QI Yong-fang,PENG You-hua

(School of Mathematics,Pingxiang University,Pingxiang,337000,Jiangxi,China)

Abstract:One new theorem for Caputo fractional derivative and two new theorems for Caputo fractional order systems,when 1＜a＜2,are proposed in this paper.The results have proved to be useful in order to apply the fractional-order extension of Lyapunov direct method,to demonstrate the instability and the stability of many fractional order systems,which can be nonlinear and time varying.

Key words:Stability;Instability;Fractional system;Lyapunov function

§1.Introduction

Fractional differential systems have gained great importance due to their applications in physics,mechanic and engineering,etc.There has been a significant development in fractional differential systems,see[1-5].In recent years,stability analysis of fractional systems is one of the most essential and fundamental issues,and has become very popular in recent years due to its applications in the fields of science and engineering[6].The stability have to be proved using techniques developed for solving the actual problems,such as the heat transfer process[7],the effect of the frequency in induction machines[8].

Some authors have proposed Lyapunov functions to prove the stability of fractional systems with order lying in(0,1).

As early as 2010,Yan Li and YangQuan Chen introduced the class-κ functions to the fractional Lyapunov direct method and provide the fractional comparison principle,which enable us to demonstrate the stability of many fractional order systems[9].

Motivated by the article[9],Manuel A.Duarte and Norelys Aguila introduce a new lemma for Caputo fractional derivatives of special functions,and a theorem related to the fractional extension of Lyapunov direct method,for proving uniform stability in the Lyapunov sense[10].

In 2011,Norelys Aguila and Manuel A.Duarte presented a new property for Caputo fractional derivatives,the result has proved to be useful in order to apply the fractional-order extension of Lyapunov direct method[11].

Fernandez-Anaya and Nava-Antonio present a similar but more general result than the Lemma 1 of[11],which can enables us to build Lyapunov functions of greater order[12].They introduce a new way of determining the stability of a greater variety of systems when the order lying in(0,1).

It is well known that the Fractional system is commonly encountered in the actual problems.Therefore,stability analysis of Fractional systems has received much attention and various stability conditions have been obtained[8-9].On the other hand,recently some stability results about delayed fractional models have been derived with the tools of Gronwall inequality and Laplace transform[10-12].It should be noted that the fractional system with order lying in(0,1)is considered fully.However,there are few works on the stability of the fractional system with order lying in(1,2).

In the article[18],an extension of theorem to the case 1<α<2 is presented.The result shows that stability of many systems can be analyzed in a unified way by the location of the eigenvalues of matrix A in the complex plane.

Motivated by the above discussions,in this paper,we will consider the problem of stability for fractional system with order lying in(1,2).This paper presents a new theorem for Caputo fractional derivative and two new theorems for Caputo fractional order systems when 1<α<2,which allow finding a simple Lyapunov function for many fractional order systems,and the consequently instability or stability proof for them,using the fractional-order extension of the Lyapunov direct method.

The paper is organized as follows:Section 2 presents some basic concepts about fractional calculus and some useful lemmas in this work.Section 3 introduces one new theorem for Caputo fractional derivative and two new theorems for Caputo fractional order systems.Section 4 presents the conclusions of the work.

§2.Preliminaries

In this section,we introduce some basic concepts about fractional calculus and some useful lemmas,which are used throughout this paper.

definition 2.1[19]Let α＞0 and f be a real function defined on[a,b].The Riemann-Liouville fractional integral of order α is defined byaI0f ≡ f,and

definition 2.219]The Caputo derivative of fractional order α＞0 is defined bycaD0f≡f,and

where n is the smallest integer greater or equal to α.

definition 2.3[20]The Laplace transform formula for the Caputo fractional derivative is:

definition 2.4[9]The Mittag-Leffler function with two parameters is defined as

the Laplace transform of Mittag-Leffler function in two parameters is

where t≥ 0,s is the variable in Laplace domain,?(s)denotes the real part of s.

Lemma 2.1[21]If α <2,β is an arbitrary real number,η is such that<η

Lemma 2.2(ExistenceandUniquenessTheorem[21])Let f(t,x)be a real-valued continuous function,defined in the domain G,satisfying in G the Lipschitz condition with respect to x,i.e.,

such that|f(t,x)|≤M<∞for all(t,x)∈G.Let also

Then there exists in the region R(h,K)a unique and continuous solution x(t)of the problem(1)-(2).

where,

§3.Main Results

3.1 Instability Analysis of the System

In this section,a new theorem for Caputo fractional derivative is proposed,which is useful to analyze the instability of the fractional system with order lying in(1,2).

Theorem 3.1 Let x(t) ∈ R be a continuous,and x′′(t)exists,x′(t0)=0.Then,for any time instant,

Proof Proving that expression(3)is true,is equivalent to prove that

Using definition 2.2,it can be written that

So,expression(4)can be written as

In order to prove the above expression,it is equivalent to prove that

Given that x′′(t)exists,L’Hopital rule can be applied(because it results00).Then

As x′(t0)=0,so expression(5)is reduced to

Let us analyze the corresponding limit.

Example 1 Considering the following fractional boundary value problem,

where 1<α≤2,together with the boundary conditions x1(0)=x2(0)=0.

To analyze the stability of system(10),let us defines the Lyapunov function

Considering the boundary conditions x1(0)=x2(0)=0,we get immediately,the theorem 3.1 is applied,

As a result,the origin of the system is unstable.

3.2 Stability Analysis of the System

Consider the following Caputo fractional nonautonomous system:

In this section,we extend the Lyapunov direct method to the case of fractional-order systems when 1<α<2,which leads to stability.

Theorem 3.2.Let x=0 be an equilibrium point for the system(12).Let V(t,x(t))be a continuously differentiable function and locally Lipschitz with respect to x such that:

where t≥ 0,β ∈ (1,2),a1,a2,a3,a and b are arbitrary positive constants.then x=0,which is the equilibrium point of system(12),is stable.

Proof According to the equations(13)and(14)given above,it can be obtained that

There exists a nonnegative function M(t)satisfying

With the help of the Laplace transform,the equation(16)is transformed to the following equation

Since V(t,x(t))is locally Lipschitz with respect to x,it follows from definition 2.5 and take advantage of the inverse Laplace transform that the unique solution of(17)is

With the lemma 2.1 given above,the following inequality is obtained(c1＞0,c2＞0),

Combing with(13)(19)and(20),it is easy to get:

since β ∈ (1,2)and a＞ 0,it follows that

Theorem 3.3(Fractional Comparison Principle).Letc0Dβtx(t)≥c0Dβty(t),β∈(1,2)and x(0)=y(0),x′(0)=y′(0).Then x(t)≥ y(t).

Proof There exists a nonnegative function m(t)satisfying

Taking the Laplace transform of equation(11)yields

Since x(0)=y(0),x′(0)=y′(0),it then follows that

Applying the inverse Laplace transform to(12)gives,

Since m(t)is nonnegative function,the following inequality is obtained:

Example 2 Considering the following fractional boundary value problem

where α ∈ (1,2),if x0＞0,f(x,t)<0,and x=0 is the equilibrium point of system(23).Then x=0 is stable.

ProofIt follows from the Theorem 3.3 andc0Dαtx0=0＞f(x,t)=c0Dαtx(t),it is clearly that Fractional Comparison Principle can be applied because of the conditions x(0)=x0and x′(0)=0,then 0≤ x ≤ x0,therefore x=0 is stable.

§4.Conclusions

One new theorem for Caputo fractional derivative and two new theorems for Caputo fractional order systems have been proposed in this paper.The results presented are valid for 1