Page 218 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 207

a mild solution of the system (7.9) (7.10) corresponding to uAA ad g
such that
m
m
J ðx ; u Þ-E as m- 1 N:
m
m
Since fu gDA ad ; m 5 1; 2;?, fu g is a bounded subset of the separable
reflexive Banach space L p ðJ; UÞ, there exists a subsequence, relabeled as
0
0
m
w
m
fu g and u AL p ðJ; UÞ such that u ! u in L p ðJ; UÞ.
0
Since A ad is closed and convex, then by Marzur lemma u AA ad . Let
m
F
fx gCCðJ; L ðΩ; HÞÞ denote the corresponding sequence of solutions of the
p
integral equation
m
m
m
x ðtÞ 5 S α ðtÞx 0 1 Ð t S α ðt 2 sÞBðsÞu ðsÞds 1 Ð t S α ðt 2 sÞfðs; x ðsÞÞds
0 0

m
1 Ð t S α ðt 2 sÞ Ð s σðτ; x ðτÞÞdWðτÞ ds:
0 0
m
It follows from the boundedness of fu g and Theorem 7.4, one can easily
check that there exists a ρ . 0 such that
m p
EOx O # ρ; m 5 0; 1; 2; ?:
For tAJ, one has
p
0
m
m
0
EOx ðtÞ2x ðtÞO #3 p21 E: Ð t S α ðt2sÞ½fðs;x ðsÞÞ2fðs;x ðsÞފds: p
H 0
0
m
13 p21 E: Ð t S α ðt2sÞ½BðsÞu ðsÞ2BðsÞu ðsފds: p
0
m
0

13 p21 E: Ð t S α ðt2sÞ Ð s ½σðτ;x ðτÞÞ2σðτ;x ðτÞފdWðτÞ ds: p
0 0
p
p p21
0
m
#3 p21 M b Ð t pωðt2sÞ EOfðs;x ðsÞÞ2fðs;x ðsÞÞO ds
e
0
1 3p
2 1 12
0
pωðt2sÞ p 1
p21
p
p 7
6
m
0
13 p21 M 6 Ð t e C Ð t EOBðsÞu ðsÞ2BðsÞu ðsÞO ds 7
p B
@ 0 dsA 0
4 5
p
s
0
m
p p21
Ð
13 p21 M b Ð t pωðt2sÞ E: ½σðτ;x ðτÞÞ2σðτ;x ðτÞފdWðτÞ: ds
e
0 0
m
p
0
p p21
#3 p21 M b Ð t pωðt2sÞ M f EOx ðsÞ2x ðsÞO ds
e
0
! p21
0 p
m
13 p21 M p p21 ðe pωb 21Þ p21 OBu 2Bu O L p ðJ;UÞ
pω
p
2 2 3 2
p
p
0
m
p p21
e
13 p21 M b Ð t pωðt2sÞ 4 Ð s ðM σ EOx ðτÞ2x ðτÞO Þ dτ 5 ds:
c p
0 0
Thus,
0 pωb 1 p 2
m
0
supEOðΦxÞðtÞ2ðΦyÞðtÞO p H #3 p21 M b e 21 A ðM f 1b c p M σ ÞsupEOx ðtÞ2x ðtÞO p
p p21@
tAJ pω tAJ
! p21
0 p
m
13 p21 M p p21 ðe pωb 21Þ p21 OBu 2Bu O L p ðJ;UÞ
pω