The Method of Lyapunov Function and Exponential Stability of Impulsive Delay Systems with Delayed Impulses

Abstract

This paper investigates the exponential stability of general impulsive delay systems with delayed impulses. By using the Lyapunov function method, some Lyapunov-based sufficient conditions for exponential stability are derived, which are more convenient to be applied than those Razumikhin-type conditions in the literature. Their applications to linear impulsive systems with time-varying delays are also proposed, and a set of sufficient conditions for exponential stability is provided in terms of matrix inequalities. Meanwhile, two examples are discussed to illustrate the effectiveness and advantages of the results obtained.

1. Introduction

Impulsive dynamical systems have received considerable attention during the recent decades since they provide a natural framework for mathematical modeling of many real- world evolutionary processes where the states undergo abrupt changes at certain instants (see, e.g., [1–4]). On the other hand, time delays often appear in practical systems and may poorly affect the performance of a system. Stability and stabilization of such systems are of both theoretical and practical importance. Therefore, the study of time-delay systems has attracted great attention over the past few years (see, e.g., [5–10]). An area of particular interest has been the stability analysis and stabilization of impulsive delay systems (IDSs), and there is extensive literature on this field (see, e.g., [11–26]). For instance, in [11–14], the robust stability for uncertain impulsive system with time-delay was studied, and the linear matrix inequalities approach to the stability was presented. In [15–26], the Lyapunov function or Lyapunov functional coupled with the Razumikhin techniques was suggested for the exponential stability and asymptotical stability of IDSs.

In the previous works on stability and stabilization of IDSs, the impulses are assumed to take the form of Δ𝑥(𝑡_𝑘)=𝐼_𝑘(𝑡_𝑘,𝑥(𝑡⁻_𝑘)), which indicates that the state “jump” at impulse times 𝑡_𝑘 is only related to the present state variables (see, e.g., [18–26]). But, in most cases, it is more applicable that the state variables on the impulses are also related to the past states. For example, in the transmission of the impulse information, input delays are often encountered. So, compared with the nondelayed impulses described above, it is much more meaningful to model the impulses as Δ𝑥𝑡_𝑘=𝐼_𝑘𝑡_𝑘,𝑥𝑡⁻_𝑘+𝐽_𝑘𝑡_𝑘,𝑥_𝑡⁻_𝑘. (1) In fact, there have been several attempts in the literature to study the stability and control problems of a particular class of delayed impulsive systems [27, 28]. For example, Lian et al. [27] investigated the optimal control problem of linear continuous-time systems possessing delayed discrete-time controllers in networked control systems. For nonlinear impulsive systems, Khadra et al. [28] studied the impulsive synchronization problem coupled by linear delayed impulses. By using the Razumikhin techniques, some sufficient conditions for asymptotic stability and exponential stability of general IDS-DI were established in [29–32], and sufficient conditions for exponential stability of impulsive stochastic functional differential systems with delayed impulses were obtained in [33].

In this paper, we will further investigate the stability of IDS-DI. By using the Lyapunov functions, some sufficient conditions ensuring exponential stability of IDS-DI are derived, which are more convenient to be applied than those Razumikhin-type conditions in [31, 32]. Their applications to linear impulsive systems with time-varying delays are also proposed, and a set of sufficient conditions for exponential stability are derived in terms of matrix inequalities.

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^𝑛 the 𝑛-dimensional real space equipped with the Euclidean norm |⋅|. Let 𝜏>0, and let 𝑃𝐶([−𝜏,0];ℝ^𝑛)={𝜑∶[−𝜏,0]→ℝ^𝑛∣𝜑(𝑡⁺)=𝜑(𝑡) for all 𝑡∈[−𝜏,0), 𝜑(𝑡⁻) exists 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.

Consider the IDS-DI described by the state equations𝑥̇(𝑡)=𝑓𝑡,𝑥_𝑡,𝑡≠𝑡_𝑘,𝑡⩾𝑡₀,Δ𝑥𝑡_𝑘=𝐼_𝑘𝑡_𝑘,𝑥𝑡⁻_𝑘+𝐽_𝑘𝑡_𝑘,𝑥_𝑡⁻_𝑘,𝑘∈ℤ₊,𝑥_𝑡0=𝜙(𝑠),𝑠∈[−𝜏,0], (2) where 𝑥∈ℝ^𝑛, 𝑓∶ℝ₊×ℂ→ℝ^𝑛, 𝐼_𝑘∶ℝ₊×ℝ^𝑛→ℝ^𝑛, 𝐽_𝑘∶ℝ₊×ℂ→ℝ^𝑛, 𝜙∈𝑃𝐶([−𝜏,0];ℝ^𝑛), and ℂ is an open set in 𝑃𝐶([−𝜏,0];ℝ^𝑛). The fixed moments of impulse times {𝑡_𝑘,𝑘∈ℤ₊} satisfy 0⩽𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯ and 𝑡_𝑘→∞ (as 𝑘→∞) and Δ𝑥(𝑡_𝑘)=𝑥(𝑡_𝑘)−𝑥(𝑡⁻_𝑘); 𝑥_𝑡, 𝑥_𝑡−∈𝑃𝐶([−𝜏,0];ℝ^𝑛) are defined as 𝑥_𝑡=𝑥(𝑡+𝜃) and 𝑥_𝑡−=𝑥(𝑡⁻+𝜃) for 𝜃∈[−𝜏,0] respectively.

Throughout this paper, we assume that 𝑓, 𝐼_𝑘, and 𝐽_𝑘, 𝑘∈ℤ₊, satisfy the necessary conditions for the global existence and uniqueness of solutions for all 𝑡⩾𝑡₀ (refer to [3, 34, 35]). Then, for any 𝜙∈𝑃𝐶([−𝜏,0];ℝ^𝑛), there exists a unique function satisfying system (2) denoted by 𝑥(𝑡;𝑡₀,𝜙), which is continuous on the right-hand side and limitable on the left-hand side. Moreover, we assume that 𝑓(𝑡,0)≡0, 𝐼_𝑘(𝑡_𝑘,0)≡0, and 𝐽_𝑘(𝑡_𝑘,0)≡0, 𝑘∈ℤ₊, which implies that 𝑥(𝑡)≡0 is a solution of (2), which is called the trivial solution.

At the end of this section, let us introduce the following definitions.

Definition 1. A function 𝑉∶[𝑡₀−𝜏,∞)×ℝ^𝑛→ℝ₊ belongs to class 𝑣₀ if the following are satisfied. (i)𝑉is continuous on each of the sets [𝑡_𝑘−1,𝑡_𝑘)×ℝ^𝑛, and for each 𝑥,𝑦∈ℝ^𝑛, 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘∈ℤ₊, lim_{(𝑡,𝑦)→(𝑡}⁻_𝑘,𝑥)𝑉(𝑡,𝑦)=𝑉(𝑡⁻_𝑘,𝑥) exists.(ii)𝑉(𝑡,𝑥) is locally Lipschitz in 𝑥∈ℝ^𝑛, and 𝑉(𝑡,0)≡0 for all 𝑡⩾𝑡₀.

Definition 2. Given a function 𝑉∈𝑣₀, the upper right-hand Dini derivative of 𝑉 with respect to system (2) is defined by 𝐷⁺𝑉𝑡,𝜓(0)=limsup_ℎ→0+1ℎ𝑉𝑡+ℎ,𝜓(0)+ℎ𝑓𝑡,𝜓−𝑉𝑡,𝜓(0) (3) for (𝑡,𝜓)∈[𝑡₀,∞)×𝑃𝐶([−𝜏,0];ℝ^𝑛).

Definition 3. The trivial solution of system (2) or, simply, system (2) is said to be exponentially stable if there is a pair of positive constants 𝜆, 𝐶 such that, for any initial data 𝑥_𝑡0=𝜙∈𝑃𝐶([−𝜏,0];ℝ^𝑛), the solution 𝑥(𝑡;𝑡₀,𝜙) satisfies 𝑥𝑡;𝑡₀,𝜙⩽𝐶𝜙𝑒^{−𝜆(𝑡−𝑡0)},𝑡⩾𝑡₀. (4)

3. Main Results

In this section, we will analyze the exponential stability of system (2) by employing the Lyapunov functions.

Theorem 4. Assume that 𝑉∈𝑣₀ and there exist constants 𝑝>0, 𝑐₁>0, 𝑐₂>0, 𝜂₂⩾0, 𝑑_1𝑘>0, 𝑑_2𝑘⩾0, 𝑘∈ℤ₊, 𝜂₁, and 𝛿 such that (i)𝑐₁|𝑥|^𝑝⩽𝑉(𝑡,𝑥)⩽𝑐₂|𝑥|^𝑝 for all (𝑡,𝑥)∈[𝑡₀−𝜏,∞)×ℝ^𝑛;(ii)𝑉(𝑡_𝑘,𝜑(0)+𝐼_𝑘(𝑡_𝑘,𝜑(0))+𝐽_𝑘(𝑡_𝑘,𝜑))⩽𝑑_1𝑘𝑉(𝑡⁻_𝑘,𝜑(0))+𝑑_2𝑘sup_{𝜃∈[−𝜏,0]}𝑉(𝑡⁻_𝑘+𝜃,𝜑(𝜃)) for all 𝜑∈𝑃𝐶([−𝜏,0];ℝ^𝑛), 𝑘∈ℤ₊;(iii)𝐷⁺𝑉(𝑡,𝜑)⩽𝜂₁𝑉(𝑡,𝜑(0))+𝜂₂sup_{𝜃∈[−𝜏,0]}𝑉(𝑡+𝜃,𝜑(𝜃)) for all 𝜑∈𝑃𝐶([−𝜏,0];ℝ^𝑛) and 𝑡⩾𝑡₀, 𝑡≠𝑡_𝑘, 𝑘∈ℤ₊;(iv)ln(𝑑_1𝑘+𝑑_2𝑘max_{𝜃∈[−𝜏,0]}𝑒^𝜂1𝜃)⩽𝛿(𝑡_𝑘−𝑡_𝑘−1) for each 𝑘∈ℤ₊;(v)𝛿+𝜂₁+𝛾𝜂₂<0, where 𝛾=sup_𝑘∈ℤ+{𝑒^{𝛿(𝑡𝑘−𝑡𝑘−1)},1/𝑒^{𝛿(𝑡𝑘−𝑡𝑘−1)}}.Then, system (2) is exponentially stable, and the convergence rate should not be greater than 𝜆/𝑝, where 𝜆 is the unique positive solution of 𝜆+𝛿+𝜂₁+𝛾𝜂₂𝑒^𝜆𝜏=0.

Proof. Fix any initial data 𝜙∈𝑃𝐶([−𝜏,0];ℝ^𝑛), and write 𝑥(𝑡;𝑡₀,𝜙)=𝑥(𝑡) and 𝑉(𝑡,𝑥(𝑡))=𝑉(𝑡) simply. Set 𝑊(𝑡)=𝑒^{−𝜂1(𝑡−𝑡0)}𝑉(𝑡), 𝑡∈[𝑡₀−𝜏,∞). For each 𝑘∈ℤ₊, by condition (ii), we have 𝑊𝑡_𝑘=𝑒^{−𝜂1(𝑡𝑘−𝑡0)}𝑉𝑡_𝑘⩽𝑒^{−𝜂1(𝑡𝑘−𝑡0)}𝑑_1𝑘𝑉𝑡⁻_𝑘+𝑑_2𝑘sup_{[]𝜃∈−𝜏,0}𝑉𝑡⁻_𝑘+𝜃⩽𝑑_1𝑘𝑒^{−𝜂1(𝑡𝑘−𝑡0)}𝑉𝑡⁻_𝑘+𝛼𝑑_2𝑘sup_{[]𝜃∈−𝜏,0}𝑉𝑡⁻_𝑘+𝜃𝑒^{−𝜂1(𝑡𝑘+𝜃−𝑡0)}=𝑑_1𝑘𝑊𝑡⁻_𝑘+𝛼𝑑_2𝑘sup_{[]𝜃∈−𝜏,0}𝑊𝑡⁻_𝑘+𝜃, (5) where 𝛼=max_{𝜃∈[−𝜏,0]}𝑒^𝜂1𝜃.
On the other hand, by condition (iii), we know that, for any 𝑡≠𝑡_𝑘, 𝑘∈ℤ₊, 𝐷⁺𝑊(𝑡)=𝑒^{−𝜂1(𝑡−𝑡0)}−𝜂₁𝑉(𝑡)+𝐷⁺𝑉(𝑡)⩽𝜂₂𝑒^{−𝜂1(𝑡−𝑡0)}sup_{[]𝜃∈−𝜏,0}𝑉(𝑡+𝜃). (6) For 𝑡∈[𝑡₀,𝑡₁), integrating inequality (6) from 𝑡₀ to 𝑡, we obtain 𝑊(𝑡)⩽𝑊𝑡₀+^𝑡_𝑡0𝜂₂𝑒^{−𝜂1(𝑠−𝑡0)}sup_{[]𝜃∈−𝜏,0}𝑉(𝑠+𝜃)𝑑𝑠. (7) This implies that 𝑊𝑡⁻₁⩽𝑊𝑡₀+^𝑡1_𝑡0𝜂₂𝑒^{−𝜂1(𝑠−𝑡0)}sup_{[]𝜃∈−𝜏,0}𝑉(𝑠+𝜃)𝑑𝑠. (8) For 𝑡∈[𝑡₁,𝑡₂), by the same method, together with (5) and (8), we have 𝑊(𝑡)⩽𝑊𝑡₁+^𝑡_𝑡1𝜂₂𝑒^{−𝜂1(𝑠−𝑡0)}sup_{[]𝜃∈−𝜏,0}𝑉(𝑠+𝜃)𝑑𝑠⩽𝑑₁₁𝑊𝑡⁻₁+𝛼𝑑₂₁sup_{[]𝜃∈−𝜏,0}𝑊𝑡⁻₁++𝜃_𝑡𝑡1𝜂₂𝑒^{−𝜂1(𝑠−𝑡0)}sup_{[]𝜃∈−𝜏,0}𝑉(𝑠+𝜃)𝑑𝑠⩽𝑑₁₁𝑊𝑡₀+^𝑡1_𝑡0𝜂₂𝑒^{−𝜂1(𝑠−𝑡0)}sup_{[]𝜃∈−𝜏,0}𝑉(𝑠+𝜃)𝑑𝑠+𝛼𝑑₂₁𝑊𝑡₀+sup_{[]𝜃∈−𝜏,0}^𝑡1_𝑡^+𝜃₀𝜂₂𝑒^{−𝜂1(𝑠−𝑡0)}×sup_{[]𝜃∈−𝜏,0}𝑉(𝑠+𝜃)𝑑𝑠+^𝑡_𝑡1𝜂₂𝑒^{−𝜂1(𝑠−𝑡0)}sup_{[]𝜃∈−𝜏,0}𝑉()⩽𝑑𝑠+𝜃𝑑𝑠₁₁+𝛼𝑑₂₁𝑊𝑡₀+𝑑₁₁+𝛼𝑑₂₁^𝑡1_𝑡0𝜂₂𝑒^{−𝜂1(𝑠−𝑡0)}sup_{[]𝜃∈−𝜏,0}𝑉()+𝑠+𝜃𝑑𝑠_𝑡𝑡1𝜂₂𝑒^{−𝜂1(𝑠−𝑡0)}sup_{[]𝜃∈−𝜏,0}𝑉(𝑠+𝜃)𝑑𝑠. (9) By induction, we have, for 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘∈ℤ₊, 𝑊(𝑡)⩽𝑊𝑡_{0𝑡0<𝑡𝑖⩽𝑡}𝑑_1𝑖+𝛼𝑑_2𝑖+^𝑡_{𝑡0𝑠<𝑡𝑖⩽𝑡}𝑑_1𝑖+𝛼𝑑_2𝑖𝜂₂𝑒^{−𝜂1(𝑠−𝑡0)}sup_{[]𝜃∈−𝜏,0}𝑉(𝑠+𝜃)𝑑𝑠. (10) Thus, for 𝑡>𝑡₀, we get 𝑉(𝑡)⩽𝑉𝑡₀𝑒^{𝜂1(𝑡−𝑡0)}_{𝑡0<𝑡𝑖⩽𝑡}𝑑_1𝑖+𝛼𝑑_2𝑖+^𝑡_{𝑡0𝑠<𝑡𝑖⩽𝑡}𝑑_1𝑖+𝛼𝑑_2𝑖𝜂₂𝑒^{𝜂1(𝑡−𝑠)}sup_{[]𝜃∈−𝜏,0}𝑉(𝑠+𝜃)𝑑𝑠. (11)
Let 𝑡_𝑗1,𝑡_𝑗2,…,𝑡_𝑗𝑚 be impulse points in (𝑠,𝑡], 𝑡>𝑠. In view of condition (iv), we get _{𝑠<𝑡𝑖⩽𝑡}𝑑_1𝑖+𝛼𝑑_2𝑖=𝑑_1𝑗1+𝛼𝑑_2𝑗1𝑑_1𝑗2+𝛼𝑑_2𝑗2⋯𝑑_1𝑗𝑚+𝛼𝑑_2𝑗𝑚⩽𝑒^{𝛿(𝑡𝑗1−𝑡𝑗1−1)}𝑒^{𝛿(𝑡𝑗2−𝑡𝑗1)}⋯𝑒^{𝛿(𝑡𝑗𝑚−𝑡𝑗𝑚−1)}=𝑒^{𝛿(𝑡𝑗𝑚−𝑡𝑗1−1)}=𝑒^{𝛿(𝑡−𝑠)}𝑒^{𝛿(𝑡𝑗𝑚−𝑡)}𝑒^{𝛿(𝑠−𝑡𝑗𝑙−1)}⩽𝛾𝑒^{𝛿(𝑡−𝑠)}, (12) where 𝑡_𝑗1−1 is the first impulsive point before 𝑡_𝑗1 and satisfies 𝑡_𝑗1−1<𝑠. Submitting this into inequality (11), then, for 𝑡>𝑡₀, 𝑉(𝑡)⩽𝛾𝑒^{(𝜂1+𝛿)(𝑡−𝑡0)}𝑉𝑡₀+^𝑡_𝑡0𝛾𝜂₂𝑒^{(𝜂1+𝛿)(𝑡−𝑠)}sup_{[]𝜃∈−𝜏,0}𝑉(𝑠+𝜃)𝑑𝑠. (13) Let Φ(𝜆)=𝜆+𝜂₁+𝛿+𝛾𝜂₂𝑒^𝜆𝜏. Then, condition (v) implies Φ(0)<0. Moreover, Φ(+∞)=+∞, and Φ^′(𝜆)=1+𝜏𝛾𝜂₂𝑒^𝜆𝜏>0. Hence, Φ(𝜆)=0 has a unique positive solution 𝜆. Next, we claim that 𝑉(𝑡)⩽𝛾sup_{[]𝜃∈−𝜏,0}𝑉𝑡₀+𝜃𝑒^{−𝜆(𝑡−𝑡0)},𝑡⩾𝑡₀−𝜏. (14) Obviously, 𝑉(𝑡)⩽sup_{[]𝜃∈−𝜏,0}𝑉𝑡₀+𝜃⩽𝛾sup_{[]𝜃∈−𝜏,0}𝑉𝑡₀+𝜃𝑒^{−𝜆(𝑡−𝑡0)},𝑡∈𝑡₀−𝜏,𝑡₀. (15) So we only need to prove (14) for 𝑡>𝑡₀. Suppose not; then there exists a 𝑡^∗∈(𝑡₀,+∞) such that 𝑉𝑡^∗>𝛾sup_{[]𝜃∈−𝜏,0}𝑉𝑡₀+𝜃𝑒^{−𝜆(𝑡∗−𝑡0)},𝑉(𝑡)⩽𝛾sup_{[]𝜃∈−𝜏,0}𝑉𝑡₀+𝜃𝑒^{−𝜆(𝑡−𝑡0)},𝑡∈𝑡₀−𝜏,𝑡^∗. (16)(17) Thus, from (13), (17), and Φ(𝜆)=0, we see that 𝑉𝑡^∗⩽𝛾sup_{[]𝜃∈−𝜏,0}𝑉𝑡₀+𝜃𝑒^{(𝜂1+𝛿)(𝑡∗−𝑡0)}+𝛾^𝑡∗_𝑡0𝜂₂𝑒^{(𝜂1+𝛿)(𝑡∗−𝑠)}sup_{[]𝜃∈−𝜏,0}𝑉(𝑠+𝜃)𝑑𝑠⩽𝛾sup_{[]𝜃∈−𝜏,0}𝑉𝑡₀+𝜃𝑒^{(𝜂1+𝛿)(𝑡∗−𝑡0)}+𝛾^𝑡∗_𝑡0𝛾𝜂₂𝑒^𝜆𝜏𝑒^{(𝜂1+𝛿)(𝑡∗−𝑠)}𝑒^{−𝜆(𝑠−𝑡0)}sup_{[]𝜃∈−𝜏,0}𝑉𝑡₀+𝜃𝑑𝑠=𝛾sup_{[]𝜃∈−𝜏,0}𝑉𝑡₀+𝜃𝑒^{−𝜆(𝑡∗−𝑡0)}, (18) which is a contradiction. Therefore, (14) holds.
Then, it follows from (14) and condition (i) that 𝑐₁|𝑥(𝑡)|^𝑝⩽𝑉(𝑡)⩽𝛾sup_{[]𝜃∈−𝜏,0}𝑉𝑡₀+𝜃𝑒^{−𝜆(𝑡−𝑡0)}⩽𝛾𝑐₂𝜙^𝑝𝑒^{−𝜆(𝑡−𝑡0)},𝑡⩾𝑡₀, (19) which implies that |𝑥(𝑡)|⩽𝐶𝜙𝑒^{(−𝜆/𝑝)(𝑡−𝑡0)},𝑡⩾𝑡₀, (20) where 𝐶=(𝛾𝑐₂/𝑐₁)^1/𝑝. This completes the proof.

Remark 5. The parameters 𝑑_1𝑘 and 𝑑_2𝑘 in condition (ii) describe the influence of impulses on the stability of the underlying continuous systems. Conditions (iv) and (v) in Theorem 4 show that the system will be stable if the impulses frequency and amplitude are suitably related to the increase or decrease of the continuous flows.

Remark 6. It is well known that the Razumikhin techniques are very effective in the study of stability problems for ordinary and functional differential systems. However, when we use the Razumikhin techniques, we need to choose an appropriate minimal class of functionals relative to which the derivative of the Lyapunov function or Lyapunov functional is estimated, which is not entirely convenient. In this sense, Theorem 4 is more convenient to be applied than those Razumikhin-type theorems in [31, 32].

Let 𝐽_𝑘≡0 in system (2); then we have the following IDS (see, e.g., [19–24]): 𝑥̇(𝑡)=𝑓𝑥_𝑡,𝑡,𝑡≠𝑡_𝑘,𝑡⩾𝑡₀,Δ𝑥𝑡_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,𝑡_𝑘,𝑘∈ℤ₊,𝑥_𝑡0=𝜙(𝑠),𝑠∈[−𝜏,0]. (21) For system (21), we have the following results by Theorem 4.

Theorem 7. Assume that 𝑉∈𝑣₀ and there exist constants 𝑝>0, 𝑐₁>0, 𝑐₂>0, 𝜂₂⩾0, 𝑑_𝑘>0, 𝑘∈ℤ₊, 𝜂₁, and 𝛿 such that (i)𝑐₁|𝑥|^𝑝⩽𝑉(𝑡,𝑥)⩽𝑐₂|𝑥|^𝑝 for all (𝑡,𝑥)∈[𝑡₀−𝜏,∞)×ℝ^𝑛;(ii)𝑉(𝑡_𝑘,𝜑(0))+𝐼_𝑘(𝑡_𝑘,𝜑(0))⩽𝑑_𝑘𝑉(𝑡⁻_𝑘,𝜑(0)) for all 𝜑∈𝑃𝐶([−𝜏,0];ℝ^𝑛), 𝑘∈ℤ₊;(iii)𝐷⁺𝑉(𝑡,𝜑)⩽𝜂₁𝑉(𝑡,𝜑(0))+𝜂₂sup_{𝜃∈[−𝜏,0]}𝑉(𝑡+𝜃,𝜑(𝜃)) for all 𝜑∈𝑃𝐶([−𝜏,0];ℝ^𝑛) and 𝑡⩾𝑡₀, 𝑡≠𝑡_𝑘, 𝑘∈ℤ₊;(iv)ln𝑑_𝑘⩽𝛿(𝑡_𝑘−𝑡_𝑘−1) for each 𝑘∈ℤ₊;(v)𝛿+𝜂₁+𝛾𝜂₂<0, where 𝛾=sup_𝑘∈ℤ+{𝑒^{𝛿(𝑡𝑘−𝑡𝑘−1)},1/𝑒^{𝛿(𝑡𝑘−𝑡𝑘−1)}}.Then, system (21) is exponentially stable for any time delay 𝜏∈(0,∞), and the convergence rate should not be greater than 𝜆/𝑝, where 𝜆 is the unique positive solution of 𝜆+𝛿+𝜂₁+𝛾𝜂₂𝑒^𝜆𝜏=0.
In particular, if one takes 𝑑_𝑘≡𝑑 for all 𝑘∈ℤ₊, then suppose the impulsive instances 𝑡_𝑘 satisfy Δ_sup=sup_𝑘∈ℤ+𝑡_𝑘−𝑡_𝑘−1<∞,Δ_inf=inf_𝑘∈ℤ+𝑡_𝑘−𝑡_𝑘−1>0. (22)

For system (21), Theorem 7 yields the following result.

Theorem 8. Assume that 𝑉∈𝑣₀ and there exist constants 𝑝>0, 𝑐₁>0, 𝑐₂>0, 𝑑>0, 𝜚>0, 𝜂₂⩾0, and 𝜂₁ such that (i)𝑐₁|𝑥|^𝑝⩽𝑉(𝑡,𝑥)⩽𝑐₂|𝑥|^𝑝 for all (𝑡,𝑥)∈[𝑡₀−𝜏,∞)×ℝ^𝑛;(ii)𝑉(𝑡_𝑘,𝜑(0))+𝐼_𝑘(𝑡_𝑘,𝜑(0))⩽𝑑𝑉(𝑡⁻_𝑘,𝜑(0)) for all 𝜑∈𝑃𝐶([−𝜏,0];ℝ^𝑛), 𝑘∈ℤ₊;(iii)𝐷⁺𝑉(𝑡,𝜑)⩽𝜂₁𝑉(𝑡,𝜑(0))+𝜂₂sup_{𝜃∈[−𝜏,0]}𝑉(𝑡+𝜃,𝜑(𝜃)) for all 𝜑∈𝑃𝐶([−𝜏,0];ℝ^𝑛) and 𝑡⩾𝑡₀, 𝑡≠𝑡_𝑘, 𝑘∈ℤ₊;(iv)𝜂₁+𝜂₂𝑔(𝑑)+ln𝑑/𝜚<0, where 𝑔(𝑑)=𝑑^Δsup/𝜚 if 𝑑>1; 𝑔(𝑑)=1 if 𝑑=1; 𝑔(𝑑)=1/𝑑^Δinf/𝜚 if 𝑑<1.Then, for any 𝜏∈(0,∞), when 𝑑>1, system (21) is exponentially stable with impulse time sequences that satisfy Δ_inf⩾𝜚; when 𝑑=1, system (21) is exponentially stable with any impulse time sequences; when 𝑑<1, system (21) is exponentially stable with impulse time sequences that satisfy Δ_sup⩽𝜚.

Proof. We just need to apply Theorem 7 with 𝛿=ln𝑑/𝜚 and 𝛾=𝑔(𝑑).

Remark 9. When 𝑑>1, the Lyapunov function 𝑉 may jump up along the state trajectories of system (21) at impulse times 𝑡_𝑘. Thus, the impulses may be viewed as disturbances; that is, they potentially destroy the stability of continuous system. In this case, it is required that the impulses do not occur too frequently.

Remark 10. When 𝑑<1, the Lyapunov function 𝑉 may jump down along the state trajectories of system (21) at impulse times 𝑡_𝑘. Thus, the impulses may be treated as a stabilizing factor; that is, they may be used to stabilize an unstable continuous system. In this case, the impulses must take place frequently enough, and their amplitude must be suitably related to the growth rate of 𝑉.

Remark 11. When 𝑑=1, both the continuous dynamics and the discrete dynamics are stable, so the system can preserve exponential stability regardless of how often or how seldom impulses occur.

4. Applications and Example

Consider the following linear impulsive systems with time-varying delays: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝑟(𝑡)),𝑡≠𝑡_𝑘,𝑡⩾𝑡₀,Δ𝑥𝑡_𝑘=𝑥𝑡_𝑘−𝑥𝑡⁻_𝑘=𝐶_𝑘𝑥𝑡⁻_𝑘+𝐷_𝑘𝑥𝑡⁻_𝑘−𝑟𝑡_𝑘,𝑘∈ℤ₊,𝑥_𝑡0=𝜙(𝑠),𝑠∈[−𝜏,0], (23) where 𝑥(𝑡)∈ℝ^𝑛 is the system state vector, 𝐴, 𝐵, 𝐶_𝑘, and 𝐷_𝑘 are 𝑛×𝑛 matrices, and 𝑟∶ℝ₊→[0,𝜏] with 𝜏<∞ is the time-varying delay.

Theorem 12. Assume that there exist a matrix 𝑃>0 and several constants 𝑑₁>0, 𝑑₂⩾0, 𝜂₂⩾0, and 𝜂₁ such that (i)the following matrix inequalities hold: −𝑑₁𝑃0𝐼+𝐶_𝑘^𝑇𝑃∗−𝑑₂𝑃𝐷^𝑇_𝑘𝑃∗∗−𝑃⩽0,𝑃𝐴+𝐴^T𝑃−𝜂₁𝑃𝑃𝐵∗−𝜂₂𝑃⩽0; (24)(25)(ii)ln(𝑑₁+𝑑₂max_{𝜃∈[−𝜏,0]}𝑒^𝜂1𝜃)⩽𝛿(𝑡_𝑘−𝑡_𝑘−1);(iii)𝛿+𝜂₁+𝜂₂𝛾<0, where 𝛾=sup_𝑘∈ℤ+{𝑒^{𝛿(𝑡𝑘−𝑡𝑘−1)},1/𝑒^{𝛿(𝑡𝑘−𝑡𝑘−1)}}.Then, system (23) is exponentially stable, and the convergence rate should not be greater than 𝜆/2, where 𝜆 is the unique positive solution of 𝜆+𝛿+𝜂₁+𝛾𝜂₂𝑒^𝜆𝜏=0.

Proof. Let 𝑉(𝑡,𝑥)=𝑥^𝑇𝑃𝑥, 𝐼_𝑘(𝑡_𝑘,𝑥(𝑡⁻_𝑘))=𝐶_𝑘𝑥(𝑡⁻_𝑘), and 𝐽_𝑘(𝑡_𝑘,𝑥_𝑡⁻_𝑘)=𝐷_𝑘𝑥(𝑡⁻_𝑘−𝑟(𝑡_𝑘)). Then, (24) combined with Schur complement yields −𝑑₁𝑃00−𝑑₂𝑃+𝐼+𝐶_𝑘𝐷_𝑘^𝑇𝑃𝐼+𝐶_𝑘𝐷_𝑘⩽0. (26) Thus, for 𝜑∈𝑃𝐶([−𝜏,0];ℝ^𝑛), 𝑡=𝑡_𝑘, 𝑘∈ℤ₊, using the second equation of (23), we get 𝑉𝑡_𝑘,𝜑(0)+𝐼_𝑘𝑡_𝑘,𝜑(0)+𝐽_𝑘𝑡_𝑘=𝜑(0)𝜑𝑡,𝜑−𝑟_𝑘^𝑇𝐼+𝐶_𝑘𝐷_𝑘^𝑇𝑃𝐼+𝐶_𝑘𝐷_𝑘×𝜑(0)𝜑−𝑟𝑡_𝑘⩽𝜑(0)𝜑−𝑟𝑡_𝑘^𝑇𝑑₁𝑃00𝑑₂𝑃𝜑(0)𝜑−𝑟𝑡_𝑘⩽𝑑₁𝑉𝑡⁻_𝑘,𝜑(0)+𝑑₂sup_{[]𝜃∈−𝜏,0}𝑉𝑡⁻_𝑘+𝜃,𝜑(𝜃). (27)
In view of (25), for 𝜑∈𝑃𝐶([−𝜏,0];ℝ^𝑛) and 𝑡≠𝑡_𝑘, 𝑘∈ℤ₊, we have 𝐷⁺𝑉=𝜑(0)𝜑((𝑡𝑡,𝜑−𝑟))^𝑇𝑃𝐴+𝐴^𝑇𝜑(0)𝜑((𝑡⩽𝑃𝑃𝐵∗0−𝑟))𝜑(0)𝜑(−𝑟(𝑡))^𝑇𝜂₁𝑃0∗𝜂₂𝑃𝜑(0)𝜑(−𝑟(𝑡))⩽𝜂₁𝑉𝑡,𝜑(0)+𝜂₂sup_{[]𝜃∈−𝜏,0}𝑉𝑡+𝜃,𝜑(𝜃). (28) Consequently, the conclusion follows from Theorem 4 immediately, and the proof is complete.

Consider the special case 𝐷_𝑘≡0; from Theorem 7 and using the similar method in the proof of Theorem 12, we can obtain the following results.

Theorem 13. Let 𝐷_𝑘≡0, Δ_sup=sup_𝑘∈ℤ+{𝑡_𝑘−𝑡_𝑘−1}<∞, and Δ_inf=inf_𝑘∈ℤ+{𝑡_𝑘−𝑡_𝑘−1}>0. Assume that there exist a matrix 𝑃>0 and several constants 𝑑>0, 𝜚>0, 𝜂₂⩾0, and 𝜂₁ such that (i)the following matrix inequalities hold: 𝐼+𝐶_𝑘^𝑇𝑃𝐼+𝐶_𝑘⩽𝑑𝑃,𝑃𝐴+𝐴^𝑇𝑃−𝜂₁𝑃𝑃𝐵∗−𝜂₂𝑃⩽0; (29)(ii)𝜂₁+𝜂₂𝑔(𝑑)+ln𝑑/𝜚<0, where 𝑔(𝑑)=𝑑^Δsup/𝜚 if 𝑑>1; 𝑔(𝑑)=1 if 𝑑=1; 𝑔(𝑑)=1/𝑑^Δinf/𝜚 if 𝑑<1.Then, for any 𝜏∈(0,∞), when 𝑑>1, system (23) is exponentially stable with impulse time sequences that satisfy Δ_inf⩾𝜚; when 𝑑=1, system (23) is exponentially stable with any impulse time sequences; when 𝑑<1, system (23) is exponentially stable with impulse time sequences that satisfy Δ_sup⩽𝜚.

Example 14. Consider the following first-order impulsive delayed neural network: 𝑥̇(𝑡)=−𝑎𝑥(𝑡)+𝑏₀𝑓(𝑥(𝑡))+𝑏₁𝑓(𝑥(𝑡−𝜏)),𝑡≠𝑡_𝑘,𝑡⩾𝑡₀,Δ𝑥𝑡_𝑘=𝑐𝑥𝑡⁻_𝑘+𝑑𝑥𝑡⁻_𝑘−𝜏,𝑘∈ℤ₊, (30) where 𝑎=2.5, 𝑏₀=−1.5, 𝑏₁=0.5, 𝑐=0.2, 𝑑=0.1, and 𝜏=0.05. 𝑓(𝑥)=tanh(𝑥), and 𝑡_𝑘=𝜎𝑘, 𝑘∈ℤ₊, 𝜎>0, is a constant. It is easy to see that system without impulses is exponentially stable and the impulses are destabilizing since 𝑐,𝑑>0. Choose 𝑉(𝑡,𝑥)=𝑥²/2; then we obtain that conditions (i) and (ii) of Theorem 4 hold with 𝑐₁=𝑐₂=1/2, 𝑝=2, 𝑑_1𝑘=(1+𝑐)(1+𝑐+𝑑)=1.56, and 𝑑_2𝑘=𝑑(1+𝑐+𝑑)=0.13.
For 𝑡≠𝑡_𝑘, by simple calculation, we have 𝐷⁺𝑉(𝑡,𝑥(𝑡))=−𝑎𝑥²(𝑡)+𝑏₀𝑥(𝑡)𝑓(𝑥(𝑡))+𝑏₁𝑥(𝑡)𝑓(𝑥(𝑡−𝜏))⩽−𝑎+𝑏₀−12𝑏₁𝑥²(𝑡)+12𝑏₁𝑥²(𝑡−𝜏)=−2𝑎+2𝑏₀−𝑏₁𝑉(𝑡,𝑥(𝑡))+𝑏₁𝑉(𝑡,𝑥(𝑡−𝜏)), (31) which implies that condition (iii) of Theorem 4 holds with 𝜂₁=−(2𝑎+2𝑏₀−𝑏₁)=−1.5 and 𝜂₂=𝑏₁=0.5. Note that 𝑡_𝑘=𝜎𝑘, so we can take 𝛿=ln(𝑑_1𝑘+𝑑_2𝑘𝑒^−𝜂1𝜏)/(𝑡_𝑘−𝑡_𝑘−1)=ln(1.56+0.13𝑒^0.075)/𝜎 and 𝛾=𝑑_1𝑘+𝑑_2𝑘𝑒^−𝜂1𝜏=1.56+0.13𝑒^0.075. Then, by Theorem 4, system (30) is exponentially stable over any impulse time sequences satisfying 𝑡_𝑘−𝑡_𝑘−1⩾𝜎>ln(1.56+0.13𝑒^0.075)/[1.5−0.5(1.56+0.13𝑒^0.075)] = 0.4298.

Remark 15. Since the system without impulses is exponentially stable and the impulses are destabilizing, the existing results in [29, 32] cannot be applied to (30). The Razumikhin-type theorem in [31] is also not convenient to be applied to this system since it is not easy to find an appropriate constant 𝑞>1 to satisfy the Razumikhin-type condition.

Example 16. Consider system (23) with the following parameters: ,𝐶𝐴=210.82,𝐵=1−0.20.31_𝑘≡−0.300−0.3,𝐷_𝑘≡0.2000.2,𝑘∈ℤ₊, (32) and 𝑟(𝑡)=8+0.02[sin(𝑡)], where [⋅] denotes the integer function. Let 𝜏=8.2, 𝑑₁=0.49, 𝑑₂=0.04, 𝜂₁=7, 𝜂₂=1, 𝛾=0.53, 𝛿=−7.9360, and 𝑡_𝑘−𝑡_𝑘−1⩽0.08. It is easy to verify that 𝜂₁+𝜂₂𝑑𝛾+𝛿=−0.4060<0,ln₁+𝑑₂max_{[]𝜃∈−𝜏,0}𝑒^𝜂1𝜃=ln0.53=−0.6349<−7.9360×0.08⩽𝛿𝑡_𝑘−𝑡_𝑘−1. (33) Solving the linear matrix inequalities (24) and (25) in Theorem 12, we obtain the following feasible solution 𝑃=^0.35070.0332_0.03320.3076. Then, by Theorem 12, we know the given system is exponentially stable.

Remark 17. In Example 16, the impulses are used to stabilize an unstable system. In this case, the impulses must be frequent enough, and their amplitude must be suitably related to the growth rate of the continuous flow.

5. Conclusions

This paper has studied the exponential stability of impulsive delay systems in which the state variables on the impulses are related to the time delay. By using the Lyapunov function method, some criteria on the exponential stability are established. Moreover, the stability criteria obtained are applied to linear impulsive systems with time-varying delays, and a set of sufficient conditions for exponential stability is provided in terms of matrix inequalities. The obtained results improve and complement some recent works. Two examples have been given to illustrate the effectiveness and advantages of the results obtained.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61273126, 11226247, and 11301004), the Anhui Provincial Nature Science Foundation (1308085QA15, 1308085MA01), the Key Natural Science Foundation (KJ2009A49), the Foundation of Anhui Education Bureau (KJ2012A019), and the 211 Project of Anhui University (32030018 and 33010205).