Stability of impulsive functional differential equations

Nonlinear Analysis 68 (2008) 3665–3678www.elsevier.com/locate/na

Stability of impulsive functional differential equationsI

Yu Zhang, Jitao Sun∗

Department of Mathematics, Tongji University, 200092, PR China

Received 10 October 2006; accepted 19 April 2007

Abstract

In this paper the stability of impulsive functional differential equations in which the state variables on the impulses are related tothe time delay is studied. By using Lyapunov functions and Razumikhin techniques, some criteria of stability, asymptotic stabilityand practical stability for impulsive functional differential equations in which the state variables on the impulses are related to thetime delay are provided. Some examples are also presented to illustrate the efficiency of the results obtained.c© 2007 Elsevier Ltd. All rights reserved.

MSC: 34A37; 34K20; 93D20

Keywords: Impulsive functional differential equation; Time delay; Stability; Lyapunov function; Razumikhin technique

1. Introduction

Many evolution processes are characterized by the fact that at certain moments of time they experience a changeof state abruptly. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the formof impulses. It is known that many biological phenomena involving threshold, bursting rhythm models in medicineand biology, optimal control models in economics, pharmacokinetics and frequency modulated systems do exhibitimpulse effects. In recent years, many results have been obtained in analysis of systems with impulse effect ordesign of control systems via impulsive control laws [1–13, and references therein]. In [1], some basic theoriesincluding the stability theory of impulsive differential equations are given. Functional differential equations have awide application in our society, so it is important to study them. There are some results on functional differentialequations [14–16, and references therein]. Some authors have also considered the impulsive functional differentialequations and have got some results [4,5, and references therein]. In [4], Lyapunov–Razumikhin stability theoremsfor functional differential equations are extended to impulsive functional differential equations. In [5], some newRazumikhin type theorems for impulsive functional differential equations are presented. Since time delay exists inmany fields of our society, systems with time delay have received significant attention in the last few years [2,6,17,18,21, and references therein]. In [21], stability theorems for impulsive equations with infinite delay are obtained. Wecan easily see that in the previous works about impulsive functional differential equations the authors always suppose

I This work is supported by the National Natural Science Foundation of China (60474008).∗ Corresponding author. Fax: +86 21 65982341.

E-mail address: [email protected] (J. Sun).

0362-546X/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2007.04.009

http://www.elsevier.com/locate/na

mailto:[email protected]

http://dx.doi.org/10.1016/j.na.2007.04.009

3666 Y. Zhang, J. Sun / Nonlinear Analysis 68 (2008) 3665–3678

that the state variables on the impulses are only related to the present state variables. But in most cases, it is moreapplicable that the state variables on the impulses that we add are also related to the former state variables. But thereare rare results about these impulsive functional differential equations in which the state variables on the impulses arerelated to the time delay.

In many cases, some well-designed, asymptotically stable control schemes cannot work as expected. One reason isthat the domain of attraction is too small. A way to overcome this problem is to use practical stability. The practicalstability only needs to stabilize a system into a region of phase space. So it has a significant practice. In recentyears, the theory of practical stability has developed rather intensively [1,6,19,20, and references therein]—in [20]practical stability of nonlinear systems has been fully developed—but it is rare to find practical stability results forthese impulsive functional differential equations in which the state variables on the impulses are related to the timedelay.

In this paper, we consider the stability of impulsive functional differential equations in which the state variableson the impulses are related to the time delay. By using Lyapunov functions and the Razumikhin technique, we obtainsome results on stability, asymptotic stability and practical stability for impulsive functional differential equations inwhich the state variables on the impulses are related to the time delay. Some examples are also presented in this paperto illustrate the efficiency of the results obtained.

This paper is organized as follows. In Section 2, we introduce some basic definitions and notation. In Section 3, weget some criteria for stability, asymptotic stability and practical stability of impulsive functional differential equationsin which the state variables on the impulses are related to the time delay. Finally, concluding remarks are given inSection 4.

2. Preliminaries

Consider the following impulsive functional differential equations in which the state variables on the impulses arerelated to the time delay:

x(t) = f (t, xt ), t ≥ t0, t 6= τk

x(τk) = Ik(x(τ−

k ))+ Jk(x(τ−

k − τ)), k ∈ N (1)

where x ∈ Rn , f ∈ C[R+× D, Rn

], Ik, Jk ∈ C[Rn, Rn], k = 1, 2, . . ., D is an open set in PC([−τ, 0], Rn),

where τ = const. > 0, PC([−τ, 0], Rn) denotes the set of piecewise right continuous functions φ : [−τ, 0] → Rn

with the sup-norm |φ| = sup−τ≤s≤0 ‖φ(s)‖, where ‖ · ‖ is a norm in Rn . For each t ≥ t0, xt ∈ PC([−τ, 0], Rn) isdefined by xt (s) = x(t + s),−τ ≤ s ≤ 0, 0 = τ0 < τ1 < τ2 < · · · < τk < · · ·, τk → ∞ for k → ∞, τ > 0,x(t+) = lims→t+ x(s), and x(t−) = lims→t− x(s). Let R+

τ = [−τ,∞).A function x(t) is called a solution of (1) with the initial condition

xσ = ϕ (2)

where σ ≥ t0 and ϕ ∈ PC([−τ, 0], Rn), if it satisfies both (1) and (2).Throughout this paper we let the following hypotheses hold:

(H1) For t ∈ [σ − τ, σ ], the solution x(t; σ, ϕ) coincides with the function ϕ(t − σ).(H2) For each function x(s) : [σ − τ,∞) → Rn , which is continuous everywhere except at the points {τk} at which

x(τ+

k ), x(τ−

k ) exist and x(τ+

k ) = x(τk), f (t, xt ) is continuous for almost all t ∈ [σ,∞) and at the discontinuouspoints f is right continuous.

(H3) f (t, φ) is Lipschitzian in φ in each compact set in PC([−τ, 0], Rn).(H4) The functions Ik, Jk, k = 1, 2, . . ., are such that if x ∈ D, Ik 6= 0 and Jk 6= 0, then Ik(x)+ Jk(x(t − τ)) ∈ D.(H5) f (t, 0) ≡ 0, Ik(0) ≡ 0 and Jk(0) ≡ 0, k = 1, 2, . . ., so that x(t) ≡ 0 is a solution of (1), which we call the zero

solution.

Under the conditions (H1)–(H5), there exists a unique solution of Eq. (1) through (σ, ϕ). (The proof of it is similarto that in [2], so we omit it.)

We denote the solution of impulsive functional differential Eqs. (1) and (2) by x(t; σ, ϕ). Let J (σ, ϕ) denote themaximal interval of the type [σ − τ, β) in which x(t; σ, ϕ) is defined.

Y. Zhang, J. Sun / Nonlinear Analysis 68 (2008) 3665–3678 3667

We using the following notation:

S(ρ) = {x ∈ Rn: ‖x‖ < ρ},

PC(ρ) = {φ ∈ PC([−τ, 0], Rn) : |φ| < ρ},

Γ n= {h ∈ C[R+

× Rn, R+] : ∀t ∈ R+, inf

xh(t, x) = 0},

Γ nτ = {h ∈ C[R+

τ × Rn, R+] : ∀t ∈ R+

τ , infx

h(t, x) = 0}.

We introduce the following definitions:

Definition 1. Let h0 ∈ Γ nτ , φ ∈ PC([−τ, 0], Rn); for any t ∈ R+, h0(t, φ) is defined by

h0(t, φ) = sup−τ≤θ≤0

h0(t + θ, φ(θ)).

Definition 2. The zero solution of the system (1) is said to be:

(D1) stable, if for any σ ≥ t0 and ε > 0, there exists a δ = δ(σ, ε) > 0 such that ϕ ∈ PC(δ) implies that‖x(t; σ, ϕ)‖ < ε, t ≥ σ ;

(D2) uniformly stable, if δ in (D1) is independent of σ ;(D3) asymptotically stable, if (D1) holds and for any σ ≥ t0, there exists some δ = δ(σ ) > 0 such that if ϕ ∈ PC(δ),

then limt→∞ x(t; σ, ϕ) = 0;(D4) uniformly asymptotically stable, if (D2) holds and there exists some δ > 0 such that for any ε > 0, there exists

some T = T (δ, ε) > 0 such that if ϕ ∈ PC(δ), then ‖x(t; σ, ϕ)‖ ≤ ε for t ≥ σ + T .

Definition 3. Let h0 ∈ Γ nτ , h ∈ Γ n ; the system (1) is said to be

(A1) (h0, h)-practically stable, if given (u, v) with 0 < u < v, we have that h0(σ, ϕ) < u implies that h(t, x(t)) < v,t ≥ σ , for some σ ∈ R+;

(A2) (h0, h)-uniformly practically stable if (A1) holds for all σ ∈ R+.

Definition 4 ([1]). The function V : [t0,∞)× S(ρ) → R+ belongs to class v0 if:

(1) the function V is continuous on each of the sets [τk−1, τk)× S(ρ) and for all t ≥ t0, V (t, 0) ≡ 0;(2) V (t, x) is locally Lipschitzian in x ∈ S(ρ);(3) for each k = 1, 2, . . ., there exist finite limits

lim(t,y)→(τ−

k ,x)V (t, y) = V (τ−

k , x),

lim(t,y)→(τ+

k ,x)V (t, y) = V (τ+

k , x),

with V (τ+

k , x) = V (τk, x) satisfied.

Definition 5 ([1]). Let V ∈ v0; D+V is defined as

D+V (t, x(t)) = limh→0+

sup1h

{V (t + h, x(t + h))− V (t, x(t))}.

3. Main results

Now we consider the impulsive functional differential equations in which the state variables on the impulses arerelated to the time delay (1).

Let the sets K , K1, K2 be defined as

K = {w ∈ C(R+, R+) : strictly increasing and w(0) = 0},

K1 = {ϕ ∈ C(R+, R+) : increasing and ϕ(s) < s for > 0},

K2 = {ϕ ∈ C(R+, R+) : increasing}.


3.1. Uniform stability of the impulsive functional differential equations

In this part, we consider the uniform stability of the impulsive functional differential equations in which the statevariables on the impulses are related to the time delay (1). We have the following two theorems about the uniformstability of the system (1):

Theorem 1. Assume that there exist functions a, b ∈ K , V ∈ v0 such that

(i) a(‖x‖) ≤ V (t, x) ≤ b(‖x‖), for all (t, x) ∈ [t0 − τ,∞)× S(ρ);(ii) D+V (t, x) < 0;

(iii) V (τk, Ik(x(τ−

k )) + Jk(x(τ−

k − τ))) ≤1+bk

2 [V (τ−

k , x(τ−

k )) + V (τ−

k − τ, x(τ−

k − τ))], where bk ≥ 0, and∑∞

k=1 bk < ∞.

Then the zero solution of (1) is uniformly stable.

Proof. Since∑

∞

k=1 bk < ∞, it follows that∏

∞

k=1(1 + bk) = M ; obviously 1 ≤ M < ∞.For any ε > 0, there exists a δ = δ(ε) > 0 such that δ < b−1(

a(ε)M ). We will prove that if ϕ ∈ PC(δ) then

‖x(t; σ, ϕ)‖ < ε for t ≥ σ . Let x(t) = x(t; σ, ϕ) denote the solution through (σ, ϕ).Let σ ∈ [τm−1, τm) for some m ∈ N . Then, we will prove that

V (t, x(t)) ≤ b(δ), σ ≤ t < τm . (3)

Obviously, for t ∈ [σ − τ, σ ], there exists an s ∈ [−τ, 0] such that t = σ + s; then

V (t, x(t)) = V (σ + s, x(σ + s)) ≤ b(‖x(σ + s)‖) ≤ b(‖ϕ(s)‖) ≤ b(δ).

So if inequality (3) does not hold, then there exists an r ∈ (σ, τm) such that

V (r , x(r)) > b(δ),

V (t, x(t)) ≤ b(δ), t ∈ [σ − τ, r).

D+V (r , x(r)) ≥ 0.

This contradicts condition (ii), so (3) holds.In view of inequality (3) and condition (iii), we have

V (τm, x(τm)) = V (τm, Im(x(τ−m ))+ Jm(x(τ

−m − τ)))

≤1 + bm

2[V (τ−

m , x(τ−m ))+ V (τ−

m − τ, x(τ−m − τ))] ≤ (1 + bm)b(δ).

Next we prove that

V (t, x(t)) ≤ (1 + bm)b(δ), τm ≤ t < τm+1. (4)

If this does not hold, then there exists an s ∈ (τm, τm+1) such that

V (s, x(s)) > (1 + bm)b(δ),

V (t, x(t)) ≤ (1 + bm)b(δ), t ∈ [σ − τ, s).

D+V (s, x(s)) ≥ 0.

This contradicts condition (ii), so (4) holds.In view of inequality (4) and condition (iii), we have

V (τm+1, x(τm+1)) = V (τm+1, Im+1(x(τ−

m+1))+ Jm+1(x(τ−

m+1 − τ)))

≤1 + bm+1

2[V (τ−

m+1, x(τ−

m+1))+ V (τ−

m+1 − τ, x(τ−

m+1 − τ))]

≤ (1 + bm+1)(1 + bm)b(δ).

By simple induction, we can prove, in general, that for k = 0, 1, 2, . . .,

V (t, x(t)) ≤ (1 + bm+k) · · · (1 + bm)b(δ), τm+k ≤ t < τm+k+1.

V (τm+k+1, x(τm+k+1)) ≤ (1 + bm+k+1)(1 + bm+k) · · · (1 + bm)b(δ).


This together with inequality (3) yields

V (t, x(t)) ≤ Mb(δ), t ≥ σ.

From this and condition (i) we have

a(‖x(t)‖) ≤ V (t, x(t)) ≤ Mb(δ) < a(ε), t ≥ σ.

So

‖x(t)‖ < ε, t ≥ σ.

The zero solution of (1) is uniformly stable.The proof of Theorem 1 is completed. �

Theorem 2. Assume that there exist functions a, b ∈ K , V ∈ v0, g1, g2 ∈ K2, g = g1 + g2 and g ∈ K1 such that

(i) a(‖x‖) ≤ V (t, x) ≤ b(‖x‖), for all (t, x) ∈ [t0 − τ,∞)× S(ρ);(ii) D+V (t, x) ≤ p(t)c(V (t, x(t))), for all t 6= τk whenever g−1(V (t, x(t))) ≥ V (t + s, x(t + s)), s ∈ [−τ, 0],

where p, c : [t0 − τ,∞) → R+, locally integrable;(iii) V (τk, Ik(x(τ

−

k ))+ Jk(x(τ−

k − τ))) ≤ g1(V (τ−

k , x(τ−

k )))+ g2(V (τ−

k − τ, x(τ−

k − τ))), and there exists an A > 0

such that∫ τk+1τk

p(s)ds < A and∫ g−1(q)

qds

c(s) ≥ A.

Then the zero solution of (1) is uniformly stable.

Proof. For any ε > 0, there exists a δ = δ(ε) > 0 such that δ < b−1(g(a(ε))). We will prove that if ϕ ∈ PC(δ) then‖x(t; σ, ϕ)‖ < ε for t ≥ σ . Let x(t) = x(t; σ, ϕ) denote the solution through (σ, ϕ).

Let σ ∈ [τm−1, τm) for some m ∈ N . Then, we will prove that

V (t, x(t)) ≤ g−1(b(δ)), σ ≤ t < τm . (5)

Obviously, for t ∈ [σ − τ, σ ], there exists an s ∈ [−τ, 0] such that t = σ + s; then

V (t, x(t)) = V (σ + s, x(σ + s)) ≤ b(‖x(σ + s)‖) ≤ b(‖ϕ(s)‖) ≤ b(δ).

So if inequality (5) does not hold, then there exists an s ∈ (σ, τm) such that

V (s, x(s)) > g−1(b(δ)) > b(δ) ≥ V (σ, x(σ )).

From the continuity of V (t, x(t)) at [σ, τm), it follows that there exists an s1 ∈ (σ, s) such that

V (s1, x(s1)) = g−1(b(δ)),

V (t, x(t)) ≤ g−1(b(δ)), t ∈ [σ − τ, s1].

and also, there exists an s2 ∈ [σ, s1) such that

V (s2, x(s2)) = b(δ),

V (t, x(t)) ≥ b(δ), t ∈ [s2, s1].

Therefore, for t ∈ [s2, s1] and −τ ≤ s ≤ 0, we have

V (t + s, x(t + s)) ≤ g−1(b(δ)) ≤ g−1(V (t, x(t))).

In view of condition (ii), we get

D+V (t, x(t)) ≤ p(t)c(V (t, x(t))), s2 ≤ t ≤ s1.

Integrate this inequality over (s2, s1); we have by condition (iii)∫ V (s1,x(s1))

V (s2,x(s2))

du

c(u)≤

∫ s1

s2

p(s)ds ≤

∫ τm

τm−1

p(s)ds < A.


On the other hand,∫ V (s1,x(s1))

V (s2,x(s2))

du

c(u)=

∫ g−1(b(δ))

b(δ)

du

c(u)≥ A.

This is a contradiction, so (5) holds.In view of inequality (5) and condition (iii), we have


−m − τ)))

≤ g1(V (τ−m , x(τ−

m )))+ g2(V (τ−m − τ, x(τ−

m − τ)))

≤ g1(g−1(b(δ)))+ g2(g

−1(b(δ))) = g(g−1(b(δ))) = b(δ).

Next, we prove that

V (t, x(t)) ≤ g−1(b(δ)), τm ≤ t < τm+1. (6)

If this does not hold, then there exists an r ∈ (τm, τm+1) such that

V (r , x(r)) > g−1(b(δ)) > b(δ) ≥ V (τm, x(τm)).

From the continuity of V (t, x(t)) at [τm, τm+1), it follows that there exists an r1 ∈ (τm, r) such that

V (r1, x(r1)) = g−1(b(δ)),

V (t, x(t)) ≤ g−1(b(δ)), t ∈ [σ − τ, r1].

and also, there exists an r2 ∈ [τm, r1) such that

V (r2, x(r2)) = b(δ),

V (t, x(t)) ≥ b(δ), t ∈ [r2, r1].

Therefore, for t ∈ [r2, r1] and −τ ≤ s ≤ 0, we have

V (t + s, x(t + s)) ≤ g−1(b(δ)) ≤ g−1(V (t, x(t))).

In view of condition (ii), we get

D+V (t, x(t)) ≤ p(t)c(V (t, x(t))), r2 ≤ t ≤ r1.

Integrate this inequality over (r2, r1); we have by condition (iii)∫ V (r1,x(r1))

V (r2,x(r2))

du

c(u)≤

∫ r1

r2

p(s)ds ≤

∫ τm+1

τm

p(s)ds < A.

On the other hand,∫ V (r1,x(r1))

V (r2,x(r2))

du

c(u)=

∫ g−1(b(δ))

b(δ)

du

c(u)≥ A.

This is a contradiction, so (6) holds.In view of inequality (6) and condition (iii), we have

V (τm+1, x(τm+1)) = V (τm+1, Im+1(x(τ−

m+1))+ Jm+1(x(τ−

m+1 − τ)))

≤ g1(V (τ−

m+1, x(τ−

m+1)))+ g2(V (τ−

m+1 − τ, x(τ−

m+1 − τ)))

≤ g1(g−1(b(δ)))+ g2(g

−1(b(δ))) = g(g−1(b(δ))) = b(δ).

By simple induction, we can prove, in general, that for i = 0, 1, 2, . . .,

V (t, x(t)) ≤ g−1(b(δ)), τm+i ≤ t < τm+i+1.

V (τm+i+1, x(τm+k+1)) ≤ b(δ). (7)


Since b(δ) < g−1(b(δ)), by inequalities (5) and (7), we have

V (t, x(t)) ≤ g−1(b(δ)), t ≥ σ.

From this and condition (i) we have

a(‖x(t)‖) ≤ V (t, x(t)) ≤ g−1(b(δ)) < a(ε), t ≥ σ.

So

‖x(t)‖ < ε, t ≥ σ.

The zero solution of (1) is uniformly stable.The proof of Theorem 2 is completed. �

3.2. Uniform asymptotic stability of the impulsive functional differential equations

In this part, we consider the uniform asymptotic stability of the impulsive functional differential equations in whichthe state variables on the impulses are related to the time delay (1). We have the following theorem about the uniformasymptotic stability of the system (1):

Theorem 3. Assume that there exist functions a, b ∈ K , V ∈ v0, g1, g2 ∈ K2, g = g1 + g2 and g ∈ K1 such that:

(i) Conditions (i)–(ii) of Theorem 2 hold.(ii) V (τk, Ik(x(τ

−

k ))+ Jk(x(τ−

k −τ))) ≤ g1(V (τ−

k , x(τ−

k )))+g2(V (τ−

k −τ, x(τ−

k −τ))); let r = supk∈N {τk −τk−1} <

∞,M1 = supt≥0∫ t+r

t p(s)ds < ∞, and M2 = infq>0∫ q

g(q)ds

c(s) > M1.

Then the impulsive functional differential equation in which the state variables on the impulses are related to thetime delay (1) is uniformly asymptotically stable.

Proof. Obviously the conditions of this theorem imply that the conditions of Theorem 2 hold, so the impulsivefunctional differential equation (1) is uniformly stable. This implies that for given η > 0, there exists a δ > 0,such that g−1(b(δ)) = a(η), and ϕ ∈ PC(δ) implies that ‖x‖ < η holds for t ≥ σ . Moreover

V (t, x(t)) ≤ g−1(b(δ)) = a(η), t ≥ σ − τ. (8)

Let σ ∈ [τm−1, τm) for some m ∈ N . Now, let ε > 0 and assume without loss of generality that ε < η. DefineM = M(ε) = sup{

1c(s) |g(a(ε)) ≤ s ≤ b(η)}, and note that 0 < M < ∞.

For a(ε) ≤ q ≤ b(η), we have g(a(ε)) ≤ g(q) < q ≤ b(η), so M2 ≤∫ q

g(q)ds

c(s) ≤ M(q − g(q)), from which we

obtain g(q) ≤ q −M2M < q − d , where d = d(ε) > 0 is chosen such that d < M2−M1

M .Let N = N (ε) be the smallest positive integer for which b(η) < a(ε) + Nd and define T = T (ε) =

r + (N − 1)(r + τ), we will prove that if ϕ ∈ PC(δ), then ‖x(t)‖ < ε for t ≥ σ + T .Given 0 < A ≤ b(η) and j ∈ N , we will show that:

(a) if V (t, x(t)) ≤ A for t ∈ [τ j − τ, τ j ), then V (t, x(t)) ≤ A for t ≥ τ j ;(b) if in addition A ≥ a(ε), then V (t, x(t)) ≤ A − d for t ≥ τ j .

First, we prove (a).If (a) does not hold, then there exists some t ≥ τ j , V (t, x(t)) > A. Then let t∗ = inf{t ≥ τ j |V (t, x(t)) > A}. Thus

for some k ∈ N and k ≥ j, t∗ ∈ [τk, τk+1). Since by condition (ii), V (τk, x(τk)) = V (τk, Ik(x(τ−

k )) + Jk(x(τ−

k −

τ))) ≤ g1(V (τ−

k , x(τ−

k ))) + g2(V (τ−

k − τ, x(τ−

k − τ))) ≤ g1(A) + g2(A) = g(A) < A, then t∗ ∈ (τk, τk+1).Moreover, V (t∗, x(t∗)) = A and V (t, x(t)) ≤ A for t ∈ [τ j − τ, t∗].

Let t = sup{t ∈ [τk, t∗]|V (t, x(t)) ≤ g(A)}. Since V (t∗, x(t∗)) = A > g(A), then t ∈ [τk, t∗), V (t, x(t)) =

g(A), and V (t, x(t)) ≥ g(A) for t ∈ [t, t∗].Thus, for t ∈ [t, t∗], s ∈ [−τ, 0], we have g(V (t + s, x(t + s))) ≤ g(A) ≤ V (t, x(t)). So from condition (i), for

t ∈ [t, t∗] the following inequality holds:

D+V (t, x(t)) ≤ p(t)c(V (t, x(t))).


We get∫ V (t∗,x(t∗))

V (t,x(t))ds

c(s) ≤∫ t∗

t p(s)ds ≤∫ τk+1τk

p(s)ds ≤ M1. But at the same time∫ V (t∗,x(t∗))

V (t,x(t))ds

c(s) =∫ A

g(A)ds

c(s) ≥

M2 > M1. This is a contradiction, so (a) holds.Next, we prove (b).Assume for the sake of the contradiction that there exists some t ≥ τ j , V (t, x(t)) > A − d. Then define

r1 = inf{t ≥ τ j |V (t, x(t)) > A − d} and let k ≥ j be chosen so that r1 ∈ [τk, τk+1).Since a(ε) ≤ A ≤ b(η), then g(A) < A − d . So from (a), V (τk, x(τk)) = V (τk, Ik(x(τ

−

k )) + Jk(x(τ−

k − τ))) ≤

g1(V (τ−

k , x(τ−

k )))+ g2(V (τ−

k − τ, x(τ−

k − τ))) ≤ g1(A)+ g2(A) = g(A) < A − d. Thus r1 ∈ (τk, τk+1). MoreoverV (r1, x(r1)) = A − d, and for t ∈ [τk, r1), V (t, x(t)) ≤ A − d.

Let r = sup{t ∈ [τk, r1]|V (t, x(t)) ≤ g(A)}; since V (r1, x(r1)) = A − d > g(A) ≥ V (τk, x(τk)), thenr ∈ [τk, r1), V (r , x(r)) = g(A), and V (t, x(t)) ≥ g(A) for t ∈ [r , r1]. Thus, for t ∈ [r , r1], s ∈ [−τ, 0], from(a) we have g(V (t + s, x(t + s))) ≤ g(A) ≤ V (t, x(t)). So from condition (i), for t ∈ [r , r1] the following inequalityholds:

D+V (t, x(t)) ≤ p(t)c(V (t, x(t))).

Then, we obtain the inequality∫ V (r1,x(r1))

V (r ,x(r))

ds

c(s)≤ M1.

But on the other hand,∫ V (r1,x(r1))

V (r ,x(r))

ds

c(s)=

∫ A−d

g(A)

ds

c(s)=

∫ A

g(A)

ds

c(s)−

∫ A

A−d

ds

c(s).

Since a(ε) ≤ A ≤ b(η), we have g(a(ε)) ≤ g(A) < A − d < A ≤ b(η). Thus 1c(s) ≤ M for A − d ≤ s ≤ A.

So, we get∫ V (r1,x(r1))

V (r ,x(r))

ds

c(s)≥ M2 −

∫ A

A−dMds = M2 − d M > M2 + M1 − M2 = M1.

This is a contradiction, so (b) holds.We define the indices k(i) for i = 1, 2, . . . , N as follows. Let k(1) = m, and for i = 2, . . . , N , let k(i) be chosen so

that τk(i)−1 < τk(i−1) + τ ≤ τk(i) .Then from condition (ii), we have τk(1) = τm ≤ τm−1 + r ≤ σ + r and for i = 2, . . . , N , τk(i) ≤ τk(i)−1 + r <

τk(i−1) + r + τ . Combining these inequalities gives us

τk(N ) ≤ σ + r + (r + τ)(N − 1) = σ + T .

We claim that for each i = 1, 2, . . . , N , V (t, x(t)) ≤ b(η)−id for t ≥ τk(i) . Since by (8) V (t, x(t)) ≤ a(η) ≤ b(η)for t ∈ [σ − τ, τk(1)), then by setting A = b(η) in our earlier argument (b), we get V (t, x(t)) ≤ b(η)− d for t ≥ τk(1) ,which establishes the base case. We now proceed by induction and assume V (t, x(t)) ≤ b(η) − jd for t ≥ τk( j) forsome 1 ≤ j ≤ N − 1. Let A = b(η)− jd; then a(ε) ≤ A ≤ b(η). Since τk( j) ≤ τk( j+1) − τ , then V (t, x(t)) ≤ A fort ∈ [τk( j+1) − τ, τ( j+1)) and so V (t, x(t)) ≤ A − d = b(η)− ( j + 1)d for t ≥ τk( j+1) . So we have proved our claim byinduction.

When j = N − 1, we get

V (t, x(t)) ≤ b(η)− Nd < a(ε), t ≥ τk(N ) .

Since σ + T ≥ τk(N ) , by condition (i), we get

‖x(t)‖ ≤ ε, t ≥ σ + T .

Thus the impulsive functional differential equation in which the state variables on the impulses are related to thetime delay (1) is uniformly asymptotically stable.

The proof of Theorem 3 is completed. �


3.3. Uniform practical stability of impulsive functional differential equations

In this part, we consider the uniform practical stability of the impulsive functional differential equations in whichthe state variables on the impulses are related to the time delay (1). We have the following two theorems about theuniform practical stability of the system (1):

Theorem 4. Assume the following conditions hold:

(i) 0 < u < v are given.(ii) h0, h ∈ Γ n, h(t, x) ≤ φ(h0(t, xt )) with φ ∈ K , whenever h0(t, xt ) < u.

(iii) There exists a function V ∈ v0 such that β(h(t, x)) ≤ V (t, x) ≤ α(h0(t, x)) for (t, x) ∈ [t0 − τ,∞) × S(ρ),where α, β ∈ K , h0 ∈ Γ n

τ .(iv) V (t, x(t)) ≥ sup{V (t + s, x(t + s)) : s ∈ [−τ, 0]} implies that D+V (t, x(t)) < 0.(v) For all k ∈ Z+ and x ∈ S(ρ), V (τk, Ik(x(τ

−

k ))+ Jk(x(τ−

k −τ))) ≤1+ck

2 [V (τ−

k , x(τ−

k ))+V (τ−

k −τ, x(τ−

k −τ))],where ck ≥ 0 and

∑∞

k=1 ck < ∞.(vi) φ(u) < v, Mα(u) < β(v), where

∏∞

k=1(1 + ck) = M.

Then the impulsive functional differential equation in which the state variables on the impulses are related to thetime delay (1) with respect to (u, v) is (h0, h)-uniformly practically stable.

Proof. From Section 2, we know that for any σ ∈ R+, there is a unique solution of problem (1) through (σ, ϕ). Since∑∞

k=1 ck < ∞, it follows that 1 ≤ M < ∞.Let σ ∈ [τm−1, τm) for some m ∈ N . If (σ, xσ ) ∈ R+

× PC([−τ, 0], Rn) such that h0(σ, xσ ) < u. Then byconditions (ii) and (vi),

h(σ, x(σ )) ≤ φ(h0(σ, xσ )) < φ(u) < v.

We then prove that

V (t, x(t)) ≤ Mα(u), t ≥ σ. (9)

For any t ∈ [σ − τ, σ ], there exists a θ ∈ [−τ, 0] such that t = σ + θ ; then from Definition 1, we know that fort ∈ [σ − τ, σ ],

h0(t, x(t)) = h0(σ + θ, x(σ + θ)) = h0(σ + θ, xσ (θ)) ≤ h0(σ, xσ ) < u.

V (t, x(t)) ≤ α(h0(t, x)) ≤ α(h0(σ, xσ )) < α(u). (10)

Next, we prove that

V (t, x(t)) ≤ α(u), σ ≤ t < τm . (11)

If this does not hold, then there exists an s ∈ [σ, τm) such that

V (s, x(s)) > α(u) > V (σ, x(σ )).

Let s = inf{t |V (t, x(t)) > α(u), t ∈ [σ, τm)}; then V (s, x(s)) = α(u), D+V (s, x(s)) ≥ 0, and from (10),V (s + s, x(s + s)) ≤ α(u) = V (s, x(s)) for s ∈ [−τ, 0]. By condition (iv), we have D+V (s, x(s)) < 0. Thisis a contradiction, so (11) holds.

By condition (v) and (11) we have


−m − τ)))

≤1 + cm

2[V (τ−

m , x(τ−m ))+ V (τ−

m − τ, x(τ−m − τ))] ≤ (1 + cm)α(u).

Next, we prove that

V (t, x(t)) ≤ (1 + cm)α(u), τm ≤ t < τm+1. (12)

If this does not hold, then there exists an r ∈ [τm, τm+1) such that

V (r , x(r)) > (1 + cm)α(u).


Let r = inf{t |V (t, x(t)) > (1 + cm)α(u), t ∈ [τm, τm+1)}; then V (r , x(r)) = (1 + cm)α(u), D+V (r , x(r)) ≥ 0, andfrom (10) and (11) V (r + s, x(r + s)) ≤ (1 + cm)α(u) = V (r , x(r)) for s ∈ [−τ, 0]. By condition (iv), we haveD+V (r , x(r)) < 0. This is a contradiction, so (12) holds.

By condition (v) and (12) we have

V (τm+1, x(τm+1)) = V (τm+1, Im+1(x(τ−

m+1))+ Jm+1(x(τ−

m+1 − τ)))

≤1 + cm+1

2[V (τ−

m+1, x(τ−

m+1))+ V (τ−

m+1 − τ, x(τ−

m+1 − τ))]

≤ (1 + cm+1)(1 + cm)α(u).

By arguments similar to those used before, we can prove that for k = 1, 2, . . .,

V (t, x(t)) ≤ (1 + cm)(1 + cm+1) · · · (1 + cm+k)α(u), τm+k ≤ t < τm+k+1.

Together with (11), this yields that

V (t, x(t)) ≤ Mα(u), t ≥ σ.

By condition (vi), we have

V (t, x(t)) ≤ Mα(u) < β(v), t ≥ σ.

So, from condition (iii), we get

h(t, x(t)) ≤ β−1(V (t, x(t))) < β−1(β(v)) = v, t ≥ σ.

Thus, the impulsive functional differential equation in which the state variables on the impulses are related to thetime delay (1) with respect to (u, v) is (h0, h)-uniformly practically stable.


Theorem 5. Assume the following conditions hold:

(i) (i)–(iii) of Theorem 4 hold.(ii) There exist functions ψ1, ψ2 ∈ K2, ψ = ψ1 + ψ2 and ψ ∈ K1 such that for any solution x(t) of problem (1),

ψ−1(V (t, x(t))) > sup{V (t + s, x(t + s)) : s ∈ [−τ, 0]}, implies that D+V (t, x(t)) ≤ g(t)w(V (t, x(t))), whereg, w : [t0 − τ,∞) → R+, locally integrable. Also, for all k ∈ Z+ and x ∈ S(ρ), V (τk, Ik(x(τ

−

k ))+ Jk(x(τ−

k −

τ))) ≤ ψ1(V (τ−

k , x(τ−

k )))+ ψ2(V (τ−

k − τ, x(τ−

k − τ))).(iii) φ(u) < v, α(u) < ψ(β(v)).

(iv) There exists a constant A > 0 such that∫ τkτk−1

g(s)ds < A, and for any q > 0,∫ ψ−1(q)

qdsw(s) ≥ A.

Then the impulsive functional differential equation in which the state variables on the impulses are related to thetime delay (1) with respect to (u, v) is (h0, h)-uniformly practically stable.

Proof. From Section 2, we know that for any σ ∈ R+, there is a unique solution of problem (1) through (σ, ϕ).Let σ ∈ [τm−1, τm) for some m ∈ N . Suppose (σ, xσ ) ∈ R+

× PC([−τ, 0], Rn) such that h0(σ, xσ ) < u. Thenby conditions (i) and (iii),

h(σ, x(σ )) ≤ φ(h0(σ, xσ )) < φ(u) < v.

We will prove that

V (t, x(t)) ≤ ψ−1(α(u)), t ≥ σ. (13)

For any t ∈ [σ − τ, σ ], there exists a θ ∈ [−τ, 0], such that t = σ + θ ; then from Definition 1, we know that fort ∈ [σ − τ, σ ],

h0(t, x(t)) = h0(σ + θ, x(σ + θ)) = h0(σ + θ, xσ (θ)) ≤ h0(σ, xσ ) < u.

Since ψ ∈ K1, from condition (i), we have for t ∈ [σ − τ, σ ],

V (t, x(t)) ≤ α(h0(t, x)) ≤ α(h0(σ, xσ )) < α(u) < ψ−1(α(u)). (14)


Firstly, we prove that

V (t, x(t)) ≤ ψ−1(α(u)), σ ≤ t < τm . (15)

If this does not hold, then there exists an s ∈ [σ, τm) such that

V (s, x(s)) > ψ−1(α(u)) > α(u) > V (σ, x(σ )).

Let s = inf{t |V (t, x(t)) > ψ−1(α(u)), t ∈ [σ, τm)}; then V (s, x(s)) = ψ−1(α(u)), and since V (σ, x(σ )) <ψ−1(α(u)), we have s > σ , and for s < t ≤ s, V (t, x(t)) > ψ−1(α(u)). From (14) and the definitionof s, we also have for σ − τ ≤ t ≤ s, V (t, x(t)) ≤ ψ−1(α(u)). Since α(u) < ψ−1(α(u)), V (σ, x(σ )) <α(u), V (s, x(s)) = ψ−1(α(u)), and V (t, x(t)) is continuous in [σ, τm), it follows that there exists an s1 ∈ [σ, s),such that V (s1, x(s1)) = α(u) and for s1 ≤ t < s, V (t, x(t)) ≥ α(u).

Therefore, from inequality (14), for t ∈ [s1, s] and s ∈ [−τ, 0], we have

V (t + s, x(t + s)) ≤ ψ−1(α(u)) ≤ ψ−1(V (t, x(t))).

In view of condition (ii), we have for t ∈ [s1, s],

D+V (t, x(t)) ≤ g(t)w(V (t, x(t))).

Integrate this inequality over (s1, s); by condition (iv) we have∫ V (s,x(s))

V (s1,x(s1))

dx

w(x)≤

∫ s

s1

g(t)dt ≤

∫ τm

σ

g(t)dt < A.

At the same time,∫ V (s,x(s))

V (s1,x(s1))

dx

w(x)=

∫ ψ−1(α(u))

α(u)

dx

w(x)≥ A.

This is a contradiction, so (15) holds.By condition (ii) and (15) we have


−m − τ)))

≤ ψ1(V (τ−m , x(τ−

m )))+ ψ2(V (τ−m − τ, x(τ−

m − τ)))

≤ ψ1(ψ−1(α(u)))+ ψ2(ψ

−1(α(u))) = ψ(ψ−1(α(u))) = α(u).

Next, we prove that

V (t, x(t)) ≤ ψ−1(α(u)), τm ≤ t < τm+1. (16)

If this does not hold, then there exists an r ∈ [τm, τm+1) such that

V (r , x(r)) > ψ−1(α(u)) > α(u) > V (τm, x(τm)).

Let r = inf{t |V (t, x(t)) > ψ−1(α(u)), t ∈ [τm, τm+1)}; then V (r , x(r)) = ψ−1(α(u)), and since V (τm, x(τm)) <

ψ−1(α(u)), we have r > τm , and for r < t ≤ r , V (t, x(t)) > ψ−1(α(u)). From (14) and (15) and the definitionof r , we also have for σ − τ ≤ t ≤ r , V (t, x(t)) ≤ ψ−1(α(u)). Since α(u) < ψ−1(α(u)), V (τm, x(τm)) <

α(u), V (r , x(r)) = ψ−1(α(u)), and V (t, x(t)) is continuous in [τm, τm+1), it follows that there exists an s1 ∈ [τm, r),such that V (r1, x(r1)) = α(u) and for r1 ≤ t < r , V (t, x(t)) ≥ α(u).

Therefore, from inequality (14) and (15), for t ∈ [r1, r ] and s ∈ [−τ, 0], we have

V (t + s, x(t + s)) ≤ ψ−1(α(u)) ≤ ψ−1(V (t, x(t))).

In view of condition (ii), we have for t ∈ [r1, r ],

D+V (t, x(t)) ≤ g(t)w(V (t, x(t))).

Integrate this inequality over (r1, r); by condition (iv) we have∫ V (r ,x(r))

V (r1,x(r1))

dx

w(x)≤

∫ r

r1

g(t)dt ≤

∫ τm+1

τm

g(t)dt < A.


At the same time,∫ V (r ,x(r))

V (r1,x(r1))

dx

w(x)=

∫ ψ−1(α(u))

α(u)

dx

w(x)≥ A.

This is a contradiction, so (16) holds.By condition (ii) and (16) we have

V (τm+1, x(τm+1)) = V (τm+1, Im+1(x(τ−

m+1))+ Jm+1(x(τ−

m+1 − τ)))

≤ ψ1(V (τ−

m+1, x(τ−

m+1)))+ ψ2(V (τ−

m+1 − τ, x(τ−

m+1 − τ)))

≤ ψ1(ψ−1(α(u)))+ ψ2(ψ

−1(α(u))) = ψ(ψ−1(α(u))) = α(u).

By arguments similar to those used before, we can prove that for k = 1, 2, . . .,

V (t, x(t)) ≤ ψ−1(α(u)), τm+k−1 ≤ t < τm+k .

V (τm+k, x(τm+k)) ≤ α(u). (17)

Since α(u) < ψ−1(α(u)), it follows by inequalities (15) and (17) and conditions (iii), (i) that

V (t, x(t)) ≤ ψ−1(α(u)) < β(v),

h(t, x(t)) ≤ β−1(V (t, x(t))) < β−1(β(v)) = v, t ≥ σ.

So the impulsive functional differential equation in which the state variables on the impulses are related to the timedelay (1) with respect to (u, v) is (h0, h)-uniformly practically stable.


Example 1. Consider the following impulsive functional differential equations in which the state variables on theimpulses are related to the time delay:

x(t) = ax(t)+ bx(t − τ), t 6= τk

x(τk) = cx(τ−

k )+ dx(τ−

k − τ). (18)

where x ∈ R, τ > 0, a, b, c, d > 0 and c + d < 1.If τk − τk−1 <

−2(c+d)2 ln(c+d)(2a+b)(c+d)2+b

, then the zero solution of (18) is uniformly stable.

Let V =12 x2, g1(x) = c(c + d)x, g2(x) = d(c + d)x , A = −2 ln(c + d); by using Theorem 2 we can get this result.


x(t) = −y3(t) sin(x(t − 1))− 5x(t)+ y2(t − 1), t 6= τk

y(t) = x(t) sin(x(t − 1))−52

y(t)+ y(t − 1), t 6= τk

x(τk) =13

x(τ−

k ), y(τk) =

√5

6y(τ−

k − 1), k ∈ N .

(19)

If (8√

2 − 10)(τk − τk−1) < ln 4, then the zero solution of (19) is uniformly stable.

Let V (x, y) =12 x2

+14 y4, g1(s) =

19 s, g2(s) =

536 s, g(s) = g1(s)+g2(s) =

14 s, p(t) = 8

√2−10, c(s) = s, A =

ln 4 > 0; then all conditions of Theorem 2 are satisfied, and the zero solution of (19) is uniformly stable.


x(t) = −a(t)x(t)+ b(t)x(t − τ)+ x(t)sat(u(t)), t 6= τk

x(τk) = cx(τ−

k )+ dx(τ−

k − τ), k ∈ N . (20)


where x ∈ Rn , τ > 0, c > 0, d > 0, c + d < 1, a(t), b(t) ∈ C[R+, R+], a(t) ≥ a > 0, b(t) ≤ b,

(1 +1

(c+d)2)b + 2K − 2a > 0, sat(u(t)) is a saturating nonlinear control, which is defined by

sat(u(t)) =

K , u(t) ≥ K > 0;u(t), − K < u(t) < K ;−K , u(t) ≤ −K < 0.

where K is a constant satisfying K > 0.Denote x ∈ Rn as x = (x1, x2, . . . , xn).Let h(t, x) = ‖x‖1 =

∑ni=1 |xi |, h0(t, x) = ‖x‖∞ = max1≤i≤n |xi |. From the definition of h0, we know that

h0(t, xt ) = sup−τ≤θ≤0 h0(t + θ, x(t + θ)) = sup−τ≤θ≤0 ‖x(t + θ)‖∞ = |xt |∞.For given (u, v), with 0 < u < c+d

n v, suppose the following assumptions hold:

(H1) tk − tk−1 <−2 ln(c+d)

−2a+2K+(1+1

(c+d)2)b

.

(H2) |xt |∞ < u implies that for any s ∈ [−τ, 0], ‖x(t)‖1 <1

c+d ‖x(t + s)‖1 holds.

Then (20) is (h0, h)-uniformly practically stable with respect to (u, v).

Let V (t, x(t)) = xT (t)x(t), ψ1(t) = (c2+ cd)t, ψ2(t) = (cd + d2)t, ψ(t) = (c + d)2t, β(x) =

1n x2, α(x) =

nx2, w(t) = t, g(t) = −2a + 2K + (1 +1

(c+d)2)b, φ(t) =

nc+d t, A = −2 ln(c + d) > 0; by using Theorem 5 we can

get this result.

4. Conclusion

In this paper, the stability of impulsive functional differential equations in which the state variables on the impulsesare related to the time delay is considered. By using Lyapunov functions and Razumikhin techniques, some resultsare provided. Stability theorems for impulsive differential equations have been extended to impulsive functionaldifferential equations in which the state variables on the impulses are related to the time delay. Some examples arealso presented to illustrate the efficiency of the results obtained. It should be noted that this is the first time that thestability of impulsive functional differential equations in which the state variables on the impulses are related to thetime delay has been studied. We can see that impulses do continue to the system’s stability behavior. In the future,we will do some further research on impulsive functional differential equations in which the state variables on theimpulses are related to the time delay.

Acknowledgements

The authors would like to thank the associate editor and the anonymous reviewer for their constructive commentsand suggestions for improving the quality of the paper.

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Stability of impulsive functional differential equations