# Impulsive Exponential Stabilization of Functional Differential Systems with Infinite Delay

Authors: Xiaoli Sun; Xiaodi Li
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

By using the Razumikhin technique and Lyapunov functions, we investigated the impulsive exponential stabilization of functional differential systems with infinite delay. A new result on the exponential stabilization by impulses is gained. Our result shows that impulses can make unstable systems stable. A numerical example is given to illustrate the feasibility of the result.


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## Body

## 1. Introduction

Recently, there are many results of impulsive stability for delay systems as impulses can make unstable systems stable and stable systems unstable after impulse effects; see [1–21] and references therein. The problem of stabilizing the solutions by imposing proper impulsive control for delayed system now attracts more and more authors' attentions; see [22–26]. For example, in [22, 26], the authors have investigated impulsive stabilization of second-order differential equations with finite delay. The main tools used are Lyapunov functionals, stability theory, and control by impulses. In [23], by employing the Razumikhin technique and Lyapunov functions, several global exponential stability criteria are established for general impulsive differential equations with finite delay. However, not much has been developed in the direction of the stabilization theory of impulsive functional differential systems, especially for infinite delays of impulsive functional differential systems. This is due to some theoretical and technical difficulties; see [14, 16–21, 24]. In [24], Luo and Shen studied impulsive stabilization of functional differential equations with infinite delay. By using Lyapunov functions and Razumikhin techniques, some Razumikhin type theorems on uniform asymptotical stability are obtained. However, to the best of the authors' knowledge, there is little work on the impulsive exponential stabilization of functional differential systems with infinite delay.The aim of this work is to establish a criterion on the impulsive exponential stabilization of functional differential systems with infinite delay by using Lyapunov functions and the Razumikhin technique. Our result shows that functional differential equations with infinite delay may be exponentially stabilized by impulses. Moreover, to some degree, the result we obtained is less conservative and more feasible than that given in [23].This paper is organized as follows. In Section2, we introduce some notations and definitions. Section 3 is devoted to the main results, and a numerical example is given to demonstrate the effectiveness of our result. In the last section, concluding remarks are given in Section 4.


## 2. Preliminaries

Letℝ denote the set of real numbers,ℝ+ the set of nonnegative real numbers, and ℝn the n-dimensional real space equipped with the Euclidean norm ∥·∥. For any t≥t0≥0>-α≥-∞, let f(t,x(s)) where s∈[t-α,t] or f(t,x(·)) be a Volterra type functional. In the case when α=+∞, the interval [t-α,t] is understood to be replaced by (-∞,t].Consider the following impulsive functional differential systems:(2.1)x′(t)=f(t,x(·)),t≥t0,t≠tk,Δx|t=tk=Ik(tk,xtk-),k=1,2…,
where the impulse times tk satisfy 0≤t0<t1<⋯<tk<⋯,limk→+∞tk=+∞, and x′ denotes the right-hand derivative of x. f∈C([tk-1,tk)×ℂ,ℝn),f(t,0)=0. ℂ is an open set in PC([-α,0],ℝn), where PC([-α,0],ℝn)={ψ:[-α,0]→ℝn is continuous everywhere except at finite number of points tk, at wh{ich ψ(tk+) and ψ(tk-) exist and ψ(tk+)=ψ(tk)}. Define PCB(t)={x∈ℂ:xisbounded}. For ψ∈PCB(t), the norm of ψ is defined by ∥ψ∥=sup-α≤θ≤0|ψ(θ)|. For any σ≥0, let PCBδ(σ)={ψ∈PCB(σ):∥ψ∥<δ}. Let K1={a∈C(ℝ+,ℝ+)∣a(0)=0 and a(t)>0 for t>0 and a is strictly increasing in t}.For eachk=1,2…,Ik(t,x)∈C([t0,∞)×ℝn,ℝn),Ik(t,0)=0, and for any ρ>0, there exists a ρ1>0(0<ρ1<ρ) such that x∈S(ρ1) implies that x+Ik(tk,x)∈S(ρ), where S(ρ)={x:∥x∥<ρ,x∈ℝn}.For any givenσ≥t0, system (2.1) is supplemented with initial conditions of the form
(2.2)xσ=ϕ,
where ϕ∈PC([-α,0],ℝn).In this paper, we assume that the solution for the initial problem (2.1)-(2.2) does exist and is unique which will be written in the form x(t,σ,ϕ); see [1, 4, 13]. Since f(t,0)=0,I(t,0)=0,k=1,2,…, then x(t)=0 is a solution of (2.1)-(2.2), which is called the trivial solution. In this paper, we always assume that the solution x(t,σ,ϕ) of (2.1)-(2.2) can be continued to ∞ from the right of σ.We introduce some definitions as follows.Definition 2.1 (see [1]).
The functionV:[-α,∞)×ℂ→ℝ+ belongs to class v0 if
(i)
V is continuous on each of the sets [tk-1,tk)×ℂ and lim(t,φ)→(tk-,ψ)V(t,φ)=V(tk-,ψ) exists;(ii)
V(t,x) is locally Lipschitzian in x and V(t,0)≡0.Definition 2.2 (see [1]).
LetV∈v0, for any (t,ψ)∈[tk-1,tk)×ℂ, the upper right-hand Dini derivative of V(t,x) along the solution of (2.1)-(2.2) is defined by
(2.3)D+V(t,ψ(0))=lim suph→0+{V(t+h,ψ(0)+hf(t,ψ))-V(t,ψ(0))}h.Definition 2.3 (see [1]).
Assume thatx(t)=x(t,σ,ϕ) is the solution of (2.1)-(2.2) through (σ,ϕ). Then the trivial solution of (2.1)-(2.2) is said to be exponentially stable if for any ε>0,σ≥t0, there exist constants λ>0 and δ=δ(ε)>0 such that ϕ∈PCBδ(σ) implies ∥x(t)∥<ε·e-λ(t-σ),t≥σ.


## 3. Main Results

In this section, we shall develop Lyapunov-Razumikhin methods and establish some theorems which provide sufficient conditions for exponential stability of the trivial solution of (2.1)-(2.2).Theorem 3.1.
Assume that there exist functionsV∈v0,w∈K1 and positive constants  p,c1,η,λ,βk≥0,k∈ℤ+ and  γ∈(1,M⋆],M⋆>1 such that the following conditions hold:
(i)
c1∥x∥η≤V(t,x)≤w(∥x∥),(t,x)∈[-α,∞)×S(ρ),(ii)
for anyσ≥t0 and ψ∈PC([-α,0],S(ρ)), if γeλ(t-σ)V(t,ψ(0))≥V(t+θ,ψ(θ)), -α≤θ≤0,t≠tk,k∈ℤ+, then
(3.1)D+V(t,ψ(0))≤pV(t,ψ(0)),(iii)
for all(tk,ψ)∈R+×PC([-α,0],S(ρ1)),V(tk,ψ(0)+Ik(tk,ψ))≤βkV(tk-,ψ(0)),k∈ℤ+, where βk satisfies ∏k=1∞max{βkM⋆,1}<∞,(iv)
tk-tk-1<lnγ/(p+λ),k∈ℤ+,then the trivial solution of (2.1)-(2.2) is exponentially stable with the approximate exponential convergence rate λ/η.Proof.
From condition (iii), there exists constantM>0 such that
(3.2)∏k=1∞max{βkM⋆,1}≤M.
For any ε∈(0,ρ1), we choose δ=δ(ε)>0 such that w(δ)≤(M⋆M)-1min{c1εη,ε}. Then, for any σ≥t0, let x(t)=x(t,σ,ϕ) be a solution of (2.1)-(2.2) through (σ,ϕ). We shall show for any ϕ∈PCBδ(σ)(3.3)∥x(t)∥<ε·e-(λ/η)(t-σ),t≥σ.
For convenience, suppose V(t)=V(t,x(t)) and
(3.4)Ṽ(σ)=max{sup-α≤θ≤0V(σ+θ,ψ(θ)),M⋆V(σ)}.
Since ϕ∈PCBδ(σ), we have Ṽ(σ)≤M⋆w(δ).
Suppose thatσ∈[tl-1,tl),l∈ℤ+. Next we prove for t∈[σ,tl)∪[tk,tk+1),k≥l,
(3.5)V(t)≤Ṽ(σ)(∏s=lkmax{βsM⋆,1})e-λ(t-σ),t≥σ.
First, it is clear that for t∈[σ-α,σ](3.6)c1∥x(t)∥η≤V(t)≤Ṽ(σ)≤M⋆w(δ)≤M-1min{c1εη,ε}<c1εη,
which implies ∥x(t)∥<ε<ρ1,t∈[σ-α,σ].
We next claim that (3.5) holds for all t∈[σ,tl), that is,
(3.7)V(t)≤Ṽ(σ)e-λ(t-σ),t∈[σ,tl).
In order to do this, let
(3.8)Γ(t)={V(t)eλ(t-σ),t≥σ,V(t),σ-α≤t≤σ,
then it is obvious that Γ(t)≥V(t) for all t≥σ-α.
Next we proveΓ(t)≤Ṽ(σ) for t∈[σ,tl), which implies that (3.5) holds for t∈[σ,tl). Suppose that this assertion is false, then there exists some t∈[σ,tl) such that Γ(t)>Ṽ(σ).
Let(3.9)t⋆=inf{t∈[σ,tl)∣Γ(t)≥Ṽ(σ)},
then in view of Γ(σ)=V(σ)≤M⋆-1Ṽ(σ)<Ṽ(σ), we get
(3.10)t⋆∈(σ,tl),Γ(t⋆)=Ṽ(σ),Γ(t)<Ṽ(σ),t∈[σ,t⋆).
Considering (3.6), we also obtain
(3.11)V(t)≤Γ(t)<Ṽ(σ)∀t∈[σ-α,t⋆).
On the other hand, considering γ∈(1,M⋆], we get
(3.12)Γ(t⋆)=Ṽ(σ)>1γṼ(σ),Γ(σ)=V(σ)<1M⋆Ṽ(σ)≤1γṼ(σ).
So we can define
(3.13)t⋆⋆=sup{t∈[σ,t⋆]∣Γ(t)≤1γṼ(σ)},
which implies that
(3.14)t⋆⋆∈[σ,t⋆),1γṼ(σ)=Γ(t⋆⋆),1γṼ(σ)<Γ(t),t∈(t⋆⋆,t⋆].
It follows from (3.11) that
(3.15)γV(t)eλ(t-σ)=γΓ(t)≥Ṽ(σ)≥V(s),σ-α≤s≤t,t∈[t⋆⋆,t⋆].
Using condition (ii), the inequality D+V(t)≤pV(t) holds for all t∈[t⋆⋆,t⋆]. Hence, we obtain for t∈[t⋆⋆,t⋆](3.16)D+Γ(t)=D+V(t)eλ(t-σ)+λV(t)eλ(t-σ)=eλ(t-σ)(D+V(t)+λV(t))≤eλ(t-σ)(pV(t)+λV(t))=V(t)eλ(t-σ)(p+λ)≤(p+λ)Γ(t).
From (iv), we define τ=̇maxk∈ℤ+{tk-tk-1}, then τ(p+λ)<lnγ.
Thus, we have(3.17)∫t⋆⋆t⋆dΓ(s)Γ(s)=∫Γ(t⋆⋆)Γ(t⋆)dss=∫(1/γ)Ṽ(σ)Ṽ(σ)dss≥lnγ>τp+λτ.
However,
(3.18)∫Γ(t⋆⋆)Γ(t⋆)dss≤∫t⋆⋆t⋆(p+λ)ds≤∫t⋆⋆t⋆⋆+τ(p+λ)ds=τp+λτ.
This is a contradiction. Therefore, we obtain Γ(t)≤Ṽ(σ),t∈[σ,tl), which implies that (3.5) holds for all t∈[σ,tl).
Meanwhile, we get fort∈[σ,tl)(3.19)c1∥x∥η≤V(t)≤Ṽ(σ)e-λ(t-σ)≤Ṽ(σ)<M⋆w(δ)≤M-1min{c1εη,ε}<c1εη,
which implies that x(tl-)∈S(ρ1),x(tl)∈S(ρ).
Furthermore, note that(3.20)V(tl)≤βlV(tl-)≤βlṼ(σ)e-λ(tl-σ),
we next prove
(3.21)V(t)≤max{βlM⋆,1}Ṽ(σ)e-λ(t-σ),t∈[tl,tl+1),
which is equal to prove
(3.22)Γ(t)≤max{βlM⋆,1}Ṽ(σ),t∈[tl,tl+1).
Suppose that this assertion is not true, then there exists some t∈[tl,tl+1) such that Γ(t)>max{βlM⋆,1}Ṽ(σ).
Let(3.23)t*=inf{t∈[tl,tl+1)∣Γ(t)≥max{βlM⋆,1}Ṽ(σ)}.
Then we know
(3.24)t*∈(tl,tl+1),Γ(t*)=max{βlM⋆,1}Ṽ(σ),Γ(t)<max{βlM⋆,1}Ṽ(σ),t∈[tl,t*).
Also, in view of the fact Γ(t)<Ṽ(σ) for t∈[σ-α,tl), we obtain
(3.25)Γ(t)<max{βlM⋆,1}Ṽ(σ)∀t∈[σ-α,t*).
Note that γ∈(1,M⋆], then we have
(3.26)Γ(t*)=max{βlM⋆,1}Ṽ(σ)>1γmax{βlM⋆,1}Ṽ(σ),(3.27)Γ(tl)≤βlṼ(σ)e-λ(tl-σ)<βlṼ(σ)≤1γβlM⋆Ṽ(σ)≤1γmax{βlM⋆,1}Ṽ(σ).
So we can define
(3.28)t**=sup{t∈[tl,t*]∣Γ(t)≤1γmax{βlM⋆,1}Ṽ(σ)}.
Then, we obtain
(3.29)t**∈[tl,t*),Γ(t**)=1γmax{βlM⋆,1}Ṽ(σ),Γ(t)>1γmax{βlM⋆,1}Ṽ(σ),t∈(t**,t*].
Thus, combining (3.25), we get
(3.30)γV(t)eλ(t-σ)=γΓ(t)≥max{βlM⋆,1}Ṽ(σ)≥Γ(s)≥V(s),σ-α≤s≤t,t∈[t**,t*].
Similarly, by assumptions (ii), (iv), as the proof of (3.16), we can obtain
(3.31)D+Γ(t)≤(p+λ)Γ(t),t∈[t**,t*].
Consequently, we have
(3.32)∫t**t*dΓ(s)Γ(s)=∫Γ(t**)Γ(t*)dss=∫(1/γ)max{βlM⋆,1}Ṽ(σ)max{βlM⋆,1}Ṽ(σ)dss≥lnγ>pτ+λτ,
where τ=̇maxk∈ℤ+{tk-tk-1}.
However, we note(3.33)∫t**t*dΓ(s)Γ(s)=∫Γ(t**)Γ(t*)dss≤∫t**t**+τ(p+λ)ds=pτ+λτ,
which is a contradiction. So (3.21) holds.
Note that(3.34)V(tl+1)≤βl+1V(tl+1-)≤βl+1max{βlM⋆,1}Ṽ(σ)e-λ(tl+1-σ).
Similarly, we can prove
(3.35)V(t)≤max{βl+1M⋆·max{βlM⋆,1},max{βlM⋆,1}}Ṽ(σ)e-λ(t-σ),t∈[tl+1,tl+2),
that is,
(3.36)V(t)≤max{βlM⋆,1}max{βl+1M⋆,1}Ṽ(σ)e-λ(t-σ),t∈[tl+1,tl+2).
By simple induction hypothesis, we may prove that for t∈[σ,tl)∪[tk,tk+1),k≥l,
(3.37)V(t)≤Ṽ(σ)(∏s=lkmax{βsM⋆,1})e-λ(t-σ),t≥σ.
Hence, (3.5) holds.
Under the help of conditions (i), (iii), and the definition ofṼ(σ), we arrive at
(3.38)c1∥x∥η≤V(t)≤Ṽ(σ)(∏s=lkmax{βsM⋆,1})e-λ(t-σ)≤w(δ)M⋆(∏s=lkmax{βsM⋆,1})e-λ(t-σ)≤w(δ)M⋆Me-λ(t-σ)≤min{c1εη,ε}e-λ(t-σ)≤c1εηe-λ(t-σ),t≥σ,
which implies that
(3.39)∥x(t)∥<ε·e-(λ/η)(t-σ),t≥σ.
Therefore, (3.3) holds. The proof of Theorem 3.1 is therefore complete.Remark 3.2.
In Theorem3.1, the impulsive condition is not straightforward to be verified. To simplify the result, we introduce some more testable conditions.Corollary 3.3.
Assume that there exist functionsV∈v0,w∈K1, and positive constants M⋆,p,c1,η,λ,βk≥0,k∈ℤ+ and  γ∈(1,eM⋆] such that conditions (i), (ii), (iv) in Theorem 3.1 hold, moreover, suppose that
(iii′)
for all (tk,ψ)∈R+×PC([-α,0],S(ρ1)),V(tk,ψ(0)+Ik(tk,ψ))≤βkV(tk-,ψ(0)),k∈ℤ+, where βk≥e-M⋆ and there exists a constant M>0 such that
(3.40)∑k=lnlnβk+(n-l+1)M⋆<M∀n≥l,l∈ℤ+,
then the trivial solution of (2.1)-(2.2) is exponentially stable with the approximate exponential convergence rate λ/η.Especially, letγ=M⋆,βk=1/γ>0 in Theorem 3.1, then we can obtain the following results whose proof is similar and thus omitted.Corollary 3.4.
Assume that there exist functionsV∈v0,w∈K1 and constants p,c1,η,γ>1,λ>0 such that the following conditions hold:
(i)
c1∥x∥η≤V(t,x)≤w(∥x∥),(t,x)∈[-α,∞)×S(ρ);(ii)
for anyσ≥t0 and ψ∈PC([-α,0],S(ρ)), if γeλ(t-σ)V(t,ψ(0))≥V(t+θ,ψ(θ)), -α≤θ≤0,t≠tk,k∈ℤ+, then D+V(t,ψ(0))≤pV(t,ψ(0));(iii)
for all(tk,ψ)∈R+×PC([-α,0],S(ρ1)),V(tk,ψ(0)+Ik(tk,ψ))≤(1/γ)V(tk-,ψ(0)),k∈ℤ+;(iv)
tk-tk-1<lnγ/(p+λ),k∈ℤ+,then the trivial solution of (2.1)-(2.2) is exponentially stable with the approximate exponential convergence rate λ/η.Remark 3.5.
Ifα<+∞,then the exponential stability of system (2.1)-(2.2) has been investigated extensively in [23] under the following assumptions (where the definition of p,α⋆,c1,c,λ,τ,dk, see [23]):
(i)
c1∥x∥p≤V(t,x)≤c2∥x∥p, for any t∈ℝ+ and x∈ℝn;(ii)
D+V(t,φ(0))≤cV(t,φ(0)), for all t∈[tk-1,tk),k∈ℤ+, whenever qV(t,φ(0))≤V(t+s,φ(s)) for s∈[-τ,0], where q≥max{eα⋆c,e2λα⋆} is a constant;(iii)
V(tk,φ(0)+Ik(tk,φ))≤dkV(tk,φ(0)), where dk>0,∀k∈ℤ+, are constants;(iv)
τ≤tk-tk-1≤α⋆ and ln(dk)+(λ+c)α⋆<-λ(tk+1-tk),k∈ℤ+.It is easily seen that these conditions are more restrictive than ones given in Theorem 3.1. For example, let dk=d(constant), tk+1-tk=1, then it is necessary that condition ln(d)<-(λ+c)α⋆-λ holds (see [23]). Note in our Corollary 3.4, we only require that ln(d)<-λ-c, where here γ=1/d,p=c in Theorem 3.1. Moreover, we see that condition tk+1-tk>τ is not necessary in our results, which are milder than the restrictions in [23].Remark 3.6.
To author's knowledge, there is little work on exponential stability of impulsive differential systems with infinite delay withD+V≥0. Our result allows for significant increases in V between impulses as long as the decreases of V at impulse times balance it properly, which shows that differential equations with infinite delay may be exponentially stabilized by impulses.In the following, an example is given to demonstrate the effectiveness of our result.Example 3.7.
Consider the following equations:(3.41)x′(t)=(15+118e-0.1t)x(t)-115∫-∞0es-0.1tx(t+s)ds,t≥0,t≠k,x(k)=1ex(k-),k=1,2…,xσ=ϕ.
Choose V(t)=x2, we easily observe η=2,c1=1,w(s)=s2,tk-tk-1=̇τ=1. Let γ=e,λ=0.2,p=2/5+2/18+(2e)/15. It is easy to check that τ(p+λ)≈0.37<1=lnγ.
Suppose thatσ=t0=0, then in view of Corollary 3.3, γeλ(t-σ)V(t,ψ(0))≥V(t+θ,ψ(θ)),θ≤0, implies that ee0.1t|x(t)|>|x(t+θ)|,θ≤0.
Hence,(3.42)D+V(t,x(·))=2x(t){(15+118e-0.1t)x(t)-115∫-∞0es-0.1tx(t+s)ds}≤2x2(t){(15+118e-0.1t)+115∫-∞0es-0.1tee0.1tds}≤2x2(t){(15+118e-0.1t)+e15∫-∞0esds}≤2x2(t){15+118+e15}≤pV(t).
Besides, V(k)=x2(k)=(1/e)x2(k-)=(1/γ)V(k-).
By Corollary3.3, the trivial solution of (3.41) is exponentially stable with the approximate exponential convergence rate 0.1. Taking initial values: ϕ=10e0.2t,t≤0. The numerical simulations are illustrated in Figures 1 and 2.Figure 1Figure 2Remark 3.8.
From above example, we see that the trivial solution of system (3.41) without impulses is unstable. However, after impulsive control, the trivial solution becomes exponentially stable. This implies that differential systems with infinite delay may be exponentially stabilized by impulses and impulses can make unstable systems stable.


## 4. Conclusions

In this paper, we study exponential stability of impulsive differential systems with infinite delay. A new sufficient criterion ensuring exponential stability is gained by using the Razumikhintechnique and Lyapunov functions. Our result shows that differential equations with infinite delay may be exponentially stabilized by impulses. Also, the result here (withα<+∞) is discussed from the point of view of its comparison with the earlier result. An example is given to illustrate the feasibility of the result.



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