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

and Applied Analysis 3 Since ∑m i=1 B i φ(t − τ i ) + D α [∑ m i=1 C i φ(t − τ i )] is continuous on [0, τ 1 ], from [3], we know that there is a unique continuous solution for system (1) on [−τ, τ 1 ], which is denoted as x 1 (t), t ∈ [−τ, τ 1 ]. Furthermore, x 1 (t) can be expressed by the following form:


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][2][3][4][5][6][7][8][9][10][11][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.
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: where ( ) denotes an order Caputo fractional derivative of ( ), 0 < < 1, , , are × constant matrices, ( = 1, 2, . . . , ) are real constants with 0 < 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][35][36][37][38][39][40][41][42][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.
For the sake of convenience, some notations are introduced firstly. Throughout this paper, det( ) represents the determinant of matrix , ( ) denotes the spectrum of matrix , [ ] represents the spectral radius of matrix , and arg( ( )) stands for the principal argument of ( ) defined on (− , ]. (a) Riemann-Liouville's fractional integral of order > 0 for a function : R + → R is given by where Γ(⋅) is Euler's gamma function.
(c) Caputo's fractional derivative of order for a function : R + → R is defined as where 0 ≤ −1 ≤ < , ∈ Z + . Here, is still written as .
(d) The Laplace transform of a function ( ) is defined as where C denotes the complex plane and ( ) isdimensional vector-valued function. For − 1 ≤ < , it follows from [1][2][3][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 = 1, 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 integerorder delay differential systems [13] to fractional-order neutral differential systems. Thus, when ∈ [0, 1 ], system (1) is given by Since [3], we know that there is a unique continuous solution for system (1) on [− , 1 ], which is denoted as 1 ( ), ∈ [− , 1 ]. Furthermore, 1 ( ) can be expressed by the following form: For ∈ [ 1 , 2 1 ], system (1) is given by Assume According to the mathematical induction, we know that system (1) has a unique continuous solution on [0, 1 ], = 1, 2, . . .. Now, for any > 0, we assert that system (1) has a unique continuous solution on [0, ]. In fact, three cases are discussed as follows.
Case 2. When 0 < − ( + 1) 1 < 1 , we only need to prove that system (1) has a unique continuous solution on , and we can use the similar proof to obtain the conclusion.
Case 3. When − ( + 1) 1 > 1 , we can repeat the above process until the condition of Case 2 is satisfied.
Note that is an arbitrary positive real number; then, we know that system (1) has a unique continuous solution on [0, +∞). Therefore, the proof is completed.
Next, we discuss the characteristic equation and delayindependent globally asymptotic stability of system (1).
From [1][2][3][4], the Laplace transform of Caputo fractionalorder derivative ( ) is given as follows: Applying the Laplace transform on both sides of system (1) yields Abstract and Applied Analysis thus, we obtain where is the characteristic matrix of system (1). Multiplying on both sides of (18) yields By means of the final-value theorem of Laplace transform [45] and  (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, when = 1 and 1 = 2 = ⋅ ⋅ ⋅ = = 0, the characteristic equation is an algebraic equation of , and (22) only has roots distributed in the complex plane. However, the characteristic equation det[Δ( , )] = 0 has countably infinite roots with = 1 and some > 0 (see [13]). For 0 < < 1 and > 0 ( = 1, 2, . . . , ), 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. (1) In this section, we derive the sufficient conditions of delayindependent globally asymptotic stability for system (1). Applying the algebraic method, we investigate the distribution of roots for equation det[Δ( , )] = 0 in any neighborhood of the infinity and find a positive number > 0 such that any characteristic root satisfies R ( ) < − < 0, where R ( ) represents the real part of the complex number . Theorem 6. The zero solution ( ) ≡ 0 of system (1) is delayindependent globally asymptotically stable if the following conditions are satisfied:
Suppose that there exists a sequence of roots { } of the characteristic equation (21) whose real parts are not uniformly bounded away from zero; that is, R ( ) < 0 and R ( ) → 0 as → +∞. Note that any eigenvalue [( − ) −1 ∑ =1 ( + )] is a continuous function of for R ( ) ≥ 0; then, it follows from ( 2 ) that Then, equality (27) implies that When the positive integer is large enough, there exist a positive constant * (0 < * < ) and a characteristic root such that |R ( )| is sufficiently small, R ( ) < 0 and max 1≤ ≤ For R ( ) = 0, from (28) and (29), we have Choosing large enough yields Therefore, for R ( ) < 0 and R ( ) → 0 as → +∞, one can obtain which contradicts the assumption that { } is a sequence of roots of the characteristic equation (21). In view of Lemma 4, the proof is completed.
Next, the asymptotic stability criteria for two special cases of system (1) are presented.
An application of the results in [23,24] yields the following conclusion.

An Illustrative Example
The following example is presented to illustrate the effectiveness and applicability of the proposed stability criteria.
It is not difficult to verify that Thus, conditions ( 1 ), ( 3 ), and ( 4 ) are satisfied. Therefore, it follows from Theorem 7 that the zero solution ( ) ≡ 0 of system (1) with the coefficient matrices (52) is delayindependent 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 solution ( ) ≡ 0 of system (1) with (52) is delayindependent globally asymptotically stable.

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 ( 1 ), ( 4 ), and ( 5 ). An application of Theorem 7 yields that the zero solution ( ) ≡ 0 of 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.