Page 117 - IJOCTA-15-3

Trajectory controllability of integro-differential system of fractional orders in Hilbert spaces
By using this control, (16)-(17) becomes, In this case, trajectory controllability of the
system (11)-(12) can be verified by the same ap-
c ν c ν
D y(ϱ) =Ay(ϱ) + D z(ϱ) − Az(ϱ) proach discussed in Theorem 1 using the Lips-
ϱ
ϱ
ϱ chitz continuity of G and h and some additional
Z
− G ϱ, z(ϱ), g(ϱ, α, z(α))dα
assumptions on B, g, and f given in Assumptions
0
ϱ [II] of Section 2.
Z
+ G ϱ, y(ϱ), g(ϱ, α, y(α))dα ,
0 4. Examples
ν ∈ (1, 2]
′
y(0) =y 0 and y (0) = z 0 . In this section, we present numerical examples to
support and illustrate the theoretical results.
Fixing w(ϱ) = y(ϱ) − z(ϱ), one can get Example 1. Assume that the follow-
c ν ing nonlinear integrodifferential system with
D w(ϱ) =Aw(ϱ)
ϱ
ϱ
controlb(ϱ, v) = v|v|.
Z
ν
+ G ϱ, y(ϱ), g(ϱ, α, y(α))dα c D y(ϱ) =a(ϱ)y(ϱ) + B(ϱ, v(ϱ))
ϱ
0 Z ϱ
ϱ + sin y(ϱ) + 3 x(α)dα , 0 < ν ≤ 1,
Z
− G ϱ, z(ϱ), g(ϱ, α, z(α))dα , 0
0
(18)
ν ∈ (1, 2],
y(0) = y 0 . (19)
′
w(0) =0 and w (0) = 0. The control B(ϱ, v) is continuous and coer-
By using the semigroup theory, the solution of the cive. It is easy to show that G and h fulfill all the
above equation may be given by requirements of the Theorem 1. Therefore, the
system (18) is T-controllable.
Z ϱ
w(ϱ) = P ν (ϱ − α) G α, y(α), Example 2. Assume that Ω = (0, 1) be the
0 bounded domain in R n with smooth boundary
α ∂Ω. Assume that the subsequent fractional sys-
Z
g(α, β, y(β))dβ − G α, z(α),
0 tem
2
Z α c ν ∂ y 1 2
g(α, β, z(β))dβ dα D y(ϱ) = ∂ϱ 2 + v(y, ϱ) + [sin y(ϱ)
ϱ
2
0
+ sin y(ϱ)] in Ω × (0, b), 0 < ν ≤ 1,
Thus,
(20)
Mb ν−1 Z ϱ
∥w(ϱ)∥ ≤ δ 1 ∥y(α) − z(α)∥
Γ(ν) 0 z(y, 0) = 0 in Ω, (21)
α

Z

+ δ 2
g(α, β, x(β))dβ

z(y, ϱ) = 0 in ∂Ω × (0, b). (22)
0
α 2 2
Z

Let us define A : L (0, 1) → L (0, 1) by
− g(α, β, z(β))dβ
dα ′′

Aw = w , where
0
′
Mb ν−1 Z ϱ D(A) = {w ∈ Y : w, w are absolutely continuous ,
≤ (δ 1 ∥x(s) − z(s)∥ + δ 2 γ∥x(α)
Γ(ν) 0
− z(α)∥)dα.
Hence, w(0) = w(1) = 0}
and
Mb ν−1
∥y(ϱ) − z(ϱ)∥ ≤ (δ 1 + δ 2 γ) +∞
Γ(ν) X 2
Aw = k < w, w k > w k , w ∈ D(A),
ϱ
Z
∥y(α) − z(α)∥dα. k=1
0 1/2
where w k (α) = (2/π) sin kα, k = 1, 2, 3 · · · is
Using “Grownwall’s inequality,” we get the orthogonal set of eigenfunctions of A. Here,
2
< w, w k > stands for the L inner product. Addi-
∥y(ϱ) − z(ϱ)∥ = 0.
tionally, A generates a strongly continuous semi-
Therefore, y(ϱ) = z(ϱ) for all ϱ ∈ I. This proves group {S(ϱ), ϱ ≥ 0} in Y presented as
T-controllability of (16)-(17). +∞
X
Case-II: If the control appears nonlinearly S(ϱ)w = exp(−k ϱ) < w, w k > w k , w ∈ Y,
2
in (11)-(12):
k=1
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