# The Asymptotic Behavior for a Class of Impulsive Delay Differential Equations

Authors: Zhichun Yang
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

This paper is concerned with asymptotical behavior for a class of impulsive delay differential equations. The new criteria for determining attracting sets and attracting basin of the impulsive system are obtained by developing the properties of quasi-invariant sets. Examples and numerical simulations are given to illustrate the effectiveness of our results. In addition, we show that the impulsive effects may play a key role to these asymptotical properties even though the solutions of corresponding nonimpulsive systems are unbounded.


---

## Body

## 1. Introduction

Impulsive delay differential equations have attracted increasing interests since time delays and impulsive effects commonly exist in many fields such as population dynamics, automatic control, drug administration, and communication networks [1–4]. In past two decades, its asymptotical behaviors such as stability and attractivity of the equilibrium point or periodical solutions have been deeply studied for impulsive functional differential equations (see, [5–18]).However, under impulsive perturbation, the solutions may not be attracted to an equilibrium point or periodical trajectory but to some bounded region. In this case, it is interesting to investigate the attracting set and attracting basin, that is, the region attracting the solutions and the range in which initial values vary when remaining the attractivity for impulsive delay differential equations. In [19], Xu and Yang first give the method to estimate global attracting set and invariant set for impulsive delayed systems by developing delayed differential inequalities. The techniques are further developed to study global attractivity for some complex impulsive systems such as impulsive neutral differential equations [20, 21] and impulsive stochastic systems [22]. But the techniques and methods given in the existing publications are invalid for determining locally attracting set and attracting basin for impulsive delay differential equations.In this paper, our objective is to mainly discuss the asymptotical behavior on (locally) attracting set and its attracting basin for a class of impulsive delay differential equations. Based on the quasi-invariant properties, we estimate the existence range of attracting set and attracting basin of the impulsive delay systems by solving algebraic equations and employing differential inequality technique. Examples are given to illustrate the effectiveness of our method and show that the asymptotic behavior of the impulsive systems may be different from one of the corresponding continuous systems.


## 2. Preliminaries

LetN be the set of all positive integers, Rn the space of n-dimensional real column vectors, and Rm×n the set of m×n real matrices. For A,B∈Rm×n or A,B∈Rn, A≥B(A≤B,A>B,A<B) means that each pair of corresponding elements of A and B satisfies the inequality “≥(≤,>,<).” R+n={x∈Rn∣x≥0}, E=(1,1,…,1)T∈Rn, and I denotes an n×n unit matrix.Letτ>0 and t0<t1<t2<⋯ be the fixed points with limk→∞tk=∞ (called impulsive moments).C
[
X
,
Y
] denotes the space of continuous mappings from the topological space X to the topological space Y. Let C=ΔC[[-τ,0],Rn] especially.Morever,PC=Δ{ϕ:[-τ,0]→Rn∣ϕ(t+)=ϕ(t) for t∈[-τ,0), ϕ(t-) exists for t∈(-τ,0], ϕ(t-)=ϕ(t) for all but at most a finite number of points t∈(-τ,0]}. PC is a space of piecewise right-hand continuous functions which is a nature extension of the phrase space C.We definePC[[t0,∞),Rn]=Δ{ψ:[t0,∞)→Rn∣ψ(t) is continuous at t≠tk, ψ(tk+) and ψ(tk-) exist, ψ(tk)=ψ(tk+), for k∈N}.Forx∈Rn, A∈Rn×n, ϕ∈C or ϕ∈PC, we define
(1)[x]+=(|x1|,|x2|,…,|xn|)T,[A]+=(|aij|)n×n,[ϕ]τ+=(∥ϕ1∥τ,∥ϕ2∥τ,…,∥ϕn∥τ),
where ∥ϕi∥τ=sups∈[-τ,0]∥ϕi(s)∥ and ∥·∥ is an norm in Rn.In this paper, we will consider a impulsive delay differential equations:(2)x˙(t)=Ax(t)+f(t,xt),t≠tk,t≥t0,Δx=Bx(tk-)+Ik(tk-,x(tk-)),k∈N,
where x˙(t) denotes the right-hand derivative of x(t), Δx=x(tk+)-x(tk-), x(tk)=x(tk+), A = diag{a1,a2,…,an}, B = diag{b1,b2,…,bn}, f∈C[[tk-1,tk)×PC,Rn], and the limit lim(t,ϕ)→(tk-,φ)f(t,ϕ)=f(tk-,φ) exists, Ik∈C[[t0,∞)×Rn,Rn], and xt∈PC is defined by xt(s)=x(t+s),s∈[-τ,0].A functionx(t):[t0-τ,∞)→Rn is called to be a solution of (2) through (t0,ϕ), if x(t)∈PC[[t0,∞),Rn] as t≥t0, and satisfies (2) with the initial condition
(3)x(t0+s)=ϕ(s),s∈[-τ,0],ϕ∈PC.
Throughout the paper, we always assume that for any ϕ∈PC, system (2) has at least one solution through (t0,ϕ), denoted by x(t,t0,ϕ) or xt(t0,ϕ) (simply x(t) and xt if no confusion should occur), where xt(t0,ϕ)=x(t+s,t0,ϕ)∈PC, s∈[-τ,0].In this paper, we need the following definitions involving attracting set, attracting basin, the quasi-invariant set of impulsive systems, and monotonous vector functions.Definition 1.
The setS⊂PC is called to be an attracting set of (2), and D⊂PC is called an attraction basin of S, if for any initial value ϕ∈D, the solution xt(t0,ϕ) converges to S as t→+∞. That is,
(4)dist(xt(t0,ϕ),S)⟶0,ast⟶+∞,
where dist(φ,S)=infψ∈S dist(φ,ψ), dist(φ,ψ)=sups∈[-τ,0]∥φ(s)-ψ(s)∥, for φ∈PC.Definition 2.
The setD⊂PC is called to be a positive quasi-invariant set of (2), if there is a positive diagonal matrix L = diag{li} such that for any initial value ϕ∈D, the solutions xt(t0,ϕ) satisfy Lxt(t0,ϕ)∈D, for t≥t0. When L=I(identity matrix) especially, the set D is called positively invariant.Definition 3.
LetΩ⊂Rn. The vector function F(x):Ω→Rn is called to be monotonically nondecreasing in Ω, if for any x′,x′′∈Ω, x′≤x′′ implies F(x′)≤F(x′′).


## 3. Main Results

In this paper, we always make the following assumptions.(H1)
There exist nonnegative constantsθ,ϱ such that 0<θ≤tk-tk-1≤ϱ, for k∈N.(H2)
[f(t,φ)]+≤p([φ]τ+) for t≥t0 and φ∈PC, where the vector function p(·):R+n→R+n is continuous and monotonically nondecreasing in R+n.(H3)
[Ik(t,x)]+≤q([x]+), for t≥t0, k∈N and x∈Rn, where the vector function q(·):R+n→R+n is continuous and monotonically nondecreasing in R+n.To obtain attractivity, we first give the quasi-invariant properties of (2).Theorem 4.
Assume that in addition to(H1)–(H3), there is a vector z*≥0 such that
(5)p(Mz*)+W[I-e-Wθ]-1q(Mz*)-Wz*<0,
where W = diag{w1,…,wn}, M = diag{m1,…,mn}, wi>0,mi≥1,i=1,2,…,n, are defined by
(6)wi={-ai-ln|1+bi|ϱ,if0<|1+bi|<1,-ai-ln|1+bi|θ,if|1+bi|≥1,mi={1|1+bi|,if0<|1+bi|<1,1,if|1+bi|≥1.
Then, the set D={ϕ∈PC∣[ϕ]τ+≤z*} is a positive quasi-invariant set of (2). When M=I especially, D is a positive invariant set of (2).Proof.
Letx(t)=x(t,t0,ϕ) be a solution of (2) through (t0,ϕ). It is easily verified that the following formula for the variation of parameters is valid:
(7)x(t)=K(t,t0)ϕ(0)+∫t0t‍K(t,s)f(s,xs)ds+∑t0<tk≤t‍K(t,tk)Ik(tk-,x(tk-)),t≥t0,
where K(t,s) is the Cauchy matrix of linear impulsive system
(8){y˙(t)=Ay(t),t≠tk,Δy(tk+)=By(tk-),k∈N.
According to the representation of the Cauchy matrix (see page 74 [2]),
(9)K(t,s)=eA(t-s)∏s<tk≤t‍(I+B),t≥s≥t0.
Since 0<θ≤tk-tk-1≤ϱ, for k∈N, we obtain the following estimate:
(10)∏s<tk≤t‍|1+bi|≤{|1+bi|((t-s)/ϱ)-1=1|1+bi|e(ln∣1+bi∣/ϱ)(t-s),if0<|1+bi|<1,|1+bi|(t-s)/θ=e(ln∣1+bi∣/θ)(t-s),if|1+bi|≥1.
In terms of the definition of M and W,
(11)[K(t,s)]+≤Me-W(t-s),t≥s≥t0.
By (7) and (11) and the assumptions (H2) and (H3), then
(12)[x(t)]+≤Me-W(t-t0)[ϕ]τ++M∫t0t‍e-W(t-s)p([xs]τ+)ds[x(t)]+≤+M∑t0<tk≤t‍e-W(t-tk)q([x(tk-)]+),t≥t0.
Since tk-tk-1≥θ>0 and W = diag{w1,…,wn}>0, we have
(13)∑t0<tk≤t‍e-W(t-tk)=∑t0<tk≤t‍e-W(t-tk)[I-e-Wθ][I-e-Wθ]-1∑t0<tk≤t‍e-W(t-tk)≤∑t0<tk≤t‍e-W(t-tk)[I-e-W(tk-tk-1)][I-e-Wθ]-1∑t0<tk≤t‍e-W(t-tk)=∑t0<tk≤t‍[e-W(t-tk)-e-W(t-tk-1)][I-e-Wθ]-1∑t0<tk≤t‍e-W(t-tk)≤[I-e-W(t-t0)][I-e-Wθ]-1.
From the strict inequality (5), there is an enough small number ε>0 such that
(14)p(Mz)+W[I-e-Wθ]-1q(Mz)-Wz<0,z=Δz*+εE>0.
In the following, we will prove that [ϕ]τ+<z implies
(15)[x(t)]+=[x(t,t0,ϕ)]+<Mz,t≥t0.
Otherwise, from the piecewise continuity of x(t), there must be an integer i and t*>t0 such that
(16)|xi(t*)|≥mizi,(17)[x(t)]+≤Mz,t0≤t<t*.
By using (12), (13), (14), (17), W>0, and the monotonicity of p(·),q(·), we can get
(18)[x(t*)]+≤e-W(t*-t0)M[ϕ]τ++M∫t0t*‍e-W(t*-s)p(Mz)ds[x(t*)]+≤+M∑t0<tk≤t*‍e-W(t*-tk)q(Mz)[x(t*)]+<e-W(t*-t0)Mz+M(I-e-W(t*-t0))W-1p(Mz)[x(t*)]+≤+M(I-e-W(t*-t0))[I-e-Wθ]-1q(Mz)[x(t*)]+=e-W(t*-t0)MW-1[x(t*)]+≤×[Wz-p(Mz)-W[I-e-Wθ]-1q(Mz)][x(t*)]+≤+W-1Mp(Mz)+M[I-e-Wθ]-1q(Mz)[x(t*)]+<MW-1[Wz-p(Mz)-W[I-e-Wθ]-1q(Mz)][x(t*)]+≤+W-1Mp(Mz)+M[I-e-Wθ]-1q(Mz)[x(t*)]+=Mz.
This contradicts (16), and so (15) holds. Letting ε→0, from (15), we have for any ϕ∈D (i.e., [ϕ]τ+≤z*),
(19)[x(t,t0,ϕ)]+≤Mz*,thatis,[M-1x(t,t0,ϕ)]+≤z*,t≥t0.
Therefore, the set D={ϕ∈PC∣[ϕ]τ+≤z*} is a positive quasi-invariant set of (2). When M=I especially, D is a positive invariant set of (2). The proof is complete.Based on the obtained quasi-invariant set, we have the followingTheorem 5.
Let(20)Δ(z)=p(z)+W[I-e-Wθ]-1q(z)-M-1Wz,z∈R+n.
Assume that all conditions in Theorem 4 hold. Define
(21)Ω1={z∈R+n∣Δ(Mz)<0},Ω2={z∈R+n∣Δ(z)<0},Ω3={z∈R+n∣Δ(z)≥0},Ω1*=⋃z*∈Ω1‍{z∈R+n∣z≤z*},Ω2*=⋃z*∈Ω2‍{z∈R+n∣z≤z*}.
Then, S={ϕ∈PC∣[ϕ]τ+∈Ω2*∩Ω3} is an attracting set of (2) and D={ϕ∈PC∣[ϕ]τ+∈Ω1*} is the attracting basin of S.Proof.
From (5) and the definitions of the above sets, then z*∈Ω1, Mz*∈Ω2, 0∈Ω1*, 0∈Ω2*, 0∈Ω3. Obviously, Ω1,Ω2,Ω1*,Ω2*,Ω3 and Ω2*∩Ω3 are nonempty, and so the definitions of the sets of S and D are valid. For any ϕ∈D, there is a z*∈Ω1 satisfying [ϕ]τ+≤z*. According to Theorem 4, we obtain
(22)[x(t)]+=[x(t,t0,ϕ)]+≤Mz*∈Ω2,∀t≥t0.
That is,
(23)σ=Δlimsupt→∞[x(t)]+∈Ω2*.
Then, for any given ε>0, there is a T1>t0 such that
(24)[x(t)]+≤εE+σ,t≥T1.
In light of W = diag{wi}>0, for the above ε>0 and T1, we can find an enough large T2>0 such that
(25)∫T2∞‍e-Wsds≤εI,∑t0<tk≤T1‍e-W(t-tk)≤εI,t>T2.
Using (12), (13), (22), (24), and (25), we have for t≥τ+T1+T2,
(26)[x(t)]+≤e-W(t-t0)M[ϕ]τ++∫t0t‍e-W(t-s)Mp([xs]τ+)ds[x(t)]+≤+∑t0<tk≤t‍Me-W(t-tk)q([x(tk-)]+)[x(t)]+≤e-W(t-t0)M[ϕ]τ+[x(t)]+≤+{∫t0t-T2‍+∫t-T2t‍}e-W(t-s)Mp([xs]τ+)ds[x(t)]+≤+{∑t0<tk≤T1‍+∑T1<tk≤t‍}Me-W(t-tk)q([x(tk-)]+)[x(t)]+≤e-W(t-t0)Mz*+∫T2∞‍e-WsMp(Mz*)ds[x(t)]+≤+∫t-T2t‍e-W(t-s)Mp(εE+σ)ds[x(t)]+≤+M∑t0<tk≤T1‍e-W(t-tk)q(Mz*)[x(t)]+≤+M∑T1<tk≤t‍e-W(t-tk)q(εE+σ)[x(t)]+≤e-W(t-t0)Mz*+εMp(Mz*)[x(t)]+≤+(I-e-WT2)W-1Mp(εE+σ)+εMq(Mz*)[x(t)]+≤+M(I-e-W(t-T1))[I-e-Wθ]-1q(εE+σ)[x(t)]+≤e-W(t-t0)Mz*+εM[p(Mz*)+q(Mz*)][x(t)]+≤+W-1Mp(εE+σ)+M[I-e-Wθ]-1q(εE+σ).
This implies that
(27)σ=limsupt→+∞[x(t)]+≤εM[p(Mz*)+q(Mz*)]+W-1Mp(εE+σ)+M[I-e-Wθ]-1q(εE+σ).
Letting ϵ→0+, then
(28)σ≤W-1Mp(σ)+M[I-e-Wθ]-1q(σ).
That is, Δ(σ)≥0 and σ∈Ω3. Thus,
(29)σ∈Ω2*∩Ω3.
From the definition of σ and S, dist (xt(t0,ϕ),S)→0 as t→+∞. The proof is complete.From the above theorems, we can obtain sufficient conditions ensuring global attractivity and stability in the following corollaries.Corollary 6.
Assume that (H1)–(H3) hold with
(30)p([φ]τ+)=P[φ]τ++μ,P=(pij)n×n≥0,μ=(μ1,…,μn)T≥0,q([x]+)=Q[x]++ν,Q=(qij)n×n≥0,ν=(ν1,…,νn)T≥0.
If the spectral radius
(31)ρ(Λ)<1,whereΛ=W-1MP+M[I-e-θW]-1Q,
then D={ϕ∈PC∣[ϕ]τ+≤Z=Δ(I-Λ)-1W-1(μ+W[I-e-θW]-1ν)} is a positive quasi-invariant set of (2), and S={ϕ∈PC∣[ϕ]τ+≤(I-Λ)-1W-1M(μ+W[I-e-θW]-1ν)} is a global attracting set of (2).Proof.
Sincep(z)=Pz+μ and q(z)=Qz+ν, we directly calculate
(32)Δ(z)=M-1W(Λ-I)z+μ+W[I-e-θW]-1ν,Δ(Mz)=W(Λ-I)z+μ+W[I-e-θW]-1ν.
Without loss of generality, we assume that μ,ν>0. Since ρ(Λ)<1, (I-Λ)-1 exists and (I-Λ)-1≥0 (see [23]), and so Z>0. For any κ>0, we take z*=(1+κ)Z>0 in Theorem 4 and verify the condition (5):
(33)Δ(Mz*)=-κ(μ+W[I-e-θW]-1ν)<0.
According to Theorem 4, when κ→0, we deduce that D is a positive quasi-invariant set of (2). Furthermore, by (33),
(34)(1+κ)Z∈Ω1,(1+κ)MZ∈Ω2.
From the arbitrariness of κ, we obtain Ω1*=Ω2*=R+n. Moreover,
(35)Ω3={z∈R+n∣Δ(z)≥0}Ω3={z∈R+n∣M-1W(Λ-I)z+μ+W[I-e-θW]-1ν≥0}Ω3={z∈R+n∣(I-Λ)z≤W-1M(μ+W[I-e-θW]-1ν)}Ω3⊂{{(μ+W[I-e-θW]-1ν)}z∈R+n∣z≤(I-Λ)-1W-1M×(μ+W[I-e-θW]-1ν)}.
It follows from Theorem 5 that S′={ϕ∈PC∣[ϕ]τ+∈Ω3} is a global attracting set of (2) and S is also a global attracting set due to S′⊂S. The proof is complete.Corollary 7.
Assume that all conditions in Corollary6 hold with μ=ν=0. Then, the zero solution x(t)=0 of (2) is globally asymptotically stable.


## 4. Illustrative Examples

The following illustrative examples will demonstrate the effectiveness of our results and also show the different asymptotical behaviors between the impulsive system and the corresponding continuous system.Example 8.
Consider a scalar nonlinear impulsive delay system(36)x˙(t)=0.2x(t)+0.2x2(t-1)+0.1,t≠tk,k∈N,t≥t0=0,Δx=-0.6x(tk-)+0.1x2(tk-)+0.1sin(etk),tk=tk-1+0.15.
According to Theorems 4 and 5, we have θ=ϱ=0.15, A=0.2, B=-0.6, M=2.5, W≐5.9086, p(z)=0.2z2+0.1, q(z)=0.1z2+0.1, Δ(z)≐1.2052z2-2.3634z+1.1052, and so
(37)Ω1={z∈R+∣Δ(Mz)<0}=(0.3079,0.4765),Ω1*=[0,0.4765],Ω2={z∈R+∣Δ(z)<0}=(0.7698,1.1913),Ω2*=[0,1.1913],Ω3={z∈R+∣Δ(z)≥0}=[0,0.7698]∪[1.1913,+∞),Ω2*∩Ω3=[0,0.7698].
Thus, S={ϕ∈PC∣[ϕ]τ+≤0.7698} is an attracting set of (36), and D={ϕ∈PC∣[ϕ]τ+≤0.4765} is an attracting basin of S. However, solutions of the corresponding continuous system (i.e., Δx=0 in (36)) may be unbounded. Taking the initial condition ϕ(s)=0.2, s∈[-1,0], Figure 1 shows the different asymptotic behavior between the solution of (36) with no impulse and one with impulses.The trajectory of (36) with: (a) no impulse (i.e., Δx=0) and (b) impulses.
(a)
(b)Example 9.
Consider a 2-dimensional impulsive delay system(38)x˙1(t)=x1(t)+0.5sin(x1(t-1))-0.4x2(t-1)-0.5,t≥0,x˙2(t)=-4x2(t)-0.5x1(t-1)+0.4cos(x2(t-1))+0.5,t≠tk,Δx1=-0.5x1(tk-)+0.1cos(x1(tk-))+0.5sin(etk),tk=0.1k,Δx2=0.1x2(tk-)+0.2sin(x2(tk-))-0.5cos(etk),k∈N.
According to Corollary 6, we have θ=ϱ=0.1, A = diag{1,-4}, B = diag{-0.5,0.1}, M = diag{2,1}, W = diag{5.9315,3.0469}, p(z)=Pz+μ, q(z)=Qz+ν, Λ=W-1MP+M[I-e-θW]-1Q, where
(39)P=(0.50.40.50.4),Q=(0.1000.2),μ=ν=(0.50.5),Λ=(0.61560.13490.16410.8928),
and so ρ(Λ)=0.9575<1. Therefore, D={ϕ∈PC∣[ϕ]τ+≤(1.7105,2.5228)T} is a positive quasi-invariant set of (38), and S={ϕ∈PC∣[ϕ]τ+≤(1.8637,2.8720)T} is a global attracting set of (38). Figure 2 shows the asymptotic properties of solutions of (38) under the different initial conditions.Figure 2
Global attracting set of (38).



---

*Source: 494067-2013-04-10.xml*