Page 216 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 205

Now,
p
EOðΦxÞðtÞO p H #4 p21 EOS α ðtÞx 0 O 14 p21 E: Ð 0 t S α ðt2sÞ Ð 0 s σðτ;xðτÞÞdWðτÞ ds: p

p
14 p21 E: Ð t S α ðt2sÞBðsÞuðsÞds: 14 p21 E: Ð t S α ðt2sÞfðs;xðsÞÞds: p
0 0
p
p p21
p
p pωb
e
#4 p21 M e EOx 0 O 14 p21 M b Ð 0 t pωðt2sÞ E: Ð 0 s σðτ;xðτÞÞdWðτÞ: ds
2 1 3p
1 12
pωðt2sÞ p 1
0
p21 Ð p7
6
p
Ð
p
14 p21 M OBO 6 t e ds A t EOuðsÞO ds 7
p @
4 0 0 5
p p21
p
e
14 p21 M b Ð 0 t pωðt2sÞ EOfðs;xðsÞÞO ds
p
2 2
2 3
p
p
p
Ð
p pωb
Ð
p p21
#4 p21 M e EOx 0 O 14 p21 M b t pωðt2sÞ 4 s ðEOσðτ;xðτÞÞO 0 Þ dτ 5 ds
e
c p
0 0 L
2
3 p
! p21 p21
2
p p
14 p21 p p 6 p21 ðe pωb 7
pω 21Þ OuO L p ðJ;UÞ5
M OBO 4
p p21
p
e
14 p21 M b Ð t pωðt2sÞ EOfðs;xðsÞÞO ds
0
p
2 2
2 3
p
s
Ð
p p21
p pωb
p
Ð
p
#4 p21 M e EOx 0 O 14 p21 M b t pωðt2sÞ 4 ð ^ M σ ð11EOxðτÞO ÞÞ dτ 5 ds
e
c p
0 0
! p21
p
14 p21 M OBO p p21 ðe pωb 21Þ p21 OuO p L p ðJ;UÞ
pω
p p21
p
e
14 p21 M b Ð t pωðt2sÞ ^ M f ð11EOxðsÞO Þds
0
Thus,
! p21
p p21 p pωb p p21 p p p21 pωb p21 p
sup E:ðΦxÞðtÞ: # 4 M e EOx 0 O 1 4 M OBO pω ðe 21Þ OuO
tAJ H L p ðJ;UÞ
0 1 p
e pωb 2 1 2
^
p
1 4 p21 M b A ð ^ M f 1 b c p M σ Þ sup ð1 1 EOxðtÞO Þ
pω tAJ
p p21@
it follows from the fact that uðUÞ; fðU; UÞ are continuous, Φ is well-defined
on C. Moreover, for x; yAC and tAJ, one has
EOðΦxÞðtÞ 2 ðΦyÞðtÞO p # 2 p21 E: Ð t S α ðt2sÞ½fðs; xðsÞÞ2fðs; yðsÞފds: p
H 0

1 2 p21 E: Ð t S α ðt2sÞ Ð s ½σðτ; xðτÞÞ2σðτ; yðτÞފdWðτÞ ds: p
0 0
p p21
p
# 2 p21 M b Ð t e pωðt2sÞ EOfðs; xðsÞÞ 2 fðs; yðsÞÞO ds
0
p
p p21
1 2 p21 M b Ð t e pωðt2sÞ E: Ð s ½σðτ; xðτÞÞ2σðτ; yðτÞފdWðτÞ: ds
0 0
p
p p21
# 2 p21 M b Ð t e pωðt2sÞ M f EOxðsÞ 2 yðsÞO ds
0
p
2 2
2 3
p
s
t
Ð
Ð
p p21
p
1 2 p21 M b e pωðt2sÞ 4 ðM σ EOxðτÞ2yðτÞO Þ dτ 5 ds
c p
0 0