Page 246 - Mathematical Techniques of Fractional Order Systems

Controllability of Fractional Chapter | 8 235

Let ðX; dÞ be a metric space induced from ðX; :U:Þ be a square normed
space. Consider H d :PðXÞ 3 PðXÞ-R 1 , fNg given by

H d ðA; BÞ 5 maxf sup dða; BÞ; sup dðA; bÞg
aAA bAB
where
dðA; bÞ 5 inf dða; bÞ; dða; BÞ 5 inf dða; bÞ:
aAA bAB
Then ðP bd;cl ðXÞ; H d Þ is a metric space and ðP cl ðXÞ; H d Þ is a generalized
metric space (see Kisielewicz, 1991).

Definition 8.7: A multivalued operator Φ:X-P cl ðXÞ is called

1. γ 2 Lipschitz if and only if there exists γ . 0 such that
H d ðΦðxÞ; ΦðyÞÞ # γdðx; yÞ
for each x; yAX;
2. a contraction if and only if it is γ 2 Lipschitz with γ , 1:

Lemma 8.4: (Bohnenblust and Karlin, 1950). Let X be a Banach space and
KAP cl;cv ðXÞ and suppose that the operator Φ:K-P cl;cv ðKÞ is upper semi-
continuous and the set ΦðKÞ is relatively compact in X: Then Φ has a fixed
point in K:

Lemma 8.5: (Covitz and Nadler, 1970). Let ðX; dÞ be a complete metric
space. If Φ:X-P cl ðXÞ is a contraction, then Φ has fixed points.
8.3 MAIN RESULTS
In this section, the controllability criteria are discussed for the considered
system (8.1). In order to prove the controllability results assume the follow-
ing hypotheses hold
n
n
(H1) The multivalued map F:J 3 R -PðR Þ be an L 2 2 Carathe ´odory
function and satisfies the following conditions: for each tAJ; the function
n
n
n
Fðt; UÞ:R -P bd;cl;cv ðR Þ is u.s.c. and for each xAR the function FðU; xÞ is
n
measurable; for each fixed xAR the set
n
N F;x 5 fvAL 2 ðJ; R Þ:vðtÞAFðt; xðtÞÞ for a:e tAJg
is nonempty. There exists a positive function ϕ :J-R 1 such that
ρ
n o
2
ρ
sup E:vðtÞ: :vðtÞAFðt; xðtÞÞ # ϕ ðtÞ
for a.e tAJ and
ϕ ðtÞ
ρ
lim inf 5 ϒ , N:
ρ-N ρ