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

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.

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: ^𝐶𝐷^𝛼𝑥(𝑡)−^𝑚_𝑖=1𝐶_𝑖𝑥𝑡−𝜏_𝑖=𝐴𝑥(𝑡)+^𝑚_𝑖=1𝐵_𝑖𝑥𝑡−𝜏_𝑖,𝑥(𝑡)(𝑡)[],𝑡≥0,=𝜑,𝑡∈−𝜏,0 (1) where^𝐶𝐷^𝛼𝑥(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<𝜏₁≤𝜏₂≤⋯≤𝜏_𝑚=𝜏, the initial function𝜑∈𝐂¹([−𝜏,0],ℝ^𝑛), and𝐂¹([−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 Section 2, 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 𝐷^−𝑞𝑓(𝑡)=1Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑓(𝑠)𝑑𝑠,𝑡>0, (2) whereΓ(⋅)is Euler’s gamma function.(b)Riemann-Liouville’s fractional derivative of order 𝑞 for a function𝑓∶ℝ⁺→ℝ^𝑛is given by ^RL𝐷^𝑞𝑓(𝑡)=1Γ𝑑𝑚−𝑞𝑑𝑡^𝑚𝑚𝑡₀(𝑡−𝑠)^{𝑚−𝑞−1}𝑓(𝑠)𝑑𝑠, (3)  where0≤𝑚−1≤𝑞<𝑚,𝑚∈ℤ⁺.(c)Caputo’s fractional derivative of order 𝑞 for a function𝑓∶ℝ⁺→ℝ^𝑛is defined as ^𝐶𝐷^𝑞𝑓(𝑡)=1Γ𝑚−𝑞^𝑡₀(𝑡−𝑠)^{𝑚−𝑞−1}𝑓^(𝑚)(𝑠)𝑑𝑠, (4) where0≤𝑚−1≤𝑞<𝑚, 𝑚∈ℤ⁺. Here,^𝐶𝐷^𝑞is still written as𝐷^𝑞.(d)The Laplace transform of a function𝑓(𝑡)is defined as 𝐹(𝑠)=£𝑓(𝑡)=₀^+∞𝑒^−𝑠𝑡𝑓(𝑡)𝑑𝑡,𝑠∈ℂ, (5) where ℂ denotes the complex plane and𝑓(𝑡)is𝑛-dimensional vector-valued function. For𝑚−1≤𝑞<𝑚, it follows from [1–4] that £𝐷^𝑞𝑓(𝑡)=𝑠^𝑞£𝑓(𝑡)−^𝑚−1_𝑘=0𝑠^{𝑞−𝑘−1}𝑓^(𝑘)(0). (6)  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 𝐸_𝑞,𝛽(𝑧)=^+∞_𝑘=0𝑧^𝑘Γ𝑘𝑞+𝛽,𝑞>0,𝛽>0,𝑧∈ℂ. (7) In particular, for𝛽=1, the Mittag-Leffler function in one parameter is given by 𝐸_𝑞(𝑧)∶=𝐸_𝑞,1(𝑧)=^+∞_𝑘=0𝑧^𝑘Γ1+𝑘𝑞,𝑞>0,𝑧∈ℂ. (8)

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,𝜏₁], system (1) is given by 𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡)+^𝑚_𝑖=1𝐵_𝑖𝜑𝑡−𝜏_𝑖+𝐷^𝛼𝑚_𝑖=1𝐶_𝑖𝜑𝑡−𝜏_𝑖. (9) Since∑^𝑚_𝑖=1𝐵_𝑖𝜑(𝑡−𝜏_𝑖)+𝐷^𝛼[∑^𝑚_𝑖=1𝐶_𝑖𝜑(𝑡−𝜏_𝑖)]is continuous on[0,𝜏₁], from [3], we know that there is a unique continuous solution for system (1) on [−𝜏,𝜏₁], which is denoted as 𝑥₁(𝑡), 𝑡∈[−𝜏,𝜏₁]. Furthermore,𝑥₁(𝑡)can be expressed by the following form: 𝑥₁(𝑡)=𝜑(𝑡)[],𝐸,𝑡∈−𝜏,0_𝛼𝐴𝑡^𝛼𝜑(0)+^𝑡₀(𝑡−𝜃)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝜃)^𝛼×^𝑚_𝑖=1𝐵_𝑖𝜑𝜃−𝜏_𝑖+𝐷^𝛼𝑚_𝑖=1𝐶_𝑖𝜑𝜃−𝜏_𝑖𝑑𝜃,𝑡∈0,𝜏₁. (10)
For𝑡∈[𝜏₁,2𝜏₁], system (1) is given by 𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡)+^𝑚_𝑖=1𝐵_𝑖𝑥₁𝑡−𝜏_𝑖+𝐷^𝛼𝑚_𝑖=1𝐶_𝑖𝑥₁𝑡−𝜏_𝑖. (11) Similarly, since∑^𝑚_𝑖=1𝐵_𝑖𝑥₁(𝑡−𝜏_𝑖)+𝐷^𝛼[∑^𝑚_𝑖=1𝐶_𝑖𝑥₁(𝑡−𝜏_𝑖)]is continuous on[−𝜏,2𝜏₁], we obtain that𝑥₂(𝑡)is a unique continuous solution of system (1) on[−𝜏,2𝜏₁] and 𝑥₂(𝑡)=𝑥₁(𝑡),𝑡∈−𝜏,𝜏₁,𝐸_𝛼𝐴𝑡^𝛼𝜑(0)+^𝑡₀(𝑡−𝜃)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝜃)^𝛼×^𝑚_𝑖=1𝐵_𝑖𝑥₁𝜃−𝜏_𝑖+𝐷^𝛼𝑚_𝑖=1𝐶_𝑖𝑥₁𝜃−𝜏_𝑖𝑑𝜃,𝑡∈𝜏₁,2𝜏₁. (12)
Assume that system (1) has a unique solution𝑥_𝑘(𝑡)on[(𝑘−1)𝜏₁,𝑘𝜏₁]. For𝑡∈[𝑘𝜏₁,(𝑘+1)𝜏₁], system (1) is given by 𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡)+^𝑚_𝑖=1𝐵_𝑖𝑥_𝑘𝑡−𝜏_𝑖+𝐷^𝛼𝑚_𝑖=1𝐶_𝑖𝑥_𝑘𝑡−𝜏_𝑖. (13) Similarly, since∑^𝑚_𝑖=1𝐵_𝑖𝑥_𝑘(𝑡−𝜏_𝑖)+𝐷^𝛼[∑^𝑚_𝑖=1𝐶_𝑖𝑥_𝑘(𝑡−𝜏_𝑖)]is continuous on[𝑘𝜏,(𝑘+1)𝜏], we obtain that𝑥_𝑘+1(𝑡)is a unique continuous solution of system (1) on[𝑘𝜏₁,(𝑘+1)𝜏₁] and 𝑥_𝑘+1(𝑡)=𝑥_𝑘(𝑡),𝑡∈−𝜏,𝑘𝜏₁,𝐸_𝛼𝐴𝑡^𝛼𝜑(0)+^𝑡₀(𝑡−𝜃)^𝛼−1𝐸_𝛼,𝛼𝐴(𝑡−𝜃)^𝛼×^𝑚_𝑖=1𝐵_𝑖𝑥_𝑘𝜃−𝜏_𝑖+𝐷^𝛼𝑚_𝑖=1𝐶_𝑖𝑥_𝑘𝜃−𝜏_𝑖𝑑𝜃,𝑡∈𝑘𝜏₁,(𝑘+1)𝜏₁. (14)
According to the mathematical induction, we know that system (1) has a unique continuous solution on[0,𝑘𝜏₁],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)𝜏₁=𝑇, we know that the assertion is true.
Case  2. When0<𝑇−(𝑘+1)𝜏₁<𝜏₁, we only need to prove that system (1) has a unique continuous solution on[𝜏₁,𝑇]. For𝑡∈[𝜏₁,𝑇], we denote𝑡₁=𝑡−𝜏₁∈[0,𝑇−𝜏₁]⊂[0,(𝑘+1)𝜏₁], and we can use the similar proof to obtain the conclusion.
Case  3. When𝑇−(𝑘+1)𝜏₁>𝜏₁, 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. Lemma 1 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(⋅)∈𝐂¹([−𝜏,0],ℝ^𝑛), its analytic solution𝑥(𝑡)satisfieslim_𝑡→+∞𝑥(𝑡)=0for all the time delays0<𝜏₁≤𝜏₂≤⋯≤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^𝛼𝑓(𝑡)is given as follows: £𝐷^𝛼𝑓(𝑡)=𝑠^𝛼£𝑓(𝑡)−𝑠^𝛼−1𝑓(0),0<𝛼<1. (15) Applying the Laplace transform on both sides of system (1) yields 𝑠^𝛼£𝑥(𝑡)−^𝑚_𝑖=1𝐶_𝑖𝑥𝑡−𝜏_𝑖−𝑠^𝛼−1𝜑(0)+𝑠^𝑚𝛼−1_𝑖=1𝐶_𝑖𝜑−𝜏_𝑖=𝐴£[𝑥(𝑡)]+^𝑚_𝑖=1𝐵_𝑖£𝑥𝑡−𝜏_𝑖. (16) Note that £𝑥𝑡−𝜏_𝑖=𝑒^{−𝑠𝜏𝑖}£[𝑥(𝑡)]+𝑒^{−𝑠𝜏𝑖0}_−𝜏𝑖𝑒^−𝑠𝑡𝜑(𝑡)𝑑𝑡,𝑖=1,2,…,𝑚; (17) thus, we obtain Δ𝑠,𝜏_𝑖£[𝑥(𝑡)]=𝑠^𝛼−1𝜑(0)−𝑠^𝑚𝛼−1_𝑖=1𝐶_𝑖𝜑−𝜏_𝑖+^𝑚_𝑖=1𝐵_𝑖𝑒^{−𝑠𝜏𝑖0}_−𝜏𝑖𝑒^−𝑠𝑡𝜑(𝑡)𝑑𝑡+𝑠^𝛼𝑚_𝑖=1𝐶_𝑖𝑒^{−𝑠𝜏𝑖0}_−𝜏𝑖𝑒^−𝑠𝑡𝜑(𝑡)𝑑𝑡, (18) where Δ𝑠,𝜏_𝑖=𝑠^𝛼𝐼−𝐴−^𝑚_𝑖=1𝐵_𝑖𝑒^{−𝑠𝜏𝑖}−𝑠^𝛼𝑚_𝑖=1𝐶_𝑖𝑒^{−𝑠𝜏𝑖} (19) is the characteristic matrix of system (1). Multiplying𝑠on both sides of (18) yields Δ𝑠,𝜏_𝑖{𝑠£[𝑥(𝑡)]}=𝑠𝑠^𝛼−1𝜑(0)−𝑠^𝑚𝛼−1_𝑖=1𝐶_𝑖𝜑−𝜏_𝑖+^𝑚_𝑖=1𝐵_𝑖𝑒^{−𝑠𝜏𝑖0}_−𝜏𝑖𝑒^−𝑠𝑡𝜑(𝑡)𝑑𝑡+𝑠^𝛼𝑚_𝑖=1𝐶_𝑖𝑒^{−𝑠𝜏𝑖0}_−𝜏𝑖𝑒^−𝑠𝑡𝜑(𝑡)𝑑𝑡. (20) 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 det𝑠^𝛼𝐼−𝐴−^𝑚_𝑖=1𝐵_𝑖𝑒^{−𝑠𝜏𝑖}−𝑠^𝛼𝑚_𝑖=1𝐶_𝑖𝑒^{−𝑠𝜏𝑖}=0 (21) 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=1and𝜏₁=𝜏₂=⋯=𝜏_𝑚=0, the characteristic equation det𝜆𝐼−𝐴−^𝑚_𝑖=1𝐵_𝑖+𝜆𝐶_𝑖=0 (22) 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 ℜ𝑒(𝑠)<−𝛿<0, where ℜ𝑒(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: 𝐻₁arg(𝜎(𝐴))>𝛼𝜋2,𝐻₂𝜌^𝑚_𝑖=1𝐶_𝑖𝐻<1,₃sup_{(𝑠)ℜ𝑒≥0}𝜌𝑠^𝛼𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝑠^𝛼𝐶_𝑖<1. (23)

Proof. Note that|arg(𝜎(𝐴))|>𝛼𝜋/2; then, all the roots of equationdet(𝜆𝐼−𝐴)=0satisfy|arg(𝜆)|>𝛼𝜋/2. Let𝑠^𝛼=𝜆; then, all the roots of equationdet(𝑠^𝛼𝐼−𝐴)=0satisfy|arg(𝑠)|>𝜋/2; that is, ℜ𝑒(𝑠)<0. Therefore, matrix𝐴is invertible and(𝑠^𝛼𝐼−𝐴)⁻¹is well defined when|arg(𝜎(𝐴))|>𝛼𝜋/2and ℜ𝑒(𝑠)≥0.
For ℜ𝑒(𝑠)≥0, it follows from the characteristic polynomial of system (1) that detΔ𝑠,𝜏_𝑖=det𝑠^𝛼𝐼−𝐴−^𝑚_𝑖=1𝐵_𝑖𝑒^{−𝑠𝜏𝑖}−𝑠^𝛼𝑚_𝑖=1𝐶_𝑖𝑒^{−𝑠𝜏𝑖}=det𝑠^𝛼𝐼−𝐴det𝐼−𝑠^𝛼𝐼−𝐴⁻¹×^𝑚_𝑖=1𝐵_𝑖+𝑠^𝛼𝐶_𝑖𝑒^{−𝑠𝜏𝑖}=det𝑠^𝛼×𝐼−𝐴_𝑛𝑗=11−𝜆_𝑗𝑠^𝛼𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝑠^𝛼𝐶_𝑖𝑒^{−𝑠𝜏𝑖}. (24) Then, we have 𝜆_𝑗𝑠^𝛼𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝑠^𝛼𝐶_𝑖𝑒^{−𝑠𝜏𝑖}≤𝜌𝑠^𝛼𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝑠^𝛼𝐶_𝑖𝑒^−𝑠𝜏<1,𝑗=1,2,…,𝑛. (25) Combining (24) and (25), we havedetΔ(𝑠,𝜏_𝑖)≠0for ℜ𝑒(𝑠)≥0and𝜏_𝑖>0,𝑖=1,2,…,𝑚; that is, if, conditions(𝐻₁)and(𝐻₃)are satisfied, then the characteristic equation (21) implies that ℜ𝑒(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, ℜ𝑒(𝑠_𝑛)<0and ℜ𝑒(𝑠_𝑛)→0as𝑛→+∞. Note that any eigenvalue𝜆_𝑗[(𝑠^𝛼𝐼−𝐴)⁻¹∑^𝑚_𝑖=1(𝐵_𝑖+𝑠^𝛼𝐶_𝑖)]is a continuous function of𝑠for ℜ𝑒(𝑠)≥0; then, it follows from(𝐻₂)that 𝜌𝑠^𝛼𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝑠^𝛼𝐶_𝑖⟶𝜌^𝑚_𝑖=1𝐶_𝑖<1,𝑠⟶+∞. (26) Hence,|𝜆_𝑗[(𝑠^𝛼𝐼−𝐴)⁻¹∑^𝑚_𝑖=1(𝐵_𝑖+𝑠^𝛼𝐶_𝑖)]|reach the maximum value for ℜ𝑒(𝑠)≥0. From condition(𝐻₃), there exists a positive constant𝜀such that sup_{(𝑠)ℜ𝑒≥0}𝜌𝑠^𝛼𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝑠^𝛼𝐶_𝑖=sup_{(𝑠)ℜ𝑒≥0}max_{1≤𝑗≤𝑛}𝜆_𝑗𝑠^𝛼𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝑠^𝛼𝐶_𝑖=1−𝜀. (27) Then, equality (27) implies that sup_{ℜ𝑒(𝜁)=0}𝜌𝜁^𝛼𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝜁^𝛼𝐶_𝑖≤1−𝜀. (28) When the positive integer𝑛is large enough, there exist a positive constant𝜀^∗(0<𝜀^∗<𝜀)and a characteristic root𝑠_𝑛such that|ℜ𝑒(𝑠_𝑛)|is sufficiently small, ℜ𝑒(𝑠_𝑛)<0and max_{1≤𝑗≤𝑛}𝜆_𝑗𝑠^𝛼_𝑛𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝑠^𝛼_𝑛𝐶_𝑖−sup_{(𝜁)ℜ𝑒=0}𝜌𝜁^𝛼𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝜁^𝛼𝐶_𝑖<𝜀^∗. (29) For ℜ𝑒(𝜁)=0, from (28) and (29), we have 𝜆_𝑗𝑠^𝛼_𝑛𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝑠^𝛼_𝑛𝐶_𝑖≤sup_{(𝜁)ℜ𝑒=0}𝜌𝜁^𝛼𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝜁^𝛼𝐶_𝑖+𝜀^∗≤1−𝜀+𝜀^∗<1,𝑗=1,2,…,𝑛. (30) Choosing𝑛large enough yields 𝜆_𝑗𝑠^𝛼_𝑛𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝑠^𝛼_𝑛𝐶_𝑖𝑒^{−𝑠𝜏𝑖}<1,𝑗=1,2,…,𝑛. (31) Therefore, for ℜ𝑒(𝑠_𝑛)<0and ℜ𝑒(𝑠_𝑛)→0as𝑛→+∞, one can obtain detΔ𝑠_𝑛,𝜏_𝑖=det𝑠^𝛼_𝑛𝐼−𝐴−^𝑚_𝑖=1𝐵_𝑖𝑒^{−𝑠𝑛𝜏𝑖}−𝑠^𝛼_𝑛^𝑚_𝑖=1𝐶_𝑖𝑒^{−𝑠𝑛𝜏𝑖}≠0, (32) 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: 𝐻₁arg(𝜎(𝐴))>𝛼𝜋2𝐻,0<𝛼<1,₄𝜌^𝑚_𝑖=1𝜉_𝑖𝐶_𝑖<1,𝜉_𝑖∈ℂ,𝜉_𝑖𝐻≤1,𝑖=1,2,…,𝑚,₅𝜌𝑠^𝛼𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝑠^𝛼𝐶_𝑖𝜉_𝑖(𝑠)<1,ℜ𝑒=0,𝜉_𝑖∈ℂ,𝜉_𝑖=1,𝑖=1,2,…,𝑚. (33)

Proof. According to condition(𝐻₁)and the proof of Theorem 6, we know that matrix𝐴is invertible and(𝑠^𝛼𝐼−𝐴)⁻¹is well defined when|arg(𝜎(𝐴))|>𝛼𝜋/2and ℜ𝑒(𝑠)=0.
On the one hand, any eigenvalue𝜆[(𝑠^𝛼𝐼−𝐴)⁻¹(∑^𝑚_𝑖=1𝜉_𝑖𝐵_𝑖+𝑠^𝛼∑^𝑚_𝑖=1𝜉_𝑖𝐶_𝑖)]is an algebraic function of(𝜉₁,𝜉₂,…,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 𝜌𝑠^𝛼𝐼−𝐴^−1𝑚_𝑖=1𝜉_𝑖𝐵_𝑖+𝑠^𝛼𝑚_𝑖=1𝜉_𝑖𝐶_𝑖(𝑠)<1,ℜ𝑒=0,𝜉_𝑖∈ℂ,𝜉_𝑖=1, (34) is equivalent to 𝜌𝑠^𝛼𝐼−𝐴^−1𝑚_𝑖=1𝜉_𝑖𝐵_𝑖+𝑠^𝛼𝑚_𝑖=1𝜉_𝑖𝐶_𝑖(𝑠)<1,ℜ𝑒=0,𝜉_𝑖∈ℂ,𝜉_𝑖≤1. (35) In view of the expression of(𝑠^𝛼𝐼−𝐴)⁻¹(∑^𝑚_𝑖=1𝜉_𝑖𝐵_𝑖+𝑠^𝛼∑^𝑚_𝑖=1𝜉_𝑖𝐶_𝑖), as ℜ𝑒(𝑠)=0and𝑠→∞, one can obtain 𝜌𝑠^𝛼𝐼−𝐴^−1𝑚_𝑖=1𝜉_𝑖𝐵_𝑖+𝑠^𝛼𝑚_𝑖=1𝜉_𝑖𝐶_𝑖⟶𝜌^𝑚_𝑖=1𝜉_𝑖𝐶_𝑖<1,∀𝜉_𝑖∈ℂ,𝜉_𝑖≤1. (36) It follows from (36) that sup_{(𝑠)ℜ𝑒=0}𝜌𝑠^𝛼𝐼−𝐴^−1𝑚_𝑖=1𝜉_𝑖𝐵_𝑖+𝑠^𝛼𝑚_𝑖=1𝜉_𝑖𝐶_𝑖≤1,∀𝜉_𝑖∈ℂ,𝜉_𝑖≤1. (37) Applying the maximum modulus principle on the unbounded region{𝑠|𝑠∈ℂ,ℜ𝑒(𝑠)≥0}, then (35) implies that 𝜌𝑠^𝛼𝐼−𝐴^−1𝑚_𝑖=1𝜉_𝑖𝐵_𝑖+𝑠^𝛼𝑚_𝑖=1𝜉_𝑖𝐶_𝑖(𝑠)<1,ℜ𝑒≥0,𝜉_𝑖∈ℂ,𝜉_𝑖≤1. (38)
On the other hand, suppose that there exists some𝑠₀∈{𝑠|𝑠∈ℂ,ℜ𝑒(𝑠)≥0}such that det𝑠^𝛼₀𝐼−𝐴−^𝑚_𝑖=1𝐵_𝑖𝑒^{−𝑠0𝜏𝑖}−𝑠^𝛼₀^𝑚_𝑖=1𝐶_𝑖𝑒^{−𝑠0𝜏𝑖}𝑠=0,₀∈{𝑠|𝑠∈ℂ,ℜ𝑒(𝑠)≥0}. (39) Choose𝜉⁰_𝑖=𝑒^{−𝑠0𝜏𝑖}such that|𝜉⁰_𝑖|≤1,𝑖=1,2,…,𝑚. It follows from (39) that 1∈𝜎𝑠^𝛼₀𝐼−𝐴^−1𝑚_𝑖=1𝐵_𝑖𝑒^{−𝑠0𝜏𝑖}+𝑠^𝛼₀^𝑚_𝑖=1𝐶_𝑖𝑒^{−𝑠0𝜏𝑖},𝑠₀∈{𝑠|𝑠∈ℂ,ℜ𝑒(𝑠)≥0}, (40) which contradicts inequality (38). Therefore, if conditions(𝐻₁), (𝐻₄), and(𝐻₅)are satisfied, then the characteristic equation (21) implies that ℜ𝑒(𝑠)<0.
The rest of the proof is similar to that of Theorem 6; then, the conclusion holds.

According to the proof of Theorem 7, 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: 𝐻₁arg(𝜎(𝐴))>𝛼𝜋2𝐻,0<𝛼<1,₄𝜌^𝑚_𝑖=1𝜉_𝑖𝐶_𝑖<1,𝜉_𝑖∈ℂ,𝜉_𝑖𝐻≤1,𝑖=1,2,…,𝑚,₆𝜌𝑠^𝛼𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝑠^𝛼𝐶_𝑖𝜉_𝑖(𝑠)<1,ℜ𝑒≥0,𝜉_𝑖∈ℂ,𝜉_𝑖≤1,𝑖=1,2,…,𝑚. (41)

Assume that|arg(𝜎(𝐴))|>𝛼𝜋/2,0<𝛼<1. Define the following matrices: 𝐿=(𝐼−𝐴)^𝑚−1_𝑖=1𝐵_𝑖+𝐶_𝑖,𝑀=(𝐼−𝐴)^𝑚−1_𝑖=1𝐵_𝑖−𝐶_𝑖,𝑁=(𝐼−𝐴)⁻¹(𝐼+𝐴). (42) Let 𝑠^𝛼=1−𝑧(𝑠)1+𝑧,ℜ𝑒≥0,0<𝛼<1; (43) 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: 𝐻₁arg(𝜎(𝐴))>𝛼𝜋2𝐻,0<𝛼<1,₂𝜌^𝑚_𝑖=1𝐶_𝑖𝐻<1,₇sup_|𝑧|≤1𝜌(𝐼−𝑧𝑁)⁻¹(𝐿+𝑧𝑀)<1. (44)

Proof. From Theorem 6, we only need to prove that the following equality holds: 𝑠^𝛼𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖+𝑠^𝛼𝐶_𝑖=(𝐼−𝑧𝑁)⁻¹(),(𝑠)|𝑧|𝐿+𝑧𝑀ℜ𝑒≥0,≤1. (45)
In fact, it is easy to obtain 𝐼−𝑧𝑁=𝐼−1−𝑠^𝛼1+𝑠^𝛼(𝐼−𝐴)⁻¹(=𝐼+𝐴)1+𝑠^𝛼𝐼−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)⁻¹𝑠^𝛼𝐼−𝐴1+𝑠^𝛼−1. (46) For ℜ𝑒(𝑠)≥0, it follows from(𝐻₁)and (43) that|𝑧|≤1; then, matrix(𝐼−𝑧𝑁)is invertible. Hence, 𝐿+𝑧𝑀=(𝐼−𝐴)^𝑚−1_𝑖=1𝐵_𝑖+𝐶_𝑖+1−𝑠^𝛼1+𝑠^𝛼(𝐼−𝐴)^𝑚−1_𝑖=1𝐵_𝑖−𝐶_𝑖=(𝐼−𝐴)^−1𝑚_𝑖=1𝐵_𝑖+𝐶_𝑖+1−𝑠^𝛼1+𝑠^𝛼𝑚_𝑖=1𝐵_𝑖−𝐶_𝑖=(𝐼−𝐴)^−1𝑚_𝑖=1𝐵_𝑖+𝐶_𝑖1+𝑠^𝛼+1−𝑠^𝛼𝑚_𝑖=1𝐵_𝑖−𝐶_𝑖1+𝑠^𝛼−1=2(𝐼−𝐴)^𝑚−1_𝑖=1𝐵_𝑖+𝑠^𝛼𝐶_𝑖1+𝑠^𝛼−1. (47) 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: 𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡)+^𝑚_𝑖=1𝐵_𝑖𝑥𝑡−𝜏_𝑖𝑥(𝑡)(𝑡)[],,𝑡≥0,=𝜑,𝑡∈−𝜏,0 (48)  where𝐷^𝛼𝑥(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<𝜏₁≤𝜏₂≤⋯≤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 Lemma 1, 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: 𝐻₁arg(𝜎(𝐴))>𝛼𝜋2𝐻,0<𝛼<1,₈𝜌𝑠^𝛼𝐼−𝐴^𝑚−1_𝑖=1𝐵_𝑖𝜉_𝑖(𝑠)<1,ℜ𝑒=0,𝜉_𝑖∈ℂ,𝜉_𝑖=1. (49)

(ii) When𝜏_𝑖=0,𝑖=1,2,…,𝑚, system (1) reduces to Caputo fractional-order linear autonomous differential systems: 𝐼−^𝑚_𝑖=1𝐶_𝑖𝐷^𝛼𝑥(𝑡)=𝐴+^𝑚_𝑖=1𝐵_𝑖𝑥(𝑡)𝑥(0),𝑡≥0,=𝑥₀, (50) where𝐷^𝛼𝑥(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: 𝐻₉det𝐼−^𝑚_𝑖=1𝐶_𝑖𝐻≠0,₁₀arg(𝜎(𝐺))>𝛼𝜋2,𝐺=𝐼−^𝑚_𝑖=1𝐶_𝑖⁻¹𝐴+^𝑚_𝑖=1𝐵_𝑖. (51)

Remark 12. Whendet[𝐼−∑^𝑚_𝑖=1𝐶_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 𝐴=−424−10,𝐵₁=,𝐵01−23₂=3−120,𝐶₁=23−1613−16,𝐶₂=−1616−13231,𝛼=2,𝜏₁=1,𝜏₂=2,𝜏=2. (52) The initial function is given by𝜑(𝑡)=𝑡, 𝑡∈[−2,0]. Let Θ=𝑠^𝛼𝐼−𝐴⁻¹𝜉₁𝐵₁+𝜉₂𝐵₂+𝑠^𝛼𝜉₁𝐶₁+𝑠^𝛼𝜉₂𝐶₂,ℜ𝑒(𝑠)=0,𝜉₁=𝜉₂=1. (53) By computation, the eigenvalues of matrixΘgive 𝜆₁(Θ)=𝑠^𝛼+6⁻¹𝜉₁+2𝜉₂+12𝑠^𝛼𝜉₁+13𝑠^𝛼𝜉₂,𝜆₂(Θ)=𝑠^𝛼+8⁻¹2𝜉₁+𝜉₂+13𝑠^𝛼𝜉₁+12𝑠^𝛼𝜉₂. (54) Note that𝜎(𝐴)={−6,−8}; then, we have arg(𝜎(𝐴))=𝜋>𝛼𝜋21,𝛼=2. (55) It is not difficult to verify that 𝜌𝜉₁𝐶₁+𝜉₂𝐶₂<1,𝜉₁≤1,𝜉₂𝜌[Θ](𝑠)𝜉≤1,<1,ℜ𝑒=0,₁=1,𝜉₂=1. (56) Thus, conditions(𝐻₁),(𝐻₃), and (𝐻₄)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 det𝑠^𝛼𝐼−𝐴−^𝑚_𝑖=1𝐵_𝑖𝑒^{−𝑠𝜏𝑖}−𝑠^𝛼𝑚_𝑖=1𝐶_𝑖𝑒^{−𝑠𝜏𝑖}=0. (57) 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 (𝐻₁), (𝐻₄), and (𝐻₅). 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is dedicated to Professor Zuxiu Zheng on the occasion of his 80th birthday and the 50th anniversary of his research work. The authors are very grateful to the associate editor, Professor Anna Mercaldo, and the two anonymous reviewers for their helpful and valuable comments and suggestions which improved the quality of the paper. This work is supported by the National Natural Science Foundation of China under Grant nos. 11072059 and 61272530, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant nos. 20110092110017 and 20130092110017, the Natural Science Foundation of Jiangsu Province of China under Grant no. BK2012741, and the Programs of Educational Commission of Anhui Province of China under Grant nos. KJ2011A197 and KJ2013Z186.