Stability of a Class of Fractional-Order Nonlinear Systems

In this letter stability analysis of fractional order nonlinear systems is studied. Some new sufficient conditions on the local (globally) asymptotic stability for a class of fractional order nonlinear systems with order 0 < α < 2 are proposed by using properties of Mittag-Leffler function and the Gronwall inequality. And the corresponding stabilization criteria are also given. The numerical simulations of two systems with order 0 < α < 1 and two systems with order 1 < α < 2 illustrate the effectiveness and universality of the proposed approach.


Introduction
During the last decade the fractional calculus has gained importance in both theoretical and applied aspects of several branches of science and engineering.There are two essential differences between integer order derivation and fractional order derivation.Firstly, the integer order derivative indicates a variation or certain attribute at particular time for a mechanical or physical process, while the fractional order derivative is concerned with the whole time domain.Secondly, the integer order derivative describes the local properties of a certain position, while the fractional order derivative is related to the whole space for a physical process.Then many physical systems are well characterized by the fractional order state equations [1][2][3][4], such as fractional order Lotka-Volterra equation [1] in biological systems, fractional order Schödinger equation [2] in quantum mechanics, fractional order Langevin equation [3] in anomalous diffusion, and fractional order oscillator equation [4] in damping vibration.
However there are several open problems in this area.Stability of fractional order systems is one of the most fundamental and important issues.On the other hand, because fractional differential operators are nonlocal and have weakly singular kernels, some methods in dealing with interorder systems cannot be simply extended to fractionalorder methods.To the best of knowledge, the stability of fractional-order nonlinear systems is still relatively few.Reference [5][6][7] investigated the necessary and sufficient stability conditions for linear fractional order differential equations and linear time-delayed fractional differential equations.The stability of -dimensional linear fractional order differential systems with order 1 <  < 2 has already been studied in [8].However, only under some special circumstances or in certain cases, the practical problems may be regarded as linear systems.Therefore, stability of nonlinear system is of great significance, and it also has important value in application.In [9], the stability of fractional nonlinear timedelay systems for Caputo's derivative are investigated, and two theorems for Mittag-Leffler stability of the fractional order nonlinear time-delay systems are proved.In [10], the authors proposed the finite-time stabilization of a class of multistate time delay of fractional nonlinear systems.In [11,12], the authors studied the stability of fractional nonlinear dynamic systems using Lyapunov direct method with the introductions of Mittag-Leffler stability and generalized Mittag-Leffler stability notions.In [13], the authors studied fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks.In [14], some new sufficient conditions ensuring asymptotical stability of fractional-order nonlinear system with delay are proposed firstly.
In this paper the stability of nonlinear fractional order nonlinear system is studied.And by using the Gronwall inequality and the properties of Mittag-Leffler function, we proposed some new sufficient conditions on the local (globally) asymptotic stability for a class of fractional order nonlinear systems with order 0 <  < 2. And the corresponding stabilization criteria are also given.Finally, four numerical simulation examples have illustrated the effectiveness and universality of the proposed methods.

Fractional Order Derivative and
Mittag-Leffler Function
Definition 1 (see [15]).The fractional integral   −  of function () is defined as follows: where fractional order  > 0, and Definition 2 (see [15]).The Caputo derivative with order  of function () is given as where  − 1 <  < ,  ∈  + .The formulas for Laplace transform of the Caputo fractional derivative      () have the following form [16]: where  − 1 ≤  < , and As a generalization of the exponential function which is frequently used in the solutions of integer-order systems, the Mittag-Leffler function is frequently used in the solutions of fractional systems.The definition and properties are given in the following.Definition 3 (see [17]).The Mittag-Leffler function is given as where  > 0 and  ∈ C.
The generalization of Mittag-Leffler function with two parameters is wildly used and defined as follows: where  > 0,  > 0, and  ∈ C.

Properties of Mittag-Leffler Functions and the Gronwall
Inequality.In this section, we give the Gronwall inequality and some important properties of the Mittag-Leffler functions which are used in the following.
Lemma 5 (see [15]).Considering the Laplace transform of Mittag-Leffler function with two parameters, we have where  and  are, respectively, the variables in the time domain and Laplace domain, R() stands for the real part of ,  ∈ R, and L{⋅} denotes the Laplace transform.
Proof.The proof of this Lemma can be found in [15].
where A ∈ R × .
Proof.The proof of this Lemma can be found in [18].

Stability and Stabilization of Fractional Order Nonlinear
System with Order 0 <  < 1.Firstly, we consider the Caputo fractional nonlinear systems [16,21] with the initial condition x 0 = x(0), where x() = ( 1 (),  2 (), . . .,   ())  ∈ R  denotes the state vector of the system,  ∈ (0, 1) is the order of the fractional-order derivative, f : R  → R  defines a nonlinear vector field in the -dimensional vector space, and Ax() and g(x()) denote the linear and nonlinear parts of f(x()), respectively.If f(x * ) = 0, the constant x * is called the equilibrium point of Caputo fractional nonlinear system (12).Without loss of generality, we suppose the equilibrium point is x = 0.
Proof.Applying the Laplace transform on ( 12), we have that is, where X() is the Laplace transform of x(), I is an  ×  identity matrix, and L{⋅} denotes the Laplace transform.By using the Laplace inverse transform, we obtain the solution of ( 16), It follows from Lemma 7 that there exist constants  1 > 0 and  2 > 0 such that Since matrix A is stable, there is a constant Based on the condition (2) ‖g(x())‖ = ‖x()‖, that is, where ‖x()‖ < .And ( − )  <   −   when 0 <  < 1 and  > , then Multiplying the inequality by    , we will get Applying Lemma 8 (Gronwall inequality) to ( 20), we have Then Therefore, when  → ∞, ‖x()‖ → 0 for  > Γ(), which implies that the system ( 12) is asymptotically stable.
The controlled fractional order nonlinear system with linear feedback control input is given as where u() = Kx() is the linear feedback control input, A = A + K, and the feedback gain matrix K ∈ R × needs to be determined.Therefore, our aim is to design a suitable feedback gain matrix K such that the controlled system is local (globally) asymptotically stable.
Proof.The proof is similar to that of Theorem 9.
Proof.The proof is similar to that of Theorem 10.
Proof.The proof is similar to that of Theorem 14.
system, fractional order Chen system, fractional order Lü system, fractional order Liu system, and so forth [22].Therefore, Theorems 9-16 can be used as the criteria to control chaos in a class of fractional-order systems.Compared with nonlinear control methods, the advantage of linear control lies in reducing control cost and is easy to implement.
Remark 18.The obtained sufficient conditions could be applied to a class of fractional order hyperchaotic systems [23][24][25].On the one hand, complex multiscroll chaotic systems have garnered much attention in recent years.J. H. Lü has done a large amount of remarkable work.In fact, the sufficient conditions could be applied to a class of complex multiscroll chaotic systems, which could also generate a complex four-scroll chaotic attractor.

Four Illustrative Examples
In this section, we apply the proposed method in stabilizing a fractional order Chen system, Chua system, Lü system, and Liu system to verify its effectiveness and universality.

Stabilization of Fractional Order Chaotic Chen System.
The fractional order Chen system [21,22] with order  = 0.95 can be de described by When the parameters are chosen as  = 35,  = 3,  = 28, and  = 0.95, system (46) exhibits the chaotic behavior, as shown in Figure 1.We consider system (46) as form ( 12) where ) . ( Adding control input u() = Kx() to system (47), the controlled system can be rewritten as  0    x() = Ax() + g(x()).It is easy to demonstrate that g(x()) satisfies lim that is, ‖g(x())‖ = ‖x()‖.The feedback gain matrix is selected as which satisfies the conditions Re(eig(A)) < 0 and  = −max Re(eig(A)) = 2 > Γ() = 1.0315 in Theorem 11.The simulation result is shown in Figure 2, which shows that the zero solution of the controlled system is asymptotically stable.

Stabilization of Fractional
Order Chaotic Lü System.The fractional order Lü [27] system with order  = 1.09 can be de described by When the parameters are chosen as  = 36,  = 3,  = 20, and  = 1.09, system (56) exhibits the chaotic behavior, as shown in Figure 5.

Conclusion
Stability of the nonlinear dynamical systems is important for scientists and engineers.Fractional dynamic systems were used intensively during the last decade in order to describe the behavior of complex systems in physical and engineering.
In this paper the stabilization of nonlinear fractional order dynamic system is studied.And by using the Gronwall inequality and the properties of Mittag-Leffler function, we proposed some new sufficient conditions on the local (globally) asymptotic stability for a class of fractional order nonlinear systems.Finally the corresponding stabilization criteria are also given.Four numerical simulation examples have illustrated the effectiveness and universality of the proposed methods.