Page 158 - Mathematical Techniques of Fractional Order Systems

146 Mathematical Techniques of Fractional Order Systems

T T T T
5 τ m x ðtÞA 1 x ðt 2 τðtÞÞA τ Z A 0 xðtÞ 1 A τ xðt 2 τðtÞÞ
0
t ð5:56Þ
ð
2 _ xðsÞZ _ xðsÞds
t2τ m
On the other hand, by adding the left sides of (5.10) and (5.11) to the
time derivative of the Lyapunov functional candidates VðtÞ, where
VðtÞ 5 V 1 ðtÞ 1 V 2 ðtÞ 1 V 3 ðtÞ, yields
ð N
dVðtÞ T T
#2 2 ωμðωÞz ðω; tÞPzðω; tÞdω 1 2x ðtÞPðA 0 xðtÞ 1 A τ xðt 2 τðtÞÞÞ
dt 0
T T
1 x ðtÞQxðtÞ 2 ð1 2 @τ m Þx ðt 2 τðtÞÞQxðt 2 τðtÞÞ
T T T T
1 τ m x ðtÞA 1 x ðt 2 τðtÞÞA τ ZðA 0 xðtÞ 1 A τ xðt 2 τðtÞÞÞ
0
ð t ð t
T
T
2 _ xðsÞZ _ xðsÞ 1 τ m η ðtÞΓηðtÞ 2 η ðtÞΓηðtÞdsds
t2τ m t2τðtÞ
ð t
T
T
1 2ðx ðtÞY 1 x ðt 2 τðtÞÞT Þ 3 xðtÞ 2 xðt 2 τðtÞ 2 _ xðsÞds
t2τðtÞ
ð5:57aÞ
N
ð
T
T
#2 2 ωμðωÞz ðω; tÞPzðω; tÞdω 1 ηðtÞ ϒ 1 ηðtÞ
0
ð t ð5:57bÞ
T
2 ηðt; sÞ Wηðt; sÞds
t2τðtÞ
where

Φ 11 Φ 12
ϒ 1 5 ð5:58aÞ
Φ 21 Φ 22
2 3
Γ 11 Γ 12 2Y
Ω 1 5 Γ T 12 Γ 22 2T 5 ð5:58bÞ
4
0 0 Z
T
T T T
ηðt; sÞ 5 x ðtÞ x ðt2τðtÞÞ _ x ðtÞ ð5:58cÞ
and
T
Φ 11 5 2PA 0 1 τ m A ZA 0 1 2Y 1 τ m Γ 11 1 Q
0
T
Φ 12 5 2PA τ 1 τ m A ZA τ 2 2Y 1 τ m Γ 12
0
T T
Φ 21 5 τ m A ZA 0 1 τ m Γ 1 2T
τ
12
T
τ
Φ 22 52 Λ 1 Q 2 2T 1 τ m A ZA τ 1 τ m Γ 22
As a result, sufficient conditions are given such that the inequalities
VðtÞ . 0 and dVðtÞ , 0 are fulfilled for any ηðtÞ 6¼ 0 and ηðt; sÞ 6¼ 0. These two
dt
inequalities are equivalent to ϒ 1 , 0 and Ω 1 $ 0.