Page 230 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 219

0
0
there exists a positive number δ such that EjðΠ 2 zÞj # δ , which means that
Π 2 ðzÞAB δ 0. Now, to prove that Π 2 maps bounded sets into equicontinuous
0
sets in C .Let t 1 ; t 2 AJ; t 2 . t 1 , and consider B q as a bounded set as in Step
b
1. Let zAB q , then if E . 0 and E # t 1 # t 2 , one has
EjðΠ 2 zÞðt 2 Þ 2 ðΠ 2 zÞðt 1 Þj # 3E Ð t 1 2E ½S ðt 2 2sÞ2S ðt 1 2sފðΠ 1 zÞðsÞds 2
2
0
0

0
2
Ð
t 1 0 0
t 1 2ε ½S ðt 2 2sÞ2S ðt 1 2sފðΠ 1 zÞðsÞds
1 3E
2

Ð
t 2 0
1 3E S ðt 2 2sÞðΠ 1 zÞðsÞds
t 1
# 3ðt 1 2 EÞδ Ð t 1 2E jS ðt 2 2 sÞ 2 S ðt 1 2 sÞj ds
2
0
0
0
1 3Eδ Ð t 1 2ε jS ðt 2 2 sÞ 2 S ðt 1 2 sÞj 2
t 1
0
0
2
1 3ðt 2 2 t 1 Þδ Ð t 2 jS ðt 2 2 sÞj ds
0
t 1
as t 2 -t 1 and ε sufficiently small, the right-hand side of the above inequality
tends to zero. By the Arzela Ascoli theorem it suffices to show that Π 2
maps B q into a precompact set in H. Let 0 , t , b be fixed and let ε be a
real number satisfying 0 , ε , t. For zAB q ,
ð t2E
0 0
ðΠ 2E zÞðtÞ 5 S ðEÞ S ðt 2 s 2 EÞðΠ 1 zÞðsÞds
0
0
since S ðtÞ is a compact operator for t . 0, the set Z E ðtÞ 5 fðΠ 2E zÞðtÞ: zAB q g
is precompact in H for every E; 0 , E , t. Furthermore
EjðΠ 2 zÞðtÞ 2 ðΠ 2E zÞðtÞj # 2E Ð t2E S ðt2sÞ½ðΠ 1 zÞðsÞ2ðΠ 1E zÞðsފds 2
2
0

0
1 2E Ð t S ðt2sÞðΠ 1 zÞðsÞds 2
0
t2ε

2 Ð t2E 2
# 2Oϕ O 1 ðt 2 EÞ EjðΠ 1 zÞðsÞ 2 ðΠ 1E zÞðsÞj ds
A L 0
1 2Oϕ O 1 E Ð t EjðΠ 1 zÞðsÞj ds
2
2
A L t2ε
# 2Oϕ O 1 ðt2εÞ Ξ 1 2Oϕ O 1 ε δ
2
2
2
2
A L A L
so ZðtÞ 5 fðΠ 2 zÞðtÞ: zAB q g is precompact in H. Hence Π 2 is completely
continuous.
^
Step 3: To prove that there exists an open set UCCðJ; HÞ with z=2λΠðzÞ
^
for λAð0; 1Þ and yA@U. Let λAð0; 1Þ and
λ ð t ð t
zðtÞ 5λðΠzÞðtÞ5λφð0Þ1 ðt2sÞ α21 BðsÞuðsÞds1λ fðs;z s 1y s Þds
ΓðαÞ 0 0
λ ð t ð s
1 ðt2sÞ α21 σðτ;z τ 1y τ ÞdWðτÞ ds
ΓðαÞ 0 2N
2
λ ð s
1λ S ðt2sÞ φð0Þ1 ðs2τÞ α21 BðτÞuðτÞdτ
Ð
t 0
0
4
ΓðαÞ 0
3
τ
s

ð
ð
1 Ð s fðτ;z τ 1y τ Þdτ1 1 ðs2τÞ α21 σðη;z η 1y η ÞdWðηÞ dτ 5 ds:
0
ΓðαÞ 0 2N