Page 215 - Mathematical Techniques of Fractional Order Systems

204 Mathematical Techniques of Fractional Order Systems

The following hypotheses are considered to prove the main results

(H 8 ) A generates an t α21 ; 1 -regularized family S α ðtÞ such that there
ΓðαÞ
ωt
exist ω . 0 and M . 0 such that OS α ðtÞO # Me ; ’tAJ:
(H 9 ) The function f:J 3 H-H satisfies the following
(i) for each tAJ, fðt; UÞ:H-H is continuous and for each xAH,
fðU; xÞ:J-H is strongly measurable;
^
(ii) there exist constants M f . 0; M f . 0 such that
p
p
EOfðt; xÞ 2 fðt; yÞO # M f EOx 2 yO ; ’x; yAH
H
H
p
p
and EOfðt; xÞO # ^ M f ð1 1 EOxO Þ; ’xAH:
H
H
(H 10 ) The function σ:J 3 H-LðK; HÞ satisfies the following
(i) for each tAJ; σðt; UÞ:H-LðK; HÞ is continuous and for each
xAH; σðU; xÞ:J-LðK; HÞ is strongly measurable;
^
(ii) there exist constants M σ . 0; M σ . 0 such that
p
p
EOσðt; xÞ 2 σðt; yÞO # M σ EOx 2 yO ; ’x; yAH
H
H
p
p
and EOσðt; xÞO # ^ M σ ð1 1 EOxO Þ; ’xAH:
H H
(H 11 ) Let uAU be the control function and the operator
BAL N ðJ; LðU; HÞÞ; OBO stands for the norm of the operator B.
U
(H 12 ) The multivalued map A:J-2 f[g has closed, convex, and
bounded values. AðUÞ is graph measurable and AðUÞDΣ, where Σ is a
bounded subset of U.
7.2.2.2 Existence of Mild Solution
Theorem 7.4: Suppose that the hypotheses ðH 8 Þ 2 ðH 12 Þ hold, then the
fractional control problem (7.9) (7.10) has a unique mild solution on J pro-
vided that
e pωb 2 1 p
p p21
2 p21 M b ðM f 1 b 2 c p M σ Þ , 1: ð7:14Þ
pω
Proof: Consider the map Φ on C defined by

ðΦxÞðtÞ 5 S α ðtÞx 0 1 Ð 0 t S α ðt 2 sÞBðsÞuðsÞds 1 Ð 0 t S α ðt 2 sÞfðs; xðsÞÞds
1 Ð t S α ðt 2 sÞ Ð s σðτ; xðτÞÞdWðτÞ ds:

0 0