# Asymptotic Stability of a Class of Impulsive Delay Differential Equations

Authors: G. L. Zhang; M. H. Song; M. Z. Liu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper is concerned with a class of linear impulsive delay differential equations. Asymptotic stability of analytic solutions of this kind of equations is studied by the property of delay
differential equations without impulsive perturbations. New numerical methods for this kind
of equations are constructed. The convergence and asymptotic stability of the methods
for this kind of equations are studied.


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## Body

## 1. Introduction

Impulsive differential equations arise widely in the study of medicine, biology, economics, and engineering, and so forth In recent years, theory of impulsive differential delay equations (IDDEs) has been an object of active research (see [1–18] and the reference therein). The results about the existence and uniqueness of IDDEs have been studied in [2, 3, 10]. And the stability of IDDEs have attracted increasing interest in both theoretical research and practical applications (see [1, 3, 4, 6–18] and the reference therein). In this paper, we use the property of delay differential equations without impulsive perturbations to study asymptotic stability of analytic solutions of a class of linear impulsive delay differential equations.There are a few papers on numerical methods of impulsive differential equations. In [5], Covachev et al. obtained a convergent difference approximation for a nonlinear impulsive ordinary system in a Banach space. In [9, 19], the authors studied the stability of Runge-Kutta methods for linear impulsive ordinary differential equations. In [20], Ding et al. studied the convergence property of an Euler method for IDDEs. In this paper, we construct new numerical methods for a class of linear impulsive delay differential equations. The convergence and asymptotic stability of the methods for this kind of equations are studied.


## 2. Stability of Analytic Solutions of a Class of Linear IDDEs

In this paper, we consider the following equation:(2.1)x'(t)=px(t)+qx(t-τ),t≥0,t≠kτ,k=0,1,2,…,x(t)=rx(t-),t=kτ,k=0,1,2,…,x(t)=Φ(t),t∈[-τ,0),
where r≠0, τ>0, p and q are real constants, Φ is a continuous function on [-τ,0), and x'(t) denotes the right-hand derivative of x(t).Definition 2.1.
The zero solution of (2.1) is said to be asymptotically stable, if
(2.2)limt→∞x(t)=0,
where x(t) is the solution of (2.1) for any initial function Φ∈C([-τ,0),R).We defineΦ(0)=limt→0-Φ(t) and the delay differential equations without impulsive perturbations as follows:
(2.3)y'(t)=(lnrτ+p)y(t)+qy(t-τ),t≥0,y(t)=Ψ(t)=rt/τ+1Φ(t),t∈[-τ,0].
In the rest of the paper, {t} is defined as {t}=t-[t] where [·] denotes the greatest integer function.Theorem 2.2.
Assume thatx(t) is the solution of (2.1) and y(t)=r{t/τ}x(t),t∈[-τ,∞), then y(t) is the solution of (2.3). On the other hand, assume that y(t) is the solution of (2.3) and x(t)=r-{t/τ}x(t),t∈[-τ,∞), then x(t) is the solution of (2.1).Proof.
(i) We prove thaty(t) is continuous if x(t) is the solution of (2.1). Obviously,
(2.4)y(t)=r{t/τ}x(t)=rt/τ-kx(t)
is continuous on [kτ,(k+1)τ),k=-1,0,1,…. We know that y(t)=Ψ(t)=rt/τ+1Φ(t),t∈[-τ,0] is continuous. Therefore,
(2.5)y(0-)=limt→0-rt/τ+1x(t)=rx(0-)=rΦ(0-),y(0)=x(0)=rx(0-)=rΦ(0-).
Hence, y(0)=y(0-). So y(t) is continuous at t=0. It follows from
(2.6)y((k+1)τ)=r{((k+1)τ)/τ}x((k+1)τ)=r{k+1}x((k+1)τ)=x((k+1)τ)=rx((k+1)τ-),y((k+1)τ-)=limt→(k+1)τ-r{t/τ}x(t)=rx((k+1)τ-),
that y(t) is continuous at t=(k+1)τ,k=0,1,2,…. Hence y(t) is continuous on t∈[-τ,∞).
(ii) We prove thaty(t) is the solution of (2.3) if x(t) is the solution of (2.1). For t∈[kτ,(k+1)τ),k=0,1,…, we have
(2.7)y′(t)=rt/τ-kx(t)lnrτ+rt/τ-kx'(t)=rt/τ-kx(t)lnrτ+rt/τ-k(px(t)+qx(t-τ))=(p+lnrτ)(rt/τ-kx(t))+q(r((t-τ)/τ)-(k-1)x(t-τ))=(p+lnrτ)y(t)+qy(t-τ).
Because y'(t) denotes the right-hand derivative of y(t), y'(t)=(p+lnr/τ)y(t)+qy(t-τ), hence y(t) is the solution of (2.3).
(iii) We prove thatx(t) is the solution of (2.1) if y(t) is the solution of (2.3). For t∈[kτ,(k+1)τ),k=0,1,…,
(2.8)x′(t)=-r-(t/τ-k)y(t)lnrτ+r-(t/τ-k)y'(t)=-r-(t/τ-k)y(t)lnrτ+r-(t/τ-k)((p+lnrτ)y(t)+qy(t-τ))=p(r-(t/τ-k)y(t))+q(r-(t/τ-k)y(t-τ))=p(r-(t/τ-k)y(t))+q(r-((t-τ)/τ-(k-1))y(t-τ))=px(t)+qx(t-τ).
Obviously,
(2.9)x((k+1)τ)=r-{k+1}y((k+1)τ)=y((k+1)τ),x((k+1)τ-)=limt→(k+1)τ-r-{t/τ}y(t)=r-1y((k+1)τ).
So x((k+1)τ)=rx((k+1)τ-),k=0,1,…. Consequently, x(t) is the solution of (2.1).By Theorem2.2, we obtain the following corollary.Corollary 2.3.
Assume thatlimt→∞x(t)=0, where x(t) is the solution of (2.1). Then limt→∞y(t)=0, where r{t/τ}x(t),t∈[-τ,∞), and y(t) also is the solution of (2.3). On the other hand, assume that limt→∞y(t)=0, where y(t) is the solution of (2.3). Then limt→∞x(t)=0, where x(t)=r-{t/τ}x(t),t∈[-τ,∞), and x(t) also is the solution of (2.1).Lemma 2.4 (see [21]).
Supposeα0=max{Reλ:λ-(lnr/τ+p)-qe-λτ=0} and y(t) is the solution of (2.3). Then, for any α>α0, there is a constant K=K(α) such that
(2.10)|y(t)|≤Keαt∥Ψ∥τ,t≥0,∥Ψ∥τ=sup-τ≤θ≤0|rθ/τ+1Φ(θ)|.Theorem 2.5.
Assume that|r|exp((p+|q|)τ)<1. Then the zero solution of (2.1) is asymptotically stable.Proof.
First we prove thatα0<0, where
(2.11)α0=max{ℜλ:λ-(lnrτ+p)-qe-λτ=0}.
Suppose that α0≥0. Then there exist a λ0 such that ℜλ0≥0 and λ0-(lnr/τ+p)-qe-λ0τ=0. So
(2.12)ℜλ0≤(ln|r|τ+p)+|q|e-Reλ0τ≤ln|r|τ+p+|q|.
And |r|exp((p+|q|)τ)<1 implies that
(2.13)ln|r|τ+p+|q|<0.
Hence (2.12) and (2.13) imply that ℜλ0<0, which is a contradiction.
Consequently,α0<0. By Lemma 2.4, we know that limt→∞y(t)=0, where y(t) the solution of (2.3). By Corollary 2.3, we know that limt→∞x(t)=0, where x(t) is the solution of (2.1).Corollary 2.6.
Assume that there is a constantλ>0, such that |r|<eλτ and p+|q|eλτ≤-λ. Then the zero solution of (2.1) is asymptotically stable.Proof.
Obviously,p+|q|≤p+|q|eλτ≤-λ<0. And because |r|<eλτ, so
(2.14)|r|e(p+|q|)τ≤|r|e-λτ<1.
By Theorem 2.5, we have that the zero solution of (2.1) is asymptotically stable.Similarly, we can get the following corollary.Corollary 2.7.
Assume that there is a constantλ>0, such that |r|<e-λτ and p+|q||r|-1≤λ. Then the zero solution of (2.1) is asymptotically stable.


## 3. Numerical Solutions of IDDEs

We consider the Runge-Kutta methods for (2.3) as follows:
(3.1)yn+1=yn+h∑i=1v‍bi((lnrτ+p)Yni+qYn-mi),Yni=yn+h∑j=1v‍aij((lnrτ+p)Ynj+qYn-mj),
where h=1/m, is a given stepsize with integer m>1. v is referred to as the number of stages. The weights bi, the abscissas ci=∑i=1v‍aij, and the matrix A=[ai,j]i,j=1v will be denoted by (A,b,c). yn=rnh/τ+1Φ(nh), n=-m,-m+1,…,0, and yn is an approximation of y(nh), n=0,1,….Define(3.2)xn=ℜ(r-{n/m}yn).
Then xn is the numerical solution of (2.1).Theorem 3.1.
Assume that the Runge-Kutta method(A,b,c) is of order p1. Then the order of (3.1) and (3.2) for (2.1) is also p1, when Ψ∈Cp1([-τ,0],R).Proof.
In [22], it has been proved that the order of (3.1) is p1, if the Runge-Kutta method (A, b, c) is of order p1. And for any fixed T>0, and any tn=nh≤T, we have |x(tn)-xn|≤(x(tn)-xn)2+(Im(r-{n/m}yn))2=|x(tn)-r-{n/m}yn|=|r-{tn/τ}y(tn)-r-{n/m}yn|=|r-{n/m}||y(tn)-yn|≤C|y(tn)-yn|=O(hp1), where C=max{1,|r|-1}. Hence the order of (3.1) and (3.2) is p1.By (3.2), we know that xn→0 and Im(r-{n/m}yn)→0, n→∞ if and only if yn→0,n→∞. The definition of P-stability region and P-stable has been given in many papers, for example, [23]. The P-stability region of (3.1) is the set Sp of pairs of numbers (β,γ),β=h(lnr/τ+p),γ=hq, such that the numerical solution {yn}n≥0 satisfies
(3.3)limn→∞yn=0
for all constant delays τ and all initial function Ψ.The Runge-Kutta method (3.1) is P-stable, if
(3.4)Sp⊇{(β,γ)∈C2:Reβ+|γ|<0}.
Therefore we obtain the following theorem.Theorem 3.2.
If|r|exp((p+|q|)τ)<1, then the solutions of (2.3) and (2.1) are asymptotically stable. In addition, if the Runge-Kutta method is P-stable, then the process (3.1) preserves asymptotically stable of (2.3); consequently, the process (3.1)-(3.2) preserves asymptotically stable of (2.1).


## 4. Numerical Experiments

Consider the following IDDE:(4.1)x'(t)=-4x(t)+x(t-1),t≥0,t≠k,k=0,1,2,…,x(t)=-2x(t-),t=k,k=0,1,2,…,x(t)=Φ(t),t∈[-1,0).
The analytic solution of (4.1) (see Figure 1) as Φ(t)≡1,t∈[-1,0) is
(4.2)x(t)={1,t∈[-1,0),-94e-4t+14,t∈[0,1),(92e-4-94t+2716)e-4(t-1)+116,t∈[1,2),-2((92e-4-92+2716)e-4+116),t=2.
Note that ϵ0.5, ϵ1, ϵ1.5, and ϵ2 are the errors of the analytic solution and the numerical solution (3.1) (3.2) for (4.1) at t=0.5,1,1.5,and2, respectively.Figure 1
The analytic solution of (4.1) and the real part analytic solution of (4.3).In Tables1 and 2, we have listed the errors at t=0.5,1.1.5 and  2 of the explicit  Euler method and the Trapezoid method.Table 1
Stepsize
ϵ
0.5
ϵ
1
ϵ
1.5
ϵ
2
h
=
1
/
50
0.007237306
0.013957231
0.009751641
0.000998634
h
=
1
/
100
0.003480570
0.006575988
0.004535880
0.000421508
h
=
1
/
200
0.001706846
0.003192398
0.002188540
0.000193107
h
=
1
/
400
0.000845200
0.0015728959
0.001075069
0.000092351
Ratio
2.045995126
2.070658147
2.086056611
2.214324691Table 2
Stepsize
ϵ
0.5
ϵ
1
ϵ
1.5
ϵ
2
h
=
1
/
50
3.50207
e
-
4
2.52492
e
-
4
2.21015
e
-
4
2.45482
e
-
4
h
=
1
/
100
8.74678
e
-
5
6.30749
e
-
5
5.52490
e
-
5
6.13071
e
-
5
h
=
1
/
200
2.18617
e
-
5
1.57657
e
-
5
1.38119
e
-
5
1.53228
e
-
5
h
=
1
/
400
5.46510
e
-
6
3.94124
e
-
6
3.45297
e
-
6
3.83045
e
-
6
Ratio
4.00167
4.00133
4.00015
4.00181Obviously,|r|e(p+|q|)τ=2e-4+1=2e-3<1, so the zero solution of (4.1) is asymptotically stable by Theorem 2.5. Assume that x(t) is the solution of (4.1) and y(t)=(-2){t}x(t), for t∈[-1,∞). Then y(t) is the solution of the following equation:
(4.3)y'(t)=(ln(-2)-4)y(t)+y(t-1),t≥0,t≠k,k=0,1,2,…,y(t)=(-2)t+1Φ(t),t∈[-1,0).
Because ℜ(ln(-2)-4)+1<0, we know that y(t)→0 as t→∞ (see Figure 1 ).The numerical process (3.1) and (3.2) for (4.1) is asymptotically stable as (3.2) is the Trapezoid method (see Figure 2) and the implicit Euler method (see Figure 3).Figure 2
The numerical solution of (4.1), when (3.1) is the Trapezoid method as h= 1/100.Figure 3
The numerical solution of (4.1), when (3.1) is the implicit Euler method as h= 1/100.



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