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Controllability of Single-valued Chapter | 7 209
0
abstract space C h . The functions f :J 3 C h -H and σ:J 3 C h -L ðK; HÞ are
2
0
the appropriate functions, where L ðK; HÞ denotes the space of all Q-Hilbert-
2
Schmidt operators from K into H.
7.3.1 Preliminaries
The abstract phase space C h is defined by
1
(
2
2
C h 5 ψ:ð2N; 0-H:for any a . 0; EjψðθÞj is bounded and measurable
1 )
2
Ð 0 ^ 2
function on½ 2 a; 0 with hðsÞ sup EjψðθÞj ds , N :
2N
s # θ # 0
If C h is endowed with the norm
ð 0 1
^
5 hðsÞ sup 2 2
OψO C h EjψðθÞj ds; for all ψAC h
2N s # θ # 0
Þ is a Banach space [Li and Liu (2007)].
then ðC h ; OUO C h
Let Cðð2N; b; HÞ be the space of all continuous H-valued stochastic
process fξðtÞ; tAð2N; bg. Let C b 5 fx:xACðð2N; b; HÞ; x 0 5 φAC h g. Set
OUO b to be a seminorm defined by
1
2 2
1 sup EjxðsÞj ; xAC b :
OxO b 5 Ox 0 O C h
sA½0;b
Lemma 7.4: (Li and Liu, 2007) Assume that xAC b , then for all tAJ; x t AC h .
Moreover,
1 1
2 2 2 2
l EjxðtÞj # Ox t O C h # l sup EjxðsÞj 1 Ox 0 O C h
sA½0;t
Ð 0 ^
where l 5 hðsÞds , N.
2N
Consider the following fractional stochastic integro-differential equation
ð t
c α 12α
D xðtÞ 5 AxðtÞ 1 BðtÞuðtÞ 1 J fðt; x t Þ 1 σðs; x s ÞdWðsÞ
t t
2N
the above equation is equivalent to the following integral equation
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
