Stability Analysis for Fractional-Order Linear Singular Delay Differential Systems

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.

The fractional dynamics system is a recent focus of interest to many researchers .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 fractionalorder differential systems [18][19][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 fractionalorder delay differential systems (see [21][22][23][24][25][26][27][28][29][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 fractionalorder 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][4][5][6][7][8][9][10] and fractional-order delay differential systems [21][22][23][24][25][26][27][28][29][30], the stability theory of fractionalorder 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: and the Caputo fractional-order singular neutral delay differential system where 0 <  < 1; () ∈ R  is the state vector;   () denotes an  order Caputo fractional-order derivative of (); matrices , , , ,  ∈ R × and matrices ,  ∈ R × are singular with rank() =  < , rank() =  < ;  ∈ R + is the time delay; and ,  are both the consistent initial functions.Different from the methods in [18][19][20][21][22][23][24][25][26][27][28][29][30][31][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  ∈ 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 delayindependently 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.

Preliminaries
In this section, we recall some definitions of fractional calculus (see [11][12][13][14]) and preliminary facts used in the paper.
According to the mathematical induction, we know that system (2) has a unique solution on [0, +∞).Therefore, the proof is completed.
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 () ≡ 0 is the zero solution of system (1) and system (2), respectively.Definition 8. Assume that (, ) (or (, )) is regular; then the zero solution () ≡ 0 of system (1) (or (2)) is called delay-independently asymptotically stable if, for any consistent initial function for any delay parameter  > 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 (, ) is regular, then there exist two nonsingular matrices ,  ∈ R × , such that system (1) is equivalent to canonical system as follows: where 0 <  < 1,  1 ∈ R  ,  2 ∈ R − , and  is nilpotent whose nilpotent index is denoted by ] = ind(, ), that is,  ] = 0,  ]−1 ̸ = 0, and In particular, when  = 0 and ind(, ) = 1, system (1) is equivalent to the following canonical system without delay: In addition, taking the Laplace transform on both sides of system (1) yields where 0 <  < 1.It follows from the properties of integral that Thus, we obtain where is the characteristic polynomial of system (1).

Stability Criteria
In this section, we derive the sufficient conditions of the delay-independently asymptotic stability for system (1) and system (2), respectively.
(H 1 ) All the eigenvalues s of matrix  1 +  1 satisfy (H 2 ) For any  ∈ R,  ∈ R + , and  = √ −1, then Proof.For any  ∈ R + , we only need to prove that all the roots of equation Δ(, ) = 0 lie in open left-half complex plane.When  = 0 and ind(, ) = 1, system (1) is reduced to the slow fractional-order subsystem and the fast subsystem Thus, the asymptotic stability of system (18) entirely depends on  1 () irrespective of  2 ().
From condition (H 1 ), any root of equation det( Let  =   , then we have  =  1/ .It follows from inequality (27) that where  is just the root of the following characteristic equation of the slow fractional-order subsystem By computation of the determinants, we have det Taking into account ( 28), (29), and (30), one can get that all the roots of equation det(   −  − ) = 0 have negative real parts.Hence, it follows from (H 1 ) that system ( 18) is asymptotically stable.On the other hand, since rank() = , the characteristic polynomial of system (1) can be expressed as where   ( = 0, 1, 2, . . ., ) are the polynomial of the exponential function  − , and their coefficients are composed of the elements of matrices , , .
According to the proof of Theorem 9, we immediately have the following result.
Note that condition (H 2 ) in Theorem 9 is a transcendental inequality, which is not convenient to use.Then we give an improved form as follows.
(H 3 ) All the eigenvalues s of matrix  1 +  1 satisfy (H 4 ) For any  ∈ R, all the eigenvalues of the complex function matrix have negative real parts, that is, R[(())] < 0.
Next, we further consider the delay-independently asymptotic stability of system (2).The characteristic polynomial of system (2) can be represented as Theorem 12. Assume that (, ) is regular and ind(, ) = 1; then system (2) is delay-independently asymptotically stable if the following conditions simultaneously hold. det 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.

An Illustrative Example
The following example is presented to illustrate the usefulness of the proposed theoretical results.
In fact, when  = 0, the solution of (50) is Therefore, the zero solution of system (66) is asymptotically stable for any  ∈ R + ; that is, system (66) is delayindependently asymptotically stable.

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.