# On the Novel Finite-Time Stability Results for Uncertain Fractional Delay Differential Equations Involving Noninstantaneous Impulses

Authors: Qien Li; Danfeng Luo; Zhiguo Luo; Quanxin Zhu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, we mainly study the finite-time stability for a kind of uncertain fractional-order delay differential equations with noninstantaneous impulses. By using the Lyapunov functions along with the generalized Grönwall inequality, we present the new stability results for the considered equations. Finally, two examples are given to demonstrate the effectiveness of our theoretical results.


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## Body

## 1. Introduction

Over the past two decades, fractional-order systems (FOSs) have been intensively studied due to its wide applications to various fields, especially in memory storage, dynamics mechanic, and communication engineering. As we know, stability theory is one of the most crucial themes in the qualitative theory of dynamical systems, and we can see the monographs [1, 2] and the papers [3–7] for Ulam–Hyers stability, [8–11] for Lipschitz stability, [12–14] for asymptotic stability, and for other forms of stability. Further, for more details about some other properties of the solutions, we can see [15–18]. In this paper, we mainly study the finite-time stability, which has been investigated by some researchers. For an extensive collection of such results, we refer the readers to the related literatures, such as the papers [19–23]. In detail, in [19], Wu et al. studied the finite-time stability of Caputo delta fractional linear difference equations with the aid of Grönwall inequality. In [20], D. F. Luo and Z. G. Luo considered the uniqueness and finite-time stability of solutions for a kind of fractional-order nonlinear difference equations with time-varying delays and improved the results in [19]. In order to explore the finite-time stability of fractional differential equations, Li and Wang [21] introduced a concept of delayed Mittag-Leffler type matrix function and studied the system in the following form:(1)DC0+αyx=Byx−τ,yx∈ℝn,x∈J≔0,T,τ>0,yx=φx,−τ≤x≤0.The authors finally presented the finite-time stability results by virtue of delayed Mittag-Leffler type matrix.Phat and Thanh [22] considered the following nonlinear FOSs with time-varying delay and nonlinear disturbance:(2)Dαxt=Axt+Bxt−ht+Dwt+ft,xt,xt−ht,wt,t≥0,xθ=φθ,θ∈−h,0.Based on a generalized Grönwall inequality, a delay-dependent sufficient condition for robust finite-time stability of such systems was established in terms of the Mittag-Leffler function.To research the influence of impulses on finite-time stability of fractional differential system, D. F. Luo and Z. G. Luo in [23] investigated the following ψ-Hilfer fractional differential equation with time-varying delays and noninstantaneous impulses:(3)DHtα,β;ψxt=Atxt+Btxt−ht+ft,xt,xt−ht,t∈J1=sk,tk+1∩t0,T,k=0,1,…,p,xt=ϕkt,xtΓγΓ2−γ,t∈J2=tk,sk∩t0,T,k=1,2,…,p,xsk=xsk+=xsk−,k=1,2,…,p,xt=ϕtΓγΓ2−γ,t∈J3=−h,t0,where J=−h,T, DHtα,β;ψ⋅ is the ψ-Hilfer fractional derivative of order α∈0,1 and type 0≤β≤1, with respect to function ψ, γ=α+β1−α, xt is the quantity of state mapping the interval J to ℝ, At, Bt:ℝ⟶ℝ are bounded operators, f:J×ℝ×ℝ⟶ℝ is continuous and ϕk:tk,sk∩t0,T×ℝ⟶ℝ, k=1,2,…,p, and ht is a continuous delay function satisfying 0≤ht≤h, t∈J, ϕt is a delayed function. The authors deduced that the solutions of system (3) had finite-time stability under some new criteria and by applying the generalized Grönwall inequality.In this paper, we first assume two increasing finite sequences of pointstii=1p+1 and sii=0p are given, such that s0=0<t1<s1<t2<⋯<sp−1<tp<sp<tp+1=T, where T∈ℝ+ is given and p is a natural number. Inspired by the papers mentioned, we consider the following uncertain fractional-order delay differential equations with noninstantaneous impulses:(4)DCtαxt=ft,xt,xt−τt+ΔAt,x,t∈J1=0,t1∪⋃k=1psk,tk+1,xt=ϕkt,Tktxtk−,t∈J2=⋃k=1ptk,sk,xt=φt,t∈−τ,0,where J=−τ,T, DCtαxt is the Caputo fractional derivative of order α∈0,1, and τt is a continuous function, such that 0≤τt≤τ, τ∈ℝ+, ∀t∈J. Let f:J×ℝ2⟶ℝ be a given function to be specified later, and the uncertainty of this system denoted by ΔAt,x, which is bounded in J×ℝ. Tkt:Ctk,sk;ℝ⟶ℝ are continuous maps for t∈tk,sk, ϕk∈Ctk,sk×ℝ;ℝ are noninstantaneous impulsive functions for all k∈1,2,…,p, and φ:−τ,0⟶ℝ is a continuous function.Remark 1 [23].
Iftk=sk, k=1,2,…,p, then system (4) reduces to an impulsive differential equation. In this case, at any point of instantaneous impulse tk, the amount of jump of the solution xt is given by Ik=ϕkt,Tktxtk−.
The main objective in this paper is to extend the work in [24, 25] to fractional order and develop the results in [21, 22] under noninstantaneous impulsive effects, where impulsive action starts at an arbitrary fixed point and remains active on a finite time interval, which is very different from the classical instantaneous impulsive case that changes are relatively short compared to the overall duration of the whole process. Our main motivation for doing this paper is to research the effect of noninstantaneous impulses on finite-time stability, and to do this, we introduce two class-K functions, γ and β, that satisfy some certain conditions, which can guarantee the finite-time stability. The difficulty of the whole process is how to construct γ and β that meet the particular properties we wanted.
This article is organized as follows. In Section2, we introduce some known definitions and lemmas, which are useful to our following works. Section 3 is devoted to researching the finite-time stability for fractional system (4). To explain the results clearly, we finally provide two examples in Section 4.


## 2. Preliminaries

In this section, we plan to introduce some basic definitions and lemmas which are used throughout this paper.Definition 1 [23].
One-parameter Mittag-Leffler function: the Mittag-Leffler function is given by the following series:(5)Eμz=∑k=0∞zkΓμk+1,where μ∈ℂ, ℜμ>0, and Γz is a gamma function given by(6)Γz=∫0∞e−ttz−1dt,ℜz>0. In particular, if μ=1, we have E1z=∑j=0∞zj/Γj+1=∑j=0∞zj/j!=ez.Lemma 1 [22].
Generalized Grönwall inequality: suppose thatα>0, b∈ℝ, at is a nonnegative function locally integrable on b,T, gt is a nonnegative, nondecreasing continuous function defined on b,T, ut is a nonnegative locally integrable function on b,T satisfying the inequality, the we get(7)ut≤at+gt∫btt−τα−1uτdτ,then(8)ut≤at+∫bt∑k=1∞gtΓαkΓαkt−ταk−1aτdτ,b≤t<T.
Moreover, ifat is a nondecreasing function on b,T, then for ∀t≥b, we have ut≤atEαgtΓαt−bα, where Eα⋅ is the Mittag-Leffler function defined by Definition 1.Definition 2 [26].
The Riemann–Liouville fractional integral of orderα>0 for a continuous function g:0,+∞⟶ℝ is defined as(9)Itαgt=1Γα∫0tt−sα−1gsds,t>0,where Itα denotes the Riemann–Liouville fractional integral of order α and Γ⋅ is Euler’s Gamma function satisfying Γα+1=αΓα.Definition 3 [26].
The fractional-order derivative in the Caputo sense for a continuous functiongt:0,+∞⟶ℝ is defined as(10)DCtαgt=1Γn−α∫0tgnst−sα−n+1ds,t>0,where n is the smallest integer greater than or equal to α.Lemma 2 [5].
Ifxt∈Cn0,T, then the following is satisfied:(11)ItβDtβxt=xt−∑k=0n−1xk0k!tk,where β∈n−1,n, n∈N. In particular, xt∈C10,T and β∈0,1, then ItβDtβxt=xt−x0.Definition 4.
A continuous functionx:0,t⟶0,+∞ is said to belong to class-K if it is strictly increasing and x0=0.
Now, we introduce a space:(12)PCJ,ℝ=x:J⟶ℝ:x∈Ctk,tk+1,ℝ,and there existxtk−,xtk+,k=1,2,…,p,withxtk−=xtk,with the norm xPC≔supxt:t∈J, J=−τ,T. Obviously, PCJ,ℝ is a Banach space.Definition 5 [23].
The fractional system (4) is finite-time stable w.r.t. δ,σ,J, δ<σ, if and only if(13)φPC≤δ⟹xPC≤σ,∀t∈J.Lemma 3.
Let0<α≤1, as for any t∈J, x satisfies the fractional system (4) if and only if x satisfies the following Volterra integral equations:(14)xt=φt,t∈−τ,0,1Γα∫0tt−ξα−1fξ,xξ,xξ−τξ+ΔAξ,xξdξ+φ0,t∈0,t1,ϕkt,Tktxtk−,t∈⋃k=1ptk,sk,1Γα∫sktt−ξα−1fξ,xξ,xξ−τξ+ΔAξ,xξdξ+ϕksk,Tkskxtk−,t∈⋃k=1psk,tk+1.Proof.
The proof process can be referred to [23].


## 3. Main Results

LetB+J denote the set of all nonnegative bounded functions on interval J. To establish the main results, we introduce the following assumptions:H1 For any t∈J, assume that the uncertainty term ΔAt,x satisfies the following condition:(15)ItαΔAt,x≤Gt,x,where Gt,x is nonnegative continuous function.H2 Assume that the nonlinear function f satisfies: there exists a positive function lft∈B+J such that(16)ft,x1,y1−ft,x2,y2PC≤lft⋅x1−x2PC+y1−y2PC,where t∈J, x1,x2,y1,y2∈PCJ,ℝ, and ft,0,0=0.H3 For x,y∈PCJ,ℝ, there exists a positive sequence λk:k=1,2,…,p, such that(17)ϕkt,x−ϕkt,yPC≤λkx−yPC,and ϕkt,0=0 holds.Throughout this paper, we always assume the following:(18)supξ∈0,tlfξ=Lf,λ=maxλk:k=1,2,…,p.We denote(19)Mk,r=supTktx:t∈∪k=1ptk,sk,x∈C⋃k=1ptk,sk;ℝ,x≤r.Based on the known results [21–23], we can clearly conclude that this fractional system (4) has at least one solution under some conditions. Now, we plan to investigate the specified stability.Theorem 1.
Suppose the validity of conditionsH1–H3 and there exist two positive real numbers δ and σ such that δ<σ, and φPC≤δ, then fractional system (4) has finite-time stability on J provided that the condition(20)∫0T∑k=1∞2LfkΓαkT−ξαk−1dξ≤σM+MG−1,holds, where M=maxλMk,r,φ0PC, and MG=supt∈JGt,xPC.Proof.
The proof process can be referred to Theorem 4.1 in [23].
Now, we need to take Lyapunov function approach to deal with the finite-time stability. In order to accomplish this work, we first consider the following uncertain fractional-order delay differential equations without any impulses:(21)DCtαxt=ft,xt,xt−τt+ΔAt,x,t∈0,T,xt=φt,t∈−τ,0.Lemma 4.
Fractional system (21) has finite-time stability provided that there exists a continuously differentiable nonnegative function Vt,x, a positive nondecreasing continuous function ht, and two nonnegative continuous functions gt,x, Ψt,x defined on J, and two class-K functions, γ and β, such that for any t∈0,T,(1)
γxt≤Vt,x≤βxt(2)
DCtαVt,x≤htVt,x+gt,x(3)
Itαgt,x≤Ψt,x(4)
δ and σ defined as in Definition 5 meet the following inequality: γ−1βδ+Ψt,x+∫0t∑k=1∞hk−1t/Γαkt−ξαk−1βδ+Ψξ,xξhξdξ≤σ(5)
Vt,xtht and V0,x0+Ψt,xtht are both locally integrable on 0,TProof.
The analysis of Definition 2 shows that operatorItα is nondecreasing and linear. We apply the fractional integral operator Itα to the both sides of condition 2 and by Lemma 2, then we have(22)Vt,x−V0≤1Γα∫0tt−ξα−1hξVξ,xξdξ+Itαgt,x,where V0=V0,x0≤βx0≤βφ≤βδ. Obviously, we can further get by condition 3 the following:(23)Vt,x≤V0+Ψt,x+1Γα∫0tt−ξα−1hξVξ,xξdξ.
Then(24)Vt,xtht≤V0+Ψt,xtht+htΓα∫0tt−ξα−1hξVξ,xξdξ.
Letut=Vt,xtht, at=V0+Ψt,xtht, and by condition (5), we know that ut and at satisfy the conditions in Lemma 1. Hence, one can obtain(25)ut≤at+∫0t∑k=1∞hktΓαkt−ξαk−1aξdξ,which implies that(26)Vt,x≤V0+Ψt,x+∫0t∑k=1∞hk−1tΓαkt−ξαk−1V0+Ψξ,xξhξdξ.
According to the condition1, we can get(27)xt≤γ−1V0+Ψt,x+∫0t∑k=1∞hk−1tΓαkt−ξαk−1V0+Ψξ,xξhξdξ≤γ−1βδ+Ψt,x+∫0t∑k=1∞hk−1tΓαkt−ξαk−1βδ+Ψξ,xξhξdξ≤σ,which implies that system (21) has finite-time stability. The proof of this Lemma is completed.Theorem 2.
Suppose the validity ofH1–H3 and Tkt:Ctk,sk;ℝ⟶ℝ are continuous linear maps for t∈tk,sk, such that Mk=supTkt:t∈∪k=1ptk,sk, k=1,2,…,p, and there exist two positive real numbers δ and σ such that δ<σ and φPC≤δ, then fractional system (4) has finite-time stability if conditions (1)–(3) and (5) in Lemma 4 and(28)βM∗+Ψt,x+∫0T∑k=1∞hk−1tΓαkt−ξαk−1βM∗+Ψξ,xξhξdξ≤γσ,hold, where M∗=maxδ,λM¯⋅ϒT,δ, M¯=maxMk,k=1,2,…,p, ϒt,δ=maxϒkt,δ,k=1,2,…,p, and ϒkt,δ is defined in the following expression (49).Proof.
For∀t∈−τ,0, we have xPC=φPC≤δ<σ, and it is obvious that system (4) has finite-time stability on −τ,0.
As∀t∈0,t1, the proof process is similar to the proof in Lemma 4.
As∀t∈t1,s1, we also apply the fractional integral operator Itα on the both sides of condition (2), then we have(29)Vt,x−Vt1+≤1Γα∫t1+tt−ξα−1hξVξ,xξdξ+Itαgt,x,where Vt1+ is called the initial value on t1,s1, and(30)Vt1+=Vt1+,xt1+≤βxt1+PC=βϕ1t1+,T1t1+xt1−PC≤βλ1M1xt1−PC.
According to expression (14), for any t∈0,t1, one can obtain(31)xtPC≤1Γα∫0tt−ξα−1fξ,xξ,xξ−τξ+ΔAξ,xξPCdξ+φ0PC≤LfΓα∫0tt−ξα−1xξPC+xξ−τξPCdξ+ItαΔAt,xtPC+φ0PC≤φ0PC+MG+LfΓα∫0tt−ξα−1xξPC+xξ−τξPCdξ,where MG is defined in the same way as in Theorem 1. We refer to the idea in [23] and set ut=supθ∈−τ,txθPC, and ∀t∈0,t1, then xξPC≤uξ, xξ−τξPC≤uξ, and ∀ξ∈0,t. From (31), it follows (32)xtPC≤φ0PC+MG+2LfΓα∫0tt−ξα−1uξdξ.
Noting that for allθ∈0,t, we can get(33)xθPC≤φ0PC+MG+2LfΓα∫0θθ−ξα−1uξdξ≤φ0PC+MG+2LfΓα∫0tt−ξα−1uξdξ.
Hence, we obtain the following result:(34)ut=supθ∈−τ,txθPC≤maxsupθ∈−τ,0xθPC,supθ∈0,txθPC≤maxδ,φ0PC+MG+2LfΓα∫0tt−ξα−1uξdξ.
By the generalized Grönwall inequality in Lemma 1, we have(35)ut≤maxδ,φ0PC+MG1+∫0t∑k=1∞2LfkΓαkt−ξαk−1dξ,which implies that, for any t∈0,t1,(36)xtPC≤ut≤maxδ,φ0PC+MG1+∫0t∑k=1∞2LfkΓαkt−ξαk−1dξ≔ϒ1t,δ.
From (30) and (36), we can conclude that(37)Vt1+≤βλ1M1ϒ1t1−,δ.
By using the same idea in Lemma 4, we can easily get(38)xtPC≤γ−1βλ1M1ϒ1t1−,δ+Ψt,x+∫0T∑k=1∞hk−1tΓαkt−ξαk−1βλ1M1ϒ1t1−,δ+Ψξ,xξhξdξ≤σ,and we can see the fractional system (4) has finite-time stability on t1,s1 by the Definition 5.
As∀t∈s1,t2, we also apply the fractional integral operator Itα on the side of condition (2), then we have(39)Vt,x−Vs1≤1Γα∫s1tt−ξα−1hξVξ,xξdξ+Itαgt,x,where Vs1 is called the initial value, and(40)Vs1=Vs1,xs1≤βxs1PC=βϕ1s1,T1s1xt1−PC≤βλ1M1xt1−PC≤βλ1M1ϒ1t1−,δ.
Therefore, we easily obtain by Lemma 4 the following:(41)xtPC≤γ−1βλ1M1ϒ1t1−,δ+Ψt,x+∫​0T∑k=1∞hk−1tΓαkt−ξαk−1βλ1M1ϒ1t1−,δ+Ψξ,xξhξdξ≤σ,and we can see the fractional system (4) has finite-time stability on s1,t2 by the Definition 5.
As∀t∈t2,s2, we have the initial value:(42)Vt2+=Vt2+,xt2+≤βxt2+PC=βϕ2t2+,T2t2+xt2−PC≤βλ2M2xt2−PC.
By the Volterra integral equation (14), for all t∈s1,t2, we have(43)xtPC≤ϕ1s1,T1s1xt1−PC+LfΓα∫s1tt−ξα−1xξPC+xξ−τξPCdξ+ItαΔAξ,xξPC≤λ1M1ϒ1t1−,δ+MG+LfΓα∫s1tt−ξα−1xξPC+xξ−τξPCdξ.
Using the same approach in (36), one can get(44)xtPC≤maxδ,λ1M1ϒ1t1−,δ+MG1+∫0t∑k=1∞2LfkΓαkt−ξαk−1dξ≔ϒ2t,δ.
By (42) and (44), we can obtain(45)Vt2+≤βλ2M2ϒ2t2−,δ.
Therefore, we easily obtain by Lemma 4 the following:(46)xtPC≤γ−1βλ2M2ϒ2t2−,δ+Ψt,x+∫0T∑k=1∞hk−1tΓαkt−ξαk−1βλ2M2ϒ2t2−,δ+Ψξ,xξhξdξ≤σ,and we can see that the fractional system (4) has finite-time stability on t2,s2 by the Definition 5.
And so on and so forth, we get the following results on the other intervals. Ast∈tk,sk, k=3,4,…,p, we have(47)xtPC≤γ−1βλkMkϒktk−,δ+Ψt,x+∫0T∑k=1∞hk−1tΓαkt−ξαk−1βλkMkϒktk−,δ+Ψξ,xξhξdξ≤σ,which implies that the fractional system (1.4) has finite-time stability on tk,sk, k=3,4,…,p by the Definition 5.
Ast∈sk,tk+1, k=2,3,…,p, we have(48)xtPC≤γ−1βλkMkϒktk−,δ+Ψt,x+∫0T∑k=1∞hk−1tΓαkt−ξαk−1βλkMkϒktk−,δ+Ψξ,xξhξdξ≤σ,(49)ϒkt,δ=maxδ,φ0PC+MG1+∫0t∑k=1∞2LfkΓαkt−ξαk−1dξ,k=1,maxδ,λ1M1ϒ1t1−,δ+MG1+∫0t∑k=1∞2LfkΓαkt−ξαk−1dξ,k=2,…maxδ,λp−1Mp−1ϒp−1tp−1−,δ+MG1+∫0t∑k=1∞2LfkΓαkt−ξαk−1dξ,k=p.
Therefore, we can see that the fractional system (4) has finite-time stability on tk,sk by the Definition 5. The proof of this theorem is completed.Remark 2.
System (4) will be an instantaneous impulsive differential equation under tk=sk, k=1,2,…,p, then the finite-time stability is affected by the number of impulses, which has been a very important and meaningful research topic, and investigated by some experts [27–29]. The results in this literature are new and can improve some referred conclusions.


## 4. Examples

In this section, we will present the following two examples to illustrate our main results.Example 1.
Assume thatα=0.5, and let 0,T=0,10, τt=2cost∈0,2, f=0.1⋅sinxt+cosxt−2, ϕkt,x=1/5sinxt, and Tktxtk−=∫tkt1/20sinxξdξ. Let e≈2.718 be a natural constant, φt=e−t, t∈−2,0, then φPC≈7.39, λ=0.2, Lf=0.1, Mk,r=0.5, and M=1, where k=1,2,…,p. Consider δ=7.40, and by Mathematica software, we have(50)∫0T∑k=1∞2LfkΓαkT−ξαk−1dξ=1.43.
To make sure that inequality (20) is true, and δ<σ, we only need to make MG not greater than 2.05, which is easy to be done. Let ΔAt,x=0.1cost+xt, ∀t∈−2,10, and MG=1.85. All conditions in Theorem 1 are satisfied, and system (4) has finite-time stability on −2,10.Example 2.
All the conditions are the same as in above example unless otherwise stated. Let the Lyapunov function beVt,x=x, then one can obtain(51)DCtαVt,x=0.1⋅sinxt+cosxt−2+0.1cost+xt≤0.1⋅sinxt+0.1+0.1cost+xt.
Assume thatht=0.1, gt,x=0.1+0.1cost+xt. It is obvious that(52)DCtαVt,x≤htVt,x+gt,x.
Letγxt=x/2, βxt=3x/2, and M¯=0.1. By calculating, we can choose Ψt,x=0.11t1/2+0.11t3/2+0.11t1/2cosx≤4.17. By Mathematica software, we have(53)βλM¯⋅ϒT,δ+Ψt,x+∫0T∑k=1∞hk−1TΓαkT−ξαk−1βλM¯⋅ϒT,δ+Ψξ,xξhξdξ=6.53,and we choose σ≥12.70. All the conditions in Theorem 2 can be verified. Hence, the fractional system (4) has finite-time stability.Remark 3.
Since there are few papers researching the finite-time stability of solutions for the uncertain fractional-order nonlinear differential equation involving time-varying delays and noninstantaneous impulses, one can see that all the results in [21, 22, 24, 25, 27–29] cannot directly be applicable to the examples mentioned above to obtain the results. This implies that the results in this paper are essentially new.


## 5. Conclusions

In this paper, we are concerned with a class of Caputo fractional-order differential equations. The addressed equation has noninstantaneous impulsive effects, which are quite different from the related references discussed in the literature [21, 22]. The nonlinear fractional-order differential system studied in the present paper is more generalized and more practical. By the definition of finite-time stability, the easily verifiable sufficient conditions have been provided to determine the stability conclusions. Finally, two typical numerical examples of the stability have been presented at the end of this paper to illustrate the effectiveness and feasibility of the proposed criterion. Consequently, this paper shows theoretically and numerically that some related references known in the literature can be enriched and complemented.An interesting extension of our study would be to discuss finite-time stability for the fractional differential equations with discontinuous right-hand side. This topic will be the subject of a forthcoming paper.



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