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Controllability of Fractional Chapter | 8 231
ð t
q
C D xðtÞ AA D xðtÞ1 BuðtÞ 1 Fðt;xðtÞÞ1 Gðs;xðsÞÞdwðsÞ; tA 0;T : 5 J;
p
C
½
0
0
xð0Þ 5 x 0 ; x ð0Þ 5 x 0
0
ð8:1Þ
where C D q and C D p denote Caputo fractional derivatives of order
0 , p # 1 , q # 2; A and B are matrices of dimensions n 3 n and n 3 m ,
n
m
respectively, xAR ; uAR are the state and control vectors. The nonlinear
functions f; G are appropriate functions to be defined later.
The chapter is organized as follows: In Section 8.2, some essential results
on the basic definitions of fractional integral and derivatives, Lemmas, pro-
positions, and some hypotheses are given to obtain the controllability results
successfully. In Section 8.3, the controllability results for the fractional sys-
tem (8.1) are studied under the fixed point theorems. In Section 8.4, a discus-
sion and future work is given. Finally, conclusions are drawn in Section 8.5.
Notations: Throughout this chapter, ðΩ; F; PÞ denotes the complete prob-
ability space with a right continuous and complete filtration fF ; tAJg (F t
t
n o
H
the σ 2 algebra generated by the random variables W ; sA½0; t and
ðsÞ
n
P 2 null set) and satisfying F CF: Let L 2 ðΩ; F ; R Þ be the Hilbert space of
t
T
n
all F 2 measurable square-integrable random variables with values in R :
T
F n
Let L ðJ; R Þ be the Hilbert space of all square integrable and F 2 measur-
2 t
n
n
able processes with values in R : Let B 5 CðJ; R Þ be the Banach space.
n
Denote the class of R 2 valued stochastic processes fξðtÞ:tAJg which are
F 2 adapted and have a finite second moments, i.e.,
t
1
2 2
:ξ: 5 sup EjξðtÞj , N:
t
For convenience, define the following notations
2 2
n 1 5 sup:E q2p ðAt q2p Þx 0 : ; n 2 5 sup:At q2p E q2p;q2p11 ðAt q2p Þx 0 : ;
2
2
21 2
n 3 5 sup:tE q2p;2 ðAt q2p Þx : ; n 4 5 sup:E q2p;q ðAðt2sÞ q2p Þ: ; l 5 :W : :
0
0
8.2 PRELIMINARIES
Let R n be the n-dimensional Euclidean space, R 1 5 ð0; NÞ; f ðtÞ be a
suitable function and the fractional order q . 0; with n 2 1 , q # n; nAℕ:
Then the following results are well known (for more details, see Miller and
Ross, 1993; Oldham and Spanier, 1974; Samko et al., 1993; Sabatier et al.,
2007; Kilbas et al., 2006; Kexue and Jigen, 2011).
Caputo fractional derivative is defined by
C
q
n
ð D fÞðtÞ 5 ðI n2q D fÞðtÞ;
01
