Page 214 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 203

where
Ð N 2λt ^ Ð N 2λt
^ uðλÞ 5 e BðtÞuðtÞdt; fðλÞ 5 e fðt; xðtÞÞdt;
0 0
^ gðλÞ 5 Ð N 2λt Ð t σðs; xðsÞÞdWðsÞ dt

e
0 0
It follows from (7.11) that
α
fðλÞ 1 λ
ðλ I 2 AÞ^ xðλÞ 5 λ α21 x 0 1 λ α21 ^ uðλÞ 1 λ α21 ^ α21 ^ σðλÞ
α
α
21
21
^ xðλÞ 5 λ α21 ðλ I2AÞ x 0 1 λ α21 ðλ I2AÞ uðλÞ ð7:12Þ
^
α
α
21 ^
1 λ α21 ðλ I2AÞ fðλÞ 1 λ α21 ðλ I2AÞ σðλÞ
21
^
one needs to find the Laplace transformable and strongly continuous family
^
α
of bounded linear operators, say S α ðtÞ such that S α ðλÞ 5 λ α21 ðλ I2AÞ : In
21
other words, one searches for the scalar functions aðtÞ and kðtÞ such that
^ 21
kðλÞ 1 α21 α 21
I2A 5 λ ðλ I2AÞ : ð7:13Þ
^
^ aðλÞ aðλÞ
In order to have the identity (7.13), we necessarily have ^ aðλÞ 5 λ 1 α and
^
1
kðλÞ 5 . By using the inverse Laplace transformation, one obtains
λ
aðtÞ 5 t α21 and kðtÞ 5 1. It can be concluded that the appropriate family S α ðtÞ
ΓðαÞ
corresponds to an ða; kÞ-regularized family with aðtÞ & kðtÞ as precisely
defined above.
From (7.12), one has
^ ^ ^ ^ ^
^ xðλÞ 5 S α ðλÞx 0 1 S α ðλÞ^ uðλÞ 1 S α ðλÞfðλÞ 1 S α ðλÞ^σðλÞ
by employing the inverse Laplace transformation on both sides of the above
equation, one can get
xðtÞ 5 S α ðtÞx 0 1 Ð t S α ðt 2 sÞBðsÞuðsÞds 1 Ð t S α ðt 2 sÞf ðs; xðsÞÞds
0 0
Ð t Ð s
1 S α ðt 2 sÞ σðτ; xðτÞÞdWðτÞ ds:
0 0
Definition 7.10: An H-valued stochastic process fxðtÞ; tAJg is said to be a
mild solution of the fractional Cauchy problem (7.9) (7.10) if
(i) xðtÞ is F t -adapted and measurable for all t $ 0 .
(ii) xðtÞ is continuous on J almost surely and the following stochastic inte-
gral is verified
xðtÞ 5 S α ðtÞx 0 1 Ð t S α ðt 2 sÞBðsÞuðsÞds 1 Ð t S α ðt 2 sÞf ðs; xðsÞÞds
0 0

1 Ð t S α ðt 2 sÞ Ð s σðτ; xðτÞÞdWðτÞ ds:
0 0