Page 233 - Mathematical Techniques of Fractional Order Systems
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222 Mathematical Techniques of Fractional Order Systems
where
2 2 2
Θ 5 8b OnO N ðb Oϕ O 1 1ÞΛ f ðq Þ
A
2α 2 2α12
16b 1 16Oϕ O 1 b
1 2 A L ½ M R 1 TrðQÞOmO N Λ σ ðq Þ
Γ ðα 1 1Þ
by (7.18) there exists a M such that Oq O 6¼ M . Let us set
^
^
^
U 5 fxACðJ; HÞ: OyO , M g, from the choice of U, there is no yA@U such
that y 5 λΠðyÞ for some λAð0; 1Þ. Consequently, by the nonlinear alternative
^
of Leray-Schauder type, one can deduce that Π has a fixed point xAU,
which is a solution of (7.15) and (7.16) on ð2N; b.
7.3.1.2 Existence of Optimal Control
Consider the following Lagrange problem ðPÞ:
0
0
Find a control u AA ad such that J ðu Þ # J ðuÞ; for all uAA ad where
ð b
J ðuÞ 5 E Lðt; x ; x ðtÞ; uðtÞÞdt
u
u
t
0
u
and x denotes the mild solution of (7.15) and (7.16) corresponding to the
control uAA ad . For the existence of solutions of Lagrange problem P, one
shall introduce the following hypothesis
(H 20 )
1. The functional L:J 3 C h 3 H 3 U-R , fNg is F t - measurable.
2. Lðt; U; U; UÞ is sequentially lower semicontinuous on C h 3 H 3 U for
almost all tAJ.
3. Lðt; x; y; UÞ is convex on U for each xAC h ; yAH and all tAJ.
4. There exists constants d; e $ 0; j . 0; μ is nonnegative and
p
1
2
2
μAL ðJ; RÞ such that Lðt; x; y; uÞ $ μðtÞ 1 dOxO 1 eEjyj 1 jOuO :
C h U
Theorem 7.7: Let the hypotheses ðH 14 Þ 2 ðH 20 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 ; x ðtÞ; u ðtÞÞdt # J ðuÞ; for all uAA ad :
0
0
0
0
t
0
Proof: If inffJ ðuÞjuAA ad g 5 N, there is nothing to prove. Without loss of
generality, assume that inffJ ðuÞjuAA ad g 5 ^ E ,1 N: Using ðH 20 Þ, one has
^ E .2N. By the definition of infimum there exists a minimizing sequence
m
m
feasible pair fðx ; u ÞgCP ad , where
P ad 5 ðx; uÞ: x is a mild solution of the system ð7:15Þ ð7:16Þ
corresponding touAA ad
