Page 203 - Mathematical Techniques of Fractional Order Systems

192 Mathematical Techniques of Fractional Order Systems

kðtÞ 5 1. It can be concluded that the appropriate family S α ðtÞ corresponds to
an ða; kÞ-regularized family with aðtÞ & kðtÞ as precisely defined above.
^
α
By setting S α ðλÞ 5 λ α21 ðλ I2AÞ 21 in (7.4) and using the inverse
Laplace transformation, one can get
xðtÞ 5 S α ðtÞφð0Þ 1 Ð t S α ðsÞx 1 ds 1 Ð t S α ðt 2 sÞBuðsÞds
0 0
Ð t Ð s
1 S α ðt 2 sÞFs; x s ; gðs; τ; x τ Þdτ ds
0 0
1 Ð t S α ðt 2 sÞ Ð s σðs; τ; x τ ÞdWðτÞ ds:

0 0
From the above observations, one can define the mild solution for frac-
tional stochastic differential inclusions (7.1) (7.2) as follows
Definition 7.8: An H-valued stochastic process fxðtÞ; tAð2N; bŠg is said
to be a mild solution of the fractional stochastic differential inclusion
(7.1) (7.2) 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 ðtÞφð0Þ 1 Ð t S α ðsÞx 1 ds 1 Ð t S α ðt 2 sÞBuðsÞds 1 Ð t S α ðt 2 sÞf ðsÞds
0 0 0
Ð t Ð s
1 S α ðt 2 sÞ σðs; τ; x τ ÞdWðτÞ ds; fAN F;x :
0 0
Let x b ðx 0 ; uÞ be the state value of the fractional order system (7.1) (7.2)
at terminal time b corresponding to the control u and the initial value
x 0 5 φðtÞAB. Set

Rðb; x 0 Þ 5 fx b ðx 0 ; uÞð0Þ: uðUÞAL 2 ðJ; UÞg
which is called the reachable set of the system (7.1) (7.2) at terminal time
b. Its closure in H is denoted by Rðb; x 0 Þ.

Definition 7.9: The fractional order system (7.1) (7.2) is said to be approxi-
mately controllable on J, if Rðb; x 0 Þ 5 H.

It is convenient at this point to introduce the operators
b Ð b

Π fUg 5 0 S α ðb 2 sÞBB S ðb 2 sÞEfUjF s gds
α
0
b
b 21
and Rðε; Π Þ 5 ðεI1Π Þ for E . 0
0 0

where B denotes the adjoint of B and S ðtÞ is the adjoint of S α ðtÞ.It is
α
b
straightforward that the operator Π is a linear bounded operator.
0
Lemma 7.3: (Mahmudov and Zorlu, 2003) The linear system corresponding
to the fractional order system (7.1) (7.2) is approximately controllable on J if
b 21
1
and only if εðεI1Π Þ -0 as ε-0 in the strong operator topology.
0