# Asymptotical Stability of Nonlinear Fractional Differential System with Caputo Derivative

Authors: Fengrong Zhang; Changpin Li; YangQuan Chen
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

This paper deals with the stability of nonlinear fractional differential systems equipped with the Caputo derivative. At first, a sufficient condition on asymptotical stability is established by using a Lyapunov-like function. Then, the fractional differential inequalities and comparison method are applied to the analysis of the stability of fractional differential systems. In addition, some other sufficient conditions on stability are also presented.


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## Body

## 1. Introduction

Fractional calculus is more than 300 years old, but it did not attract enough interest at the early stage of development. In the last three decades, fractional calculus has become popular among scientists in order to model various physical phenomena with anomalous decay, such as dielectric polarization, electrode-electrolyte polarization, electromagnetic waves, and viscoelastic systems [1]. Recent advances in fractional calculus have been reported in [2].Recently, stability of fractional differential systems has attracted increasing interest. In 1996, Matignon [3] firstly studied the stability of linear fractional differential systems with the Caputo derivative. Since then, many researchers have done further studies on the stability of linear fractional differential systems [4–11]. For the nonlinear fractional differential systems, the stability analysis is much more difficult and only a few are available.Some authors [12, 13] studied the following nonlinear fractional differential system:(1.1)CD0,tqx(t)=f(t,x(t)),
with initial values x(0)=x0(0),…,x(m-1)(0)=x0(m-1), where m-1<q≤m. They discussed the continuous dependence of solution on initial conditions and the corresponding structural stability by applying Gronwall's inequality. In [14] the authors dealt with the following fractional differential system: (1.2)D0,tqx(t)=f(t,x(t)),
where 0<q≤1, 𝔇0,tq denotes either the Caputo, or the Riemann-Liouville fractional derivative operator. They proposed fractional Lyapunov's second method and firstly extended the exponential stability of integer order differential systems to the Mittag-Leffler stability of fractional differential systems. Moreover, the pioneering work on the generalized Mittag-Leffler stability and the generalized fractional Lyapunov direct method was proposed in [15].In this paper, we further study the stability of nonlinear fractional differential systems with Caputo derivative by utilizing a Lyapunov-like function. Taking into account the relation between asymptotical stability and generalized Mittag-Leffler stability, we are able to weaken the conditions assumed for the Lyapunov-like function. In addition, based on the comparison principle of fractional differential equations [16, 17], we also study the stability of nonlinear fractional differential systems by utilizing the comparison method. Our contribution in this paper is that we have relaxed the condition of the Lyapunov-like function and that we have further studied the stability. The present paper is organized as follows. In Section 2, some definitions and lemmas are introduced. In Section 3, sufficient conditions on asymptotical stability and generalized Mittag-Leffler stability are given. The comparison method is applied to the analysis of the stability of fractional differential systems in Section 4. Conclusions are included in the last section.


## 2. Preliminaries and Notations

Let us denote byℝ+ the set of nonnegative real numbers, by ℝ the set of real numbers, and by ℤ+ the set of positive integer numbers. Let 0<q<1 and set Cq([t0,T],ℝ)={f∈C((t0,T],ℝ),(t-t0)qf(t)∈C([t0,T],ℝ)}, and Cq([t0,T]×Ω,ℝ)={f(t,x(t))∈C((t0,T]×Ω,ℝ),(t-t0)qf(t,x(t))∈C([t0,T]×Ω,ℝ)}, where C((t0,t],ℝ) denotes the space of continuous functions on the interval (t0,t].Let us first introduce several definitions, results, and citations needed here with respect to fractional calculus which will be used later. As to fractional integrability and differentiability, the reader may refer to [18].Definition 2.1.
The fractional integral with noninteger orderq≥0 of function x(t) is defined as follows:
(2.1)Dt0,t-qx(t)=1Γ(q)∫t0t(t-τ)q-1x(τ)dτ,
where Γ(·) is the Gamma function.Definition 2.2.
The Riemann-Liouville derivative with orderq of function x(t) is defined as follows:
(2.2)RLDt0,tqx(t)=1Γ(m-q)dmdtm∫t0t(t-τ)m-q-1x(τ)dτ,
where m-1≤q<m and m∈ℤ+.Definition 2.3.
The Caputo derivative with noninteger orderq of function x(t) is defined as follows:
(2.3)CDt0,tqx(t)=1Γ(m-q)∫t0t(t-τ)m-q-1x(m)(τ)dτ,
where m-1<q<m and m∈ℤ+.Definition 2.4.
The Mittag-Leffler function is defined by(2.4)Eα(z)=∑k=0∞zkΓ(kα+1),
where α>0, z∈ℝ. The two-parameter Mittag-Leffler function is defined by
(2.5)Eα,β(z)=∑k=0∞zkΓ(kα+β),
where α>0 and β∈ℝ, z∈ℝ.
ClearlyEα(z)=Eα,1(z). The following definitions are associated with the stability problem in the paper.Definition 2.5.
The constantxeq is an equilibrium of fractional differential system 𝔇t0,tqx(t)=f(t,x) if and only if f(t,xeq)=𝔇t0,tqx(t)|x(t)=xeq for all t>t0, where 𝔇t0,tq means either the Caputo or the Riemann-Liouville fractional derivative operator.Throughout the paper, we always assume thatxeq=0.Definition 2.6 (see [15]).
The zero solution of𝔇t0,tqx(t)=f(t,x(t)) with order q∈(0,1) is said to be stable if, for any initial value x0, there exists an ε>0 such that ∥x(t)∥≤ε for all t>t0. The zero solution is said to be asymptotically stable if, in addition to being stable, ∥x(t)∥→0 as t→+∞.Definition 2.7.
Let𝔹⊂ℝn be a domain containing the origin. The zero solution of 𝔇t0,tqx(t)=f(t,x(t)) is said to be Mittag-Leffler stable if
(2.6)‖x(t)‖≤{m(x0)Eq(-λ(t-t0)q)}b,
where t0 is the initial time and x0 is the corresponding initial value, q∈(0,1), λ≥0,b>0, m(0)=0, m(x)≥0, and m(x) is locally Lipschitz on x∈𝔹⊂ℝn with the Lipschitz constant ℒ0.Definition 2.8.
Let𝔹⊂ℝn be a domain containing the origin. The zero solution of 𝔇t0,tqx(t)=f(t,x(t)) is said to be generalized Mittag-Leffler stable if
(2.7)‖x(t)‖≤{m(x0)(t-t0)-γEq,1-γ(-λ(t-t0)q)}b,
where t0 is the initial time and x0 is the corresponding initial value, q∈(0,1), -q<γ≤1-q, λ≥0,b>0, m(0)=0,m(x)≥0, and m(x) is locally Lipschitz on x∈𝔹⊂ℝn with the Lipschitz constant ℒ0.Remark 2.9.
Mittag-Leffler stability and generalized Mittag-Leffler stability both belong to algebraical stability, which also imply asymptotical stability (see [15]).Definition 2.10.
A functionα(r) is said to belong to class-𝒦 if α:ℝ+→ℝ+ is continuous function such that α(0)=0 and it is strictly increasing.Definition 2.11 (see [19]).
The class-𝒦 functions α(r) and β(r) are said to be with local growth momentum at the same level if there exist r1>0, ki>0(i=1,2) such that k1α(r)≥β(r)≥k2α(r) for all r∈[0,r1]. The class-𝒦 functions α(r) and β(r) are said to be with global growth momentum at the same level if there exist ki>0(i=1,2) such that k1α(r)≥β(r)≥k2α(r) for all r∈ℝ+.It is useful to recall the following lemmas for our developments in the sequel.Lemma 2.12 (see [20]).
Letv,w∈C1-q([t0,T],ℝ) be locally Hölder continuous for an exponent 0<q<ν≤1, h∈C([t0,T]×ℝ,ℝ) and(i)
RLDt0,tqv(t)≤h(t,v(t)),(ii)
RLDt0,tqw(t)≥h(t,w(t)), t0<t≤T,with nonstrict inequalities (i) and (ii), where v0=Γ(q)v(t)(t-t0)1-q|t=t0 and w0=Γ(q)w(t)(t-t0)1-q|t=t0. Suppose further that h satisfies the standard Lipschitz condition
(2.8)h(t,x)-h(t,y)≤L(x-y),x≥y,L>0.
Then, v0≤w0 implies v(t)≤w(t), t0<t≤T.Remark 2.13.
In Lemma2.12, if we replace RLDt0,tq by CDt0,tq, but other conditions remain unchanged, then the same result holds.Lemma 2.14 (see [16]).
Letv,w∈C1-q([t0,T],ℝ), h∈C([t0,T]×ℝ,ℝ) and (i)
v(t)≤(v0/Γ(q))(t-t0)q-1+(1/Γ(q))∫t0t(t-s)q-1h(s,v(s))ds,(ii)
w(t)≥(w0/Γ(q))(t-t0)q-1+(1/Γ(q))∫t0t(t-s)q-1h(s,w(s))ds,where t0<t≤T, v0=Γ(q)v(t)(t-t0)1-q|t=t0, w0=Γ(q)w(t)(t-t0)1-q|t=t0, and 0<q<1. Assume that both inequalities are nonstrict and h(t,x) is nondecreasing in x for each t. Further, suppose that h satisfies the standard Lipschitz condition
(2.9)h(t,x)-h(t,y)≤L(x-y),x≥y,L>0.
Then, v0≤w0 implies v(t)≤w(t), t0<t≤T.Remark 2.15.
In Lemmas2.12 and 2.14, T can be +∞.


## 3. Stability of Nonlinear Fractional Differential Systems

Let us consider the following nonlinear fractional differential system [14, 15]: (3.1)CDt0,tqx(t)=f(t,x(t)),
with the initial condition x0=x(t0), where f:[t0,∞)×Ω→ℝn is piecewise continuous in t and Ω⊂ℝn is a domain that contains the equilibrium point xeq=0, 0<q<1. Here and throughout the paper, we always assume there exists a unique solution x(t)∈C1[t0,∞) to system (3.1) with the initial condition x(t0).Recently, Li et al. [14, 15] investigated the Mittag-Leffler stability and the generalized Mittag-Leffler stability (the asymptotic stability) of system (3.1) by using the fractional Lyapunov's second method, where the following theorem has been presented.Theorem 3.1.
Letxeq=0 be an equilibrium point of system (3.1) with t0=0, and let 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
(3.2)α1‖x‖a≤V(t,x(t))≤α2‖x‖ab,(3.3)CD0,tpV(t,x(t))≤-α3‖x‖ab,
where t≥0, x∈𝔻, p∈(0,1), and α1, α2, α3, a, and b are arbitrary positive constants. Then xeq=0 is Mittag-Leffler stable (locally asymptotically stable). If the assumptions hold globally on ℝn, then xeq=0 is globally Mittag-Leffler stable (globally asymptotically stable).In the following, we give a new proof for Theorem3.1.Proof of Theorem3.1.
From (3.2) and (3.3), we can get
(3.4)CD0,tpV(t,x(t))≤-α3α2V(t,x(t)).
Obviously, for the initial value V(0,x(0)), the linear fractional differential equation
(3.5)CD0,tpV(t,x(t))=-α3α2V(t,x(t))
has a unique solution V(t,x(t))=V(0,x(0))Ep((-α3/α2)tp).
Taking into account Remark2.13 and the relationship between (3.4) and (3.5), we obtain
(3.6)V(t,x(t))≤V(0,x(0))Ep(-α3α2tp),
where Ep((-α3/α2)tp) is a nonnegative function [21]. Substituting (3.6) in (3.2) yields
(3.7)‖x(t)‖≤[V(0,x(0))α1Ep(-α3α2tp)]1/a,
where Ep((-α3/α2)tp)→0(t→+∞) from the asymptotic expansion of Mittag-Leffler function [22]. Hence the proof is completed.According to the above results, we have the following theorem.Theorem 3.2.
Letxeq=0 be an equilibrium point of system (3.1), and let 𝔻⊂ℝn be a domain containing the origin. Assume that there exist a continuously differentiable function V(t,x(t)):[t0,∞)×𝔻→ℝ+ and class-𝒦 function α satisfying
(3.8)V(t,x(t))≥α(‖x‖),(3.9)CDt0,tpV(t,x(t))≤0,
where x∈𝔻, p∈(0,1). Then xeq=0 is locally stable. If the assumptions hold globally on ℝn, then xeq=0 is globally stable.Proof.
Proceeding the same way as that in the proof of Theorem3.1, it follows from (3.9) that V(t,x(t))≤V(t0,x(t0)). Again taking into account (3.8), one can get
(3.10)‖x(t)‖≤α-1(V(t0,x(t0))),
where t≥t0. Therefore, the equilibrium point xeq=0 is stable. So the proof is finished.In the above two theorems, the stronger requirements on functionV have been assumed to ensure the existence of CDt0,tpV(t,x(t)). This undoubtedly increases the difficulty in choosing the function V(t,x(t)). In fact, we can weaken the continuously differential function V(t,x(t)) as V(t,x(t))∈C1-p([t0,∞)×𝔻,ℝ+). Here we give the corresponding results.Theorem 3.3.
Letxeq=0 be an equilibrium point of system (3.1), and let 𝔻⊂ℝn be a domain containing the origin, V(t,x(t))∈C1-p([t0,∞)×𝔻,ℝ+). Assume there exists a class-𝒦 function α such that
(3.11)V(t,x(t))≥α(‖x‖),(3.12)RLDt0,tpV(t,x(t))≤0,
where t>t0≥0, x∈𝔻, and p∈(0,1). Then xeq=0 is locally asymptotically stable. If the assumptions hold globally on ℝn, then xeq=0 is globally asymptotically stable.Proof.
Note that the linear fractional differential equation(3.13)RLDt0,tpV(t,x(t))=0
has a unique solution V(t,x(t))=(V0/Γ(p))(t-t0)p-1 for initial value V0=Γ(p)V(t,x(t))(t-t0)1-p|t=t0.
Taking into account Lemma2.12 and the relationship between (3.12) and (3.13), we obtain
(3.14)V(t,x(t))≤V0Γ(p)(t-t0)p-1.
Substituting (3.14) into (3.11) gives
(3.15)‖x(t)‖≤α-1(V0Γ(p)(t-t0)p-1)⟶0(t⟶+∞),
from the definition of class-𝒦. This completes the proof.Corollary 3.4.
Letxeq=0 be an equilibrium point of system (3.1), let 𝔻⊂ℝn be a domain containing the origin, and let V(t,x(t))∈C1-p([t0,∞)×𝔻,ℝ+) be locally Lipschitz with respect to x. Assume V(t,0)=0,
(3.16)V(t,x(t))≥a‖x‖b,RLDt0,tpV(t,x(t))≤0,
where t>t0≥0, x∈𝔻, p∈(0,1), and a, b are arbitrary positive constants. Then xeq=0 is generalized Mittag-Leffler stable. If the assumptions hold globally on ℝn, then xeq=0 is globally generalized Mittag-Leffler stable.Proof.
In Theorem3.3, by replacing α(∥x∥) by a∥x∥b, we can get
(3.17)‖x(t)‖≤{V0a(t-t0)p-1Ep,p(0⋅(t-t0)p)}1/b,
so the conclusion holds.Theorem 3.5.
Letxeq=0 be an equilibrium point of system (3.1), let 𝔻⊂ℝn be a domain containing the origin, and let V(t,x(t))∈C1-p([t0,∞)×𝔻,ℝ+) be locally Lipschitz with respect to x. Assume (i)
there exist class-𝒦 functions αi(i=1,2,3) having global growth momentum at the same level and satisfying
(3.18)α1(‖x‖)≤V(t,x(t))≤α2(‖x‖),RLDt0,tpV(t,x(t))≤-α3(‖x‖),(ii)
there existsa>0 such that α1(r) and ra have global growth momentum at the same level,where t>t0≥0, x∈𝔻, and p∈(0,1). Then xeq=0 is locally generalized Mittag-Leffler stable. If the assumptions hold globally on ℝn, then xeq=0 is globally generalized Mittag-Leffler stable.Proof.
It follows from condition (i) that there existsk1>0 such that
(3.19)RLDt0,tpV(t,x(t))≤-α3(‖x‖)≤-k1α2(‖x‖)≤-k1V(t,x(t)).
On the other hand, the linear fractional differential equation
(3.20)RLDt0,tpV(t,x(t))=-k1V(t,x(t))
has a unique solution
(3.21)V(t,x(t))=V0Γ(p)(t-t0)p-1⋅Ep,p(-k1(t-t0)p),
for the initial value V0=Γ(p)V(t,x(t))(t-t0)1-p|t=t0.
Using (3.19), (3.20), and Lemma 2.12, we obtain
(3.22)α1(‖x‖)≤V(t,x(t))≤V0Γ(p)(t-t0)p-1⋅Ep,p(-k1(t-t0)p),
where Ep,p(-k1(t-t0)p) is a nonnegative function [23, 24].
In addition, using condition (ii), one gets(3.23)(k2‖x‖)a≤α1(‖x‖),
for all x∈𝔻, where k2>0.
Substituting (3.23) into (3.22), we finally obtain
(3.24)‖x(t)‖≤{V0k2aΓ(p)(t-t0)p-1Ep,p(-k1(t-t0)p)}1/a.
Hence, the zero solution of system (3.1) is locally generalized Mittag-Leffler stable. If the assumptions hold globally on ℝn, then xeq=0 is globally generalized Mittag-Leffler stable. The proof is completed.Remark 3.6.
The nonnegative functionEp,p(-k1(t-t0)p) tends to zero as t approaches infinity from the asymptotic expansion of two-parameter Mittag-Leffler function [22], so the zero solution of system (3.1) satisfying the conditions of Theorem 3.5 is also asymptotically stable.


## 4. The Comparison Results on the Stability

It is well known that the comparison method is an effective way in judging the stability of ordinary differential systems. In this section, we will discuss similar results on the stability of fractional differential systems by using the comparison method.In what follows, we consider system (3.1) with f(t,0)=0 and the scalar fractional differential equation (4.1)RLDt0,tqu(t)=g(t,u),u0=Γ(q)u(t)(t-t0)1-q|t=t0,
where the initial value u0∈ℝ+, u(t)∈C1-q([t0,∞),ℝ), g∈C([t0,∞)×ℝ,ℝ) is Lipschitz in u and g(t,0)=0, 0<q<1. Also, we assume there exists a unique solution u(t)(t≥t0) for system (4.1) with the initial value u0.Theorem 4.1.
For system (3.1), let xeq=0 be an equilibrium point of system (3.1), and let Ω⊂ℝn be a domain containing the origin. Assume that there exist a Lyapunov-like function V∈C1-q([t0,∞)×Ω,ℝ+) and a class-𝒦 function α such that V(t,0)=0, V(t,x)≥α(∥x∥), and V(t,x) satisfies the inequality
(4.2)RLDt0,tqV(t,x)≤g(t,V(t,x)),(t,x)∈[t0,∞)×Ω.
Suppose further that g(t,x) is nondecreasing in x for each t. (i)
If the zero solution of (4.1) is stable, then the zero solution of system (3.1) is stable;(ii)
if the zero solution of (4.1) is asymptotically stable, then the zero solution of system (3.1) is asymptotically stable, too.Proof.
Letx(t)=x(t,t0,x0) denote the solution of system (3.1) with initial value x0∈Ω. Along the solution curve x(t), V(t,x(t) can be written as V(t) and
(4.3)V(t)≤V0Γ(q)(t-t0)q-1+1Γ(q)∫t0t(t-s)q-1g(s,V(s))ds,
where V0=Γ(q)V(t)(t-t0)1-q|t=t0. Applying the fractional integral operator Dt0,t-q to both sides of (4.1) leads to
(4.4)u(t)=u0Γ(q)(t-t0)q-1+1Γ(q)∫t0t(t-s)q-1g(s,u(s))ds.
Now, taking u0=V0 and applying Lemma 2.14 to inequalities (4.3) and (4.4), one has V(t)≤u(t), t>t0.(i)
If the zero solution of (4.1) is stable, then for any initial value u0≥0, there exists ϵ>0 such that |u(t)|<ϵ for all t>t0. Therefore, taking into account V(t,x(t))≥α(∥x∥), one gets
(4.5)α(‖x(t)‖)≤V(t,x)≤u(t)<ϵ,
that is, ∥x(t)∥<α-1(ϵ), and the zero solution of system (3.1) is stable.(ii)
One can directly derive(4.6)α(‖x(t)‖)≤V(t,x)≤u(t)<ϵ
from the same argument in (i). Then, taking the limit to both sides of (4.6) and combining with the definition of class-𝒦 function, one can obtain limt→+∞∥x(t)∥=0.
The proof is thus finished.Remark 4.2.
In Theorem4.1 and system (4.1), if we replace order q by p∈(0,1), but other conditions remain unchanged, then the result in Theorem 4.1 still holds.Especially, if the class-𝒦 function α(∥x∥) in Theorem 4.1 and ∥x∥a have global growth momentum at the same level, then we can have similar comparison result on the generalized Mittag-Leffler stability as follows.Theorem 4.3.
For system (3.1), let xeq=0 be an equilibrium of system (3.1), and let Ω⊂ℝn be a domain containing the origin. Assume that there exists a Lyapunov-like function V∈C1-q([t0,∞)×Ω,ℝ+) such that V(t,0)=0, V(t,x)≥k∥x∥a, and V(t,x) is locally Lipschitz in x and satisfies the inequality
(4.7)RLDt0,tqV(t,x)≤g(t,V(t,x)),(t,x)∈[t0,∞)×Ω,
where k>0, a>0. Suppose further that g(t,x) is nondecreasing in x for each t. Then the zero solution of system (3.1) is also locally generalized Mittag-Leffler stable if the zero solution of (4.1) is locally generalized Mittag-Leffler stable. In addition, if the assumptions hold globally on ℝn, then the globally generalized Mittag-Leffler stability of zero solution of (4.1) implies the globally generalized Mittag-Leffler stability of zero solution of system (3.1).Proof.
First, from Definition2.8, if the zero solution of (4.1) is generalized Mittag-Leffler stable, then there exist λ≥0, b>0, -q<γ≤1-q such that
(4.8)|u(t)|≤{m(u0)(t-t0)-γEq,1-γ(-λ(t-t0)q)}b,
where m(0)=0, m(x)≥0 and m(x) is locally Lipschitz in x with Lipschitz constant ℒ0.
Takingu0=V0=Γ(q)V(t,x)(t-t0)1-q|t=t0 and noting that V(t,x)≤u(t) holds from Theorem 4.1, then taking into account (4.8) and V(t,x)≥k∥x∥a, we obtain
(4.9)k‖x(t)‖a≤V(t,x)≤{m(u0)(t-t0)-γEq,1-γ(-λ(t-t0)q)}b.
Furthermore,
(4.10)‖x(t)‖≤{m(Γ(q)V(t,x(t0))(t-t0)1-q|t=t0)k1/b⋅(t-t0)-γEq,1-γ(-λ(t-t0)q)}b/a.
Let M(x)=m(Γ(q)V(t,x)(t-t0)1-q|t=t0)/k1/b. Then it follows that
(4.11)‖x(t)‖≤{M(x(t0))(t-t0)-γEq,1-γ(-λ(t-t0)q)}b/a,
where M(0)=m(Γ(q)V(t,0)(t-t0)1-q|t=t0)/k1/b=0 due to V(t,0)=0. It is obvious that M(x) is a nonnegative function from m(x),V(t,x)≥0 and k>0. In addition, M(x) is locally Lipschitz in x since m(x) and V(t,x) are locally Lipschitz in x. So, the zero solution of system (3.1) is generalized Mittag-Leffler stable. The proof is completed.


## 5. Conclusion

In this paper, we have studied the stability of the zero solution of nonlinear fractional differential systems with the Caputo derivative and the commensurate order0<q<1 by using a Lyapunov-like function. Compared to [15], we weaken the continuously differential function V(t,x) as V(t,x)∈C1-p([t0,∞)×𝔻,ℝ+). Sufficient conditions on generalized Mittag-Leffler stability and asymptotical stability are derived. Meanwhile, comparison method is applied to the analysis of the stability of fractional differential systems by fractional differential inequalities.



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