Page 225 - Mathematical Techniques of Fractional Order Systems
P. 225
214 Mathematical Techniques of Fractional Order Systems
Proof: Consider the operator Φ:C b -C b defined by
8
φðtÞ; tAð2N; 0
> ð t
1
>
>
φð0Þ 1 BðsÞuðsÞds
> α21
>
> ðt2sÞ
>
ΓðαÞ 0
>
>
>
>
> ð t ð s ð t
1
>
1 σðτ; x τ ÞdWðτÞ ds 1 fðs; x s Þds
> α21
>
> ðt2sÞ
>
ΓðαÞ 0 2N 0
<
ðΦxÞðtÞ 5
"
> 1 ð s
> Ð t α21
1 S ðt 2 sÞ φð0Þ 1
> 0 BðτÞuðτÞdτ
> ðs2τÞ
> 0
>
> ΓðαÞ 0
>
>
> #
> τ
> ð s ð
> 1
> Ð s α21
> 1 fðτ; x τ Þdτ 1 ðs2τÞ σðη; x η ÞdWðηÞ dτ ds; tAJ:
>
> 0
:
ΓðαÞ 0 2N
Let yðUÞ:ð2N; b-H be the function defined by
yðtÞ 5 φðtÞ; if tAð2N; 0 then y 0 5 φ, we define the function z by
0; if tAJ
0; if tAð2N; 0
zðtÞ 5
zðtÞ; if tAJ:
If xðtÞ satisfies
1 ð t
xðtÞ 5 φð0Þ 1 ðt2sÞ α21 BðsÞuðsÞds
ΓðαÞ
0
1 ð t ð s ð t
1 ðt2sÞ α21 σðτ; x τ ÞdWðτÞ ds 1 fðs; x s Þds
ΓðαÞ 0 2N 0
" ð s ð s
Ð t 1 α21
1 S ðt 2 sÞ φð0Þ 1 ðs2τÞ BðτÞuðτÞdτ 1 fðτ; x τ Þdτ
0
0
ΓðαÞ 0 0
1 ð s α21 ð τ #
1 ðs2τÞ σðη; x η ÞdWðηÞ dτ ds;
ΓðαÞ 0 2N
one can decompose it as xðtÞ 5 zðtÞ 1 yðtÞ; tAJ which implies x t 5 z t 1 y t ,
tAJ and the function zðUÞ satisfies z 0 5 0 and
1 ð t
zðtÞ5φð0Þ1 ðt2sÞ α21 BðsÞuðsÞds
ΓðαÞ 0
1 ð t α21 ð s ð t
1 ðt2sÞ σðτ;z τ 1y τ ÞdWðτÞ ds1 fðs;z s 1y s Þds
ΓðαÞ 0 2N 0
" ð s
Ð t 1 α21
1 S ðt2sÞ φð0Þ1 ðs2τÞ BðτÞuðτÞdτ
0
0
ΓðαÞ 0
1 ð s ð τ #
s
Ð
1 fðτ;z τ 1y τ Þdτ 1 ðs2τÞ α21 σðη;z η 1y η ÞdWðηÞ dτ ds:
0
ΓðαÞ 0 2N
