Page 248 - Mathematical Techniques of Fractional Order Systems

Controllability of Fractional Chapter | 8 237

One needs to show that Φ satisfies all the conditions of Lemma 8.4.For
the sake of convenience, subdivide the proof into four steps.
Step 1. For each xAB; Φ is convex.
In fact, if Ψ 1 ; Ψ 2 AΦðxÞ; then for each tAJ; there exists v 1 ; v 2 AN F;x such
that
Ψ i ðtÞ5E q2p ðAt q2p Þx 0 2At q2p E q2p;q2p11 ðAt q2p Þx 0 1tE q2p;2 ðAt q2p Þx 0
0
"
t
ð
1 t2s q21 E q2p;q ðAðt2sÞ q2p ÞB B E q2p;q ðA ðT2sÞ q2p ÞW 21

0
(
ðEx 1 2E q2p ðAT q2p Þx 0 1AT q2p E q2p;q2p11 ðAT q2p Þx 0 2TE q2p;2 ðAT q2p Þx 0
0
ð T
q21 q2p
2 ðT2sÞ E q2p;q ðAðT2sÞ Þ
0
)#
ð s ð t
3 v i ðsÞ1 Gðθ;xðθÞÞdwðθÞ ds ðsÞds1 ðt2sÞ q21 E q2p;q ðAðt2sÞ q2p Þ
0 0
ð s
3 v i ðsÞ1 Gðθ;xðθÞÞdwðθÞ ds; i51;2:
0
Let 0 # λ # 1; then for each tAJ; one can have

½λΨ 1 1ð12λÞΨ 2 ŠðtÞ5E q2p ðAt q2p Þx 0 2At q2p E q2p;q2p11 ðAt q2p Þx 0 1tE q2p;2 ðAt q2p Þx 0 0
"
ð t

1 ðt2sÞ q21 E q2p;q ðAðt2sÞ q2p ÞB B E q2p;q ðA ðT2sÞ q2p ÞW 21

0
(
Ex 1 2E q2p ðAT q2p Þx 0
1AT q2p E q2p;q2p11 ðAT q2p Þx 0 2TE q2p;2 ðAT q2p Þx 0 0

ð T
2 ðT2sÞ q21 E q2p;q ðAðT2sÞ q2p Þ
0
" # )#
ð s
3 ðλv 1 ðsÞ1ð12λÞv 2 ðsÞÞ1 Gðθ;xðθÞÞdwðθÞ ds ðsÞds
0
ð t
1 ðt2sÞ q21 E q2p;q ðAðt2sÞ q2p Þ
0

ð s
3 ðλv 1 ðsÞ1ð12λÞv 2 ðsÞÞ1 Gðθ;xðθÞÞdwðθÞ ds:
0