Mittag–Leffler stability of fractional order nonlinear dynamic systems

Automatica 45 (2009) 1965–1969

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Mittag–Leffler stability of fractional order nonlinear dynamic systemsI Yan Li a , YangQuan Chen b,∗ , Igor Podlubny c a

Institute of Applied Math, School of Mathematics and System Sciences, Shandong University, Jinan 250100, PR China

b

Center for Self-Organizing and Intelligent Systems (CSOIS), Electrical and Computer Engineering Department, Utah State University, Logan, UT 84322-4160, USA

c

Department of Applied Informatics and Process Control, Faculty BERG, Technical University of Kosice, B. Nemcovej 3, 04200 Kosice, Slovak Republic

article

info

Article history: Received 14 April 2008 Received in revised form 18 March 2009 Accepted 4 April 2009 Available online 13 May 2009

abstract In this paper, we propose the definition of Mittag–Leffler stability and introduce the fractional Lyapunov direct method. Fractional comparison principle is introduced and the application of Riemann–Liouville fractional order systems is extended by using Caputo fractional order systems. Two illustrative examples are provided to illustrate the proposed stability notion. Published by Elsevier Ltd

Keywords: Fractional order dynamic system Nonautonomous system Fractional Lyapunov direct method Mittag–Leffler stability Fractional comparison principle

1. Introduction In nonlinear systems, Lyapunov’s direct method (also called the Lyapunov’s second method) provides a way to analyze the stability of a system without explicitly solving the differential equations. The method generalizes the idea, which shows the system is stable if there exist some Lyapunov function candidates for the system. The Lyapunov direct method is a sufficient condition to show the stability of nonlinear systems, which means the system may still be stable, even if one cannot find a Lyapunov function candidate to conclude the system stability property. As far as the motivation of this paper is concerned, we note that many systems exhibit the fractional phenomena, such as motions in complex media/environments, random walk of bacteria in fractal substance and the chemotaxi behavior and food seeking of microbes (Cohen, Golding, Ron, & Ben-Jacob, 2001), etc. These phenomena are always related to the complexity and heredity of systems due to the fractional properties of system components,

I This work was completed while Y. Li visited the Center for Self-Organizing and Intelligent Systems (CSOIS), Utah State University from August 2007 to August 2008. Part of this paper was presented at the Third IFAC Workshop on Fractional Derivative and Applications (FDA08), November 5-7, 2008, Ankara, Turkey. This paper was recommended for publication in revised form by Associate Editor Wei Kang under the direction of Editor André L. Tits. ∗ Corresponding author. Tel.: +1 435 797 0148; fax: +1 435 797 3054. E-mail addresses: [email protected], [email protected] (Y. Chen). URL: http://fractionalcalculus.googlepages.com/ (Y. Chen).

0005-1098/$ – see front matter. Published by Elsevier Ltd doi:10.1016/j.automatica.2009.04.003

such as the fractional viscoelastic material, the fractional circuit element and the fractal structure, etc (Bagley & Torvik, 1983a,b). In particular, the memristor (a contraction for memory resistor), which is said to be the missing circuit element (Chua, 1971), shows some hereditary properties. Allowing for the fact that the fractional calculus itself is a kind of convolution, the memristor is naturally likely to be linked to fractional calculus. Finally, it is possible that, in the future, there will be more fractional order dynamic systems in micro/nano scales. Recently, fractional calculus was introduced to the stability analysis of nonlinear systems, for example Momani and Hadid (2004), Zhang, Li, and Chen (2005), Chen (2006), Tarasov (2007), Sabatier (2008) and Li, Chen, Podlubny, and Cao (2008), where integer-order methods of stability analysis were extended to fractional order dynamic systems. However, as pointed out in Chen (2006), the decay of generalized energy of a dynamic system does not have to be exponential for the system to be stable. The energy decay actually can be of any rate, including power law decay. For extending the application of fractional calculus in nonlinear systems, we propose the Mittag–Leffler stability and the fractional Lyapunov direct method with a view to enrich the knowledge of both system theory and fractional calculus. Meanwhile, the fact that computation becomes faster and memory becomes cheaper makes the application of fractional calculus, in reality, possible and affordable (Chen, 2006). This work is motivated by the simple fact, as also indicated in Chen (2006), that the generalized energy of a system does not have to decay exponentially for the system to be stable in the sense of Lyapunov.

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Y. Li et al. / Automatica 45 (2009) 1965–1969

3. Fractional nonautonomous systems

Our contributions of this paper include:

• The study of the fractional Lyapunov direct method and the Mittag–Leffler stability of nonautonomous systems.

• The extension of the application of Riemann–Liouville fractional order systems by using Caputo fractional order systems.

α

• The fractional comparison principle and several other fractional inequalities extend the application of fractional calculus. 2. Fractional calculus 2.1. Caputo and Riemann–Liouville fractional derivatives Fractional calculus plays an important role in modern science (Chen & Moore, 2002; Podlubny, 1999a,b; Sabatier, Agrawal, & Tenreiro Machado, 2007; Tarasov, 2007; Xu & Tan, 2006). In this paper, we use both Riemann–Liouville and Caputo fractional order operators as our main tools, which are shown in Podlubny (1999b). The uniform formula of a fractional integral with α ∈ (0, 1) is defined as −α

a Dt

f (t ) =

0 (α)

f (τ )

t

Z

1

(t − τ )1−α

a

dτ ,

(1)

where f (t ) is an arbitrary integrable function, a Dt−α is the fractional integral of order α on [a, t ], and 0 (·) denotes the Gamma function. For an arbitrary real number p, the Riemann–Liouville and Caputo fractional derivatives are defined, respectively, as p a Dt f

(t ) =

d[p]+1 h

−([p]−p+1)

a Dt

dt [p]+1

−([p]−p+1)

p

and Ca Dt f (t ) = a Dt



f (t )

i

d[p]+1 dt

(2)



f (t ) , [p]+1

(3)

2.2. Mittag–Leffler type functions Similar to the exponential function frequently used in the solutions of integer-order systems, a function frequently used in the solutions of fractional order systems is the Mittag–Leffler function, defined as Eα ( z ) =

zk

k=0

0 (kα + 1)

t0 Dt

x(t ) = f (t , x)

(7)

with initial condition x(t0 ), where D denotes either Caputo or Riemann–Liouville fractional operator, α ∈ (0, 1), f : [t0 , ∞] × Ω → Rn is piecewise continuous in t and locally Lipschitz in x on [t0 , ∞]× Ω , and Ω ∈ Rn is a domain that contains the origin x = 0. The equilibrium point of (7) is defined as follows: Definition 1. The constant x0 is an equilibrium point of fractional dynamic system (7), if and only if f (t , x0 ) = t0 Dtα x0 . Without loss of generality, let the equilibrium point be x = 0. 3.1. Lipschitz condition and Caputo fractional nonautonomous systems The fact that f (t , x) is locally bounded and is locally Lipschitz in x implies the uniqueness and existence of the solution to the fractional order system (7) (Podlubny, 1999b). In the next part of this subsection, we study the relationship between the Lipschitz condition and the fractional nonautonomous system (7). Lemma 2. For the real-valued continuous f (t , x) in (7), we have kt0 Dt−α f (t , x(t ))k ≤ t0 Dt−α kf (t , x(t ))k, where α ≥ 0 and k · k denotes an arbitrary norm. Proof. It follows from (1) and the properties of norm that

Z t

1 f (τ , x(τ ))

kt0 Dt f (t , x(t ))k = dτ

1 −α 0 (α) t0 (t − τ ) Z t 1 kf (τ , x(τ ))k ≤ dτ = t0 Dt−α kf (t , x(t ))k.  0 (α) t0 (t − τ )1−α −α

where [p] stands for the integer part of p, D and C D denote the Riemann–Liouville and Caputo fractional derivatives, respectively. The geometric and physical interpretation of fractional-order integration and differentiation was suggested in Podlubny (2002), and in Podlubny and Heymans (2006) it was shown that there is a physical interpretation for the initial values expressed in terms of Riemann–Liouville derivatives. When the Caputo derivatives are used, the interpretation of initial values is the same as in the classical integer-order case.

∞ X

Consider the Caputo or Riemann–Liouville fractional nonautonomous system (Podlubny, 1999a; Tarasov, 2007)

,

(4)

where α > 0. The Mittag–Leffler function with two parameters has the following form:

(8)

Theorem 3. In (7), let t0 Dtα = Ct0 Dαt . If x = 0 is an equilibrium point of system (7), f is Lipschitz on x with Lipschitz constant l and is piecewise continuous with respect to t, then the solution of (7) satisfies kx(t )k ≤ kx(t0 )kEα (l(t − t0 )α ), where α ∈ (0, 1). Proof. By applying the fractional integral operator t0 Dt−α to both sides of (7) (Li & Deng, 2007), it follows from Lemma 2 and the Lipschitz condition that

|kx(t )k − kx(t0 )k| ≤ kx(t ) − x(t0 )k = kt0 Dt−α f (t , x(t ))k ≤ t0 Dt−α kf (t , x(t ))k ≤ lt0 Dt−α kx(t )k. There exists a nonnegative function M (t ) satisfying

kx(t )k − kx(t0 )k = l t0 Dt−α kx(t )k − M (t ).

(9)

By applying the Laplace transform (L{·}) to (9), it follows that

(6)

kx(t0 )ksα−1 − sα M (s) , (10) sα − l where kx(s)k = L{kx(t )k}. Applying the inverse Laplace transform to (10) gives kx(t )k = kx(t0 )kEα (l(t − t0 )α ) − M (t ) ∗ [t −1 Eα,0 (l(t − t0 )α )], where ∗ denotes the convolution operator dE (l(t −t )α ) and t −1 Eα,0 (l(t − t0 )α ) = α dt 0 ≥ 0.1 It then follows that kx(t )k ≤ kx(t0 )kEα (l(t − t0 )α ). 

where t and s are, respectively, the variables in the time domain and Laplace domain, R(s) denotes the real part of s, λ ∈ R and L{·} stands for the Laplace transform.

1 This inequality can be derived directly from the definition of Mittag–Leffler function (4).

Eα,β (z ) =

∞ X

zk

k=0

0 (kα + β)

,

(5)

where α > 0 and β > 0. For β = 1, we have Eα (z ) = Eα,1 (z ). Also, E1,1 (z ) = ez . Moreover, the Laplace transform of Mittag–Leffler function in two parameters is

L{t β−1 Eα,β (−λt α )} =

sα−β sα

+λ

,

1

(R(s) > |λ| α ),

kx(s)k =

Y. Li et al. / Automatica 45 (2009) 1965–1969

Samko, 2001) are nonnegative functions, it follows that V (t ) ≤ V (0)Eβ (−α3 α2−1 t β ). Substituting this equation into (12) yields

4. Mittag–Leffler stability Lyapunov stability provides an important tool for stability analysis in nonlinear systems. In fact, stability issues have been extensively covered by Lyapunov and there are several tests associated with this name. We primarily consider what is often called Lyapunov’s direct method, which involves finding a Lyapunov function candidate for a given nonlinear system. If such a function exists, the system is stable. Applying Lyapunov’s direct method, one can search for an appropriate function. Note that the Lyapunov direct method is a sufficient condition, which means if one cannot find a Lyapunov function candidate to conclude the system stability property, the system may still be stable and one cannot claim the system is not stable. In this paper, we extend the Lyapunov direct method by considering incorporating fractional order operators. That is, the nonlinear dynamic systems themselves could be fractional orders, and the evolution of the Lyapunov function could be a time-fractional order. Let us first define the stability in the sense of Mittag–Leffler. Definition 4 (Definition of the Mittag–Leffler Stability). The solution of (7) is said to be Mittag–Leffler stable if

kx(t )k ≤ {m[x(t0 )]Eα (−λ(t − t0 )α )}b ,

(11)

where t0 is the initial time, α ∈ (0, 1), λ > 0, b > 0, m(0) = 0, m(x) ≥ 0, and m(x) is locally lipschitz on x ∈ B ∈ Rn with Lipschitz constant m0 . 5. Fractional order extension of Lyapunov direct method By using the Lyapunov direct method, we can get the asymptotic stability of the corresponding systems. In this section, we extend the Lyapunov direct method to the case of fractional order systems, which leads to the Mittag–Leffler stability. Theorem 5. Let x = 0 be an equilibrium point for the system (7) and D ⊂ Rn be a domain containing the origin. Let V (t , x(t )) : [0, ∞)× D → R be a continuously differentiable function and locally Lipschitz with respect to x such that

α1 kxka ≤ V (t , x(t )) ≤ α2 kxkab ,

(12)

C β 0 Dt V

(13)

(t , x(t )) ≤ −α3 kxkab ,

where t ≥ 0, x ∈ D, β ∈ (0, 1), α1 , α2 , α3 , a and b are arbitrary positive constants. Then x = 0 is Mittag–Leffler stable. If the assumptions hold globally on Rn , then x = 0 is globally Mittag–Leffler stable. β

Proof. It follows from Eqs. (12) and (13) that C0 Dt V (t , x(t )) ≤ − αα3 V (t , x(t )). There exists a nonnegative function M (t ) satisfying 2

C β 0 Dt V

(t , x(t )) + M (t ) = −α3 α2−1 V (t , x(t )).

(14)

Taking the Laplace transform of (14) gives β

s V (s) − V (0)s

β−1

+ M (s) = −α3 α2 V (s), −1

(15)

where nonnegative constant V (0) = V (0, x(0)) and V (s) =

L{V (t , x(t ))}. It then follows that V (s) =

V (0)sβ−1 −M (s) . If x α sβ + α3 2

(0) = 0,

namely V (0) = 0, the solution to (7) is x = 0. If x(0) 6= 0, V (0) > 0. Because V (t , x) is locally Lipschitz with respect to x, it follows from the fractional uniqueness and existence theorem (Podlubny, 1999b) and the inverse Laplace transform that α the unique solution of (14) is V (t ) = V (0)Eβ (− α3 t β ) − M (t ) ∗

h

t

β−1

i

α Eβ,β (− α3 t β ) 2

2

. Since both t

1967

β−1

and

α Eβ,β (− α3 t β ) 2

(Miller &

h

α

V (0)

i 1a

E (− α3 t β ) α1 β 2

V (0)

> 0 for x(0) 6= 0. Let h i 1a α V (0,x(0)) V (0) ≥ 0, then we have kx(t )k ≤ mEβ (− α3 t β ) , m= α = α 1 1 2 where m = 0 holds if and only if x(0) = 0. Because V (t , x) is locally Lipschitz with respect to x and V (0, x(0)) = 0 if and only if V (0,x(0)) is also Lipschitz with respect x(0) = 0, it follows that m = α1 to x(0) and m(0) = 0, which imply the Mittag–Leffler stability of kx(t )k ≤

system (7).

, where

α1



Lemma 6. Let β ∈ (0, 1) and M (0) ≥ 0, then C β 0 Dt M

β

(t ) ≤ 0 Dt M (t ),

where D and C D are the Riemann–Liouville and the Caputo fractional operators, respectively. Proof. By using the composition of fractional operators (Li & Deng, β β−1 d 2007; Podlubny, 1999b), we have C0 Dt M (t ) = 0 Dt M (t ) = dt β

M (t ) −

β

M (t ).

0 Dt 0 Dt

β

M (0)t −β 0 (1−β)

. Because, β ∈ (0, 1) and M (0) ≥ 0, C0 Dt M (t ) ≤



Theorem 7. Assume that the assumptions in Theorem 5 are satβ β isfied except replacing C0 Dt by 0 Dt , then we have kx(t )k ≤

h

V (0)

i 1a

α

E (− α3 t β ) α1 β 2

.

Proof. It follows from Lemma 6 and V (t , x) β

C β 0 Dt V

β

≥ 0 that C0 Dt β (t , x(t )) ≤ 0 Dt

V (t , x(t )) ≤ 0 Dt V (t , x(t )), which implies V (t , x(t )) ≤ −α3 kxkab . Following the same proof in Theorem 5 yields kx(t )k ≤

h

V (0)

i 1a

α

E (− α3 t β ) α1 β 2

.



Theorem 8. For the fractional order system (7), where t0 Dtα = 0 Dαt , f (t , x) satisfies Lipschitz condition with Lipschitz constant l > 0. Suppose that there exists a Lyapunov candidate V (t , x) satisfying

α1 kxka ≤ V (t , x) ≤ α2 kxk, V˙ (t , x) ≤ −α3 kxk,

(16) (17)

where a, α1 , α2 , α3 are positive constants and V˙ (t , x) = have kx(t )k ≤

h

V (0,x(0))

α1



α

E1−α − α 3l t 1−α 2

i1/a

dV (t ,x) . dt

We

.

Proof. It follows from (16) and (17), Theorem 3 and Lemma 2 that −α −α C 1−α −1 ˙ V (t , x) = 0 D−α 0 Dt t V (t , x) ≤ −α30 Dt kxk ≤ −α3 l 0 Dt −α α−1 −1 −1 kf (t , x)k ≤ −α3 l k0 Dt f (t , x)k = −α3 l kxk, where [0 Dt x(t )]t =0 = 0. Therefore, the conclusion can be obtained by using Theorem 5.  6. Fractional Lyapunov direct method by using the class-K functions In this section, the class-K functions are applied to the analysis of fractional Lyapunov direct method. Definition 9. A continuous function α : [0, t ) → [0, ∞) is said to belong to class-K if it is strictly increasing and α(0) = 0 (Khalil, 2002). Lemma 10 (Fractional Comparison Principle). Let x(0) = y(0) and C β C β 0 Dt x(t ) ≥ 0 Dt y(t ), where β ∈ (0, 1). Then x(t ) ≥ y(t ).

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Y. Li et al. / Automatica 45 (2009) 1965–1969

β

β

Proof. It follows from C0 Dt x(t ) ≥ C0 Dt y(t ) that there exists a nonnegative function m(t ) satisfying C β 0 Dt x

β

(t ) = m(t ) + C0 Dt y(t ).

(18) β

β−1

Taking the Laplace transform of Eq. (18) yields s X (s)− s x(0) = M (s) + sβ Y (s) − sβ−1 y(0), where M (s) = L{m(t )}. It follows from x(0) = y(0) that X (s) = s−β M (s) + Y (s). Applying the inverse Laplace transform to the previous equation gives x(t ) = −β 0 Dt m(t ) + y(t ). It follows from m(t ) ≥ 0 and (1) that x(t ) ≥ y(t ).  Theorem 11. Let x = 0 be an equilibrium point for the nonautonomous fractional order system (7). Assume that there exists a Lyapunov function V (t , x(t )) and class-K functions αi (i = 1, 2, 3) satisfying

α1 (kxk) ≤ V (t , x) ≤ α2 (kxk),

(19)

C β 0 Dt V

(20)

(t , x(t )) ≤ −α3 (kxk),

where β ∈ (0, 1). Then the equilibrium point of system (7) is asymptotically stable. Proof. It follows from (19) and (20) that C β 0 Dt V

≤ −α3 (α2−1 (V )).

(21)

It then follows from Lemma 10 and V (t , x) ≥ 0 that V (t , x(t )) ≤ V (0, x(0)). Case 1: Suppose there exists a constant t1 ≥ 0 satisfying V (t1 , x(t1 )) = 0, it follows from (19) that x(t1 ) = 0. It then follows from x = 0 is the equilibrium point of system (7) that x(t ) = 0 for t ≥ t1 . Case 2: Assume that there exists a positive constant ε such that V (t , x) ≥ ε for t ≥ 0. Then it follows from V (t , x) ≤ V (0, x(0)) that

Example 13. Consider the following fractional order dynamic system C α 0 Dt x

(t ) = −

t

Z

g (τ , x) dτ + h(t , x),

where g (t , x) ≥ 0, x(0) > 0, α ∈ (0, 1) and h(t , x) equal 0 or Rt g (τ , x) dτ for x 6= 0 or x = 0, respectively. The equilibrium 0 point x = 0 is asymptotically stable.

Rt

Proof. It follows from − 0 g (τ , 0) dτ + h(t , 0) = 0 and Definition 1 that x = 0 is the equilibrium point of system (23). Let the Lyapunov candidate be V (x) = x2 , it follows from x(0) > 0 and x = 0 is the equilibrium point that dV (x)/dx = 2x ≥ 0, where the equality holds ifhand only if xi = 0. Moreover, it follows from g (t , x) ≥ 0 that d

t ≥ 0.

(22)

Substituting (22) into (21) gives −α3 (α2 (V (t , x))) ≤ −α3 (α2 −1

(ε)) = − α3 (α2−1 (ε)) . g (0)

α3 (α2−1 (ε)) V (0, x(0)) V (0,x(0))

−1

≤ −lV (t , x), where 0 < l = β

It then follows that C0 Dt V (t , x(t )) ≤ −α3 (α2−1 (V (t , x))) ≤ −lV (t , x). Following the same proof in Theorem 5 gives V (t , x) ≤ V (0, x(0))Eβ (−lt β ), which contradicts the assumption that V (t , x) ≥ ε . Based on the discussions in both Case 1 and Case 2, we have V (t , x) tends to zero as t → ∞. It follows from (19) that limt →∞ x(t ) = 0.  Theorem 12. If the assumptions in Theorem 11 are satisfied except β β replacing C0 Dt by 0 Dt , then we have limt →∞ x(t ) = 0. Proof. It follows from Lemma 6 and V (t , x) β

C β 0 Dt V

β

≥ 0 that C0 Dt β (t , x(t )) ≤ 0 Dt

V (t , x(t )) ≤ 0 Dt V (t , x(t )), which implies V (t , x(t )) ≤ −α3 (kxk). Following the same proof in Theorem 11 gives limt →∞ x(t ) = 0.  7. Two illustrative examples The following illustrative examples are provided to show the usefulness of the Mittag–Leffler stability notion.

Rt

g (τ , x) dτ /dt ≥ 0. Therefore, 0 h R i t dV (x) = 2xC0 D1t −α − 0 g (τ , x) dτ + h(t , x) ≤ 0, where equality dt holds if and only if x(t ) = 0. If there exists a positive constant  > 0, such that kx(t )k ≥  holds for all t ≥ 0. It follows from V (x) = x2 = kxk2 is bounded that there exists a positive constant α3 satisfying dVdt(x) ≤ −α3 kxk2 . It then follows from the Lyapunov  √ −1α t direct method (Khalil, 2002) that kx(t )k ≤ x(0)e 2 3 , which contradicts the assumption kx(t )k ≥  for all t ≥ 0. Therefore, the equilibrium point x = 0 is asymptotically stable.  Example 14. For the Caputo fractional order system C α 0 Dt x

(t ) = −x3 (t ),

(24)

where α ∈ (0, 1) and x(0) 6= 0 is the initial condition. The equilibrium point x = 0 is asymptotically stable. Proof. Let the Lyapunov candidate be V (x) = x4 , we have V˙ (x(t )) = 4x3 (t )˙x(t ), where x˙ denotes the derivative of x with respect to t. For an arbitrary positive constant  , we obtain t +

Z

x3 (τ )˙x(τ )dτ = t

0 < ε ≤ V (t , x) ≤ V (0, x(0)),

(23)

0

 1 4 x (t + ) − x4 (t ) . 4

(25)

Moreover, by applying Riemann–Liouville fractional order operator 0 D1t −α to the system (24), it follows from the Liebnitz’ Rule that

Z −1 d t (t − τ )α−1 x3 (τ )dτ 0 (α) dt 0   Z t −1 x3 (0) 1−α = + (t − τ )α−2 x3 (τ )dτ . (26) 0 (α) (t − τ )1−α τ =t 0 (α) 0

x˙ (t ) = −0 D1t −α x3 (t ) =

It then follows from x(0) 6= 0 and x = 0 is the equilibrium point that x(0)x(t ) > 0 for all t ∈ (0, +∞). It can be proved that x(0)x(t ) ≤ x2 (0) for all t ≥ 0, where equality holds if and only if t = 0. By multiplying x(0) to both sides of (26), it follows from −x4 (0)t α−1

x(0)x(t ) ≤ x2 (0) that x(0)˙x(t ) ≤ < 0. Applying the 0 (α) previous equation and x ( 0 ) x ( t ) > 0 for all t ∈ R t + 3   (0, +∞) to (25) yield t x (τ )˙x(τ )dτ = 14 x4 (t + ) − x4 (t ) < 0. Therefore, V (x) = x4 (t ) is a decreasing function. Suppose there exists a positive constant ξ satisfying x(0)x(t ) ≥ ξ for all t ≥ 0, we have C α 0 Dt V

= C0 Dtα−1 4x3 x˙ ≤ ≤

4ξ 3 x4 (0)

C α−1 x 0 Dt

(0)˙x

−4ξ 3 C α−1 α−1 −4ξ 3 4 3 D t = − 4 ξ = x (0) ≤ −α3 V , 0 (α) 0 t x4 (0)

where α3 =

4ξ 3 x4 (0)

> 0. It follows from Theorem 5 that limt →∞ V (x(t )) = limt →∞ x4 (t ) = 0 which contradicts the assumption

Y. Li et al. / Automatica 45 (2009) 1965–1969 1

comments on our paper. Y. Li is supported by the State Scholarship Fund of China (Grant No.LiuJinChu[2007]3020-2007102037). I. Podlubny was supported by grants APVV LPP-0283-06 and VEGA 1/4058/07. Y. Chen and I. Podlubny were funded in part by the National Academy of Sciences under the Collaboration in Basic Science and Engineering Program/Twinning Program supported by Contract No. INT-0002341 from the National Science Foundation.

α=0.1 α=0.5 α=0.9

0.9

1969

0.8 0.7 0.6 0.5 0.4

References

0.3 0.2 0.1

0

10

20

30

40

50

60

70

80

90

100

Fig. 1. The solution of Caputo fractional order system (24) for α = 0.1, 0.5 and 0.9, where x(0) = 1.

x(0)x(t ) > ξ . Therefore, the equilibrium point x = 0 is asymptotically stable. Finally, for α = 0.1, 0.5 and 0.9, the solution (x(t )) of system (24) is shown in Fig. 1, which is plotted by using Matlab/Simulink.  8. Conclusion and future work In this paper, we studied the stabilization of nonlinear fractional order dynamic systems. We discussed fractional nonautonomous systems and the application of the Lipschitz condition to fractional order systems. We proposed the definition of Mittag–Leffler stability and the fractional Lyapunov direct method, which enriched the knowledge of both the system theory and the fractional calculus. We introduced the fractional comparison principle. We partly extended the application of Riemann–Liouville fractional order systems by using fractional comparison principle and Caputo fractional order systems. Two illustrative examples were provided to demonstrate the applicability of the proposed approach. Our future works include the generalized Mittag–Leffler stability of fractional nonautonomous systems and the search for fractional Lyapunov functions. Acknowledgments We would like to thank Mr. Y. Cao and Dr. C. P. Li for discussions, and the reviewers and the Associate Editor for their useful

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