# Stability and Stabilization for a Class of Semilinear Fractional Differential Systems

Authors: Badawi Hamza Elbadawi Ibrahim; Qixiang Dong; Zhengdi Zhang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

This 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. The 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.


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## Body

## 1. Introduction

Over the past several decades, fractional calculus has attracted much attention from scientists and engineers. This 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 [1–4] 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]. There 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 [8–10]. The modelling and stability of the water jet mixed-flow pump fractional-order shafting system has also been studied [11]. The 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 siding-mode control approach was used to investigate the stabilization and synchronization problems of the nonlinear fractional-order system (e.g., [14–18]). 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 [19–24], 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. Theoretically, a stability theorem is developed with the property of convolution and the asymptotic properties of Mittag-Leffler functions. Therefore, 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. Thus, 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 argλiA>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.The remainder of this paper is organized as follows. In Section2, 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.


## 2. Preliminaries

In this section, we state some definitions and results that are going to be used in our investigations.Definition 1.
(see [3]). The Caputo fractional derivative of order α of function x(t) is defined as(1)Dtαt0Cxt=1Γn−α∫t0tt−τn−α−1xnτdτ,where Γ(·) is the gamma function, i.e., Γz=∫0∞e−ttz−1dt, and n is an integer satisfying n − 1 < α ≤ n.
The Laplace transform of the Caputo fractional derivativeDtαt0Cxt is(2)∫0∞e−stDtαt0Cxtdt=sαXs−∑k=0n−1sα−k−1xkt0,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. The definitions and properties are therefore given as follows.Definition 2.
(see [3]). The Mittag-Leffler function is defined as(3)Eαz=∑k=0∞zkΓαk+1,Reα>0,z∈C.
The Mittag-Leffler function with two parameters is defined as(4)Eα,βz=∑k=0∞zkΓαk+β,Reα>0,β∈R,z∈C.
It is easy to see thatEα(z) = Eα,1(z) and E1(z) = E1,1(z) = ez. The Laplace transform of Mittag-Leffler function is formulated as(5)∫0∞e−sttαk−β−1Eα,βk±atαdt=k!sα−βsα∓ak+1,Res>a1/α.Definition 3.
(see [2]). For A ∈ Cn×n, the matrix Mittag-Leffler function is defined by(6)Eα,βA=∑k=0∞AkΓαk+β,β∈C,Reα>0.Lemma 1.
(see [26]). The following properties hold:(i)
There exist finite real constantsM1 ≥ 1, M2 ≥ 1, M3 ≥ 1 such that for any 0 < α < 1,(7)Eα,1Atα≤M1eAtα,Eα,αAtα≤M2eAtα,whereA denotes matrix and ⋅ denotes any vector or induced matrix norm.(ii)
Ifα ≥ 1, then for β = 1, 2, α,(8)Eα,βAtα≤M3eAtα.Definition 4.
(see [7]). The constant x0 ∈ Rn is an equilibrium point of the Caputo dynamic system Dtαt0Cxt=ft,xt if and only if f(t, x0) = 0.
Without loss of generality, we may assume that the equilibrium point isx0 = 0, representing the origin of Rn (See [7]). Hence, in the rest of this paper, we always assume that the nonlinear function f satisfies f(t, 0) = 0.Definition 5.
The zero solution ofDtαt0Cxt=ft,xt, with order 0 < α ≤ 1(1 < α < 2) is said to be stable if for any initial values xk(k = 0)(xk(k = 0, 1)) and any ϵ > 0 there exists t0 > 0 such that ∥x(t)∥ < ϵ for all t > t0. The zero solution is said to be asymptotically stable if limt⟶∞∥x(t)∥ = 0.
The following property of convolution plays a key role in the proof of the main results.Lemma 2.
(see [27]). Let 1 < p, q < ∞ satisfy 1/p+1/q=1. If f ∈ Lp(R, X) and g ∈ Lq(R, X), then f∗g ∈ C0(R, X). Here, X is a Banach space, f∗g denotes the convolution of the functions f and g, i.e.,f∗gt=∫0tft−τgτdτ for t ≥ 0, and C0(R, X) = {f ∈ C(R, X): f vanishes at infinity}.


## 3. Existence Results

In this section, we consider the following system of fractional differential equation:(9)Dtα0Cxt=Axt+ft,xt,where xt=x1t,x2t,…,xntT∈Rn denotes the state vector of the system, α ∈ (0, 2) is the order of the fractional-order derivative, f: [0, +∞) × Rn ⟶ Rn defines a nonlinear vector field in the n-dimensional vector space, and A ∈ Rn×n is a constant matrix. Here after, we assume that λi(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).


### 3.1. Stability for the Case 0 <α ≤ 1

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) = x0, can be expressed as(10)xt=Eα,1Atαx0+∫0tt−τα−1Eα,αAt−ταfτ,xτdτ.The existence and uniqueness of solutions to this problem are widely studied [1–3].Theorem 1.
The zero solution of system (9) is locally asymptotically stable if the following conditions are satisfied:(1)
The matrixA is stable.(2)
There is a functiong ∈ Lq(0, ∞) such that(11)ft,xt≤gt,whereq is a positive constant satisfying 1/p + 1/q = 1 and p>1/α.Proof.
Let the initial condition bex(0) = x0. Then, the solution of (9) is given by (10). This is obtained from condition (1) that argλiA>απ/2. It then follows from Lemma 1 that there exist constants M1 > 0 and M2 > 0 such that(12)xt≤M1eAtαx0+M2∫0tt−τα−1eAt−ταfτ,xτdτ.
Because the matrixA is stable, there exist constants M > 0 and ω > 0 such that(13)eAt≤Me−ωt.
Substituting it into (12) and using condition (2), one has(14)xt≤MM1e−ωtαx0+MM2∫0tt−τα−1e−ωt−ταfτ,xτdτ≤MM1e−ωtαx0+MM2∫0tt−τα−1e−ωt−ταgτdτ.
Now, we define a functionφ: [0, +∞) ⟶ [0, +∞) as(15)φu=uα−1e−ωuα.
Because 0 <α ≤ 1 and p > 1/α, a simple computation shows that φ ∈ Lp(0, +∞). On the other hand, from condition (2) we know that g ∈ Lq(0, +∞). Then, it follows from Lemma 2 that φ∗g ∈ C0[0, +∞), which implies that(16)limt⟶+∞∫0tt−τα−1e−ωt−ταgτdτ=0.
This, combined with (14), shows that limt⟶+∞∥x(t)∥ = 0; hence, Theorem 1 is proved.


### 3.2. Stability for the Case 1 <α ≤ 2

When 1 <α ≤ 2, the solution to equation (9), with initial condition x(0) = x0 and x′(0) = x1, satisfies the following integral equation:(17)xt=Eα,1Atαx0+tEα,2Atαx1+∫0tt−τα−1Eα,αAt−ταfτ,xτdτ.Because 1 <α ≤ 2, from Lemma 1 (ii), there exists M3 > 0 such that(18)Eα,βAtα≤M3eAtα,for β = 1, 2, or α. We first prove an existence and uniqueness result via the principle of Banach contraction.Theorem 2.
LetT > 0 be arbitrary, the matrix A be stable, and f be Lipschitz w.r.t. the second variable, i.e., there exists a constant L > 0 such that(19)ft,u−ft,v≤Lu−v,for each t ∈ [0, T] and u, v ∈ Rn. If(20)L<αωMM3,where M, M3, and ω are constants appeared in (13) and (8), then equation (9), with the initial condition x(0) = x0 and x′(0) = x1, has a unique solution in [0, T].Proof.
We transform the problem into a fixed-point problem. LetX = C([0, T], Rn) be the Banach space of all continuous functions from [0, T] into Rn, endowed with the norm ∥x∥ = supt∈[0,T]∥x(t)∥. We define an operator Q: X ⟶ X as(21)Qxt=Eα,1Atαx0+tEα,2Atαx1+∫0tt−τα−1Eα,αAt−ταfτ,xτdτ,for t ∈ [0, T]. Then, equation (9) has a unique solution if and only if Q has a unique fixed point. Taking x, y ∈ C([0, T], Rn) arbitrarily and t ∈ [0, T]. We obtain(22)Qxt−Qyt≤∫0tt−τα−1Eα,αAt−ταfτ,xτ−fτ,yτdτ≤M3∫0tt−τα−1eAt−ταfτ,xτ−fτ,yτdτ≤MM3Lx−y∫0tt−τα−1e−ωt−ταdτ≤MM3L1−e−ωtααωx−y≤MM3αωLx−y.
It follows that(23)Qx−Qy≤MM3αωLx−y.
Condition (20) shows that MM3/αωL<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). The proof is completed.
Similar to Theorem1, we now prove the stability of equation (9) for 1 < α ≤ 2.Theorem 3.
Suppose that the conditions in Theorem2 hold. Then, the zero solution of system (9) is locally asymptotically stable, if the following conditions are satisfied:(1)
argλiA>απ/2, i = 1, 2, …, n.(2)
There is a functiong ∈ Lq(0, ∞) such that(24)ft,xt≤gt,whereq is a positive constant satisfying 1/p + 1/q = 1 and p > 1.Proof.
Suppose that the initial condition isx(0) = x0 and x′(0) = x1, then the solution of (9) has the following form:(25)xt=Eα,1Atαx0+tEα,2Atαx1+∫0tt−τα−1Eα,αAt−ταfτ,xτdτ,for t > 0. Because 1 < α ≤ 2, it follows from (25) and condition (2) that(26)xt≤M3eAtαx0+M3eAtαx1t+M3∫0tt−τα−1eAt−ταfτ,xτdτ≤M3eAtαx0+M3eAtαx1t+M3∫0tt−τα−1eAt−ταgτdτ,for all t > 0. From condition (1) we know that the matrix A is stable. Hence, the inequality (13) holds. Therefore, we obtain(27)xt≤MM3e−ωtαx0+MM3e−ωtαx1t+MM3∫0tt−τα−1e−ωt−ταgτdτ,for all t > 0. Similar to the proof of Theorem 1, we define the function φ: [0, +∞) ⟶ [0, +∞) as(28)φu=uα−1e−ωuα.
Because 1 <α ≤ 2, it is easy to see that φ ∈ Lp(0, +∞) for any p > 1. Because of Lemma 2 and condition (2), we obtain φ∗g ∈ C0[0, +∞). Therefore, limt⟶+∞∥x(t)∥ = 0. The proof is completed.Remark 1.
Theorems1 and 3 give us a simple procedure for determining stability of the fractional-order nonlinear system with Caputo derivative of order 0 < α < 2. If the nonlinear term f(t, x(t)) fulfills condition (2), then the exact solution need not be reached. Importantly, it is required to calculate matrix A’s eigenvalues and test their arguments. If argλiA>απ/2, i = 1, 2, …, n, we conclude that the origin is stable asymptotically.


### 3.3. 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:(29)Dtα0Cxt=Axt+ft,xt+But,where x, A and f are as in system (9), B ∈ Rn×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:(30)Dtα0Cxt=Axt+ft,xt+But=Axt+ft,xt+BKxt=A+BKxt+ft,xt=A˜xt+ft,xt.Suppose that (A, B) is controllable, then the feedback gain K can be chosen such that system (30) is asymptotically stable.Theorem 4.
If 0 <α < 2, feedback gain K is chosen such that the following conditions hold:(i)
argλiA˜>απ/2,i=1,2,…,n, where λiA˜ is the eigenvalue of matrix A˜.(ii)
f(t, x(t)) satisfies ∥f(t, x(t))∥ ≤ g(t), where g ∈ Lq(0, ∞).(iii)
The matrixA˜ is stable. Then, the controlled system (29) is locally asymptotically stable.Proof.
The proof of Theorem4 is similar to that of Theorems 1 and 3.Remark 2.
The nonlinear term of Chaotic fractional-order systems satisfies ∥f(t, x(t))∥ ≤ g(t), where g ∈ Lq(0, ∞), i.e., the hyperchaotic fractional-order novel system. Therefore, in a large class of generalized fractional-order chaotic or hyperchaotic systems, Theorem 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.


## 3.1. Stability for the Case 0 <α ≤ 1

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) = x0, can be expressed as(10)xt=Eα,1Atαx0+∫0tt−τα−1Eα,αAt−ταfτ,xτdτ.The existence and uniqueness of solutions to this problem are widely studied [1–3].Theorem 1.
The zero solution of system (9) is locally asymptotically stable if the following conditions are satisfied:(1)
The matrixA is stable.(2)
There is a functiong ∈ Lq(0, ∞) such that(11)ft,xt≤gt,whereq is a positive constant satisfying 1/p + 1/q = 1 and p>1/α.Proof.
Let the initial condition bex(0) = x0. Then, the solution of (9) is given by (10). This is obtained from condition (1) that argλiA>απ/2. It then follows from Lemma 1 that there exist constants M1 > 0 and M2 > 0 such that(12)xt≤M1eAtαx0+M2∫0tt−τα−1eAt−ταfτ,xτdτ.
Because the matrixA is stable, there exist constants M > 0 and ω > 0 such that(13)eAt≤Me−ωt.
Substituting it into (12) and using condition (2), one has(14)xt≤MM1e−ωtαx0+MM2∫0tt−τα−1e−ωt−ταfτ,xτdτ≤MM1e−ωtαx0+MM2∫0tt−τα−1e−ωt−ταgτdτ.
Now, we define a functionφ: [0, +∞) ⟶ [0, +∞) as(15)φu=uα−1e−ωuα.
Because 0 <α ≤ 1 and p > 1/α, a simple computation shows that φ ∈ Lp(0, +∞). On the other hand, from condition (2) we know that g ∈ Lq(0, +∞). Then, it follows from Lemma 2 that φ∗g ∈ C0[0, +∞), which implies that(16)limt⟶+∞∫0tt−τα−1e−ωt−ταgτdτ=0.
This, combined with (14), shows that limt⟶+∞∥x(t)∥ = 0; hence, Theorem 1 is proved.


## 3.2. Stability for the Case 1 <α ≤ 2

When 1 <α ≤ 2, the solution to equation (9), with initial condition x(0) = x0 and x′(0) = x1, satisfies the following integral equation:(17)xt=Eα,1Atαx0+tEα,2Atαx1+∫0tt−τα−1Eα,αAt−ταfτ,xτdτ.Because 1 <α ≤ 2, from Lemma 1 (ii), there exists M3 > 0 such that(18)Eα,βAtα≤M3eAtα,for β = 1, 2, or α. We first prove an existence and uniqueness result via the principle of Banach contraction.Theorem 2.
LetT > 0 be arbitrary, the matrix A be stable, and f be Lipschitz w.r.t. the second variable, i.e., there exists a constant L > 0 such that(19)ft,u−ft,v≤Lu−v,for each t ∈ [0, T] and u, v ∈ Rn. If(20)L<αωMM3,where M, M3, and ω are constants appeared in (13) and (8), then equation (9), with the initial condition x(0) = x0 and x′(0) = x1, has a unique solution in [0, T].Proof.
We transform the problem into a fixed-point problem. LetX = C([0, T], Rn) be the Banach space of all continuous functions from [0, T] into Rn, endowed with the norm ∥x∥ = supt∈[0,T]∥x(t)∥. We define an operator Q: X ⟶ X as(21)Qxt=Eα,1Atαx0+tEα,2Atαx1+∫0tt−τα−1Eα,αAt−ταfτ,xτdτ,for t ∈ [0, T]. Then, equation (9) has a unique solution if and only if Q has a unique fixed point. Taking x, y ∈ C([0, T], Rn) arbitrarily and t ∈ [0, T]. We obtain(22)Qxt−Qyt≤∫0tt−τα−1Eα,αAt−ταfτ,xτ−fτ,yτdτ≤M3∫0tt−τα−1eAt−ταfτ,xτ−fτ,yτdτ≤MM3Lx−y∫0tt−τα−1e−ωt−ταdτ≤MM3L1−e−ωtααωx−y≤MM3αωLx−y.
It follows that(23)Qx−Qy≤MM3αωLx−y.
Condition (20) shows that MM3/αωL<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). The proof is completed.
Similar to Theorem1, we now prove the stability of equation (9) for 1 < α ≤ 2.Theorem 3.
Suppose that the conditions in Theorem2 hold. Then, the zero solution of system (9) is locally asymptotically stable, if the following conditions are satisfied:(1)
argλiA>απ/2, i = 1, 2, …, n.(2)
There is a functiong ∈ Lq(0, ∞) such that(24)ft,xt≤gt,whereq is a positive constant satisfying 1/p + 1/q = 1 and p > 1.Proof.
Suppose that the initial condition isx(0) = x0 and x′(0) = x1, then the solution of (9) has the following form:(25)xt=Eα,1Atαx0+tEα,2Atαx1+∫0tt−τα−1Eα,αAt−ταfτ,xτdτ,for t > 0. Because 1 < α ≤ 2, it follows from (25) and condition (2) that(26)xt≤M3eAtαx0+M3eAtαx1t+M3∫0tt−τα−1eAt−ταfτ,xτdτ≤M3eAtαx0+M3eAtαx1t+M3∫0tt−τα−1eAt−ταgτdτ,for all t > 0. From condition (1) we know that the matrix A is stable. Hence, the inequality (13) holds. Therefore, we obtain(27)xt≤MM3e−ωtαx0+MM3e−ωtαx1t+MM3∫0tt−τα−1e−ωt−ταgτdτ,for all t > 0. Similar to the proof of Theorem 1, we define the function φ: [0, +∞) ⟶ [0, +∞) as(28)φu=uα−1e−ωuα.
Because 1 <α ≤ 2, it is easy to see that φ ∈ Lp(0, +∞) for any p > 1. Because of Lemma 2 and condition (2), we obtain φ∗g ∈ C0[0, +∞). Therefore, limt⟶+∞∥x(t)∥ = 0. The proof is completed.Remark 1.
Theorems1 and 3 give us a simple procedure for determining stability of the fractional-order nonlinear system with Caputo derivative of order 0 < α < 2. If the nonlinear term f(t, x(t)) fulfills condition (2), then the exact solution need not be reached. Importantly, it is required to calculate matrix A’s eigenvalues and test their arguments. If argλiA>απ/2, i = 1, 2, …, n, we conclude that the origin is stable asymptotically.


## 3.3. 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:(29)Dtα0Cxt=Axt+ft,xt+But,where x, A and f are as in system (9), B ∈ Rn×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:(30)Dtα0Cxt=Axt+ft,xt+But=Axt+ft,xt+BKxt=A+BKxt+ft,xt=A˜xt+ft,xt.Suppose that (A, B) is controllable, then the feedback gain K can be chosen such that system (30) is asymptotically stable.Theorem 4.
If 0 <α < 2, feedback gain K is chosen such that the following conditions hold:(i)
argλiA˜>απ/2,i=1,2,…,n, where λiA˜ is the eigenvalue of matrix A˜.(ii)
f(t, x(t)) satisfies ∥f(t, x(t))∥ ≤ g(t), where g ∈ Lq(0, ∞).(iii)
The matrixA˜ is stable. Then, the controlled system (29) is locally asymptotically stable.Proof.
The proof of Theorem4 is similar to that of Theorems 1 and 3.Remark 2.
The nonlinear term of Chaotic fractional-order systems satisfies ∥f(t, x(t))∥ ≤ g(t), where g ∈ Lq(0, ∞), i.e., the hyperchaotic fractional-order novel system. Therefore, in a large class of generalized fractional-order chaotic or hyperchaotic systems, Theorem 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.


## 4. Applications

In this section, we apply the obtained results to some semilinear systems to illustrate the effectiveness of the theory.Example 1.
Consider the fractional nonlinear system with Caputo derivative:(31)Dtα0Cx1t=3x1+7x2+2x3+e−w1tsinx2x3,Dtα0Cx2t=−2x1−4x2+4x3+e−w2tsinx22,Dtα0Cx3t=e−w3tsinx1x2−x3,where wi > 0, i = 1, 2, 3. This system can be rewritten as (9), where(32)A=372−2−4400−1,ft,x=e−w1tsinx2x3e−w2tsinx22e−w3tsinx1x2.
It is easy to verify that(33)ft,x≤e−2w1tsin2x2x3+e−2w2tsin2x22+e−2w3tsin2x1x2≤sin2x2x3+sin2x22+sin2x1x2e−w0t≤3e−w0t∈Lq0,+∞,where w0=maxw1,w2,w3. So, condition (2) in Theorems 1 and 3 is satisfied. The eigenvalues of A are λ1,2 = −0.5000 ± 1.3229i and λ3 = −1. According to Theorems 1 and 3, if α < 1.2300, then the zero solution of (31) is asymptotically stable. For the initial values (x1(0), x2(0), x3(0)) = (0.1, −0.2, 0.3), simulation results are displayed in Figures 1–3. Figures 1 and 2 show that the zero solution of system (31) is asymptotically stable with α = 1.21 and α = 1.22, respectively. Figure 3 also shows that the zero solution of system (31) is unstable with α = 1.23.Figure 1
System (31) is asymptotically stable with α = 1.21.
(a)
(b)
(c)Figure 2
System (31) is asymptotically stable with α = 1.22.
(a)
(b)
(c)Figure 3
System (31) is unstable with α = 1.23.
(a)
(b)
(c)Example 2.
Consider the following fractional nonlinear system:(34)Dtα0Cx1t=−x1+e−w1tsinx2x3,Dtα0Cx2t=x3,Dtα0Cx3t=x1−x2−x3−e−w3tsinx1x2.
The system can be rewritten as (9), where(35)A=−1000011−1−1,ft,x=e−w1tsinx2x30−e−w3tsinx1x2.
Obviously,(36)ft,xt≤e−w0:=gt∈Lq.
The eigenvalues ofA are λ1,2=−1/2±3i/2 and λ3 = −1. According to Theorems 1 and 3, if α<4/3, then the zero solution of (32) is asymptotically stable.
Simulation results with initial values (x1(0), x2(0), x3(0)) = (−0.01, −0.02, 0.03) are displayed in Figures 4–6. Figures 4 and 5 show that the zero solution of system (34) is asymptotically stable with α = 1.1 and α = 1.3, respectively. Figure 6 further shows that the zero solution of system (34) is unstable when α = 1.34.Figure 4
System (34) is asymptotically stable with α = 1.1.
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(c)Figure 5
System (34) is asymptotically stable with α = 1.3.
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(c)Figure 6
System (34) is unstable with α = 1.34.
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(c)Example 3.
The fractional-order novel hyperchaotic system can be written as(37)Dtα0Cx1t=ax2−x1,Dtα0Cx2t=−bx1+e−w2tsinx1x3+x4,Dtα0Cx3t=−2e−w3tsinx12−2e−w3tsinx22−cx3−x4,Dtα0Cx4t=−dx1,where a, b, c, and d are some parameters. System (37) can be rewritten as (9) if we write(38)A=−aa00−b00100−c−1−d000,ft,xt=0e−w2tsinx1x3−2e−w3tsinx12−2e−w3tsinx220.
It is easy to see thatf(t, x(t)) satisfies(39)ft,xt≤e−2w2tsin2x1x3+8e−2w3tsin2x12+8e−2w3tsin2x22≤sin2x1x3+8sin2x12+8sin2x22e−w0t≤5e−w0t∈Lq.
Whena = 10, b = 40, c = 2.5, and d = 10, system (37) displays a chaotic attractor, and we construct the linear state feedback controller as (9). For simplicity, let B = (1,1,1,1)T. According to Theorem 4, feedback gain can be obtained by argλiA˜>απ/2. By a simple calculation, if we choose K = (2, −8, −3.5, −1), then the eigenvalues of A˜ are (−17.3028, −0.11769, −2.7898 ± 10.7230i). Hence, argλiA˜>0.5πα if α < 1.6203. Thus, the conditions of Theorem 4 are satisfied. The zero solution of the controlled system is asymptotically stable with α = 1.14 and α = 1.15, respectively. The results of simulation with initial values (x1(0), x2(0), x3(0), x4(0)) = (−0.01, −0.02, −0.03, 0.04) are shown in Figures 7–10.Figure 7
Asymptotical stabilization of the fractional-order novel hyperchaotic system (37) with α = 1.14 and feedback gain K = (2, −8, −3.5, −1).
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(d)Figure 8
Attractor of the fractional-order novel hyperchaotic system with orderα = 1.14 (a = 10, b = 40, c = 2.5, and d = 10).
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(d)Figure 9
Asymptotical stabilization of the fractional-order novel hyperchaotic system (37) with α = 1.15 and feedback gain K = (2, −8, −3.5, −1).
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(d)Figure 10
Attractor of the fractional-order novel hyperchaotic system with orderα = 1.15 (a = 10, b = 40, c = 2.5, and d = 10).
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## 5. Conclusions

The stability of nonlinear dynamical systems is important for scientists and engineers. Therefore, 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. Three illustrative examples were provided to demonstrate the applicability of the proposed approach.



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