Page 206 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 195

t Ð t ε ^ Ð t
Ð
yðtÞ 5 S α ðsÞx 1 ds 1 S α ðt 2 sÞBu ðs; φ 1 yÞds 1 S α ðt 2 sÞfðsÞds
0 0 0

Ð t Ð s ^
1 S α ðt 2 sÞ σðs; τ; φ 1 y τ ÞdWðτÞ ds; tAJ; fAN F;x
0 0 τ
where
n
ε ^ b 21 Ð b ~

u ðs; φ 1 yÞ 5 B S ðb 2 sÞðEI1Π Þ E~ x b 1 0 φðsÞdWðsÞ 2 S α ðbÞφð0Þ
α
0
2 Ð b S α ðsÞx 1 ds 2 Ð b S α ðb 2 sÞfðsÞds
0 0
Ð s o
^
b
Ð
2 S α ðb 2 sÞ σðs; τ; φ 1 y τ ÞdWðτÞ ds ; fAN F;x :
0 0 τ
0
0
Set B 5 fyAB b ; y 0 5 0ABg. For any yAB , we have
b
b
1 1

OyO b 5 Oy 0 O B 1 sup EOyðsÞO 2 2 5 sup EOyðsÞO 2 2
sA½0;bŠ sA½0;bŠ
0
thus ðB ; OUO b Þ is a Banach space.
b
0
2
For any positive number r, set B r 5 fyAB : OyO # rg, then for each r,
b b
0
B r is clearly a bounded closed convex set in B , and for yAB r , we have
b
^ 2 2 ^ 2
Oy t 1 φ O # 2ðOy t O 1 Oφ O Þ
t B B t B

^
^
2
2
2
# 4 sup EOyðsÞO 1 Oy 0 O 1 sup EOφðsÞO 1 Oφ O 2
sA½0;tŠ B sA½0;tŠ 0 B
2 2ωb 2 2
# 4 r 1 M e EOφð0ÞO 1 4OφO 5 4r 1 r ;

H B
2
2
2 2ωb

where r 5 4M e EOφð0ÞO 1 4OφO .
H B
0
0
0
Define the multivalued map Ψ:B -PðB Þ by Ψy the set of ρAB such that
b b b
8
0; tAð2N; 0Š
>
>
Ð t Ð t Ð t
< ε ^
ρðtÞ 5 0 S α ðsÞx 1 ds 1 0 S α ðt 2 sÞBu ðs; φ 1 yÞds 1 0 S α ðt 2 sÞfðsÞds

>
> Ð t Ð s ^
: 1 S α ðt 2 sÞ σðs; τ; φ 1 y τ ÞdWðτÞ ds; tAJ
0 0 τ
ε
where fAN F;x . Obviously the operator Φ has a fixed point if and only if the
operator Ψ has a fixed point.
Step 1: For each yAB r ; Ψ is convex.
Let ρ ; ρ AΨy then there exist f 1 ; f 2 AN F;x such that for each tAJ,
2
1
one has
^
ε
ρ ðtÞ 5 Ð 0 t S α ðsÞx 1 ds 1 Ð 0 t S α ðt 2 sÞBu ðs; φ 1 yÞds 1 Ð 0 t S α ðt 2 sÞf i ðsÞds
i

Ð t Ð s ^
1 S α ðt 2 sÞ σðs; τ; φ 1 y τ ÞdWðτÞ ds; i 5 1; 2:
0 0 τ