Page 234 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 223

m
m
m
m
such that J ðx ; u Þ-^ ε as m-N: Since fu gDA ad ; m 5 1; 2; ?; fw g is
p
a bounded subset of the separable reflexive Banach space L ðJ; UÞ, there
w 0
m
p
m
0
exists a subsequence, relabeled as fu g and u AL ðJ; UÞ such that u ! u
p
in L ðJ; UÞ.
0
Since A ad is closed and convex, then by Marzur lemma u AA ad . Let
m
fx gCC b denote the corresponding sequence of solutions of the integral
equation
8
φðtÞ; tAð2N;0Š
> ð t ð t
1
>
>
φð0Þ1 BðsÞu ðsÞds1
> α21 m m
> ðt2sÞ fðs;x Þds
> s
>
ΓðαÞ
> 0 0
>
>
>
1
> ð t ð s
>
> α21 m
> 1 ðt2sÞ σðτ;x ÞdWðτÞ ds
> τ
>
< ΓðαÞ 0 2N
m
x ðtÞ5 2
s s
> ð ð
m
> Ð t 1
1 S ðt 2sÞ φð0Þ1 BðτÞu ðτÞdτ 1
> 0 α21 m fðτ;x Þdτ
> 4 ðs2τÞ
> 0 τ
ΓðαÞ
>
> 0 0
>
>
>
1
> ð s ð τ
>
1
> α21 m
>
η
> ðs2τÞ σðη;x ÞdWðηÞ dτ ds; tAJ
>
> ΓðαÞ 2N
: 0
m 2
by Theorem 7.6, there exists a ρ . 0 such that Ejx j # ρ; m 5 0; 1; 2; ?:
m
m
Let x ðtÞ 5 z ðtÞ 1 yðtÞ. For tAJ,
2
2
E z ðtÞ2z ðtÞ #6E 1 Ð t ðt2sÞ α21 BðsÞu ðsÞ2BðsÞu ðsÞ ds

m

0
m

0

ΓðαÞ 0
16E Ð t ½fðs;z 1y s Þ2fðs;z 1y s ފds 2
m
0
0 s s
2
16E 1 Ð t ðt2sÞ α21 Ð s ½σðτ;z 1y τ Þ2σðτ;z 1y τ ފdWðτÞ ds
0
m

ΓðαÞ 0 2N τ τ

2
16E Ð t 0 1 Ð s ðs2τÞ α21 BðτÞu ðτÞ2BðτÞu ðτÞ dτ ds

m
0
S ðt2sÞ
0 ΓðαÞ 0
16E Ð t 0 Ð s ½fðτ;z 1y τ Þ2fðτ;z 1y τ ފdτ ds 2
0

m
S ðt2sÞ
0 0 τ τ

"
ð s
16E Ð t 0 1 ðs2τÞ α21
S ðt2sÞ
0
ΓðαÞ 0

# 2
6
Ð τ X
3 ½σðη;z 1y η Þ2σðη;z 1y η ފdWðηÞ dτ ds #6 I i :
m
0

2N η η

i51