Page 224 - Mathematical Techniques of Fractional Order Systems
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Controllability of Single-valued Chapter | 7 213
Definition 7.15: An H-valued stochastic process fxðtÞ; tAð2N; bg is said
to be a mild solution of the problem (7.15) (7.16) if
xðtÞ is measurable and F t adapted for t $ 0;
2
xðtÞ 5 φðtÞ on ð2N; 0 satisfying OφO , N;
Ð t α21 C h
ðt2sÞ xðsÞdsADðAÞ for all tAJ ; and
0
1 ð t 1 ð t
xðtÞ 5 φð0Þ 1 ðt2sÞ α21 AxðsÞds 1 ðt2sÞ α21 BðsÞuðsÞds
ΓðαÞ 0 ΓðαÞ 0
Ð t 1 ð t α21 ð s
1 fðs; x s Þds 1 ðt2sÞ σðτ; x τ ÞdWðτÞ ds:
0
ΓðαÞ 0 2N
Suppose that there exists a resolvent set fSðtÞg t $ 0 which is differentiable
and the functions f; σ are continuous, then by Lemma 7.5 (iii), if x is a
mild solution of (7.15 7.16), then
8
φðtÞ; tAð2N;0
>
> ð t ð t ð s
> 1 1
φð0Þ1 BðsÞuðsÞds1 σðτ;x τ ÞdWðτÞ ds
> α21 α21
>
> ðt2sÞ ðt2sÞ
>
> 2N
ΓðαÞ 0 ΓðαÞ 0
>
>
>
> "
< ð s
xðtÞ5 1 Ð t fðs;x s Þds1 Ð t S ðt 2sÞ φð0Þ1 1 ðs2τÞ α21 BðτÞuðτÞdτ
0
> 0 0
ΓðαÞ 0
>
>
>
> #
> τ
> ð s ð
> Ð s 1
> α21
> 1 fðτ;x τ Þdτ 1 ðs2τÞ σðη;x η ÞdWðηÞ dτ ds; tAJ:
> 0
>
: ΓðαÞ 0 2N
Theorem 7.6: If hypotheses ðH 14 Þ 2 ðH 17 Þ hold, then the fractional order
control problem (7.15) (7.16) has a mild solution on ð2N; b, provided
that, there exists a constant M . 0 such that
M
. 1 ð7:18Þ
p21
2αp22 p21 p 2 2
2
2
2
2
4l 8Ejφð0Þj 1 8OBO b p ðb 1 1ÞOuO 1 Θ 1 4OφO
αp21 L p ðJ;UÞ C h
where
2 3
2α 2 2α12
16b 1 16Oϕ O 1 b
Θ 5 8b OnO N ðb Oϕ O 1 1ÞΛ f ðM Þ 1 4 2 A L 5
2
2
2
A
Γ ðα 1 1Þ
3 M R 1 TrðQÞOmO N Λ σ ðM Þ:
½
