Journal of Computational and Applied Mathematics 176 (2005) 223–229

www.elsevier.com/locate/cam

Eventual practical stability of impulsive differential equationswith time delay in terms of two measurements�

Yu Zhang, Jitao Sun∗

Department of Applied Mathematics, Tongji University, 1239, Siping Rd., Shanghai 200092, PR China

Received 3 February 2004; received in revised form 19 June 2004

Abstract

In this paper we introduce a new stability—eventual practical stability for impulsive differential equations withtime delay. By using Lyapunov functions and comparison principle, we will get some criteria of eventual practicalstability, eventual practical quasistability and strong eventual practical stability for impulsive differential equationswith time delay in terms of two measurements.© 2004 Elsevier B.V. All rights reserved.

Keywords:Impulsive differential equation; Time delay; Eventual practical stability; Lyapunov function

1. Introduction

In recent years, significant progress has been made in the theory of impulsive differential[1,4–12]andreferences therein. Since practical stability only needs to stabilize a system into a region of phase space, ithas been widely used in application.And the theory of practical stability has developed rather intensively[10,1–3]and references therein. In Ref.[10,1], the authors have gotten some results for practical stabilityof impulsive differential equations. In Ref.[5], the author has obtained some results for eventual stabilityof impulsive differential equations. In Ref.[2], the authors have obtained some results for the practicalstability of finite delay differential systems in terms of two measures, but there does not exist impulses.However, in the present paper, we will introduce a new stability for impulsive differential equation

with time delay—eventual practical stability.And we will consider this stability for impulsive differential

� This work was supported by NNSF and Shanghai City Natural Science Foundation of China under Grant 03ZR14095.∗ Corresponding author. Tel.: +86-21-5103-0747; fax: +86-21-6598-2341.E-mail address:sunjt@sh163.net(J. Sun).

0377-0427/$ - see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.cam.2004.07.014

224 Y. Zhang, J. Sun / Journal of Computational and Applied Mathematics 176 (2005) 223–229

equations with time delay in terms of two measurements. By using Lyapunov functions and comparisonprinciple, we will get some results.This paper is organized as follows. In Section 2, we introduce some basic definitions and notations.

In Section 3, we get some criteria for eventual practical stability, eventual practical quasistability andstrong eventual practical stability for impulsive differential equations with time delay in terms of twomeasurements. Finally, concluding remarks are given in Section 4.

2. Preliminaries

Consider the following impulsive differential equations with time delay:

x(t) = f (t, x(t), x(t − �)), t > t0, t �= �k,

x(t0 + t) = �(t), t ∈ [−�,0],�x(t)

�= x(t+) − x(t) = Ik(x(t)), t = �k, k = 1,2, . . . , (1)

wheret ∈ R+, � = const>0, f ∈ C[R+ × � × �,Rn], � is a domain inRn containing the origin,� ∈ C[[−�,0],�], f (t,0,0) ≡ 0, Ik(0)=0, 0= �0< �1< �2< · · ·< �k < · · ·, �k → ∞ for k → ∞, andx(t+) = lims→t+x(s), x(t

−) = lims→t−x(s). The functionsIk : Rn → Rn, k = 1,2, . . ., are such that if‖x‖<H andIk(x) �= 0, then‖x + Ik(x)‖<H , whereH = const>0.Denote by PC([−�,0],Rn) the set of piecewise left continuous functions� : [−�,0] → Rn with the

sup-norm|�| = sup−�� s�0‖�(s)‖, where‖ · ‖ is a norm inRn, andR� = [−�,∞).Throughout this paper we let the following hypotheses hold:(H1) For each functionx(s) : [t0 − �,∞) → Rn, which is continuous everywhere except the points

�k at whichx(�+k ) andx(�

−k ) exist andx(�

−k ) = x(�k), f (t, x(t), x(t − �)) is continuous for almost all

t ∈ [t0,∞) and at the discontinuous pointsf is left continuous.(H2) f (t,�) is Lipschitzian in� in each compact set in PC([−�,0],Rn).Under the conditions (H1) and (H2), there is a unique solution of Eq. (1) through(t0,�).Together with Eq. (1), we consider the following comparison system:

u(t) = g(t, u), t > t0, t �= �k,

u(t0 + 0) = u0,

�u(�k) = u(�+k ) − u(�k) = Jk(u(�k)) k = 1,2, . . . , (2)

whereg : R+ × G → Rm, Jk : G → Rm, k = 1,2, . . . ,G is a domain inRm containing the origin,u0 ∈ G. We denoteJ+(t0, u0) the maximal interval of the type[t0, w) in which the solution of system(2) is defined. Under the upper hypotheses, we know thatJ+(t0, u0) = [t0,∞) [1].LetS(�) = {x ∈ Rn : ‖x‖< �},K = {a ∈ C[R+, R+] : a(t) is monotone strictly increasing anda(0) = 0},�n = {h ∈ C[R+ × Rn, R+] : ∀t ∈ R+, inf xh(t, x) = 0},�n

� = {h ∈ C[R+� × Rn, R+] : ∀t ∈ R�, inf xh(t, x) = 0},

PC(�) = {� ∈ PC([−�,0],Rn) :| � | < �}.

Y. Zhang, J. Sun / Journal of Computational and Applied Mathematics 176 (2005) 223–229 225

We have the following definitions:

Definition 1 (Kou and Zhang[2] ). Let h0 ∈ �n� , � ∈ PC([−�,0],Rn), for any t ∈ R+, h0(t,�) is

defined by

h0(t,�) = sup−����0

h0(t + �,�(�)).

Definition 2. Let h0 ∈ �n� , h ∈ �n, system (1) is said to be

(A1) (h0, h)—eventually practically stable if for given(u, v) with 0<u<v, and somet0 ∈ R+, thereexists a�(u, v)>0, such thath0(t0,�)<u implies thath(t, x(t))< v, t� t0��(u, v);(A2) (h0, h)—uniformly eventually practically stable if (A1) holds for allt0 ∈ R+;(A3) (h0, h)—eventually practically quasistable if for given(u, v, T ) with u>0, v >0, T >0, and

somet0 ∈ R+, there exists a�(u, v)>0, such thath0(t0,�)<u implies thath(t, x(t))< v, t� t0 +T , t0��(u, v);(A4) (h0, h)—uniformly eventually practically quasistable if (A3) holds for allt0 ∈ R+;(A5) (h0, h)—strongly eventually practically stable if both (A1) and (A3) hold;(A6) (h0, h)—strongly uniformly eventually practically stable if both (A2) and (A4) hold.

Definition 3 (Yang[10] ). The functionV : [t0,∞) × S(�) → R+ belongs to classv0 if

(1) the functionV is continuous on each of the sets[�k−1, �k) × S(�) and for allt� t0,V (t,0) ≡ 0;(2) V (t, x) is locally Lipschitzian inx ∈ S(�);(3) for eachk = 1,2, . . ., there exist finite limits

lim(t,y)→(�k,x)

V (t, y) = V (�−k , x).

Definition 4 (Bainov and Stamova[1] ). Let V ∈ v0,D−V is defined as

D−V (t, x(t)) = lim→0−

inf1

{V (t + , x(t + ) + f (t, x(t), x(t − �))) − V (t, x(t))}.

Definition 5 (Bainov and Stamova[1] ). The functiong : (t0,∞) × G → Rm (G ⊂ Rm) is said to bequasimonotone increasing in(t0,∞) × G if for each pair of points(t, u) and(t, v) from (t0,∞) × G

and for j ∈ {1,2, . . . , m} the inequalitygj (t, u)�gj (t, v) holds wheneveruj = vj andui �vi , fori = 1,2, . . . , m, i �= j , i.e., for anyt ∈ (t0,∞) fixed and anyj ∈ {1,2, . . . , m} the functiongj (t, u) isnondecreasing with respect to(u1, . . . , uj−1, uj+1, . . . , um).

3. Main results

First, we introduce a lemma that has been given in Ref.[1].

Lemma 1 (Bainov and Stamova[1] ). Let the following conditions hold:

1. The functiong is quasimonotone increasing, continuous in the set(�k, �k+1] × G, k = 1,2, . . ., andr : J+(t0, u0) × Rm is the maximal solution of the Eq.(2);

226 Y. Zhang, J. Sun / Journal of Computational and Applied Mathematics 176 (2005) 223–229

2. The functionsk : G → Rm,k(u) = u + Jk(u), k = 1,2, . . . are monotone increasing inG;3. For eachk ∈ N andv ∈ G there exists the limitlim(t,u)→(t,v)g(t, u);4. The functionV ∈ v0 is such thatV (t0,�)�u0 and letx(t)= x(t, t0,�) denote the solution of Eq. (1).5. The inequality

D−V (t, x(t))�g(t, V (t, x(t))), t �= �k,

V (t+, x(t) + Ik(x(t)))�k(V (t, x(t))), t = �k

is valid for eacht� t0 and any functionx ∈ C[[t0,∞),�) for which

V (t + s, x(t + s))�V (t, x(t)) s ∈ [−�,0].Then

V (t, x(t, t0,�))�r(t, t0, u0) t ∈ J+(t0, u0).

Now, when we consider the eventual practical stability of system (1), we have the following results:

Theorem 1. Let the following conditions hold:

(i) The conditions of Lemma1 hold.(ii) 0<u<v are given.(iii) h0 ∈ �n

� , h ∈ �n, andh(t, x)��(h0(t, x)) with � ∈ K wheneverh0(t, x)<u.(iv) V ∈ v0 and there exist�, � ∈ K such that

�(h(t, x))�V (t, x)��(h0(t, x)).

(v) �(u)< v and�(u)< �(v).Then

(a) if system(2) is uniformly eventually practically stable with respect to(�(u), �(v)), then system(1) is(h0, h)-uniformly eventually practically stable with respect to(u, v);

(b) if system(2) is strongly uniformly eventually practically stable with respect to(�(u), �(v)), then thesystem(1) is (h0, h)-strongly uniformly eventually practically stable with respect to(u, v).

Proof. (a) For given(u, v) with 0<u<v, since the comparison system (2) is uniformly eventuallypractically stable with respect to(�(u), �(v)), then for anyt0 ∈ R+, there exists a�(u, v)>0 such thatu0< �(u) impliesu(t, t0, u0)< �(v), t� t0��(u, v), whereu(t, t0, u0) is any solution of system (2).For any(t0,�) ∈ R+ × PC([−�,0], Rn) such thath0(t0,�)<u, we have from condition (iii) and (v)

h(t0,�)��(h0(t0,�))��(u)< v.

We then have

h(t, x(t))< v ∀t� t0��(u, v). (3)

If this is not the case, then there exists as1> t0 such that

h(s1, s1)�v and h(t, x(t))< vt ∈ [t0, s1).

Y. Zhang, J. Sun / Journal of Computational and Applied Mathematics 176 (2005) 223–229 227

Let V (t0,�) = u0, then from condition (iv) we have

V (t0,�) = u0��(h0(t0,�))< �(u)

which implies thatu(t, t0, u0)< �(v), t� t0��(u, v).Since the conditions of Lemma 1 hold, then

V (t, x(t))�r(t, t0, u0), t� t0,

wherer(t, t0, u0) is the maximal solution of (2), this together with condition (iv), we have the followingcontradiction:

�(v)��(h(s1, x(s1)))�V (s1, x(s1))�r(s1, t0, u0)< �(v).

So inequality (3) holds. So system (1) is(h0, h)—uniformly eventually practically stable with respectto (u, v).(b) For given(u, v, T ) with 0<u<v, T >0, since the comparison system (2) is strongly uniformly

eventually practically stable with respect to(�(u), �(v)), by what we have proved in (a), system (1) is(h0, h)—uniformly eventually practically stable. Thus for anyt0 ∈ R+, there exists a�1(u, v)>0, suchthath0(t0,�)<u implies

h(t, x(t))< v, t� t0��1(u, v). (4)

Since the comparison system (2) is uniformly eventually practically quasistable with respect to(�(u),�(v)), then for anyt0 ∈ R+, there exists a�2(u, v)>0 such thatu0< �(u) impliesu(t, t0, u0)< �(v),t� t0 + T , t0��2(u, v), whereu(t, t0, u0) is any solution of system (2).Let �(u, v) =max{�1(u, v), �2(u, v)}, we will prove that

h(t, x(t))�v, t� t0 + T , t0��(u, v). (5)

Let V (t0,�) = u0, then from condition (iv) we have

V (t0,�) = u0��(h0(t0,�))< �(u)

which implies thatu(t, t0, u0)< �(v), t� t0��2(u, v).Since the conditions of Lemma 1 hold, then

V (t, x(t))�r(t, t0, u0), t� t0,

wherer(t, t0, u0) is the maximal solution of system (2), this together with condition (iv) we have

�(h(t, x(t)))�V (t, x(t))�r(t, t0, u0)��(v), t� t0 + T , t0��(u, v)

from which we know that inequality (5) holds.Thus, system (1) is(h0, h)—strongly uniformly eventually practically stable with respect to

(u, v). �

Theorem 2. Let condition(i), (iii) and(iv) of Theorem1 be satisfied and

(iv) (u, v, T ) with u>0, v >0, T >0 are given;(v) �(u)< v.

228 Y. Zhang, J. Sun / Journal of Computational and Applied Mathematics 176 (2005) 223–229

Then if system(2) is uniformly eventually practically quasistable with respect to(�(u), �(v)), thensystem(1) is (h0, h)-uniformly eventually practically quasistable with respect to(u, v).

Proof. For given(u, v, T ) with u>0, v >0, T >0, since the comparison system (2) is uniformly even-tually practically quasistable with respect to(�(u), �(v)), then for anyt0 ∈ R+, there exists a�(u, v)>0such thatu0< �(u) impliesu(t, t0, u0)< �(v), t� t0 + T , t0��(u, v), whereu(t, t0, u0) is any solutionof system (2).For any(t0,�) ∈ R+ × PC([−�,0], Rn) such thath0(t0,�)<u, we have from condition (iii) of

Theorem 1 and (v)

h(t0,�)��(h0(t0,�))��(u)< v.

We then have

h(t, x(t))< v t� t0 + T , t0��(u, v). (6)

Let V (t0,�) = u0, then from condition (iv) we have

V (t0,�) = u0��(h0(t0,�))< �(u)

which implies thatu(t, t0, u0)< �(v), t� t0 + T , t0��(u, v).Since the conditions of Lemma 1 hold, then

V (t, x(t))�r(t, t0, u0), t� t0,

wherer(t, t0, u0) is the maximal solution of the system (2), this together with condition (iv) of Theorem1, we have the following inequality:

�(h(t, x(t)))�V (t, x(t))�r(t, t0, u0)< �(v), t� t0 + T , t0��(u, v).

So inequality (6) holds.System (1) is(h0, h)—uniformly eventually practically quasistablewith respectto (u, v). �

4. Conclusion

In this paper, we have introduced a new stability—eventual practical stability for impulsive differentialequations with time delay. By using Lyapunov functions and comparison principle, we have obtainedsome results for eventual practical stability, eventual practical quasistability and strong eventual practicalstability of this system.We can see that impulses do contribute to the system’s practical stability behavior.

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