Page 219 - Mathematical Techniques of Fractional Order Systems

208 Mathematical Techniques of Fractional Order Systems

which implies that there exists a constant N . 0 such that
p
0 p
sup E:x ðtÞ2x ðtÞ: # N :Bu 2Bu : for tAJ
0
m
m

L p ðJ;UÞ
tAJ
where
p21
3 p21 M p p21 ðe pωb 21Þ p21
pω
N 5

p
p p21 e
1 2 3 p21 M b pωb pω 2 1 ðM f 1 b 2c p M σ Þ
pωb p
p p21 e
and 3 p21 M b pω 2 1 ðM f 1 b 2c p M σ Þ , 1.
s
0
Since B is strongly continuous, we have :Bu 2Bu : p ! 0
m
L p ðJ;UÞ
s
m
0 p
as m-N thus, we have EOx 2 x O ! 0as m-N, this yields that
s
m
0
x ! x in CðJ; L p ðΩ; HÞÞ as m-N. Note that ðH 13 Þ implies the assump-
tions of Balder (1987) are satisfied. Hence by Balder’s theorem, one can

conclude that ðx; uÞ-E Ð b Lðt; xðtÞ; uðtÞÞdt is sequentially lower semicon-
0
tinuous in the strong topology of L 1 ðJ; HÞ. Since L p ðJ; UÞCL 1 ðJ; UÞ, J is
weakly lower semicontinuous on L p ðJ; UÞ, and since by ðH 13 ÞðivÞ; J .2N,
0
J attains its infimum at u AA ad that is

ð b ð b
E5 lim E Lðt;x ðtÞ;u ðtÞÞdt $E Lðt;x ðtÞ;u ðtÞÞdt 5J ðx ;u Þ$E:
0
m
0
0
0
m
m-N
0 0
7.3 CONTROLLABILITY RESULT OF SINGLE-VALUED
FRACTIONAL STOCHASTIC DIFFERENTIAL EQUATION BY
USING ANALYTIC RESOLVENT OPERATORS
This section investigates the solvability and optimal controls for fractional
stochastic integro-differential equations with infinite delay in Hilbert space
by using analytic resolvent operators. Some suitable conditions are estab-
lished to guarantee the existence of mild solutions with the help of Leray-
Schauder nonlinear alternative fixed point theorem. Then the existence of
optimal control is investigated for the corresponding Lagrange problem.
Consider the following form of fractional stochastic integro-differential
equation with infinite delay
ð t
c α 12α
D xðtÞ 5 AxðtÞ 1 BðtÞuðtÞ 1 J fðt; x t Þ 1 σðs; x s ÞdWðsÞ; tAJ ð7:15Þ
t t
2N
xðtÞ 5 φðtÞAC h ð7:16Þ
12α
where 0 , α , 1, J t is the ð1 2 αÞ order Riemann Liouville fractional
integral operator, u is a given control function, it takes values from separable
reflexive Hilbert space U. B is a linear operator from U into H. The history
x t :Ω-C h is defined by x t ðθÞ 5 fxðt 1 θÞ; θAð2N; 0Šg, which belongs to an