Page 211 - Mathematical Techniques of Fractional Order Systems
P. 211
200 Mathematical Techniques of Fractional Order Systems
^
n Ð t 2 Ð t 2
E
EOρðtÞ 2 ρ ðtÞO # 4 E: t2E S α ðsÞx 1 ds: 1 E: t2E S α ðt2sÞBu ðs; φ1yÞds:
2
E
t
1 E: Ð t2E S α ðt2sÞfðsÞds: 2
^
2 o
t
1 E: Ð t2E S α ðt2sÞ Ð 0 s σðs; τ; φ 1y τ ÞdWðτÞ ds: :
τ
Hence, there exist relatively compact sets arbitrarily close to the set
VðtÞ 5 fðΨyÞðtÞ:yAB r g, and the set VðtÞ is relatively compact in H for all
tA½0;b. Since, it is compact at t 5 0, hence VðtÞ is relatively compact in H
for all tA½0;b.
n
ðnÞ
Step 5: Ψ has a closed graph. Let y -y as n-N; ρ AΨy , for each
ðnÞ
n
n
y AB q and ρ -ρ as n-N. We prove that ρ AΨy . Since ρ AΨy , there
ðnÞ
ðnÞ
exists f ðnÞ AN F;y ðnÞ such that
8
0; tAð2N; 0
>
>
< Ð t Ð t E ^ ðnÞ Ð t ðnÞ
n
ρ ðtÞ 5 0 S α ðsÞx 1 ds 1 0 S α ðt 2 sÞBu ðs; φ 1 y Þds 1 0 S α ðt 2 sÞf ðsÞds
> Ð t Ð s ^
1 σðs; τ; φ 1 y ÞdWðτÞ ds; tAJ:
> ðnÞ
0 0 τ
: S α ðt 2 sÞ τ
one must show that there exists f AN F;y such that for each tAJ
8
0; tAð2N; 0
>
^
>
E
Ð t Ð t Ð t
< S α ðsÞx 1 ds 1 S α ðt 2 sÞBu ðs; φ 1 y Þds 1
ρ ðtÞ 5 0 0 0 S α ðt 2 sÞf ðsÞds
> Ð t Ð s ^
1
>
τ
: S α ðt 2 sÞ σðs; τ; φ 1 y ÞdWðτÞ ds; tAJ:
0 0 τ
Now, for every tAJ, one has
^
E
n
:ρ ðtÞ 2 Ð t S α ðsÞx 1 ds 2 Ð t S α ðt 2 sÞBu ðs; φ 1 y Þds
ðnÞ
0 0
Ð t Ð t Ð s ^
2 S α ðt 2 sÞf ðnÞ ðsÞds 2 S α ðt 2 sÞ σðs; τ; φ 1 y ÞdWðτÞ ds
ðnÞ
0 0 0 τ τ
Ð t Ð t
^
E
2 ρ ðtÞ 2 S α ðsÞx 1 ds 2 S α ðt 2 sÞBu ðs; φ 1 y Þds
0 0
Ð s 2
^
t
t
T
Ð
2 S α ðt2sÞf ðsÞds2 S α ðt2sÞ σðs; τ; φ 1y ÞdWðτÞ ds : -0
Ð
0 0 0 τ τ b
as n-N. Consider the linear continuous operator Θ:L 2 ðJ; HÞ-CðJ; HÞ
Ð t Ð t
ðΘρÞðtÞ 5 S α ðt 2 sÞfðsÞds 1 S α ðt 2 sÞBB S ðb 2 tÞ
0 0 α
Ð b b 21
3 ðEI1Π Þ S α ðb 2 τÞfðτÞdτ ds
0 τ
clearly it follows from Lemma 7.1 that Θ3N F;x is a closed graph. Also from
the definition of Θ, one has that for every tAJ
^
ε
n
ρ ðtÞ 2 Ð t S α ðsÞx 1 ds 2 Ð t S α ðt 2 sÞBu ðs; φ 1 y Þds
ðnÞ
0 0
Ð s
^
t
Ð
2 S α ðt 2 sÞ σðs; τ; φ 1 y ÞdWðτÞ dsAΘðN
ðnÞ
0 0 τ τ F;y ðnÞÞ
