Page 208 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 197

2
^
ε
~
2
r
2
x
EOu ðs;φ1y ÞO # 5OBO M 2 E:E~ b 1 ð b φðsÞdWðsÞ: 1EOS α ðbÞφð0ÞO 2
ε 2 0
2
1E: Ð b S α ðsÞx 1 ds: 1E: Ð b S α ðb2sÞfðsÞds: 2
0 0

2
^
1E: Ð b S α ðb2sÞ Ð s σðs;τ;φ 1y τ ÞdWðτÞ ds:
0 0 τ
2
5OBO M 2 ð b
~
2 2ωb
2
2
# 2EO~ x b O 12 EOφðsÞO ds1M e EOφð0ÞO 2
E 2 0
0 1
e 2ωb 21
2 2 Ð b 2ωðb2sÞ 2
1M bx 1 @ A 1M b e EOfðsÞO ds
2ω 0
s
^
2
2
Ð
1M b Ð b 2ωðb2sÞ E: σðs;τ;φ 1y τ ÞdWðτÞ: ds
e
τ
0 0
(
2
5OBO M 2 ð b
~
2
2
2 2ωb
# 2EO~ x b O 12 EOφðsÞO ds1M e EOφð0ÞO 2
ε 2 0
0 1
e 2ωb 21
2 2 Ð b 2ωðb2sÞ ^ ^ 2
1M bx 1 @ A 1M b e ðμðrÞ1ðd1βÞEOφ 1y s O Þds
2ω 0 B
s
)
2
^
s
Ð
2
e
1M b Ð b 2ωðb2sÞ TrðQÞ E:σðs;τ;φ 1y τ Þ: dτds
τ
0 0 Q
(
2
5OBO M 2 ð b
~
2 2ωb
2
2
# 2EO~ x b O 12 EOφðsÞO ds1M e EOφð0ÞO 2
ε 2 0
0 1 0 1
e 2ωb 21 e 2ωb 21
2 2 ^
1M bx 1 @ A 1M bðμðrÞ1ðd1βÞð4r1r ÞÞ A
@
2ω 2ω
)
^
2
2
1M b Ð b 2ωðb2sÞ TrðQÞmðsÞΛ σ ðOφ 1y s O Þds
e
0 s B
(
2
5OBO M 2 ð b
~
2
2 2ωb
2
# 2EO~ b O 12 EOφðsÞO ds1M e EOφð0ÞO 2
x
ε 2 0
0 1 0 1
e 2ωb 21 e 2ωb 21
2 2 ^
1M bx 1 @ A 1M bðμðrÞ1ðd1βÞð4r1r ÞÞ A
@
2ω 2ω
)
ð b
2 2ωðb2sÞ
1M bTrðQÞsupmðsÞ e Λ σ ð4r1r Þds :5Ξ
tAJ 0
thus from (7.8), one has
0 1
(
4
4
20OBO M b e 2ωb 2 1 ð b
~
2 2ωb
2
2
I 2 # @ A 2EO~ x b O 1 2 EOφðsÞO ds 1 M e EOφð0ÞO 2
ε 2 2ω 0
0 1 0 1
e 2ωb 2 1 e 2ωb 2 1
2 2 ^
1 M bx 1 @ A 1 M bðμðrÞ 1 ðd 1 βÞð4r 1 r ÞÞ A
@
2ω 2ω
)
2
e
1 M bTrðQÞ sup tAJ mðtÞ Ð 0 b 2ωðb2sÞ Λ σ ð4r 1 r Þds