Stability and Stabilization for a Class of Semilinear Fractional Differential Systems

-is paper considers a class of semilinear fractional-order systems with Caputo derivative. New conditions ensuring asymptotic stability and stabilization of fractional systems with the fractional order between 0 and 2 are proposed. -e analysis is based on a property of convolution and asymptotic properties of Mittag-Leffler functions. Some numerical examples are provided to illustrate the feasibility and validity of the proposed approach.


Introduction
Over the past several decades, fractional calculus has attracted much attention from scientists and engineers. is is because fractional differential equations have proven to be effective in modelling many physical phenomena and have been applied in different science and engineering fields. Significant contributions have been proposed in fractional differential equations both in theory and applications. For example, see [1][2][3][4] and references therein.
In recent years, the stability of fractional-order systems has gained increasing interest due to its importance in control theory. Also, stability theory is an important topic in the study of differential equations. In 1996, Matignon studied the stability of linear fractional differential equations in [5], which is regarded as the first work in this area. Li et al. investigated the Mittag-Leffler stability of nonlinear fractional dynamic systems in [6] and suggested the Lyapunov direct method for nonlinear fractional-order stability systems [7].
ere has been more literature on stability of dynamic fractional-order systems, in which important and sufficient conditions were discussed for the stability of linear and linear time-delay fractional differential equations as stated in [8][9][10]. e modelling and stability of the water jet mixed-flow pump fractional-order shafting system has also been studied [11]. e stability of fractional-order nonlinear systems with 0 < α < 1 was derived in [6,12,13], according to the Lyapunov approach. Based on the uncertain Takagi-Sugeno fuzzy model, the stability problems of nonlinear fractional-order systems were studied, whereas the sidingmode control approach was used to investigate the stabilization and synchronization problems of the nonlinear fractional-order system (e.g., [14][15][16][17][18]). As noted, a growing number of scientists are dedicated to the stability of fractional systems, with most of the above findings concentrating only on nonlinear fractional systems of 0 < α < 1.
In [19][20][21][22][23][24], the authors studied the stability of fractional nonlinear systems with order 0 < α ≤ 2. Various sufficient conditions for asymptotic stability (local or global) are obtained by using Mittag-Leffler function, Laplace transform, and the generalized Gronwall inequality. In summary, the authors of [19,20,23,25] conducted studies on the stability relying on a class of commensurate and incommensurate fractional-order systems together with fractionally controlled systems with linear feedback inputs. By using Mittag-Leffler, Laplace transforms, and Gronwall-Bellman lemma, Zhang et al. [21] discussed the stability of n-dimensional nonlinear fractional order.
In this paper, we discuss the stability of a class of semilinear fractional differential systems with the fractional order between 0 and 2. eoretically, a stability theorem is developed with the property of convolution and the asymptotic properties of Mittag-Leffler functions. erefore, based on this theory of stability, a basic criterion for stabilizing a class of nonlinear fractional-order systems is derived, in which control parameters can be selected through the linear control theory pole placement technique. Our results give us a simple method for determining the stability of nonlinear fractional systems with Caputo derivative with order 0 < α ≤ 2. Compared with the abovementioned stability method, the conditions we have proposed for the nonlinear component f(t, x(t)) are new and much simpler to test. us, there is no need to arrive at an exact solution if only the nonlinear term satisfies certain conditions. What is needed is to calculate the eigenvalues of the linear coefficient matrix A and test that ‖arg(λ i (A))‖ > (απ/2) is satisfied by its arguments. In addition, the results obtained can be used to stabilize the class of fractional-order nonlinear systems by means of a linear state feedback controller. e remainder of this paper is organized as follows. In Section 2, we recall some definitions and lemmas that will be used in the analysis. In Section 3, the main results, together with the stability and stabilization of the equilibrium points are presented. Section 4 is devoted to some numerical simulation examples that illustrate the validation and effectiveness of the theory. We conclude our work in Section 5.

Preliminaries
In this section, we state some definitions and results that are going to be used in our investigations.
e Laplace transform of the Caputo fractional deriv- where n is an integer such that n − 1 < α ≤ n. Similar to the exponential function, the function frequently used in the fractional differential equations is the Mittag-Leffler function. e definitions and properties are therefore given as follows.
Definition 2 (see [3]). e Mittag-Leffler function is defined as e Mittag-Leffler function with two parameters is defined as It is easy to see that E α (z) � E α,1 (z) and E 1 (z) � E 1,1 (z) � e z . e Laplace transform of Mittag-Leffler function is formulated as Definition 3 (see [2]). For A ∈ C n×n , the matrix Mittag-Leffler function is defined by Lemma 1 (see [26]). The following properties hold: where A denotes matrix and ‖ · ‖ denotes any vector or induced matrix norm.
(ii) If α ≥ 1, then for β � 1, 2, α, Definition 4 (see [7]). e constant x 0 ∈ R n is an equilibrium point of the Without loss of generality, we may assume that the equilibrium point is x 0 � 0, representing the origin of R n (See [7]). Hence, in the rest of this paper, we always assume that the nonlinear function f satisfies f(t, 0) � 0.
is said to be stable if for any initial values x k (k � 0)(x k (k � 0, 1)) and any ϵ > 0 there exists t 0 > 0 such that ‖x(t)‖ < ϵ for all t > t 0 . e zero solution is said to be asymptotically stable if lim t⟶∞ ‖x(t)‖ � 0. e following property of convolution plays a key role in the proof of the main results.

Existence Results
In this section, we consider the following system of fractional differential equation: where . , x n (t)) T ∈ R n denotes the state vector of the system, α ∈ (0, 2) is the order of the fractional-order derivative, f: [0, +∞) × R n ⟶ R n defines a nonlinear vector field in the n-dimensional vector space, and A ∈ R n×n is a constant matrix. Here after, we assume that λ i (i � 1, 2, . . ., n) are the eigenvalues of the matrix A and that f(t, 0) � 0, i.e., 0 is an equilibrium point of system (9).

Stability for the Case
We first present a stability result for the case 0 < α ≤ 1. In this case, the solution to equation (9), with initial condition x(0) � x 0 , can be expressed as e existence and uniqueness of solutions to this problem are widely studied [1][2][3].

Theorem 1.
e zero solution of system (9) is locally asymptotically stable if the following conditions are satisfied: where q is a positive constant satisfying 1/p + 1/q � 1 and p > (1/α).
for each t ∈ [0, T] and u, v ∈ R n . If where M, M 3 , and ω are constants appeared in (13) and (8), then equation (9) for t ∈ [0, T]. en, equation (9) has a unique solution if and only if Q has a unique fixed point. Taking x, y ∈ C([0, T], R n ) arbitrarily and t ∈ [0, T]. We obtain It follows that Condition (20) shows that (MM 3 /αω)L < 1, which implies that Q is a contraction. Hence, we deduce by the principle of Banach contraction that Q has a unique fixed point, which is the unique solution to equation (9). e proof is completed.
Similar to eorem 1, we now prove the stability of equation (9)   en, the zero solution of system (9) is locally asymptotically stable, if the following conditions are satisfied: where q is a positive constant satisfying 1/p + 1/q � 1 and p > 1.

Stabilization of a Class of Fractional-Order Semilinear
System. In this subsection, we propose the stabilization theory of a class of fractional-order semilinear controlled systems. We consider the controlled systems of the following form: where x, A and f are as in system (9), B ∈ R n×n is the input matrix, and u is the control input. If u is chosen to be a linear state feedback control, i.e., u � Kx for some feedback gains K, then system (29) becomes a closed-loop system: , x(t)).

(30)
Suppose that (A, B) is controllable, then the feedback gain K can be chosen such that system (30) is asymptotically stable.

4
Journal of Function Spaces Theorem 4. If 0 < α < 2, feedback gain K is chosen such that the following conditions hold: Proof. e proof of eorem 4 is similar to that of eorems 1 and 3.

Remark 2.
e nonlinear term of Chaotic fractional-order systems satisfies ‖f(t, x(t))‖ ≤ g(t), where g ∈ L q (0, ∞), i.e., the hyperchaotic fractional-order novel system. erefore, in a large class of generalized fractional-order chaotic or hyperchaotic systems, eorem 4 can be applied to control chaotic. Of all control methods, linear feedback control is particularly attractive and has been widely extended to practical implementation due to its ease among configuration and implementation.

Applications
In this section, we apply the obtained results to some semilinear systems to illustrate the effectiveness of the theory.

Example 3.
e fractional-order novel hyperchaotic system can be written as where a, b, c, and d are some parameters. System (37) can be rewritten as (9)

Conclusions
e stability of nonlinear dynamical systems is important for scientists and engineers. erefore, in this paper, we studied a class of semilinear fractional-order systems with Caputo derivative by using properties of convolution and Mittag-Leffler function methods. We introduced the fractional comparison principle for Caputo fractional-order systems, which enriched the knowledge of both system theory and fractional calculus. We established a new sufficient condition of the asymptotic stability of zero solution for a class of fractional-order semilinear systems with order 0 < α < 2.
ree illustrative examples were provided to demonstrate the applicability of the proposed approach.
Data Availability e data used in the examples were originally taken from MATLAB, and they are available from the corresponding author upon request.

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