Page 212 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 201

ðnÞ

since y -y for some f AN F;y , it follows from Lemma 7.1 that
^
E

ρ ðtÞ 2 Ð t S α ðsÞx 1 ds 2 Ð t S α ðt 2 sÞBu ðs; φ 1 y Þds

0 0
Ð s
^
t
2 S α ðt 2 sÞ σðs; τ; φ 1 y ÞdWðτÞ dsAΘðN F;y Þ:
Ð

0 0 τ τ
Therefore Ψ has a closed graph.
It can be concluded from Step 1 Step 5 together with the Arzela Ascoli
theorem that Ψ is a compact multivalued map, u.s.c with convex closed
values. As a consequence of Lemma 7.2, one can deduce that Ψ has a fixed
point, which is a mild solution of the fractional order system (7.1) (7.2).
Theorem 7.3: Suppose that the hypotheses ðH 1 Þ 2 ðH 7 Þ hold, then the nonlin-
ear multivalued fractional stochastic differential equation (7.1) (7.2) is
approximately controllable on J.
ε
E
Proof: Let x ðUÞ be a fixed point of Φ in B r .By Theorem 7.2 any fixed
E
point of Φ is a mild solution of (7.1) (7.2) under the control function
E ε
uðs; x Þ and by stochastic Fubini theorem it satisfies, that for some f AN F;x E
h i
E b 21 Ð b ~ Ð t
x ðbÞ 5 ~ x b 2 EðEI1Π Þ E~ x b 1 φðsÞdWðsÞ 2 S α ðsÞx 1 ds
0 0 0
E
b 21
1 E Ð t ðEI1Π Þ S α ðt 2 sÞf ðsÞds
0 s
b 21
E

1 E Ð t ðEI1Π Þ S α ðt 2 sÞ Ð s σðs; τ; x ÞdWðτÞ ds
0 s 0 τ
Moreover, by hypothesis ðH 7 Þ on F & σ and the Dounford Pettis theo-
E
E
rem, one has that the sequences ff ðsÞg and fσðs; τ; x Þg are weakly compact,
τ
respectively, in L 2 ðJ; HÞ and L 2 ðLðK; HÞÞ, so they are subsequences still
E E
τ
denoted by ff ðsÞg and fσðs; τ; x Þg that are weakly converges to say f and σ ,
respectively, in L 2 ðJ; HÞ and L 2 ðLðK; HÞÞ. Thus, one has

E 2 b 21 2 b 21 Ð b S α ðsÞx 1 ds : 2
~
EOx ðbÞ 2 ~ x b O # 7 EOEðEI1Π Þ x b O 1 E:EðEI1Π Þ
0 0 0
2
b 21 ~
1 E Ð b OEðEI1Π Þ φðsÞO 2 ds
0 0 L 2 ðK;HÞ
b 21
E
1 E: Ð b EðEI1Π Þ S α ðb2sÞ½f ðsÞ2fðsފds: 2
0 s
b 21
1 E: Ð b EðEI1Π Þ S α ðb2sÞfðsÞds: 2
0 s
b 21
1 E: Ð b EðEI1Π Þ S α ðb 2 sÞ
0 s
Ð s E 2
3 ½σðs; τ; x Þ2σðs; τ; x τ ފdWðτÞ ds:
0 τ
2
b 21
1 E: Ð b EðEI1Π Þ S α ðb2sÞ Ð s σðs; τ; x τ ÞdWðτÞ ds: :

0 s 0
On the other hand by hypothesis ðH 6 Þ, for all 0 # s # b the operator
b 21
b 21
EðEI1Π Þ -0 strongly as E-0 1 and moreover OEðEI1Π Þ O # 1. It
0 0
follows from Lebesgue dominated convergence theorem and the compactness
E
1
2
of S α ðtÞ that EOx ðbÞ 2 ~ x b O -0as E-0 . This proves the approximate con-
trollability of the multivalued fractional differential equation (7.1) (7.2).