b(t) (k(s)u(s)ds, btEQ, ( 1.20)

thE

u(t) ~betHc(to)exp[ ( b(r )k(r )dr]

+ r~c'(s)exp[ fs b(r)k(r)dr]ds), tltEQ. (1.21)

Mil ri)ng va ung d1!ngBd di Gronwall-Bellman Hoang Thanh Long

trang do crt) =art)

b(t) .

Chung minh djnh Iy 1.6.

Chia hai v6 cua (1.20) cho bet), ap dlJng dinh ly 1.5.(0)

1.2.7 Djnh Iy 1.7 (Xem[10], tr.191-192).

Cho u(t), art) la cac ham lien tl:lc,khong am tren n Gid sa K(t,s) 2!O, gi6i nQi v6i to::{s::{t ::{t]va K(t,s) =0 v6i to::{t < s ::{t] Ne'u

u(t)~a(t)+ rl K(t,s)u(s)ds, \?tEn,

thl

u(t) ~ rp(t), \?tEn,

trong do rp(t)la nghi~m cua phuong trlnh

rp(t)=art) + rl K(t,s)rp(s)ds.

Dinh ly 1.8, 1.9 sau day duqc ap dlJng ra't hi~u qua trong vi~c khao sat cac bai loan 6n dinh No la h~ qua cua dinh ly 1.5.

1.2.8 Djnh Iy 1.8 (Xem [3]).

Cho u(t) la ham lien tl:lc,khong am tren n va thoa man bat dcing thac u(t) ~ exp[ -art - ta)]u(ta)

+ r (au(s) + b)exp[-a(t-s)]ds, \?tEn,

trong do a, 0 < a, 0 < b la cac hling so: Khi do, ta co:

u(t) ~ exp[ -( a - a)(t - ta)]u(ta)

+b(a - at! [1- exp[ -(a - a)(t - 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

Mil nganh : 1.01.01

MlJrQngva u'ngdlJng Bii d€ Gronwall-Bellman Hoang Thanh Long

c6 thS chung minh nhl1 sail:

B~t

x(t) =u(t)exp( at)

Khi d6, tu (1.24), ta dl1Qc:

( 1.26)

x(t) S Keto)+ i~ [axes) + bexp(as)]ds,

Ap dl;lng dinh 1:91.5, ta dl1Qc:

( 1.27)

x(t) S x(to)exp[a(t - to)] + bexp(at) rt exp[(a - a)s]ds

Jto

S x(to)exp[a(t - to)]

+ b(a - arl exp(at){ exp[(a - a)t] - exp[(a - a)to]}' (1.28)

V~y u(t) S u(to)exp[ -(a - a)(t - to)]

+ b(a - arl{l- exp[-(a - a)(t - to)]} (D)

1.2.9 Djnh Iy 1.9.

Cha u(t), art), b(t) fa cac ham lien tl;lc, khong am tren Q Ne'u

u(t) S exp[ -art - to)]u(to)

+ 1: [a( s)u( s) + b( s)] exp[ -art - s)]ds ,\ftED., (1.29)

trang do a fa hang so: thEta co:

u(t) S u(to)exp[ -art - to) + rt a( s)ds]

Jto

+ rtob(s)exp[-a(t-s) + fa(r)dr]ds,\ftED s (1.30)

Chung minh djnh Iy 1.9 Tl1ong tl! chung minh dinh 1:91.8.(D)

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,