Page 205 - Mathematical Techniques of Fractional Order Systems

194 Mathematical Techniques of Fractional Order Systems

7.2.1.2 Approximate Controllability Result
Theorem 7.2: Suppose that the hypotheses ðH 1 Þ ðH 5 Þ are satisfied then the
system (7.1) (7.2) has a mild solution on J, provided that

2ωb
e 2 1 ^
2
4M b δ 1 4ðd 1 βÞ 1 ΔTrðQÞ sup mðtÞ
2ω tAJ
ð7:6Þ
4 4 2ωb
5OBO M bðe 2 1Þ
1 1 , 1
2ωε 2
Proof: Let B b be the space of all functions x:ð2N; bŠ-H such that x 0 AB
and the restriction x: ½0;bŠ-H is continuous. Let OUO b be a seminorm in B b
defined by

1
OxO b 5 Ox 0 O B 1 sup EOxðsÞO 2 2 :

sA½0;bŠ
ε ε
For any E . 0, consider the operator Φ :B b -PðB b Þ defined by Φ x, the
set of ρAB b such that

φðtÞ; tAð2N;0Š;
8
>
<
Ð t Ð t ε Ð t
ρðtÞ5 S α ðtÞφð0Þ1 0 S α ðsÞx 1 ds1 0 S α ðt2sÞBu ðs;xÞds1 0 S α ðt2sÞfðsÞds
>
1 S α ðt2sÞ
: Ð t Ð s
0 0 σðs;τ;x τ ÞdWðτÞ ds;tAJ; fAN F;x
where
n
E b 21 Ð b ~
u ðs; xÞ 5 B S ðb 2 sÞðEI1Π Þ E~ x b 1 0 φðsÞdWðsÞ 2 S α ðbÞφð0Þ
α
0
b Ð b
2 S α ðsÞx 1 ds 2 S α ðb 2 sÞfðsÞds
Ð
0 0
o
Ð b Ð s
2 S α ðb 2 sÞ σðs; τ; x τ ÞdWðτÞ ds ; fAN F;x :
0 0
For φAB, define

^
φðtÞ 5 φðtÞ; tAð2N; 0Š
S α ðtÞφð0Þ; tAJ
^
^
then φAB b . Let xðtÞ 5 φðtÞ 1 yðtÞ; 2N , t # b. It is easy to check that x
satisfies (7.1) (7.2) if and only if y 0 5 0 and