Asymptotic Stability of a Class of Impulsive Delay Differential Equations

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.

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: 𝑥^′(𝑡)(𝑡)()𝑥(𝑡)(𝑡=𝑝𝑥+𝑞𝑥𝑡−𝜏,𝑡≥0,𝑡≠𝑘𝜏,𝑘=0,1,2,…,=𝑟𝑥⁻)𝑥(𝑡)(𝑡)[),,𝑡=𝑘𝜏,𝑘=0,1,2,…,=Φ,𝑡∈−𝜏,0 (2.1) 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 lim_𝑡→∞𝑥(𝑡)=0, (2.2) where 𝑥(𝑡) is the solution of (2.1) for any initial function Φ∈𝐶([−𝜏,0),𝑅).

We define Φ(0)=lim_𝑡→0⁻Φ(𝑡) and the delay differential equations without impulsive perturbations as follows: 𝑦^′(𝑡)=ln𝑟𝜏𝑦(𝑡)()𝑦(𝑡)(𝑡)+𝑝+𝑞𝑦𝑡−𝜏,𝑡≥0,=Ψ=𝑟^𝑡/𝜏+1Φ(𝑡),𝑡∈[−𝜏,0]. (2.3) In the rest of the paper, {𝑡} is defined as {𝑡}=𝑡−[𝑡] where [⋅] denotes the greatest integer function.

Theorem 2.2. Assume that 𝑥(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 that 𝑦(𝑡) is continuous if 𝑥(𝑡) is the solution of (2.1). Obviously, 𝑦(𝑡)=𝑟^{𝑡/𝜏}𝑥(𝑡)=𝑟^{𝑡/𝜏−𝑘}𝑥(𝑡) (2.4) is continuous on [𝑘𝜏,(𝑘+1)𝜏),𝑘=−1,0,1,…. We know that 𝑦(𝑡)=Ψ(𝑡)=𝑟^𝑡/𝜏+1Φ(𝑡),𝑡∈[−𝜏,0] is continuous. Therefore, 𝑦(0⁻)=lim_𝑡→0−𝑟^𝑡/𝜏+1𝑥(𝑡)=𝑟𝑥(0⁻)=𝑟Φ(0⁻),𝑦(0)=𝑥(0)=𝑟𝑥(0⁻)=𝑟Φ(0⁻). (2.5) Hence, 𝑦(0)=𝑦(0⁻). So 𝑦(𝑡) is continuous at 𝑡=0. It follows from 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⁻)=lim_{𝑡→(𝑘+1)𝜏−}𝑟^{𝑡/𝜏}𝑥(𝑡)=𝑟𝑥((𝑘+1)𝜏⁻), (2.6) 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 that 𝑦(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 𝑦^′(𝑡)=𝑟^{𝑡/𝜏−𝑘}𝑥(𝑡)ln𝑟𝜏+𝑟^{𝑡/𝜏−𝑘}𝑥^′(𝑡)=𝑟^{𝑡/𝜏−𝑘}𝑥(𝑡)ln𝑟𝜏+𝑟^{𝑡/𝜏−𝑘}(𝑡)(=𝑝𝑥+𝑞𝑥𝑡−𝜏)𝑝+ln𝑟𝜏𝑟^{𝑡/𝜏−𝑘}𝑥(𝑡)+𝑞𝑟^{((𝑡−𝜏)/𝜏)−(𝑘−1)}𝑥()=𝑡−𝜏𝑝+ln𝑟𝜏𝑦(𝑡)+𝑞𝑦(𝑡−𝜏). (2.7) 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 that 𝑥(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,…, 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+ln𝑟𝜏𝑦(𝑡)(𝑟+𝑞𝑦𝑡−𝜏)=𝑝^{−(𝑡/𝜏−𝑘)}𝑦(𝑡)+𝑞𝑟^{−(𝑡/𝜏−𝑘)}𝑦()𝑟𝑡−𝜏=𝑝^{−(𝑡/𝜏−𝑘)}𝑦(𝑡)+𝑞𝑟^{−((𝑡−𝜏)/𝜏−(𝑘−1))}𝑦(𝑡−𝜏)=𝑝𝑥(𝑡)+𝑞𝑥(𝑡−𝜏). (2.8) Obviously, 𝑥((𝑘+1)𝜏)=𝑟^−{𝑘+1}𝑦)𝜏))𝜏),𝑥)𝜏((𝑘+1=𝑦((𝑘+1((𝑘+1⁻)=lim_{𝑡→(𝑘+1)𝜏−}𝑟^{−{𝑡/𝜏}}𝑦(𝑡)=𝑟⁻¹𝑦((𝑘+1)𝜏). (2.9) So 𝑥((𝑘+1)𝜏)=𝑟𝑥((𝑘+1)𝜏⁻),𝑘=0,1,…. Consequently, 𝑥(𝑡) is the solution of (2.1).

By Theorem 2.2, we obtain the following corollary.

Corollary 2.3. Assume that lim_𝑡→∞𝑥(𝑡)=0, where 𝑥(𝑡) is the solution of (2.1). Then lim_𝑡→∞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 lim_𝑡→∞𝑦(𝑡)=0, where 𝑦(𝑡) is the solution of (2.3). Then lim_𝑡→∞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]). Suppose 𝛼₀=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 𝑦(𝑡)≤𝐾𝑒^𝛼𝑡‖Ψ‖_𝜏,𝑡≥0,‖Ψ‖_𝜏=sup_{−𝜏≤𝜃≤0}𝑟^𝜃/𝜏+1Φ(𝜃). (2.10)

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 that 𝛼₀<0, where 𝛼₀=maxℜ𝜆∶𝜆−ln𝑟𝜏+𝑝−𝑞𝑒^−𝜆𝜏=0. (2.11) Suppose that 𝛼₀≥0. Then there exist a 𝜆₀ such that ℜ𝜆₀≥0 and 𝜆₀−(ln𝑟/𝜏+𝑝)−𝑞𝑒^−𝜆0𝜏=0. So ℜ𝜆₀≤ln|𝑟|𝜏+𝑝+𝑞𝑒^{−Re𝜆0𝜏}≤ln|𝑟|𝜏+𝑝+𝑞. (2.12) And |𝑟|exp((𝑝+|𝑞|)𝜏)<1 implies that ln|𝑟|𝜏+𝑝+𝑞<0. (2.13) Hence (2.12) and (2.13) imply that ℜ𝜆₀<0, which is a contradiction.
Consequently, 𝛼₀<0. By Lemma 2.4, we know that lim_𝑡→∞𝑦(𝑡)=0, where 𝑦(𝑡) the solution of (2.3). By Corollary 2.3, we know that lim_𝑡→∞𝑥(𝑡)=0, where 𝑥(𝑡) is the solution of (2.1).

Corollary 2.6. Assume that there is a constant 𝜆>0, such that |𝑟|<𝑒^𝜆𝜏 and 𝑝+|𝑞|𝑒^𝜆𝜏≤−𝜆. Then the zero solution of (2.1) is asymptotically stable.

Proof. Obviously, 𝑝+|𝑞|≤𝑝+|𝑞|𝑒^𝜆𝜏≤−𝜆<0. And because |𝑟|<𝑒^𝜆𝜏, so |𝑟|𝑒^{(𝑝+|𝑞|)𝜏}≤|𝑟|𝑒^−𝜆𝜏<1. (2.14) 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 constant 𝜆>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: 𝑦_𝑛+1=𝑦_𝑛+ℎ^𝑣_𝑖=1𝑏_𝑖ln𝑟𝜏+𝑝𝑌^𝑖_𝑛+𝑞𝑌^𝑖_𝑛−𝑚,𝑌^𝑖_𝑛=𝑦_𝑛+ℎ^𝑣_𝑗=1𝑎_𝑖𝑗ln𝑟𝜏+𝑝𝑌^𝑗_𝑛+𝑞𝑌^𝑗_𝑛−𝑚, (3.1) where ℎ=1/𝑚, is a given stepsize with integer 𝑚>1. 𝑣 is referred to as the number of stages. The weights 𝑏_𝑖, the abscissas 𝑐_𝑖=∑^𝑣_𝑖=1𝑎_𝑖𝑗, and the matrix 𝐴=[𝑎_𝑖,𝑗]^𝑣_𝑖,𝑗=1 will be denoted by (𝐴,𝑏,𝑐). 𝑦_𝑛=𝑟^{𝑛ℎ/𝜏+1}Φ(𝑛ℎ), 𝑛=−𝑚,−𝑚+1,…,0, and 𝑦_𝑛 is an approximation of 𝑦(𝑛ℎ), 𝑛=0,1,….

Define 𝑥_𝑛=ℜ𝑟^{−{𝑛/𝑚}}𝑦_𝑛. (3.2) Then 𝑥_𝑛 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∈𝐶^𝑝1([−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 𝑡_𝑛=𝑛ℎ≤𝑇, we have |𝑥(𝑡_𝑛)−𝑥_𝑛|≤(𝑥(𝑡_𝑛)−𝑥_𝑛)²+(Im(𝑟^{−{𝑛/𝑚}}𝑦_𝑛))²=|𝑥(𝑡_𝑛)−𝑟^{−{𝑛/𝑚}}𝑦_𝑛|=|𝑟^{−{𝑡𝑛/𝜏}}𝑦(𝑡_𝑛)−𝑟^{−{𝑛/𝑚}}𝑦_𝑛|=|𝑟^{−{𝑛/𝑚}}||𝑦(𝑡_𝑛)−𝑦_𝑛|≤𝐶|𝑦(𝑡_𝑛)−𝑦_𝑛|=𝑂(ℎ^𝑝1), where 𝐶=max{1,|𝑟|⁻¹}. Hence the order of (3.1) and (3.2) is 𝑝₁.

By (3.2), we know that 𝑥_𝑛→0 and Im(𝑟^{−{𝑛/𝑚}}𝑦_𝑛)→0, 𝑛→∞ if and only if 𝑦_𝑛→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 𝑆_𝑝 of pairs of numbers (𝛽,𝛾),𝛽=ℎ(ln𝑟/𝜏+𝑝),𝛾=ℎ𝑞, such that the numerical solution {𝑦_𝑛}_𝑛≥0 satisfies lim_𝑛→∞𝑦_𝑛=0 (3.3) for all constant delays 𝜏 and all initial function Ψ.

The Runge-Kutta method (3.1) is P-stable, if 𝑆_𝑝⊇𝛽,𝛾∈𝐶²∶Re𝛽+𝛾<0. (3.4) 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: 𝑥^′(𝑡)(𝑡)()𝑥(𝑡)(𝑡=−4𝑥+𝑥𝑡−1,𝑡≥0,𝑡≠𝑘,𝑘=0,1,2,…,=−2𝑥⁻)𝑥(𝑡)(𝑡)[).,𝑡=𝑘,𝑘=0,1,2,…,=Φ,𝑡∈−1,0 (4.1) The analytic solution of (4.1) (see Figure 1) as Φ(𝑡)≡1,𝑡∈[−1,0) is 𝑥(𝑡)=[),−91,𝑡∈−1,04𝑒^−4𝑡+14[),9,𝑡∈0,12𝑒⁻⁴−94𝑡+27𝑒16^{−4(𝑡−1)}+1[),916,𝑡∈1,2−22𝑒⁻⁴−92+27𝑒16⁻⁴+116,𝑡=2. (4.2) Note that 𝜖_0.5, 𝜖₁, 𝜖_1.5, 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 Tables 1 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 

Table 2 

Obviously, |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: 𝑦^′(𝑡)=(())𝑦(𝑡)()𝑦(𝑡)=()ln−2−4+𝑦𝑡−1,𝑡≥0,𝑡≠𝑘,𝑘=0,1,2,…,−2^𝑡+1Φ(𝑡),𝑡∈[−1,0). (4.3) 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.

Acknowledgment

The authors wish to thank referees for valuable comments. The research was supported by the NSF of China no. 11071050.