Page 217 - Mathematical Techniques of Fractional Order Systems

206 Mathematical Techniques of Fractional Order Systems

Thus,
pωb
p p21 p p21 e 2 1 p p
sup EOðΦxÞðtÞ 2 ðΦyÞðtÞO # 2 M b ðM f 1 b 2 c p M σ Þ sup EOxðtÞ 2 yðtÞO
H
tAJ pω tAJ
which implies by inequality (7.14) that Φ is a contraction and hence by con-
traction mapping principle there exists a unique fixed point x of Φ, which is
a mild solution of the fractional control problem (7.9) (7.10).

7.2.2.3 Existence of Optimal Control Result
Consider the following Lagrange problem ðPÞ:
0
Find a control u AA ad such that
0
J ðu Þ # J ðuÞ; ’ uAA ad
where

b
ð
J ðuÞ 5 E Lðt; x ðtÞ; uðtÞÞdt
u
0
u
and x denotes the mild solution of (7.9) (7.10) corresponding to the control
uAA ad . For the existence of solution for problem ðPÞ, one shall introduce
the following hypothesis
(H 13 )
(i) The functional L:J 3 H 3 U-R , fNg is F t - measurable;
(ii) Lðt; U; UÞ is sequentially lower semicontinuous on H 3 U for almost
all tAJ;
(iii) L(t,x,˙ c) is convex on U for each xAH and almost all tAJ;
1
(iv) there exist constants d $ 0; e . 0; μ is nonnegative and μAL ðJ; RÞ
such that
p
p
Lðt; x; uÞ $ μðtÞ 1 dEOxO 1 EOuO :
H U
Theorem 7.5: Let the hypotheses ðH 8 Þ 2 ðH 13 Þ hold. Suppose that B is a
strongly continuous operator. Then the Lagrange problem ðPÞ admits at least
0
one optimal pair, i.e., there exists an admissible control u AA ad such that
ð b
J ðu Þ 5 E Lðt; x ðtÞ; u ðtÞÞdt # J ðuÞ; ’ uAA ad :
0
0
0
0
Proof: If inffJ ðuÞjuAA ad g 51 N, there is nothing to prove. Without
loss of generality, one can assume that inffJ ðuÞjuAA ad g 5 E ,1 N:
Using ðH 13 Þ, one has E .2N. By the definition of infimum there exists a

m
m
minimizing sequence feasible pair fðx ; u ÞgCP ad , where P ad 5 ðx; uÞ: x is