Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 243863, 8 pages doi:10.1155/2008/243863 Research Article Strict Stability Criteria for Impulsive Functional Differential Systems Kaien Liu 1 and Guowei Yang 2 1 School of Mathematics, Qingdao University, Q ingdao, Shandong 266071, China 2 School of Automation Engineering, Qingdao University, Qingdao, Shandong 266071, China Correspondence should be addressed to Kaien Liu, kaienliu@yahoo.com.cn Received 4 September 2007; Accepted 18 November 2007 Recommended by Alexander Domoshnitsky By using Lyapunov functions and Razumikhin techniques, the strict stability of impulsive func- tional differential systems is investigated. Some comparison theorems are given by virtue of differ- ential inequalities. The corresponding theorems in the literature can be deduced from our results. Copyright q 2008 K. Liu and G. Yang. 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 Since time-delay systems are frequently encountered in engineering, biology, economy, and other disciplines, it is significant to study these systems 1. On the other hand, because many evolution processes in nature are characterized by the fact that at certain moments of time they experience an abrupt change of state, the study of dynamic systems with impulse effects has been assuming greater importance 2–4. It is natural to expect that the hybrid systems which are called impulsive functional differential systems can represent a truer framework for mathematical modeling of many real world phenomena. Recently, several papers dealing with stability problem for impulsive functional differential systems have been published 5–10. The usual stability concepts do not give any information about the rate of decay of the solutions, and hence are not strict concepts. Consequently, strict-stability concepts have been defined and criteria for such notions to hold are discussed in 11. Till now, to the best of our knowledge, only the following very little work has been done in this direction 12–15. In this paper, we investigate strict stability for impulsive functional differential systems. The paper is organized as follows. In Section 2, we introduce some basic definitions and nota- tions. In Section 3, we first give two comparison lemmas on differential inequalities. Then, by these lemmas, a comparison theorem is obtained and several direct results are deduced from it. An example is also given to illustrate the advantages of our results. 2 Journal of Inequalities and Applications 2. Preliminaries We consider the following impulsive functional differential system: x  tf  t, x t  ,t /  τ k , x  τ k   x  τ k  − x  τ − k   I k  x  τ − k  ,k∈ Z  , 2.1 where Z  is the set of all positive integers, f : R  × D → R n , D is an open set in PC −τ,0, R n ,hereR  0, ∞,τ > 0, and PC−τ,0, R n {φ : −τ,0 → R n ,φt is con- tinuous everywhere except for a finite number of points  t at which φ  t   and φ  t −  exist and φ  t  φ  t}. I k : Sρ 0  → R n for each k ∈ Z  ,whereSρ 0 {x ∈ R n : x <ρ 0 , · denotes the norm of vector in R n },0 τ 0 ≤ τ 1 <τ 2 < ··· <τ k < ··· with τ k →∞as k →∞and x  t denotes the right-hand derivative of xt.Foreacht ∈ R  , x t ∈ PCis defined by x t sxts, −τ ≤ s ≤ 0. For φ ∈ PC, |φ| 1  sup −τ≤s≤0 φs, |φ| 2  inf −τ≤s≤0 φs. We assume that ft, 0 ≡ 0 and I k 0 ≡ 0, so that xt ≡ 0 is a solution of 2.1, which we call the zero solution. Let t 0 ∈ τ m−1 ,τ m  for some m ∈ Z  and ϕ ∈ D, a function xt : t 0 − τ, β → R n β ≤∞ is said to be a solution of 2.1 with the initial condition x t 0  ϕ, 2.2 if it is continuous and satisfies the differential equation x  tft, x t  in each t 0 ,τ m , τ i ,τ i1 ,i m, m  1, ,andatt  τ i it satisfies xτ i I i xτ − i . Throughout this paper, we always assume the following conditions hold to ensure the global existence and uniqueness of solution of 2.1 through t 0 ,ϕ. H 1  f is continuous on τ k−1 ,τ k  × D for each k ∈ Z  and for all k ∈ Z  and ϕ ∈ D,the limits lim t,φ→τ − k ,ϕ ft, φfτ − k ,ϕ exist. H 2  ft, φ is Lipschitzian in φ in each compact set in D. H 3  I k x ∈ CSρ 0 , R n  for all k ∈ Z  and there exists ρ 0 ≤ ρ such that x ∈ Sρ 0  implies that x  I k x ∈ Sρ for all k ∈ Z  . The function V t, x : R  × R n → R  belongs to class V 0 if the following hold. A 1  V is continuous on each of the sets τ k−1 ,τ k  × R n and for each x ∈ R n and k ∈ Z  , lim t,y→τ − k ,x V t, yV τ − k ,x exists. A 2  V t, x is locally Lipschitzian in x ∈ R n and for t ∈ R  ,Vt, 0 ≡ 0. Let V ∈ V 0 , D  V along the solution xt of 2.1 is defined as D  V  t, xt   lim δ→0  sup 1 δ  V  t  δ, xt  δ  − V  t, xt  . 2.3 Let us introduce the following notations for further use: i K 0  {au ∈ CR  , R   : increasing and a00}; ii K  {au ∈ K 0 : strictly increasing}; iii K 1  {au ∈ K 0 : au ≤ u and au > 0foru>0}; K. Liu and G. Yang 3 iv K 2  {au ∈ K : au ≥ u}; v PC 1 ρ{φ ∈ PC−τ,0, R n  : |φ| 1 <ρ}; vi PC 2 θ{φ ∈ PC−τ,0, R n  : |φ| 2 >θ>0}. Definition 2.1. The zero solution of 2.1 is said to be strictly stable SS, if for any t 0 ∈ R  and ε 1 > 0, there exists a δ 1  δ 1 t 0 ,ε 1  > 0 such that ϕ ∈ PC 1 δ 1  implies xt; t 0 ,ϕ <ε 1 for t ≥ t 0 , and for every 0 <δ 2 ≤ δ 1 , there exists an 0 <ε 2 <δ 2 such that ϕ ∈ PC 2  δ 2  implies ε 2 <   x  t; t 0 ,ϕ    ,t≥ t 0 . 2.4 Definition 2.2. The zero solution of 2.1 is said to be strictly uniformly stable SUS,ifδ 1 ,δ 2 , and ε 2 in SS are independent of t 0 . Remark 2.3. If in SS or SUS, ε 2  0, we obtain nonstrict stabilities, that is, the usual stability or uniform stability, respectively. Moreover, strict stability immediately implies that the zero solution is not asymptotically stable. The preceding notions imply that the motion remains in the tube like domains. To ob- tain sufficient conditions for such stability concepts to hold, it is necessary to simultaneously obtain both lower and upper bounds of the derivative of Lyapunov function. Thus, we need to consider the following two auxiliary systems: v   g 1 t, v,t /  τ k , v  τ k   φ k  v  τ − k  , v  t 0   v 0 ≥ 0, 2.5 and u   g 2 t, u,t /  τ k , u  τ k   ψ k  u  τ − k  , u  t 0   u 0 ≥ 0, 2.6 where g 1 ,g 2 ∈ CR  × R  , R,g 1 t, u ≤ g 2 t, u, g 1 t, 0 ≡ g 2 t, 0 ≡ 0, φ k ,ψ k : R  → R  , φ k u ≤ ψ k u for each k ∈ Z  . From the theory of impulsive differential systems 2,weobtainthat ρ  t; t 0 ,v 0  ≤ γ  t; t 0 ,u 0  ,t≥ t 0 whenever v 0 ≤ u 0 , 2.7 where ρt; t 0 ,v 0  and γt; t 0 ,u 0  are the minimal and maximal solutions of 2.5, 2.6, respec- tively. The corresponding definitions of strict stability of the auxiliary systems 2.5, 2.6 are as follows. Definition 2.4. The zero solutions of comparison systems 2.5, 2.6, as a system, are said to be strictly stable SS ∗ ,ifforanyt 0 ∈ R  and ε 1 > 0, there exist a δ 1  δ 1 t 0 ,ε 1 ,δ 2  δ 2 t 0 ,ε 1 , and ε 2  ε 2 t 0 ,ε 1  satisfying 0 <ε 2 <δ 2 <δ 1 <ε 1 such that ε 2 <ρ  t; t 0 ,v 0  ≤ γ  t; t 0 ,u 0  <ε 1 ,t≥ t 0 , provided δ 2 <v 0 ≤ u 0 <δ 1 . 2.8 Definition 2.5. The zero solutions of comparison systems 2.5,2.6, as a system , a re said to be strictly uniformly stable SUS ∗ ,ifδ 1 ,δ 2 ,andε 2 in SS ∗  are independent of t 0 . 4 Journal of Inequalities and Applications 3. Main results We first give two Razumikhin-type comparison lemmas on differential inequalities. Lemma 3.1. Assume that i g 1 ,g 2 ∈ CR  × R  , R, − g 1 t, ·,g 2 t, · ∈ K 0 for each t; ii there exists m i : R  → R  i  1, 2,wherem i ti  1, 2 are continuous on τ k−1 ,τ k  and lim t→τ − k m i tm i τ − k i  1, 2 exist, k ∈ Z  , satisfying g 1  t, m 1 t  ≤ D  m 1 t, D  m 2 t ≤ g 2  t, m 2 t  . 3.1 Then ρt ≤ m 1 t if inf −τ≤s≤0 m 1  t 0  s  ≥ v 0 , 3.2 m 2 t ≤ γt if sup −τ≤s≤0 m 2  t 0  s  ≤ u 0 , 3.3 where ρtρt; t 0 ,v 0  and γtγt; t 0 ,u 0  are the minimal and maximal solutions of systems 3.4 and 3.5, respectively, v   g 1 t, v, v  t 0   v 0 ≥ 0, 3.4 u   g 2 t, u, u  t 0   u 0 ≥ 0. 3.5 Proof. First, we prove that 3.2 holds. Otherwise, there exist t 0 ≤ t 1 <t 2 such that a ρt 1 m 1 t 1 , b m 1 t  s ≥ m 1 t,s∈ −τ, 0,t∈ t 1 ,t 2 , and c ρt 2  <m 1 t 2 . By a, b,andii, applying the classical comparison theorem, we have ρt ≤ m 1 t,t∈  t 1 ,t 2  , 3.6 which contradicts c.So3.2 is correct. Equation 3.3 can be proved in the same way as above. Then Lemma 3.1 holds. Lemma 3.2. Assume that (i) in Lemma 3.1 holds. Suppose further that ii there exists V 1 ∈ V 0 satisfying φ k  V 1  τ − k ,x  ≤ V 1  τ k ,x I k x  ,k∈ Z  , 3.7 where φ k ∈ K 1 , and for any solution xt of 2.1, V 1 t  s, xt  s ≥ V 1 t, xt,s∈ −τ,0, implies that g 1  t, V 1  t, xt  ≤ D  V 1  t, xt  ; 3.8 K. Liu and G. Yang 5 iii there exists V 2 ∈ V 0 satisfying V 2  τ k ,x I k x  ≤ ψ k  V 2  τ − k ,x  ,k∈ Z  , 3.9 where ψ k ∈ K 2 , and for any solution xt of 2.1, V 2 t  s, xt  s ≤ V 2 t, xt,s∈ −τ,0, implies that D  V 2 t, xt ≤ g 2 t, V 2 t, xt. 3.10 Then ρt ≤ V 1  t, xt  if inf −τ≤s≤0 V 1  t 0  s, x  t 0  s  ≥ v 0 , 3.11 V 2  t, xt  ≤ γt if sup −τ≤s≤0 V 2  t 0  s, x  t 0  s  ≤ u 0 , 3.12 where ρtρt; t 0 ,v 0  and γtγt; t 0 ,u 0  are the minimal and maximal solutions of 2.5, 2.6, respectively. Proof. Assume t 0 ∈ τ m−1 ,τ m ,m∈ Z  .First,weprovethat3.11 holds for t ∈ t 0 ,τ m ,thatis ρt ≤ V 1  t, xt  ,t∈  t 0 ,τ m  . 3.13 Let m 1 tV 1 t, xt, t ≥ t 0 . Equation 3.13 holds obviously by Lemma 3.1 for t ∈ t 0 ,τ m .By ii, V 1 τ m ,xτ m  ≥ φ m V 1 τ − m ,xτ − m  ≥ φ m ρτ − m   ρτ m . The same proof as for t ∈ t 0 ,τ m  leads to ρt ≤ V 1  t, xt  ,t∈  τ m ,τ m1  . 3.14 By induction, 3.11 is correct. Similarly, 3.12 can be proved by using Lemma 3.1 and assump- tion iii. Using Lemma 3.2, we can easily get the following theorem about strict stability proper- ties of 2.1. Theorem 3.3. Assume that all the conditions of Lemma 3.2 hold. Suppose further that there exist func- tions a i ,b i ∈ K, i  1, 2, such that iv b i x ≤ V i t, x ≤ a i x for x ∈ Sρ. Then the strict stability properties of comparison systems 2.5, 2.6 imply the corresponding strict stability properties of zero solution of 2.1. Proof. First, let us prove strict stability o f the zero solution of 2.1. Suppose that 0 <ε 1 <ρ 0 and t 0 ∈ R  are given. Assume that SS ∗  holds. Then, given b 2 ε 1  > 0, there exists  δ 1   δ 1 t 0 ,ε 1 ,  δ 2   δ 2 t 0 ,ε 1 ,andε 2  ε 2 t 0 ,ε 1  satisfying 0 < ε 2 <  δ 2 <  δ 1 <b 2 ε 1  such that ε 2 <ρt ≤ γt <b 2  ε 1  provided  δ 2 <v 0 ≤ u 0 <  δ 1 ,t≥ t 0 . 3.15 By iv, there exist 0 <δ 2 <δ 1 <ε 1 such that for s ∈ −τ, 0, V i  t 0  s, x  ∈ PC 2   δ 2  ∩ PC 1   δ 1  provided δ 2 < x <δ 1 ,i 1, 2. 3.16 Next, choose ε 2  ε 2 t 0 ,ε 1  > 0 such that a 1 ε 2  ≤ ε 2 and ε 2 <δ 2 . We claim that with the choices of ε 2 ,δ 2 ,andδ 1 , the zero solution of 2.1 is strictly stable. That means that if xtxt; t 0 ,ϕ is any solution of 2.1, ϕ ∈ PC 2 δ 2  ∩ PC 1 δ 1  implies that ε 2 < xt <ε 1 ,t≥ t 0 .Ifnot,we have either of the following alternatives. 6 Journal of Inequalities and Applications Case 1. There exists a t 1 ∈ τ r ,τ r1  such that ε 2 ≥   x  t 1    . 3.17 Then clearly xt <ρ 0 ,t 0 ≤ t ≤ t 1 . Thus, by Lemma 3.2, i and ii imply that ρt ≤ V 1 t, xt provided v 0 ≤ inf s∈−τ,0 V 1  t 0  s, x  t 0  s  ,t∈  t 0 ,t 1  . 3.18 Using 3.15–3.18 and iv,weget a 1 ε 2  ≥ a 1 xt 1  ≥ V 1 t 1 ,xt 1  ≥ ρt 1  > ε 2 ≥ a 1 ε 2 , 3.19 which is a contradiction. Case 2. There exists a  t 2 ∈ τ s ,τ s1  such that ε 1 ≤   x   t 2    , 3.20   xt   <ε 1 ,t 0 ≤ t<τ s . 3.21 By H 3 , 3.21 yields   x  τ s       x  τ − s   I s  x  τ − s    <ρ. 3.22 Because of 3.20 and 3.22, there exists a t 2 ∈ τ s ,  t 2  such that ε 1 ≤   x  t 2    <ρ. 3.23 By Lemma 3.2, i and iii imply that V 2 t, xt ≤ γt provided sup s∈−τ,0 V 2  t 0  s, x  t 0  s  ≤ u 0 ,t∈  t 0 ,t 2  . 3.24 From 3.15, 3.23, 3.24,andiv, we have the following contradiction: b 2  ε 1  ≤ b 2    x  t 2     ≤ V 2  t 2 ,xt 2   ≤ γ  t 2  <b 2  ε 1  . 3.25 We, therefore, obtain the strict stability of the zero solution of 2.1. If we assume that the zero solutions of comparison systems 2.5, 2.6 are SUS ∗ , since  δ 1 ,  δ 2 are independent of t 0 ,we obtain, because of iv, δ 1 and δ 2 in 3.16 are independent of t 0 , and hence, SUS of 2.1 holds. Using Theorem 3.3, we can get two direct results on strictly uniform stability of zero solution of 2.1 and the first one is Theorem 3.3 in 15. Corollary 3.4. In Theorem 3.3, suppose that g 1 ≡ g 2 ≡ 0, φ k u1 − c k u, ψ k u1  d k u, k ∈ Z  ,where0 ≤ c k < 1,  ∞ k1 c k < ∞,andd k ≥ 0,  ∞ k1 d k < ∞. Then the zero solution of 2.1 is strictly uniformly stable. K. Liu and G. Yang 7 Corollary 3.5. In Theorem 3.3, suppose that g 1 t, u−M  1 tu, g 2 t, uM  2 tu,whereM  i t ∈ C R  , R  ,i 1, 2, and M i t,i 1, 2 are bounded, φ k u and ψ k u,k∈ Z  arejustthesameasin Corollary 3.4. Then the zero solution of 2.1 is strictly uniformly stable. Proof. Under the given hypotheses, it is easy to obtain the solutions of 2.5 and 2.6: vtv 0  t 0 ≤τ k ≤t  1 − c k  exp  −  M 1 t − M 1  t 0  , utu 0  t 0 ≤τ k ≤t  1  d k  exp  M 2 t − M 2 t 0   . 3.26 Since M i t, i  1, 2, are bounded, there exist two positive constants B 1 ,B 2 such that |M 1 t|≤ B 1 , |M 2 t|≤B 2 . Also, since  ∞ k1 c k < ∞,  ∞ k1 d k < ∞, it follows that  ∞ k1 1 − c k N and  ∞ k1 1d k M, obviously 0 <N≤ 1, 1 ≤ M<∞.Givenε 1 > 0, choose δ 1  M −1 exp−2B 2 ε 1 and for 0 <δ 2 <δ 1 , choose ε 2  δ 2 N exp−2B 1 . Then, if δ 2 <v 0 ≤ u 0 <δ 1 ,wehave ε 2 <vt ≤ ut <ε 1 . 3.27 That is, the zero solutions of 2.5, 2.6 are strictly uniformly stable. Hence, by Theorem 3.3, the zero solution of 2.1 is strictly uniformly stable. Example 3.6. Consider the system x  t−atxtbtxt − τ,t /  τ k ,t≥ 0, x  τ k   I k  x  τ − k  ,k∈ Z  , 3.28 where at,bt are continuous on R  ,bt ≥ 0,I k x ∈ CR, R. Assume that −1/1  t 2  ≤ −atbt ≤ 1/1  t 2 , 1 − c k x 2 ≤ x  I k x 2 ≤ 1  d k x 2 with 0 ≤ c k < 1,  ∞ k1 c k < ∞,and d k ≥ 0,  ∞ k1 d k < ∞. Let V 1 t, xV 2 t, xV x1/2x 2 ,then  1 − c k  V x 1 2  1 − c k  x 2 ≤ V  x  I k x   1 2  x  I k x  2 ≤ 1 2  1  d k  x 2   1  d k  V x. 3.29 For any solution xt of 3.28 such that V xt  s ≥ V xt,s∈ −τ, 0, we have D  V  xt   −atx 2 tbtxtxt − τ ≥  − atbt  x 2 t ≥− 2 1  t 2 V  xt  , 3.30 and if V xt  s ≤ V xt,s∈ −τ, 0,wehave D  V  xt   −atx 2 tbtxtxt − τ ≤  − atbt  x 2 t ≤ 2 1  t 2 V  xt  . 3.31 By Corollary 3.5, the zero solution of 2.1 is strictly uniformly stable. 8 Journal of Inequalities and Applications Acknowledgments This project is supported by the National Natural Science Foundation of China 60673101 and the Natural Science Foundation of Shandong Province Y2007G30. The authors are grateful to the referees for their helful comments. References 1 J. 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Inequalities and Applications Volume 2008, Article ID 243863, 8 pages doi:10.1155/2008/243863 Research Article Strict Stability Criteria for Impulsive Functional Differential Systems Kaien Liu 1 and. called impulsive functional differential systems can represent a truer framework for mathematical modeling of many real world phenomena. Recently, several papers dealing with stability problem for impulsive. for impulsive functional differential equations,” Applied Mathematics and Computation, vol. 125, no. 2–3, pp. 375–386, 2002. 9 Y. Xing and M. Han, “A new approach to stability of impulsive functional