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Controllability of Fractional Chapter | 8 233
xðtÞ 5 E q2p ðAt q2p Þx 0 2 At q2p E q2p;q2p11 ðAt q2p Þx 0 1 tE q2p;2 ðAt q2p Þx 0
0
ð t ð s
1 ðt2sÞ q21 E q2p;q ðAðt2sÞ q2p Þ BuðsÞ 1 vðsÞ 1 Gðθ; xðθÞÞdwðθÞ ds:
0 0
The system (8.2) is controllable on J if and only if the controllability
Grammian matrix
T
ð
W 5 ðT2sÞ q21 ½E q2p;q ðAðT2sÞ q2p ÞB½E q2p;q ðAðT2sÞ q2p ÞB ds
0
is positive definite, for some T . 0 (see, Sakthivel et al., 2013). Here B
denotes the transpose of B:
0
Lemma 8.1: (Mahmudov, 2003) Let G:J 3 Ω-L be a strongly measurable
2
mapping such that
ð T
p
E:GðtÞ: 0 dt , N:
L
0 2
Then
ð t ð t
p
E: p E:GðsÞ: 0ds;
GðsÞdwðsÞ: # L G
L
0 0 2
for all tAJ and p $ 2; where L G is the constant involving p and T:
Let ðX; :U:Þ be a Banach space. Denote P cl ðXÞ 5 fYAPðXÞ:Y is closedg;
P bd ðXÞ 5 fYAPðXÞ:Y is boundedg; P cp ðXÞ 5 fYAPðXÞ:Y is compactg, and
P cp;cv ðXÞ 5 fYAPðXÞ:Y is compact and convexg.
For more details on multivalued maps readers can refer to the books
(Aubin and Cellina, 1984; Gorniewicz, 1999; Hu and Papageorgiou, 2013;
Kisielewicz, 1991).
Definition 8.1: A multivalued map Φ:X-PðXÞ is convex (closed)
valued if ΦðxÞ is convex (closed) for all xAX: Φ is bounded on
bounded sets if ΦðBÞ 5 , xAB ΦðxÞ is bounded in X for all
BAP bd ðXÞði:e:; sup sup :y: , NÞ:
xAByAΦðxÞ
Definition 8.2: Φ is called upper semicontinuous (u.s.c.) on X if for each
x 0 AX; the set Φðx 0 Þ is a nonempty closed subset of X; and if for each open
set N of X containing Φðx 0 Þ; there exists an open neighborhood N 0 of x 0
such that ΦðN 0 ÞDN:
Definition 8.3: Φ is said to be completely continuous if ΦðBÞ is relatively
compact for every BAP bd ðXÞ: Φ has a fixed point if there is xAX such that
xAΦðxÞ: The fixed point set of the multivalued operator Φ will be denoted by
FixΦ:
