# Studying the Stability of theψ-Hilfer Fractional Differential System

Authors: Jinping Yang; Zhiqiang Li
Journal: Complexity
 (2022)

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## Abstract

This paper devotes to the study on the stability and decay of solution to fractional differential system involving theψ-Hilfer fractional derivative of order α∈0,1 and type β∈0,1. We first derive the solution of linear system by using the generalized Laplace transform, which can be represented by the form of Mittag-Leffler function. Then, in terms of the asymptotic expansion of the Mittag-Leffler function, stability properties of linear system are analyzed in more detail. Finally, we construct a linearization theorem and determine the stability near the equilibrium for the autonomous nonlinear differential system with the ψ-Hilfer derivative.


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## Body

## 1. Introduction

In the last five decades, fractional differential equations (FDEs) have attracted great attention from researchers since FDEs may acquire more accurate results than the integer-order differential equations, especially for the description of materials and processes charactering memorial and hereditary properties [1–4]. Some authors studied works on existence, uniqueness, and data dependence of solution to FDEs [5–11], while another scholars focused on stability results of FDEs (including linear and nonlinear cases) with different types of fractional derivatives [12–18].The first motivation on the stability problem of FDEs came from Matignon [19], where he discussed stability of linear FDEs in the Caputo sense with derivative order α∈0,1. After that, the stability analysis for the case of Riemann-Liouville derivative was presented by Qian et al. [20] based on the eigenvalue of system matrix. Recently, Li and Li investigated the stability of linear FDEs involving the Hadamard and Caputo-Hadamard derivatives [16], as well as the ψ-Riemann-Liouville and ψ-Caputo derivatives [17].The stability mentioned above for linear FDEs is a Lyapunov stability, however, we mainly consider the structural stability for nonlinear FDEs. The most key question is that a linearization theorem has to be established. In 2013, Li and Ma [15] presented such result and investigated the local stability of nonlinear FDEs with Caputo derivative of order α∈0,1 by applying the linearization theorem. Very recently, Li and Sarwar constructed new linearization approximations for nonlinear FDEs with the Riemann-Liouville and Hadamard derivatives [21]. Meanwhile, Li and Li established similar conclusions in the cases of ψ-Riemann-Liouville and ψ-Caputo derivatives and discussed the local stability of nonlinear system [17]. In this work, we will study the stability and decay of solution to fractional differential system with more general fractional derivative (i.e., ψ-Hilfer fractional derivative). The motivation in the paper is to investigate fractional differential system in a generic way so that it is possible to unify and recover most of the previous results concerning fractional differential system with different fractional derivatives due to the appropriate choice of the kernel ψ in the ψ-Hilfer fractional derivative. Before the discussion, we first introduce several basic concepts and lemmas used in the coming section.Definition 1 (see [17]).
Leta,b−∞≤a<b≤∞ denote a finite or infinite interval and α>0. Suppose that ψt is a strictly monotone increasing function, having continuous derivative ψ′t and satisfying ψt⟶+∞ as t⟶+∞. For a given function ft, we define the ψ-fractional integral of order αas follows,(1)ΨDa,t−αft=1Γα∫atψt−ψτα−1fτψ′τdτ,t>a.Definition 2 (see [17]).
Leta,b−∞≤a<b≤∞ denote a finite or infinite interval and n−1<α<n∈ℕ. Suppose that ψt∈C1a,b is a strictly monotone increasing function satisfying ψ′t≠0 and ψt⟶+∞ as t⟶+∞. For a given function ft, the ψ-Riemann–Liouville fractional derivative of order α is defined as(2)ΨDa,tαft=1Γn−αδψn∫atψt−ψτn−α−1fτψ′τdτ,t>a,where δψn=1/ψ′td/dtn, δψn=δψδψn−1 and δψ0=I.Definition 3 (see [17]).
Leta,b−∞≤a<b≤∞ denote a finite or infinite interval and n−1<α<n∈ℕ. Suppose that ψt∈Cna,b is a strictly monotone increasing function satisfying ψ′t≠0 and ψt⟶+∞ as t⟶+∞. For a given function ft, the ψ-Caputo fractional derivative of order α is denoted by(3)CψDa,tαft=1Γn−α∫atψt−ψτn−α−1δψnfτψ′τdτ,t>a,where δψnft=1/ψ′td/dtnft.In the following, we introduce theψ-Hilfer fractional derivative which interpolates between the ψ-Riemann–Liouville derivative and the ψ-Caputo one.Definition 4 (see [22]).
Leta,b−∞≤a<b≤∞ denote a finite or infinite interval. Let n−1<α<n∈ℕ and 0≤β≤1. Suppose that ψt∈Cna,b is a strictly monotone increasing function satisfying ψ′t≠0 and ψt⟶+∞ as t⟶+∞. For a given function ft, the ψ-Hilfer fractional derivative of order α and type β is defined by(4)HψDa,tα,βft=ψDa,t−βn−αδψnψDa,t−1−βn−αft,t>a.Remark 1.
It is worth mentioning that theψ-Hilfer fractional derivative is just the ψ-Riemann–Liouville derivative if β=0 and is transformed into the ψ-Caputo derivative if β=1.Definition 5 (see [17]).
The generalized Laplace transform of a given functionft, defined on a,+∞a∈ℝ, is denoted as(5)Fs=ℒψft=∫a∞e−sψt−ψaftψ′tdt,s∈ℂ.According to Definition 5, we readily obtained that(6)ℒψHψDa,tα,βft=sαℒlψft−∑k=0n−1sn−k−1−βn−αδψkψDa,t−1−βn−αfa,where n−1<α<n∈ℕ and 0≤β≤1.In order to denote the solution of theψ-Hilfer linear fractional differential system, we need the Mittag-Leffler function which is defined as follows.Definition 6 (see [3]).
Letμ>0 and ν∈ℂ. The two-parameter Mittag-Leffler function is given by(7)Eμ,νz=∑k=0∞zkΓμk+ν,z∈ℂ.The following two properties regarding the Mittag-Leffler function will be used frequently in the next section.Lemma 1 (see [17]).
The generalized Laplace transform of the Mittag-Leffler function can be represented as(8)ℒψψt−ψaμk+ν−1Eμ,νk±λψt−ψaμ=k!sμ−νsμ∓λk+1,where Res>λ1/μ and Eμ,νky=dk/dykEμ,νy.Lemma 2 (see [3]).
Let0<μ<2, ν∈ℂ, and θ∈ℝ satisfy the relation πμ/2<θ<minπ,πμ. Then we have the following asymptotic expansions, for any integer p≥1,(9)Eμ,νz=1μz1−ν/μexpz1/μ−∑k=1pz−kΓν−μk+Oz−1−p,z⟶∞,
ifargz≤θ, and(10)Eμ,νz=−∑k=1pz−kΓν−μk+Oz−1−p,z⟶∞,
ifθ≤argz≤π.Finally, let us introduce the definitions of the equilibrium and stability.Definition 7.
The originxeq=0 is said to an equilibrium of the ψ-Hilfer fractional differential system(11)HψDa,tα,βxt=ft,x,0<α<1,0≤β≤1,if ft,0≡0 for any t≥a when β=1 and for any t>a when 0≤β<1.Remark 2.
For the case0≤β<1, we remark that the starting point a is not reachable since the solution xt⟶∞ as t⟶a+, see also [12]. This is different from the case β=1.In this paper we will assume the equilibriumxeq=0.Definition 8.
Let0<α<1 and 0≤β≤1. Then the zero solution of system (11) is called:(1)
stable if(i)
Whenβ=1, then for any initial value ψDa,t−1−β1−αxt|t=a=xa, i.e., xt|t=a=xa, there exists an ϵ>0 such that xt<ϵ for all t≥a;(ii)
When0≤β<1, then for any initial value ψDa,t−1−β1−αxt|t=a=xa, there exists an ϵ>0 and a˜>a such that xt<ϵ for all t≥a˜.(2)
asymptotically stable iflimt⟶+∞xt=0.The rest of the paper is outlined as follows. In Section2, we study the stability and decay of solution to linear fractional differential system with ψ-Hilfer derivative in the sense of Lyapunov stability. The local stability of nonlinear differential system involving ψ-Hilfer fractional derivative is discussed by using the linearization theorem in Section 3. Conclusion remarks is given in the last section. The letter C appeared in the paper will represent a generic positive constant, which may take different values at different places.


## 2. Stability of Linear System

In this section, we analyze the stability to the following linear system with theψ-Hilfer fractional derivative,(12)HψDa,tα,βxt=Axt,0<α<1,0≤β≤1,t>a,A∈ℝn×n,with xt=x1t,x2t,…,xntT and the initial value ψDa,t−1−β1−αxt|t=a=xa=xa1,xa2,…,xanT.Next, we will discuss the stability and decay of solution to the linear system (12) under different dimensions in detail, especially for the case of low dimension, i.e., n=1,2,3.


### 2.1. 1-D Case

Ifn=1, system (12) can be written as(13)HψDa,tα,βxt=λxt,0<α<1,0≤β≤1,t>a,λ∈ℝ,where the initial value is ψDa,t−1−β1−αxt|t=a=xa.According to the formulas (6) and (8), the solution of system (13) may be represented as(14)xt=xaψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα.From the solution (14), the following stability results hold. Throughout this paper, we denote(15)h=2,β=0,1,0<β≤1.Theorem 1.
Let0<α<1 and 0≤β≤1.(1)
Ifλ<0, then system (13) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
Ifλ=0, then:(i)
Whenβ=1, then system (13) is stable but not asymptotically stable.(ii)
When0≤β<1, then system (13) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α.(3)
Ifλ>0, then system (13) is unstable.Proof.
(1)
Ifλ<0, using equality (10) yields(16)Eα,β−11−α+1λψt−ψaα=−∑k=1pλψt−ψaα−kΓβ−11−α+1−αk+Oλψt−ψaα−1−p,t⟶∞.Whenβ=0, one has(17)Eα,β−11−α+1λψt−ψaα=−1λ2Γ−αψt−ψa−2α+Oλψt−ψaα−3,t⟶∞.If0<β≤1, it follows that(18)Eα,β−11−α+1λψt−ψaα=−1λΓβ1−αψt−ψa−α+Oλψa−ψaα−2,t⟶∞.Therefore,(19)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα⟶0,t⟶∞,which displays that system (13) is asymptotically stable, and its solution decays to zero astψ−ψaβ−11−α−hα.(2)
Ifλ=0, the solution (14) is reduced to(20)xt=ψt−ψaβ−11−αΓβ−11−α+1xa.Obviously, system (13) is stable but not asymptotically stable when β=1 and is asymptotically stable when 0≤β<1, where the decay rate of the solution is Oψt−ψaβ−11−α.(3)
Whenλ>0, by means of (9), one deduces that(21)Eα,β−11−α+1λψt−ψaα=1αλψt−ψaα1−β1−α/αexpλ1/αψt−ψa−∑k=1pλψt−ψaα−kΓ1−αk+Oλψt−ψaα−1−p,t⟶∞.
As a result,(22)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα⟶∞,t⟶∞.Remark 3.
Ifβ=1 and β=0, then the ψ-Hilfer derivative recovers to the ψ-Caputo and ψ-Riemann-Liouville derivatives respectively and the results of Theorem 1 are just consistent with those given by Theorem 1 in [17]. In particularly, we get the stability results for the cases of Caputo and Riemann-Liouville derivatives when taking ψt=t and a=0, which have been established in [19, 20].


### 2.2. 2-D Case

For the casen=2, system (12) is given by(23)HψDa,tα,βxt=Axt,0<α<1,0≤β≤1,t>a,A∈ℝ2×2,where the initial values are ψDa,t−1−β1−αxit|t=a=xai,i=1,2.As what has been done in [16, 17], here we will directly consider(24)A=λσ,A=λ1λ.Case 1.
A=λσ. Recall from (6) and (8) that the solution of system (23) is expressed as(25)x1t=xa1ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα,x2t=xa2ψt−ψaβ−11−αEα,β−11−α+1σψt−ψaα.Theorem 2.
Let0<α<1 and 0≤β≤1. Suppose that λ,σ∈ℝ.(1)
If bothλ<0 and σ<0, then system (23) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
Ifλ=0,σ<0, or σ=0,λ<0, or λ=σ=0, then:(i)
Whenβ=1, then system (23) is stable but not asymptotically stable.(ii)
When0≤β<1, then system (23) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α.(3)
If at least one ofλ and σ is greater than zero, then system (23) is unstable.Whenλ,σ∈ℂ, it is easy to know that they are conjugate in this case.Theorem 3.
Let0<α<1 and 0≤β≤1. Assume that λ=σ¯∈ℂ.(1)
Ifargλ>πα/2, then system (23) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
Ifargλ=πα/2, then system (23) is stable but not asymptotically stable.(3)
Ifargλ<πα/2, then system (23) is unstable.Proof.
(1)
Ifargλ>πα/2, it follows from (10) that(26)Eα,β−11−α+1λψt−ψaα=−∑k=1pλψt−ψaα−kΓβ−11−α+1−αk+Oλψt−ψaα−1−p,t⟶∞.We further conclude that(27)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα=−1λ2Γ−αψt−ψaβ−11−α−2α+Oλψt−ψaα−3,t⟶∞,ifβ=0, and(28)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα=−1λΓβ1−αψt−ψaβ−11−α−α+Oλψt−ψaα−2,t⟶∞,if0<β≤1. Consequently,(29)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα⟶0,t⟶∞,which implies that system (23) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α−hα.(2)
Ifargλ=πα/2, taking λ=rexp±iαπ/2r>0 and exploiting (9) lead to(30)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα=1αλψt−ψaα1−β1−α/αψt−ψaβ−11−αexpλ1/αψt−ψa−∑k=1pλ−kψt−ψaβ−11−α−αkΓβ−11−α+1−αk+Oλ−1−pψt−ψaβ−11−α−α−αp=1αr1−β1−α/αexp±i1−β1−α2π+r1/αψt−ψa−∑k=1pr−kψt−ψaβ−11−α−αkΓβ−11−α+1−αkexp±iαkπ2+Or−1−pψt−ψaβ−11−α−α−αp,t⟶∞.Thus,(31)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα⟶1αr1−β1−α/α,t⟶∞.This shows that system (23) is stable but not asymptotically stable.(3)
Ifargλ<πα/2, letting λ=rexpiθr>0,θ<πα/2 and using (9) once more, we have(32)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα=1αr1−β1−α/αexpr1/αψt−ψacosθα×expiθ1−β1−αα+r1/αψt−ψasinθα−∑k=1pr−kψt−ψaβ−11−α−αkΓβ−11−α+1−αkexp−iθk+Or−1−pψt−ψaβ−11−α−α−αp,t⟶∞.
Owing toθ<πα/2, we find expr1/αψt−ψacosθ/α⟶∞ as t⟶∞. As a consequence,(33)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα⟶∞,t⟶∞.
This indicates that system (23) is unstable. The proof is now complete.Case 2.
A=λ1λ. By means of (6) and (8), the solution of system (23) is denoted by(34)x1t=xa1ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα+xa21αψt−ψaα+β−11−αEα,α+β−11−αλψt−ψaα−β−11−αEα,α+β−11−α+1λψt−ψaα,x2t=xa2ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα.
In the situation,λ must be a real number and the following holds.Theorem 4.
Let0<α<1 and 0≤β≤1. Suppose λ∈ℝ.(1)
Ifλ<0, then system (23) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
Ifλ=0, then:(I)
Whenβ=1, then system (23) is unstable.(II)
When0≤β<1, one has:(i)
When0<α<1−β/2−β, then system (23) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α+α.(ii)
Whenα=1−β/2−β, then system (23) is stable but not asymptotically stable.(iii)
When1−β/2−β<α<1, then system (23) is unstable.(3)
Ifλ>0, then system (23) is unstable.Proof.
(1)
Whenλ<0, according to equality (10), we arrive at(35)1αψt−ψaα+β−11−αEα,α+β−11−αλψt−ψaα−β−11−αEα,α+β−11−α+1λψt−ψaα=1λ2Γβ−11−α+1−αψt−ψaβ−11−α−α+Oλ−3ψt−ψaβ−11−α−2α,t⟶∞,if0<β≤1, and(36)1αψt−ψaα+β−11−αEα,α+β−11−αλψt−ψaα−β−11−αEα,α+β−11−α+1λψt−ψaα=2λ3Γβ−11−α+1−2αψt−ψaβ−11−α−2α+Oλ−4ψt−ψaβ−11−α−3α,t⟶∞,ifβ=0. Consequently,(37)|1αψt−ψaα+β−11−αEα,α+β−11−αλψt−ψaα−β−11−αEα,α+β−11−α+1λψt−ψaα|⟶0,t⟶∞.Combining (19) and (37), we know that system (23) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α−hα.(2)
Whenλ=0, the solution (34) is reduced to(38)x1t=xa1ψt−ψaβ−11−αΓβ−11−α+1+xa2ψt−ψaα+β−11−αΓα+β−11−α+1,x2t=xa2ψt−ψaβ−11−αΓβ−11−α+1..Ifβ=1, by the first equality in (38), we immediately see that system (23) is unstable. If 0≤β<1, by (38), it is easy to check that: when0<α<1−β/2−β, system (23) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α+α; when α=1−β/2−β, system (23) is stable but not asymptotically stable; when 1−β/2−β<α<1, system (23) is unstable.(3)
Whenλ>0, we find that system (23) is unstable by using the solution (34). The proof is completed.


### 2.3. 3-D Case

Ifn=3, system (12) may write as(39)HψDa,tα,βxt=Axt,0<α<1,0≤β≤1,t>a,A∈ℝ3×3,with the initial valuesψDa,t−1−β1−αxit|t=a=xai,i=1,2,3. Same as [16, 17], we can directly investigate the matrix A in the following three cases:(40)A=λ1λ2λ3,A=λ11λ1λ2,A=λ11λ11λ1.Case 1:A=λ1λ2λ3. Similar to (14) or (25), the solution of system (39) is(41)x1t=xa1ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα,x2t=xa2ψt−ψaβ−11−αEα,β−11−α+1λ2ψt−ψaα,x3t=xa3ψt−ψaβ−11−αEα,β−11−α+1λ3ψt−ψaα.We first discuss the case for allλi∈ℝ,i=1,2,3.Theorem 5.
Let0<α<1 and 0≤β≤1. Assume that λi∈ℝ,i=1,2,3.(1)
If allλi<0,i=1,2,3, then system (39) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
If at least one ofλii=1,2,3 equals to zero and others are less than zero, then:(i)
Whenβ=1, then system (39) is stable but not asymptotically stable.(ii)
When0≤β<1, then system (39) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α.(3)
If at least one ofλii=1,2,3 is greater than zero, then system (39) is unstable.If there exists someλi which is a complex number, without loss of generality, we assume that λ1∈ℝ and λ2=λ¯3∈ℂ.Theorem 6.
Let0<α<1 and 0≤β≤1. Assume that λ1∈ℝ and λ2=λ¯3∈ℂ.(1)
Ifλ1<0 and argλ2>πα/2, then system (39) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
Ifλ1=0, argλ2>πα/2 and 0≤β<1, then system (39) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α.(3)
Ifλ1=0, argλ2>πα/2 and β=1, or λ1≤0 and argλ2=πα/2, then system (39) is stable but not asymptotically stable.(4)
Ifλ1>0 and λ2,λ3∈ℂ, or λ1∈ℝ and argλ2<πα/2, then system (39) is unstable.Case 2:A=λ11λ1λ2. It follows from (6) and (8) that the solution of system (39) is(42)x1t=xa1ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα+xa21αψt−ψaα+β−11−αEα,α+β−11−αλψt−ψaα−β−11−αEα,α+β−11−α+1λψt−ψaα,x2t=xa2ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα,x3t=xa3ψt−ψaβ−11−αEα,β−11−α+1λ2ψt−ψaα.In this case, we know that bothλ1 and λ2 are real numbers.Theorem 7.
Let0<α<1 and 0≤β≤1. Assume λ1,λ2∈ℝ.(1)
If bothλ1<0 and λ2<0, then system (39) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
Ifλ1<0 and λ2=0, then:(I)
Whenβ=1, then system (39) is stable but not asymptotically stable.(II)
When0≤β<1, then system (39) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α.(3)
Ifλ1=0 and λ2≤0, then:(I)
Whenβ=1, then system (39) is unstable.(II)
When0≤β<1, one has:(i)
When0<α<1−β/2−β, then system (39) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α+α.(ii)
Whenα=1−β/2−β, then system (39) is stable but not asymptotically stable.(iii)
When1−β/2−β<α<1, then system (39) is unstable.(4)
Ifλ1>0 and λ2∈ℝ, or λ2>0 and λ1∈ℝ, then system (39) is unstable.Case 3:A=λ11λ11λ1. In terms of (6) and (8), the solution of system (39) reads as(43)x1t=xa1ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα+xa21αψt−ψaα+β−11−αEα,α+β−11−αλψt−ψaα−β−11−αEα,α+β−11−α+1λψt−ψaα+xa312α2ψt−ψa2α+β−11−αEα,β−11−α+2α−1λ1ψt−ψaα−2β−31−αEα,β−11−α+2αλ1ψt−ψaα+β−11−αβ−11−α+αEα,β−11−α+2α+1λ1ψt−ψaα,x2t=xa2ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα+xa31αψt−ψaα+β−11−αEα,α+β−11−αλψt−ψaα−β−11−αEα,α+β−11−α+1λψt−ψaα,x3t=xa3ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα.In such a case,λ1 is a real number, and we can derive the following theorem.Theorem 8.
Let0<α<1 and 0≤β≤1. Suppose λ1∈ℝ.(1)
Ifλ1<0, then system (39) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
Ifλ1=0, then:(I)
Whenβ=1, then system (39) is unstable.(II)
When0≤β<1, one has:(i)
When0<α<1−β/3−β, then system (39) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α+2α.(ii)
Whenα=1−β/3−β, then system (39) is stable but not asymptotically stable.(iii)
When1−β/3−β<α<1, then system (39) is unstable.(3)
Ifλ1>0, then system (39) is unstable.Proof.
(1)
Considerλ1<0. According to (10), we have, for β=0,(44)Eα,β−11−α+2α−1λ1ψt−ψaα−2β−31−αEα,β−11−α+2αλ1ψt−ψaα+β−11−αβ−11−α+αEα,β−11−α+2α+1λ1ψt−ψaα=−6α2Γβ−11−α−2α+1λ1ψt−ψaα−4+Oλ1ψt−ψaα−5,t⟶∞and for0<β≤1,(45)Eα,β−11−α+2α−1λ1ψt−ψaα−2β−31−αEα,β−11−α+2αλ1ψt−ψaα+β−11−αβ−11−α+αEα,β−11−α+2α+1λ1ψt−ψaα=−2α2Γβ−11−α−α+1λ1ψt−ψaα−3+Oλ1ψt−ψaα−4,t⟶∞.Thus there holds(46)12α2ψt−ψa2α+β−11−αEα,β−11−α+2α−1λ1ψt−ψaα−2β−31−αEα,β−11−α+2αλ1ψt−ψaα+β−11−αβ−11−α+αEα,β−11−α+2α+1λ1ψt−ψaα⟶0.ast⟶∞. Together with (19) and (37), we show that system (39) is asymptotically stable, and the solution decays to zero like ψt−ψaβ−11−α−hα.(2)
Whenλ1=0, the solution (43) is translated to(47)x1t=xa1ψt−ψaβ−11−αΓβ−11−α+1+xa2ψt−ψaα+β−11−αΓα+β−11−α+1+xa3ψt−ψa2α+β−11−αΓ2α+β−11−α+1,x2t=xa2ψt−ψaβ−11−αΓβ−11−α+1+xa3ψt−ψaα+β−11−αΓα+β−11−α+1,x3t=xa3ψt−ψaβ−11−αΓβ−11−α+1.Similar to the proof of (2) in Theorem4, it can be easily known that the conclusions are true.(3)
For the caseλ1>0, from the solution (43), it is clear that system (39) is unstable. The proof of Theorem 8 is now complete.


### 2.4.n-D case n>3

In this subsection, we will study the stability of system (12) with n-dimensional case. Similar to [16, 17], we directly investigate the following two cases:(48)A=λ1λ2⋱λn,A=J1J2⋱Jν,where the dimension nj of Jordan block Jj is greater than or equal to 2 for j=1,2,…,ν.Case 1:A=λ1λ2⋱λn. Applying (6) and (8), the solution of system (12) is stated as(49)x1t=xa1ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα,x2t=xa2ψt−ψaβ−11−αEα,β−11−α+1λ2ψt−ψaα,⋯xnt=xanψt−ψaβ−11−αEα,β−11−α+1λnψt−ψaα.When allλii=1,2,…,n belong to real numbers, we have the following theorem.Theorem 9.
Let0<α<1 and 0≤β≤1. Assume that λi∈ℝ,i=1,2,…,n.(1)
If allλi<0 for i=1,2,…,n, then system (12) is asymptotically stable, and its solution decays to zero as ψtψaβ−11−α−hα.(2)
If at least one ofλii=1,2,…,n equals to zero and others are less than zero, then:(i)
Whenβ=1, then system (12) is stable but not asymptotically stable.(ii)
When0≤β<1, then system (12) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α.(3)
If at least one ofλii=1,2,…,n is greater than zero, then system (12) is unstable.If there are someλi,s which are complex numbers and others are real numbers, without loss of generality, we may assume that λ1,λ2,…,λ2k−1,λ2k∈ℂ, and λ2k+1,…, λn∈ℝ, where k is from 1 to n/2 at most.Theorem 10.
Let0<α<1 and 0≤β≤1. Assume that λ1,λ2,…,λ2k−1,λ2k∈ℂ, λ2k+1,…, λn∈ℝ, and λ1=λ¯2,λ3=λ¯4,…,λ2k−1=λ¯2k.(1)
If allargλl>πα/2l=1,2,…,2k and all λm<0m=2k+1,…,n, then system (12) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
If at least one ofλmm=2k+1,…,n equals to zero and others are less than zero, and all argλl>πα/2l=1,2,…,2k, and 0≤β<1, then system (12) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α.(3)
If at least one ofλmm=2k+1,…,n equals to zero and others are less than zero, and all argλl>πα/2l=1,2,…,2k, and β=1, or at least one of argλll=1,2,…,2k equals to πα/2 and others are greater than πα/2, and all λm≤0m=2k+1,…,n, then system (12) is stable but not asymptotically stable.(4)
If at least one ofλmm=2k+1,…,n is greater than zero and λl∈ℂl=1,2,…,2k, or at least one of argλll=1,2,…,2k is less than πα/2 and λm∈ℝm=2k+1,…,n, then system (12) is unstable.Case 2:A=J1J2⋱Jν, where Jj=λj1λj1⋱⋱λj1λjnj×nj, λj∈ℂ, nj≥2, and ∑j=1νnj=n.Theorem 11.
Let0<α<1 and 0≤β≤1. Assume that λj∈ℂ,j=1,2,…,ν.(1)
If allargλj>πα/2 for j=1,2,…,ν, then system (12) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
If at least one ofargλjj=1,2,…,ν is less than or equal to πα/2, or if there exists some λi=0 and β=1, and others belong to arbitrary complex numbers, then system (12) is unstable.(3)
If there is someλi=0 and 0≤β<1, and all other eigenvalues satisfy argλj>πα/2j=1,2,…,i−1,i+1,…,ν, then:(i)
When0<α<1−β/ni−β, then system (12) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α+ni−1α.(ii)
Whenα=1−β/ni−β, then system (12) is stable but not asymptotically stable.(iii)
When1−β/ni−β<α<1, then system (12) is unstable.Proof.
It is enough to consider one of Jordan blockJj,j=1,2,…,ν. Without loss of generality we will investigate Jordan block J1. In such a case, the solution is written as(50)x1t=xa1ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα+xa2ψt−ψaβ−11−α∂∂λ1Eα,β−11−α+1λ1ψt−ψaα+xa312!ψt−ψaβ−11−α∂2∂λ12Eα,β−11−α+1λ1ψt−ψaα+⋯+xan1ψt−ψaβ−11−αn1−1!∂n1−1∂λ1n1−1Eα,β−11−α+1λ1ψt−ψaα,x2t=xa2ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα+xa3ψt−ψaβ−11−α∂∂λ1Eα,β−11−α+1λ1ψt−ψaα+⋯+xan1ψt−ψaβ−11−αn1−2!∂n1−2∂λ1n1−2Eα,β−11−α+1λ1ψt−ψaα⋯xn1−1t=xan1−1ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα+xan1ψt−ψaβ−11−α∂∂λ1Eα,β−11−α+1λ1ψt−ψaα,xn1t=xan1ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα.(1)
We first prove the caseargλ1>πα/2. Noting that (10), it holds that, for m=0,1,2,…,n1−1,(51)1m!∂m∂λ1mEα,β−11−α+1λ1ψt−ψaα=1m!∂m∂λ1m−∑k=1pλ1ψt−ψaα−kΓβ−11−α+1−αk+Oλ1ψt−ψaα−1−p=−∑k=1p−k−k−1⋯−k−m+1m!Γβ−11−α+1−αkλ1−k−mψt−ψa−αk+Oλ1−1−p−mψt−ψaα−1−p=∑k=1p−1m+1k+m−1!λ1−k−mk−1!m!Γβ−11−α+1−αkψt−ψa−αk+Oλ1−1−p−mψt−ψaα−1−p,t⟶∞.Then we conclude that, form=0,1,2,…,n1−1,(52)1m!ψt−ψaβ−11−α∂m∂λ1mEα,β−11−α+1λ1ψt−ψaα=−1m+1m+1λ12+mΓβ−11−α+1−2αψt−ψaβ−11−α−2α+Oλ1−3−mψt−ψaβ−11−α−3α,t⟶∞,ifβ=0, and(53)1m!ψt−ψaβ−11−α∂m∂λ1mEα,β−11−α+1λ1ψt−ψaα=−1m+1λ11+mΓβ−11−α+1−αψt−ψaβ−11−α−α+Oλ1−2−mψt−ψaβ−11−α−2α,t⟶∞,if0<β≤1. Hence it holds that(54)1m!ψt−ψaβ−11−α∂m∂λ1mEα,β−11−α+1λ1ψt−ψaα⟶0,t⟶∞,which indicates the solution (50) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α−hα. Therefore, if all argλj>πα/2 for j=1,2,…,ν, then system (12) is asymptotically stable, and its solution decays to zero as Oψt−ψaβ−11−α−hα.(2)
We now turn to the proof of caseargλ1≤πα/2. Observe that (9), one deduces that(55)∂∂λ1Eα,β−11−α+1λ1ψt−ψaα=∂∂λ11αλ11−β1−α/αψt−ψa1−β1−αexpλ11/αψt−ψa−∑k=1pλ1ψt−ψaα−kΓβ−11−α+1−αk+Oλ1ψt−ψaα−1−p=1αψt−ψa1−β1−α1−β1−ααλ11−β1−α−α/α+λ11−β1−α/αψt−ψa1αλ11−α/αexpλ11/αψt−ψa−∑k=1p−kλ1−k−1ψt−ψa−αkΓβ−11−α+1−αk+Oλ1−2−pψt−ψa−1+pαast⟶∞. Furthermore, taking λ1=rexp±iαπ/2 yields that(56)ψt−ψaβ−11−α∂∂λ1Eα,β−11−α+1λ1ψt−ψaα=1α2r2−β1−α/αψt−ψaexp±ir1/αψt−ψa+2−β1−α2π+1−β1−αα2r1−β1−α−α/αexp±ir1/αψt−ψa+1−β1−α−α2π+∑k=1pkr−k−1ψt−ψaβ−11−α−αkΓβ−11−α+1−αkexp±ik+1απ2+Or−2−pψt−ψaβ−11−α−1+pα,t⟶∞.Therefore we have(57)ψt−ψaβ−11−α∂∂λ1Eα,β−11−α+1λ1ψt−ψaα≥1α2r2−β1−α/αψt−ψa⟶∞,t⟶∞.That is to say, the solution (50) is unstable. If argλ1<πα/2, obviously, the solution (50) is also unstable. Hence, if at least one of argλjj=1,2,…,ν is less than or equal to πα/2, then system (12) is unstable.According to (2) of Theorem 11 in [17], if there is some λi=0 and β=1, and others belong to arbitrary complex numbers, then system (12) is also unstable.(3)
It remains to show the caseλ1=0. Note that, for m=0,1,2,…,n1−1,(58)ψt−ψaβ−11−α1m!∂m∂λ1mEα,β−11−α+1λ1ψt−ψaα|λ1=0=1Γβ−11−α+1+αmψt−ψaαm+β−11−α.
Then the solution (50) is reduced to(59)x1t=ψt−ψaβ−11−αΓβ−11−α+1xa1+ψt−ψaα+β−11−αΓβ−11−α+1+αxa2+⋯+ψt−ψan1−1α+β−11−αΓβ−11−α+1+n1−1αxan1,x2t=ψt−ψaβ−11−αΓβ−11−α+1xa2+ψt−ψaα+β−11−αΓβ−11−α+1+αxa3+⋯+ψt−ψan1−2α+β−11−αΓβ−11−α+1+n1−2αxan1,⋯xn1−1t=ψt−ψaβ−11−αΓβ−11−α+1xan1−1+ψt−ψaα+β−11−αΓβ−11−α+1+αxan1,xn1t=ψt−ψaβ−11−αΓβ−11−α+1xan1.
We claim that, the solution (59) is asymptotically stable and it decays to zero as ψt−ψan1−1α+β−11−αif0<α<1−β/n1−β; the solution (59) is stable but not asymptotically stable if α=1−β/n1−β; the solution (59) is unstable if 1−β/n1−β<α<1. Therefore, if there exists some λi=0, and all argλj>πα/2j=1,2,…,i−1,i+1,…,ν, then the claimed conclusions hold. This finishes the proof of Theorem 11.The following theorem presents the results of more general case.Theorem 12.
Let0<α<1 and 0≤β≤1.(1)
If all the eigenvaluesλA of the matrix A satisfy argλA>πα/2, then system (12) is asymptotically stable and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
If the zero eigenvalue of the matrixA has the same algebraic and geometric multiplicities and the rest eigenvalues satisfy argλA>πα/2, then:(i)
Whenβ=1, then system (12) is stable but not asymptotically stable.(ii)
When0≤β<1, then system (12) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α.(3)
If the eigenvalues of the matrixA satisfying argλA=πα/2 have the same algebraic and geometric multiplicities, and the rest eigenvalues satisfy argλA>πα/2, then system (12) is stable but not asymptotically stable.(4)
If at least one eigenvalueλA of the matrix A satisfies argλA<πα/2, and/or if the zero eigenvalue of the matrix A has the different algebraic and geometric multiplicities and β=1, and/or if the eigenvalues λA of the matrix A satisfying argλA=πα/2 have the different algebraic and geometric multiplicities, then system (12) is unstable.(5)
If the zero eigenvalue of the matrixA has the different algebraic and geometric multiplicities, and 0≤β<1, and the rest eigenvalues satisfy argλA>πα/2. Suppose that the algebraic multiplicity of the zero eigenvalue is nk,2≤nk≤n,nk,n∈ℕ. Then one has:(i)
When0<α<1−β/nk−β, then system (12) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α+nk−1α.(ii)
Whenα=1−β/nk−β, then system (12) is stable but not asymptotically stable.(iii)
When1−β/nk−β<α<1, then system (12) is unstable.Remark 4.
For linear fractional differential system, we can derive its analytic solution by using Laplace transform. Then, according to asymptotic expansion of Mittag-Leffler function, the stability and decay of the solution can be established. However, such a method is not suitable for nonlinear system and we have to apply linearization theorem to study local stability of system, which is the main objective in the next section.


## 2.1. 1-D Case

Ifn=1, system (12) can be written as(13)HψDa,tα,βxt=λxt,0<α<1,0≤β≤1,t>a,λ∈ℝ,where the initial value is ψDa,t−1−β1−αxt|t=a=xa.According to the formulas (6) and (8), the solution of system (13) may be represented as(14)xt=xaψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα.From the solution (14), the following stability results hold. Throughout this paper, we denote(15)h=2,β=0,1,0<β≤1.Theorem 1.
Let0<α<1 and 0≤β≤1.(1)
Ifλ<0, then system (13) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
Ifλ=0, then:(i)
Whenβ=1, then system (13) is stable but not asymptotically stable.(ii)
When0≤β<1, then system (13) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α.(3)
Ifλ>0, then system (13) is unstable.Proof.
(1)
Ifλ<0, using equality (10) yields(16)Eα,β−11−α+1λψt−ψaα=−∑k=1pλψt−ψaα−kΓβ−11−α+1−αk+Oλψt−ψaα−1−p,t⟶∞.Whenβ=0, one has(17)Eα,β−11−α+1λψt−ψaα=−1λ2Γ−αψt−ψa−2α+Oλψt−ψaα−3,t⟶∞.If0<β≤1, it follows that(18)Eα,β−11−α+1λψt−ψaα=−1λΓβ1−αψt−ψa−α+Oλψa−ψaα−2,t⟶∞.Therefore,(19)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα⟶0,t⟶∞,which displays that system (13) is asymptotically stable, and its solution decays to zero astψ−ψaβ−11−α−hα.(2)
Ifλ=0, the solution (14) is reduced to(20)xt=ψt−ψaβ−11−αΓβ−11−α+1xa.Obviously, system (13) is stable but not asymptotically stable when β=1 and is asymptotically stable when 0≤β<1, where the decay rate of the solution is Oψt−ψaβ−11−α.(3)
Whenλ>0, by means of (9), one deduces that(21)Eα,β−11−α+1λψt−ψaα=1αλψt−ψaα1−β1−α/αexpλ1/αψt−ψa−∑k=1pλψt−ψaα−kΓ1−αk+Oλψt−ψaα−1−p,t⟶∞.
As a result,(22)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα⟶∞,t⟶∞.Remark 3.
Ifβ=1 and β=0, then the ψ-Hilfer derivative recovers to the ψ-Caputo and ψ-Riemann-Liouville derivatives respectively and the results of Theorem 1 are just consistent with those given by Theorem 1 in [17]. In particularly, we get the stability results for the cases of Caputo and Riemann-Liouville derivatives when taking ψt=t and a=0, which have been established in [19, 20].


## 2.2. 2-D Case

For the casen=2, system (12) is given by(23)HψDa,tα,βxt=Axt,0<α<1,0≤β≤1,t>a,A∈ℝ2×2,where the initial values are ψDa,t−1−β1−αxit|t=a=xai,i=1,2.As what has been done in [16, 17], here we will directly consider(24)A=λσ,A=λ1λ.Case 1.
A=λσ. Recall from (6) and (8) that the solution of system (23) is expressed as(25)x1t=xa1ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα,x2t=xa2ψt−ψaβ−11−αEα,β−11−α+1σψt−ψaα.Theorem 2.
Let0<α<1 and 0≤β≤1. Suppose that λ,σ∈ℝ.(1)
If bothλ<0 and σ<0, then system (23) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
Ifλ=0,σ<0, or σ=0,λ<0, or λ=σ=0, then:(i)
Whenβ=1, then system (23) is stable but not asymptotically stable.(ii)
When0≤β<1, then system (23) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α.(3)
If at least one ofλ and σ is greater than zero, then system (23) is unstable.Whenλ,σ∈ℂ, it is easy to know that they are conjugate in this case.Theorem 3.
Let0<α<1 and 0≤β≤1. Assume that λ=σ¯∈ℂ.(1)
Ifargλ>πα/2, then system (23) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
Ifargλ=πα/2, then system (23) is stable but not asymptotically stable.(3)
Ifargλ<πα/2, then system (23) is unstable.Proof.
(1)
Ifargλ>πα/2, it follows from (10) that(26)Eα,β−11−α+1λψt−ψaα=−∑k=1pλψt−ψaα−kΓβ−11−α+1−αk+Oλψt−ψaα−1−p,t⟶∞.We further conclude that(27)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα=−1λ2Γ−αψt−ψaβ−11−α−2α+Oλψt−ψaα−3,t⟶∞,ifβ=0, and(28)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα=−1λΓβ1−αψt−ψaβ−11−α−α+Oλψt−ψaα−2,t⟶∞,if0<β≤1. Consequently,(29)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα⟶0,t⟶∞,which implies that system (23) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α−hα.(2)
Ifargλ=πα/2, taking λ=rexp±iαπ/2r>0 and exploiting (9) lead to(30)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα=1αλψt−ψaα1−β1−α/αψt−ψaβ−11−αexpλ1/αψt−ψa−∑k=1pλ−kψt−ψaβ−11−α−αkΓβ−11−α+1−αk+Oλ−1−pψt−ψaβ−11−α−α−αp=1αr1−β1−α/αexp±i1−β1−α2π+r1/αψt−ψa−∑k=1pr−kψt−ψaβ−11−α−αkΓβ−11−α+1−αkexp±iαkπ2+Or−1−pψt−ψaβ−11−α−α−αp,t⟶∞.Thus,(31)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα⟶1αr1−β1−α/α,t⟶∞.This shows that system (23) is stable but not asymptotically stable.(3)
Ifargλ<πα/2, letting λ=rexpiθr>0,θ<πα/2 and using (9) once more, we have(32)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα=1αr1−β1−α/αexpr1/αψt−ψacosθα×expiθ1−β1−αα+r1/αψt−ψasinθα−∑k=1pr−kψt−ψaβ−11−α−αkΓβ−11−α+1−αkexp−iθk+Or−1−pψt−ψaβ−11−α−α−αp,t⟶∞.
Owing toθ<πα/2, we find expr1/αψt−ψacosθ/α⟶∞ as t⟶∞. As a consequence,(33)ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα⟶∞,t⟶∞.
This indicates that system (23) is unstable. The proof is now complete.Case 2.
A=λ1λ. By means of (6) and (8), the solution of system (23) is denoted by(34)x1t=xa1ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα+xa21αψt−ψaα+β−11−αEα,α+β−11−αλψt−ψaα−β−11−αEα,α+β−11−α+1λψt−ψaα,x2t=xa2ψt−ψaβ−11−αEα,β−11−α+1λψt−ψaα.
In the situation,λ must be a real number and the following holds.Theorem 4.
Let0<α<1 and 0≤β≤1. Suppose λ∈ℝ.(1)
Ifλ<0, then system (23) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
Ifλ=0, then:(I)
Whenβ=1, then system (23) is unstable.(II)
When0≤β<1, one has:(i)
When0<α<1−β/2−β, then system (23) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α+α.(ii)
Whenα=1−β/2−β, then system (23) is stable but not asymptotically stable.(iii)
When1−β/2−β<α<1, then system (23) is unstable.(3)
Ifλ>0, then system (23) is unstable.Proof.
(1)
Whenλ<0, according to equality (10), we arrive at(35)1αψt−ψaα+β−11−αEα,α+β−11−αλψt−ψaα−β−11−αEα,α+β−11−α+1λψt−ψaα=1λ2Γβ−11−α+1−αψt−ψaβ−11−α−α+Oλ−3ψt−ψaβ−11−α−2α,t⟶∞,if0<β≤1, and(36)1αψt−ψaα+β−11−αEα,α+β−11−αλψt−ψaα−β−11−αEα,α+β−11−α+1λψt−ψaα=2λ3Γβ−11−α+1−2αψt−ψaβ−11−α−2α+Oλ−4ψt−ψaβ−11−α−3α,t⟶∞,ifβ=0. Consequently,(37)|1αψt−ψaα+β−11−αEα,α+β−11−αλψt−ψaα−β−11−αEα,α+β−11−α+1λψt−ψaα|⟶0,t⟶∞.Combining (19) and (37), we know that system (23) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α−hα.(2)
Whenλ=0, the solution (34) is reduced to(38)x1t=xa1ψt−ψaβ−11−αΓβ−11−α+1+xa2ψt−ψaα+β−11−αΓα+β−11−α+1,x2t=xa2ψt−ψaβ−11−αΓβ−11−α+1..Ifβ=1, by the first equality in (38), we immediately see that system (23) is unstable. If 0≤β<1, by (38), it is easy to check that: when0<α<1−β/2−β, system (23) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α+α; when α=1−β/2−β, system (23) is stable but not asymptotically stable; when 1−β/2−β<α<1, system (23) is unstable.(3)
Whenλ>0, we find that system (23) is unstable by using the solution (34). The proof is completed.


## 2.3. 3-D Case

Ifn=3, system (12) may write as(39)HψDa,tα,βxt=Axt,0<α<1,0≤β≤1,t>a,A∈ℝ3×3,with the initial valuesψDa,t−1−β1−αxit|t=a=xai,i=1,2,3. Same as [16, 17], we can directly investigate the matrix A in the following three cases:(40)A=λ1λ2λ3,A=λ11λ1λ2,A=λ11λ11λ1.Case 1:A=λ1λ2λ3. Similar to (14) or (25), the solution of system (39) is(41)x1t=xa1ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα,x2t=xa2ψt−ψaβ−11−αEα,β−11−α+1λ2ψt−ψaα,x3t=xa3ψt−ψaβ−11−αEα,β−11−α+1λ3ψt−ψaα.We first discuss the case for allλi∈ℝ,i=1,2,3.Theorem 5.
Let0<α<1 and 0≤β≤1. Assume that λi∈ℝ,i=1,2,3.(1)
If allλi<0,i=1,2,3, then system (39) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
If at least one ofλii=1,2,3 equals to zero and others are less than zero, then:(i)
Whenβ=1, then system (39) is stable but not asymptotically stable.(ii)
When0≤β<1, then system (39) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α.(3)
If at least one ofλii=1,2,3 is greater than zero, then system (39) is unstable.If there exists someλi which is a complex number, without loss of generality, we assume that λ1∈ℝ and λ2=λ¯3∈ℂ.Theorem 6.
Let0<α<1 and 0≤β≤1. Assume that λ1∈ℝ and λ2=λ¯3∈ℂ.(1)
Ifλ1<0 and argλ2>πα/2, then system (39) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
Ifλ1=0, argλ2>πα/2 and 0≤β<1, then system (39) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α.(3)
Ifλ1=0, argλ2>πα/2 and β=1, or λ1≤0 and argλ2=πα/2, then system (39) is stable but not asymptotically stable.(4)
Ifλ1>0 and λ2,λ3∈ℂ, or λ1∈ℝ and argλ2<πα/2, then system (39) is unstable.Case 2:A=λ11λ1λ2. It follows from (6) and (8) that the solution of system (39) is(42)x1t=xa1ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα+xa21αψt−ψaα+β−11−αEα,α+β−11−αλψt−ψaα−β−11−αEα,α+β−11−α+1λψt−ψaα,x2t=xa2ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα,x3t=xa3ψt−ψaβ−11−αEα,β−11−α+1λ2ψt−ψaα.In this case, we know that bothλ1 and λ2 are real numbers.Theorem 7.
Let0<α<1 and 0≤β≤1. Assume λ1,λ2∈ℝ.(1)
If bothλ1<0 and λ2<0, then system (39) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
Ifλ1<0 and λ2=0, then:(I)
Whenβ=1, then system (39) is stable but not asymptotically stable.(II)
When0≤β<1, then system (39) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α.(3)
Ifλ1=0 and λ2≤0, then:(I)
Whenβ=1, then system (39) is unstable.(II)
When0≤β<1, one has:(i)
When0<α<1−β/2−β, then system (39) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α+α.(ii)
Whenα=1−β/2−β, then system (39) is stable but not asymptotically stable.(iii)
When1−β/2−β<α<1, then system (39) is unstable.(4)
Ifλ1>0 and λ2∈ℝ, or λ2>0 and λ1∈ℝ, then system (39) is unstable.Case 3:A=λ11λ11λ1. In terms of (6) and (8), the solution of system (39) reads as(43)x1t=xa1ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα+xa21αψt−ψaα+β−11−αEα,α+β−11−αλψt−ψaα−β−11−αEα,α+β−11−α+1λψt−ψaα+xa312α2ψt−ψa2α+β−11−αEα,β−11−α+2α−1λ1ψt−ψaα−2β−31−αEα,β−11−α+2αλ1ψt−ψaα+β−11−αβ−11−α+αEα,β−11−α+2α+1λ1ψt−ψaα,x2t=xa2ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα+xa31αψt−ψaα+β−11−αEα,α+β−11−αλψt−ψaα−β−11−αEα,α+β−11−α+1λψt−ψaα,x3t=xa3ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα.In such a case,λ1 is a real number, and we can derive the following theorem.Theorem 8.
Let0<α<1 and 0≤β≤1. Suppose λ1∈ℝ.(1)
Ifλ1<0, then system (39) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
Ifλ1=0, then:(I)
Whenβ=1, then system (39) is unstable.(II)
When0≤β<1, one has:(i)
When0<α<1−β/3−β, then system (39) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α+2α.(ii)
Whenα=1−β/3−β, then system (39) is stable but not asymptotically stable.(iii)
When1−β/3−β<α<1, then system (39) is unstable.(3)
Ifλ1>0, then system (39) is unstable.Proof.
(1)
Considerλ1<0. According to (10), we have, for β=0,(44)Eα,β−11−α+2α−1λ1ψt−ψaα−2β−31−αEα,β−11−α+2αλ1ψt−ψaα+β−11−αβ−11−α+αEα,β−11−α+2α+1λ1ψt−ψaα=−6α2Γβ−11−α−2α+1λ1ψt−ψaα−4+Oλ1ψt−ψaα−5,t⟶∞and for0<β≤1,(45)Eα,β−11−α+2α−1λ1ψt−ψaα−2β−31−αEα,β−11−α+2αλ1ψt−ψaα+β−11−αβ−11−α+αEα,β−11−α+2α+1λ1ψt−ψaα=−2α2Γβ−11−α−α+1λ1ψt−ψaα−3+Oλ1ψt−ψaα−4,t⟶∞.Thus there holds(46)12α2ψt−ψa2α+β−11−αEα,β−11−α+2α−1λ1ψt−ψaα−2β−31−αEα,β−11−α+2αλ1ψt−ψaα+β−11−αβ−11−α+αEα,β−11−α+2α+1λ1ψt−ψaα⟶0.ast⟶∞. Together with (19) and (37), we show that system (39) is asymptotically stable, and the solution decays to zero like ψt−ψaβ−11−α−hα.(2)
Whenλ1=0, the solution (43) is translated to(47)x1t=xa1ψt−ψaβ−11−αΓβ−11−α+1+xa2ψt−ψaα+β−11−αΓα+β−11−α+1+xa3ψt−ψa2α+β−11−αΓ2α+β−11−α+1,x2t=xa2ψt−ψaβ−11−αΓβ−11−α+1+xa3ψt−ψaα+β−11−αΓα+β−11−α+1,x3t=xa3ψt−ψaβ−11−αΓβ−11−α+1.Similar to the proof of (2) in Theorem4, it can be easily known that the conclusions are true.(3)
For the caseλ1>0, from the solution (43), it is clear that system (39) is unstable. The proof of Theorem 8 is now complete.


## 2.4.n-D case n>3

In this subsection, we will study the stability of system (12) with n-dimensional case. Similar to [16, 17], we directly investigate the following two cases:(48)A=λ1λ2⋱λn,A=J1J2⋱Jν,where the dimension nj of Jordan block Jj is greater than or equal to 2 for j=1,2,…,ν.Case 1:A=λ1λ2⋱λn. Applying (6) and (8), the solution of system (12) is stated as(49)x1t=xa1ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα,x2t=xa2ψt−ψaβ−11−αEα,β−11−α+1λ2ψt−ψaα,⋯xnt=xanψt−ψaβ−11−αEα,β−11−α+1λnψt−ψaα.When allλii=1,2,…,n belong to real numbers, we have the following theorem.Theorem 9.
Let0<α<1 and 0≤β≤1. Assume that λi∈ℝ,i=1,2,…,n.(1)
If allλi<0 for i=1,2,…,n, then system (12) is asymptotically stable, and its solution decays to zero as ψtψaβ−11−α−hα.(2)
If at least one ofλii=1,2,…,n equals to zero and others are less than zero, then:(i)
Whenβ=1, then system (12) is stable but not asymptotically stable.(ii)
When0≤β<1, then system (12) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α.(3)
If at least one ofλii=1,2,…,n is greater than zero, then system (12) is unstable.If there are someλi,s which are complex numbers and others are real numbers, without loss of generality, we may assume that λ1,λ2,…,λ2k−1,λ2k∈ℂ, and λ2k+1,…, λn∈ℝ, where k is from 1 to n/2 at most.Theorem 10.
Let0<α<1 and 0≤β≤1. Assume that λ1,λ2,…,λ2k−1,λ2k∈ℂ, λ2k+1,…, λn∈ℝ, and λ1=λ¯2,λ3=λ¯4,…,λ2k−1=λ¯2k.(1)
If allargλl>πα/2l=1,2,…,2k and all λm<0m=2k+1,…,n, then system (12) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
If at least one ofλmm=2k+1,…,n equals to zero and others are less than zero, and all argλl>πα/2l=1,2,…,2k, and 0≤β<1, then system (12) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α.(3)
If at least one ofλmm=2k+1,…,n equals to zero and others are less than zero, and all argλl>πα/2l=1,2,…,2k, and β=1, or at least one of argλll=1,2,…,2k equals to πα/2 and others are greater than πα/2, and all λm≤0m=2k+1,…,n, then system (12) is stable but not asymptotically stable.(4)
If at least one ofλmm=2k+1,…,n is greater than zero and λl∈ℂl=1,2,…,2k, or at least one of argλll=1,2,…,2k is less than πα/2 and λm∈ℝm=2k+1,…,n, then system (12) is unstable.Case 2:A=J1J2⋱Jν, where Jj=λj1λj1⋱⋱λj1λjnj×nj, λj∈ℂ, nj≥2, and ∑j=1νnj=n.Theorem 11.
Let0<α<1 and 0≤β≤1. Assume that λj∈ℂ,j=1,2,…,ν.(1)
If allargλj>πα/2 for j=1,2,…,ν, then system (12) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
If at least one ofargλjj=1,2,…,ν is less than or equal to πα/2, or if there exists some λi=0 and β=1, and others belong to arbitrary complex numbers, then system (12) is unstable.(3)
If there is someλi=0 and 0≤β<1, and all other eigenvalues satisfy argλj>πα/2j=1,2,…,i−1,i+1,…,ν, then:(i)
When0<α<1−β/ni−β, then system (12) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α+ni−1α.(ii)
Whenα=1−β/ni−β, then system (12) is stable but not asymptotically stable.(iii)
When1−β/ni−β<α<1, then system (12) is unstable.Proof.
It is enough to consider one of Jordan blockJj,j=1,2,…,ν. Without loss of generality we will investigate Jordan block J1. In such a case, the solution is written as(50)x1t=xa1ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα+xa2ψt−ψaβ−11−α∂∂λ1Eα,β−11−α+1λ1ψt−ψaα+xa312!ψt−ψaβ−11−α∂2∂λ12Eα,β−11−α+1λ1ψt−ψaα+⋯+xan1ψt−ψaβ−11−αn1−1!∂n1−1∂λ1n1−1Eα,β−11−α+1λ1ψt−ψaα,x2t=xa2ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα+xa3ψt−ψaβ−11−α∂∂λ1Eα,β−11−α+1λ1ψt−ψaα+⋯+xan1ψt−ψaβ−11−αn1−2!∂n1−2∂λ1n1−2Eα,β−11−α+1λ1ψt−ψaα⋯xn1−1t=xan1−1ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα+xan1ψt−ψaβ−11−α∂∂λ1Eα,β−11−α+1λ1ψt−ψaα,xn1t=xan1ψt−ψaβ−11−αEα,β−11−α+1λ1ψt−ψaα.(1)
We first prove the caseargλ1>πα/2. Noting that (10), it holds that, for m=0,1,2,…,n1−1,(51)1m!∂m∂λ1mEα,β−11−α+1λ1ψt−ψaα=1m!∂m∂λ1m−∑k=1pλ1ψt−ψaα−kΓβ−11−α+1−αk+Oλ1ψt−ψaα−1−p=−∑k=1p−k−k−1⋯−k−m+1m!Γβ−11−α+1−αkλ1−k−mψt−ψa−αk+Oλ1−1−p−mψt−ψaα−1−p=∑k=1p−1m+1k+m−1!λ1−k−mk−1!m!Γβ−11−α+1−αkψt−ψa−αk+Oλ1−1−p−mψt−ψaα−1−p,t⟶∞.Then we conclude that, form=0,1,2,…,n1−1,(52)1m!ψt−ψaβ−11−α∂m∂λ1mEα,β−11−α+1λ1ψt−ψaα=−1m+1m+1λ12+mΓβ−11−α+1−2αψt−ψaβ−11−α−2α+Oλ1−3−mψt−ψaβ−11−α−3α,t⟶∞,ifβ=0, and(53)1m!ψt−ψaβ−11−α∂m∂λ1mEα,β−11−α+1λ1ψt−ψaα=−1m+1λ11+mΓβ−11−α+1−αψt−ψaβ−11−α−α+Oλ1−2−mψt−ψaβ−11−α−2α,t⟶∞,if0<β≤1. Hence it holds that(54)1m!ψt−ψaβ−11−α∂m∂λ1mEα,β−11−α+1λ1ψt−ψaα⟶0,t⟶∞,which indicates the solution (50) is asymptotically stable, and the decay rate is Oψt−ψaβ−11−α−hα. Therefore, if all argλj>πα/2 for j=1,2,…,ν, then system (12) is asymptotically stable, and its solution decays to zero as Oψt−ψaβ−11−α−hα.(2)
We now turn to the proof of caseargλ1≤πα/2. Observe that (9), one deduces that(55)∂∂λ1Eα,β−11−α+1λ1ψt−ψaα=∂∂λ11αλ11−β1−α/αψt−ψa1−β1−αexpλ11/αψt−ψa−∑k=1pλ1ψt−ψaα−kΓβ−11−α+1−αk+Oλ1ψt−ψaα−1−p=1αψt−ψa1−β1−α1−β1−ααλ11−β1−α−α/α+λ11−β1−α/αψt−ψa1αλ11−α/αexpλ11/αψt−ψa−∑k=1p−kλ1−k−1ψt−ψa−αkΓβ−11−α+1−αk+Oλ1−2−pψt−ψa−1+pαast⟶∞. Furthermore, taking λ1=rexp±iαπ/2 yields that(56)ψt−ψaβ−11−α∂∂λ1Eα,β−11−α+1λ1ψt−ψaα=1α2r2−β1−α/αψt−ψaexp±ir1/αψt−ψa+2−β1−α2π+1−β1−αα2r1−β1−α−α/αexp±ir1/αψt−ψa+1−β1−α−α2π+∑k=1pkr−k−1ψt−ψaβ−11−α−αkΓβ−11−α+1−αkexp±ik+1απ2+Or−2−pψt−ψaβ−11−α−1+pα,t⟶∞.Therefore we have(57)ψt−ψaβ−11−α∂∂λ1Eα,β−11−α+1λ1ψt−ψaα≥1α2r2−β1−α/αψt−ψa⟶∞,t⟶∞.That is to say, the solution (50) is unstable. If argλ1<πα/2, obviously, the solution (50) is also unstable. Hence, if at least one of argλjj=1,2,…,ν is less than or equal to πα/2, then system (12) is unstable.According to (2) of Theorem 11 in [17], if there is some λi=0 and β=1, and others belong to arbitrary complex numbers, then system (12) is also unstable.(3)
It remains to show the caseλ1=0. Note that, for m=0,1,2,…,n1−1,(58)ψt−ψaβ−11−α1m!∂m∂λ1mEα,β−11−α+1λ1ψt−ψaα|λ1=0=1Γβ−11−α+1+αmψt−ψaαm+β−11−α.
Then the solution (50) is reduced to(59)x1t=ψt−ψaβ−11−αΓβ−11−α+1xa1+ψt−ψaα+β−11−αΓβ−11−α+1+αxa2+⋯+ψt−ψan1−1α+β−11−αΓβ−11−α+1+n1−1αxan1,x2t=ψt−ψaβ−11−αΓβ−11−α+1xa2+ψt−ψaα+β−11−αΓβ−11−α+1+αxa3+⋯+ψt−ψan1−2α+β−11−αΓβ−11−α+1+n1−2αxan1,⋯xn1−1t=ψt−ψaβ−11−αΓβ−11−α+1xan1−1+ψt−ψaα+β−11−αΓβ−11−α+1+αxan1,xn1t=ψt−ψaβ−11−αΓβ−11−α+1xan1.
We claim that, the solution (59) is asymptotically stable and it decays to zero as ψt−ψan1−1α+β−11−αif0<α<1−β/n1−β; the solution (59) is stable but not asymptotically stable if α=1−β/n1−β; the solution (59) is unstable if 1−β/n1−β<α<1. Therefore, if there exists some λi=0, and all argλj>πα/2j=1,2,…,i−1,i+1,…,ν, then the claimed conclusions hold. This finishes the proof of Theorem 11.The following theorem presents the results of more general case.Theorem 12.
Let0<α<1 and 0≤β≤1.(1)
If all the eigenvaluesλA of the matrix A satisfy argλA>πα/2, then system (12) is asymptotically stable and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
If the zero eigenvalue of the matrixA has the same algebraic and geometric multiplicities and the rest eigenvalues satisfy argλA>πα/2, then:(i)
Whenβ=1, then system (12) is stable but not asymptotically stable.(ii)
When0≤β<1, then system (12) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α.(3)
If the eigenvalues of the matrixA satisfying argλA=πα/2 have the same algebraic and geometric multiplicities, and the rest eigenvalues satisfy argλA>πα/2, then system (12) is stable but not asymptotically stable.(4)
If at least one eigenvalueλA of the matrix A satisfies argλA<πα/2, and/or if the zero eigenvalue of the matrix A has the different algebraic and geometric multiplicities and β=1, and/or if the eigenvalues λA of the matrix A satisfying argλA=πα/2 have the different algebraic and geometric multiplicities, then system (12) is unstable.(5)
If the zero eigenvalue of the matrixA has the different algebraic and geometric multiplicities, and 0≤β<1, and the rest eigenvalues satisfy argλA>πα/2. Suppose that the algebraic multiplicity of the zero eigenvalue is nk,2≤nk≤n,nk,n∈ℕ. Then one has:(i)
When0<α<1−β/nk−β, then system (12) is asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α+nk−1α.(ii)
Whenα=1−β/nk−β, then system (12) is stable but not asymptotically stable.(iii)
When1−β/nk−β<α<1, then system (12) is unstable.Remark 4.
For linear fractional differential system, we can derive its analytic solution by using Laplace transform. Then, according to asymptotic expansion of Mittag-Leffler function, the stability and decay of the solution can be established. However, such a method is not suitable for nonlinear system and we have to apply linearization theorem to study local stability of system, which is the main objective in the next section.


## 3. Stability of Nonlinear System

In this part, we study the stability of the autonomous nonlinear differential system with theψ-Hilfer derivative, that is,(60)HψDa,tα,βxt=fx,0<α<1,0≤β≤1,t>a,with the initial value ψDa,t−1−β1−αxt|t=a=xa, where xt,fx∈ℝn, and fx satisfies the Lipschitz condition.We present the following equivalence between system (60) and a nonlinear Volterra integral equation.Lemma 3.
Let0<α<1 and 0≤β≤1. Suppose that the function fx is continuous. Then system (60) is equivalent to a nonlinear Volterra integral equation of the second kind of the form(61)xt=ψt−ψaβ−11−αΓβ−11−α+1xa+1Γα∫atψt−ψτα−1fxτψ′τdτ.Proof.
The proof of this lemma can be referred to that of Theorem13 in [10].To investigate the stability of system (60), we need to introduce the following some notations.Definition 9.
The origin is called a hyperbolic equilibrium of system (60) if the following conditions hold.(1)
The origin is an equilibrium of system (60);(2)
All the eigenvaluesλf′0 of the linearized matrix f′0 satisfy λf′0≠0 and argλf′0≠πα/2.Definition 10.
([15]) Assume that fx and gy are continuous vector fields defined on V⊆ℝn and W⊆ℝn, and they generate flows ϕt,f:V⟶V and ϕt,g:W⟶W, respectively. The vector fields fx and gy are said to be locally topologically equivalent if there is a homeomorphism h:V⟶W such that h∘ϕt,fx=ϕt,g∘hx for x∈δx^,r⊂V and ∀x^∈V. The vector fields fx and gy are called globally topologically equivalent if the homeomorphism h is valid for the whole space V.Based on the above several definitions, it is easy to derive the following linearization theorem.Theorem 13.
Let0<α<1 and 0≤β≤1. If the origin O is a hyperbolic equilibrium of (60), then vector field fx is topologically equivalent with its linearization vector field f′0x in the neighborhood δ0 of the origin O.Proof.
Ifβ=1, the proof is just that of Theorem 13 in [17]. For 0≤β<1, the proof of the theorem is very similar to those of Theorems 2 and 3 in [21] and the details are left to the reader as exercise. □By means of linearization theorem, that is Theorem13, we may obtain the local stability to system (60) provided that the equilibrium of system (60) is a hyperbolic equilibrium.Theorem 14.
Let0<α<1 and 0≤β≤1. Suppose that the zero solution of system (60) is a hyperbolic equilibrium.(1)
If all the eigenvaluesλf′0 of Jacobian matrix f′0 satisfy argλf′0>πα/2, then system (60) is locally asymptotically stable, and its solution decays to zero as ψt−ψaβ−11−α−hα.(2)
If at least one eigenvalueλf′0 of Jacobian matrix f′0 satisfies argλf′0<πα/2, then system (60) is unstable.Remark 5.
If the Jacobian matrixf′0 has critical eigenvalues, i.e., λf′0=0, and/or argλf′0=πα/2, then we can not determine whether system (60) is (locally) stable or unstable in this case.


## 4. Conclusion Remarks

This paper discusses the stability and decay of the solution of theψ-Hilfer fractional differential systems with the initial condition. We investigate the Lyapunov stability and decay rates of the solution of linear system in great detail. In accordance with the linearization theorem established, the local stability of the autonomous nonlinear system is presented provided that the equilibrium of system is a hyperbolic equilibrium. From the results we have obtained, they extend the conclusions of the paper [17] which corresponds to cases β=1 (ψ-Caputo case) and β=0 (ψ-Riemann-Liouville case). In particular, if ψt=t and β=1,0, then the results are just those given by the papers [19, 20]; if ψt=logt and β=1,0, then the conclusions are consist with those achieved in the paper [16].



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*Source: 5437691-2022-07-16.xml*