Page 202 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 191

(i) ^ aðλÞ 6¼ 0 and 1 AρðAÞ for all λ . ω
^ ^ aðλÞ 21
(ii) HðλÞ: 5 kðλÞðI2 ^ aðλÞAÞ satisfies the estimates
Mn!
ðnÞ
OH ðλÞO # n11 ; λ . ω; nAℕ:
ðλ2ωÞ
Let A be a closed linear operator and fS α ðtÞg t $ 0 be an exponentially
bounded and strongly continuous operator family in LðHÞ such that the
^
Laplace transform S α ðλÞ exists for λ . ω. It is proved in Lizama (2000) that
S α ðtÞ is an ða; kÞ-regularized family with generator A if and only if for every
21
λ . ω; ðI2 ^ aðλÞAÞ exists in LðHÞ and
^ 21 ð N
kðλÞ 1 2λs
HðλÞx 5 I2A x 5 e S α ðsÞxds; xAH:
^
^ aðλÞ aðλÞ 0
Consider the following form of FSDEs
ð t ð t
c α c α21
D xðtÞ5AxðtÞ1 D BuðtÞ1Ft;x t ; gðt;s;x s Þds 1 σðt;s;x s ÞdWðsÞ
t t
0 0
by taking the Laplace transform on both sides of the above equation,
we have

^
α
λ ^ xðλÞ 2 λ α21 xð0Þ 2 λ α22 0 1 ^ uðλÞ 1 FðλÞ 1 ^σðλÞ ð7:3Þ
x ð0Þ 5 A^ xðλÞ 1
λ 12α
where

^ uðλÞ 5 Ð N 2λt BuðtÞdt; σðλÞ 5 Ð N 2λt Ð t σðt; s; x s ÞdWðsÞ dt
^
e
e
0 0 0
^

^ xðλÞ 5 Ð N 2λt xðtÞdt; FðλÞ 5 Ð N 2λt Ft; x t ; Ð t gðt; s; x s Þds dt

e
e
0 0 0
it follows from (7.3) that
α
ðλ I 2 AÞ^ xðλÞ 5 λ α21 φð0Þ 1 λ α22 x 1 1 λ α21 ^ uðλÞ 1 λ α21 ^ α21 ^ σðλÞ
FðλÞ 1 λ
α
α
21
21
^ xðλÞ 5 λ α21 ðλ I2AÞ φð0Þ 1 λ α22 ðλ I2AÞ x 1
^
α
21
1 λ α21 ðλ I2AÞ ½^ uðλÞ 1 FðλÞ 1 ^σðλފ
ð7:4Þ
now, one needs to find the Laplace transformable and strongly continuous
family of bounded linear operators, say S α ðtÞ such that
α
^
S α ðλÞ 5 λ α21 ðλ I2AÞ : In other words, one searches for the scalar func-
21
tions aðtÞ and kðtÞ such that
^ 21
kðλÞ 1 α21 α 21
I2A 5 λ ðλ I2AÞ : ð7:5Þ
^
^ aðλÞ aðλÞ
In order to have the identity (7.5), one necessarily has ^ aðλÞ 5 λ 1 α and
^
α21
t
kðλÞ 5 . By using inverse Laplace transformation, one gets aðtÞ 5 ΓðαÞ and
1
λ