Stability analysis of fractional-order systems with the Riemann–Liouville derivative

Abstract

In this paper, the stability of fractional-order systems with the Riemann–Liouville derivative is discussed. By applying the Mittag-Leffler function, generalized Gronwall inequality and comparison principle to fractional differential systems, some sufficient conditions ensuring stability and asymptotic stability are given.

Keywords:

• asymptotic stability
• generalized Gronwall inequality
• Riemann–Liouville derivative
• fractional differential system

1. Introduction

Fraction calculus has more than 300 years history. With the development of science and engineering applications, fractional calculus has become one of the most hottest topics. Up to now, many fractional results have been presented which are very useful (Debnath, 2004; Miller & Ross, 1993; Podlubny, 1999; Samko, Kilbas, & Marichev, 1993; Zhang & Li, 2011).

Stability analysis is the most fundamental for studying fractional differential equations. Recently, many stability results of fractional-order systems are interesting in physical systems, so more and more stability results have been found, see, for instance, Ahn and Chen (2008), Ahmed, EI-Saka, and EI-Saka (2007), Deng, Li, and Liu, (2007), Li, Chen, and Podlubny (2009), Li and Zhang (2011), Miller and Ross (1993), Moze, Sabatier, and Oustaloup (2007), Odibat (2010), Qian, Li, Agarwal, and Wong (2010), Radwan, Soliman, Elwakil, and Sedeek (2009), Sabatier, Moze, and Farges (2010), Samko et al. (1993), Tavazoei and Haeri (2009), Wen, Wu, and Lu, (2008) and Zhang and Li (2011). These stability results are mainly concerned with the linear fractional differential system. For example, in Matignon (1996), a sufficient and necessary condition on asymptotic stability of linear fractional differential system with order 0<α<1 was first given. Then some other research on the stability of fractional-order systems appeared. Of course, there also exist fractional-order systems with order lying in (1, 2). In Zhang and Li (2011), authors dealt with the following fractional differential system: where denotes either the Caputo or the Riemann–Liouville fractional derivative operator. They analysed stability of the above fractional differential system by applying Gronwall's inequality (Corduneanu, 1971) and related results.

In this paper, three conditions about B(t) are given as follows:

• (I)   is bounded;

• (II)   is bounded;

• (III)  .

Under these conditions, the stability and asymptotic stability of nonautonomous linear fractional differential systems with the Riemann–Liouville derivative are analysed by using generalized Gronwall's inequality, some properties of the Mittag-Leffler function and relevant results. From the results derived in this paper, we can also analyse the stability of these nonlinear systems in the future.

This paper is organized as follows. In Section 2 some necessary definitions and lemmas are recalled, which will be used later. The main results are presented in Section 3. Finally, some conclusions are drawn in Section 4.

2. Preliminaries

In this section, the most commonly used definitions and results are stated, which will be used later.

Definition 2.1

The Riemann–Liouville fractional derivative with order α of function x(t) is defined as where is the Gamma function.

The Laplace transform of the Riemann–Liouville fractional derivative is

Definition 2.2

The Mittag-Leffler function with two parameters is defined as where . When β=1, one has , furthermore, E_{1, 1}(z)=e^z.

The Laplace transform of the Mittag-Leffler function is

Definition 2.3

The zero solution of with order is said to be stable if, for any initial values , there exists such that for all t>t₀. The zero solution is said to be asymptotically stable if, in addition to being stable, as t→+∞.

lemma 1

If and is an arbitrary real number, μ satisfies , and C>0 is a real constant, then where , spec(A) denotes the eigenvalues of matrix A and denotes the l₂ norm.

lemma 2

If and is an arbitrary complex number and μ satisfies , then for an arbitrary integer p≥1, the following expansions hold: with and with and .

Especially, in Zhang and Li (2011) it has been obtained that the matrix is bounded, i.e. for some M_k>0.

lemma 3

Suppose α>0, a(t) is a nonnegative locally integrable function on 0≤t<T (some T≤∞) and g(t) is a nonnegative and nondecreasing continuous function defined on (constant), and suppose u(t) is nonnegative and locally integrable on 0≤t<T with on this interval, then

Moreover, if a(t) is a nondecreasing function on [0, T), then

lemma 4

Suppose that g(t) and u(t) are continuous on and r≥0 are two constants, if then

3. Stability of nonautonomous linear fractional differential systems

3.1. Fractional-order α:0<α<1

Consider the nonautonomous fractional system (1) with the initial condition (2) where x∈Rⁿ, matrix is a continuous t matrix.

Theorem 1

Suppose and is bounded, i.e. , where M, N>0, then the solution of Equation (1) is asymptotically stable.

Proof By the Laplace transform and the inverse Laplace transform, the solution of Equations (1) with (2) can be written as then we can obtain From the boundedness, we can obtain (3) Multiplying by e^{γ t} both sides of Equation (3), we have Let , then according to Lemma 4, one has (4) Multiplying by e^{−γ t} both sides of Equation (4), we can obtain then , so . That is, the solution of Equation (1) is asymptotically stable.

3.2. Fractional-order α:1<α<2

Consider the following fractional-order system: (5) with the initial conditions (6) where x∈Rⁿ, matrix is a continuous matrix.

Theorem 2

If the eigenvalues of matrix A satisfy and is bounded, i.e. for some M>0, then the zero solution of Equation (5) is asymptotically stable.

Proof By the Laplace transform and the inverse Laplace transform, the solution of Equations (5) with (6) can be written as then we can obtain where L, M, M₀, M₁>0 such that Based on Lemmas 2 and 3, we can obtain When . That is, the solution of Equation (5) is asymptotically stable.

Remark 1 Suppose the Caputo derivative takes the place of the Riemann–Liouville derivative in Equation (1) and all other assumed conditions remain the same, then the conclusions of Theorem 2 still hold.

Theorem 3

If all eigenvalues of matrix A satisfy is nondecreasing and , then the zero solution is asymptotically stable.

Proof By the Laplace transform and the inverse Laplace transform, the solution of Equations (5) with (6) can be written as then one can obtain (7) Then where L, M₀, M₁>0 such that Multiplying by on both sides of Equation (7), one obtains Applying Lemma 3 leads to Then Since , then as t→∞, so is bounded, i.e. ∃N, such that .

We also can obtain the following expression from the solution: where Since , then When t→∞, . So the solution of Equation (5) is asymptotically stable.

Remark 2 Suppose the Caputo derivative takes the place of the Riemann–Liouville derivative in Equation (5) and all other assumed conditions remain the same, then the conclusion is stable.

4. Conclusions

In this paper, we have studied the stability and asymptotic stability of the nonautonomous linear differential system with the Riemann–Liouville fractional derivative and established the corresponding stability results of its zero solution. By using the Laplace transform, Mittag-Leffler function, the generalized Gronwall inequality, some sufficient conditions ensuring the stability and asymptotic stability of the perturbed linear fractional differential system with the Riemann–Liouville fractional derivative were given.

Acknowledgements

We would like to thank Ranchao Wu and Yanfen Lu for discussions, and the reviewers and the associate editor for their useful comments on our paper. Ranchao Wu is supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China under grant 20093401120001, the Natural Science Foundation of Anhui Province under grant 11040606M12 and the 211 project of Anhui University under grant KJJQ1102.