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Controllability of Single-valued Chapter | 7 225

Thus,
8
2p22 2αp22
< p p
m 0 m 0 2
2 1 p21
E z ðtÞ2z ðtÞ # 6 2 αp21 b :Bu 2Bu : p ðJ;UÞ

L
Γ ðαÞ
:
2α
t
Ð Ð t m 0 2 2b M R
1 bM f ðρÞ Oz 2 z O ds 1
0 0 s s C h 2
Γ ðα 1 1Þ
α ð t
2b TrðQÞM σ ðρÞ α21 m 0 2
1 ðt2sÞ Oz 2 z O ds
2 s s C h
αΓ ðαÞ 0
2p22 2αp1p22
Oϕ O 2 p p
0 2
p21
m
1 2 A αp21 b :Bu 2Bu : p
Γ ðαÞ L ðJ;UÞ
2 2α12
t
A
2 Ð Ð s m 0 2 2Oϕ O b
1 Oϕ O bM f ðρÞ Oz 2 z O dτds 1
A 0 0 τ τ 2
Γ ðα 1 1Þ
9
t
2 α11
2Oϕ O b TrðQÞM σ ðρÞ ð ð s =
0 2
m
1 A ðs2τÞ α21 Oz 2 z O dτds
2 τ τ C h
αΓ ðαÞ 0 0 ;
^
which implies that there exists a constant N . 0 such that
m 0 ^ m 0 2
2
for tAJ
sup E z ðtÞ2z ðtÞ # N:Bu 2Bu : p
L ðJ;UÞ
tAJ
m 0 2 s
since B is strongly continuous, one has :Bu 2Bu : p ! 0as m-N
L ðJ;UÞ
s
0
2
m
thus, Ejz ðtÞ 2 z ðtÞj ! 0as m-N, which is equivalent to
m 0 m s 0
Ox 2 x O C b -0as m-N, this yields that x ! x in C b as m-N. Note
that ðH 20 Þ implies the assumptions of Balder (1987). Hence by Balder’s theo-
rem, one can conclude that ðx; x t ; uÞ-E Ð t Lðt; x t ; xðtÞ; uðtÞÞdt is sequentially

0
1
lower semicontinuous in the strong topology of L ðJ; HÞ. Since,
p
1
p
L ðJ; UÞCL ðJ; UÞ, J is weakly lower semicontinuous on L ðJ; UÞ, and since
0
by ðH 20 ÞðivÞ; J .2N, J attains its infimum at u AA ad , i.e.,
ð b ð b
^ E5 lim E Lðt;x ;x ðtÞ;u ðtÞÞdt $E Lðt;x ;x ðtÞ;u ðtÞÞdt 5J ðu Þ$^ E:
0
0
0
0
m
m
m
m-N 0 t 0 t
This completes the proof.
7.4 CONCLUSION
This chapter studied the controllability results of different classes of single-
valued and multivalued FSDEs. In the first part of this chapter, the approxi-
mate controllability of some class of fractional stochastic integro-differential
inclusion, and solvability and existence of optimal control of FSDEs of order