Page 210 - Mathematical Techniques of Fractional Order Systems

Controllability of Single-valued Chapter | 7 199

n Ð t 1 2E 2
EOρðt 2 Þ 2 ρðt 1 ÞO # 12 E: ½S α ðt 2 Þ2S α ðt 1 ފx 1 ds:
2
0
2
1 E: Ð t 1 ½S α ðt 2 Þ2S α ðt 1 ފx 1 ds: 1 E: Ð t 2 S α ðt 2 Þx 1 ds: 2
t 1 2E t 1
^
E
1 E: Ð 0 t 1 2E ½S α ðt 2 2sÞ2S α ðt 1 2sފBu ðs; φ1yÞds: 2
^
E
1 E: Ð t 1 ½S α ðt 2 2sÞ2S α ðt 1 2sފBu ðs; φ1yÞds: 2
t 1 2E
2
^
1 E: Ð t 2 S α ðt 2 2sÞBu ðs; φ1yÞds: 1 E: Ð t 1 2E ½S α ðt 2 2sÞ2S α ðt 1 2sފfðsÞds: 2
E
t 1 0
2
1 E: Ð t 1 2E ½S α ðt 2 2sÞ2S α ðt 1 2sފfðsÞds: 1 E: Ð t 1 t 2 S α ðt 2 2sÞfðsÞds: 2
t 1

^
1 E: Ð t 1 2E ½S α ðt 2 2sÞ2S α ðt 1 2sފ Ð s σðs; τ; φ τ 1y τ ÞdWðτÞ ds: 2
0 0

^
1 E: Ð t 1 Ð s σðs; τ; φ τ 1y τ ÞdWðτÞ ds: 2
t 1 2E ½S α ðt 2 2sÞ2S α ðt 1 2sފ 0
o
1 E: Ð t 2 S α ðt 2 2sÞ Ð s σðs; τ; ^ φ τ 1y τ ÞdWðτÞ ds: 2
t 1 0
n
t 1 2E
Ð 2 2
# 12 ðt 1 2 EÞ OS α ðt 2 Þ 2 S α ðt 1 ÞO EOx 1 O ds
0
2
2
2
1 E Ð t 1 OS α ðt 2 Þ 2 S α ðt 1 ÞO EOx 1 O ds 1 ðt 2 2 t 1 ÞM 2 Ð t 2 e 2ωt EOx 1 O ds
t 1 2E t 1
2
1 ðt 1 2 EÞOBO 2 Ð t 1 2E OS α ðt 2 2 sÞ 2 S α ðt 1 2 sÞO Ξds
0
2
1 EOBO 2 Ð t 1 OS α ðt 2 2 sÞ 2 S α ðt 1 2 sÞO Ξds 1 ðt 2 2 t 1 ÞM OBO 2 Ð t 2 e 2ωt Ξds
2
t 1 2E t 1
t 1 2E
Ð 2
1 ðt 1 2 EÞ OS α ðt 2 2 sÞ 2 S α ðt 1 2 sÞO ðμðrÞ 1 ð ^ d 1 βÞð4r 1 r ÞÞds
0
2
1 E Ð t 1 OS α ðt 2 2 sÞ 2 S α ðt 1 2 sÞO ðμðrÞ 1 ð ^ d 1 βÞð4r 1 r ÞÞds

t 1 2E

1 ðt 2 2 t 1 ÞM 2 Ð t 2 e 2ωt ðμðrÞ 1 ð ^ d 1 βÞð4r 1 r ÞÞds
t 1
ð t 1 2E
2

1 ðt 1 2 EÞTrðQÞ sup mðtÞ OS α ðt 2 2 sÞ 2 S α ðt 1 2 sÞO Λ σ ð4r 1 r Þds
tAJ 0
ð
t 1
2

1 ETrðQÞ sup mðtÞ OS α ðt 2 2 sÞ 2 S α ðt 1 2 sÞO Λ σ ð4r 1 r Þds
tAJ t 1 2E
ð
t 2
2 e 2ωt Λ σ ð4r 1 r Þds

1 ðt 2 2 t 1 ÞM TrðQÞ sup mðtÞ
tAJ t 1
for E sufficiently small, we can verify that the right-hand side of the above
inequality tends to zero as t 2 -t 1 . Furthermore, the compactness of S α ðtÞ
implies the continuity in the uniform operator topology. Thus, Ψ maps B r
into an equicontinuous family of functions.
Step 4: Next, one needs to prove that VðtÞ 5 fðΨyÞðtÞ; yAB r g is relatively
compact in H. Obviously VðtÞ is relatively compact for t 5 0. Let 0 , t # b
be fixed and E be a real number satisfying 0 , ε , t. For yAB r , define
t2E Ð t2E E ^ Ð t2E
ρ 5 0 S α ðsÞx 1 ds 1 0 S α ðt 2 sÞBu ðs; φ 1 yÞds 1 0 S α ðt 2 sÞfðsÞds
Ð
ε
t2E Ð s ^
1 S α ðt 2 sÞ σðs; τ; φ 1 y τ ÞdWðτÞ ds
Ð
0 0 τ
where fAN F;x . Since S α ðtÞ is a compact operator, the set V E ðtÞ 5 fρ ðtÞ:yAB r g
E
is relatively compact in H for each E,0 , E , t. Moreover, for each 0 , E , t,