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Stability Analysis of Fractional Differential Systems with Order Lying in (1, 2)
Advances in Difference Equations volume 2011, Article number: 213485 (2011)
Abstract
The stability of dimensional linear fractional differential systems with commensurate order and the corresponding perturbed systems is investigated. By using the Laplace transform, the asymptotic expansion of the MittagLeffler function, and the Gronwall inequality, some conditions on stability and asymptotic stability are given.
1. Introduction
Fractional calculus has a long history with more than three hundred years [1–3]. Up to now, it has been proved that fractional calculus is very useful. Many mathematical models of real problems arising in various fields of science and engineering were established with the help of fractional calculus, such as viscoelastic systems, dielectric polarization, electrodeelectrolyte polarization, and electromagnetic waves [4–7].
Recently, the stability theory of fractional differential equations (FDEs) is of main interest in physical systems. Moreover, some stability results have been found [8–17]. These stability results are almost about the linear fractional differential systems with commensurate order (i.e., the fractional derivative order has to be an integer multiple of minimal fractional order [18]). For example, a necessary and sufficient condition on asymptotic stability of linear fractional differential system with order was first given in [9]. Then, some literatures on the stability of linear fractional differential systems with order have been appeared [11–15]. However, not all the fractional differential systems have fractional orders in (0, 1). There exist fractional models which have fractional orders lying in (1, 2), for example, superdiffusion [19]. Hence, the stability of linear fractional differential systems with order has also been considered by using the conversion methods and transfer function [8, 10]. Almost all of the above literatures dealt with the fractional differential systems with Caputo derivative. Recently, Qian et al. [16] have investigated the stability of fractional differential systems with RiemannLiouville derivative whose order lies in (0, 1) in details. It is worth mentioning that not all of the stability conditions are parallel to the corresponding classical integerorder differential equations because of nonlocality and weak singularities of fractional calculus. For example, the solution to an autonomous fractional differential equation cannot define a dynamical system in the sense of semigroup [20]. Of course, some of the mathematical tools for the integerorder differential equation can be applied to fractional kinetics. In [20], the authors first define the Lyapunov exponents for fractional differential system then determine their bounds, where the basic ideas and techniques are borrowed from [21, 22].
In this paper, we study the stability of autonomous linear fractional differential systems, nonautonomous linear fractional differential systems, and the corresponding perturbed systems with order by using the properties of MittagLeffler functions and the Gronwall inequality.
The paper is organized as follows. In Section 2, we first recall some definitions and lemmas used throughout the paper. In Section 3, the stability analysis is presented for autonomous linear fractional differential systems with order . The stability of nonautonomous linear fractional differential systems and the corresponding perturbed systems are studied in Sections 4 and 5, respectively. Conclusions and comments are included in Section 6.
2. Preliminaries
Let us denote by the set of real numbers, denote by the set of positive real numbers, denote by the set of positive integer numbers, and denote by the set of complex numbers.
In this section, we recall the most commonly used definitions and properties of fractional derivatives, MittagLeffler functions, and their asymptotic expansions.
Definition 2.1.
The RiemannLiouville derivative with order of function is defined as follows:
where , is the Gamma function.
Definition 2.2.
The Caputo derivative with order of function is defined as follows:
where .
Their Laplace transforms for are given as follows [23]:
Definition 2.3.
The MittagLeffler function is defined by
where the real part of , that is, , . The twoparameter MittagLeffler function is defined by
where and , .
One can see from the above equations. By analogy with (2.6), for , we introduce a matrix MittagLeffler function defined by [24]
The following definitions of stability are introduced.
Definition 2.4.
The constant is an equilibrium of fractional differential system if and only if for all , where the operator denotes either or .
Without loss of generality, let the equilibrium be , we introduce the following definition.
Definition 2.5.
The zero solution of with order is said to be stable if, for any initial values (), there exists such that for all . The zero solution is said to be asymptotically stable if, in addition to being stable, as .
It is useful to recall the following asymptotic formulas for our developments in the sequel.
Lemma 2.6.
If , is an arbitrary complex number and is an arbitrary real number such that
then for an arbitrary integer , the following expansions hold:
with and
with , .
Proof.
These results were proved in [23].
Especially, taking into account the Lemma 2.6 and derivatives of the MittagLeffler function, we obtain
with , and
with , , .
Lemma 2.7 (see [25]).
If and , is an arbitrary real number, satisfies , and is a real constant, then
where , denotes the eigenvalues of matrix and denotes the norm.
Lemma 2.8 (Jordan Decomposition [26]).
Let be a square complex matrix, then there exists an invertible matrix such that
where the are the Jordan blocks of with the eigenvalues of on the diagonal. The Jordan blocks are uniquely determined by .
Lemma 2.9 (see [27]).
If
where all the functions involved are continuous on , , and , then satisfies
If, in addition, is nondecreasing, then
3. Stability of Autonomous Linear Fractional Differential Systems
3.1. The RiemannLiouville Derivative Case
In this subsection, we consider the following system of fractional differential equations:
with the initial conditions
where , matrix , and . Then, by analyzing the solutions of the above initial value problem (3.1)(3.2), one can find the following result.
Theorem 3.1.
The autonomous fractional differential system (3.1) with RiemannLiouville derivative and the initial conditions (3.2) is asymptotically stable iff . In this case, the components of the state decay towards 0 like . Moreover, the system (3.1) is stable iff either it is asymptotically stable, or those critical eigenvalues which satisfy have the same algebraic and geometric multiplicities.
Proof.
Applying the Laplace transform, we can get the solution of (3.1)(3.2),
Firstly, we study the properties of the elements of matrixes , . With regard to matrix , there exists an invertible matrix , such that
from Lemma 2.8, where the Jordan block
, is the eigenvalue of matrix and . Substituting (3.4) into , we yield
where . The matrix can be written as follows by computing
where the operator is given as follows:
The nonzero elements of can be described uniformly as

(i)
If , then
(3.10)
It is obvious that () for and . Thus, ().

(ii)
If , three cases will be considered separately.
Case 1 ().
If and , then
That is to say, () from the asymptotic expansion (2.12) and (). Moreover, the components of the state decay towards 0 like . Taking into account the entire function , we also get the boundedness of (; ).
Case 2 ().
If and , from the asymptotic expansion (2.11), we have
then
because of , that is, .
So, ().
Case 3 ().
Let , where is the modulus of and .
Firstly, suppose that the critical eigenvalue has the same algebraic and geometric multiplicities, that is, the matrix is a diagonal matrix, then, according to (3.7), we have
If , we have the diagonal elements of matrix (3.14) () from the asymptotic expansion (2.11). So, the solution of (3.1) is stable in this case.
Next, suppose that the algebraic multiplicity of critical eigenvalue is not equal to the geometric multiplicity, that is, the matrix is a Jordan block matrix, and matrix is the same as (3.7), then the nondiagonal elements of can be evaluated from (3.12) as follows:
So,
that is, as .
According to the above discussions, the proof is completed.
Remark 3.2.

(1)
If , then system (3.1) is not stable.

(2)
If has zero eigenvalue, system (3.1) is not stable.

(3)
If has critical eigenvalue(s) , that is, , and the arguments of the rest eigenvalues in absolute values are greater than , then system (3.1) is not stable provided that has different geometric and algebraic multiplicities.
3.2. The Caputo Derivative Case
Now, we consider the fractional differential system with Caputo derivative
under the initial conditions
where , , and are as in Section 3.1. Then, one can get the following theorem.
Theorem 3.3.
The autonomous fractional differential system (3.17) with Caputo derivative and initial conditions (3.18) is asymptotically stable iff . In this case, the components of the state decay towards 0 like . Moreover, the system (3.17) is stable iff either it is asymptotically stable, or those critical eigenvalues which satisfy have the same algebraic and geometric multiplicities.
Proof.
This theorem can be proved in the same manner as that in the proof of Theorem 3.1, so it is omitted here.
4. Stability of Nonautonomous Linear Fractional Differential Systems
4.1. The RiemannLiouville Derivative Case
We will consider a nonautonomous fractional differential system with RiemannLiouville derivative
under the initial conditions
where , matrix , is a continuous matrix, and . The main results of this subsection are derived as follows.
Theorem 4.1.
If the matrix such that , the critical eigenvalues which satisfy have the same algebraic and geometric multiplicities, and is bounded, then the zero solution of (4.1) is stable.
Proof.
Applying the Laplace transform, we can get the solution of (4.1)(4.2),
From the proof of Theorem 3.1, the matrix is bounded for . Therefore, there exist positive numbers , such that (). Now, we can get the estimate of solution
Applying the Gronwall inequality (2.17) leads to
Thus, we derive that is bounded according to the condition , that is, the zero solution of (4.1) is stable. The proof is completed.
Similarly, we can derive the following conclusion.
Theorem 4.2.
If the matrix such that , , and (, ) for , then the zero solution of (4.1) is asymptotically stable.
Proof.
From the proof of Theorem 3.1, the following expression is valid:
where such that and . Moreover, from (4.4) and (2.17), one has
Substituting (4.7) into (4.6), we have
where . It follows from the condition (, ) for that there exists a constant , such that and
So, the zero solution of (4.1) is asymptotically stable.
4.2. The Caputo Derivative Case
In this subsection, we consider a nonautonomous fractional differential system with Caputo derivative
under the initial conditions
where , , and are as in Section 4.1, is a continuously differentiable matrix. We can get the solution of (4.10)(4.11) by using the Laplace transform and Laplace inverse transform
The main stability results of this subsection are derived as follows.
Theorem 4.3.
If the matrix such that , , the critical eigenvalues which satisfy have the same algebraic and geometric multiplicities, and is bounded, then the zero solution of (4.10) is stable.
Proof.
The proof line is similar to that of Theorem 4.1.
Theorem 4.4.
If the matrix such that , , and (, ) for , then the zero solution of (4.10) is asymptotically stable.
Proof.
From the solution (4.12) and Lemma 2.7, we can directly get
where and , such that . Furthermore, there exists a constant such that
that is,
Substituting (4.15) into (4.13) gives
It follows from the condition (, ) for that there exists a constant , such that and
So, the zero solution of (4.10) is asymptotically stable.
5. Stability of the Perturbed Systems
In this section, we only study the perturbed system of a linear fractional differential system with RiemannLiouville derivative
under the initial conditions
where , matrix , and . is a continuous function in which ; moreover, fulfils the Lipschitz condition with respect to . Then, the unique solution of (5.1)(5.2) can be written as
The following theorem can be proved by the same argument used in the proof of Theorem 4.1.
Theorem 5.1.
If the matrix such that , , the critical eigenvalues which satisfy have the same algebraic and geometric multiplicities. Moreover, suppose that there exists a positive function which satisfies the following conditions:
(i) is bounded,
(ii),
then the zero solution of (5.1) is stable.
Theorem 5.2.
If the matrix such that , , , and suppose that the function satisfies uniformly
then the zero solution of (5.1) is asymptotically stable.
Proof.
According to the proof of Theorem 3.1 and Lemma 2.7, we have
where such that and . Taking into account the condition (5.4), there exists a constant , such that
Then,
where . Applying the Gronwall inequality (2.16) to (5.7) yields
So, the zero solution of (5.1) is asymptotically stable due to . The proof is thus finished.
6. Conclusion
It is well know that many physical phenomena having memory and genetic characteristics can be described by using the fractional differential systems. Especially, the fractional differential systems with order have recently gained an increasing attention [23, 28–31]. It should be noted that [29, 30] are earlier and interesting work on fractional interval systems. Motivated by the above research activities, in this paper, we have studied the stability of linear fractional differential systems and the corresponding perturbed systems with RimannLiouville derivative and Caputo derivative for the commensurate order . The main analytic tools used in this paper are the MittagLeffler function and the Gronwall inequality. For the autonomous linear fractional differential systems with order , the necessary and sufficient conditions on stability and asymptotic stability are given, which are almost the same as those with the fractional derivative order . But the components of the state decay towards 0 like , which is different from the case with Caputo derivative order . For the nonautonomous linear fractional differential systems, we have derived some sufficient conditions on stability and asymptotic stability. We have further given the asymptotic stability results of the perturbed systems with order .
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Acknowledgments
The authors wish to thank Professors D. Baleanu and Juan J. Trujillo for their kind invitation to submit our paper to their special issue on Fractional Models and their Applications. They also thank the anonymous reviewers of this paper for their careful reading and invaluable correction suggestions. The present work was partially supported by the National Natural Science Foundation of China under Grant no. 10872119 and the the Key Disciplines of Shanghai Municipality under Grant no. S30104.
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Zhang, F., Li, C. Stability Analysis of Fractional Differential Systems with Order Lying in (1, 2). Adv Differ Equ 2011, 213485 (2011). https://doi.org/10.1155/2011/213485
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Keywords
 Fractional Order
 Fractional Derivative
 Fractional Calculus
 Fractional Differential Equation
 Jordan Block