Page 232 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 221

2
2
2
But Oz s 1 y s O # 4l sup EjzðsÞj 1 4OφO 2 as proved in Step 1. If
C h sA½0;tŠ C h

one lets q ðtÞ be the right-hand side of the above inequality then,
2

Oz s 1 y s O # q ðtÞ; tAJ and therefore (7.19) becomes
C h
2αp22 ! 2p22
8OBO 2 p p
2 2 p21 2
EjzðtÞj # 8Ejφð0Þj 1 2 b αp21 OuO
Γ ðαÞ L p ðJ;UÞ
2α
Ð t 16b M R
1 8b nðsÞΛ f ðq ðsÞÞds 1

0 2
Γ ðα 1 1Þ
α ð t
16b TrðQÞ α21 2 2 2
1 2 ðt2sÞ mðsÞΛ σ ðq ðsÞÞds 1 8b Oϕ O 1Ejφð0Þj

A L
αΓ ðαÞ 0
2αp22 ! 2p22 ð7:20Þ
8b Oϕ O 1 OBO 2 p p
2
2
p21
1 A L b αp21 OuO 2
2
Γ ðαÞ L p ðJ;UÞ
2
16Oϕ O 1 b 2α12 M R
3
1 8b Oϕ O 2 Ð t nðsÞΛ f ðq ðsÞÞds 1 A L

A L 1 0 2
Γ ðα 1 1Þ
2
16Oϕ O 1 b 2α11 TrðQÞ ð t

1 A L mðsÞΛ σ ðq ðsÞÞds
2
Γ ðα 1 1Þ 0

using (7.20) in the definition of q ðsÞ, one has
2
q ðtÞ 5 4l sup EjzðsÞj 1 4OφO 2
2

C h
sA½0;tŠ
2
2αp22 ! 2p22
8OBO 2 p p
8Ejφð0Þj 1 2 b αp21 OuO
2 6 2 p21 2
# 4l 4
Γ ðαÞ L p ðJ;UÞ
2α
Ð t 16b M R

1 8b nðsÞΛ f ðq ðsÞÞds 1
0 2
Γ ðα 1 1Þ
α ð t
16b TrðQÞ α21

1 2 ðt2sÞ mðsÞΛ σ ðq ðsÞÞds
αΓ ðαÞ 0
2αp22 ! 2p22
8b Oϕ O 1 OBO 2 p p
2
2
2
2
2
1 8b Oϕ O 1 Ejφð0Þj 1 A L b p21 OuO 2
A L 2 αp21 L p ðJ;UÞ
Γ ðαÞ
2
1 8b Oϕ O 2 Ð t nðsÞΛ f ðq ðsÞÞds 1 16Oϕ O 1 b 2α12 M R
A L
3

2
A L 1 0
Γ ðα 1 1Þ
3
2
16Oϕ O 1 b 2α11 TrðQÞ ð t
1 A L mðsÞΛ σ ðq ðsÞÞds 1 4OφO 2

5
2
Γ ðα 1 1Þ 0 C h
consequently
Oq O

# 1
2p22

2αp22 p21 p
OBO 2
2
2
2
4l 8Ejφð0Þj 1 8 2 b p ðb 1 1ÞOuO 2 1 Θ 1 4OφO 2
Γ ðαÞ αp21 L p ðJ;UÞ C h