# Asymptotic Stability of Caputo Type Fractional Neutral Dynamical Systems with Multiple Discrete Delays

Authors: Hai Zhang; Daiyong Wu; Jinde Cao
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We discuss the delay-independent asymptotic stability of Caputo type
fractional-order neutral differential systems with multiple discrete delays. Based on the algebraic
approach and matrix theory, the sufficient conditions are derived to ensure the asymptotic stability
for all time-delay parameters. By applying the stability criteria, one can avoid solving the roots
of transcendental equations. The results obtained are computationally flexible and convenient.
Moreover, an example is provided to illustrate the effectiveness and applicability of the proposed
theoretical results.


---

## Body

## 1. Introduction

Fractional calculus is regarded as a generalization of the classical integer-order calculus to arbitrary order. Since the fractional-order derivative has nonlocal property and weakly singular kernels, it provides an excellent tool for the description of memory and hereditary properties of dynamical processes. Recently, it has gained increasing interests from researchers in various areas and has become one of the central subjects [1–12]. For more details on fractional calculus theory, one can see the monographs of Miller and Ross [1], Podlubny [2], Kilbas et al. [3], and Diethelm [4]. Lakshmikantham et al. [5] and Baleanu et al. [6] have elaborated the theory of fractional-order dynamics systems.Stability is an important performance metric for dynamic systems. Meanwhile, time delay has an important effect on the stability and performance of dynamic systems. In the past few decades, numerous results on the stability problem of integer-order delay differential systems have been obtained (see [13–20] and references therein). Recently, there are some results on the stability of fractional-order differential systems [21–33]. For example, Matignon [23, 24] discussed the asymptotic stability of linear fractional-order autonomous systems. In terms of comparison principle [25, 26] and Lyapunov direct method [27], Li et al. [25, 27] obtained the Mittag-Leffler stability theorems of fractional-order systems. Linear matrix inequality (LMI) method [28] and variational Lyapunov method [29] were also used to investigate the stability of fractional-order systems. Wang et al. [30] investigated Hyers-Ulam-Rassias stability of a certain fractional differential equation by means of the fixed point theorem. Moreover, Rivero et al. [31], Li and Zhang [32], and Choi et al. [33] summarized and reviewed the developments and advances in stability of fractional-order dynamical systems, respectively.It is worth pointing out that the notable contributions have been made to the stability of fractional-order delay differential systems [34–44]. Many methods have been applied to discuss various types of stability problems for fractional-order delay dynamical systems, such as Gronwall integral inequality method [34–36], final-value theorem of Laplace transform [37], Lyapunov functional method [38, 39], analytical and numerical methods [40], fixed point theorems [41–43], and semigroup theory [42].In this paper, motivated by the aforementioned works, we are devoted to discussing the delay-independent asymptotic stability for linear Caputo fractional neutral differential difference system with multiple discrete delays as follows:(1)
C
D
α
[
x
(
t
)
-
∑
i
=
1
m
C
i
x
(
t
-
τ
i
)
]
=
A
x
(
t
)
+
∑
i
=
1
m
B
i
x
(
t
-
τ
i
)
,
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
m
m
t
≥
0
,
x
(
t
)
=
φ
(
t
)
,
t
∈
[
-
τ
,
0
]
,
whereC
D
α
x
(
t
)denotes anαorder Caputo fractional derivative ofx
(
t
),0
<
α
<
1, A
,
B
i
,
C
iaren
×
nconstant matrices,τ
i
(
i
=
1,2
,
…
,
m
)are real constants with0
<
τ
1
≤
τ
2
≤
⋯
≤
τ
m
=
τ, the initial functionφ
∈
C
1
(
[
-
τ
,
0
]
,
R
n
), andC
1
(
[
-
τ
,
0
]
,
R
n
)denotes space of continuously differentiable functions mapping the interval[
-
τ
,
0
]intoR
n.Compared to integer-order differential systems, the research on the stability of fractional dynamical systems is still at the stage of exploiting and developing. Different from the methods in [34–43], we apply the algebraic approach and matrix theory to establish the delay-independent asymptotic stability criteria for system (1), which do not contain information on delays. We establish the sufficient conditions which ensure that all the roots of characteristic equation lie in open left-half complex plane and are uniformly bounded away from the imaginary axis. At the same time, by applying these stability criteria, one can avoid solving the roots of the transcendental equations. The results obtained are computationally flexible and efficient.The rest of this paper is organized as follows. In Section2, we present some definitions, notations, and lemmas related to the main results. In Section 3, the sufficient conditions of the delay-independent asymptotic stability for system (1) are derived based on the algebraic approach and matrix theory. In Section 4, an example is provided to illustrate the effectiveness and applicability of the proposed criteria. Finally, some concluding remarks are drawn in Section 5.


## 2. Preliminaries

In this section, we present some definitions of fractional calculus (see [1–4]), notations, and lemmas used in the paper.For the sake of convenience, some notations are introduced firstly. Throughout this paper,det
⁡
(
A
) represents the determinant of matrix A, σ
(
A
) denotes the spectrum of matrixA,ρ
[
A
]represents the spectral radius of matrixA, and arg
(
σ
(
A
)
) stands for the principal argument ofσ
(
A
)defined on(
-
π
,
π
].(a)
Riemann-Liouville’s fractional integral of orderq
>
0 for a functionf
:
R
+
→
R
nis given by
(2)
D
-
q
f
(
t
)
=
1
Γ
(
q
)
∫
0
t
‍
(
t
-
s
)
q
-
1
f
(
s
)
d
s
,
t
>
0
,
whereΓ
(
·
)is Euler’s gamma function.
(b)
Riemann-Liouville’s fractional derivative of orderq for a functionf
:
R
+
→
R
nis given by
(3)
RL
D
q
f
(
t
)
=
1
Γ
(
m
-
q
)
d
m
d
t
m
∫
0
t
‍
(
t
-
s
)
m
-
q
-
1
f
(
s
)
d
s
,
where0
≤
m
-
1
≤
q
<
m
,
m
∈
Z
+.
(c)
Caputo’s fractional derivative of orderq for a functionf
:
R
+
→
R
nis defined as
(4)
C
D
q
f
(
t
)
=
1
Γ
(
m
-
q
)
∫
0
t
‍
(
t
-
s
)
m
-
q
-
1
f
(
m
)
(
s
)
d
s
,
where0
≤
m
-
1
≤
q
<
m, m
∈
Z
+. Here,C
D
qis still written asD
q.
(d)
The Laplace transform of a functionf
(
t
)is defined as
(5)
F
(
s
)
=
£
[
f
(
t
)
]
=
∫
0
+
∞
‍
e
-
s
t
f
(
t
)
d
t
,
s
∈
C
,
where C denotes the complex plane andf
(
t
)isn-dimensional vector-valued function. Form
-
1
≤
q
<
m, it follows from [1–4] that
(6)
£
[
D
q
f
(
t
)
]
=
s
q
£
[
f
(
t
)
]
-
∑
k
=
0
m
-
1
‍
s
q
-
k
-
1
f
(
k
)
(
0
)
.
The following Mittag-Leffler function plays an important role in the study on fractional-order differential systems, which is considered as a natural generalization of the exponential function.
(e)
The Mittag-Leffler function in two parameters is defined as(7)
E
q
,
β
(
z
)
=
∑
k
=
0
+
∞
‍
z
k
Γ
(
k
q
+
β
)
,
q
>
0
,
β
>
0
,
z
∈
C
.
In particular, forβ
=
1, the Mittag-Leffler function in one parameter is given by
(8)
E
q
(
z
)
:
=
E
q
,
1
(
z
)
=
∑
k
=
0
+
∞
‍
z
k
Γ
(
1
+
k
q
)
,
q
>
0
,
z
∈
C
.Applying the method of steps [34], we obtain the following lemma which generalizes well-known results of integer-order delay differential systems [13] to fractional-order neutral differential systems.Lemma 1.
For system (1), there exists a unique continuous solution on[
0
,
+
∞
).Proof.
For system (1), on the interval[
-
τ
,
0
],x
(
t
)
=
φ
(
t
). Thus, whent
∈
[
0
,
τ
1
], system (1) is given by
(9)
D
α
x
(
t
)
=
A
x
(
t
)
+
∑
i
=
1
m
‍
B
i
φ
(
t
-
τ
i
)
+
D
α
[
∑
i
=
1
m
‍
C
i
φ
(
t
-
τ
i
)
]
.
Since∑
i
=
1
m
B
i
φ
(
t
-
τ
i
)
+
D
α
[
∑
i
=
1
m
C
i
φ
(
t
-
τ
i
)
]is continuous on[
0
,
τ
1
], from [3], we know that there is a unique continuous solution for system (1) on [
-
τ
,
τ
1
], which is denoted as x
1
(
t
), t
∈
[
-
τ
,
τ
1
]. Furthermore,x
1
(
t
)can be expressed by the following form:
(10)
x
1
(
t
)
=
{
φ
(
t
)
,
t
∈
[
-
τ
,
0
]
,
E
α
(
A
t
α
)
φ
(
0
)
+
∫
0
t
‍
(
t
-
θ
)
α
-
1
E
α
,
α
[
A
(
t
-
θ
)
α
]
m
×
{
∑
i
=
1
m
‍
B
i
φ
(
θ
-
τ
i
)
+
D
α
[
∑
i
=
1
m
‍
C
i
φ
(
θ
-
τ
i
)
]
}
d
θ
,
t
∈
[
0
,
τ
1
]
.
Fort
∈
[
τ
1
,
2
τ
1
], system (1) is given by
(11)
D
α
x
(
t
)
=
A
x
(
t
)
+
∑
i
=
1
m
‍
B
i
x
1
(
t
-
τ
i
)
+
D
α
[
∑
i
=
1
m
‍
C
i
x
1
(
t
-
τ
i
)
]
.
Similarly, since∑
i
=
1
m
B
i
x
1
(
t
-
τ
i
)
+
D
α
[
∑
i
=
1
m
C
i
x
1
(
t
-
τ
i
)
]is continuous on[
-
τ
,
2
τ
1
], we obtain thatx
2
(
t
)is a unique continuous solution of system (1) on[
-
τ
,
2
τ
1
] and
(12)
x
2
(
t
)
=
{
x
1
(
t
)
,
t
∈
[
-
τ
,
τ
1
]
,
E
α
(
A
t
α
)
φ
(
0
)
+
∫
0
t
‍
(
t
-
θ
)
α
-
1
E
α
,
α
[
A
(
t
-
θ
)
α
]
mm
×
{
∑
i
=
1
m
‍
B
i
x
1
(
θ
-
τ
i
)
+
D
α
[
∑
i
=
1
m
‍
C
i
x
1
(
θ
-
τ
i
)
]
}
d
θ
,
t
∈
[
τ
1
,
2
τ
1
]
.
Assume that system (1) has a unique solutionx
k
(
t
)on[
(
k
-
1
)
τ
1
,
k
τ
1
]. Fort
∈
[
k
τ
1
,
(
k
+
1
)
τ
1
], system (1) is given by
(13)
D
α
x
(
t
)
=
A
x
(
t
)
+
∑
i
=
1
m
‍
B
i
x
k
(
t
-
τ
i
)
+
D
α
[
∑
i
=
1
m
‍
C
i
x
k
(
t
-
τ
i
)
]
.
Similarly, since∑
i
=
1
m
B
i
x
k
(
t
-
τ
i
)
+
D
α
[
∑
i
=
1
m
C
i
x
k
(
t
-
τ
i
)
]is continuous on[
k
τ
,
(
k
+
1
)
τ
], we obtain thatx
k
+
1
(
t
)is a unique continuous solution of system (1) on[
k
τ
1
,
(
k
+
1
)
τ
1
] and
(14)
x
k
+
1
(
t
)
=
{
x
k
(
t
)
,
t
∈
[
-
τ
,
k
τ
1
]
,
E
α
(
A
t
α
)
φ
(
0
)
+
∫
0
t
‍
(
t
-
θ
)
α
-
1
E
α
,
α
[
A
(
t
-
θ
)
α
]
m
×
{
∑
i
=
1
m
‍
B
i
x
k
(
θ
-
τ
i
)
+
D
α
[
∑
i
=
1
m
‍
C
i
x
k
(
θ
-
τ
i
)
]
}
d
θ
,
m
m
m
m
t
∈
[
k
τ
1
,
(
k
+
1
)
τ
1
]
.
According to the mathematical induction, we know that system (1) has a unique continuous solution on[
0
,
k
τ
1
]
,
k
=
1,2
,
….
Now, for anyT
>
0, we assert that system (1) has a unique continuous solution on[
0
,
T
]. In fact, three cases are discussed as follows.
Case  1. When(
k
+
1
)
τ
1
=
T, we know that the assertion is true.
Case  2. When0
<
T
-
(
k
+
1
)
τ
1
<
τ
1, we only need to prove that system (1) has a unique continuous solution on[
τ
1
,
T
]. Fort
∈
[
τ
1
,
T
], we denotet
1
=
t
-
τ
1
∈
[
0
,
T
-
τ
1
]
⊂
[
0
,
(
k
+
1
)
τ
1
], and we can use the similar proof to obtain the conclusion.
Case  3. WhenT
-
(
k
+
1
)
τ
1
>
τ
1, we can repeat the above process until the condition of Case 2 is satisfied.
Note thatTis an arbitrary positive real number; then, we know that system (1) has a unique continuous solution on[
0
,
+
∞
)
.Therefore, the proof is completed.Remark 2.
Lemma1 ensures the existence and uniqueness of solution for system (1) on[
0
,
+
∞
). Evidently, Caputo’s fractional derivative of a constant is equal to zero; then,x
(
t
)
≡
0is the zero solution of system (1).Definition 3.
The zero solutionx
(
t
)
≡
0of system (1) is called delay-independent globally asymptotically stable if, for any initial functionφ
(
·
)
∈
C
1
(
[
-
τ
,
0
]
,
R
n
), its analytic solutionx
(
t
)satisfieslim
⁡
t
→
+
∞
x
(
t
)
=
0for all the time delays0
<
τ
1
≤
τ
2
≤
⋯
≤
τ
m.
Next, we discuss the characteristic equation and delay-independent globally asymptotic stability of system (1).
From [1–4], the Laplace transform of Caputo fractional-order derivativeD
α
f
(
t
)is given as follows:
(15)
£
[
D
α
f
(
t
)
]
=
s
α
£
[
f
(
t
)
]
-
s
α
-
1
f
(
0
)
,
0
<
α
<
1
.
Applying the Laplace transform on both sides of system (1) yields
(16)
s
α
£
[
x
(
t
)
-
∑
i
=
1
m
‍
C
i
x
(
t
-
τ
i
)
]
-
s
α
-
1
φ
(
0
)
+
s
α
-
1
∑
i
=
1
m
‍
C
i
φ
(
-
τ
i
)
=
A
£
[
x
(
t
)
]
+
∑
i
=
1
m
‍
B
i
£
[
x
(
t
-
τ
i
)
]
.
Note that
(17)
£
[
x
(
t
-
τ
i
)
]
=
e
-
s
τ
i
£
[
x
(
t
)
]
+
e
-
s
τ
i
∫
-
τ
i
0
‍
e
-
s
t
φ
(
t
)
d
t
,
mmmmmmmmmmi
i
=
1,2
,
…
,
m
;
thus, we obtain
(18)
Δ
(
s
,
τ
i
)
£
[
x
(
t
)
]
=
s
α
-
1
φ
(
0
)
-
s
α
-
1
∑
i
=
1
m
‍
C
i
φ
(
-
τ
i
)
+
∑
i
=
1
m
‍
B
i
e
-
s
τ
i
∫
-
τ
i
0
‍
e
-
s
t
φ
(
t
)
d
t
+
s
α
∑
i
=
1
m
‍
C
i
e
-
s
τ
i
∫
-
τ
i
0
‍
e
-
s
t
φ
(
t
)
d
t
,
where
(19)
Δ
(
s
,
τ
i
)
=
[
s
α
I
-
A
-
∑
i
=
1
m
‍
B
i
e
-
s
τ
i
-
s
α
∑
i
=
1
m
‍
C
i
e
-
s
τ
i
]
is the characteristic matrix of system (1). Multiplyingson both sides of (18) yields
(20)
Δ
(
s
,
τ
i
)
{
s
£
[
x
(
t
)
]
}
=
s
{
s
α
-
1
φ
(
0
)
-
s
α
-
1
∑
i
=
1
m
‍
C
i
φ
(
-
τ
i
)
mmmmmmmmmmm
+
∑
i
=
1
m
‍
B
i
e
-
s
τ
i
∫
-
τ
i
0
‍
e
-
s
t
φ
(
t
)
d
t
mmmmmmmmmmm
+
s
α
∑
i
=
1
m
‍
C
i
e
-
s
τ
i
∫
-
τ
i
0
‍
e
-
s
t
φ
(
t
)
d
t
}
.
By means of the final-value theorem of Laplace transform [45] and Definition 3, if all the roots of characteristic equationdet
⁡
[
Δ
(
s
,
τ
i
)
]
=
0lie in open left-half complex plane and are uniformly bounded away from the imaginary axis, then the zero solution of system (1) is delay-independent globally asymptotically stable.
Therefore, we immediately have the following conclusion.Lemma 4.
If all the roots of characteristic equation(21)
det
⁡
[
s
α
I
-
A
-
∑
i
=
1
m
‍
B
i
e
-
s
τ
i
-
s
α
∑
i
=
1
m
‍
C
i
e
-
s
τ
i
]
=
0
lie in open left-half complex plane and are uniformly bounded away from the imaginary axis, then the zero solution of system (1) is delay-independent globally asymptotically stable.Remark 5.
As we know, whenα
=
1andτ
1
=
τ
2
=
⋯
=
τ
m
=
0, the characteristic equation
(22)
det
⁡
[
λ
I
-
A
-
∑
i
=
1
m
‍
(
B
i
+
λ
C
i
)
]
=
0
is an algebraic equation ofλ, and (22) only hasnroots distributed in the complex plane. However, the characteristic equationdet
⁡
[
Δ
(
s
,
τ
i
)
]
=
0has countably infinite roots withα
=
1and someτ
i
>
0(see [13]). For0
<
α
<
1andτ
i
>
0
(
i
=
1,2
,
…
,
m
)
, it is very difficult to solve the roots of the transcendental equation (21) in practice. Based on these considerations, we are devoted to establishing the algebraic stability criteria of system (1) in the next section.


## 3. Stability Criteria for System (1)

In this section, we derive the sufficient conditions of delay-independent globally asymptotic stability for system (1). Applying the algebraic method, we investigate the distribution of roots for equationdet
⁡
[
Δ
(
s
,
τ
i
)
]
=
0in any neighborhood of the infinity and find a positive numberδ
>
0such that any characteristic rootssatisfies R
e
(
s
)
<
-
δ
<
0
, where R
e
(
s
)represents the real part of the complex numbers.Theorem 6.
The zero solutionx
(
t
)
≡
0of system (1) is delay-independent globally asymptotically stable if the following conditions are satisfied:
(23)
(
H
1
)
|
arg
(
σ
(
A
)
)
|
>
α
π
2
,
m
(
H
2
)
ρ
[
∑
i
=
1
m
‍
C
i
]
<
1
,
m
(
H
3
)
sup
⁡
R
e
(
s
)
≥
0
ρ
[
(
s
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
s
α
C
i
)
]
<
1
.Proof.
Note that|
arg
(
σ
(
A
)
)
|
>
α
π
/
2; then, all the roots of equationdet
⁡
(
λ
I
-
A
)
=
0satisfy|
arg
(
λ
)
|
>
α
π
/
2. Lets
α
=
λ; then, all the roots of equationdet
⁡
(
s
α
I
-
A
)
=
0satisfy|
arg
(
s
)
|
>
π
/
2; that is, R
e
(
s
)
<
0. Therefore, matrixAis invertible and(
s
α
I
-
A
)
-
1is well defined when|
arg
(
σ
(
A
)
)
|
>
α
π
/
2and R
e
(
s
)
≥
0.
ForR
e
(
s
)
≥
0, it follows from the characteristic polynomial of system (1) that
(24)
det
⁡
Δ
(
s
,
τ
i
)
=
det
⁡
[
s
α
I
-
A
-
∑
i
=
1
m
‍
B
i
e
-
s
τ
i
-
s
α
∑
i
=
1
m
‍
C
i
e
-
s
τ
i
]
=
det
⁡
[
s
α
I
-
A
]
det
⁡
[
∑
i
=
1
m
‍
I
-
(
s
α
I
-
A
)
-
1
m
m
m
m
m
m
m
m
m
m
×
∑
i
=
1
m
‍
(
B
i
+
s
α
C
i
)
e
-
s
τ
i
]
=
det
⁡
[
s
α
I
-
A
]
×
∏
j
=
1
n
‍
(
1
-
λ
j
[
(
s
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
s
α
C
i
)
e
-
s
τ
i
]
)
.
Then, we have
(25)
|
λ
j
[
(
s
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
s
α
C
i
)
e
-
s
τ
i
]
|
≤
ρ
[
(
s
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
s
α
C
i
)
]
|
e
-
s
τ
|
<
1
,
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
j
=
1,2
,
…
,
n
.
Combining (24) and (25), we havedet
⁡
Δ
(
s
,
τ
i
)
≠
0for R
e
(
s
)
≥
0andτ
i
>
0
,
i
=
1,2
,
…
,
m; that is, if, conditions(
H
1
)and(
H
3
)are satisfied, then the characteristic equation (21) implies that R
e
(
s
)
<
0.
Suppose that there exists a sequence of roots{
s
n
}of the characteristic equation (21) whose real parts are not uniformly bounded away from zero; that is, R
e
(
s
n
)
<
0and R
e
(
s
n
)
→
0asn
→
+
∞. Note that any eigenvalueλ
j
[
(
s
α
I
-
A
)
-
1
∑
i
=
1
m
(
B
i
+
s
α
C
i
)
]is a continuous function ofsfor R
e
(
s
)
≥
0; then, it follows from(
H
2
)that
(26)
ρ
[
(
s
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
s
α
C
i
)
]
⟶
ρ
[
∑
i
=
1
m
‍
C
i
]
<
1
,
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
n
m
s
⟶
+
∞
.
Hence,|
λ
j
[
(
s
α
I
-
A
)
-
1
∑
i
=
1
m
(
B
i
+
s
α
C
i
)
]
|reach the maximum value for R
e
(
s
)
≥
0. From condition(
H
3
), there exists a positive constantεsuch that
(27)
sup
⁡
R
e
(
s
)
≥
0
ρ
[
(
s
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
s
α
C
i
)
]
=
sup
⁡
R
e
(
s
)
≥
0
max
⁡
1
≤
j
≤
n
|
λ
j
[
(
s
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
s
α
C
i
)
]
|
=
1
-
ε
.
Then, equality (27) implies that
(28)
sup
⁡
R
e
(
ζ
)
=
0
ρ
[
(
ζ
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
ζ
α
C
i
)
]
≤
1
-
ε
.
When the positive integernis large enough, there exist a positive constantε
*
(
0
<
ε
*
<
ε
)and a characteristic roots
nsuch that|
R
e
(
s
n
)
|is sufficiently small, R
e
(
s
n
)
<
0and
(29)
|
max
⁡
1
≤
j
≤
n
λ
j
[
(
s
n
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
s
n
α
C
i
)
]
-
sup
⁡
R
e
(
ζ
)
=
0
ρ
[
(
ζ
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
ζ
α
C
i
)
]
|
<
ε
*
.
For R
e
(
ζ
)
=
0, from (28) and (29), we have
(30)
|
λ
j
[
(
s
n
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
s
n
α
C
i
)
]
|
≤
sup
⁡
R
e
(
ζ
)
=
0
ρ
[
(
ζ
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
ζ
α
C
i
)
]
+
ε
*
≤
1
-
ε
+
ε
*
<
1
,
j
=
1,2
,
…
,
n
.
Choosingnlarge enough yields
(31)
|
λ
j
[
(
s
n
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
s
n
α
C
i
)
e
-
s
τ
i
]
|
<
1
,
j
=
1,2
,
…
,
n
.
Therefore, for R
e
(
s
n
)
<
0and R
e
(
s
n
)
→
0asn
→
+
∞, one can obtain
(32)
det
⁡
Δ
(
s
n
,
τ
i
)
=
det
⁡
[
s
n
α
I
-
A
-
∑
i
=
1
m
‍
B
i
e
-
s
n
τ
i
mmmn
-
s
n
α
∑
i
=
1
m
‍
C
i
e
-
s
n
τ
i
]
≠
0
,
which contradicts the assumption that{
s
n
}is a sequence of roots of the characteristic equation (21). In view of Lemma 4, the proof is completed.Theorem 7.
The zero solutionx
(
t
)
≡
0of system (1) is delay-independent globally asymptotically stable if the following conditions are satisfied:
(33)
(
H
1
)
|
arg
(
σ
(
A
)
)
|
>
α
π
2
,
0
<
α
<
1
,
(
H
4
)
ρ
[
∑
i
=
1
m
‍
ξ
i
C
i
]
<
1
,
ξ
i
∈
C
,
|
ξ
i
|
≤
1
,
i
=
1,2
,
…
,
m
,
(
H
5
)
ρ
[
(
s
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
s
α
C
i
)
ξ
i
]
<
1
,
R
e
(
s
)
=
0
,
ξ
i
∈
C
,
|
ξ
i
|
=
1
,
i
=
1,2
,
…
,
m
.Proof.
According to condition(
H
1
)and the proof of Theorem 6, we know that matrixAis invertible and(
s
α
I
-
A
)
-
1is well defined when|
arg
(
σ
(
A
)
)
|
>
α
π
/
2and R
e
(
s
)
=
0.
On the one hand, any eigenvalueλ
[
(
s
α
I
-
A
)
-
1
(
∑
i
=
1
m
ξ
i
B
i
+
s
α
∑
i
=
1
m
ξ
i
C
i
)
]is an algebraic function of(
ξ
1
,
ξ
2
,
…
,
ξ
m
)with|
ξ
i
|
<
1and is continuous with|
ξ
i
|
≤
1
,
i
=
1,2
,
…
,
m
.An application of the maximum modulus principle yields that
(34)
ρ
[
(
s
α
I
-
A
)
-
1
(
∑
i
=
1
m
‍
ξ
i
B
i
+
s
α
∑
i
=
1
m
‍
ξ
i
C
i
)
]
<
1
,
aaaaaaaaaaai
R
e
(
s
)
=
0
,
ξ
i
∈
C
,
|
ξ
i
|
=
1
,
is equivalent to
(35)
ρ
[
(
s
α
I
-
A
)
-
1
(
∑
i
=
1
m
‍
ξ
i
B
i
+
s
α
∑
i
=
1
m
‍
ξ
i
C
i
)
]
<
1
,
aaaaaaaaaaaa
R
e
(
s
)
=
0
,
ξ
i
∈
C
,
|
ξ
i
|
≤
1
.
In view of the expression of(
s
α
I
-
A
)
-
1
(
∑
i
=
1
m
ξ
i
B
i
+
s
α
∑
i
=
1
m
ξ
i
C
i
), as R
e
(
s
)
=
0ands
→
∞, one can obtain
(36)
ρ
[
(
s
α
I
-
A
)
-
1
(
∑
i
=
1
m
‍
ξ
i
B
i
+
s
α
∑
i
=
1
m
‍
ξ
i
C
i
)
]
⟶
ρ
[
∑
i
=
1
m
‍
ξ
i
C
i
]
<
1
,
mmmmmmmmmmmmmmmmmnmm
∀
ξ
i
∈
C
,
|
ξ
i
|
≤
1
.
It follows from (36) that
(37)
sup
⁡
R
e
(
s
)
=
0
ρ
[
(
s
α
I
-
A
)
-
1
(
∑
i
=
1
m
‍
ξ
i
B
i
+
s
α
∑
i
=
1
m
‍
ξ
i
C
i
)
]
≤
1
,
mmmmmmmmmmmmmmm
∀
ξ
i
∈
C
,
|
ξ
i
|
≤
1
.
Applying the maximum modulus principle on the unbounded region{
s
|
s
∈
C
,
R
e
(
s
)
≥
0
}, then (35) implies that
(38)
ρ
[
(
s
α
I
-
A
)
-
1
(
∑
i
=
1
m
‍
ξ
i
B
i
+
s
α
∑
i
=
1
m
‍
ξ
i
C
i
)
]
<
1
,
aaaaaaaaaaaa
R
e
(
s
)
≥
0
,
ξ
i
∈
C
,
|
ξ
i
|
≤
1
.
On the other hand, suppose that there exists somes
0
∈
{
s
|
s
∈
C
,
R
e
(
s
)
≥
0
}such that
(39)
det
⁡
[
s
0
α
I
-
A
-
∑
i
=
1
m
‍
B
i
e
-
s
0
τ
i
-
s
0
α
∑
i
=
1
m
‍
C
i
e
-
s
0
τ
i
]
=
0
,
m
m
m
m
m
m
m
m
n
m
s
0
∈
{
s
|
s
∈
C
,
R
e
(
s
)
≥
0
}
.
Chooseξ
i
0
=
e
-
s
0
τ
isuch that|
ξ
i
0
|
≤
1
,
i
=
1,2
,
…
,
m. It follows from (39) that
(40)
1
∈
σ
[
(
s
0
α
I
-
A
)
-
1
(
∑
i
=
1
m
‍
B
i
e
-
s
0
τ
i
+
s
0
α
∑
i
=
1
m
‍
C
i
e
-
s
0
τ
i
)
]
,
mmmmmmmmmmi
s
0
∈
{
s
|
s
∈
C
,
R
e
(
s
)
≥
0
}
,
which contradicts inequality (38). Therefore, if conditions(
H
1
), (
H
4
), and(
H
5
)are satisfied, then the characteristic equation (21) implies that R
e
(
s
)
<
0.
The rest of the proof is similar to that of Theorem6; then, the conclusion holds.According to the proof of Theorem7, we have the result as follows.Corollary 8.
The zero solutionx
(
t
)
≡
0of system (1) is delay-independent globally asymptotically stable if the following conditions are satisfied:
(41)
(
H
1
)
|
arg
(
σ
(
A
)
)
|
>
α
π
2
,
0
<
α
<
1
,
(
H
4
)
ρ
[
∑
i
=
1
m
‍
ξ
i
C
i
]
<
1
,
ξ
i
∈
C
,
|
ξ
i
|
≤
1
,
i
=
1,2
,
…
,
m
,
(
H
6
)
ρ
[
(
s
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
s
α
C
i
)
ξ
i
]
<
1
,
R
e
(
s
)
≥
0
,
ξ
i
∈
C
,
|
ξ
i
|
≤
1
,
i
=
1,2
,
…
,
m
.Assume that|
arg
(
σ
(
A
)
)
|
>
α
π
/
2
,
0
<
α
<
1. Define the following matrices:
(42)
L
=
(
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
C
i
)
,
M
=
(
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
-
C
i
)
,
N
=
(
I
-
A
)
-
1
(
I
+
A
)
.
Let
(43)
s
α
=
1
-
z
1
+
z
,
R
e
(
s
)
≥
0
,
0
<
α
<
1
;
then, we have|
z
|
≤
1.Theorem 9.
The zero solutionx
(
t
)
≡
0of system (1) is delay-independent globally asymptotically stable if the following conditions are satisfied:
(44)
(
H
1
)
|
arg
(
σ
(
A
)
)
|
>
α
π
2
,
0
<
α
<
1
,
(
H
2
)
ρ
[
∑
i
=
1
m
‍
C
i
]
<
1
,
(
H
7
)
sup
⁡
|
z
|
≤
1
ρ
[
(
I
-
z
N
)
-
1
(
L
+
z
M
)
]
<
1
.Proof.
From Theorem6, we only need to prove that the following equality holds:
(45)
(
s
α
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
s
α
C
i
)
=
(
I
-
z
N
)
-
1
(
L
+
z
M
)
,
aaaaaaaaaaaaaaaaaaaaaaaaaii
R
e
(
s
)
≥
0
,
|
z
|
≤
1
.
In fact, it is easy to obtain(46)
I
-
z
N
=
[
I
-
1
-
s
α
1
+
s
α
(
I
-
A
)
-
1
(
I
+
A
)
]
=
[
(
1
+
s
α
)
I
-
(
1
-
s
α
)
(
I
-
A
)
-
1
(
I
+
A
)
]
(
1
+
s
α
)
-
1
=
(
I
-
A
)
-
1
[
(
I
-
A
)
(
1
+
s
α
)
-
(
1
-
s
α
)
(
I
+
A
)
]
×
(
1
+
s
α
)
-
1
=
2
(
I
-
A
)
-
1
(
s
α
I
-
A
)
(
1
+
s
α
)
-
1
.
For R
e
(
s
)
≥
0, it follows from(
H
1
)and (43) that|
z
|
≤
1; then, matrix(
I
-
z
N
)is invertible. Hence,
(47)
L
+
z
M
=
[
(
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
C
i
)
+
1
-
s
α
1
+
s
α
(
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
-
C
i
)
]
=
(
I
-
A
)
-
1
[
∑
i
=
1
m
‍
(
B
i
+
C
i
)
+
1
-
s
α
1
+
s
α
∑
i
=
1
m
‍
(
B
i
-
C
i
)
]
=
(
I
-
A
)
-
1
[
∑
i
=
1
m
‍
(
B
i
+
C
i
)
(
1
+
s
α
)
mmmmmnn
+
(
1
-
s
α
)
∑
i
=
1
m
‍
(
B
i
-
C
i
)
]
(
1
+
s
α
)
-
1
=
2
(
I
-
A
)
-
1
∑
i
=
1
m
‍
(
B
i
+
s
α
C
i
)
(
1
+
s
α
)
-
1
.
Combining (46) and (47) yields that (45) holds. Therefore, we complete the proof.Next, the asymptotic stability criteria for two special cases of system (1) are presented.(i)
WhenC
i
=
0
,
i
=
1,2
,
…
,
m, system (1) reduces to Caputo fractional-order linear retarded type differential difference systems with multiple delays as follows:
(48)
D
α
x
(
t
)
=
A
x
(
t
)
+
∑
i
=
1
m
‍
B
i
x
(
t
-
τ
i
)
,
t
≥
0
,
x
(
t
)
=
φ
(
t
)
,
t
∈
[
-
τ
,
0
]
,
whereD
α
x
(
t
)denotes anαorder Caputo fractional derivative ofx
(
t
),0
<
α
<
1,A
,
B
iaren
×
nconstant matrices,τ
i
(
i
=
1,2
,
…
,
m
)are real constants with0
<
τ
1
≤
τ
2
≤
⋯
≤
τ
m
=
τ,φ
∈
C
(
[
-
τ
,
0
]
,
R
n
), andC
(
[
-
τ
,
0
]
,
R
n
)denotes space of continuous functions mapping the interval[
-
τ
,
0
]intoR
n.Similar to Lemma1, if the initial functionφ
∈
C
(
[
-
τ
,
0
]
,
R
n
), then there exists a unique continuous solution for system (48) on[
0
,
+
∞
).Corollary 10.
The zero solutionx
(
t
)
≡
0of system (48) is delay-independent globally asymptotically stable if the following conditions are satisfied:
(49)
(
H
1
)
|
arg
(
σ
(
A
)
)
|
>
α
π
2
,
0
<
α
<
1
,
(
H
8
)
ρ
[
(
s
α
I
-
A
)
-
1
∑
i
=
1
m
‍
B
i
ξ
i
]
<
1
,
mmi
R
e
(
s
)
=
0
,
ξ
i
∈
C
,
|
ξ
i
|
=
1
.(ii)
Whenτ
i
=
0
,
i
=
1,2
,
…
,
m, system (1) reduces to Caputo fractional-order linear autonomous differential systems:
(50)
[
I
-
∑
i
=
1
m
‍
C
i
]
D
α
x
(
t
)
=
[
A
+
∑
i
=
1
m
‍
B
i
]
x
(
t
)
,
t
≥
0
,
x
(
0
)
=
x
0
,
whereD
α
x
(
t
)denotes anαorder Caputo fractional derivative ofx
(
t
),0
<
α
<
1, and A
,
B
i
,
C
iaren
×
nconstant matrices.An application of the results in [23, 24] yields the following conclusion.Corollary 11.
The zero solutionx
(
t
)
≡
0of system (50) is globally asymptotically stable if the following conditions are satisfied:
(51)
(
H
9
)
det
⁡
[
I
-
∑
i
=
1
m
‍
C
i
]
≠
0
,
(
H
10
)
|
arg
(
σ
(
G
)
)
|
>
α
π
2
,
m
m
m
m
m
G
=
(
I
-
∑
i
=
1
m
‍
C
i
)
-
1
(
A
+
∑
i
=
1
m
‍
B
i
)
.Remark 12.
Whendet
⁡
[
I
-
∑
i
=
1
m
C
i
]
=
0, system (50) reduces to linear fractional singular differential system. The stability analysis of linear fractional singular (delay) differential systems will become our future investigative works.


## 4. An Illustrative Example

The following example is presented to illustrate the effectiveness and applicability of the proposed stability criteria.Example 1.
Consider system (1) with
(52)
A
=
[
-
4
2
4
-
10
]
,
B
1
=
[
0
1
-
2
3
]
,
B
2
=
[
3
-
1
2
0
]
,
C
1
=
[
2
3
-
1
6
1
3
-
1
6
]
,
C
2
=
[
-
1
6
1
6
-
1
3
2
3
]
,
α
=
1
2
,
τ
1
=
1
,
τ
2
=
2
,
τ
=
2
.
The initial function is given byφ
(
t
)
=
t, t
∈
[
-
2,0
]. Let
(53)
Θ
=
(
s
α
I
-
A
)
-
1
(
ξ
1
B
1
+
ξ
2
B
2
+
s
α
ξ
1
C
1
+
s
α
ξ
2
C
2
)
,
mmmmmmmmmmii
R
e
(
s
)
=
0
,
|
ξ
1
|
=
|
ξ
2
|
=
1
.
By computation, the eigenvalues of matrixΘgive
(54)
λ
1
(
Θ
)
=
(
s
α
+
6
)
-
1
(
ξ
1
+
2
ξ
2
+
1
2
s
α
ξ
1
+
1
3
s
α
ξ
2
)
,
λ
2
(
Θ
)
=
(
s
α
+
8
)
-
1
(
2
ξ
1
+
ξ
2
+
1
3
s
α
ξ
1
+
1
2
s
α
ξ
2
)
.
Note thatσ
(
A
)
=
{
-
6
,
-
8
}; then, we have
(55)
|
arg
(
σ
(
A
)
)
|
=
π
>
α
π
2
,
α
=
1
2
.
It is not difficult to verify that
(56)
ρ
[
ξ
1
C
1
+
ξ
2
C
2
]
<
1
,
|
ξ
1
|
≤
1
,
|
ξ
2
|
≤
1
,
ρ
[
Θ
]
<
1
,
R
e
(
s
)
=
0
,
|
ξ
1
|
=
1
,
|
ξ
2
|
=
1
.
Thus, conditions(
H
1
)
,
(
H
3
), and (
H
4
)are satisfied. Therefore, it follows from Theorem 7 that the zero solutionx
(
t
)
≡
0of system (1) with the coefficient matrices (52) is delay-independent globally asymptotically stable.
In fact, the characteristic equation of system (1) with the coefficient matrices (52) can be expressed as
(57)
det
⁡
[
s
α
I
-
A
-
∑
i
=
1
m
‍
B
i
e
-
s
τ
i
-
s
α
∑
i
=
1
m
‍
C
i
e
-
s
τ
i
]
=
0
.
Obviously, the characteristic equation (57) includes the transcendental terms. It is very difficult that one precisely solves the roots of (57). An application of Theorem 7 yields that the zero solutionx
(
t
)
≡
0of system (1) with (52) is delay-independent globally asymptotically stable.


## 5. Conclusions

In this paper, the delay-independent asymptotic stability of linear fractional-order linear neutral differential systems with multiple discrete delays has been discussed. We have synchronously taken into account the factors of such systems including Caputo’s fractional-order derivative, state delays. The asymptotic stability criteria have been derived based on the algebraic approach and matrix theory, which ensure the asymptotic stability for all time-delay parameters. By applying these stability criteria, one can avoid solving the roots of transcendental equations. The results obtained are computationally flexible and efficient. In fact, the characteristic equation of system (1) with (52) includes the transcendental terms. Generally, it is very difficult that one precisely solves the roots of characteristic equation. In Example 1, we analyse the distribution of characteristic roots when the coefficient matrices satisfy the appropriate conditions. We only need to check the spectrum range under conditions (H
1), (H
4), and (H
5). An application of Theorem 7 yields that the zero solutionx
(
t
)
≡
0of system (1) is delay-independent globally asymptotically stable. Example 1 shows that Theorem 7 is computationally flexible and efficient. The stability analysis of linear fractional singular (delay) differential systems will become our future investigative works.



---

*Source: 138124-2014-07-20.xml*