Research Article | Open Access

# Stability Analysis for Fractional-Order Linear Singular Delay Differential Systems

**Academic Editor:**Guanghui Wen

#### 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], -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: and the Caputo fractional-order singular neutral delay differential system where ; is the state vector; denotes an order Caputo fractional-order derivative of ; matrices and matrices are singular with rank, ; is the time delay; and , 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 . 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 for a function is defined as
where is the Gamma function.

*Definition 2 (see [12]). *The Riemann-Liouville’s fractional derivative of order for a function is defined as
where , .

*Definition 3 (see [12]). *The Caputo’s fractional derivative of order for a function is defined as
where , . For the sake of convenience, is still written as .

*Definition 4 (see [12]). *The Mittag-Leffler function in two parameters is defined as
where , and and denotes the complex plane. In particular, for , the Mittag-Leffler function in one parameter is defined as

For , the Laplace transform of Caputo fractional-order derivative is given as follows:

Throughout this paper, let be the space of all continuous functions mapping the interval into , and denotes the space of all continuous differential functions mapping the interval into . The consistent initial function and .

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 , the matrix pair is called regular if is not identically zero, where .

Lemma 6. *Assume that is regular, and is the consistent initial function, then system (2) has a unique solution on .*

*Proof. *For system (2), on the interval , . Thus, when , system (2) is given by
that is,
Since is regular, and is continuous on , by applying Theorem 3.2 in [32], we obtain that is a unique solution of system (2) on .

For , system (2) is given by
that is,
Since is continuous on , similarly, we obtain that is a unique solution of system (2) on .

Assume that system (2) has a unique solution on . For , system (2) is given by
that is,
Since is continuous on , similarly, we obtain that is a unique solution of system (2) on .

According to the mathematical induction, we know that system (2) has a unique solution on . Therefore, the proof is completed.

Lemma 7 (see [32]). *Assume that is regular and is the consistent initial function; then system (1) has a unique solution on .*

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 is the zero solution of system (1) and system (2), respectively.

*Definition 8. *Assume that (or ) is regular; then the zero solution of system (1) (or (2)) is called delay-independently asymptotically stable if, for any consistent initial function (or ), its analytic solution satisfies
for any delay parameter .

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 , such that system (1) is equivalent to canonical system as follows: where , , , and is nilpotent whose nilpotent index is denoted by , that is, , , and

In particular, when and , 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 . It follows from the properties of integral that Thus, we obtain where 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 is regular, and ; then system (1) is delay-independently asymptotically stable if the following conditions simultaneously hold.*(H_{1})*All the eigenvalues s of matrix satisfy
*(H_{2})*For any , , and , then
*

*Proof. *For any , we only need to prove that all the roots of equation lie in open left-half complex plane.

When and , system (1) is reduced to the slow fractional-order subsystem
and the fast subsystem
Thus, the asymptotic stability of system (18) entirely depends on irrespective of .

From condition , any root of equation satisfies
Let , then we have . 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
Taking into account (28), (29), and (30), one can get that all the roots of equation have negative real parts. Hence, it follows from that system (18) is asymptotically stable.

On the other hand, since , the characteristic polynomial of system (1) can be expressed as
where are the polynomial of the exponential function , and their coefficients are composed of the elements of matrices , , .

Let represent the real part of the complex number . In (31), for any , if we assume that , then we have and are bounded. Denote
When and , the polynomial has the following estimates:
Thus, for and , equation has no root.

In addition, from condition , we know that the roots of equation are distributed in open left-half complex plane. When increases from zero, the roots of equation are distributed in right-half complex plane if and only if there exists some such that passes through the imaginary axis. Yet, this is impossible by condition . Therefore, all the roots of equation lie in open left-half complex plane.

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

Corollary 10. *Assume that is regular and ; then system (1) is delay-independently asymptotically stable if the following conditions simultaneously hold.*(H_{1})′*All the roots of equation
* *have negative real parts.*(H_{2})*For any , and , then
*

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.

Theorem 11. *Assume that is regular and ; then system (1) is delay-independently asymptotically stable if the following conditions simultaneously hold.*(H_{3})*All the eigenvalues s of matrix satisfy
*(H_{4})*For any , all the eigenvalues of the complex function matrix
**have negative real parts, that is, .*

*Proof. *From Theorem 9, we only need to prove condition is equivalent to .

In fact, it follows from and Corollary 10 that matrix is nonsingular; then we have
Therefore,

Obviously, if , then inequality (39) holds. Now, let ; then we have
From inequality (40), we know that is not the eigenvalue of matrix , here . In addition, it is easy to know
Then inequality (40) is equivalent to
When , we have
but
Therefore, inequality (40) is equivalent to
Accordingly, we have
which shows that condition implies that condition 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

Theorem 12. *Assume that is regular and ; then system (2) is delay-independently asymptotically stable if the following conditions simultaneously hold.*(H_{5})*All the roots of equation
* *have negative real parts.*(H_{6})*For any and , then
*

*Proof. *Let ; then system (2) can be written as
Take
then system (18) is equivalent to
Hence, system (2) is delay-independently asymptotically stable if and only if system (52) is delay-independently asymptotically stable. Since
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 is regular and ; then system (2) is delay-independently asymptotically stable if the following conditions simultaneously hold.*(H_{7})*All the roots of equation
* *have negative real parts.*(H_{8})*For any , all the eigenvalues of the complex function matrix
* *have negative real parts, that is, .*

*Proof. *We only need to prove that condition is equivalent to . From condition , we know that the matrix is nonsingular. Then, it follows from that
Then we obtain

Obviously, if , then inequality (58) holds. Now, let ; then we have
From inequality (59), we know that is not the eigenvalue of matrix
where . In addition, it is easy to know
Then inequality (59) is equivalent to
When , we have
but
Therefore, inequality (59) is equivalent to
which shows that condition implies that condition 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
It can be written as
Then we have
It follows from equation that
Moreover, . Thus, condition and condition are satisfied. From Theorem 11, we know that system (66) is delay-independently asymptotically stable.

In fact, when , the solution of (50) is
When , the solution of (50) is
Here , then . When , we have . From [25], there exists a positive constant , such that for
When , one can obtain
Therefore, the zero solution of system (66) is asymptotically stable for any ; 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.

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Copyright © 2014 Hai Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.