ieee/caa journal of automatica sinica 1 stability...

IEEE/CAA JOURNAL OF AUTOMATICA SINICA 1

Stability analysis of a class of nonlinear fractionaldifferential systems with Riemann-Liouville

derivativeRuoxun Zhang, Shiping Yang, Shiwen Feng

Abstract—This paper investigates the stability of n-dimensionalnonlinear fractional differential systems with Riemann-Liouvillederivative. By using the Mittag-Leffler function, Laplace trans-form and the Gronwall-Bellman lemma, one sufficient conditionis attained for the asymptotical stability of a class of nonlinearfractional differential systems whose order lies in (0, 2). Accord-ing to this theory, if the nonlinear term satisfies some conditions,then the stability condition for nonlinear fractional differentialsystems is the same as the ones for corresponding linear systems.Several examples are provided to illustrate the applications of ourresult.

Index Terms—Stability, Nonlinear fractional differential sys-tem, Riemann-Liouville derivative

I. INTRODUCTION

IN this paper, we consider the stability of n-dimensionalnonlinear fractional differential systems with Riemann-Liouville derivative:

RL0 D

αt x(t) = Ax(t) + f(x(t)) (1)

where 0 < α < 2, x(t) ∈ Rn×1is the state vector,RL0 D

αt x(t)denotes Riemann-Liouville’s fractional derivative

with the lower limit 0 for the function x(t), A ∈ Rn×nis theconstant parameter matrix andf(x(t)) ∈ Rn×1is a nonlinearfunction vector.

In the last 30 years, fractional calculus has attracted at-tention of many mathematicians, physicists and engineers.Significant contributions have been made to both the theoryand applications of fractional differential equations (see [1]and references there in). Also, fractional differential equationshave recently been proved to be valuable tools in modelingof many physical phenomena in various ?elds of science andengineering.

Recently, the stability of fractional differential systems hasattracted increasing interest due to its importance in controltheory. In 1996, Matignon [2] firstly studied the stabilityof linear fractional differential systems. Since then, manyresearchers have studied further on the stability of linear

This article has been accepted for publication in a future issue of thisjournal, but has not been fully edited. Content may change prior to finalpublication.

This article was recommended by Associate Editor Antonio Visioli.Ruoxun Zhang and Shiwen Feng are with the College of Teacher Edu-

cation, Xingtai University, Xingtai 054001, China (email: [email protected],[email protected])

Shiping Yang is with the College of Physics and Information Engineer-ing, Hebei Normal University, Shijiazhuang 050016,China (email: [email protected])

Digital Object Identifier 10.1109/JAS.2016.7510199

fractional differential systems [3− 5]. The stability analysis ofnonlinear fractional differential systems is much more difficultand only a few available. For example, Li et al. investigated theMittag-Leffler stability of fractional order nonlinear dynamicsystems [6] and proposed Lyapunov direct method to checkstability of fractional order nonlinear dynamic systems [7].Wen et al. [8] and Zhou et al [9] considered the stabilityof nonlinear fractional differential systems. In [10], Zhanget al proposed a single state adaptive-feedback controller forstabilization of three-dimensional fractional-order chaotic sys-tems. Based on the theory of Linear Matrix Inequality (LMI),Faieghi et al [11] proposed a simple controller for stabilizationof a class of fractional-order chaotic systems. Wang et al.present the Ulam -Hyers stability for fractional Langevinequations [12], and Ulam- Hyers-Mittag-Lef?er stability forfractional delay differential equations [13]. The methods whichthey proposed for stability of a class of fractional differentialequations provide us with a very useful method for studyingHyers–Ulam stable system. That is, one does not have to reachthe exact solution. What is required is to get a function whichsatis?es a suitable approximation inequality.

Note that these papers on the stability of the frac-tional differential systems mainly concentrated on fractionalorderαlying in (0, 1). Recently, in Ref [14], Zhang et alconsidered the stability of nonlinear fractional differentialsystems with Caputo derivative whose order lies in (0, 2).In this paper, we study the stability of the nonlinear fractionaldifferential systems with Riemann-Liouville derivative whoseorder lies in (0, 2). By using the Mittag-Leffler function,Laplace transform and the Gronwall-Bellman lemma, a sta-bility theorem is proven theoretically. The stability conditionshave no restriction on the norm of the linear parameter matrixA. The paper is outlined as follows. In section II, somedefinitions and lemmas are introduced. In section III, thestability of a class of nonlinear fractional differential systemswith commensurate order 0 < α < 2 is investigated. Thesimulation and conclusions are included in section IV and V,respectively.

II. PRELIMINARIES

Definition 2.1 [15]. The Riemann-Liouville derivative withαof function x(t) is defined as follows

RLt0 D

αt x(t) =

1Γ(n−α)

dn

dtn

∫ tt0

x(τ)(t−τ)α−n+1 dτ,

(n− 1 ≤ α < n) (2)

2 IEEE/CAA JOURNAL OF AUTOMATICA SINICA

The Laplace transform of the Riemann-Liouville fractionalderivative RLt0 D

αt x(t) is

∫ ∞0

e−stRLt0 Dαt x(t)dt =

sαX(t)−n−1∑

k=0

sk[Dα−k−1x(t)]t=t0 (3)

Definition 2.2 [15]. The two-parameter Mittag- Lefflerfunction is defined as

Eα, β(z) =∞∑

k=0

zk

Γ(αk + β), (Reα > 0, β ∈ C, z ∈ C). (4)

The Laplace transform of Mittag-Leffler function can be foundto be ∫ ∞

0

e−sttαk−β−1E(k)α,β(±atα)dt =k!sα−β

(sα ∓ a)k+1 , (R(s) > |a|1n ). (5)

Definition 2.3 [16]. By analogy with Definition 2.2, forA ∈ Cn×n, a matrix Mittag-Leffler function is defined as:Eα,β(A) =

∞∑k=0

Ak

Γ(αk+β) , (β ∈ C, R(α) > 0)Lemma 2.1 [8]. If A ∈ Cn×n, 0 < α < 2, β is an arbitrary

real number, µ is such that πα2 < µ < min{π, πα} and C1isa real constant, then

||Eα,β(A)|| ≤ C11 + ||A|| , (6)

where µ ≤ arg(λ(A)) ≤ π, λ(A) denotes the eigenvalues ofmatrix A and || · ||denotes the l2-norm.

Lemma 2.2 [17]. (Gronwall-Bellman lemma) If

ϕ(t) ≤ h(t) +∫ t

t0

g(τ)ϕ(τ)dτ, t0 ≤ t ≤ t1. (7)

where g(t), h(t) and ϕ(t) are continuous on [t0, t1], t1 →∞,t0 ≤ t ≤ t1 and g(t) ≥ 0. Then ϕ(t) satisfies

φ(t) ≤ h(t) +∫ t

t0

h(τ)g(τ) exp[∫ t

τ

g(s)ds]dτ. (8)

In addition, if h(t) is nondecreasing, then

φ(t) ≤ h(t) exp[∫ t

t0

g(s)ds]dτ. (9)

III. STABILITY THEORY OF N-DIMENSIONAL NONLINEARFRACTIONAL DIFFERENTIAL SYSTEMS

In this section, based on the definition and lemma in section2, we present the stability theorem for a class of nonlinearfractional differential systems such as system (1).

Theorem 1. Consider the system (1). Let λi(A) (i =1, 2, · · · , n)be the eigenvalues of matrix A. If

1) | arg(λi(A))| > απ/2;2) The nonlinear function f(x(t))satisfies

lim||x(t)||→0

||f(x(t))||||x(t)|| = 0. (10)

Then the zero solution of (1) is locally asymptotically stable.Proof. a) The case 0 < α < 1In this case, the initial condition is

RL0 D

α−1t x(t)|t=0 = x0 (11)

Taking Laplace transform of (1), we have

X(s) = (Isα −A)−1(x0 + L[f(x(t))]) (12)where I is an n× n identity matrix.

Then taking inverse Laplace transform for (12), it yields

x(t) = x0tα−1Eα,α(Atα)+∫ t0

(t− τ)α−1Eα,α(A(t− τ)α)f(x(τ))dτ (13)

By the condition (10), there exist C0 > 0 and δ > 0, suchthat

||f(x(t))|| < α||A||2C0

||x(t)||as||x(t)|| < δ (14)

From (14) and Lemma 2.1 , (13) gives

||x(t)|| ≤ C0||x0||tα−1

1 + ||Atα|| +∫ t0

||(t− τ)α−1||C01 + ||A(t− τ)α||

α||A||2C0

||x(τ)||dτ =C0||x0||tα−11 + ||A||tα +

∫ t0

α||A||(t− τ)α−12(1 + ||A||(t− τ)α) ||x(τ)||dτ

According to Lemma 2.2, we obtain

||x(t)|| ≤ C0||x0||tα−1

1 + ||A||tα +∫ t

0

C0||x0||τα−11 + ||A||τα

× α||A||(t− τ)α−1

2(1 + ||A||(t− τ)α)exp

( ∫ tτ

α||A||(t− s)α−12(1 + ||A||(t− s)α)ds

)dτ

=C0||x0||

t1−α + ||A||t+

∫ t0

αC0||x0||τα−1||A||(t− τ)α−12(1 + ||A||τα)(1 + ||A||(t− τ)α)0.5 dτ

≤ C0||x0||t1−α + ||A||t

+∫ t

0

αC0||x0||τα−1||A||(t− τ)α−12(1 + ||A||τα)0.5(1 + ||A||(t− τ)α)0.5 dτ

≤ C0||x0||t1−α + ||A||t

+ 0.5αC0||x0||∫ t

0

τ0.5α−1(t− τ)0.5α−1dτ

≤ C1||x0||t1−α + ||A||t

+ 0.5αC0||x0||Γ(0.5α)Γ(0.5α)Γ(α) tα−1 → 0

as t →∞.So, the zero solution of (1) is locally asymptotically stable.

ZHANG et al.: STABILITY ANALYSIS OF A CLASS OF NONLINEAR FRACTIONAL· · · 3

2) The case 1 < α < 2In this case, the initial condition is

RL0 D

α−kt x(t)|t=0 = xk−1, (k = 1, 2) (15)

We can get the solution of (1) with the initial condition (15)by using the Laplace transform and Laplace inverse transform:

x(t) = x0tα−1Eα ,α(Atα) + tα−2 x1Eα,α−1(Atα)

+∫ t

0

(t− τ)α−1Eα,α(A(t− τ)α)f(x(τ))dτ (16)

By the condition (10), there exist C0 > 0 and δ > 0, suchthat

||f(x(t))|| < (α− 1)||A||2C0

||x(t)||as||x(t)|| < δ. (17)

From (17) and Lemma 2.1 , (16) gives

||x(t)|| ≤ C0||x0||tα−1

1 + ||Atα|| +C1||x1||tα−21 + ||Atα||

+∫ t

0

||(t− τ)α−1||C01 + ||A(t− τ)α||

(α− 1)||A||2C0

||x(τ)||dτ

=C0||x0||tα−11 + ||A||tα +

C1||x1||tα−21 + ||A||tα

+∫ t

0

||(t− τ)α−1||1 + ||A||(t− τ)α

(α− 1)||A||2

||x(τ)||dτ (18)

where C1 > 0. According to Lemma 2.2, we obtain

||x(t)| ≤ C0||x0||τα−1

1 + ||A||τα +

C1||x1||τα−21 + ||A||τα +

∫ t0

(C0||x0||τα−11 + ||A||τα

+C1||x1||τα−21 + ||A||τα

) (α− 1)||A||(t− τ)α−12(1 + ||A||(t− τ)α)

× exp( ∫ t

τ

(α− 1)||A||(t− s)α−12(1 + ||A||(t− s)α) ds

)dτ

=C0||x0||tα−11 + ||A||tα +

C1||x1||tα−21 + ||A||tα

+∫ t

0

(C0||x0||tα−11 + ||A||tα +

C1||x1||tα−21 + ||A||tα

)

× (α− 1)||A||(t− τ)α−1

2(1 + ||A||(t− τ)α)1−α−12αdτ

≤ C0||x0||tα−1

1 + ||A||tα +C1||x1||tα−21 + ||A||tα

+∫ t

0

C0||x0||τα−1(1 + ||A||τα)0.75

× (α− 1)||A||(t− τ)α−1

2(1 + ||A||(t− τ)α)1−α−12αdτ

+∫ t

0

C1||x1||τα−2(1 + ||A||τα)α−12α

× (α− 1)||A||(t− τ)α−1

2(1 + ||A||(t− τ)α)1−α−12αdτ

≤ C0||x0||tα−1

1 + ||A||tα +C1||x1||tα−21 + ||A||tα

+(α− 1)C0||x0||2||A||0.25+1/(2α)

∫ t0

τ0.25α−1(t− τ)0.5α−1.5dτ

+(α− 1)C1||x1||

2

∫ t0

τ0.5α−1.5(t− τ)0.5α−1.5dτ

≤ C0||x0||||A||t +C1||x1||||A||t2

+(α− 1)C0||x0||2||A||0.25+1/(2α)

Γ(0.25α)Γ(0.5α− 0.5)Γ(0.75α− 0.5)t0.75(2−α)

+(α− 1)C1||x1||

2Γ(0.5α)Γ(0.5α− 0.5)

Γ(α− 0.5)t2−α→ 0 as t →∞. (19)

So, the zero solution of (1) is locally asymptotically stable.Remark 1. The nonlinear term of many fractional or-

der chaotic systems satisfy (12). For example, fractional-order Lorenz system [17], fractional-order Chen system [18],fractional-order Lü system [19], fractional-order Liu system[20], fractional-order Arneodo system [21], fractional-orderChua system [22] and fractional-order hyperchaotic Chensystem [23] etc. So, Theorem 1 can be applicable to controlchaos in a large class of generalized fractional-order chaoticor hyperchaotic systems via a linear feedback controller. SeeExample 3 in Section 4.

Remark 2. Theorem 1 provides us with a simple procedurefor determining the stability of the fractional order nonlinearsystems with Caputo derivative with order 0 < α < 2. If thenonlinear term f(x(t))satisfies Eq.(10), then one does not haveto reach the exact solution. What is required is to calculatethe eigenvalues of the matrix A, and test their arguments. If| arg(λi(A))| > απ/2for all i, we conclude that the origin isasymptotically stable.

IV. THREE ILLUSTRATIVE EXAMPLES

The following illustrative examples are provided to showthe effectiveness of the stability theorem. When numerically


solving fractional differential equations, we adopt the methodintroduced in [24].

Example 1. Consider the nonlinear fractional differentialsystems

RL0 D

αt x1 = x1 + x2 + x3 + x2x3

RL0 D

αt x2 = −x1 + x2 − x3 + x22

RL0 D

αt x3 = x1x2 − x3

(20)

System (20) can be rewritten as (1), in which

A =

1 1 1−1 1 −10 0 −1

, f(x(t)) =

x2x3x22x1x2

(21)

Obviously, it is easy to verify that

lim||x(t)||→0

||f(x(t)||||x(t)|| = lim||x(t)||→0

√(x2x3)2 + x42 + (x1x2)2√

x21 + x22 + x

23

≤ lim||x(t)||→0

√(x2x3)2 + x42 + (x1x2)2√

x22

≤ lim||x(t)||→0

√x23 + x

22 + x

21 = 0,

which implies thatf(x(t) satisfies Condition (2) in Theorem1. By using simple calculation, the eigenvalues

of A are λ 1,2 = 1±i and λ 3 = −1. According to Theorem1, if α < 0.5, the zero solution of (20) is asymptotically stable.Simulation results are displayed in Figs. 1–3. Fig. 1 and Fig. 2show the zero solution of system (20) is asymptotically stablewith α = 0.4 and α = 0.49 , respectively. Fig. 3 shows thezero solution of the system (20) is unstable with α = 0.5.

Fig. 1. System (20) is asymptotically stable with α = 0.4

Example 2. Consider the nonlinear fractional differentialsystems

RL0 D

αt x1 = −x1 + x2x3

RL0 D

αt x2 = x3

RL0 D

αt x3 = x1 − x2 − x3 − x1x2

(22)

System (22) can be rewritten as (1), in which

A =

−1 0 00 0 11 −1 −1

, f(x(t)) =

x2x30−x1x2

(23)

Obviously, it is easy to verify that lim||x(t)||→0

||f(x(t)||||x(t)|| = 0,

which implies thatf(x(t) satisfies Condition (2) in Theorem 1.By using simple calculation, the eigenvalues of A are λ 1,2 =−1/2 ± √3i/2 and λ 3 = −1. According to Theorem 1, ifα < 4/3, the zero solution of (22) is asymptotically stable.Simulation results are displayed in Figs. 4–7. Figs.4–6 showthe zero solution of the system (22) is asymptotically stablewith α = 1.1α = 1.3 and α = 1.33 , respectively. Fig. 7shows the zero solution of the system (22) is not stable withα = 1.34.


Fig. 3. System (20) is unstable with α = 0.50




Example 3. The fractional-order hyperchaotic Chen systemcan be written as

RL0 D

αt x1 = a(x2 − x1) + x4

RL0 D

αt x2 = dx1 + cx2 − x1x3

RL0 D

αt x3 = x1x2 − bx3

RL0 D

αt x4 = x2x3 + rx4

(24)

where a, b, c,d and r are five parameters. When a = 35, b =3, c = 12, d = 7, r = 0.3 and α = 1.1 system (24) displaysa chaotic attractor, as shown in Fig. 8.


Fig. 7. System (22) is not stable with α = 1.34

System (24) can be rewritten as a controlled system:

RL0 D

αt x1 = a(x2 − x1) + x4

RL0 D

αt x2 = dx1 + cx2 − x1x3 + u1

RL0 D

αt x3 = x1x2 − bx3

RL0 D

αt x4 = x2x3 + rx4 + u2

When α = 1.1, a = 35, b = 3, c = 12, d = 7, r = 0.3, weselect the linear state feedback controller u1 = −22x2, u2 =−x4. Then, the two conditions of Theorem 1 are satisfied well.It concludes that the zero solution of the controlled system isasymptotically stable. The results of simulation are shown inFig. 9, while the feedback is activated at timet =10 s.


Fig. 8. Attractor of fractional order hyperchaotic Chen system withorder α = 1.1 (a = 35, b = 3, c = 12, d = 7, r = 0.3)

Fig. 9. Asymptotical stabilization of fractional order hyperchaoticChen system with order α = 1.1

V. CONCLUSIONS

In this paper, we have studied the local asymptotic stabilityof the zero solution of n-dimensional nonlinear fractionaldifferential systems with with Riemann-Liouville derivative.The results are obtained in terms of the Mittag-Leffler func-tion, Laplace transform and the Gronwall-Bellman lemma.Compare of the current results with the results in Ref.[14]shows that the stability condition of Riemann-Liouville frac-tional differential system is same as that for Caputo fractionaldifferential systems. Three numerical examples are given todemonstrate the effectiveness of the proposed approach.

VI. ACKNOWLEDGEMENT

This work was supported by the Natural Science Foun-dation of Hebei Province, China (No. A2015108010 andNo.A2015205161), the science research project of Hebeihigher education institutions, China (No.z2012021).

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Ruoxun Zhang received the B.S., M.S. and Ph.D.degrees in Physics and Information Engineering allfrom Hebei Normal University, Shijiazhuang, China,in 1990, 2008 and 2012 respectively. He is cur-rently an associate professor at Xingtai University.His current research interests include control andsynchronization of fractional-order chaotic systems,neural networks, adaptive control theory. E-mail:[email protected]

Shiping Yang received the B.S. degree in PhysicsDepartment from Beijing Normal University, Bei-jing, China, in 1982, and the M.S. degree in Instituteof Physics from Chinese Academy of Sciences, Bei-jing, China, in 1988. He is currently a professor anda doctoral adviser at Hebei Normal University. Hiscurrent research interests include fractional-ordernonlinear dynamic systems, chaos, neural networks,control theory. E-mail: [email protected],

Shiwen Feng received the B.S. degree in Com-puter Science from Hebei Normal University, Shi-jiazhuang, China, in 1992, and the M.S. degreein Information Science and Technology from EastChina Normal University, Shanghai, China, in 2009.He is currently an associate professor at XingtaiUniversity. His current research interests includeinformation processing, chaos, neural networks, con-trol theory. E-mail: [email protected]

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