Page 245 - Mathematical Techniques of Fractional Order Systems

234 Mathematical Techniques of Fractional Order Systems

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Definition 8.4: A multivalued map Φ:X-P cl ðR Þ is said to be measurable if
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for every vAR ; the function x/dðv; ΦðxÞÞ 5 inf :v z: : zAΦðxÞ is
measurable.
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For each xAL 2 ðJ; R Þ; define the set of selections of F by
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vAN F;x 5 fvAL 2 ðJ; R Þ:vðtÞAFðt; xðtÞÞ for a:e: tAJg:

Lemma 8.2: (Hu and Papageorgiou, 2013) Let Φ be a completely continuous
multivalued map with nonempty compact values, then Φ is u.s.c. if and only
if Φ has a closed graph (i.e., x n -x; y n -y; y n AΦðx n Þ imply yAΦðxÞ).

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Definition 8.5: A multivalued map Φ:J 3 R -PðR Þ is said to be L 2 2
Carathe ´odory if
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1. t/Φðt; xÞ is measurable for each xAR ;
2. x/Φðt; xÞ is u.s.c. for almost all tAJ;
1
1
3. for each ρ . 0; there exists ϕ AL ðJ; R Þ such that
ρ
2
2
:Φðt; xÞ: :¼ supfE:v: :vAΦðt; xÞg # ϕ ðtÞ for all :x: 2 R n # ρ and for a:etAJ:
ρ
Lemma 8.3: (Lasota and Opial, 1965) Let X be a Banach space. Let
Φ:J 3 X-P cp;cv ðXÞ be a L 2 2 Carathe ´odory multivalued map with N F;x 6¼ φ
and let Λ be a linear continuous mapping from L 2 ðJ; XÞ to CðJ; XÞ; then the
operator
Λ3N F :CðJ; XÞ-P cp;cv ðCðJ; XÞÞ; x/ðΛ3N F ÞðxÞ:¼ΛðN F;x Þ
is a closed graph operator in CðJ; XÞ 3 CðJ; XÞ:

Proposition 8.1: (Castaing and Valadier, 2006) Let X be a separable
Banach space. Let F 1 ; F 2 :J-P cp ðXÞ be measurable multivalued maps, then
the multivalued map t/F 1 ðtÞ - F 2 ðtÞ is measurable.

Theorem 8.1: (Castaing and Valadier, 2006) Let X be a separable metric
space, ðΦ; LÞ be a measurable space, F is a multivalued map from Φ to com-
plete nonempty subset of X: If for each open set U in X; FðUÞ 5
ft:FðtÞ - U 6¼ φgAL; then F admits a measurable selection.

Now introduce the set of all states attainable from x 0 in time t . 0
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denoted as R t ðx 0 Þ 5 fxðt; x 0 ; uÞ:uðUÞAL 2 ðΩ; F ; R Þg; where xðt; x 0 ; uÞ is the
T
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solution of the system (8.1) corresponding to x 0 AR ; uðUÞAL 2 ðΩ; F ; R Þ:
T
Definition 8.6: The stochastic system (8.1) is completely controllable on J if
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R T ðx 0 Þ 5 L 2 ðΩ; F ; R Þ;
T
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that is if all points in L 2 ðΩ; F ; R Þ can be reached from the point x 0 at time T:
T