Research Article | Open Access

Hai Zhang, Daiyong Wu, Jinde Cao, "Asymptotic Stability of Caputo Type Fractional Neutral Dynamical Systems with Multiple Discrete Delays", *Abstract and Applied Analysis*, vol. 2014, Article ID 138124, 10 pages, 2014. https://doi.org/10.1155/2014/138124

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

**Academic Editor:**Anna Mercaldo

#### 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: wheredenotes anorder Caputo fractional derivative of,, areconstant matrices,are real constants with, the initial function, anddenotes space of continuously differentiable functions mapping the intervalinto.

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, represents the determinant of matrix , denotes the spectrum of matrix,represents the spectral radius of matrix, and stands for the principal argument ofdefined on.(a)Riemann-Liouville’s fractional integral of order for a functionis given by whereis Euler’s gamma function.(b)Riemann-Liouville’s fractional derivative of order for a functionis given by where.(c)Caputo’s fractional derivative of order for a functionis defined as where, . Here,is still written as.(d)The Laplace transform of a functionis defined as where denotes the complex plane andis-dimensional vector-valued function. For, it follows from [1–4] that 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 In particular, for, the Mittag-Leffler function in one parameter is given by

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.*

*Proof. *For system (1), on the interval,. Thus, when, system (1) is given by
Sinceis continuous on, from [3], we know that there is a unique continuous solution for system (1) on , which is denoted as , . Furthermore,can be expressed by the following form:

For, system (1) is given by
Similarly, sinceis continuous on, we obtain thatis a unique continuous solution of system (1) on and

Assume that system (1) has a unique solutionon. For, system (1) is given by
Similarly, sinceis continuous on, we obtain thatis a unique continuous solution of system (1) on and

According to the mathematical induction, we know that system (1) has a unique continuous solution on.

Now, for any, we assert that system (1) has a unique continuous solution on. In fact, three cases are discussed as follows.*Case* *1*. When, we know that the assertion is true.*Case* *2*. When, we only need to prove that system (1) has a unique continuous solution on. For, we denote, and we can use the similar proof to obtain the conclusion.*Case* *3*. When, we can repeat the above process until the condition of Case 2 is satisfied.

Note thatis an arbitrary positive real number; then, we know that system (1) has a unique continuous solution onTherefore, the proof is completed.

*Remark 2. *Lemma 1 ensures the existence and uniqueness of solution for system (1) on. Evidently, Caputo’s fractional derivative of a constant is equal to zero; then,is the zero solution of system (1).

*Definition 3. *The zero solutionof system (1) is called delay-independent globally asymptotically stable if, for any initial function, its analytic solutionsatisfiesfor all the time delays.

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 derivativeis given as follows:
Applying the Laplace transform on both sides of system (1) yields
Note that
thus, we obtain
where
is the characteristic matrix of system (1). Multiplyingon both sides of (18) yields
By means of the final-value theorem of Laplace transform [45] and Definition 3, if all the roots of characteristic equationlie 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
**
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, whenand, the characteristic equation
is an algebraic equation of, and (22) only hasroots distributed in the complex plane. However, the characteristic equationhas countably infinite roots withand some(see [13]). Forand 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 equationin any neighborhood of the infinity and find a positive numbersuch that any characteristic rootsatisfies where represents the real part of the complex number.

Theorem 6. *The zero solutionof system (1) is delay-independent globally asymptotically stable if the following conditions are satisfied:
*

*Proof. *Note that; then, all the roots of equationsatisfy. Let; then, all the roots of equationsatisfy; that is, . Therefore, matrixis invertible andis well defined whenand .

For , it follows from the characteristic polynomial of system (1) that
Then, we have
Combining (24) and (25), we havefor and; that is, if, conditionsandare satisfied, then the characteristic equation (21) implies that .

Suppose that there exists a sequence of rootsof the characteristic equation (21) whose real parts are not uniformly bounded away from zero; that is, and as. Note that any eigenvalueis a continuous function offor ; then, it follows fromthat
Hence,reach the maximum value for . From condition, there exists a positive constantsuch that
Then, equality (27) implies that
When the positive integeris large enough, there exist a positive constantand a characteristic rootsuch thatis sufficiently small, and
For , from (28) and (29), we have
Choosinglarge enough yields
Therefore, for and as, one can obtain
which contradicts the assumption thatis a sequence of roots of the characteristic equation (21). In view of Lemma 4, the proof is completed.

Theorem 7. *The zero solutionof system (1) is delay-independent globally asymptotically stable if the following conditions are satisfied:
*

*Proof. *According to conditionand the proof of Theorem 6, we know that matrixis invertible andis well defined whenand .

On the one hand, any eigenvalueis an algebraic function ofwithand is continuous withAn application of the maximum modulus principle yields that
is equivalent to
In view of the expression of, as and, one can obtain
It follows from (36) that
Applying the maximum modulus principle on the unbounded region, then (35) implies that

On the other hand, suppose that there exists somesuch that
Choosesuch that. It follows from (39) that
which contradicts inequality (38). Therefore, if conditions, , andare satisfied, then the characteristic equation (21) implies that .

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 solutionof system (1) is delay-independent globally asymptotically stable if the following conditions are satisfied:
*

Assume that. Define the following matrices: Let then, we have.

Theorem 9.

*Proof. *From Theorem 6, we only need to prove that the following equality holds:

In fact, it is easy to obtain
For , it follows fromand (43) that; then, matrixis invertible. Hence,
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) When, system (1) reduces to Caputo fractional-order linear retarded type differential difference systems with multiple delays as follows: wheredenotes anorder Caputo fractional derivative of,,areconstant matrices,are real constants with,, anddenotes space of continuous functions mapping the intervalinto.

Similar to Lemma 1, if the initial function, then there exists a unique continuous solution for system (48) on.

Corollary 10. *The zero solutionof system (48) is delay-independent globally asymptotically stable if the following conditions are satisfied:
*

(ii) When, system (1) reduces to Caputo fractional-order linear autonomous differential systems: wheredenotes anorder Caputo fractional derivative of,, and areconstant matrices.

An application of the results in [23, 24] yields the following conclusion.

Corollary 11. *The zero solutionof system (50) is globally asymptotically stable if the following conditions are satisfied:
*

*Remark 12. *When, 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
The initial function is given by, . Let
By computation, the eigenvalues of matrixgive
Note that; then, we have
It is not difficult to verify that
Thus, conditions, and are satisfied. Therefore, it follows from Theorem 7 that the zero solutionof 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
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 solutionof 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 (), (), and (). An application of Theorem 7 yields that the zero solutionof 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.

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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.