New Results on Global Exponential Stability of Impulsive Functional Differential Systems with Delayed Impulses

and Applied Analysis 3 delay. In the end, an example is provided to illustrate the effectiveness and the advantages of the results obtained. 2. Preliminaries Let R denote the set of real numbers, R the set of nonnegative real numbers, Z the set of positive integers, and R the n-dimensional real space equipped with the Euclidean norm | · |. Let τ > 0 and PC −τ, 0 ;R {φ : −τ, 0 → Rn|φ t φ t for all t ∈ −τ, 0 , φ t− exist and φ t− φ t for all, but at most a finite number of points t ∈ −τ, 0 } be with the norm ‖φ‖ sup−τ θ 0|φ θ |, where φ t and φ t− denote the right-hand and left-hand limits of function φ t at t, respectively. Denote PC t0 − τ, b ;R {φ : t0 − τ, b → R | φ is piecewise continuous} for b > t0, and PC t0−τ,∞ ;R {φ|φ| t0−τ,b ∈ PC t0−τ, b ;R for all b > t0 − τ}. Consider the IFDS in which the state variables on the impulses are related to the time delay: ẋ t f t, xt , t / tk, t t0, Δx tk Ik ( tk, x ( tk )) Jk ( tk, xt− k ) , k ∈ Z xt0 φ s , s ∈ −τ, 0 , 2.1 where x ∈ R, f : R ×C → R, Ik : R ×Rn → R, Jk : R ×C → R, φ ∈ PC −τ, 0 ;R , C is a open set in PC −τ, 0 ;R . and The fixed moments of impulse times {tk, k ∈ Z } satisfy 0 t0 < t1 < · · · < tk < · · · , tk → ∞ as k → ∞ , Δx tk x tk − x tk ; xt, xt− ∈ PC −τ, 0 ;R are defined by xt x t θ , xt− x t− θ for θ ∈ −τ, 0 , respectively. Throughout this paper, we assume that f, Ik, and Jk, k ∈ Z , satisfy the necessary conditions for the global existence and uniqueness of solutions for all t t0, see 6, 30– 33 . Then for any φ ∈ PC −τ, 0 ;R , there exists a unique function satisfying system 2.1 denoted by x t; t0, φ , which is continuous on the right-hand side and limitable on the lefthand side. Moreover, we assume that f t, 0 ≡ 0, Ik tk, 0 ≡ 0 and Jk tk, 0 ≡ 0, k ∈ Z , which imply that x t ≡ 0 is a solution of 2.1 , which is called the trivial solution. At the end of this section, let us introduce the following definitions. Definition 2.1. A function V : t0 − τ,∞ × R → R belongs to class v0 if i V is continuous on each of the sets tk−1, tk × R, and for each x ∈ R, t ∈ tk−1, tk , k ∈ Z , lim t,y → t− k ,x V t, y V tk, x exists; ii V t, x is locally Lipschitz in x ∈ R, and V t, 0 ≡ 0 for all t t0. Definition 2.2. Given a function V ∈ v0, the upper right-handDini derivative of V with respect to system 2.1 is defined by D V ( t, ψ 0 ) lim sup h→ 0 1 h [ V ( t h, ψ 0 hf ( t, ψ )) − V (t, ψ 0 )], 2.2 for t, ψ ∈ t0,∞ × PC −τ, 0 ;R . 4 Abstract and Applied Analysis Definition 2.3. The trivial solution of system 2.1 or, simply, system 2.1 is said to be globally exponentially stable if there exist positive constants α and C such that for any initial data xt0 φ ∈ PC −τ, 0 ;R , the solution x t; t0, φ satisfies ∣∣x(t; t0, φ )∣∣ C∥∥φ∥∥e−α t−t0 , t t0. 2.3 3. Main Results In this section, we shall analyze the global exponential stability of system 2.1 by employing the Razumikhin techniques and the Lyapunov functions. Theorem 3.1. Assume that there exist functions V ∈ v0, c ∈ PC t0 − τ,∞ ;R , several positive constants c1, c2, c̃, p, q, and nonnegative constants ρ1, ρ2, ρ1 ρ2 1 such that i c1|x| V t, x c2|x|, for all t, x ∈ t0 − τ,∞ × R; ii V tk, φ 0 ρ1 1 μk V tk, φ 0 ρ2 1 μk supθ∈ −τ,0 V t − k θ, φ θ , for each k ∈ Z and φ ∈ PC −τ, 0 ;R , where μk, k ∈ Z , are nonnegative constants with Σk 1μk < ∞; iii D V t, φ 0 −c t V t, φ 0 , for all t t0, t / tk, k ∈ Z , φ ∈ PC −τ, 0 ;R , whenever V t θ, φ < qV t, φ 0 , θ ∈ −τ, 0 ; iv ρ1 ρ2e < q < e , inft t0c t c̃, where infk∈Z {tk − tk−1}. Then the trivial solution of system 2.1 is globally exponentially stable and the convergence rate should not be greater than 1/p c̃ − ln q/ . Proof. Set L ∏∞ k 1 1 μk ; from the condition Σ ∞ k 1μk < ∞, we known that 1 L < ∞. Fix any initial data φ ∈ PC −τ, 0 ;R and write x t; t0, φ x t , V t, x t V t simply. From condition iv , we can choose a small enough constant γ > 0 such that e ( ρ1 ρ2e ) < q < e c̃−γ , γ < c̃. 3.1 Set q̃ qe−γτ > 1, choose M > 0 such that q̃c2 < M. Define W t e t−t0 V t . In the following, we shall show that W t LM ∥∥φ∥∥p, t t0. 3.2 In order to do so, we first prove that W t < M ∥∥φ∥∥p, t ∈ t0 − τ, t1 . 3.3 It is noted that W t0 θ c2 ∥∥φ∥∥p < 1 q̃ M ∥∥φ∥∥p < M∥∥φ∥∥p, θ ∈ −τ, 0 . 3.4 Abstract and Applied Analysis 5 So it only needs to proveand Applied Analysis 5 So it only needs to prove W t < M ∥∥φ∥∥p, t ∈ t0, t1 . 3.5 We assume, on the contrary, there exist some t ∈ t0, t1 such that W t M‖φ‖p. Set t∗ inf { t ∈ t0, t1 : W t M ∥∥φ∥∥p}. 3.6 Note that W t is continuous on t ∈ t0, t1 , then t∗ ∈ t0, t1 and W t∗ M ∥∥φ∥∥p, W t < M∥∥φ∥∥p, t ∈ t0 − τ, t∗ . 3.7