Page 252 - Mathematical Techniques of Fractional Order Systems

Controllability of Fractional Chapter | 8 241

2 q2p q2p 2
E:Ψðt 1 Þ2Ψðt 2 Þ: #9E:½E q2p ðAt Þ2E q2p ðAt ފx 0 :
1 2
q2p q2p q2p q2p 2
19E:½At E q2p;q2p11 ðAt Þ2At E q2p;q2p11 ðAt ފx 0 :
1 1 2 2
19E:½t 1 E q2p;2 ðAt q2p Þ2t 2 E q2p;2 ðAt q2p ފx : 2
0
1 2 0
t 2 2
ð
19E: ð t 2 2sÞ q21 E q2p;q ðAðt 2 2sÞ q2p ÞBu x ðsÞds:
t 1
t 1 2
ð
19E: ½ t 1 2sÞ q21 E q2p;q ðAðt 1 2sÞ q2p Þ2ðt 2 2sÞ q21 E q2p;q ðAðt 2 2sÞ q2p ފBu x ðsÞds:
ð
0
ð
t 2
19E: ð t 2 2sÞ q21 E q2p;q ðAðt 2 2sÞ q2p ÞvðsÞds: 2
t 1
t 1
ð
19E: ½ t 1 2sÞ q21 E q2p;q ðAðt 1 2sÞ q2p Þ2 t 2 2sÞ q21 E q2p;q ðAðt 2 2sÞ q2p ފvðsÞds: 2
ð
ð
0

t 2
ð ð s
19E: ð t 2 2sÞ q21 E q2p;q ðAðt 2 2sÞ q2p Þ Gðθ;xðθÞÞdwðθÞ ds: 2
t 1 0
t 1
ð
19E: ½ t 1 2sÞ q21 E q2p;q ðAðt 1 2sÞ q2p Þ2 t 2 2sÞ q21 E q2p;q ðAðt 2 2sÞ q2p ފ
ð
ð
0

ð s
3 Gðθ;xðθÞÞdwðθÞ ds: 2
0
q2p q2p 2
#9E:½E q2p ðAt Þ2E q2p ðAt ފx 0 :
1 2
19E:½At q2p E q2p;q2p11 ðAt q2p Þ2At q2p E q2p;q2p11 ðAt q2p ފx 0 : 2
1 1 2 2
0
19E:½t 1 E q2p;2 ðAt q2p Þ2t 2 E q2p;2 ðAt q2p ފx : 2
0
1
2
q ð t 2
ðt 2 2t 1 Þ q21 q2p 2 2 2
19 ð t 2 2sÞ :E q2p;q ðAðt 2 2sÞ Þ: :B: E:u x ðsÞ: ds
q
t 1
ð
t 1
2
q21 q2p q21 q2p ފ: :B: 2
ð
ð
19t 1 :½ t 1 2sÞ E q2p;q ðAðt 1 2sÞ Þ2 t 2 2sÞ E q2p;q ðAðt 2 2sÞ
0
q ð t 2
2 ðt 2 2t 1 Þ q21 q2p 2
3E:u x ðsÞ: ds19 ð t 2 2sÞ :E q2p;q ðAðt 2 2sÞ Þ: ϕ ðsÞds
ρ
q
t 1
ð
t 1
q21 q2p q21 q2p 2
ð
ð
19t 1 :½ t 1 2sÞ E q2p;q ðAðt 1 2sÞ Þ2 t 2 2sÞ E q2p;q ðAðt 2 2sÞ ÞŠ: ϕ ðsÞds
ρ
0
q ð t 2
ðt 2 2t 1 Þ q21 q2p 2 2
19 ð t 2 2sÞ :E q2p;q ðAðt 2 2sÞ Þ: TM 1 M G 11E:xðsÞ: ds
b
q
t 1
ð
t 1
q21 q2p q21 q2p 2
ð
19t 1 :½ t 1 2sÞ E q2p;q ðAðt 1 2sÞ Þ2ðt 2 2sÞ E q2p;q ðAðt 2 2sÞ ÞŠ:
0

2
3TM 1 M G 11E:xðsÞ: ds:
b