Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 289480, 12 pages
doi:10.1155/2009/289480
Research Article
Impulsive Exponential Stabilization of Functional
Differential Systems with Infinite Delay
Xiaoli Sun1 and Xiaodi Li2
1
2
Department of Mathematics and Information Science, Zaozhuang University, Zaozhuang 277100, China
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
Correspondence should be addressed to Xiaodi Li, sodymath@163.com
Received 3 June 2009; Revised 26 September 2009; Accepted 21 October 2009
Recommended by Binggen Zhang
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.
Copyright q 2009 X. Sun and X. Li. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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 secondorder 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.
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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 Section 2, 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
Let R denote the set of real numbers, R the set of nonnegative real numbers, and Rn the ndimensional real space equipped with the Euclidean norm · . For any t ≥ t0 ≥ 0 > −α ≥ −∞,
let ft, xs where s ∈ t − α, t or ft, 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:
x t
Δx|t
tk
ft, x·,
Ik tk , xt−k ,
t ≥ t 0 , t / tk ,
2.1
k
1, 2 . . . ,
where the impulse times tk satisfy 0 ≤ t0 < t1 < · · · < tk < · · · , limk → ∞ tk
∞, and x
n
denotes the right-hand derivative of x. f ∈ Ctk−1 , tk × C, R , ft, 0
0. C is an open
set in P C−α, 0, Rn , where P C−α, 0, Rn {ψ : −α, 0 → Rn is continuous everywhere
except at finite number of points tk , at which ψtk and ψt−k exist and ψtk ψtk }. Define
P CBt
{x ∈ C : x is bounded}. For ψ ∈ P CBt, the norm of ψ is defined by ψ
{ψ ∈ P CBσ : ψ < δ}. Let K1
{a ∈
sup−α≤θ≤0 |ψθ|. For any σ ≥ 0, let P CBδ σ
CR , R | a0 0 and at > 0 for t > 0 and a is strictly increasing in t}.
For each k 1, 2 . . . , Ik t, x ∈ Ct0 , ∞ × Rn , Rn , Ik t, 0 0, and for any ρ > 0, there
exists a ρ1 > 0 0 < ρ1 < ρ such that x ∈ Sρ1 implies that x Ik tk , x ∈ Sρ, where
Sρ {x : x < ρ, x ∈ Rn }.
For any given σ ≥ t0 , system 2.1 is supplemented with initial conditions of the form
xσ
φ,
2.2
where φ ∈ P C−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 xt, σ, φ; see 1, 4, 13. Since ft, 0
0, It, 0 0, k 1, 2, . . . , then xt 0 is a solution of 2.1-2.2, which is called the trivial
solution. In this paper, we always assume that the solution xt, σ, φ of 2.1-2.2 can be
continued to ∞ from the right of σ.
We introduce some definitions as follows.
Discrete Dynamics in Nature and Society
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Definition 2.1 see 1. The function V : −α, ∞ × C → R belongs to class v0 if
i V is continuous on each of the sets tk−1 , tk × C and limt,ϕ → t−k ,ψ V t, ϕ
exists;
V t−k , ψ
ii V t, x is locally Lipschitzian in x and V t, 0 ≡ 0.
Definition 2.2 see 1. Let V ∈ 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
D V t, ψ0
V t h, ψ0 hf t, ψ − V t, ψ0
.
lim sup
h
h → 0
2.3
Definition 2.3 see 1. Assume that xt
xt, 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 >
δε > 0 such that φ ∈ P CBδ σ implies
0, σ ≥ t0 , there exist constants λ > 0 and δ
xt < ε · e−λt−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 functions V ∈ 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 xη ≤ V t, x ≤ wx, t, x ∈ −α, ∞ × Sρ,
ii for any σ ≥ t0 and ψ ∈ P C−α, 0, Sρ, if γeλt−σ V t, ψ0 ≥ V t θ, ψθ, −α ≤
θ ≤ 0, t / tk , k ∈ Z , then
D V t, ψ0 ≤ pV t, ψ0 ,
3.1
iii for all tk , ψ ∈ R × P C−α, 0, Sρ1 , V tk , ψ0 Ik tk , ψ ≤ βk V t−k , ψ0, k ∈
Z , where βk satisfies ∞
k 1 max{βk M , 1} < ∞,
iv tk − tk−1 < ln γ/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 constant M > 0 such that
∞
k 1
max βk M , 1 ≤ M.
3.2
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For any ε ∈ 0, ρ1 , we choose δ δε > 0 such that wδ ≤ M M−1 min{c1 εη , ε}. Then, for
any σ ≥ t0 , let xt xt, σ, φ be a solution of 2.1-2.2 through σ, φ. We shall show for
any φ ∈ P CBδ σ
xt < ε · e−λ/ηt−σ ,
For convenience, suppose V t
V σ
3.3
t ≥ σ.
V t, xt and
sup V σ θ, ψθ , M V σ .
max
−α≤θ≤0
3.4
Since φ ∈ P CBδ σ, we have V σ ≤ M wδ.
Suppose that σ ∈ tl−1 , tl , l ∈ Z . Next we prove for t ∈ σ, tl ∪ tk , tk1 , k ≥ l,
k
V t ≤ V σ
−λt−σ
max βs M , 1 e
,
t ≥ σ.
3.5
s l
First, it is clear that for t ∈ σ − α, σ
c1 xtη ≤ V t ≤ V σ ≤ M wδ ≤ M−1 min{c1 εη , ε} < c1 εη ,
3.6
which implies xt < ε < ρ1 , t ∈ σ − α, σ.
We next claim that 3.5 holds for all t ∈ σ, tl , that is,
V t ≤ V σe−λt−σ ,
3.7
t ∈ σ, tl .
In order to do this, let
Γt
⎧
⎨V teλt−σ ,
t ≥ σ,
⎩V t,
σ − α ≤ t ≤ σ,
3.8
then it is obvious that Γt ≥ V t for all t ≥ σ − α.
Next we prove Γt ≤ V 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 Γt > V σ.
Let
t
then in view of Γσ
inf t ∈ σ, tl | Γt ≥ V σ ,
3.9
V σ ≤ M −1 V σ < V σ, we get
t ∈ σ, tl ,
Γt V σ,
Γt < V σ,
t ∈ σ, t .
3.10
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Considering 3.6, we also obtain
V t ≤ Γt < V σ
3.11
∀t ∈ σ − α, t .
On the other hand, considering γ ∈ 1, M , we get
Γt Γσ
V σ >
V σ <
1
V σ,
γ
3.12
1
1
V σ ≤ V σ.
M
γ
So we can define
t
1
sup t ∈ σ, t | Γt ≤ V σ ,
γ
3.13
which implies that
t ∈ σ, t ,
1
V σ
γ
Γt ,
1
V σ < Γt,
γ
t ∈ t , t .
3.14
It follows from 3.11 that
γΓt ≥ V σ ≥ V s,
γV teλt−σ
σ − α ≤ s ≤ t, t ∈ t , t .
3.15
Using condition ii, the inequality D V t ≤ pV t holds for all t ∈ t , t . Hence, we obtain
for t ∈ t , t D Γt
D V teλt−σ λV teλt−σ
eλt−σ D V t λV t
≤ eλt−σ pV t λV t
V teλt−σ p λ
≤ p λ Γt.
3.16
From iv, we define τ ˙ maxk∈Z {tk − tk−1 }, then τp λ < ln γ.
Thus, we have
t
dΓs
t Γs
Γt ds
Γt s
V σ
ds
≥ ln γ > τp λτ.
1/γ V σ s
3.17
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However,
Γt ds
≤
Γt s
t
p λ ds ≤
t
t τ
p λ ds
3.18
τp λτ.
t
This is a contradiction. Therefore, we obtain Γt ≤ V σ, t ∈ σ, tl , which implies that 3.5
holds for all t ∈ σ, tl .
Meanwhile, we get for t ∈ σ, tl c1 xη ≤ V t ≤ V σe−λt−σ ≤ V σ < M wδ ≤ M−1 min{c1 εη , ε} < c1 εη ,
3.19
which implies that xt−l ∈ Sρ1 , xtl ∈ Sρ.
Furthermore, note that
V tl ≤ βl V t−l ≤ βl V σe−λtl −σ ,
3.20
we next prove
V t ≤ max βl M , 1 V σe−λt−σ ,
3.21
t ∈ tl , tl1 ,
which is equal to prove
Γt ≤ max βl M , 1 V σ,
3.22
t ∈ tl , tl1 .
Suppose that this assertion is not true, then there exists some t ∈ tl , tl1 such that Γt >
max{βl M , 1}V σ.
Let
3.23
t∗ inf t ∈ tl , tl1 | Γt ≥ max βl M , 1 V σ .
Then we know
t∗ ∈ tl , tl1 ,
Γt∗ max βl M , 1 V σ,
Γt < max βl M , 1 V σ,
t ∈ tl , t∗ .
3.24
Also, in view of the fact Γt < V σ for t ∈ σ − α, tl , we obtain
Γt < max βl M , 1 V σ
∀t ∈ σ − α, t∗ .
3.25
Note that γ ∈ 1, M , then we have
1
max βl M , 1 V σ > max βl M , 1 V σ,
γ
1
1
Γtl ≤ βl V σe−λtl −σ < βl V σ ≤ βl M V σ ≤ max βl M , 1 V σ.
γ
γ
Γt∗ 3.26
3.27
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So we can define
t∗∗
1
sup t ∈ tl , t∗ | Γt ≤ max βl M , 1 V σ .
γ
3.28
1
max βl M , 1 V σ,
γ
1
Γt > max βl M , 1 V σ, t ∈ t∗∗ , t∗ .
γ
3.29
Then, we obtain
t∗∗ ∈ tl , t∗ ,
Γt∗∗ Thus, combining 3.25, we get
γV teλt−σ
γΓt
≥ max βl M , 1 V σ
3.30
≥ Γs
≥ V s,
σ − α ≤ s ≤ t, t ∈ t∗∗ , t∗ .
Similarly, by assumptions ii, iv, as the proof of 3.16, we can obtain
D Γt ≤ p λ Γt,
t ∈ t∗∗ , t∗ .
3.31
Consequently, we have
t∗
dΓs
t∗∗ Γs
Γt∗ ds
Γt∗∗ s
max{βl M ,1}V σ
ds
≥ ln γ > pτ λτ,
1/γ max{βl M ,1}V σ s
3.32
where τ ˙ maxk∈Z {tk − tk−1 }.
However, we note
t∗
dΓs
t∗∗ Γs
Γt∗ ds
≤
Γt∗∗ s
t∗∗ τ
t∗∗
p λ ds
pτ λτ,
3.33
which is a contradiction. So 3.21 holds.
Note that
V tl1 ≤ βl1 V t−l1 ≤ βl1 max βl M , 1 V σe−λtl1 −σ .
3.34
Similarly, we can prove
V t ≤ max βl1 M · max βl M , 1 , max βl M , 1 V σe−λt−σ ,
t ∈ tl1 , tl2 ,
3.35
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Discrete Dynamics in Nature and Society
that is,
V t ≤ max βl M , 1 max βl1 M , 1 V σe−λt−σ ,
t ∈ tl1 , tl2 .
3.36
By simple induction hypothesis, we may prove that for t ∈ σ, tl ∪ tk , tk1 , k ≥ l,
k
V t ≤ V σ
−λt−σ
max βs M , 1 e
,
t ≥ σ.
3.37
s l
Hence, 3.5 holds.
Under the help of conditions i, iii, and the definition of V σ, we arrive at
c1 xη ≤ V t
≤ V σ
k
−λt−σ
max βs M , 1 e
s l
≤ wδM
k
−λt−σ
max βs M , 1 e
s l
3.38
≤ wδM Me−λt−σ
≤ min{c1 εη , ε}e−λt−σ
≤ c1 εη e−λt−σ ,
t ≥ σ,
which implies that
xt < ε · e−λ/ηt−σ ,
t ≥ σ.
3.39
Therefore, 3.3 holds. The proof of Theorem 3.1 is therefore complete.
Remark 3.2. In Theorem 3.1, 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 functions V ∈ 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 × P C−α, 0, Sρ1 , V tk , ψ0 Ik tk , ψ ≤ βk V t−k , ψ0, k ∈
Z , where βk ≥ e−M and there exists a constant M > 0 such that
n
k l
ln βk n − l 1M < M
∀n ≥ l, l ∈ Z ,
3.40
then the trivial solution of 2.1-2.2 is exponentially stable with the approximate exponential
convergence rate λ/η.
Discrete Dynamics in Nature and Society
9
Especially, let γ 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 functions V ∈ v0 , w ∈ K1 and constants p, c1 , η, γ > 1, λ >
0 such that the following conditions hold:
i c1 xη ≤ V t, x ≤ wx, t, x ∈ −α, ∞ × Sρ;
ii for any σ ≥ t0 and ψ ∈ P C−α, 0, Sρ, if γeλt−σ V t, ψ0 ≥ V t θ, ψθ, −α ≤
θ ≤ 0, t / tk , k ∈ Z , then D V t, ψ0 ≤ pV t, ψ0;
iii for all tk , ψ ∈ R × P C−α, 0, Sρ1 , V tk , ψ0 Ik tk , ψ ≤ 1/γV t−k , ψ0, k ∈
Z ;
iv tk − tk−1 < ln γ/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. If α < ∞, 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 xp ≤ V t, x ≤ c2 xp , for any t ∈ R and x ∈ Rn ;
ii D V t, ϕ0 ≤ cV t, ϕ0, for all t ∈ tk−1 , tk , k ∈ Z , whenever qV t, ϕ0 ≤
V t s, ϕs for s ∈ −τ, 0, where q ≥ max{eα c , e2λα } is a constant;
iii V tk , ϕ0 Ik tk , ϕ ≤ dk V tk , ϕ0, where dk > 0, ∀k ∈ Z , are constants;
iv τ ≤ tk − tk−1 ≤ α and lndk λ cα < −λtk1 − tk , k ∈ Z .
It is easily seen that these conditions are more restrictive than ones given in Theorem 3.1.
dconstant, tk1 − tk
1, then it is necessary that condition lnd <
For example, let dk
−λ cα − λ holds see 23. Note in our Corollary 3.4, we only require that lnd < −λ − c,
where here γ 1/d, p c in Theorem 3.1. Moreover, we see that condition tk1 − 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 with D 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:
x t
1
1 0 s−0.1t
1
e−0.1t xt −
e
xt sds,
5 18
15 −∞
xk
1 √ x k− ,
e
xσ
φ.
k
1, 2 . . . ,
t ≥ 0, t / k,
3.41
10
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4500
Non-impulsive effects
4000
3500
xt
3000
2500
2000
1500
1000
500
0
0
5
10
15
20
25
30
t-axis
Figure 1
Choose V t x2 , we easily observe η 2, c1 1, ws s2 , tk − tk−1 ˙ τ 1. Let γ e, λ
√
0.2, p 2/5 2/18 2 e/15. It is easy to check that τp λ ≈ 0.37 < 1 ln γ.
0, then in view of Corollary 3.3, γeλt−σ V t, ψ0 ≥ V t Suppose that σ
t0
√ 0.1t
θ, ψθ, θ ≤ 0, implies that ee |xt| > |xt θ|, θ ≤ 0.
Hence,
D V t, x·
2xt
1 −0.1t
1
1 0 s−0.1t
e
e
xt sds
xt −
5 18
15 −∞
≤ 2x2 t
1
1
e−0.1t
5 18
1
15
0
−∞
√
es−0.1t ee0.1t ds
√ 0
1
e
1
e−0.1t es ds
5 18
15 −∞
√ 1
e
1
≤ 2x2 t
5 18 15
≤ 2x2 t
3.42
≤ pV t.
Besides, V k x2 k 1/ex2 k− 1/γV k− .
By Corollary 3.3, 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.
Remark 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.
Discrete Dynamics in Nature and Society
11
14
Impulsive effects
12
10
xt
8
6
4
2
0
0
5
10
15
20
25
30
t-axis
Figure 2
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 with
α < ∞ 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.
Acknowledgments
The authors are deeply grateful to the Associate Editor and two reviewers for their careful
reading of this paper and helpful comments, which have been very useful for improving the
quality of this paper.
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