Page 103 - IJOCTA-15-1

On a robust stability criterion in the Cattaneo–Hristov diffusion equation

Definition 2. The unperturbed solution ¯y j ≡ 0 of according to the expression (9) and (11). This
(8) is called robustly stable with respect to exter- function satisfies (2) and (3). From this last ex-
, if for all ε j > 0 there pression, we observe that
nal perturbations v j ∈ V δ j
exists η j (ε j ) > 0 such that under the condition: ∂y X
∞
≤ δ j < η j (ε j ), any other solution of (8) (x, t) = β j cos(β j x)y j (t), (15)
∂x
∥v j ∥ V δ j
< ε j . j=1
satisfies ∥y j ∥ Y δ j
and
∞
∂y X
We show that the unperturbed solution ¯y j of (8) (x, t) = sin(β j x) ˙y j (t). (16)
is robustly stable with respect to external pertur- ∂t j=1
according to Definition 2, which
bations v j ∈ V δ j We obtain the following result.
follows from the inequalities (11) and (13). In-
P ∞
δ
deed, we observe that for all ε j > 0, the inequality Lemma 1. If j=1 j is convergent with sum
< ε j is obtained after choosing equal to δ, then the norm of the function y de-
∥y j ∥ Y δ j
≤ 2Mδ,
fined in (14) admits the bound: ∥y∥ Y Ω
ε j
δ j ≤ η j (ε j ) = ( ) where
1 κ 2 2
2 max , 1 + R R κ 2
2κ 1 β 2 κ 1 M = max 2 , , 1 +
j 2π κ 1 2πκ 1 κ 1
(17)
ε j κ 2
= , = 1 + .
κ 2 κ 1
2 1 +
κ 1
Proof. We observe from (14) and (11) that
where the last equality is obtained from our as- ∞ ∞ ∞
sumption made in (4) about the thermal conduc- |y(x, t)| ≤ X |y j (t)| ≤ X δ j < R 2 X δ j .
2
tivity and elastic conductivity constants. If δ j > 0 κ 1 β j 2 π κ 1
j=1 j=1 j=1
satisfies this inequality, then every solution y j ob- P ∞
δ
If the series j=1 j is convergent, then y is
on the right-hand
tained by choosing v j ∈ V δ j P ∞
δ
side of the linear fractional differential equation bounded on Ω. We note that if j=1 j is con-
< ε j . Hence vergent, then the Weierstrass M-test implies that
(8) satisfies the inequality ∥y j ∥ Y δ j
the series on the right-hand side of (14) converges
the unperturbed solution ¯y j is robustly stable in
the sense of Definition 2. uniformly to y on Ω.
Using the expression (15) and the inequality (11)
we obtain
∞
∂y X
5. Robust stability criterion (x, t) ≤ β j |y j (t)|

∂x
j=1
In this section we obtain our main result with the ∞ ∞
X δ j R X
help of the robust stability criterion that has been ≤ < δ j .
, which will allow us to es- κ 1 β j πκ 1
obtained for the set Y δ j j=1 j=1
tablish a robust stability criterion in the set Y Ω P ∞ ∂y
If the series j=1 j is convergent, then is
δ
using Definition 1. P ∞ ∂x
δ
bounded on Ω. If the series j=1 j is conver-
Let ε > 0. We define E ε as the family of sequences gent, then the Weierstrass M-test implies that
{ε j } j∈N of positive real numbers for which the fol- the series on the right-hand side of (15) converges
P ∞
lowing equality j=1 j = ε holds. This set in- uniformly to ∂y on Ω.
ε
∂x
cludes the sequences {ε p,j } j∈N defined by ε p,j = From (16) and (12) we finally obtain
ε(1 − p)p j−1 for each p ∈ (0, 1).
∞ ∞
∂y X κ 2 X
We consider a sequence of positive real num- (x, t) ≤ | ˙y j (t)| ≤ 2 1 + δ j .

bers {δ j } j∈N and define the following function ∂t j=1 κ 1 j=1
P ∞
v(x, t) = sin(β j x)v j (t), where v j ∈ V δ j is
j=1 If the series P ∞ δ ∂y is
arbitrary for each j ∈ N. If {y j } j∈N is the se- j=1 j is convergent, then ∂t
∞
P
bounded on Ω. If the series j=1 j is conver-
δ
quence of Fourier coefficients y j that are obtained
gent, then the Weierstrass M-test implies that
as a solution of (8) when the function v j is con-
the series on the right-hand side of (15) converges
sidered on the right-hand side, then the solution ∂y
of (1) is the function uniformly to ∂t on Ω.
∞ As a conclusion from the three inequalities, it fol-
X
y(x, t) = sin(β j x)y j (t), (14) lows that ∥y∥ ≤ 2Mδ. We observe that the
Y Ω
j=1 value of M is a consequence of the assumption
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