# Tempered Mittag–Leffler Stability of Tempered Fractional Dynamical Systems

Authors: Jingwei Deng; Weiyuan Ma; Kaiying Deng; Yingxing Li
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Due to finite lifespan of the particles or boundedness of the physical space, tempered fractional calculus seems to be a more reasonable physical choice. Stability is a central issue for the tempered fractional system. This paper focuses on the tempered Mittag–Leffler stability for tempered fractional systems, being much different from the ones for pure fractional case. Some new lemmas for tempered fractional Caputo or Riemann–Liouville derivatives are established. Besides, tempered fractional comparison principle and extended Lyapunov direct method are used to construct stability for tempered fractional system. Finally, two examples are presented to illustrate the effectiveness of theoretical results.


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## Body

## 1. Introduction

Fractional derivatives were first proposed by Leibnitz soon after the more familiar classic integer order derivatives. In recent decades, the study of fractional differential systems has attracted wide attention. Compared with the classical calculus, fractional calculus can better characterize memory and hereditary properties of processes and materials. They are now used to model the dynamical evolution in the fields of physics, chemistry, biology, and so on. Fractional calculus can be most easily understood in terms of probability. The relationships among random walks, Brownian motion, and diffusion processes were given in [1]. It is more reasonable to replace classic derivatives by fractional analogues in the diffusion equation [2].Fractional calculus involves the operation of convolution with a power law function. Multiplying by an exponential factor results in tempered fractional derivatives and integrals [3], this exponential tempering has many merits both in mathematical and practical. A truncated Lévy flight was investigated to capture the natural cutoff in real physical systems [4]. Without a sharp cutoff, the tempered Lévy flight was studied as a smoother alternative [5]. Cartea and del-Castillo-Negrete [6] explored the tempered fractional diffusion equation by the tempered Lévy flight. In finance, the tempered stable process models describe price fluctuations with semiheavy tails [7–10]. Tempered fractional time derivatives can be also found in geophysics [11–13], Brownian motion [14], and so on.As in classical calculus, stability analysis is still one of the most important tasks in fractional differential system [15–20]. It is a basic feature in fractional physical and biological systems, such as Duffing oscillator [21], neural networks [22–24], and predator-prey models [25]. At present, Lyapunov method has been applied to analyze Mittag–Leffler stability of different fractional systems [26–30]. Li et al. [26, 27] obtained a series of conclusions on the Mittag–Leffler stable for nonlinear fractional equations. In [28], Mittag–Leffler stability of multiple equilibrium points of fractional recurrent neural networks was considered. In [29], a convex and positive definite function was used to analyze Mittag–Leffler stable for fractional system. In [30], the authors presented the Lyapunov stability analysis for fractional nonlinear systems with impulses.As far as we know, no paper has discussed stability analysis for tempered fractional system. Motivated by this, we think it is very necessary and meaningful to study Mittag–Leffler stability of tempered fractional dynamical systems both in theoretical research and practical application. Because tempered fractional operators combine with nonlocal, weak singularity, and exponential factors [31–33], it has many differences to fractional case in stability analysis. In this paper, tempered Mittag–Leffler stability is first proposed. It is a more appropriate concept for tempered fractional system. Tempered comparison principle and some inequalities are given for tempered fractional calculus or systems. Then, sufficient conditions for tempered Mittag–Leffler stability are provided and verified by the Lyapunov method. Finally, the theoretical results are applied to some examples.This paper is organized as follows. In Section2, some necessary definitions and lemmas are prepared. Section 3 mainly discusses the sufficient criterions ensuring Mittag–Leffler stability of the tempered fractional systems. In Section 4, two examples are presented to show the effectiveness of theoretical results. We conclude the paper with some discussions in Section 5.


## 2. Preliminaries

Tempered fractional calculus plays an important role in different fields [34, 35]. In practical application, many different tempered fractional derivatives are proposed, such as Caputo, Riemmann–Liouville, and Riesz. Some definitions and lemmas are stated below, which will be used later.Definition 1 (see [13]).
The tempered fractional integral of orderα>0 and tempered parameter λ≥0 is defined as(1)Iatα,λxt=1Γα∫ate−λt−st−sα−1xsds,where Γ presents the Euler gamma function.Definition 2 (see [3]).
The tempered fractional Caputo derivative of tempered parameterλ≥0 is defined as(2)DaCtα,λxt≔e−λtDaCtαeλtut=e−λtΓn−α∫at1t−sα−n+1dneλsxsdsnds,where n−1≤α<n,n∈ℕ, and DaCtα is the Caputo fractional derivative.Definition 3 (see [9]).
The tempered fractional Riemann–Liouville derivative of tempered parameterλ≥0 is defined as(3)Datα,λut≔e−λtDatαeλtut=e−λtΓn−αdndtn∫ateλsust−sα−n+1ds,where n−1≤α<n,n∈ℕ, and Datα is the Riemann–Liouville fractional derivative operator.
In order to study the stability of tempered fractional systems, several lemmas are needed.Lemma 1 (see [36]).
Let0<α<1,λ≥0 and t≥a, then(4)DaCtα,λxt=Datα,λxt−e−λtt−a−αΓ1−αeλtxtt=a.Lemma 2 (see [36]).
Let0<α<1,λ≥0, then(i)
Iatα,λDaCtα,λxt=xt−e−λt−axa,DaCtα,λIatα,λxt=xt(ii)
Iatα,λDatα,λxt=xt−e−λtt−aα−1/ΓαDatα,λeλtutt=a and Datα,λIatα,λxt=xtLemma 3 (see [36]).
The Laplace transform of tempered fractional integral and Caputo derivative (2) are given as(i)
ℒI0tα,λxt=λ+s−αXs(ii)
ℒD0Ctα,λxt=s+λαXs−∑k=0n−1s+λα−k−1dk/dtkeλtxtt=0, where Xs=ℒxt denotes the Laplace transform of xt


## 3. Main Results

In this section, tempered fractional comparison principles, some inequalities, and tempered Mittag–Leffler stability are derived.


### 3.1. Tempered Fractional Comparison Principles

In this section, we establish tempered fractional comparison principles.Lemma 4.
Assume thatD0Ctα,λxt≥D0Ctα,λyt, x0=y0, α∈0,1 and λ≥0, then xt≥yt.Proof.
Following fromD0Ctα,λxt≥D0Ctα,λyt, there exists a function mt≥0 such that(5)D0Ctα,λxt=mt+D0Ctα,λyt.
By Lemma3, equation (5) yields(6)s+λαXs−s+λα−1x0=Ms+s+λαYs−s+λα−1y0.
According tox0=y0, we have(7)s+λαXs=Ms+s+λαYs,Thus,(8)Xs=Ys+s+λ−αMs.
Taking the inverse Laplace transform on (8), solution of system (5) can be written as(9)xt=yt+I0tα,λmt=yt+1Γα∫0te−λt−τt−τα−1mτdτ.
According tomt≥0, therefore we obtain xt≥yt.Lemma 5.
Assume thatD0tα,λxt≥D0tα,λyt, x0=y0, and α∈0,1, then xt≥yt.Proof.
From Lemma1 and D0tα,λxt≥D0tα,λyt, we derive(10)D0Ctα,λxt+e−λtt−αΓ1−αx0≥D0Ctα,λyt+e−λtt−αΓ1−αy0.
That isD0Ctα,λxt≥D0Ctα,λyt. From Lemma 4, we obtain xt≥yt.


### 3.2. Some Inequalities

In this section, we construct some inequalities for tempered fractional derivatives or systems.From Lemma1, we could easily obtain the following lemma.Lemma 6.
The relationship betweenD0Ctα,λxt and D0tα,λxt is as follows:(11)D0Ctα,λxt≤D0tα,λxt,ifx0≥0,D0Ctα,λxt≥D0tα,λxt,ifx0≤0,where α∈0,1.Lemma 7.
Ifxt∈C10,+∞,ℝ is a continuously differentiable function, the following inequality holds:(12)D0Ctα,λxt+≤sgnxtD0Ctα,λxt,0<α<1,λ≥0,where xt+=limτ⟶t+xt.Proof.
We takeyt=eλtxt into Theorem 2 in [22](13)D0Ctαyt+≤sgnytD0Ctαyt,0<α<1,for the Caputo fractional derivative. That is,(14)D0Ctαeλtxt+≤sgnxtD0Ctαeλtxt.
Multiplying both sides of equation (14) by e−λt, it gives(15)e−λtD0Ctαeλtxt+≤sgnxte−λtD0Ctαeλtxt.
Using Definition2, we obtain (12).
Consider the following tempered fractional system(16)Dtα,λt0xt=ft,xt,0<t<T,subjects to the proper initial conditions, where α∈0,1, λ≥0, D denotes either DaCtα,λ or Datα,λ, f:t0,+∞×D⟶ℝn is piecewise continuous in t and locally Lipschitz in x, and D⊂ℝn is a domain contain the origin.Theorem 1.
For the real-valued continuous functionft,x in (16), we have(17)I0tα,λft,x≤I0tα,λft,x,where α>0,λ≥0, and ⋅ denotes an arbitrary norm.Proof.
It follows from (1) that(18)I0tα,λft,xt=1Γα∫0te−λt−st−sα−1fs,xsds≤1Γα∫0te−λt−st−sα−1fs,xsds=I0tα,λft,xt.Theorem 2.
Ifx=0 is an equilibrium point of system (16) with Dtα,λt0=Dtα,λ0C and ft,x is Lipschitz on x with Lipschitz constant l and piecewise continuous with respect to t; then, we have(19)xt≤x0e−λtEαltα.Proof.
By applying the tempered fractional integral operatorI0tα,λ to system (16), it follows from Lemma 2 and Lipschitz condition in ft,x that(20)xt−e−λtx0≤xt−e−λtx0=I0tα,λD0Ctα,λxt=I0tα,λft,xt≤I0tα,λft,xt≤lI0tα,λxt.
There exists a functionMt≥0 such that(21)xt−e−λtx0=lI0tα,λxt−Mt.
Combining with Lemma3 and Laplace transform to (21), we obtain(22)xs=s+λα−1x0−s+λαMss+λα−l,where xs=ℒxt. Taking the inverse Laplace transform to (22) gives(23)xt=x0e−λtEαltα−Mt∗e−λtt−1Eα,0ltα,where ∗ denotes the convolution operator. Obviously, e−λtt−1Eα,0ltα≥0, then inequality xt≤x0e−λtEαltα is obtained.


### 3.3. Tempered Mittag–Leffler Stability

In this section, some sufficient conditions are established for the tempered Mittag–Leffler stability of system (16).Definition 4.
If and only ifft,x-=Dtα,λt0x-, then x¯∈ℝn is an equilibrium point of tempered fractional system (16).Definition 5.
Assumex¯=0 is an equilibrium point of system (16), the solution of (16) is said to be tempered Mittag–Leffler stable if(24)xt≤mxt0e−λtEα−lt−t0αb,where α>0,λ≥0,l>0,b>0, and m0=0,mx≥0 satisfies locally Lipschitz condition.Remark 1.
Tempered Mittag–Leffler stability is a generalization of Mittag–Leffler stability. Whenλ=0, tempered Mittag–Leffler stability can be reduced to Mittag–Leffler stability [26, 27].Remark 2.
Both Mittag–Leffler stability and tempered Mittag–Leffler stability imply asymptotic stability, that is,xt⟶0 as t⟶∞.Theorem 3.
Assumex=0 be an equilibrium point for (16) and domain D⊂ℝn contains the origin. Let Vt,xt:0,+∞×D⟶ℝ be a continuously differentiable function and locally Lipschitz with respect to x, such that(25)α1xta≤Vt,xt≤α2xtab,(26)D0Ctβ,λVt,xt≤−α3xtab,where t≥0,x∈D,β∈0,1,andα1,α2,α3,a, and b are given positive constants, then x=0 is tempered Mittag–Leffler stable. If the assumptions hold globally on ℝn, then x=0 is globally tempered Mittag–Leffler stable.Proof.
It follows from equations (25) and (26) that(27)D0Ctβ,λVt,xt≤−α3α2Vt,xt.
There exists a functionMt≥0 such that(28)D0Ctβ,λVt,xt+Mt=−α3α2Vt,xt.
Taking the Laplace transform to (28) gives(29)s+λβVs−s+λβ−1V0+Ms=−α3α2Vs,where nonnegative constant V0=V0,x0 and Vs=ℒVt,xt. We rewrite this in the form(30)Vs=V0s+λβ−1−Mss+λβ+α3/α2.
Applying the inverse Laplace transform to (30), unique solution of (28) is(31)Vt=V0e−λtEβ−α3α2tβ−Mt∗e−λttβ−1Eβ,β−α3α2tβ.
Becausetβ−1 and Eβ,β−α3/α2tβ are nonnegative functions, we obtain(32)Vt≤V0e−λtEβ−α3α2tβ.
Substituting (32) into (25) satisfies(33)xt≤V0α1e−λtEβ−α3α2tβ1/a,and x0=0 if and only if V0/α1e−λt=0.
BecauseVt,x is local Lipschitz condition with respect to x and x0=0 if and only if V0,x0=0, then V0,x0/α1e−λt satisfies local Lipschitz condition about x0. So, system (16) is tempered Mittag–Leffler stable.Theorem 4.
Assume all conditions in Theorem3 are satisfied except replacing D0Ctβ,λ by D0tβ,λ, then we have(34)xt≤V0α1e−λtEβ−α3α2tβ1/a.Proof.
It follows from Lemma6 and Vt,x≥0 that(35)D0Ctβ,λVt,x≤D0tβ,λVt,x≤−α3xtab.
A similar proof method in Theorem3 shows result (35).Theorem 5.
For the tempered fractional system (10), where Dtα,λ0=Dtα,λ0, ft,x is Lipschitz on x with constant l>0, and ft,0=0, if there exists a Lyapunov candidate Vt,x yielding(36)α1xta≤Vt,xt≤α2xt,(37)V˙t,xt≤−α3xt,where α3>λα2,α1,α2,α3,a are given positive constants and V˙t,xt=dVt,x/dt, then(38)xt≤V0α1e−λtE1−α−α3−λα2α2t1−α1/a.Proof.
From (1) and (2), we find that tempered fractional derivative on Vt,x can be represented in the form(39)D0Ct1−α,λVt,x=e−λtΓα∫0t1t−s1−αdeλsVs,xsdsds=e−λtΓα∫0t1t−s1−αλeλsVs,xs+eλsV˙s,xsds=λe−λtΓα∫0teλsVs,xst−s1−αds+e−λtΓα∫0teλsV˙s,xst−s1−αds=λI0tα,λVt,xt+I0tα,λV˙t,xt.
Using (36) and (37) and Lipschitz condition on ft,x, we obtain(40)D0Ct1−α,λVt,x≤λα2I0tα,λxt−α3I0tα,λxt=−α3−λα2I0tα,λxt≤−α3−λα2l−1I0tα,λft,xt.
We can use Lemmas7 and 2 to write(41)D0Ct1−α,λVt,x≤−α3−λα2l−1I0tα,λft,xt=−α3−λα2l−1xt,where D0tα−1eλtutt=0=0. By the same proof in Theorem 3, conclusion (38) holds.


## 3.1. Tempered Fractional Comparison Principles

In this section, we establish tempered fractional comparison principles.Lemma 4.
Assume thatD0Ctα,λxt≥D0Ctα,λyt, x0=y0, α∈0,1 and λ≥0, then xt≥yt.Proof.
Following fromD0Ctα,λxt≥D0Ctα,λyt, there exists a function mt≥0 such that(5)D0Ctα,λxt=mt+D0Ctα,λyt.
By Lemma3, equation (5) yields(6)s+λαXs−s+λα−1x0=Ms+s+λαYs−s+λα−1y0.
According tox0=y0, we have(7)s+λαXs=Ms+s+λαYs,Thus,(8)Xs=Ys+s+λ−αMs.
Taking the inverse Laplace transform on (8), solution of system (5) can be written as(9)xt=yt+I0tα,λmt=yt+1Γα∫0te−λt−τt−τα−1mτdτ.
According tomt≥0, therefore we obtain xt≥yt.Lemma 5.
Assume thatD0tα,λxt≥D0tα,λyt, x0=y0, and α∈0,1, then xt≥yt.Proof.
From Lemma1 and D0tα,λxt≥D0tα,λyt, we derive(10)D0Ctα,λxt+e−λtt−αΓ1−αx0≥D0Ctα,λyt+e−λtt−αΓ1−αy0.
That isD0Ctα,λxt≥D0Ctα,λyt. From Lemma 4, we obtain xt≥yt.


## 3.2. Some Inequalities

In this section, we construct some inequalities for tempered fractional derivatives or systems.From Lemma1, we could easily obtain the following lemma.Lemma 6.
The relationship betweenD0Ctα,λxt and D0tα,λxt is as follows:(11)D0Ctα,λxt≤D0tα,λxt,ifx0≥0,D0Ctα,λxt≥D0tα,λxt,ifx0≤0,where α∈0,1.Lemma 7.
Ifxt∈C10,+∞,ℝ is a continuously differentiable function, the following inequality holds:(12)D0Ctα,λxt+≤sgnxtD0Ctα,λxt,0<α<1,λ≥0,where xt+=limτ⟶t+xt.Proof.
We takeyt=eλtxt into Theorem 2 in [22](13)D0Ctαyt+≤sgnytD0Ctαyt,0<α<1,for the Caputo fractional derivative. That is,(14)D0Ctαeλtxt+≤sgnxtD0Ctαeλtxt.
Multiplying both sides of equation (14) by e−λt, it gives(15)e−λtD0Ctαeλtxt+≤sgnxte−λtD0Ctαeλtxt.
Using Definition2, we obtain (12).
Consider the following tempered fractional system(16)Dtα,λt0xt=ft,xt,0<t<T,subjects to the proper initial conditions, where α∈0,1, λ≥0, D denotes either DaCtα,λ or Datα,λ, f:t0,+∞×D⟶ℝn is piecewise continuous in t and locally Lipschitz in x, and D⊂ℝn is a domain contain the origin.Theorem 1.
For the real-valued continuous functionft,x in (16), we have(17)I0tα,λft,x≤I0tα,λft,x,where α>0,λ≥0, and ⋅ denotes an arbitrary norm.Proof.
It follows from (1) that(18)I0tα,λft,xt=1Γα∫0te−λt−st−sα−1fs,xsds≤1Γα∫0te−λt−st−sα−1fs,xsds=I0tα,λft,xt.Theorem 2.
Ifx=0 is an equilibrium point of system (16) with Dtα,λt0=Dtα,λ0C and ft,x is Lipschitz on x with Lipschitz constant l and piecewise continuous with respect to t; then, we have(19)xt≤x0e−λtEαltα.Proof.
By applying the tempered fractional integral operatorI0tα,λ to system (16), it follows from Lemma 2 and Lipschitz condition in ft,x that(20)xt−e−λtx0≤xt−e−λtx0=I0tα,λD0Ctα,λxt=I0tα,λft,xt≤I0tα,λft,xt≤lI0tα,λxt.
There exists a functionMt≥0 such that(21)xt−e−λtx0=lI0tα,λxt−Mt.
Combining with Lemma3 and Laplace transform to (21), we obtain(22)xs=s+λα−1x0−s+λαMss+λα−l,where xs=ℒxt. Taking the inverse Laplace transform to (22) gives(23)xt=x0e−λtEαltα−Mt∗e−λtt−1Eα,0ltα,where ∗ denotes the convolution operator. Obviously, e−λtt−1Eα,0ltα≥0, then inequality xt≤x0e−λtEαltα is obtained.


## 3.3. Tempered Mittag–Leffler Stability

In this section, some sufficient conditions are established for the tempered Mittag–Leffler stability of system (16).Definition 4.
If and only ifft,x-=Dtα,λt0x-, then x¯∈ℝn is an equilibrium point of tempered fractional system (16).Definition 5.
Assumex¯=0 is an equilibrium point of system (16), the solution of (16) is said to be tempered Mittag–Leffler stable if(24)xt≤mxt0e−λtEα−lt−t0αb,where α>0,λ≥0,l>0,b>0, and m0=0,mx≥0 satisfies locally Lipschitz condition.Remark 1.
Tempered Mittag–Leffler stability is a generalization of Mittag–Leffler stability. Whenλ=0, tempered Mittag–Leffler stability can be reduced to Mittag–Leffler stability [26, 27].Remark 2.
Both Mittag–Leffler stability and tempered Mittag–Leffler stability imply asymptotic stability, that is,xt⟶0 as t⟶∞.Theorem 3.
Assumex=0 be an equilibrium point for (16) and domain D⊂ℝn contains the origin. Let Vt,xt:0,+∞×D⟶ℝ be a continuously differentiable function and locally Lipschitz with respect to x, such that(25)α1xta≤Vt,xt≤α2xtab,(26)D0Ctβ,λVt,xt≤−α3xtab,where t≥0,x∈D,β∈0,1,andα1,α2,α3,a, and b are given positive constants, then x=0 is tempered Mittag–Leffler stable. If the assumptions hold globally on ℝn, then x=0 is globally tempered Mittag–Leffler stable.Proof.
It follows from equations (25) and (26) that(27)D0Ctβ,λVt,xt≤−α3α2Vt,xt.
There exists a functionMt≥0 such that(28)D0Ctβ,λVt,xt+Mt=−α3α2Vt,xt.
Taking the Laplace transform to (28) gives(29)s+λβVs−s+λβ−1V0+Ms=−α3α2Vs,where nonnegative constant V0=V0,x0 and Vs=ℒVt,xt. We rewrite this in the form(30)Vs=V0s+λβ−1−Mss+λβ+α3/α2.
Applying the inverse Laplace transform to (30), unique solution of (28) is(31)Vt=V0e−λtEβ−α3α2tβ−Mt∗e−λttβ−1Eβ,β−α3α2tβ.
Becausetβ−1 and Eβ,β−α3/α2tβ are nonnegative functions, we obtain(32)Vt≤V0e−λtEβ−α3α2tβ.
Substituting (32) into (25) satisfies(33)xt≤V0α1e−λtEβ−α3α2tβ1/a,and x0=0 if and only if V0/α1e−λt=0.
BecauseVt,x is local Lipschitz condition with respect to x and x0=0 if and only if V0,x0=0, then V0,x0/α1e−λt satisfies local Lipschitz condition about x0. So, system (16) is tempered Mittag–Leffler stable.Theorem 4.
Assume all conditions in Theorem3 are satisfied except replacing D0Ctβ,λ by D0tβ,λ, then we have(34)xt≤V0α1e−λtEβ−α3α2tβ1/a.Proof.
It follows from Lemma6 and Vt,x≥0 that(35)D0Ctβ,λVt,x≤D0tβ,λVt,x≤−α3xtab.
A similar proof method in Theorem3 shows result (35).Theorem 5.
For the tempered fractional system (10), where Dtα,λ0=Dtα,λ0, ft,x is Lipschitz on x with constant l>0, and ft,0=0, if there exists a Lyapunov candidate Vt,x yielding(36)α1xta≤Vt,xt≤α2xt,(37)V˙t,xt≤−α3xt,where α3>λα2,α1,α2,α3,a are given positive constants and V˙t,xt=dVt,x/dt, then(38)xt≤V0α1e−λtE1−α−α3−λα2α2t1−α1/a.Proof.
From (1) and (2), we find that tempered fractional derivative on Vt,x can be represented in the form(39)D0Ct1−α,λVt,x=e−λtΓα∫0t1t−s1−αdeλsVs,xsdsds=e−λtΓα∫0t1t−s1−αλeλsVs,xs+eλsV˙s,xsds=λe−λtΓα∫0teλsVs,xst−s1−αds+e−λtΓα∫0teλsV˙s,xst−s1−αds=λI0tα,λVt,xt+I0tα,λV˙t,xt.
Using (36) and (37) and Lipschitz condition on ft,x, we obtain(40)D0Ct1−α,λVt,x≤λα2I0tα,λxt−α3I0tα,λxt=−α3−λα2I0tα,λxt≤−α3−λα2l−1I0tα,λft,xt.
We can use Lemmas7 and 2 to write(41)D0Ct1−α,λVt,x≤−α3−λα2l−1I0tα,λft,xt=−α3−λα2l−1xt,where D0tα−1eλtutt=0=0. By the same proof in Theorem 3, conclusion (38) holds.


## 4. Applications

In this section, we will give three examples to demonstrate theoretical analysis. The Adams–Bashforth–Moulton method [37] is employed for solving tempered fractional differential equations in the simulations.Example 1.
Consider the tempered fractional Riemann–Liouville system:(42)D0tα,λxt=−xt,0<α<1,λ≥0,where x0>0. The Lyapunov function candidate is chosen as Vt,x=x. From Lemma 1, we obtain(43)D0Ctα,λV=D0Ctα,λx≤D0tα,λx=−V.
By Theorem3, we have(44)xt≤x0e−λtEα−tα.
Then,x=0 is tempered Mittag–Leffler stable. To verify the result, we choose parameters as α=0.95andx0=4 and the tempered parameters as λ=2,4,6,8, respectively. The time evolution of the system states (42) is shown in Figure 1. It is presented that the solution of system (42) converges to the equilibrium point x=0. The larger the tempered parameter λ is, the faster the convergence speed becomes.Figure 1
Time evolution of system statesxt with different tempered parameters λ in Example 1.Example 2.
Consider the tempered fractional Caputo Hopfield neural networks:(45)D0Ctα,λxit=−aixit+∑j=1nbijfjxjt+Ii,where 0<α<1,λ≥0,i=1,2,…,n and n is number of units. xit is the ith state, fj is the jth activation function, bij is constant connection weight of the jth neuron on the ith neuron, ai>0 denotes the resting rate when the ith neuron disconnected, and Ii is the external inputs. Under the conditions(46)fjx≤ljx,lj>0,j=1,2,…,n,(47)ci=ai−∑j=1nbjili>0,i=1,2,…,n,system (45) is globally tempered Mittag–Leffler stable.
Letxt=x1,x2,…,xnT be any solution of system (45). We choose Lyapunov function as follows:(48)Vt,xt=xt=∑i=1nxit.
By inequalities (46) and (47) and Lemma 7, tempered fractional Caputo derivative on Vt,xt can be written as(49)D0Ctα,λVt,xt=∑i=1nD0Ctα,λxit≤∑i=1nsgnxitD0Ctα,λxit≤−ai∑i=1nxit+∑i=1n∑j=1nljbijxjt=∑i=1n−ai+∑j=1nlibjixit≤−cxt,where c=minc1.c2,…,cn. From (49) and Theorem 3, system (45) is globally tempered Mittag–Leffler stable.
To illustrate the effectiveness of Example2, in system (45), we let x=x1,x2,x3T,α=0.98, x10=5,x20=−3,x30=3,fjxj=tanhxj, and cj=6 for j=1,2,3 and(50)A=aij3×3=2−1.201.81.711.15−4.7501.1.
It is obvious that condition (47) is satisfied. Let tempered parameters λ=0,2,4,6, respectively. As shown in Figure 2, the equilibrium point x=0 is tempered Mittag–Leffler stable and the solution of system (45) converges to x=0.Figure 2
Time evolution of system statesxt with different tempered parameters λ in Example 2.
(a)(b)(c)Example 3.
Consider the following tempered fractional system:(51)D0Ctα,λx1=−2x1+sinx31+t2x1,D0Ctα,λx2=−2x2+cosx1x2,D0Ctα,λx3=x3,where 0<α<1,λ≥0 and xt=x1t,x2t,x3t∈ℝ3.
Let the Lyapunov functionVt,x=x1+x2. By Lemma 7, we obtain(52)D0Ctα,λVt,x≤sgnx1tD0Ctα,λx1t+sgnx2tD0Ctα,λx2t=sgnx1t−2x1t+sinx3t1+t2x1t+sgnx2t−2x2t+cosx1tx2t≤−2x1t+sinx3t1+t2x1t−2x2t+cosx1tx2t≤−x1t+x2t.
Then, the conditions of Theorem3 are satisfied. Hence, x=0 is globally tempered Mittag–Leffler stable with respect to x1,x2. Take α=0.9,x10=10,x20=−5,andx30=5. The numerical simulation is shown as Figure 3 with different tempered parameters λ=0,0.5,1,1.5. It is obvious x1t and x2t converge to 0. When tempered parameter of the system increase, the part solution of system converges faster.Figure 3
Time evolution of system statesxt with different tempered parameters λ in Example 3.
(a)(b)(c)


## 5. Conclusions

In this paper, we present some stability results for the tempered fractional systems. Based on the Laplace transform, we obtain the comparison principle for the tempered fractional systems. Some theorems about tempered Mittag–Leffler stability are derived, which enrich the knowledge of the system theory and the tempered fractional calculus and are helpful in characterizing the tempered fractional system models. Furthermore, we will study stability of tempered fractional systems with time-varying delays in future work.



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