Stability analysis of fractional-order linear neutral delay differential–algebraic system described by the Caputo–Fabrizio derivative

This paper is concerned with the asymptotic stability of linear fractional-order neutral delay differential–algebraic systems described by the Caputo–Fabrizio (CF) fractional derivative. A novel characteristic equation is derived using the Laplace transform. Based on an algebraic approach, stability criteria are established. The effect of the index on such criteria is analyzed to ensure the asymptotic stability of the system. It is shown that asymptotic stability is ensured for the index-1 problems provided that a stability criterion holds for any delay parameter. Also, asymptotic stability is still valid for higher-index problems under the conditions that the system matrices have common eigenvectors and each pair of such matrices is simultaneously triangularizable so that a stability criterion holds for any delay parameter. An example is provided to demonstrate the effectiveness and applicability of the theoretical results.


Introduction
Fractional calculus is attracting more and more researchers in applied sciences and engineering because of the many advantages of fractional derivatives which provide important tools in mathematical modeling related to many interdisciplinary areas, see, e.g., [20-28, 34, 40]. Any physical system can be represented more accurately through a fractional system. Also it has been found that it is more appropriate to capture the real dynamical behavior rather than classical calculus. It should be pointed out that fractional calculus has gained the popularity due to its peculiar properties and recent progress of research in this area. For more details, one can see [3,5,10,12,41]. There are different types of fractional derivatives, those of Riemann-Liouville and Caputo are the most popular in the literature [4,19,32,37,39].
In their recent work, Caputo and Fabrizio [7] introduced a new fractional-order derivative with a nonsingular kernel, hereinafter called the fractional Caputo-Fabrizio (CF) derivative. This new fractional derivative is less affected by the past compared to the Caputo fractional derivative, which may exhibit slow stabilization [1,33]. The properties and numerical aspects of the CF derivative and their corresponding fractional integrals been studied in [2,6,8,11,18,30,38]. In this paper, we are interested in linear fractional-order neutral delay differential-algebraic equations described by the CF derivative. The presence of differential and difference operators as well as the algebraic constraints makes the study of such equations more complicated than that for standard fractional delay differential equations or fractional delay algebraic equations. For this reason, recently, a great attention has been paid to fractional delay differentialalgebraic systems. One of the most important research topics of the theory of such systems is the stability analysis. However, in the literature cited above, there are only few results (see, e.g., [16,17,35,36]) on this topic. The stability of such systems has some particular properties including regularity behavior and the index which is a very important characteristic of fractional delay differential-algebraic equations but which does not need to be considered in standard differential systems. Most of the considered fractional delay differential-algebraic systems, so far, are of index one (impulsefree). However, if the index is other than one then classical results on stability fail for fractional delay differential-algebraic equation. So this paper aims to make some contribution to fill this research gap. In [29] the authors consider the asymptotic stability of linear fractional-order ordinary differential equations described by the CF derivative, whereas the authors of [31] consider the stability analysis of a linear fractional-order system with time delay, establish a characteristic equation using the Laplace transform, and provide some brief sufficient stability conditions. While being different, we extend the analysis carried out in the above cited references. We apply a spectrum-based approach to establish asymptotic stability criteria for fractional-order neutral delay differentialalgebraic equations, and the novelty of this work lies in the following aspects. Firstly, the CF definition of the fractional derivative is applied to analyze linear fractional-order differential-algebraic systems including neutral time delay and singular coefficient matrices. Secondly, by using the Laplace transform, we establish a novel characteristic equation. Thirdly, we apply an algebraic approach to establish sufficient asymptotic stability criteria ensuring that all the roots of characteristic equation lie in open left half of the complex plane. Fourthly, we extend the asymptotic stability results to a broader class of linear systems which are regular, not impulse-free, and with a non-commutative family of matrices. We consider such systems with simultaneously triangularizable matrices, where the condition on simultaneous triangularization of a pair of matrices can be extended so that the rank of their commutator is less than or equal to one. A nice consequence is that these stability criteria avoid solving the characteristic (transcendental) equation.
The rest of the paper is organized as follows: In Sect. 2 we formulate the problem and introduce notation that will be used throughout the paper. In Sect. 3 we establish the main results for the asymptotic stability criteria for fractional neutral delay differentialalgebraic equations. In Sect. 4 we provide an example to illustrate the effectiveness and applicability of the proposed criteria. Finally, some concluding remarks are given in Sect. 5.

Problem formulation and notations
We consider the linear fractional-order neutral delay differential-algebraic equations (FNDDAEs): where 0 < α < 1, y(t) is a real vector of size n (the state vector); A, B, C, E are real constant matrices of size n × n, and E is assumed to be singular with rank(E) = r < n; the positive pareameter τ is the time delay and ϕ is a consistent initial function. The notation CF D α y(t) stands for Caputo-Fabrizio fractional-order derivative of order α of y(t) defined by (see [7]) The following notations will be used throughout the paper. For a complex matrix X, det[X], σ [X], X , and ρ[X] denote respectively its determinant, spectrum, spectral norm, and spectral radius. The open left half complex plane is denoted by C -. The symbol L denotes the Laplace transform.

Main results
Applying the Laplace transform to (1), we obtain We have Then (2) leads to and after rearrangement to Setting y 0 (s) = (E -Ce -sτ )y(0), we get where Since the distribution of eigenvalues of P(s) totally determines the stability of system (1), the following definition is obvious. (1) is
Similar to that given in [42], sY (s) = 0. It immediately follows the theorem below.
, lie in open left half of the complex plane and are uniformly bounded away from the imaginary axis.

Solvability analysis of linear FNDDAEs
There is one well-known fact in the theory of delay algebraic equations that not only the stability of system (1) depends on spectral conditions but also the solvability is connected to the regularity of the matrix pair (E, A). Consequently, the solvability of FNDDAEs (1) is discussed under the regularity of the matrix pair (E, A).

Definition 3.3 Consider system (1) described by the CF derivative with the family of ma
System (1) is known as impulse-free if the pair (E, A) is impulse-free.

Definition 3.4
An initial function ϕ is called consistent with system (1) if the associated initial value problem IVP (1) has at least one solution. System (1) is called solvable (resp. regular) if, for every consistent initial function ϕ, the associated IVP has a solution (resp. has a unique solution).

Definition 3.5 ([13] (Quasi-Weierstrass form)) The matrix pencil (E, A)
is regular if and only if there exist matrices P, Q in R n×n such that where N is a nilpotent with index of nilpotency v and J ∈ R r×r is a matrix in Jordan canonical form. If nr > 0, we call v the index of the pencil (E, A) and write ind( Remark 3.2 The regularity of the pair (E, A) ensures that system (1) with τ = 0 is regular, and it further ensures the existence of a unique solution to system (1) on [0, ∞).
Now, according to [9] and by applying the method of steps, we obtain the following lemma which generalizes the results of fractional-order differential-algebraic system without delay described by the CF derivative to fractional-order neutral delay differentialalgebraic system (1). (1) is uniquely solvable on [0, +∞).

Algebraic analysis criteria for linear FNDDAEs
In this section, stability criteria are given based on an algebraic approach. In order to obtain the main results, we introduce the equivalent form of system (1) by means of the nonsingular transform. Assume that (E, A) is regular, then according to Definition 3.5 there exist two nonsingular matrices P, Q in R n×n such that system (1) is equivalent to a canonical system as follows: The stability of (1) and (4) is equivalent, so we might as well let or in particular case (when ind(E, A) = 1) in system (1). Algebraic stability criteria are given in the following lemma.
Lemma 3.2 Consider system (1) described by CF derivative with the family of matrices {E, A, B, C} defined in (5). If the following conditions are satisfied: Or Then, for all s ∈ C such that Re(s) ≥ 0, we have Proof To simplify the notation, let us denote Condition (C 2 ) can then be written sup Re(s)≥0 ρ[K] < 1 2 . We have Condition (C 1 ) ensures that sγ (E -C) -(A + B) is nonsingular and condition (C 2 ) ensures that ρ[K(e -sτ -1)] < 1 and hence I -K(e -sτ -1) is nonsingular. Therefore, (C 3 ) is satisfied. Also, let us denote Condition (C 2 ) can then be written sup Re(s)≥0 ρ[L] < 1. We have Condition (C 1 ) ensures that sγ E -A is nonsingular and condition (C 2 ) ensures that ρ[Le -sτ ] < 1 and hence I -Le -sτ is nonsingular. Therefore, (C 3 ) is satisfied.
Remark 3.4 Note that condition (C 3 ) in Lemma 3.2 is transcendental inequality, which is difficult to use, and since condition (C 2 ) implies that by applying the algebraic criteria (C 2 ), one can avoid solving the roots of transcendental equation, which is very convenient to check the stability of system (1).
Remark 3.5 Since the spectral radius of a matrix is always less than or equal to any induced norm of the matrix, Lemma 3.2 remains valid under condition (C 1 ) or (C 1 ) and respectively.

Asymptotic stability of linear FNDDAEs when ind(E, A) ≥ 1
In this section, an attempt to analyze the effect of the index of (E, A) on the second criterion (C 2 ) or (C 2 ) isgiven to ensure the asymptotic stability of system (1) for any delay parameter.
Under assumption (C 1 ) or (C 1 ), sometimes it is more convenient to check assumption (C 2 ) or (C 2 ) by using an operator norm instead of spectral radius, i.e., using assumption (C * 2 ) or (C * * 2 ). For the sake of simplicity, if C = 0 and from this point of view, a somewhat simpler condition can be given instead of (C 2 ) or (C * * 2 ) for explaining the effect of the index of (E, A) on the asymptotic stability of system (1).
Unfortunately, if the index of (E, A) is greater than 1, then the right-hand side of (C * * * 2 ) is simply zero, and the proposition does not apply. This once again confirms that for highindex problems, the coefficient matrices C and B must be highly structured so that the asymptotic stability would be preserved. So we first restrict the investigation of the algebraic criteria for the asymptotic stability to index-1 problems.
(a) The index-1 case (ind(E, A) = 1) Sufficient conditions for asymptotic stability of system (1) are given in the following theorem with special case when the index of (E, A) is 1. (6). If the following conditions are satisfied:

Theorem 3.2 Consider system (1) described by CF derivative with the family of matrices {E, A, B, C} defined in
Then system (1) is asymptotically stable for all values of the delay τ .
This implies that Re(s) < 0 for any root of the characteristic equation Q(s, τ ). We need to prove that the roots remain bound away from the imaginary axis. Suppose the contrary, then there exists a sequence of roots {s n } of the characteristic equation Q(s, τ ) whose Re(s n ) < 0 and Re(s n ) → 0 as n → +∞.

Since any eigenvalue
1)] is a continuous function of s for Re(s) ≥ 0.
Similar to [15] and from condition (C 2 ), there exists ε > 0 such that Hence, we have When the positive integer n is large enough, there exists a positive constant ε * (0 < ε * < ε), and a characteristic root s n s.t. | Re(s n )| is sufficiently small, Re(s n ) < 0 and max 0≤j≤n λ j s n s n (1α) For Re(ω) = 0, from (7) and (8), and for j = 1, 2, . . . , n, we have Choosing n large enough yields Therefore, for Re(s n ) < 0 and Re(s n ) → 0 as n → +∞, one can obtain which contradicts the assumption that {s n } is a sequence of roots of the characteristic equation. Similarly, by Lemma 3.2, if conditions (C 1 ) and (C 2 ) are satisfied, then condition (C 3 ) holds. This implies that Re(s) < 0 for any root of the characteristic equation Q(s, τ ), and in the same way we can prove that the roots remain bound away from the imaginary axis.   (b) The higher-index case (ind(E, A) > 1) Asymptotic stability of system (1) for the case that (E, A) is regular but is not impulsefree, i.e., ind(E, A) = v, v > 1 is considered. In fact, this type of systems has achieved only few results in special cases of the coefficient matrices (see, e.g., [13,16]). It has been shown that for some differential-algebraic equation, delay differential equation, and delay differential-algebraic equation, the commutativity of such coefficient matrices implies the asymptotic stability.
In this paper we aim to make some contribution to fill this research gap. We extend the case of the matrices where each pair of the matrices of the family {E, A, B, C} is simultaneously triangularizable, where E, A, B, and C are defined in (5). This class of system is much broader than the class of commutative matrices.     Notes: Once Lemma 3.5 is established, then Lemma 3.4 can be considered as its special case.

Hypothesis (H 2 ) All the matrices E, A, B, C defined in
Now, under Hypothesis (H 1 ) and Hypothesis (H 2 ), the following theorem gives the asymptotic stability of system (1), which is regular, is not impulse-free and with a noncommutative pair of matrices of the family {E, A, B, C}, where E, A, B, and C are defined in (5). According to Lemma 3.5, if the rank of their commutator is less than or equal to one, we can obtain a broader criterion for non-commutative cases. Or Then system (1) is asymptotically stable for all values of the delay τ .
Proof According to Lemma 3.2, for all s ∈ C such that Re(s) ≥ 0, Since every pair of matrices of the family {E, A, B, C} defined in (5) is simultaneously triangularizable and the matrices E, A, B, and C have k (1 ≤ k ≤ n) common eigenvector, then there exists a nonsingular matrix T ∈ R n×n such that T -1 ET , T -1 AT , T -1 BT ,and T -1 CT are upper triangular matrices.
Or equivalently, Obviously, 1 e -sτ -1 is not an eigenvalue of the matrix (s(s(1α) + α) -1 (E -C) -(A + B)) -1 (B + s(s(1α) + α) -1 C) and according to condition (C 2 ), we have With the notation of the proof of Lemma 3.2, we get Thus system (1) is asymptotically stable. In the same way, if (C 1 ) and (C 2 ) are satisfied, then it is easy to show that system (1) (5) is commutative and satisfies the following conditions: Or

Conclusions
Asymptotic stability of linear fractional-order neutral delay differential-algebraic systems described by the Caputo-Fabrizio (CF) fractional derivative has been investigated. Solvability and uniqness on [0, +∞) of such a system have been considered (Lemma 3.1). Using the Laplace transform, we have derived a new characteristic equation. This characteristic equation involves a transcendental term, which makes it difficult to use in practice and in particular to study the asymptotic stability of such a system. To overcome this difficulty, some sufficient algebraic criteria have been given to ensure the asymptotic stability (Lemma 3.2). We have successfully shown that under these algebraic criteria, asymptotic stability holds when ind(E, A) = 1 (Theorem 3.2). Morevere, when ind(E, A) > 1, we have shown that if all system matrices have k (1 ≤ k ≤ n) common eigenvectors and if every pair of matrices is simultaneously triangularizable so that the algebraic criteria hold, then this system is still asymptotically stable for any delay parameter (Theorem 3.3). The effectiveness of the theoretical results has been illustrated by a numerical example.