Page 250 - Mathematical Techniques of Fractional Order Systems

Controllability of Fractional Chapter | 8 239

ρ,E:Φðx ρ ÞðtÞ: 2
5:E q2p ðAt q2p Þx 0 2At q2p E q2p;q2p11 ðAt q2p Þx 0 1tE q2p;2 ðAt q2p Þx 0 0
ð t ð t
1 ðt2sÞ q21 E q2p;q ðAðt2sÞ q2p ÞBu ρ ðsÞds1 ðt2sÞ q21 E q2p;q ðAðt2sÞ q2p Þv ρ ðsÞds
0 0
ð t ð s
q21 q2p 2
1 ðt2sÞ E q2p;q ðAðt2sÞ Þ Gðθ;x ρ ðθÞÞdwðθÞ ds:
0 0
2
#6:E q2p ðAt q2p Þx 0 : 16:At q2p E q2p;q2p11 ðAt q2p Þx 0 : 2
ð t
2
16:tE q2p;2 ðAt q2p Þx : 16E: ðt2sÞ q21 E q2p;q ðAðt2sÞ q2p ÞBu ρ ðsÞds: 2
0
0
0
ð t
q21 q2p 2
16E: ðt2sÞ E q2p;q ðAðt2sÞ Þv ρ ðsÞds:
0
ð t ð s
q21 q2p 2
16E: ðt2sÞ E q2p;q ðAðt2sÞ Þ Gðθ;x ρ ðθÞÞdwðθÞ ds:
0 0
T 2q 2 2 2 2
#6n 1 16n 2 16n 3 136 n :B: :B : l E:x 1 : 1n 1 1n 2 1n 3
4
q 2
T 2q T 2q 2 T 2q
1 n 4 ϕ ðtÞ1 Tn 4 M 1 M G 11 supE:x ρ ðtÞ: 16 n 4 ϕ ðtÞ
ρ
ρ
b
q 2 q 2 tAJ q 2
T 2q 2
16 Tn 4 M 1 M G 11 supE:x ρ ðtÞ:
b
q 2 tAJ
0 1
T 2q 2 2 2 T 2q 2 2 2 2
#6n 1 116 n :B: :B : l A 136 n :B: :B : lE:x 1 :
@
4
4
q 2 q 2
0 1 0 1
T 2q 2 2 T 2q 2 2
2
2
16n 2 116 n :B: :B : l A 16n 3 116 n :B: :B : l A
4
@
@
4
q 2 q 2
0 1 0 1
T 2q T 2q 2 2 T 2q 2 2
2
2
16 n 4 ϕ ðtÞ 116 n :B: :B : l A 16116 n :B: :B : l A
ρ
4
@
4
@
q 2 q 2 q 2
T 2q 2
3 Tn 4 M 1 M G 11 supE:x ρ ðtÞ: :
b
q 2 tAJ
Dividing both sides of the above inequality by ρ and taking the limit as
ρ-N; one can get