Stability and Stabilization for a Class of Semilinear Fractional Differential Systems

Ibrahim, Badawi Hamza Elbadawi; Dong, Qixiang; Zhang, Zhengdi

doi:https://doi.org/10.1155/2020/3814902

Journal of Function Spaces

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Abstract Introduction Preliminaries Applications Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 3814902 | https://doi.org/10.1155/2020/3814902

Stability and Stabilization for a Class of Semilinear Fractional Differential Systems

Badawi Hamza Elbadawi Ibrahim,^1,2Qixiang Dong,³and Zhengdi Zhang¹

Academic Editor: Hugo Leiva

Received28 Nov 2019

Revised05 Feb 2020

Accepted22 Feb 2020

Published30 Mar 2020

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.

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 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 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.

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 aswhere Γ(·) is the gamma function, i.e., , and n is an integer satisfying n − 1 < α ≤ n.
The Laplace transform of the Caputo fractional derivative iswhere 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 asThe Mittag-Leffler function with two parameters is defined asIt is easy to see that E_α(z) = E_α,1(z) and E₁(z) = E_1,1(z) = e^z. The 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:(i)There exist finite real constants M₁ ≥ 1, M₂ ≥ 1, M₃ ≥ 1 such that for any 0 < α < 1, 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]). The constant x₀ ∈ Rⁿ is an equilibrium point of the Caputo dynamic system if and only if f(t, x₀) = 0.
Without loss of generality, we may assume that the equilibrium point is x₀ = 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 of , with order 0 < α ≤ 1(1 < α < 2) 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 t₀ > 0 such that ∥x(t)∥ < ϵ for all t > t₀. The zero solution is said to be asymptotically stable if lim_t⟶∞∥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 . If f ∈ L^p(R, X) and ∈ L^q(R, X), then ∈ C₀(R, X). Here, X is a Banach space, denotes the convolution of the functions f and , i.e., 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:where 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 ∈ 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).

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

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 matrix A is stable.(2)There is a function ∈ L^q(0, ∞) such that where q is a positive constant satisfying 1/p + 1/q = 1 and .

Proof. Let the initial condition be x(0) = x₀. Then, the solution of (9) is given by (10). This is obtained from condition (1) that . It then follows from Lemma 1 that there exist constants M₁ > 0 and M₂ > 0 such thatBecause the matrix A is stable, there exist constants M > 0 and ω > 0 such thatSubstituting it into (12) and using condition (2), one hasNow, we define a function φ: [0, +∞) ⟶ [0, +∞) asBecause 0 < α ≤ 1 and p > 1/α, a simple computation shows that φ ∈ L^p(0, +∞). On the other hand, from condition (2) we know that ∈ L^q(0, +∞). Then, it follows from Lemma 2 that ∈ C₀[0, +∞), which implies thatThis, combined with (14), shows that lim_t⟶+∞∥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:

Because 1 < α ≤ 2, from Lemma 1 (ii), there exists M₃ > 0 such thatfor β = 1, 2, or α. We first prove an existence and uniqueness result via the principle of Banach contraction.

Theorem 2. Let T > 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 thatfor each t ∈ [0, T] and u, ∈ Rⁿ. Ifwhere 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. Let X = C([0, T], Rⁿ) be the Banach space of all continuous functions from [0, T] into Rⁿ, endowed with the norm ∥x∥ = sup_t∈[0,T]∥x(t)∥. We define an operator Q: X ⟶ X asfor 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 obtainIt follows thatCondition (20) shows that , 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 Theorem 1, we now prove the stability of equation (9) for 1 < α ≤ 2.

Theorem 3. Suppose that the conditions in Theorem 2 hold. Then, the zero solution of system (9) is locally asymptotically stable, if the following conditions are satisfied:(1), i = 1, 2, …, n.(2)There is a function ∈ L^q(0, ∞) such that where q is a positive constant satisfying 1/p + 1/q = 1 and p > 1.

Proof. Suppose that the initial condition is x(0) = x₀ and x′(0) = x₁, then the solution of (9) has the following form:for t > 0. Because 1 < α ≤ 2, it follows from (25) and condition (2) thatfor all t > 0. From condition (1) we know that the matrix A is stable. Hence, the inequality (13) holds. Therefore, we obtainfor all t > 0. Similar to the proof of Theorem 1, we define the function φ: [0, +∞) ⟶ [0, +∞) asBecause 1 < α ≤ 2, it is easy to see that φ ∈ L^p(0, +∞) for any p > 1. Because of Lemma 2 and condition (2), we obtain ∈ C₀[0, +∞). Therefore, lim_t⟶+∞∥x(t)∥ = 0. The proof is completed.

Remark 1. Theorems 1 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 , 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: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:

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), where is the eigenvalue of matrix .(ii)f(t, x(t)) satisfies ∥f(t, x(t))∥ ≤ (t), where ∈ L^q(0, ∞).(iii)The matrix is stable. Then, the controlled system (29) is locally asymptotically stable.

Proof. The proof of Theorem 4 is similar to that of Theorems 1 and 3.

Remark 2. The nonlinear term of Chaotic fractional-order systems satisfies ∥f(t, x(t))∥ ≤ (t), where ∈ L^q(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:where > 0, i = 1, 2, 3. This system can be rewritten as (9), whereIt is easy to verify thatwhere . 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.

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Example 2. Consider the following fractional nonlinear system:The system can be rewritten as (9), whereObviously,The eigenvalues of A are and λ₃ = −1. According to Theorems 1 and 3, if , 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.

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Example 3. The fractional-order novel hyperchaotic system can be written aswhere a, b, c, and d are some parameters. System (37) can be rewritten as (9) if we writeIt is easy to see that f(t, x(t)) satisfiesWhen a = 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 . By a simple calculation, if we choose K = (2, −8, −3.5, −1), then the eigenvalues of are (−17.3028, −0.11769, −2.7898 ± 10.7230i). Hence, 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.

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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.

Data Availability

The data used in the examples were originally taken from MATLAB, and they are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by Key Program of the National Natural Science Foundation of China (Grant no. 11872189).

References

K. Diethelm, “The analysis of fractional differential equations,” Lecture Notes in Mathematics, vol. 2004, Springer, Berlin, Germany, 2019.
View at: Google Scholar
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” North-Holland Mathematics Studies, vol. 204, Elsevier Science, Amsterdam, Netherlands, 2006.
View at: Google Scholar
I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1998.
S. Abbas, M. Benchohra, and G. M. N’Guerekata, “Topics in fractional differential equations,” Developments in Mathematics, vol. 27, Springer, New York, NY, USA, 2012.
View at: Google Scholar
D. Matignon, “Stability results on fractional differential equations with application to control processing,” in Proceedings of IMACS-SMC, Lille, France, 1996.
View at: Google Scholar
Y. Li, Y. Q. Chen, and I. Podlubny, “Mittag-Leffler stability of fractional order nonlinear dynamic systems,” Automatica, vol. 45, no. 8, pp. 1965–1969, 2009.
View at: Publisher Site | Google Scholar
Y. Li, Y. Q. Chen, and I. Podlubny, “Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1810–1821, 2010.
View at: Publisher Site | Google Scholar
J. Shen and J. Lam, “Non-existence of finite-time stable equilibria in fractional-order nonlinear systems,” Automatica, vol. 50, no. 2, pp. 547–551, 2014.
View at: Publisher Site | Google Scholar
B. Wang, J. Ding, F. Wu, and D. Zhu, “Robust finite-time control of fractional-order nonlinear systems via frequency distributed model,” Nonlinear Dynamics, vol. 85, no. 4, pp. 2133–2142, 2016.
View at: Publisher Site | Google Scholar
B. Xu, D. Chen, H. Zhang, F. Wang, X. Zhang, and Y. Wu, “Hamiltonian model and dynamic analyses for a hydro-turbine governing system with fractional item and time-lag,” Communications in Nonlinear Science and Numerical Simulation, vol. 47, pp. 35–47, 2017.
View at: Publisher Site | Google Scholar
B. Xu, D. Chen, H. Zhang, and F. Wang, “The modeling of the fractional-order shafting system for a water jet mixed-flow pump during the startup process,” Communications in Nonlinear Science and Numerical Simulation, vol. 29, no. 1–3, pp. 12–24, 2015.
View at: Publisher Site | Google Scholar
N. Aguila-Camacho, M. A. Duarte-Mermoud, and J. A. Gallegos, “Lyapunov functions for fractional order systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 9, pp. 2951–2957, 2014.
View at: Publisher Site | Google Scholar
H. T. Tuan and H. Trinh, “Stability of fractional-order nonlinear systems by Lyapunov direct method,” IET Control Theory & Applications, vol. 12, no. 17, pp. 2417–2422, 2018.
View at: Publisher Site | Google Scholar
D. Y. Chen, W. L. Zhao, J. C. Sprott, and X. Y. Ma, “Application of Takagi-Sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization,” Nonlinear Dynamics, vol. 73, no. 3, pp. 1495–1505, 2013.
View at: Publisher Site | Google Scholar
R. Rakkiyappan, G. Velmurugan, and J. Cao, “Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays,” Nonlinear Dynamics, vol. 78, no. 4, pp. 2823–2836, 2014.
View at: Publisher Site | Google Scholar
D. Ding, D. Qi, and Q. Wang, “Non-linear Mittag-Leffler stabilisation of commensurate fractional-order non-linear systems,” IET Control Theory & Applications, vol. 9, no. 5, pp. 681–690, 2015.
View at: Publisher Site | Google Scholar
M. P. Aghababa, “Control of non-integer-order dynamical systems using sliding mode scheme,” Complexity, vol. 21, no. 6, pp. 224–233, 2016.
View at: Publisher Site | Google Scholar
B. Wang, D. Zhu, J. Xue, and F. Wu, “Stabilization conditions for fuzzy control of uncertain fractional order non-linear systems with random disturbances,” IET Control Theory & Applications, vol. 10, no. 6, pp. 637–647, 2016.
View at: Publisher Site | Google Scholar
L. Chen, Y. He, Y. Chai, and R. Wu, “New results on stability and stabilization of a class of nonlinear fractional-order systems,” Nonlinear Dynamics, vol. 75, no. 4, pp. 633–641, 2014.
View at: Publisher Site | Google Scholar
T. Li and Y. Wang, “Stability of a class of fractional-order nonlinear systems,” Discrete Dynamics in Nature and Society, vol. 2014, Article ID 724270, p. 14, 2014.
View at: Publisher Site | Google Scholar
R. Zhang, G. Tian, S. Yang, and H. Cao, “Stability analysis of a class of fractional order nonlinear systems with order lying in (0, 2),” ISA Transactions, vol. 56, pp. 102–110, 2015.
View at: Publisher Site | Google Scholar
S. Huang, R. Zhang, and D. Chen, “Stability of nonlinear fractional-order time varying systems,” Journal of Computational and Nonlinear Dynamics, vol. 11, no. 3, 2016.
View at: Publisher Site | Google Scholar
S. Huang and B. Wang, “Stability and stabilization of a class of fractional-order nonlinear systems for 0 < α < 2,” Nonlinear Dynamics, vol. 88, no. 2, pp. 973–984, 2017.
View at: Publisher Site | Google Scholar
S. Huang and B. Wang, “Stability and stabilization of a class of fractional-order nonlinear systems for 0 < α < 2,” Journal of Computational and Nonlinear Dynamics, vol. 13, no. 3, Article ID 031003, 2018.
View at: Publisher Site | Google Scholar
Z. Wang, D. Yang, and H. Zhang, “Stability analysis on a class of nonlinear fractional-order systems,” Nonlinear Dynamics, vol. 86, no. 2, pp. 1023–1033, 2016.
View at: Publisher Site | Google Scholar
M. D. l. Sen, “About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory,” Fixed Point Theory and Applications, vol. 2011, Article ID 867932, 19 pages, 2011.
View at: Publisher Site | Google Scholar
W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, “Vector-valued laplace transforms and cauchy problems,” Monographs in Mathematics, vol. 96, Springer, Basel, Switzerland, 2013.
View at: Google Scholar

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Copyright © 2020 Badawi Hamza Elbadawi Ibrahim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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