Finite-Time Stability of Atangana–Baleanu Fractional-Order Linear Systems

-is paper investigates a fractional-order linear system in the frame of Atangana–Baleanu fractional derivative. First, we prove that some properties for the Caputo fractional derivative also hold in the sense of AB fractional derivative. Subsequently, several sufficient criteria to guarantee the finite-time stability and the finite-time boundedness for the system are derived. Finally, an example is presented to illustrate the validity of our main results.


Introduction
e topic of fractional calculus is more than three hundred years old. e number of applications of fractional calculus rapidly grows in recent decades. Fractional differential equations (FDEs) [1][2][3][4][5][6][7] have been successfully confirmed to be useful tools in various fields such as electrical circuits, diffusion, economy, and control problem.
Recently, many authors try to find new fractional operators with different kernels in order to better describe these phenomena. In 2015, Caputo and Fabrizi [8] present a new definition of fractional derivative with exponential kernel which is called CF fractional derivatives. In 2016, based on CF fractional derivatives, Atangana and Baleanu [9] introduced another new definition of fractional derivatives called AB fractional derivatives with a nonlocal and nonsingular kernel which are built by the generalized Mittag-Leffler function. In addition, the study in [9] derived the fractional integral associate to AB derivatives by taking the inverse Laplace transform and using the convolution theorem. e detail definitions of ABR fractional derivative, ABC fractional derivative, and AB fractional integral will be introduced in Section 2.
On the other hand, many basic properties of AB fractional differintegral have already been studied in recent years [10][11][12], such as integration by parts, mean value theorem and Taylor's theorem, semigroup property, and product rule and chain rule. And some problems about existence of solution, stability, controllability, and optimal control are also studied by several researchers [13][14][15][16][17][18]. Having the advantage of nonlocal and nonsingular kernel, AB fractional derivatives have been widely applied in many fields such as diffusion equation [19], electromagnetic waves in dielectric media [20], chaos [21], and circuit model [22].
Finite-time stability (FTS) and finite-time boundedness (FTB) analysis is an important problem in control theory [23][24][25][26][27][28][29]. Since it always considers the behavior of systems in finite time, it is very useful in many practical application systems. Moreover, it should be noted that FTS and Lyapunov asymptotic stability are independent concepts. In [23], the authors have introduced the definition of FTS and FTB and given the sufficient condition for the FTB of linear timeinvariant (LTI) systems with integer order. e FTS and FTB for the fractional-order systems with α ∈ (0, 1) have been given in [24]. Motivated by the above works, this paper studies the FTS and FTB for the class of fractional-order LTI systems in the sense of this new type of AB fractional derivatives. First, we establish one new property of this new type of AB fractional derivatives. Second, we introduce the concept of FTS and FTB for the fractional-order LTI system in the sense of AB fractional derivative corresponding with integer-order system and provide the sufficient conditions which guarantee FTS and FTB for a class of fractional-order LTI systems in the sense of AB fractional derivative. ird, the problem of designing feedback controllers to keep FTS and FTB for fractional closed-loop systems in the sense of AB is studied. roughout this paper, R n denotes the n-dimensional Euclidean space, R m×n is m × n real matrix, A T is the transpose of matrix A, A − 1 is the inverse of matrix A, A > 0( < 0) indicates that the matrix A is positive (negative) definite, and λ max (A) and λ min (A) are the maximum and minimum eigenvalues of matrix A.

Preliminaries
In this section, we present the definition of AB fractional derivatives and integral.
In addition, some properties of AB fractional calculus and Mittag-Leffler function are introduced.
Definition 3 (see [9]). Let 0 < α < 1, f ∈ L 1 (0, T), and T > 0. en, the ABC fractional derivative of order α is defined by In [9,12], the authors have introduced some basic properties of AB fractional differintegral which will be used in this paper. For 0 < α < 1, the relation between AB integral operators and AB differential operators is given as e Laplace transform of AB fractional derivative is defined as follows: where In addition, the Mittag-Leffler function which plays an important role in AB fractional derivative will appear frequently in this paper. It is indispensable to introduce some more properties of the Mittag-Leffler function. e generalized Mittag-Leffler functions (two parameters) are denoted by E α,β (t) � ∞ n�0 (t n /Γ(αn + β)), α, β > 0, and . e Laplace transform of the Mittag-Leffler function is given as In [30], Wang et al. provide some properties which are useful to prove Lemma 2. It is easy to note that And for 0 < α < 1, the functions E α (·) and E α,α (·) are nonnegative, and for t � 0, we have E α (0) � 1.

Lemmas
is section will introduce two important lemmas. In [29], Ye et al. have obtained the generalized Gronwall inequality in the sense of Caputo derivative which has wide applications in fractional differential equations. On this basis, Jarad et al. [15] proposed the Gronwall inequality in the frame of AB fractional derivative. en, . (9) In [31], Norelys et al. present a new property for the Caputo fractional derivative that is , t ≥ t 0 holds for a continuous and derivable function x(t) ∈ R, and 0 < α < 1. According to this, we can prove that this property also holds in the sense of AB fractional derivative. Lemma 2. Let 0 < α < 1 and x(t) ∈ C(J; R) be a continuous and differentiable function. en, for any t ∈ J, the following inequality holds: Proof. By the definition of AB fractional derivative, we have 2 Complexity Equation (10) is equivalent to . en, equation (12) can be reformulated as Indeed, equation (14) holds obviously since E α (0) � 1 and y(t) � 0 deduce that the first term equals zero and the nonnegativity of y 2 (s), y 2 (0), E α (·), and E α,α (·) leads to the nonnegativity of the latter two terms.
From Lemma 2, we can easily obtain the following two results. Corollary 1. Let 0 < α < 1 and x(t) ∈ C(J; R n ) be a continuous and differentiable function on J. en, for any t ∈ J, the following inequality holds: Proof. Assume that x(t) � (x 1 (t), x 2 (t), . . . , x n (t)) T and x i (t) ∈ C(J, R), i � 1, 2, . . . , n. By Lemma 2, we have Corollary 2. Let U ∈ R n×n be a symmetric positive definite matrix, 0 < α < 1, and x(t) ∈ C(J; R n ) be a continuous and differentiable function on J. en, for any t ∈ J, the following inequality holds: Proof. From matrix theory, we know that there exists a nonsingular matrix V such that U � V T V. Define y(t) � Vx(t); then,

Main Results
In this section, the concept of FTS and FTB for fractionalorder LTI systems in the sense of AB fractional derivative corresponding with integer-order systems is introduced. Several sufficient conditions of FTS and FTB for such systems are given. Consider the following fractional order LTI system in the sense of AB fractional derivative: where 0 < α < 1, x(·) ∈ R n , and A ∈ R n×n is a constant matrix. (19) is said to be of finite-time stability with respect to (r 1 , r 2 , R, T), with positive scalars r 1 , r 2 , T, r 2 > r 1 , and a matrix R > 0, if

Proof. Define the function V(x(t)) � x T (t)Ux(t).
From Corollary 2 and condition equation (21), we have en, there exists a nonnegative function M(t) satisfying Taking the Laplace transform on both sides of equality (24), we obtain It can be reformulated that Applying the inverse Laplace transform, one has Due to the nonnegativity of the second term in equation (27), it is easy to obtain . (28) Taking V(x(t)) � x T (t)Ux(t) and U � R (1/2) VR (1/2) into equation (28) yields is implies that According to condition equation (22), we can conclude that For system (19), we also have the following results similar to eorem 1.
Based on system (19), we will consider the finite-time stabilization of fractional-order systems in the AB fractional derivative sense with control function: where u(t) ∈ R m is a control function and B ∈ R n×m is a constant matrix. We assume that all the state variables are available for state feedback. e problem is to design a state feedback controller u(t) � Kx(t), where K ∈ R m×n is the control gain matrix to be designed, such that the closed-loop system is where A c � A + BK, being FTS with respect to (r 1 , r 2 , R, T).

Theorem 2.
Assume that there exists a scalar c > 0, a matrix V ∈ R n×n , V > 0, and a matrix L ∈ R m×n satisfying where U � R (1/2) VR (1/2) . en, under the feedback controller Proof. Apply the state feedback controller u(t) � LU − 1 x(t) to system (33) such that the corresponding closed-loop system is By Corollary 3, it is clear that system (34) is FTS under the condition equation (35) and equation (36).
Next, we will study the problem of FTB for fractionalorder LTI systems in the frame of AB fractional derivative. Consider the following systems: where w(t) ∈ R l is the disturbance input and satisfies w T (t)w(t) ≤ d, d ≥ 0. D ∈ R n×l is a constant matrix.
Proof. Apply the state feedback controller u(t) � LU − 1 x(t) to system (53) such that the corresponding closed-loop system is By eorem 3, it is clear that system (52) is FTB under the condition equations (41), (42), and (53).

Example
An example is presented to illustrate the main results. Example 1. Consider the following fractional-order LTI system in the sense of AB fractional derivative with a disturbance defined by where the initial value x(0) � 1 0 T and the disturbance w(t) � ((sin(t) − 1)cos(t)) T . e parameters are given as c 1 � 1, c 2 � 20, d � 2, T � 10, R � I, and B(α) � 1. Applying the condition equations (40), (41), and (42) of eorem 3 and setting c � 0.05, we can obtain the feasible matrices U 1 and V as follows: (56) erefore, system (55) is FTB with respect to (1, 20, I, 10, 2). e state trajectory over 0 − 10 s with the initial state x(0) � 1 0 T is shown in Figure 1. It is easy to see that system (55) is FTB with respect to (1, 20, I, 10, 2) from Figure 2.

Conclusion
e contribution of this paper is to present one new property of AB fractional derivatives and to provide the sufficient 6 Complexity conditions which guarantee FTS and FTB for a class of fractional-order LTI systems in the sense of AB fractional derivative, as well as to study the problem of designing feedback controllers. Because of the interesting nonsingular kernel of this new fractional derivative, there is much work on this type of fractional calculus that is worth thinking about and discussing in the future.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.