# Global Exponential Stability of Impulsive Functional Differential Equations via Razumikhin Technique

Authors: Shiguo Peng; Liping Yang
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

This paper develops some new Razumikhin-type theorems on global exponential
stability of impulsive functional differential equations. Some applications
are given to impulsive delay differential equations. Compared with
some existing works, a distinctive feature of this paper is to address exponential
stability problems for any finite delay. It is shown that the functional
differential equations can be globally exponentially stabilized by impulses
even if it may be unstable itself. Two examples verify the effectiveness of
the proposed results.


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## Body

## 1. Introduction

Functional differential equations (FDEs) which include delay differential equations (DDEs) play a very important role in formulation and analysis in mechanical, electrical, control engineering and physical sciences, economic, and social sciences [1, 2]. Therefore, the theory of FDEs has been developed very quickly. The investigation for FDEs has attracted the considerable attention of researchers and many qualitative theories of FDEs have been obtained. A large number of stability criteria of FDEs have been reported.In addition to the delay effect, as is well known, impulsive effect is likely to exist in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time in the fields such as medicine and biology, economics, electronics, and telecommunications [3]. So far, a lot of interesting results on stability have been reported that have focused on the impulsive effect of FDEs (see, e.g., [4–16] and the references cited therein).In particular, several papers devoted to the study of exponential stability of impulsive functional differential equations (IFDEs) have appeared during the past years. In [14, 15], the authors have investigated exponential stability of IFDEs by using the method of Lyapunov functions and Razumikhin techniques. In [16], the authors have also studied exponential stability by using the method of Lyapunov functional. However, some results in [15, 16] imposed a restrictive condition on time delays which were less than the length of all the impulsive intervals (see, e.g., [15, Theorems 3.1-3.2] and [16, Theorem 3.1]). The aim of this paper is to establish global exponential stability criteria for IFDEs by employing the Razumikhin technique which illustrate that impulses do contribute to the stability of some IFDEs and the restrictive condition that the time delays are less than the length of all the impulsive intervals can be removed in this paper.


## 2. Preliminaries

Throughout this paper, unless otherwise specified, we use the following notations. Letℝ=(-∞,+∞), ℝ+=[0,+∞), ℕ={1,2,…}, I be the identity matrix, λmax(·) and λmin(·) be the maximal eigenvalue and the minimal eigenvalue of a matrix, respectively. If A is a vector or matrix, its transpose is denoted by AT. For x∈ℝn and A∈ℝn×n, let |x|=xTx be Euclidean vector norm, and denote the induced matrix norm by(2.1)∥A∥=supx≠0|Ax||x|=λmax(ATA).Letτ>0 and C=C([-τ,0];ℝn) denote the family of all bounded continuous ℝn-valued functions ϕ defined on [-τ,0]. PC(𝕀;ℝn)={ψ:𝕀→ℝn∣ψ(s) is continuous for all but at most countable points s∈𝕀 and at these points s∈𝕀, ψ(s+) and ψ(s-) exist and ψ(s+)=ψ(s)}, where 𝕀⊂ℝ is an interval, ψ(s+) and ψ(s-) denote the right-hand and left-hand limits of the function ψ(s) at time s, respectively. Especially, let PC≜PC([-τ,0];ℝn) with norm ∥ψ∥=sup-τ⩽s⩽0|ψ(s)|.In this paper, we consider the following IFDEs:(2.2)x′(t)=f(t,xt),t≠tk,t⩾0,Δx(tk)=x(tk+)-x(tk-)=Ik(x(tk-)),t=tk,k∈ℕ,x(s)=ϕ(s),-τ⩽s⩽0,
where f:ℝ+×PC→ℝn, xt(θ)=x(t+θ), θ∈[-τ,0]. The initial function ϕ∈PC. The impulsive function Ik∈C(ℝn;ℝn)(k∈ℕ), and the impulsive moments tk(k=1,2,…) satisfy 0=t0<t1<t2<⋯, and limk→∞tk=∞.In this paper, we assume that functionsf and Ik, k∈ℕ, satisfy all necessary conditions for the global existence and uniqueness of solutions for all t⩾t0. Denote by x(t)=x(t,t0,ϕ) the solution of (2.2) such that xt0=ϕ. For the purpose of stability in this paper, we also assume that f(t,0)=0 and Ik(0)=0, k∈ℕ. So system (2.2) admits a zero solution or trivial solution x(t,t0,0)=0. We further assume that all the solutions x(t) of (2.2) are continuous except at tk,k∈ℕ, at which x(t) is right continuous, that is, x(tk+)=x(tk),k∈ℕ.Definition 2.1.
The trivial solution of system (2.2) is said to be globally exponentially stable, if there exist numbers λ>0 and M⩾1 such that
(2.3)|x(t,t0,ϕ)|⩽M∥ϕ∥e-λt,t⩾0,
whenever ϕ∈PC.Definition 2.2.
V:[-τ,∞)×ℝn→ℝ+ is said to belong to the class ν0, if V is continuous on each of the sets [tk-1,tk)×ℝn, lim(t,y)→(tk-,x)V(t,y)=V(tk-,x) exists, V(t,x) is locally Lipschitzian in all x∈ℝn, and V(t,0)≡0 for all t≥-τ.Definition 2.3.
V:[-τ,∞)×ℝn→ℝ+ is said to belong to the class ν1, if V is continuous on each of the sets [tk-1,tk)×ℝn, V(t,0)≡0 for all t≥-τ, lim(t,y)→(tk-,x)V(t,y)=V(tk-,x) exists, Vt(t,x),Vx(t,x) are continuous, where (t,x)∈[tk-1,tk)×ℝn,k∈ℕ,
(2.4)Vt(t,x)=∂V(t,x)∂t,Vx(t,x)=(∂V(t,x)∂x1,…,∂V(t,x)∂xn).Definition 2.4.
Given a functionV:[-τ,+∞)×ℝn→ℝ+, the upper right-hand derivative of V with respect to system (2.2) is defined by
(2.5)D+V(t,φ(0))=lim suph→0+1h[V(t+h,φ(0)+hF(t,φ))-V(t,φ(0))]
for (t,φ)∈ℝ+×PC.


## 3. Razumikhin-Type Theorems

In this section, we will present some Razumikhin-type theorems on global exponential stability for system (2.2) based on the Lyapunov-Razumikhin method.Theorem 3.1.
Letρ=supk∈ℕ{tk-tk-1}<∞ and c1,c2,p,q,γ all be positive numbers, c a real number, q>1/γ>1, and c<ln(1/γ)/ρ.  Suppose that there exists a function V∈ν0 such that(i)
c1|x|p⩽V(t,x)⩽c2|x|p for all t⩾t0-τ, and x∈ℝn;(ii)
D+V(t,φ(0))⩽cV(t,φ(0)) for all t∈[tk-1,tk), k∈ℕ, whenever qV(t,φ(0))⩾V(t+θ,φ(θ)) for all θ∈[-τ,0];(iii)
V(tk,x+Ik(x))⩽γV(tk-,x), k∈ℕ, x∈ℝn.Then the trivial solution of system  (2.2) is globally exponentially stable.Proof.
For anyϕ∈PC, we denote the solution x(t,t0,ϕ) of (2.2) by x(t). Without loss of generality, we assume that ∥ϕ∥≠0.
Sinceq>1/γ and c<-(lnγ/ρ), there exist positive numbers μ and h such that
(3.1)q>eμτγ>1γ,c<c+μ<-ln(γ+h)ρ.
Set γ1=-ln(γ+h/ρ), then γ<e-γ1ρ<1.
SetW(t)=eμtV(t,x(t)), we have
(3.2)D+W(t)=μW(t)+eμtD+V(t,x(t)),t∈[tk-1,tk),k∈ℕ.
Let M̅>c2/c1γ be a fixed number. In the following, we will prove that
(3.3)W(t)<c1M̅∥ϕ∥p,t⩾t0-τ.
We first prove that
(3.4)W(t)<c1M̅∥ϕ∥p,t∈[t0-τ,t1).
It is noted that W(t0+θ)⩽c2∥ϕ∥p<γc1M̅∥ϕ∥p<c1M̅∥ϕ∥p, θ∈[-τ,0]. So, it only needs to prove that W(t)<c1M̅∥ϕ∥p for t∈(t0,t1). On the contrary, there exist some t∈(t0,t1) such that W(t)⩾c1M̅∥ϕ∥p. Set t*=inf{t∈(t0,t1):W(t)⩾c1M̅∥ϕ∥p}, then we have t*∈(t0,t1) and W(t*)⩾c1M̅∥ϕ∥p. Set t̅=sup{t∈[t0,t*):W(t)⩽γc1M̅∥ϕ∥p}. Then t̅∈(t0,t*) and W(t̅)=γc1M̅∥ϕ∥p. For t∈[t̅,t*], we have
(3.5)W(t)⩾γc1M̅∥ϕ∥p⩾γW(t+θ),∀θ∈[-τ,0].
Hence
(3.6)V(t,x(t))⩾γe-μτV(t+θ,x(t+θ))⩾1qV(t+θ,x(t+θ)),∀θ∈[-τ,0].
By condition (ii) and (3.2), it follows that for t∈[t̅,t*](3.7)D+W(t)=μW(t)+eμtD+V(t,x(t))⩽(μ+c)W(t)⩽γ1W(t).
So, we obtain W(t*)⩽W(t̅)eγ1(t*-t̅)⩽γc1M̅∥ϕ∥peγ1ρ<c1M̅∥ϕ∥p. This is a contradiction, so (3.4) holds.
Now, we assume that for somem∈ℕ,
(3.8)W(t)<c1M̅∥ϕ∥p,t∈[t0-τ,tm).
We will prove that
(3.9)W(t)<c1M̅∥ϕ∥p,t∈[tm,tm+1).
Suppose not, there exist some t∈[tm,tm+1) such that W(t)⩾c1M̅∥ϕ∥p. Set t*=inf{t∈[tm,tm+1):W(t)⩾c1M̅∥ϕ∥p}. From condition (iii) and (3.8), we have W(tm)⩽γW(tm-)⩽γc1M̅∥ϕ∥p<c1M̅∥ϕ∥p. Hence t*∈(tm,tm+1) and W(t*)=c1M̅∥ϕ∥p. Set t̅=sup{t∈[tm,t*):W(t)⩽γc1M̅∥ϕ∥p}. Then we have W(t̅)=γc1M̅∥ϕ∥p. Furthermore, we have W(t)⩾γc1M̅∥ϕ∥p⩾γW(t+θ) for t∈[t̅,t*]. Thus, (3.7) holds. Then W(t*)⩽γc1M̅∥ϕ∥peγ1ρ<c1M̅∥ϕ∥p, which yields a contradiction. Therefore, (3.9) holds.
By mathematical induction, we have(3.10)W(t)<c1M̅∥ϕ∥p,t⩾0.
Hence
(3.11)|x(t)|<M∥ϕ∥e-λt,t⩾0,
where M=M̅1/p, λ=μ/p. The proof is therefore complete.Theorem 3.2.
Letϱ=infk∈ℕ{tk-tk-1}>0 and c1,c2,p,q,γ,c all be positive numbers, q>γ⩾1, and c>lnγ/ϱ. Suppose that there exists a function V∈ν0 such that(i)
c1|x|p⩽V(t,x)⩽c2|x|p for all t⩾t0-τ, and x∈ℝn,(ii)
D+V(t,φ(0))⩽-cV(t,φ(0)) for all t∈[tk-1,tk)(k∈ℕ), whenever qV(t,φ(0))⩾V(t+θ,φ(θ)) for all θ∈[-τ,0];(iii)
V(tk,x+Ik(x))⩽γV(tk-,x), k∈ℕ, x∈ℝn.Then the trivial solution of system  (2.2) is globally exponentially stable.Proof.
Sinceq>γ and c>lnγ/ϱ, there exist positive numbers μ and h such that
(3.12)q>(γ+h)eμτ>γ,c>c-μ>ln(γ+2h)ϱ.
Set q̅=γ+h and γ2=ln(γ+2h)/ϱ, then 1⩽γ<q̅<eγ2ϱ. Set W(t)=eμtV(t,x(t)), where x(t) is defined as in the proof of Theorem 3.1.
Now, letM̅>q̅c2/c1, we will prove that (3.3) holds.
We first prove that (3.4) holds. In fact, it is noticed that W(t0+θ)⩽c2∥ϕ∥p<(1/q̅)c1M̅∥ϕ∥p<c1M̅∥ϕ∥p, for θ∈[-τ,0]. So it only needs to prove W(t)<c1M̅∥ϕ∥p for t∈(t0,t1). On the contrary, there exists t∈(t0,t1) such that W(t)⩾c1M̅∥ϕ∥p. Set t*=inf{t∈(t0,t1):W(t)⩾c1M̅∥ϕ∥p}, then we have t*∈(t0,t1). Set t̅=sup{t∈[t0,t*):W(t)⩽(1/q̅)c1M̅∥ϕ∥p}. Then t̅∈(t0,t*) and W(t̅)=(1/q̅)c1M̅∥ϕ∥p. For t∈[t̅,t*], we have
(3.13)q̅W(t)⩾c1M̅∥ϕ∥p⩾W(t+θ),∀θ∈[-τ,0].
Hence
(3.14)V(t,x(t))⩾1q̅e-μτV(t+θ,x(t+θ))⩾1qV(t+θ,x(t+θ)),∀θ∈[-τ,0].
It follows that for t∈[t̅,t*](3.15)D+W(t)⩽(μ-c)W(t)⩽-γ2W(t),
which leads to W(t*)⩽W(t̅). This is a contradiction to the fact W(t*)=c1M̅∥ϕ∥p>(1/q̅)c1M̅∥ϕ∥p=W(t̅), so (3.4) holds.
Now, we assume that for somem∈ℕ, (3.8) holds. We will prove that (3.9) holds. In order to do this, we first claim that
(3.16)W(tm-)⩽1q̅c1M̅∥ϕ∥p.
Suppose not, then we have W(tm-)>(1/q̅)c1M̅∥ϕ∥p. There are two cases to be considered.
Case 1.
W(t)>(1/q̅)c1M̅∥ϕ∥p for all t∈[tm-1,tm).
By (3.8), we have q̅W(t)>c1M̅∥ϕ∥p>W(t+θ) for θ∈[-τ,0] and t∈[tm-1,tm). Thus, we get D+W(t)⩽-γ2W(t) for t∈[tm-1,tm) which leads to W(tm-)⩽W(tm-1)e-γ2(tm-tm-1)<c1M̅∥ϕ∥pe-γ2ϱ<(1/q̅)c1M̅∥ϕ∥p. This is a contradiction.Case 2.
There is somet∈[tm-1,tm) such that W(t)⩽(1/q̅)c1M̅∥ϕ∥p.
Sett̅=sup{t∈[tm-1,tm):W(t)⩽(1/q̅)c1M̅∥ϕ∥p}. Then t̅∈[tm-1,tm) and W(t̅)=(1/q̅)c1M̅∥ϕ∥p. Since for t∈[t̅,tm), q̅W(t)⩾c1M̅∥ϕ∥p⩾W(t+θ), θ∈[-τ,0]. By (3.15), we have D+W(t)⩽0 for t∈[t̅,tm), which gives W(tm-)⩽W(t̅)=(1/q̅)c1M̅∥ϕ∥p. This is also a contradiction.
Hence, (3.16) holds. It follows from (iii) that W(tm)⩽γW(tm-)⩽(γ/q̅)c1M̅∥ϕ∥p<c1M̅∥ϕ∥p. Now, we assume that (3.9) is not true, set t*=inf{t∈[tm,tm+1):W(t)⩾c1M̅∥ϕ∥p}. Hence t*∈(tm,tm+1) and W(t*)=c1M̅∥ϕ∥p. If W(t)>(1/q̅)c1M̅∥ϕ∥p for t∈[tm,t*], set t̅=tm, otherwise, set t̅=sup{t∈[tm,t*):W(t)⩽(1/q̅)c1M̅∥ϕ∥p}. Thus, we have q̅W(t)⩾c1M̅∥ϕ∥p⩾W(t+θ) for t∈[t̅,t*]. Hence, by (3.15), D+W(t)⩽0 for t∈[t̅,t*]. Then W(t*)⩽W(t̅)<c1M̅∥ϕ∥p, which yields a contradiction. Therefore, (3.9) holds. The rest is the same as in the proof of Theorem 3.1.Remark 3.3.
By Theorems3.1 and 3.2, we can design impulsive control {Ik(x(tk-)),k∈ℕ} for the following FDEs
(3.17)x′(t)=f(t,xt),t⩾t0,xt0(s)=ϕ(s),-τ⩽s⩽0,ϕ∈C,
such that the system can be impulsively stabilized to its trivial solution. In Theorem 3.1, the constant c may be chosen as a positive number. In the stability theory of FDEs, the condition D+V(t,φ(0))⩽cV(t,φ(0)) allows the derivative of the Lyapunov function to be positive which may not even guarantee the stability of functional differential system (see, e.g., [4, 15]). However, as we can see from Theorem 3.1, impulses play an important role in making a functional differential system globally exponentially stable even if it may be unstable itself.Remark 3.4.
It is important to emphasize that, in contrast with some existing exponential stability results for IFDEs in the literature [15, 16], Theorems 3.1 and 3.2 are also valid for any finite delay. Therefore, our new results are more practically applicable than those in the literature, since the restrictive condition that the supper bound of time delay is less than the length of all the impulsive intervals is actually removed here.


## 4. Applications and Examples

Now, we will apply the general Razumikhin-type theorems established in Section3 to deal with the global exponential stability of impulsive delay differential equations (IDDEs).Consider a delay system of the form(4.1)x′(t)=F(t,x(t),x(t-δ1(t)),…,x(t-δm(t))),t≠tk,t⩾0,Δx(tk)=x(tk+)-x(tk-)=Ik(x(tk-)),t=tk,k∈ℕ,x(s)=ϕ(s),-τ⩽s⩽0,
where ϕ∈PC, δi:ℝ+→[0,τ],1⩽i⩽m, are all continuous, and(4.2)F:ℝ+×ℝn×ℝn×m→ℝn
is continuous. We also assume that (4.1) has a global solution which is again denoted by x(t)=x(t,t0,ϕ), F(t,0,…,0)=0 and Ik(0)=0,k∈ℕ.Theorem 4.1.
Letρ=supk∈ℕ{tk-tk-1}<∞, λ0 be a real number, and λ1,…,λm,c1,c2,p,γ be all positive numbers, 0<γ<1. Suppose that there exists a function V∈ν1 such that(i)
c1|x|p⩽V(t,x)⩽c2|x|p for all t⩾t0⩾-τ, and x∈ℝn;(ii)
(4.3)Vt(t,x)+Vx(t,x)F(t,x,y)⩽λ0V(t,x)+∑i=1mλiV(t-δi(t),yi)
for all t∈[tk-1,tk)(k∈ℕ),x∈ℝn,y=(y1,…,ym)∈ℝn×m;(iii)
V(tk,x+Ik(x))⩽γV(tk-,x), k∈ℕ, x∈ℝn.If λ0+∑i=1mλi/γ+lnγ/ρ<0, then the trivial solution of  (4.1) is globally exponentially stable.Proof.
Fort∈[tk-1,tk),(k∈ℕ),ϕ∈PC, let
(4.4)f(t,ϕ)=F(t,ϕ(0),ϕ(-δ1(t)),…,ϕ(-δm(t))).
Then system (4.1) becomes system (2.2), and D+V(t,ϕ(0)) becomes
(4.5)D+V(t,ϕ(0))=Vt(t,ϕ(0))+Vx(t,ϕ(0))F(t,ϕ(0),ϕ(-δ1(t)),…,ϕ(-δm(t))).
If λ0+∑i=1mλi/γ+lnγ/ρ<0, then there exists a constant q>1/γ, such that
(4.6)λ0+q∑i=1mλi+lnγρ<0.
So, if t∈[tk-1,tk)(k∈ℕ) and qV(t,φ(0))⩾V(t+θ,φ(θ)) for all θ∈[-τ,0], then
(4.7)D+V(t,φ(0))⩽λ0V(t,φ(0))+∑i=1mλiV(t-δi(t),φ(-δi(t)))⩽(λ0+q∑i=1mλi)V(t,φ(0))=cV(t,φ(0)),
where c=λ0+q∑i=1mλi<-lnγ/ρ. By Theorem 3.1, the trivial solution of (4.1) is globally exponentially stable.Corollary 4.2.
Letρ=supk∈ℕ{tk-tk-1}<∞. Assume that there exist scalar numbers αi⩾0(1⩽i⩽m),  0<α<1 and η such that
(4.8)xTF(t,x,0)⩽η|x|2,(4.9)|F(t,x,y)-F(t,x,0)|⩽∑i=1mαi|yi|,(4.10)|x+Ik(x)|⩽α|x|
for all t⩾t0,x∈ℝn, y=(y1,…,ym)∈ℝn×m, k∈ℕ.
If(4.11)η+∑i=1mαiα+lnαρ<0,
then the trivial solution of  (4.1) is globally exponentially stable.Proof.
LetV(t,x)=xTx=|x|2, then we can easily see that condition (i) of Theorem 4.1 holds.
Forx∈ℝn and y=(y1,…,ym)∈ℝn×m, from (4.8) and (4.9), we can calculate that
(4.12)Vt(t,x)+Vx(t,x)F(t,x,y)=2xTF(t,x,y)=2xTF(t,x,0)+2xT[F(t,x,y)-F(t,x,0)]⩽2η|x|2+2∑i=1mαi|x|·|yi|.
Let γ=α2, by the inequality ab⩽(a2+b2)/2, we have
(4.13)|x|·|yi|=(γ-1/4|x|)(γ1/4|yi|)⩽γ-1/2|x|2+γ1/2|yi|22.
Substituting (4.13) into (4.12), we obtain
(4.14)Vt(t,x)+Vx(t,x)F(t,x,y)⩽(2η+∑i=1mαiα)|x|2+γ1/2∑i=1mαi|yi|2=λ0|x|2+∑i=1mλi|yi|2,
where λ0=2η+∑i=1mαi/α, λi=γ1/2αi, i=1,…,m.
From (4.11), we have
(4.15)λ0+∑i=1mλiγ+lnγρ<0.
The conclusion follows from Theorem 4.1 immediately and the proof is completed.Remark 4.3.
Letϱ=infk∈N{tk-tk-1}>0. If γ⩾1, by Theorem 3.2, we can also give some other results on global exponential stability for (4.1). For the details, we omit them here.Example 4.4.
Consider a scalar nonlinear impulsive delay differential equation(4.16)x′(t)=F(t,x(t),x(t-δ(t))),t≠tk,Δx(tk)=ckx(tk-),t=tk,k∈ℕ,x(s)=ϕ(s),s∈[-τ,0]
on t⩾0, where δ:R+→[0,τ] is a continuous function, ck∈ℝ, and
(4.17)F(t,x,y)=bx-110x3-ycost,
with x=x(t),y=x(t-δ(t)), b>0.
From (4.8)-(4.9), we can see η=b, α1=1. By Corollary 4.2, if there exists a scalar number 0<α<1, such that
(4.18)|1+ck|⩽α,b+1α+lnαρ<0,
then the trivial solution of (4.16) is global exponentially stable.Example 4.5.
Consider the following linear impulsive delay system:(4.19)x′(t)=Ax(t)+Bx(t-δ(t)),t≠tk,t⩾0,Δx(tk)=Ckx(tk-),t=tk,k∈ℕ,x(s)=ϕ(s),-τ⩽s⩽0,ϕ∈PC,
where
(4.20)A=[0.10.2-0.10.20.150.300.240.1],B=[-0.120.0300.12-0.20.0500.14-0.1],Ck=[-0.5000-0.8000-0.4],δ:ℝ+→[0,τ] is a continuous function. Since λmin(A)=-0.2315 and λmax(A)=0.4388, we can not use the results in [5, 14] to determine the exponential stability. From Corollary 4.2, we can choose η=λmax((A+AT)/2)=0.4409, α1=∥B∥=0.2905, α=∥I+Ck∥=0.6, by (4.11), we obtain that if supk∈ℕ{tk-tk-1}<-αlnα/(α1+αλ1)=0.5521, then trivial solution of (4.19) is globally exponentially stable.Remark 4.6.
For (4.19), the authors in [15] chose δ(t)=(1/40)(1+e-t), τ=0.05, supk∈ℕ{tk-tk-1}⩽0.2 such that τ⩽tk-tk-1⩽0.2. However, if τ>supk∈ℕ{tk-tk-1}, the results in [15, 16] fail to determine the global exponential stability.


## 5. Conclusion

In this paper, some new Razumikhin-type theorems on global exponential stability for IFDEs are obtained by employing Lyapunov-Razumikhin technique. Some applications to IDDEs are also given. It should be mentioned that our results may allow us to develop an effective impulsive control strategy to stabilize an underlying delay dynamical system even if it may be unstable in practice, which is particularly meaningful for applications in engineering and technology. Two examples are also given to demonstrate the effectiveness of the theoretical results.



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*Source: 987372-2010-02-18.xml*