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

Abstract

By using the Lyapunov functions and the Razumikhin techniques, the exponential stability of impulsive functional differential systems with delayed impulses is investigated. The obtained results have shown that the system will stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the continuous flows, and they improve and complement ones from some recent works. An example is provided to illustrate the effectiveness and the advantages of the results obtained.

1. Introduction

There has been a growing interest in the theory of impulsive dynamical systems in the past decades because of their applications to various problems arising in communications, control technology, impact mechanics, electrical engineering, medicine, biology and so forth; see the monographs [1, 2] and the papers [3–8] and the references therein. In particular, special attention has been focused on stability and impulsive stabilization of impulsive functional differential systems (IFDSs) (see, e.g., [9–26]).

However, in these previous works on stability of IFDSs, the authors always suppose that the state variables on the impulses are only related to the present state variables. But in most cases, it is more applicable that the state variables on the impulses that we add are also related to the past ones. For example, it is more realistic in practice if the impulsive control depends on a past state due to a time lag between the time when the observation of the state is made and the time when the feedback control reaches the system.

In fact, there have been several attempts in the literature to study the stability and control problems of a particular class of IFDSs with delayed impulses (see, e.g., [27–36]). 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. studied the impulsive synchronization problem coupled by linear delayed impulses in [28]. In addition, in [29–34], the authors investigate the uniform asymptotic stability and global exponential stability of general IFDSs: Δ𝑥𝑥̇(𝑡)𝑡_𝑘=𝑓𝑡,𝑥_𝑡,𝑡≠𝑡_𝑘,𝑡⩾𝑡₀,=𝐼_𝑘𝑡^𝑘,𝑥_𝑡^−𝑘,𝑘∈ℤ⁺.(1.1) But in these stability analyses, the effects of time delay on the impulses have been ignored. For example in [31–34], the Lyapunov function was assumed to be satisfied 𝑉(𝑡_𝑘,𝜑(0)+𝐼_𝑘(𝑡_𝑘,𝜑))⩽(1+𝑑_𝑘)𝑉(𝑡^−𝑘,𝜑(0)).

Very recently, in [35], Zhang and Sun established some sufficient conditions for uniform stability, uniform asymptotical stability, and practical stability of a particular class of IFDSs with delayed impulses: Δ𝑥𝑡_𝑘𝑥̇(𝑡)=𝑓𝑡,𝑥_𝑡,𝑡≠𝑡_𝑘,𝑡⩾𝑡₀,=𝐼_𝑘𝑥𝑡^−𝑘+𝐽_𝑘𝑥𝑡^−𝑘−𝜏,𝑘∈ℤ₊.(1.2) However, their results are only valid for some specific systems due to the restrictive requirements on the continuous flows and impulsive gain. Lin et al. [36] investigated the exponential stability and uniform stability of the following more generalized IFDSs with delayed impulses: Δ𝑥𝑡_𝑘𝑥̇(𝑡)=𝑓𝑡,𝑥_𝑡,𝑡≠𝑡_𝑘,𝑡⩾𝑡₀,=𝐼_𝑘𝑥𝑡^−𝑘+𝐽_𝑘𝑥^𝑡−𝑘,𝑘∈ℤ⁺.(1.3) But those results can only been applied to the systems with stable discrete dynamics since their results need the strong condition of impulsive gain 𝑑_𝑘+𝑒_𝑘<1.

Motivated by the above discussions, in this paper, we further study the exponential stability of IFDSs with delayed impulses. Different from the previous works on exponential stability of IFDSs with/without delayed impulses [18, 31, 34, 36], we will divide the systems into two classes: the system with stable continuous dynamics and unstable discrete dynamics, the systems with unstable continuous dynamics and stable discrete dynamics. The first class of impulsive systems corresponds to the case when the continuous dynamics are subjected to impulsive perturbations, while the second class of impulsive systems corresponds to the case when impulses are employed to stabilize the unstable continuous dynamics. This idea is enlightened in part by the works Chen and Zheng [37] about the uncertain impulsive systems. By using the Lyapunov functions and the Razumikhin techniques, some global exponential stability criteria are derived. The results obtained improve and complement some recent works. It is worth mentioning that our results shown that the system will be stable if the impulses' frequency and amplitude are suitably related to the increase or decrease of the continuous flows. Moreover, some results obtained can be applied to IFDSs with any time 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^𝑛 the 𝑛-dimensional real space equipped with the Euclidean norm |⋅|. Let 𝜏>0 and PC([−𝜏,0];ℝ^𝑛)={𝜑∶[−𝜏,0]→ℝ^𝑛|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₊∣𝜑ispiecewisecontinuous} 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: Δ𝑥𝑡_𝑘𝑥̇(𝑡)=𝑓𝑡,𝑥_𝑡,𝑡≠𝑡_𝑘,𝑡⩾𝑡₀,=𝐼_𝑘𝑡_𝑘,𝑥𝑡^−𝑘+𝐽_𝑘𝑡^𝑘,𝑥_𝑡^−𝑘,𝑘∈ℤ⁺𝑥_𝑡0=𝜙(𝑠),𝑠∈[−𝜏,0],(2.1) 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), ℂ is a open set in PC([−𝜏,0];ℝ^𝑛). and The fixed moments of impulse times {𝑡_𝑘,𝑘∈ℤ₊} satisfy 0⩽𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯,𝑡_𝑘→∞ (as 𝑘→∞), Δ𝑥(𝑡_𝑘)=𝑥(𝑡_𝑘)−𝑥(𝑡^−𝑘); 𝑥_𝑡, 𝑥_𝑡−∈PC([−𝜏,0];ℝ^𝑛) are defined by 𝑥_𝑡=𝑥(𝑡+𝜃), 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, see [6, 30–33]. Then for any 𝜙∈PC([−𝜏,0];ℝ^𝑛), 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 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 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^𝑛→ℝ₊ belongs to class 𝑣₀ if (i)𝑉is continuous on each of the sets [𝑡_𝑘−1,𝑡_𝑘)×ℝ^𝑛, and for each 𝑥∈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 2.2. Given a function 𝑉∈𝑣₀, the upper right-hand Dini derivative of 𝑉 with respect to system (2.1) is defined by 𝐷⁺𝑉(𝑡,𝜓(0))=limsup_ℎ→0+1ℎ[𝑉(𝑡+ℎ,𝜓(0)+ℎ𝑓(𝑡,𝜓))−𝑉(𝑡,𝜓(0))],(2.2) for (𝑡,𝜓)∈[𝑡₀,∞)×PC([−𝜏,0];ℝ^𝑛).

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 𝑥_𝑡0=𝜙∈PC([−𝜏,0];ℝ^𝑛), 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|^𝑝⩽𝑉(𝑡,𝑥)⩽𝑐₂|𝑥|^𝑝, for all (𝑡,𝑥)∈[𝑡₀−𝜏,∞)×ℝ^𝑛;(ii)𝑉(𝑡_𝑘,𝜑(0))⩽𝜌₁(1+𝜇_𝑘)𝑉(𝑡^−𝑘,𝜑(0))+𝜌₂(1+𝜇_𝑘)sup_{𝜃∈[−𝜏,0]}𝑉(𝑡^−𝑘+𝜃,𝜑(𝜃)),  for each 𝑘∈ℤ₊ and 𝜑∈𝑃𝐶([−𝜏,0];ℝ^𝑛), where 𝜇_𝑘, 𝑘∈ℤ₊, are nonnegative constants with Σ^∞𝑘=1𝜇_𝑘<∞;(iii)𝐷⁺𝑉(𝑡,𝜑(0))⩽−𝑐(𝑡)𝑉(𝑡,𝜑(0)), for all 𝑡⩾𝑡₀, 𝑡≠𝑡_𝑘, 𝑘∈ℤ₊, 𝜑∈PC([−𝜏,0];ℝ^𝑛), whenever 𝑉(𝑡+𝜃,𝜑)<𝑞𝑉(𝑡,𝜑(0)), 𝜃∈[−𝜏,0];(iv)𝜌₁+𝜌₂𝑒𝑐̃^𝜏<𝑞<𝑒𝑐̃^𝜚, inf_{𝑡⩾0𝑥00085𝑡0}𝑐(𝑡)⩾𝑐̃, where 𝜚=inf_𝑘∈ℤ+{𝑡_𝑘−𝑡_𝑘−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−(ln𝑞/𝜚)).

Proof. Set 𝐿=∏^∞𝑘=1(1+𝜇_𝑘); from the condition Σ^∞𝑘=1𝜇_𝑘<∞, we known that 1⩽𝐿<∞. Fix any initial data 𝜙∈PC([−𝜏,0];ℝ^𝑛) 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^𝛾𝜏𝜌¹+𝜌₂𝑒𝑐̃^𝜏<𝑞<𝑒⁽𝑐̃_−𝛾)𝜚,𝛾<𝑐̃.(3.1)
Set 𝑞̃=𝑞𝑒^−𝛾𝜏>1, choose 𝑀>0 such that 𝑞̃𝑐₂<𝑀. Define 𝑊(𝑡)=𝑒^{𝛾(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<1𝑞̃𝑀‖𝜙‖^𝑝<𝑀‖𝜙‖^𝑝,𝜃∈[−𝜏,0].(3.4) 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) Define 𝑡^∗∗=sup𝑡∈𝑡₀,𝑡^∗∶𝑊(𝑡)⩽1𝑞̃𝑀‖𝜙‖^𝑝,(3.8) then 𝑡^∗∗∈(𝑡₀,𝑡^∗) and 𝑊𝑡^∗∗=1𝑞̃𝑀‖𝜙‖^𝑝,𝑊(𝑡)>1𝑞̃𝑀‖𝜙‖^𝑝,𝑡∈𝑡^∗∗,𝑡^∗.(3.9) Consequently, for all 𝑡∈[𝑡^∗∗,𝑡^∗], 𝑊(𝑡+𝜃)⩽𝑀‖𝜙‖^𝑝⩽𝑞̃𝑊(𝑡),𝜃∈[−𝜏,0],(3.10) which implies that 𝑉(𝑡+𝜃)=𝑒^{−𝛾(𝑡+𝜃−𝑡0)}𝑊(𝑡+𝜃)⩽𝑞̃𝑒^{−𝛾(𝑡+𝜃−𝑡0)}𝑊(𝑡)⩽𝑞̃𝑒^𝛾𝜏𝑉(𝑡)=𝑞𝑉(𝑡),𝜃∈[−𝜏,0].(3.11) Then it follows from condition (iii) that one has that 𝐷⁺𝑊(𝑡)=𝑒^{𝛾(𝑡−𝑡0)}𝛾𝑉(𝑡)+𝐷⁺𝑉(𝑡)⩽(𝛾−𝑐(𝑡))𝑊(𝑡),𝑡∈𝑡^∗∗,𝑡^∗,(3.12) which leads to 𝑊𝑡^∗⩽𝑊𝑡^∗∗𝑒^{∫𝑡∗𝑡∗∗(𝛾−𝑐(𝑠))d𝑠}⩽𝑊𝑡^∗∗𝑒^(𝛾−𝑐̃_)(𝑡^∗_−𝑡^∗∗₎⩽1𝑞̃𝑀‖𝜙‖^𝑝<𝑀‖𝜙‖^𝑝,(3.13) this is a contradiction. Thus (3.5) holds.
Now we assume that for some 𝑚∈ℤ₊,𝑚⩾1, 𝑊(𝑡)<𝑀_𝑚‖𝜙‖^𝑝,𝑡∈𝑡₀−𝜏,𝑡_𝑚,(3.14) where 𝑀₁=𝑀, 𝑀_𝑚=𝑀∏_{1⩽𝑖⩽𝑚−1}(1+𝜇_𝑖) for 𝑚⩾2. We will prove that 𝑊(𝑡)<𝑀_𝑚+1‖𝜙‖^𝑝,𝑡∈𝑡_𝑚,𝑡_𝑚+1.(3.15) To do this, we first claim 𝑊𝑡^−𝑚+𝜃⩽𝑒⁽𝑐̃_−𝛾)𝜏𝑞̃𝑀^𝑚‖𝜙‖^𝑝,𝜃∈[−𝜏,0).(3.16) Suppose not, then there exists 𝜃̃∈[−𝜏,0) such that 𝑊(𝑡^−𝑚+𝜃̃)>(𝑒⁽𝑐̃_−𝛾)𝜏/𝑞̃)𝑀_𝑚‖𝜙‖^𝑝. Without lose generality, we assume 𝑡_𝑚+𝜃̃∈(𝑡_𝑙−1,𝑡_𝑙], 𝑙∈ℤ₊,𝑙⩽𝑚.
There are two cases to be considered.
Case 1. 𝑊(𝑡)>(𝑒⁽𝑐̃_−𝛾)𝜏/𝑞̃)𝑀_𝑚‖𝜙‖^𝑝 over 𝑡∈[𝑡_𝑙−1,𝑡_𝑚+𝜃̃).
By assumption (3.14), for all 𝑡∈[𝑡_𝑙−1,𝑡_𝑚+𝜃̃), we get 𝑊(𝑡+𝜃)<𝑀_𝑚‖𝜙‖^𝑝<𝑒⁽𝑐̃_−𝛾)𝜏𝑀_𝑚‖𝜙‖^𝑝<𝑞̃𝑊(𝑡),𝜃∈[−𝜏,0].(3.17) Thus, by conditions (iii)-(iv) and inequalities (3.10)–(3.13), we have 𝑊𝑡^−𝑚+𝜃̃⩽𝑊𝑡_𝑙−1𝑒^(𝛾−𝑐̃_)(𝑡^−𝑚₊𝜃̃_{−𝑡𝑙−1)}<𝑀_𝑚‖𝜙‖^𝑝𝑒⁽𝑐̃_−𝛾)𝜏𝑒^(𝛾−𝑐̃_{)(𝑡𝑚−𝑡𝑙−1)}⩽𝑒⁽𝑐̃_−𝛾)𝜏𝑞^{𝑚−𝑙+1}𝑀_𝑚‖𝜙‖^𝑝<𝑒⁽𝑐̃_−𝛾)𝜏𝑞̃𝑀^𝑚‖𝜙‖^𝑝.(3.18) This is a contradiction.
Case 2. There are some 𝑡∈[𝑡_𝑙−1,𝑡_𝑚+𝜃̃) such that 𝑊(𝑡)>(𝑒⁽𝑐̃_−𝛾)𝜏/𝑞̃)𝑀_𝑚‖𝜙‖^𝑝.
In this case, define 𝑡=sup𝑡∈𝑡_𝑙−1,𝑡_𝑚+𝜃̃∶𝑊(𝑡)⩽𝑒⁽𝑐̃_−𝛾)𝜏𝑞̃𝑀^𝑚‖𝜙‖^𝑝.(3.19) Then 𝑡∈[𝑡_𝑙−1,𝑡_𝑚+𝜃̃) and 𝑊𝑡=𝑒⁽𝑐̃_−𝛾)𝜏𝑞̃𝑀^𝑚‖𝜙‖^𝑝,𝑊(𝑡)>𝑒⁽𝑐̃_−𝛾)𝜏𝑞̃𝑀^𝑚‖𝜙‖^𝑝,𝑡∈𝑡,𝑡_𝑚+𝜃̃.(3.20) So from assumption (3.14), for any 𝑡∈[𝑡,𝑡_𝑚+𝜃̃), we have 𝑊(𝑡+𝜃)<𝑀_𝑚‖𝜙‖^𝑝<𝑒⁽𝑐̃_−𝛾)𝜏𝑀_𝑚‖𝜙‖^𝑝⩽𝑞̃𝑊(𝑡),𝜃∈[−𝜏,0].(3.21) It follows from condition (iii) that 𝑊𝑡^−𝑚+𝜃̃⩽𝑊𝑡=𝑒⁽𝑐̃_−𝛾)𝜏𝑞̃𝑀^𝑚‖𝜙‖^𝑝.(3.22) This is also a contradiction. Hence, inequality (3.16) holds.
Similarly, we can prove 𝑊𝑡^−𝑚⩽1𝑞̃𝑀^𝑚‖𝜙‖^𝑝.(3.23) Then it follows from (3.16), (3.23), and condition (ii) that we obtain 𝑊𝑡_𝑚⩽⩽𝜌𝜌₁1+𝜇_𝑚𝑊𝑡^−𝑚+𝜌₂1+𝜇_𝑚𝑒^𝛾𝜏sup_{[]𝜃∈−𝜏,0}𝑊𝑡^−𝑚+𝜃₁+𝜌₂𝑒𝑐̃^𝜏𝑞̃𝑀^𝑚+1‖𝜙‖^𝑝<𝑀_𝑚+1‖𝜙‖^𝑝.(3.24) Now we suppose that (3.15) is not true, let 𝑡^∗=inf𝑡∈𝑡_𝑚,𝑡_𝑚+1∶𝑊(𝑡)⩾𝑀_𝑚+1‖𝜙‖^𝑝.(3.25) Then 𝑡^∗∈(𝑡_𝑚,𝑡_𝑚+1) and 𝑊𝑡^∗=𝑀_𝑚+1‖𝜙‖^𝑝,𝑊(𝑡)<𝑀_𝑚+1‖𝜙‖^𝑝,𝑡∈𝑡_𝑚,𝑡^∗.(3.26)
If 𝑊(𝑡)>(1/𝑞̃)𝑀_𝑚+1‖𝜙‖^𝑝 for all 𝑡∈[𝑡_𝑚,𝑡^∗], set 𝑡^∗∗=𝑡_𝑚; otherwise, let 𝑡^∗∗=sup𝑡∈𝑡_𝑚,𝑡^∗∶𝑊(𝑡)⩽1𝑞̃𝑀^𝑚+1‖𝜙‖^𝑝.(3.27) Thus for all 𝑡∈[𝑡^∗∗,𝑡^∗], we have 𝑊(𝑡+𝜃)⩽𝑀_𝑚+1‖𝜙‖^𝑝⩽𝑞̃𝑊(𝑡),𝜃∈[−𝜏,0].(3.28) It follows from condition (iii) that 𝐷⁺𝑊(𝑡)=𝑒^{𝛾(𝑡−𝑡0)}𝛾𝑉(𝑡)+𝐷⁺𝑉(𝑡)⩽(𝛾−𝑐(𝑡))𝑊(𝑡),𝑡∈𝑡^∗∗,𝑡^∗,(3.29) which implies 𝑊𝑡^∗⩽𝑊𝑡^∗∗𝑒^(𝛾−𝑐̃_)(𝑡^∗_−𝑡^∗∗₎⩽𝑊𝑡^∗∗<𝑀_𝑚+1‖𝜙‖^𝑝.(3.30) This is a contradiction. Therefore, (3.15) holds.
By mathematical induction, (3.15) holds for any 𝑚∈ℤ₊. That is, (3.2) holds, which implies that ||𝑥(𝑡)||⩽𝐶‖𝜙‖𝑒^{(−𝛾/𝑝)(𝑡−𝑡0)},𝑡⩾𝑡₀,(3.31) where 𝐶=(LM/𝑐₁)^1/𝑝. This completes the proof.

Remark 3.2. The parameters 𝜌₁ and 𝜌₂ in condition (ii) describe the influence of impulses on the stability of the underlying continuous systems. When 𝜌₁+𝜌₂⩾1, the Lyapunov function 𝑉 may jump up along the state trajectories of system (2.1) at impulsive time instant 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. Theorem 3.1 tells us to what extent we can relax the restriction on the impulses to keep the exponential stability property of the original continuous system.

Theorem 3.3. 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|^𝑝⩽𝑉(𝑡,𝑥)⩽𝑐₂|𝑥|^𝑝, for all (𝑥,𝑡)∈ℝ^𝑛×[𝑡₀−𝜏,∞);(ii)𝑉(𝑡_𝑘,𝜑(0))⩽𝜌₁(1+𝜇_𝑘)𝑉(𝑡^−𝑘,𝜑(0))+𝜌₂(1+𝜇_𝑘)sup_{𝜃∈[−𝜏,0]}𝑉(𝑡^−𝑘+𝜃,𝜑(𝜃)), for each 𝑘∈ℤ₊ and 𝜑∈𝑃([−𝜏,0];ℝ^𝑛), where 𝜇_𝑘, 𝑘∈ℤ₊, are nonnegative constants with ∑^∞𝑘=1𝜇_𝑘<∞;(iii)𝐷⁺𝑉(𝑡,𝜑(0))⩽𝑐(𝑡)𝑉(𝑡,𝜑(0)), for all 𝑡⩾𝑡₀, 𝑡≠𝑡_𝑘, 𝑘∈ℤ₊, 𝜑∈PC([−𝜏,0];ℝ^𝑛), whenever 𝑉(𝑡+𝜃,𝜑)<𝑞𝑉(𝑡,𝜑(0)), 𝜃∈[−𝜏,0];(iv)𝑞>1/(𝜌₁+𝜌₂)>𝑒𝑐̃^𝜚, 𝑐̃𝜚⩾sup_𝑡⩾𝑡0∫^{𝑡𝑡+𝜚}𝑐(𝑠)𝑑𝑠, where 𝜚=sup_𝑘∈ℤ+{𝑡_𝑘−𝑡_𝑘−1}.
Then the trivial solution of system (2.1) is globally exponentially stable for any time delay 𝜏∈(0,∞) and the convergence rate should not be greater than (1/𝑝)((ln𝑞/𝜚)−𝑐̃).

Proof . From condition (iv), we can choose a small enough constant 𝛾>0 such that 𝑞>𝑒^𝛾𝜏𝜌₁+𝜌₂𝑒^𝛾𝜏>1𝜌₁+𝜌₂𝑒^𝛾𝜏>𝑒⁽𝑐̃_+𝛾)𝜚,𝑞𝑒^−𝛾𝜏>1.(3.32) Set 𝑞̃=𝑞𝑒^−𝛾𝜏. The following proof can be completed by using the similar arguments as in the proof of Theorem 3.1, so it is omitted.

Remark 3.4. When 𝜌₁+𝜌₂<1, the Lyapunov function 𝑉 may jump down along the state trajectories of system (2.1) at impulsive time instant 𝑡_𝑘. Thus the impulses may be viewed impulsive stabilizing, that is, they may be used to stabilize the continuous system if the original continuous system is not stable. In this case, the impulses must be frequent and their amplitude must be suitably related the growth rate of 𝑉.

Remark 3.5. If 𝑐(𝑡)≡𝑐, then Theorem 3.3 becomes Theorem 3.1 in [36] with 𝑑_𝑘=𝜌₁(1+𝜇_𝑘), 𝑒_𝑘=𝜌₂(1+𝜇_𝑘), 𝑑_𝑘+𝑒_𝑘<1. Obviously, Theorem 3.3 in this paper has a wider adaptive range than those in [36].
Let 𝐽_𝑘≡0 in system (2.1), then we have the following IFDS (see [9–23, 26]): Δ𝑥𝑥̇(𝑡𝑡_𝑘)=𝑓𝑥_𝑡,𝑡,𝑡≠𝑡_𝑘,𝑡⩾𝑡₀,𝑥_𝑡0=𝐼_𝑘𝑥𝑡^−𝑘,𝑡_𝑘,𝑘∈ℤ₊,=𝜙(𝑠),𝑠∈[−𝜏,0].(3.33) For system (3.33), we have the following results by Theorems  3.1 and 3.2, respectively.

Corollary 3.6. Assume that there exist functions 𝑉∈𝑣₀, 𝑐∈PC([𝑡₀−𝜏,∞);ℝ₊), and several positive constants 𝑐₁, 𝑐₂, 𝑐̃, 𝑝, 𝑞, and a constant 𝜌⩾1 such that (i)𝑐₁|𝑥|^𝑝⩽𝑉(𝑡,𝑥)⩽𝑐₂|𝑥|^𝑝, for all (𝑥,𝑡)∈ℝ^𝑛×[𝑡₀−𝜏,∞);(ii)𝑉(𝑡_𝑘,𝜑(0))⩽𝜌(1+𝜇_𝑘)𝑉(𝑡^−𝑘,𝜑(0)), for each 𝑘∈ℤ₊ and 𝜑∈𝑃([−𝜏,0];ℝ^𝑛), where 𝜇_𝑘, 𝑘∈ℤ₊, are nonnegative constants with ∑^∞𝑘=1𝜇_𝑘<∞;(iii)𝐷⁺𝑉(𝑡,𝜑(0))⩽−𝑐(𝑡)𝑉(𝑡,𝜑(0)), for all 𝑡⩾𝑡₀, 𝑡≠𝑡_𝑘,𝑘∈ℤ₊,𝜑∈PC([−𝜏,0];ℝ^𝑛), whenever 𝑉(𝑡+𝜃,𝜑)<𝑞𝑉(𝑡,𝜑(0)), 𝜃∈[−𝜏,0];(iv)𝜌<𝑞<𝑒𝑐̃^𝜚, inf_𝑡⩾𝑡0𝑐(𝑡)⩾𝑐̃, where 𝜚=inf_𝑘∈ℤ+{𝑡_𝑘−𝑡_𝑘−1}.
Then the trivial solution of system (3.33) is globally exponentially stable for any time delay 𝜏∈(0,∞) and the convergence rate should not be greater than (1/𝑝)(𝑐̃−(ln𝑞/𝜚)).

Corollary 3.7. Assume that there exist functions 𝑉∈𝑣₀, 𝑐∈PC([𝑡₀−𝜏,∞);ℝ₊) and several positive constants 𝑐₁, 𝑐₂, 𝑐̃, 𝑝, 𝑞, and a constant 𝜌<1 such that (i)𝑐₁|𝑥|^𝑝⩽𝑉(𝑡,𝑥)⩽𝑐₂|𝑥|^𝑝, for all (𝑥,𝑡)∈ℝ^𝑛×[𝑡₀−𝜏,∞);(ii)𝑉(𝑡_𝑘,𝜑(0))⩽𝜌(1+𝜇_𝑘)𝑉(𝑡^−𝑘,𝜑(0)), for each 𝑘∈ℤ₊, 𝜑∈𝑃([−𝜏,0];ℝ^𝑛), where 𝜇_𝑘, 𝑘∈ℤ₊, are nonnegative constants with ∑^∞𝑘=1𝜇_𝑘<∞;(iii)𝐷⁺𝑉(𝑡,𝜑(0))⩽𝑐(𝑡)𝑉(𝑡,𝜑(0)), for all 𝑡⩾𝑡₀, 𝑡≠𝑡_𝑘, 𝑘∈ℤ₊, 𝜑∈PC([−𝜏,0];ℝ^𝑛), whenever 𝑉(𝑡+𝜃,𝜑)<𝑞𝑉(𝑡,𝜑(0)), 𝜃∈[−𝜏,0];(iv)𝑞>1/𝜌>𝑒𝑐̃^𝜚, 𝑐̃𝜚⩾sup_{𝑡⩾0𝑥00085𝑡0}∫^{𝑡𝑡+𝜚}𝑐(𝑠)𝑑𝑠, where 𝜚=sup_𝑘∈ℤ+{𝑡_𝑘−𝑡_𝑘−1}. Then the trivial solution of system (3.33) is globally exponentially stable for any time delay 𝜏∈(0,∞) and the convergence rate should not be greater than (1/𝑝)((ln𝑞/𝜚)−𝑐̃).

Remark 3.8. If 𝑐(𝑡)≡𝑐>0, 𝜇_𝑘≡0, 𝑘∈ℤ₊, then Theorems  3.1 and 3.2 in [25] follow from Corollaries 3.6 and 3.7, respectively.

4. Example

In this section, an example is given to show the effectiveness and advantages of our results.

Example 4.1. Consider the following IFDS (see [35, 36]): 𝑥𝑥̇(𝑡𝑡)=𝑎𝑥(𝑡)+𝑏𝑥(𝑡−𝜏),𝑡≠𝑡_𝑘,𝑡>0,_𝑘=𝑐𝑥𝑡^−𝑘+𝑑𝑥𝑡^−𝑘−𝜏,𝑘∈ℤ₊,(4.1) where 𝑥∈ℝ, 𝜏>0.
In the following, we will divide the system (4.1) into two classes to consider.
Case 1. 𝑎⩾0 and 0<|𝑐|+|𝑑|<1.

Property 1. The trivial solution of system (4.1) is globally exponentially stable with impulse time sequences that satisfy sup_𝑘∈ℤ+𝑡_𝑘−𝑡_𝑘−1<−|𝑐|+||𝑑||ln|𝑐|+||𝑑||𝑎|𝑐|+||𝑑||+||𝑏||.(4.2)

Proof . From equality (4.2), one can choose a small enough constant ℎ>0 such that sup_𝑘∈ℤ+|𝑐|+||𝑑||−ℎ>0,𝑡_𝑘−𝑡_𝑘−1<−|𝑐|+||𝑑||−ℎln|𝑐|+||𝑑||𝑎|𝑐|+||𝑑||−ℎ+||𝑏||.(4.3)
Let 𝑉(𝑡,𝑥)=|𝑥|. By calculation, we have 𝐷⁺𝑉(𝑡,𝜑(0))⩽𝑎||𝜑(0)||+||𝑏||||𝜑(−𝜏)||=𝑎𝑉(𝑡,𝜑(0))+||𝑏||𝑉(𝑡,𝜑),(4.4) for all 𝑡≠𝑡_𝑘, 𝑘∈ℤ₊ and 𝜑∈PC([−𝜏,0];ℝ). By taking 𝑝=1, 𝑐₁=𝑐₂=1, 𝜌₁=|𝑐|, 𝜌₂=|𝑑|, 𝑞=1/(|𝑐|+|𝑑|−ℎ), 𝑐̃=𝑐(𝑡)≡𝑎+(|𝑏|/(|𝑐|+|𝑑|−ℎ)), and 𝜇_𝑘≡0, 𝑘∈ℤ₊ in Theorem 3.3, it is easy to obtain Property 1.

Remark 4.2. In this case, the impulses are used to stabilize the unstable original continuous system. In [35], under assumption that 𝑎, 𝑏, 𝑐, 𝑑>0, and 𝑐+𝑑<1, Zhang and Sun obtained that system (4.1) is uniformly stable if the impulses’ instances satisfy sup_𝑘∈ℤ+𝑡_𝑘−𝑡_𝑘−1<−2(𝑐+𝑑)²ln(𝑐+𝑑)(2𝑎+𝑏)(𝑐+𝑑)²+𝑏;(4.5) Lin et al. [36] derived that system (4.1) is exponentially stable if sup_𝑘∈ℤ+𝑡_𝑘−𝑡_𝑘−1<−12(𝑐+𝑎(𝑑𝑐)+ln𝑑)(𝑐++𝑏𝑑).(4.6) Obviously, under condition 𝑎, 𝑏, 𝑐, 𝑑>0, and 𝑐+𝑑<1, we get −|𝑐|+||𝑑||ln|𝑐|+||𝑑||𝑎|𝑐|+||𝑑||+||𝑏||=−(𝑐+𝑎(𝑑𝑐)+ln𝑑)(𝑐++𝑏𝑑)>−12(𝑐+𝑎(𝑑𝑐)+ln𝑑)(𝑐++𝑏𝑑),(4.7) and one can also verify that −(𝑐+𝑑)ln(𝑐+𝑑)𝑎(𝑐+𝑑)+𝑏>−2(𝑐+𝑑)²ln(𝑐+𝑑)(2𝑎+𝑏)(𝑐+𝑑)²+𝑏.(4.8) So our results are less conservative than those in [35, 36].
Case 2. 𝑎<0 and |𝑐|+|𝑑|⩾1.

Property 2. Suppose that system’s parameters 𝑎, 𝑏, 𝑐, 𝑑 and time delay 𝜏 satisfy |𝑐|+||𝑑||²𝑒^−2𝑎𝜏<−2𝑎||+𝑏||||𝑏||.(4.9) Then the trivial solution of system (4.1) is globally exponentially stable with impulse time sequences that satisfy inf_𝑘∈ℤ+𝑡_𝑘−𝑡_𝑘−1>2𝑎𝜏−2ln|𝑐|+||𝑑||2𝑎+||𝑏||+||𝑏|||𝑐|+||𝑑||²𝑒^−2𝑎𝜏.(4.10)

Proof. From equalities (4.9) and (4.10), we can choose a small enough constant ℎ>0 such that inf|𝑐_𝑘∈ℤ+|+||𝑑||²𝑒^−2𝑎𝜏<|𝑐|+||𝑑||²𝑒^−2𝑎𝜏+ℎ<−2𝑎||+𝑏||||𝑏||,𝑡_𝑘−𝑡_𝑘−1>−ln|𝑐|+||𝑑||²𝑒^−2𝑎𝜏+2ℎ2𝑎+||𝑏||+||𝑏|||𝑐|+||𝑑||²𝑒^−2𝑎𝜏+ℎ.(4.11) Set 𝑞=(|𝑐|+|𝑑|)²𝑒^−2𝑎𝜏+ℎ, then one can conclude that 2𝑎+||𝑏||+𝑞||𝑏||<0,|𝑐|+||𝑑||²𝑒^{−(2𝑎+|𝑏|+𝑞|𝑏|)𝜏}<𝑞<−2𝑎+||𝑏||||𝑏||.(4.12)
Let 𝑉(𝑡,𝑥)=(1/2)𝑥². By calculation, we have 𝐷⁺𝑉(𝑡,𝜑(0))⩽𝑎+12||𝑏||𝜑²(0)+12||𝑏||𝜑²(−𝜏)=2𝑎+||𝑏||𝑉(𝑡,𝜑(0))+||𝑏||𝑉(𝑡,𝜑),(4.13) for all 𝑡≠𝑡_𝑘, 𝑘∈ℤ₊, and 𝜑∈PC([−𝜏,0];ℝ). By taking 𝑝=2, 𝑐₁=𝑐₂=2, 𝜌₁=|𝑐|(|𝑐|+|𝑑|), 𝜌₂=|𝑑|(|𝑐|+|𝑑|), 𝑐̃=𝑐(𝑡)≡−(2𝑎+|𝑏|+𝑞|𝑏|), and 𝜇_𝑘≡0, 𝑘∈ℤ₊ in Theorem 3.3, we can obtain Property 2.

Remark 4.3. In this case, the underlying continuous system is stable, the impulses are disturbances, which potentially destroy the stability of continuous system. So the existing results in [35, 36] are invalid for this case.

5. Conclusions

This paper has studied the exponential stability of IFDSs in which the state variables on the impulses are related to the time delay. By using the Razumikhin techniques and the Lyapunov functions, some criteria on the global exponential stability are established. The obtained results improve and complement some recent works. An example has been given to illustrate the effectiveness and the advantages of the results obtained.

Acknowledgments

This work was supported by the 211 Project of the Anhui University (32030018/33010205/KJTD002B), the Research Fund for Doctor Station of the Ministry of Education of China (20113401110001), the Key Natural Science Foundation (KJ2009A49), the Foundation of Anhui Education Bureau (KJ2012A019), and the National Natural Science Foundation of China (11126179).