Page 113 - IJOCTA-15-3

Trajectory controllability of integro-differential system of fractional orders in Hilbert spaces
We consider Y, V be Hilbert spaces with V is a Also
subspace of Y and L 2 ([0, b], Y), L 2 ([0, b], V) be the n−1
X r
ν c
ν
r
corresponding function spaces. We consider the I ( D f(ϱ)) = f(ϱ) − t f (0).”
t ϱ
fractional semilinear integrodifferential system of r=0 r!
order ν ∈ (0, 1] : n−1 1
where C ((0, b); Y) and L ((0, b); Y) represents
c ν
D y(ϱ) =Ay(ϱ) + B(ϱ, v(ϱ)) the (n − 1) time continuously differentiable and
ϱ
ϱ space of Y -valued Bochner integrable functions,
Z
+ G ϱ, y(ϱ), g(ϱ, α, y(α))dα , respectively.
0
varrho ∈ I = [0, b], ν ∈ (0, 1], Definition 4. [30] A continuous function x ∈
(1) C(I, Y) is called the mild solution of (1)-(2) along
with u ∈ L 2 (I; V) if it fulfills
y(0) = y 0 . (2)
Z ϱ
In the above system, c D ν stands for the Ca- y(ϱ) =S ν (ϱ)y 0 + (ϱ − α) ν−1 ˆ
S ν (ϱ − α)
ϱ
puto fractional derivative of order ν. The state 0

y(ϱ) assumes its values in Hilbert space Y and
the control v(ϱ) assumes values in Hilbert space B(α, v(α)) + G α, y(α), (3)
V. A : D(A) ⊂ Y → Y is the infinitesi- Z α
mal generator of a strongly continuous semigroup g(α, β, y(β))dβ dα,
0
{S(ϱ), ϱ ∈ R}. The operator B : J × V → Y
is nonlinear. Moreover, G : J × Y × Y → Y and where
Z ∞
g : ∆ × Y → Y are appropriate function defined ν
S ν (ϱ) = ζ ν (α)S(ϱ α)dα
later. ∆ = {(ϱ, α) ∈ I : 0 ≤ α ≤ ϱ ≤ b} and 0
y 0 ∈ Y. and
Z ∞
We assume that C(I, Y) : I → Y be the Ba- S ν (ϱ) = ν αζ ν (α)S(ϱ α)dα.
ν
ˆ
nach space of all continuous functions with supre- 0
mum norm given by Here
1 −1− 1 − 1
∥ψ∥ = sup ∥ψ(η)∥, ψ ∈ C(I, Y). ζ ν (α) = α ν ψ ν (α ν )
0≤η≤b ν
is a function defined on (0, ∞) satisfying ζ ν (α) ≥
Definition 1. [41] “The fractional integral of Z ∞
+ 0, ζ ν (α)dα = 1 and
order ν ∈ R with the lower limit 0 for a func- 0
+
1
tion f ∈ L (R ) is defined as
∞
1 X Γ(nν + 1)
Z ϱ n−1 −νn−1
1 ν−1 ψ ν (α) = (−1) α sin(nπν),
ν
I f(ϱ) = (ϱ − α) f(α)dα, ϱ > 0 , ν > 0 π n!
t
Γ(ν) 0 n=1
α ∈ (0, ∞).
provided the right-hand side is point-wise defined
on [0, ∞), where Γ(·) is gamma function.” Lemma 1. [42] “For any fixed t ≥ 0, the op-
ˆ
erators S ν (ϱ) and S ν (ϱ) are linear and bounded,
Definition 2. [41] “The Riemann-Liouville (R-
+ that is, for any x ∈ Y, ∥S ν (ϱ)y∥ ≤ M∥y∥ and
L) derivative of order ν ∈ R with lower limit Mν
ˆ
zero for a function f : [0, ∞) → R can be written ∥S ν (ϱ)y∥ ≤ ∥y∥, where M is a constant
Γ(1 + ν)
as
such that ∥S(ϱ)∥ ≤ M, for all ϱ ≥ 0.”
1 d n Z ϱ f(α)
L ν
D f(ϱ) = dα, For more details on the fractional systems, re-
ϱ
Γ(n − ν) dt n 0 (ϱ − α) ν+1−n 43 AK12 45
ϱ > 0, fer Bazhlekova, Miller, Podlubny and for
the detailed study of cosine and sine families, one
n − 1 < ν < n.” can relate to Travis and Webb. 46
Before discussing the T-controllability of frac-
Definition 3. [14] “The Caputo fractional
tional integrodifferential system having order
derivative of order ν for a function f ∈
1
C n−1 ((0, b); Y) ∩ L ((0, b); Y) can be written as ν ∈ (0, 1], we discuss the relations between T-
controllability and complete controllability.
1 Z ϱ
ν
c D f(ϱ) = (ϱ − α) n−ν−1 n Definition 5. The system (1)-(2) is called com-
f (α)dα
ϱ
Γ(n − ν) 0 pletely controllable provided that for every y 0 , y 1 ∈
where n − 1 < ν < n, n = [ν] + 1 denotes the Y, and fixed b, ∃ v(·) ∈ L 2 (I, V) such that y(·) ful-
integral part of the real number ν. fills y(b) = y 1 .
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