Stability Analysis for Fractional-Order Linear Singular Delay Differential Systems

Abstract

We investigate the delay-independently asymptotic stability of fractional-order linear singular delay differential systems. Based on the algebraic approach, the sufficient conditions are presented to ensure the asymptotic stability for any delay parameter. By applying the stability criteria, one can avoid solving the roots of transcendental equations. An example is also provided to illustrate the effectiveness and applicability of the theoretical results.

1. Introduction

Singular systems play important roles in modelling and analyzing a lot of practical systems, such as power systems, economic systems, chemical processes, and network analysis. Depending on these fields of applications, they are also called differential-algebraic systems, descriptor systems, semistate systems, or generalized state-space systems. Over the last two decades, many results regarding integer-order singular systems have been obtained (see [1–10]). In [3–10], various stability problems of integer-order singular systems have been considered by applying the different methods, such as linear matrix inequality (LMI) method [3, 4, 6, 7], 𝜃-method and BDF method [5], Lyapunov-Krasovskii functional method [8, 9], and Lyapunov direct method [10].

The fractional dynamics system is a recent focus of interest to many researchers [11–32]. Many practical phenomena in the fields of economics, engineering, and physics can be represented more accurately through fractional derivative formulation. The basic theory of factional calculus can be found in the monographs of Miller and Ross [11], Podlubny [12], Kilbas et al. [13], and Diethelm [14]. Moreover, Lakshmikantham et al. [15] and Baleanu et al. [16] have elaborated the theory of fractional-order dynamics systems and the recent developments.

As we all know, stability is an important performance metric for dynamic systems. Since the fractional derivative has the nonlocal property and weakly singular kernels, the analysis of stability of fractional-order systems is more complex than that of integer-order differential systems. In recent years, there are some results on the stability of fractional-order differential systems [18–20]. For example, Li et al. [18, 19] proposed the Mittag-Leffler stability of fractional-order systems based on fractional comparison principle [18] and Lyapunov direct method [19]. On the other hand, time delay has an important effect on the stability and performance of dynamic systems. It is worth mentioning that the notable contributions have been made to the stability of fractional-order delay differential systems (see [21–30]). In particular, Deng et al. [24] studied Lyapunov asymptotic stability of fractional linear delay differential systems by using the final-value theorem of Laplace transform. De la Sen [25] considered Robust stability of fractional-order linear delayed dynamic systems by means of fixed point theory. In [26], Li and Zhang presented a survey on the stability of fractional-order (delay) differential equations. Kaslik and Sivasundaram [27] investigated the asymptotic stability of linear fractional-order delay differential equations by using the analytical and numerical methods. By employing Lyapunov functional method, Sadati et al. [28] and Baleanu et al. [29] established Mittag-Leffler stability theorem and Razumikhin stability theorem for fractional-order nonlinear delay systems, respectively.

For fractional-order linear singular systems without delay, N’Doye et al. [31] studied the stabilization problems by means of LMI method. However, to the best of our knowledge, there are very few works on the stability of fractional-order linear singular delay differential systems as reported in the current literatures except [32], in which Zhang and Jiang investigated the finite-time stability of fractional-order singular delay differential systems in terms of the Gronwall integral inequality. Compared to the stability theory of integer-order singular dynamics systems [3–10] and fractional-order delay differential systems [21–30], the stability theory of fractional-order singular delay systems is not yet sufficiently elaborated.

In this paper, we are interested in the delay-independent stability of the Caputo fractional-order singular delay differential system as follows: 𝐸𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡)+()𝑥(𝑡)=𝐵𝑥𝑡−𝜏,𝑡≥0,𝜑(𝑡),−𝜏≤𝑡≤0 (1) and the Caputo fractional-order singular neutral delay differential system 𝐷^𝛼[(𝑡)((𝑡)()𝑥(𝑡)(𝑡)𝐸𝑥−𝐶𝑥𝑡−𝜏)]=𝐴𝑥+𝐵𝑥𝑡−𝜏,𝑡≥0,=𝜑,−𝜏≤𝑡≤0, (2) where 0<𝛼<1; 𝑥(𝑡)∈ℝ^𝑛 is the state vector; 𝐷^𝛼𝑥(𝑡) denotes an 𝛼 order Caputo fractional-order derivative of 𝑥(𝑡); matrices 𝐴,𝐵,𝐴,𝐵,𝐶∈ℝ^𝑛×𝑛 and matrices 𝐸,𝐸∈ℝ^𝑛×𝑛 are singular with rank(𝐸)=𝑟<𝑛, rank(𝐸)=𝑟<𝑛; 𝜏∈ℝ⁺ is the time delay; and 𝜑, 𝜑 are both the consistent initial functions.

Different from the methods in [18–32], we apply the algebraic approach to establish the delay-independently asymptotic stability criteria for system (1) and system (2). The novelty of this paper lies in the following aspects. Firstly, we synchronously take into account the factors of such systems including the Caputo’s fractional-order derivative, pseudostate delay, and singular coefficient matrices. Secondly, the algebraic approach is applied to derive the sufficient conditions of the delay-independent stability, which ensure the asymptotic stability for any delay parameter 𝜏∈ℝ⁺. Thirdly, by applying these stability criteria, one can avoid solving the roots of transcendental equations.

This paper is organized as follows. In Section 2, we introduce some definitions and preliminary facts used in the paper. In Section 3, the sufficient conditions of the delay-independently asymptotic stability for system (1) and system (2) are derived based on the algebraic approach, respectively. 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 recall some definitions of fractional calculus (see [11–14]) and preliminary facts used in the paper.

Definition 1 (see [12]). Riemann-Liouville’s fractional integral of order 𝑞>0 for a function 𝑓∶ℝ⁺→ℝ^𝑛 is defined as 𝐷^−𝑞𝑓(𝑡)=1Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑓(𝑠)𝑑𝑠,𝑡>0, (3) where Γ(⋅) is the Gamma function.

Definition 2 (see [12]). The Riemann-Liouville’s fractional derivative of order 𝑞 for a function 𝑓∶ℝ⁺→ℝ^𝑛 is defined as ^RL𝐷^𝑞𝑓(𝑡)=1Γ𝑑𝑚−𝑞𝑑𝑡^𝑚𝑚𝑡₀(𝑡−𝑠)^{𝑚−𝑞−1}𝑓(𝑠)𝑑𝑠, (4) where 0≤𝑚−1≤𝑞<𝑚, 𝑚∈ℤ⁺.

Definition 3 (see [12]). The Caputo’s fractional derivative of order 𝑞 for a function 𝑓∶ℝ⁺→ℝ^𝑛 is defined as ^𝐶𝐷^𝑞𝑓(𝑡)=1Γ𝑚−𝑞^𝑡₀(𝑡−𝑠)^{𝑚−𝑞−1}𝑓^(𝑚)(𝑠)𝑑𝑠, (5) where 0≤𝑚−1≤𝑞<𝑚, 𝑚∈ℤ⁺. For the sake of convenience, ^𝐶𝐷^𝑞 is still written as 𝐷^𝑞.

Definition 4 (see [12]). The Mittag-Leffler function in two parameters is defined as 𝐸_𝑞,𝛽(𝑧)=^+∞_𝑘=0𝑧^𝑘Γ,𝑘𝑞+𝛽 (6) where 𝑞>0, 𝛽>0 and 𝑧∈ℂ and ℂ denotes the complex plane. In particular, for 𝛽=1, the Mittag-Leffler function in one parameter is defined as 𝐸_𝑞(𝑧)=^+∞_𝑘=0𝑧^𝑘Γ1+𝑘𝑞,𝑞>0,𝑧∈ℂ. (7)
For 𝛼∈(0,1), the Laplace transform of Caputo fractional-order derivative 𝐷^𝛼𝑓(𝑡) is given as follows: £^𝑐𝐷^𝛼𝑓(𝑡)=𝑠^𝛼£𝑓(𝑡)−𝑠^𝛼−1𝑓(0). (8)
Throughout this paper, let 𝐂([−𝜏,0],ℝ^𝑛) be the space of all continuous functions mapping the interval [−𝜏,0] into ℝ^𝑛, and 𝐂^𝟏([−𝜏,0],ℝ^𝑛) denotes the space of all continuous differential functions mapping the interval [−𝜏,0] into ℝ^𝑛. The consistent initial function 𝜑(⋅)∈𝐂([−𝜏,0],ℝ^𝑛) and 𝜑(⋅)∈𝐂^𝟏([−𝜏,0],ℝ^𝑛).

As discussed by Zhang [21], applying the method of steps, we obtain the following lemmas which generalize well-known results of integer-order singular systems without delay (see [1]) to fractional-order singular delay differential systems.

Definition 5. For any given two matrices 𝐸,𝐴∈ℝ^𝑛×𝑛, the matrix pair (𝐸,𝐴) is called regular if det(𝜆^𝛼𝐸−𝐴) is not identically zero, where 𝜆∈ℂ.

Lemma 6. Assume that (𝐸,𝐴) is regular, and 𝜑(⋅)∈𝐂^𝟏([−𝜏,0],ℝ^𝑛) is the consistent initial function, then system (2) has a unique solution on [0,+∞).

Proof. For system (2), on the interval [−𝜏,0], 𝑥(𝑡)=𝜑(𝑡). Thus, when 𝑡∈[0,𝜏], system (2) is given by 𝐷^𝛼𝐸𝑥(𝑡)−𝐶𝜑(𝑡−𝜏)=𝐴𝑥(𝑡)+𝐵𝜑(𝑡−𝜏), (9) that is, 𝐸𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝜑(𝑡−𝜏)+𝐶𝐷^𝛼𝜑(𝑡−𝜏). (10) Since (𝐸,𝐴) is regular, and 𝐵𝜑(𝑡−𝜏)+𝐶𝐷^𝛼[𝜑(𝑡−𝜏)] is continuous on [0,𝜏], by applying Theorem  3.2 in [32], we obtain that 𝑥₁(𝑡) is a unique solution of system (2) on [0,𝜏].
For 𝑡∈[𝜏,2𝜏], system (2) is given by 𝐸𝐷^𝛼𝑥(𝑡)−𝐶𝑥₁(𝑡−𝜏)=𝐴𝑥(𝑡)+𝐵𝑥₁(𝑡−𝜏), (11) that is, 𝐸𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥₁(𝑡−𝜏)+𝐶𝐷^𝛼𝑥₁(𝑡−𝜏). (12) Since 𝐵𝑥₁(𝑡−𝜏)+𝐶𝐷^𝛼[𝑥₁(𝑡−𝜏)] is continuous on [𝜏,2𝜏], similarly, we obtain that 𝑥₂(𝑡) is a unique solution of system (2) on [𝜏,2𝜏].
Assume that system (2) has a unique solution 𝑥_𝑘(𝑡) on [(𝑘−1)𝜏,𝑘𝜏]. For 𝑡∈[𝑘𝜏,(𝑘+1)𝜏], system (2) is given by 𝐸𝐷^𝛼𝑥(𝑡)−𝐶𝑥_𝑘(𝑡−𝜏)=𝐴𝑥(𝑡)+𝐵𝑥_𝑘(𝑡−𝜏), (13) that is, 𝐸𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥_𝑘(𝑡−𝜏)+𝐶𝐷^𝛼𝑥_𝑘(𝑡−𝜏). (14) Since 𝐵𝑥_𝑘(𝑡−𝜏)+𝐶𝐷^𝛼[𝑥_𝑘(𝑡−𝜏)] is continuous on [𝑘𝜏,(𝑘+1)𝜏], similarly, we obtain that 𝑥_𝑘+1(𝑡) is a unique solution of system (2) on [𝑘𝜏,(𝑘+1)𝜏].
According to the mathematical induction, we know that system (2) has a unique solution on [0,+∞). Therefore, the proof is completed.

Lemma 7 (see [32]). Assume that (𝐸,𝐴) is regular and   𝜑(⋅)∈𝐂([−𝜏,0],ℝ^𝑛) is the consistent initial function; then system (1) has a unique solution on [0,+∞).

Assume that the conditions in Lemmas 6 and 7 are satisfied, which ensure the existence and uniqueness of the solutions of system (1) and system (2). From Definition 3, the Caputo’s fractional derivative of a constant is equal to zero; then 𝑥(𝑡)≡0 is the zero solution of system (1) and system (2), respectively.

Definition 8. Assume that (𝐸,𝐴) (or (𝐸,𝐴)) is regular; then the zero solution 𝑥(𝑡)≡0 of system (1) (or (2)) is called delay-independently asymptotically stable if, for any consistent initial function 𝜑(⋅)∈𝐂([−𝜏,0],ℝ^𝑛) (or 𝜑(⋅)∈𝐂^𝟏([−𝜏,0],ℝ^𝑛)), its analytic solution 𝑥(𝑡) satisfies lim_𝑡→+∞𝑥(𝑡)=0 (15) for any delay parameter 𝜏>0.

Now, we introduce the first equivalent form of system (1) by means of the nonsingular transform, which is also called the standard canonical decomposition of a singular system.

Assume that (𝐸,𝐴) is regular, then there exist two nonsingular matrices 𝑃,𝑄∈ℝ^𝑛×𝑛, such that system (1) is equivalent to canonical system as follows: ^𝑐𝐷^𝛼𝑥₁(𝑡)=𝐴₁𝑥₁(𝑡)+𝐵₁₁𝑥₁()+𝑡−𝜏𝐵₁₂𝑥₂()𝑁𝑡−𝜏,𝑡≥0,_𝑐𝐷^𝛼𝑥₂(𝑡)=𝑥₂(𝑡)+𝐵₂₁𝑥₁()+𝑡−𝜏𝐵₂₂𝑥₂()𝑥𝑡−𝜏,𝑡≥0,₁(𝑡)=𝜑₁(𝑡)𝑥,−𝜏≤𝑡≤0,₂(𝑡)=𝜑₂(𝑡),−𝜏≤𝑡≤0, (16) where 0<𝛼<1, 𝑥₁∈ℝ^𝑟, 𝑥₂∈ℝ^𝑛−𝑟, and 𝑁 is nilpotent whose nilpotent index is denoted by 𝜈=ind(𝐸,𝐴), that is, 𝑁^𝜈=0, 𝑁^𝜈−1≠0, and 𝑃𝐼𝐸𝑄=^𝑟00𝑁,𝑃𝐴𝑄=𝐴₁00𝐼_𝑛−𝑟,𝑃𝐵𝑄=𝐵₁₁𝐵₁₂𝐵₂₁𝐵₂₂,𝑄⁻¹𝑥(𝑡)=𝑥₁(𝑡)𝑥₂(𝑡). (17)

In particular, when 𝜏=0 and ind(𝐸,𝐴)=1, system (1) is equivalent to the following canonical system without delay: 𝐷^𝛼𝑥₁(𝑡)=𝐴₁+𝐵₁𝑥₁(𝑡),𝑡≥0,0=𝑥₂(𝑡)𝑥,𝑡≥0,₁(𝑡)=𝜑₁(𝑡)𝑥,−𝜏≤𝑡≤0,₂(𝑡)=𝜑₂(𝑡),−𝜏≤𝑡≤0. (18)

In addition, taking the Laplace transform on both sides of system (1) yields 𝑠^𝛼𝐸£[𝑥(𝑡)]−𝑠^𝛼−1𝜑(0)=𝐴£[𝑥(𝑡)]+𝐵£[𝑥(𝑡−𝜏)], (19) where 0<𝛼<1. It follows from the properties of integral that £[𝑥(𝑡−𝜏)]=𝑒^−𝑠𝜏£[𝑥(𝑡)]+𝑒^−𝑠𝜏0_−𝜏𝑒^−𝑠𝑡𝜑(𝑡)𝑑𝑡. (20) Thus, we obtain 𝑠^𝛼𝐸−𝐴−𝐵𝑒^−𝑠𝜏£[𝑥(𝑡)]=𝑠^𝛼−1𝜑(0)+𝐵𝑒^−𝑠𝜏0_−𝜏𝑒^−𝑠𝑡𝜑(𝑡), (21) where Δ(𝑠,𝜏)=det𝑠^𝛼𝐸−𝐴−𝐵𝑒^−𝑠𝜏 (22) is the characteristic polynomial of system (1).

3. Stability Criteria

In this section, we derive the sufficient conditions of the delay-independently asymptotic stability for system (1) and system (2), respectively.

Theorem 9. Assume that (𝐸,𝐴) is regular, and ind(𝐸,𝐴)=1; then system (1) is delay-independently asymptotically stable if the following conditions simultaneously hold.(H₁)All the eigenvalues 𝜆s of matrix 𝐴₁+𝐵₁ satisfy arg𝜆𝐴₁+𝐵₁>𝛼𝜋2. (23)(H₂)For any 𝑦∈ℝ, 𝜏∈ℝ⁺, and 𝑖=√−1, then Δ𝑖𝑦,𝜏=det𝑖𝑦^𝛼𝐸−𝐴−𝐵𝑒^{−𝑖𝑦𝜏}≠0. (24)

Proof. For any 𝜏∈ℝ⁺, we only need to prove that all the roots of equation Δ(𝑠,𝜏)=0 lie in open left-half complex plane.
When 𝜏=0 and ind(𝐸,𝐴)=1, system (1) is reduced to the slow fractional-order subsystem 𝐷^𝛼𝑥₁(𝑡)=𝐴₁+𝐵₁𝑥₁(𝑡) (25) and the fast subsystem 0=𝑥₂(𝑡). (26) Thus, the asymptotic stability of system (18) entirely depends on 𝑥₁(𝑡) irrespective of 𝑥₂(𝑡).
From condition (H₁), any root of equation det(𝜆𝐼_𝑛1−𝐴₁−𝐵₁)=0 satisfies arg(𝜆)>𝛼𝜋2. (27) Let 𝜆=𝑠^𝛼, then we have 𝑠=𝜆^1/𝛼. It follows from inequality (27) that (𝑠>𝜋arg)2, (28) where 𝑠 is just the root of the following characteristic equation of the slow fractional-order subsystem det𝑠^𝛼𝐼_𝑟−𝐴₁−𝐵₁=0. (29) By computation of the determinants, we have det𝑠^𝛼𝐸−𝐴−𝐵=det𝑃⁻¹𝑠^𝛼𝐼_𝑟−𝐴₁−𝐵₁00−𝐼_𝑛−𝑟𝑄⁻¹=(−1)^𝑛−𝑟det𝑠^𝛼𝐼_𝑟−𝐴₁−𝐵₁det𝑃⁻¹𝑄⁻¹. (30) Taking into account (28), (29), and (30), one can get that all the roots of equation det(𝑠^𝛼𝐸−𝐴−𝐵)=0 have negative real parts. Hence, it follows from (H₁) that system (18) is asymptotically stable.
On the other hand, since rank(𝐸)=𝑟, the characteristic polynomial of system (1) can be expressed as Δ(𝑠,𝜏)=𝑃₀⋅𝑠^𝛼𝑟+𝑃₁⋅𝑠^𝛼𝑟−1+⋯+𝑃_𝑟−1⋅𝑠^𝛼+𝑃_𝑟,𝑃₀≠0, (31) where 𝑃_𝑖(𝑖=0,1,2,…,𝑟) are the polynomial of the exponential function 𝑒^−𝑠𝜏, and their coefficients are composed of the elements of matrices 𝐸, 𝐴, 𝐵.
Let ℜ𝑒(𝑠) represent the real part of the complex number 𝑠. In (31), for any 𝜏∈ℝ⁺, if we assume that ℜ𝑒(𝑠)≥0, then we have |𝑒^−𝑠𝜏|≤1 and 𝑃_𝑖(𝑖=0,1,2,…,𝑟) are bounded. Denote 𝑄=max_{1≤𝑖≤𝑟}𝑃_𝑖𝑃₀,𝑀=max{1,(1+𝑟)𝑄}. (32) When |𝑠^𝛼|>𝑀 and ℜ𝑒(𝑠)≥0, the polynomial Δ(𝑠,𝜏) has the following estimates: |Δ(≥𝑃𝑠,𝜏)|₀𝑠^𝛼𝑟𝑃1−¹𝑃₀1|𝑠^𝛼|−𝑃₂𝑃₀1(𝑠^𝛼)²𝑃−⋯−^𝑟𝑃₀1(𝑠^𝛼)^𝑟≥𝑀^𝑟𝑃₀1−𝑟𝑄(1+𝑟)𝑄>0. (33) Thus, for |𝑠^𝛼|>𝑀 and ℜ𝑒(𝑠)≥0, equation Δ(𝑠,𝜏)=0 has no root.
In addition, from condition (H₁), we know that the roots of equation Δ(𝑠,0)=0 are distributed in open left-half complex plane. When 𝜏=0 increases from zero, the roots of equation Δ(𝑠,𝜏)=0 are distributed in right-half complex plane if and only if there exists some 𝑠(|𝑠|≤𝑀^1/𝛼) such that 𝑠 passes through the imaginary axis. Yet, this is impossible by condition (H₂). Therefore, all the roots of equation Δ(𝑠,𝜏)=0 lie in open left-half complex plane.

According to the proof of Theorem 9, we immediately have the following result.

Corollary 10. Assume that (𝐸,𝐴) is regular and ind(𝐸,𝐴)=1; then system (1) is delay-independently asymptotically stable if the following conditions simultaneously hold.(H₁)′All the roots of equation Δ(𝑠,0)=det𝑠^𝛼𝐸−𝐴−𝐵=0 (34) have negative real parts.(H₂)For any 𝑦∈ℝ, 𝜏∈ℝ⁺ and 𝑖=√−1, then Δ𝑖𝑦,𝜏=det𝑖𝑦^𝛼𝐸−𝐴−𝐵𝑒^{−𝑖𝑦𝜏}≠0. (35)

Note that condition (H₂) in Theorem 9 is a transcendental inequality, which is not convenient to use. Then we give an improved form as follows.

Theorem 11. Assume that (𝐸,𝐴) is regular and ind(𝐸,𝐴)=1; then system (1) is delay-independently asymptotically stable if the following conditions simultaneously hold.(H₃)All the eigenvalues 𝜆s of matrix 𝐴₁+𝐵₁ satisfy arg𝜆𝐴₁+𝐵₁>𝛼𝜋2. (36)(H₄)For any 𝑦∈ℝ, all the eigenvalues of the complex function matrix 𝐺𝑦=𝐴+𝐵−𝑖𝑦^𝛼𝐸⁻¹1𝐵−2𝐼 (37)have negative real parts, that is, ℜ𝑒[𝜆(𝐺(𝑦))]<0.

Proof. From Theorem 9, we only need to prove condition (H₄) is equivalent to (H₂).
In fact, it follows from (H₁) and Corollary 10 that matrix (𝑖𝑦)^𝛼𝐸−𝐴−𝐵 is nonsingular; then we have Δ𝑖𝑦,𝜏=det𝑖𝑦^𝛼𝐸−𝐴−𝐵𝑒^{−𝑖𝑦𝜏}=det𝑖𝑦^𝛼𝐸−𝐴−𝐵×det𝐼−𝑖𝑦^𝛼𝐸−𝐴−𝐵⁻¹𝐵𝑒^{−𝑖𝑦𝜏}−1≠0. (38) Therefore, det𝐼−𝑖𝑦^𝛼𝐸−𝐴−𝐵⁻¹𝐵𝑒^{−𝑖𝑦𝜏}−1≠0. (39)
Obviously, if 𝑦𝜏=2𝑘𝜋(𝑘=0,±1,±2,…), then inequality (39) holds. Now, let 𝑦𝜏≠2𝑘𝜋(𝑘=0,±1,±2,…); then we have 1det𝑒^{−𝑖𝑦𝜏}−1𝐼−𝑖𝑦^𝛼𝐸−𝐴−𝐵⁻¹𝐵≠0,∀𝑦𝜏≠2𝑘𝜋,𝑘=0,±1,±2,…,𝑦∈ℝ,𝜏>0. (40) From inequality (40), we know that (𝑒^{−𝑖𝑦𝜏}−1)⁻¹ is not the eigenvalue of matrix [(𝑖𝑦)^𝛼𝐸−𝐴−𝐵]⁻¹𝐵, here 𝑦𝜏≠2𝑘𝜋(𝜏>0,𝑘=0,±1,±2,…). In addition, it is easy to know 𝑒^{−𝑖𝑦𝜏}−1⁻¹1=−2+𝑖sin𝑦𝜏2.1−cos𝑦𝜏 (41) Then inequality (40) is equivalent to ℜ𝑒𝜆𝑖𝑦^𝛼𝐸−𝐴−𝐵⁻¹𝐵1≠−2,∀𝑦∈ℝ. (42) When 𝑦→∞, we have 𝜆𝑖𝑦^𝛼𝐸−𝐴−𝐵⁻¹𝐵⟶0, (43) but 𝑒^{−𝑖𝑦𝜏}−1⁻¹≥12. (44) Therefore, inequality (40) is equivalent to ℜ𝑒𝜆𝑖𝑦^𝛼𝐸−𝐴−𝐵⁻¹𝐵1>−2,∀𝑦∈ℝ. (45) Accordingly, we have ℜ𝑒𝜆𝐴+𝐵−𝑖𝑦^𝛼𝐸⁻¹𝐵<12,∀𝑦∈ℝ, (46) which shows that condition (H₂) implies that condition (H₄) holds and vice versa. The proof is completed.

Next, we further consider the delay-independently asymptotic stability of system (2). The characteristic polynomial of system (2) can be represented as ℎ(𝑠,𝜏)=det𝑠^𝛼𝐸−𝐴−𝐵𝑒^−𝑠𝜏−𝑠^𝛼𝐶𝑒^−𝑠𝜏. (47)

Theorem 12. Assume that (𝐸,𝐴) is regular and ind(𝐸,𝐴)=1; then system (2) is delay-independently asymptotically stable if the following conditions simultaneously hold.(H₅)All the roots of equation ℎ(𝑠,0)=det𝑠^𝛼𝐸−𝐴−𝐵−𝑠^𝛼𝐶=0 (48) have negative real parts.(H₆)For any 𝑦∈ℝ and 𝜏∈ℝ⁺, then ℎ𝑖𝑦,𝜏=det𝑖𝑦^𝛼𝐸−𝐴−𝐵𝑒^{−𝑖𝑦𝜏}−𝑖𝑦^𝛼𝐶𝑒^{−𝑖𝑦𝜏}≠0. (49)

Proof. Let 𝑧(𝑡)=𝐷^𝛼𝑥(𝑡)); then system (2) can be written as 𝐷^𝛼𝑥(𝑡)(𝑡),(𝑡)(𝑡)()().=𝑧0=𝐴𝑥−𝐸𝑧+𝐵𝑥𝑡−𝜏+𝐶𝑧𝑡−𝜏 (50) Take 𝑥(𝑡)=𝑥(𝑡)𝑧(𝑡),𝐸=,𝐼000𝐴=,0𝐼𝐴−𝐸𝐵=;00𝐵𝐶 (51) then system (18) is equivalent to 𝐸𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏). (52) Hence, system (2) is delay-independently asymptotically stable if and only if system (52) is delay-independently asymptotically stable. Since det𝑠^𝛼𝐸−𝐴−𝐵𝑠=det^𝛼𝑠𝐼−𝐼−𝐴−𝐵𝐸−𝐶=det^𝛼𝐸−𝐴−𝐵−𝜆𝐶,det𝑖𝑦^𝛼𝐸−𝐴−𝐵𝑒^{−𝑖𝑦𝜏}=det𝑖𝑦^𝛼𝐼−𝐼−𝐴−𝐵𝑒^{−𝑖𝑦𝜏}𝐸−𝐶𝑒^{−𝑖𝑦𝜏}=det𝑖𝑦^𝛼𝐸−𝐴−𝐵𝑒^{−𝑖𝑦𝜏}−𝑖𝑦^𝛼𝐶𝑒^{−𝑖𝑦𝜏} (53)(54) are true, by Theorem 9 and Corollary 10, it is easy to get the conclusion. The proof is completed.

Similar to the analysis of Theorem 11, we have the following result.

Theorem 13. Assume that (𝐸,𝐴) is regular and ind(𝐸,𝐴)=1; then system (2) is delay-independently asymptotically stable if the following conditions simultaneously hold.(H₇)All the roots of equation ℎ(𝑠,0)=det𝑠^𝛼𝐸−𝐴−𝐵−𝑠^𝛼𝐶=0 (55) have negative real parts.(H₈)For any 𝑦∈ℝ, all the eigenvalues of the complex function matrix 𝐻𝑦=𝐴+𝐵−𝑖𝑦^𝛼(𝐸−𝐶)⁻¹𝐵+𝑖𝑦^𝛼𝐶−12𝐼 (56) have negative real parts, that is, ℜ𝑒[𝜆(𝐻(𝑦))]<0.

Proof. We only need to prove that condition (H₈) is equivalent to (H₆). From condition (H₅), we know that the matrix (𝑖𝑦)^𝛼(𝐸−𝐶)−𝐴−𝐵 is nonsingular. Then, it follows from (H₆) that ℎ𝑖𝑦,𝜏=det𝑖𝑦^𝛼𝐸−𝐴−𝐵+𝑖𝑦^𝛼𝐶𝑒^{−𝑖𝑦𝜏}=det𝑖𝑦^𝛼(𝐸−𝐶)−𝐴−𝐵×det𝐼−𝑖𝑦^𝛼(𝐸−𝐶)−𝐴−𝐵⁻¹×𝐵+𝑖𝑦^𝛼𝐶𝑒^{−𝑖𝑦𝜏}−1≠0. (57) Then we obtain det𝐼−𝑖𝑦^𝛼(𝐸−𝐶)−𝐴−𝐵⁻¹𝐵+𝑖𝑦^𝛼𝐶𝑒^{−𝑖𝑦𝜏}−1≠0,∀𝑦𝜏≠2𝑘𝜋,𝑘=0,±1,±2,…,𝑦∈ℝ,𝜏>0. (58)
Obviously, if 𝑦𝜏=2𝑘𝜋(𝑘=0,±1,±2,…), then inequality (58) holds. Now, let 𝑦𝜏≠2𝑘𝜋(𝑘=0,±1,±2,…); then we have 1det𝑒^{−𝑖𝑦𝜏}−1𝐼−𝑖𝑦^𝛼(𝐸−𝐶)−𝐴−𝐵⁻¹𝐵+𝑖𝑦^𝛼𝐶≠0,∀𝑦𝜏≠2𝑘𝜋,𝑘=0,±1,±2,…,𝑦∈ℝ,𝜏>0. (59) From inequality (59), we know that (𝑒^{−𝑖𝑦𝜏}−1)⁻¹ is not the eigenvalue of matrix 𝑖𝑦^𝛼(𝐸−𝐶)−𝐴−𝐵⁻¹𝐵+𝑖𝑦^𝛼𝐶, (60) where 𝑦𝜏≠2𝑘𝜋(𝜏>0,𝑘=0,±1,±2,…). In addition, it is easy to know 𝑒^{−𝑖𝑦𝜏}−1⁻¹1=−2+𝑖sin𝑦𝜏2.1−cos𝑦𝜏 (61) Then inequality (59) is equivalent to ℜ𝑒𝜆𝑖𝑦^𝛼(𝐸−𝐶)−𝐴−𝐵⁻¹𝐵+𝑖𝑦^𝛼𝐶1≠−2. (62) When 𝑦→∞, we have 𝜆𝑖𝑦^𝛼(𝐸−𝐶)−𝐴−𝐵⁻¹𝐵+𝑖𝑦^𝛼𝐶⟶0, (63) but 𝑒^{−𝑖𝑦𝜏}−1⁻¹≥12. (64) Therefore, inequality (59) is equivalent to ℜ𝑒𝜆𝐴+𝐵−𝑖𝑦^𝛼(𝐸−𝐶)⁻¹𝐵+𝑖𝑦^𝛼𝐶<12, (65) which shows that condition (H₆) implies that condition (H₈) is true and vice versa. The proof is completed.

Remark 14. Compared with Theorems 9 and 12, Theorems 11 and 13 actually offer two algebraic criteria of the delay-independently asymptotic stability of delayed fractional-order linear singular differential systems, which is more concise and convenient to check the stability of such systems without solving the transcendental equations.

4. An Illustrative Example

The following example is presented to illustrate the usefulness of the proposed theoretical results.

Example 1. Consider two-dimensional Caputo fractional-order singular differential difference system 𝐷^1/3𝑥₁(𝑡)=−𝑥₁(𝑡),𝑥₂(𝑡)=12𝑥₂(𝑡−𝜏). (66) It can be written as 𝐷^1/3𝑥₁(𝑡)=−𝑥₁(𝑡),0=𝑥₂(𝑡)−12𝑥₂(𝑡−𝜏). (67) Then we have 𝐸=,1000𝐴=,−10011𝐵=000−2,𝐴₁+𝐵₁𝐺𝑦==−2,𝐴+𝐵−𝑖𝑦^𝛼𝐸⁻¹1𝐵−2−1𝐼=2030−2. (68) It follows from equation det[𝜆𝐼−𝐴₁−𝐵₁]=0 that arg(𝜆)=𝜋>𝛼𝜋2. (69) Moreover, ℜ𝑒[𝜆(𝐺(𝑦))]<0. Thus, condition (H₃) and condition (H₄) are satisfied. From Theorem 11, we know that system (66) is delay-independently asymptotically stable.
In fact, when 𝜏=0, the solution of (50) is 𝑥₁(𝑡)=𝐸_𝛼−𝑡^𝛼𝑥₁(0),𝑥₂(𝑡)=0. (70) When 𝜏≠0, the solution of (50) is 𝑥₁(𝑡)=𝐸_𝛼−𝑡^𝛼𝑥₁(0),𝑥₂(𝑡)=12^𝑘𝑥₂(𝑡−𝑘𝜏). (71) Here 𝑡−𝑘𝜏≤0, then 𝑘≥𝑡/𝜏. When 𝑡→+∞, we have 𝑘→+∞. From [25], there exists a positive constant 𝑀₁≥1, such that for 𝛼=1/3∈(0,1)𝐸_𝛼−𝑡^𝛼≤𝑀₁𝑒^−𝑡. (72) When 𝑡→+∞, one can obtain 𝑥₁(𝑡)⟶0,𝑥₂(𝑡)⟶0. (73) Therefore, the zero solution of system (66) is asymptotically stable for any 𝜏∈ℝ⁺; that is, system (66) is delay-independently asymptotically stable.

5. Conclusions

In this paper, the delay-independently asymptotic stability of delayed fractional-order linear singular differential systems has been discussed. We have synchronously taken into account the factors of such systems including Caputo fractional-order derivative, state delay, and singular coefficient matrices. In terms of the algebraic approach, some sufficient conditions are derived to ensure the asymptotic stability of the systems without solving the transcendental equations, which are very convenient to check the stability of such systems. An example is also provided to illustrate the theoretical results.

Conflict of Interests

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

Acknowledgments

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, the Programs of Educational Commission of Anhui Province of China under Grant nos. KJ2011A197 and KJ2013Z186.