Global Exponential Stability of Impulsive Functional Differential Equations via Razumikhin Technique

Abstract

This paper develops some new Razumikhin-type theorems on global exponential stability of impulsive functional differential equations. Some applications are given to impulsive delay differential equations. Compared with some existing works, a distinctive feature of this paper is to address exponential stability problems for any finite delay. It is shown that the functional differential equations can be globally exponentially stabilized by impulses even if it may be unstable itself. Two examples verify the effectiveness of the proposed results.

1. Introduction

Functional differential equations (FDEs) which include delay differential equations (DDEs) play a very important role in formulation and analysis in mechanical, electrical, control engineering and physical sciences, economic, and social sciences [1, 2]. Therefore, the theory of FDEs has been developed very quickly. The investigation for FDEs has attracted the considerable attention of researchers and many qualitative theories of FDEs have been obtained. A large number of stability criteria of FDEs have been reported.

In addition to the delay effect, as is well known, impulsive effect is likely to exist in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time in the fields such as medicine and biology, economics, electronics, and telecommunications [3]. So far, a lot of interesting results on stability have been reported that have focused on the impulsive effect of FDEs (see, e.g., [4–16] and the references cited therein).

In particular, several papers devoted to the study of exponential stability of impulsive functional differential equations (IFDEs) have appeared during the past years. In [14, 15], the authors have investigated exponential stability of IFDEs by using the method of Lyapunov functions and Razumikhin techniques. In [16], the authors have also studied exponential stability by using the method of Lyapunov functional. However, some results in [15, 16] imposed a restrictive condition on time delays which were less than the length of all the impulsive intervals (see, e.g., [15, Theorems 3.1-3.2] and [16, Theorem 3.1]). The aim of this paper is to establish global exponential stability criteria for IFDEs by employing the Razumikhin technique which illustrate that impulses do contribute to the stability of some IFDEs and the restrictive condition that the time delays are less than the length of all the impulsive intervals can be removed in this paper.

2. Preliminaries

Throughout this paper, unless otherwise specified, we use the following notations. Let ℝ=(−∞,+∞), ℝ⁺=[0,+∞), ℕ={1,2,…}, 𝐼 be the identity matrix, 𝜆_max(⋅) and 𝜆_min(⋅) be the maximal eigenvalue and the minimal eigenvalue of a matrix, respectively. If 𝐴 is a vector or matrix, its transpose is denoted by 𝐴^𝑇. For 𝑥∈ℝ^𝑛 and 𝐴∈ℝ^𝑛×𝑛, let |𝑥|=𝑥^𝑇𝑥 be Euclidean vector norm, and denote the induced matrix norm by

‖𝐴‖=sup_𝑥≠0|𝐴𝑥|=|𝑥|𝜆_max(𝐴^𝑇𝐴).(2.1)

Let 𝜏>0 and 𝐶=𝐶([−𝜏,0];ℝ^𝑛) denote the family of all bounded continuous ℝ^𝑛-valued functions 𝜙 defined on [−𝜏,0]. 𝑃𝐶(𝕀;ℝ^𝑛)={𝜓∶𝕀→ℝ^𝑛∣𝜓(𝑠) is continuous for all but at most countable points 𝑠∈𝕀 and at these points 𝑠∈𝕀, 𝜓(𝑠⁺) and 𝜓(𝑠⁻) exist and 𝜓(𝑠⁺)=𝜓(𝑠)}, where 𝕀⊂ℝ is an interval, 𝜓(𝑠⁺) and 𝜓(𝑠⁻) denote the right-hand and left-hand limits of the function 𝜓(𝑠) at time 𝑠, respectively. Especially, let 𝑃𝐶≜𝑃𝐶([−𝜏,0];ℝ^𝑛) with norm ‖𝜓‖=sup_{−𝜏⩽𝑠⩽0}|𝜓(𝑠)|.

In this paper, we consider the following IFDEs:

𝑥^′(𝑡)=𝑓(𝑡,𝑥_𝑡),𝑡≠𝑡_𝑘,𝑡⩾0,Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘)=𝐼_𝑘(𝑥(𝑡⁻_𝑘)),𝑡=𝑡_𝑘,𝑘∈ℕ,𝑥(𝑠)=𝜙(𝑠),−𝜏⩽𝑠⩽0,(2.2) where 𝑓∶ℝ⁺×𝑃𝐶→ℝ^𝑛, 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃), 𝜃∈[−𝜏,0]. The initial function 𝜙∈𝑃𝐶. The impulsive function 𝐼_𝑘∈𝐶(ℝ^𝑛;ℝ^𝑛)(𝑘∈ℕ), and the impulsive moments 𝑡_𝑘(𝑘=1,2,…) satisfy 0=𝑡₀<𝑡₁<𝑡₂<⋯, and lim_𝑘→∞𝑡_𝑘=∞.

In this paper, we assume that functions 𝑓 and 𝐼_𝑘, 𝑘∈ℕ, satisfy all necessary conditions for the global existence and uniqueness of solutions for all 𝑡⩾𝑡₀. Denote by 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝜙) the solution of (2.2) such that 𝑥_𝑡0=𝜙. For the purpose of stability in this paper, we also assume that 𝑓(𝑡,0)=0 and 𝐼_𝑘(0)=0, 𝑘∈ℕ. So system (2.2) admits a zero solution or trivial solution 𝑥(𝑡,𝑡₀,0)=0. We further assume that all the solutions 𝑥(𝑡) of (2.2) are continuous except at 𝑡_𝑘,𝑘∈ℕ, at which 𝑥(𝑡) is right continuous, that is, 𝑥(𝑡⁺_𝑘)=𝑥(𝑡_𝑘),𝑘∈ℕ.

Definition 2.1. The trivial solution of system (2.2) is said to be globally exponentially stable, if there exist numbers 𝜆>0 and 𝑀⩾1 such that |𝑥(𝑡,𝑡₀,𝜙)|⩽𝑀‖𝜙‖𝑒^−𝜆𝑡,𝑡⩾0,(2.3) whenever 𝜙∈𝑃𝐶.

Definition 2.2. 𝑉∶[−𝜏,∞)×ℝ^𝑛→ℝ⁺ is said to belong to the class 𝜈₀, if 𝑉 is continuous on each of the sets [𝑡_𝑘−1,𝑡_𝑘)×ℝ^𝑛, lim_{(𝑡,𝑦)→(𝑡}⁻_𝑘,𝑥)𝑉(𝑡,𝑦)=𝑉(𝑡⁻_𝑘,𝑥) exists, 𝑉(𝑡,𝑥) is locally Lipschitzian in all 𝑥∈ℝ^𝑛, and 𝑉(𝑡,0)≡0 for all 𝑡≥−𝜏.

Definition 2.3. 𝑉∶[−𝜏,∞)×ℝ^𝑛→ℝ⁺ is said to belong to the class 𝜈₁, if 𝑉 is continuous on each of the sets [𝑡_𝑘−1,𝑡_𝑘)×ℝ^𝑛, 𝑉(𝑡,0)≡0 for all 𝑡≥−𝜏, lim_{(𝑡,𝑦)→(𝑡}⁻_𝑘,𝑥)𝑉(𝑡,𝑦)=𝑉(𝑡⁻_𝑘,𝑥) exists, 𝑉_𝑡(𝑡,𝑥),𝑉_𝑥(𝑡,𝑥) are continuous, where (𝑡,𝑥)∈[𝑡_𝑘−1,𝑡_𝑘)×ℝ^𝑛,𝑘∈ℕ, 𝑉_𝑡(𝑡,𝑥)=𝜕𝑉(𝑡,𝑥)𝜕𝑡,𝑉_𝑥(𝑡,𝑥)=(𝜕𝑉(𝑡,𝑥)𝜕𝑥₁,…,𝜕𝑉(𝑡,𝑥)𝜕𝑥_𝑛).(2.4)

Definition 2.4. Given a function 𝑉∶[−𝜏,+∞)×ℝ^𝑛→ℝ⁺, the upper right-hand derivative of 𝑉 with respect to system (2.2) is defined by 𝐷⁺𝑉(𝑡,𝜑(0))=limsup_ℎ→0+1ℎ[𝑉(𝑡+ℎ,𝜑(0)+ℎ𝐹(𝑡,𝜑))−𝑉(𝑡,𝜑(0))](2.5) for (𝑡,𝜑)∈ℝ⁺×𝑃𝐶.

3. Razumikhin-Type Theorems

In this section, we will present some Razumikhin-type theorems on global exponential stability for system (2.2) based on the Lyapunov-Razumikhin method.

Theorem 3.1. Let 𝜌=sup_𝑘∈ℕ{𝑡_𝑘−𝑡_𝑘−1}<∞ and 𝑐₁,𝑐₂,𝑝,𝑞,𝛾 all be positive numbers, 𝑐 a real number, 𝑞>1/𝛾>1, and 𝑐<ln(1/𝛾)/𝜌. Suppose that there exists a function 𝑉∈𝜈₀ such that(i)𝑐₁|𝑥|^𝑝⩽𝑉(𝑡,𝑥)⩽𝑐₂|𝑥|^𝑝 for all 𝑡⩾𝑡₀−𝜏, and 𝑥∈ℝ^𝑛;(ii)𝐷⁺𝑉(𝑡,𝜑(0))⩽𝑐𝑉(𝑡,𝜑(0)) for all 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘∈ℕ, whenever 𝑞𝑉(𝑡,𝜑(0))⩾𝑉(𝑡+𝜃,𝜑(𝜃)) for all 𝜃∈[−𝜏,0];(iii)𝑉(𝑡_𝑘,𝑥+𝐼_𝑘(𝑥))⩽𝛾𝑉(𝑡⁻_𝑘,𝑥), 𝑘∈ℕ, 𝑥∈ℝ^𝑛.

Then the trivial solution of system (2.2) is globally exponentially stable.

Proof. For any 𝜙∈𝑃𝐶, we denote the solution 𝑥(𝑡,𝑡₀,𝜙) of (2.2) by 𝑥(𝑡). Without loss of generality, we assume that ‖𝜙‖≠0.
Since 𝑞>1/𝛾 and 𝑐<−(ln𝛾/𝜌), there exist positive numbers 𝜇 and ℎ such that 𝑒𝑞>^𝜇𝜏𝛾>1𝛾,𝑐<𝑐+𝜇<−ln(𝛾+ℎ)𝜌.(3.1) Set 𝛾₁=−ln(𝛾+ℎ/𝜌), then 𝛾<𝑒^−𝛾1𝜌<1.
Set 𝑊(𝑡)=𝑒^𝜇𝑡𝑉(𝑡,𝑥(𝑡)), we have 𝐷⁺𝑊(𝑡)=𝜇𝑊(𝑡)+𝑒^𝜇𝑡𝐷⁺𝑉(𝑡,𝑥(𝑡)),𝑡∈[𝑡_𝑘−1,𝑡_𝑘),𝑘∈ℕ.(3.2) Let 𝑀>𝑐₂/𝑐₁𝛾 be a fixed number. In the following, we will prove that 𝑊(𝑡)<𝑐₁𝑀‖𝜙‖^𝑝,𝑡⩾𝑡₀−𝜏.(3.3) We first prove that 𝑊(𝑡)<𝑐₁𝑀‖𝜙‖^𝑝,𝑡∈[𝑡₀−𝜏,𝑡₁).(3.4) It is noted that 𝑊(𝑡₀+𝜃)⩽𝑐₂‖𝜙‖^𝑝<𝛾𝑐₁𝑀‖𝜙‖^𝑝<𝑐₁𝑀‖𝜙‖^𝑝, 𝜃∈[−𝜏,0]. So, it only needs to prove that 𝑊(𝑡)<𝑐₁𝑀‖𝜙‖^𝑝 for 𝑡∈(𝑡₀,𝑡₁). On the contrary, there exist some 𝑡∈(𝑡₀,𝑡₁) such that 𝑊(𝑡)⩾𝑐₁𝑀‖𝜙‖^𝑝. Set 𝑡^∗=inf{𝑡∈(𝑡₀,𝑡₁)∶𝑊(𝑡)⩾𝑐₁𝑀‖𝜙‖^𝑝}, then we have 𝑡^∗∈(𝑡₀,𝑡₁) and 𝑊(𝑡^∗)⩾𝑐₁𝑀‖𝜙‖^𝑝. Set 𝑡=sup{𝑡∈[𝑡₀,𝑡^∗)∶𝑊(𝑡)⩽𝛾𝑐₁𝑀‖𝜙‖^𝑝}. Then 𝑡∈(𝑡₀,𝑡^∗) and 𝑊(𝑡)=𝛾𝑐₁𝑀‖𝜙‖^𝑝. For 𝑡∈[𝑡,𝑡^∗], we have 𝑊(𝑡)⩾𝛾𝑐₁𝑀‖𝜙‖^𝑝⩾𝛾𝑊(𝑡+𝜃),∀𝜃∈[−𝜏,0].(3.5) Hence 𝑉(𝑡,𝑥(𝑡))⩾𝛾𝑒^−𝜇𝜏1𝑉(𝑡+𝜃,𝑥(𝑡+𝜃))⩾𝑞𝑉(𝑡+𝜃,𝑥(𝑡+𝜃)),∀𝜃∈[−𝜏,0].(3.6) By condition (ii) and (3.2), it follows that for 𝑡∈[𝑡,𝑡^∗]𝐷⁺𝑊(𝑡)=𝜇𝑊(𝑡)+𝑒^𝜇𝑡𝐷⁺𝑉(𝑡,𝑥(𝑡))⩽(𝜇+𝑐)𝑊(𝑡)⩽𝛾₁𝑊(𝑡).(3.7) So, we obtain 𝑊(𝑡^∗)⩽𝑊(𝑡)𝑒^{𝛾1(𝑡∗−𝑡)}⩽𝛾𝑐₁𝑀‖𝜙‖^𝑝𝑒^𝛾1𝜌<𝑐₁𝑀‖𝜙‖^𝑝. This is a contradiction, so (3.4) holds.
Now, we assume that for some 𝑚∈ℕ, 𝑊(𝑡)<𝑐₁𝑀‖𝜙‖^𝑝,𝑡∈[𝑡₀−𝜏,𝑡_𝑚).(3.8) We will prove that 𝑊(𝑡)<𝑐₁𝑀‖𝜙‖^𝑝,𝑡∈[𝑡_𝑚,𝑡_𝑚+1).(3.9) Suppose not, there exist some 𝑡∈[𝑡_𝑚,𝑡_𝑚+1) such that 𝑊(𝑡)⩾𝑐₁𝑀‖𝜙‖^𝑝. Set 𝑡^∗=inf{𝑡∈[𝑡_𝑚,𝑡_𝑚+1)∶𝑊(𝑡)⩾𝑐₁𝑀‖𝜙‖^𝑝}. From condition (iii) and (3.8), we have 𝑊(𝑡_𝑚)⩽𝛾𝑊(𝑡⁻_𝑚)⩽𝛾𝑐₁𝑀‖𝜙‖^𝑝<𝑐₁𝑀‖𝜙‖^𝑝. Hence 𝑡^∗∈(𝑡_𝑚,𝑡_𝑚+1) and 𝑊(𝑡^∗)=𝑐₁𝑀‖𝜙‖^𝑝. Set 𝑡=sup{𝑡∈[𝑡_𝑚,𝑡^∗)∶𝑊(𝑡)⩽𝛾𝑐₁𝑀‖𝜙‖^𝑝}. Then we have 𝑊(𝑡)=𝛾𝑐₁𝑀‖𝜙‖^𝑝. Furthermore, we have 𝑊(𝑡)⩾𝛾𝑐₁𝑀‖𝜙‖^𝑝⩾𝛾𝑊(𝑡+𝜃) for 𝑡∈[𝑡,𝑡^∗]. Thus, (3.7) holds. Then 𝑊(𝑡^∗)⩽𝛾𝑐₁𝑀‖𝜙‖^𝑝𝑒^𝛾1𝜌<𝑐₁𝑀‖𝜙‖^𝑝, which yields a contradiction. Therefore, (3.9) holds.
By mathematical induction, we have 𝑊(𝑡)<𝑐₁𝑀‖𝜙‖^𝑝,𝑡⩾0.(3.10) Hence |𝑥(𝑡)|<𝑀‖𝜙‖𝑒^−𝜆𝑡,𝑡⩾0,(3.11) where 𝑀=𝑀^1/𝑝, 𝜆=𝜇/𝑝. The proof is therefore complete.

Theorem 3.2. Let 𝜚=inf_𝑘∈ℕ{𝑡_𝑘−𝑡_𝑘−1}>0 and 𝑐₁,𝑐₂,𝑝,𝑞,𝛾,𝑐 all be positive numbers, 𝑞>𝛾⩾1, and 𝑐>ln𝛾/𝜚. Suppose that there exists a function 𝑉∈𝜈₀ such that(i)𝑐₁|𝑥|^𝑝⩽𝑉(𝑡,𝑥)⩽𝑐₂|𝑥|^𝑝 for all 𝑡⩾𝑡₀−𝜏, and 𝑥∈ℝ^𝑛,(ii)𝐷⁺𝑉(𝑡,𝜑(0))⩽−𝑐𝑉(𝑡,𝜑(0)) for all 𝑡∈[𝑡_𝑘−1,𝑡_𝑘)(𝑘∈ℕ), whenever 𝑞𝑉(𝑡,𝜑(0))⩾𝑉(𝑡+𝜃,𝜑(𝜃)) for all 𝜃∈[−𝜏,0];(iii)𝑉(𝑡_𝑘,𝑥+𝐼_𝑘(𝑥))⩽𝛾𝑉(𝑡⁻_𝑘,𝑥), 𝑘∈ℕ, 𝑥∈ℝ^𝑛.

Then the trivial solution of system (2.2) is globally exponentially stable.

Proof. Since 𝑞>𝛾 and 𝑐>ln𝛾/𝜚, there exist positive numbers 𝜇 and ℎ such that 𝑞>(𝛾+ℎ)𝑒^𝜇𝜏>𝛾,𝑐>𝑐−𝜇>ln(𝛾+2ℎ)𝜚.(3.12) Set 𝑞=𝛾+ℎ and 𝛾₂=ln(𝛾+2ℎ)/𝜚, then 1⩽𝛾<𝑞<𝑒^𝛾2𝜚. Set 𝑊(𝑡)=𝑒^𝜇𝑡𝑉(𝑡,𝑥(𝑡)), where 𝑥(𝑡) is defined as in the proof of Theorem 3.1.
Now, let 𝑀>𝑞𝑐₂/𝑐₁, we will prove that (3.3) holds.
We first prove that (3.4) holds. In fact, it is noticed that 𝑊(𝑡₀+𝜃)⩽𝑐₂‖𝜙‖^𝑝<(1/𝑞)𝑐₁𝑀‖𝜙‖^𝑝<𝑐₁𝑀‖𝜙‖^𝑝, for 𝜃∈[−𝜏,0]. So it only needs to prove 𝑊(𝑡)<𝑐₁𝑀‖𝜙‖^𝑝 for 𝑡∈(𝑡₀,𝑡₁). On the contrary, there exists 𝑡∈(𝑡₀,𝑡₁) such that 𝑊(𝑡)⩾𝑐₁𝑀‖𝜙‖^𝑝. Set 𝑡^∗=inf{𝑡∈(𝑡₀,𝑡₁)∶𝑊(𝑡)⩾𝑐₁𝑀‖𝜙‖^𝑝}, then we have 𝑡^∗∈(𝑡₀,𝑡₁). Set 𝑡=sup{𝑡∈[𝑡₀,𝑡^∗)∶𝑊(𝑡)⩽(1/𝑞)𝑐₁𝑀‖𝜙‖^𝑝}. Then 𝑡∈(𝑡₀,𝑡^∗) and 𝑊(𝑡)=(1/𝑞)𝑐₁𝑀‖𝜙‖^𝑝. For 𝑡∈[𝑡,𝑡^∗], we have 𝑞𝑊(𝑡)⩾𝑐₁𝑀‖𝜙‖^𝑝⩾𝑊(𝑡+𝜃),∀𝜃∈[−𝜏,0].(3.13) Hence 1𝑉(𝑡,𝑥(𝑡))⩾𝑞𝑒^−𝜇𝜏1𝑉(𝑡+𝜃,𝑥(𝑡+𝜃))⩾𝑞𝑉(𝑡+𝜃,𝑥(𝑡+𝜃)),∀𝜃∈[−𝜏,0].(3.14) It follows that for 𝑡∈[𝑡,𝑡^∗]𝐷⁺𝑊(𝑡)⩽(𝜇−𝑐)𝑊(𝑡)⩽−𝛾₂𝑊(𝑡),(3.15) which leads to 𝑊(𝑡^∗)⩽𝑊(𝑡). This is a contradiction to the fact 𝑊(𝑡^∗)=𝑐₁𝑀‖𝜙‖^𝑝>(1/𝑞)𝑐₁𝑀‖𝜙‖^𝑝=𝑊(𝑡), so (3.4) holds.
Now, we assume that for some 𝑚∈ℕ, (3.8) holds. We will prove that (3.9) holds. In order to do this, we first claim that 𝑊(𝑡⁻_𝑚1)⩽𝑞𝑐₁𝑀‖𝜙‖^𝑝.(3.16) Suppose not, then we have 𝑊(𝑡⁻_𝑚)>(1/𝑞)𝑐₁𝑀‖𝜙‖^𝑝. There are two cases to be considered.
Case 1. 𝑊(𝑡)>(1/𝑞)𝑐₁𝑀‖𝜙‖^𝑝 for all 𝑡∈[𝑡_𝑚−1,𝑡_𝑚).
By (3.8), we have 𝑞𝑊(𝑡)>𝑐₁𝑀‖𝜙‖^𝑝>𝑊(𝑡+𝜃) for 𝜃∈[−𝜏,0] and 𝑡∈[𝑡_𝑚−1,𝑡_𝑚). Thus, we get 𝐷⁺𝑊(𝑡)⩽−𝛾₂𝑊(𝑡) for 𝑡∈[𝑡_𝑚−1,𝑡_𝑚) which leads to 𝑊(𝑡⁻_𝑚)⩽𝑊(𝑡_𝑚−1)𝑒^{−𝛾2(𝑡𝑚−𝑡𝑚−1)}<𝑐₁𝑀‖𝜙‖^𝑝𝑒^−𝛾2𝜚<(1/𝑞)𝑐₁𝑀‖𝜙‖^𝑝. This is a contradiction.Case 2. There is some 𝑡∈[𝑡_𝑚−1,𝑡_𝑚) such that 𝑊(𝑡)⩽(1/𝑞)𝑐₁𝑀‖𝜙‖^𝑝.
Set 𝑡=sup{𝑡∈[𝑡_𝑚−1,𝑡_𝑚)∶𝑊(𝑡)⩽(1/𝑞)𝑐₁𝑀‖𝜙‖^𝑝}. Then 𝑡∈[𝑡_𝑚−1,𝑡_𝑚) and 𝑊(𝑡)=(1/𝑞)𝑐₁𝑀‖𝜙‖^𝑝. Since for 𝑡∈[𝑡,𝑡_𝑚), 𝑞𝑊(𝑡)⩾𝑐₁𝑀‖𝜙‖^𝑝⩾𝑊(𝑡+𝜃), 𝜃∈[−𝜏,0]. By (3.15), we have 𝐷⁺𝑊(𝑡)⩽0 for 𝑡∈[𝑡,𝑡_𝑚), which gives 𝑊(𝑡⁻_𝑚)⩽𝑊(𝑡)=(1/𝑞)𝑐₁𝑀‖𝜙‖^𝑝. This is also a contradiction.
Hence, (3.16) holds. It follows from (iii) that 𝑊(𝑡_𝑚)⩽𝛾𝑊(𝑡⁻_𝑚)⩽(𝛾/𝑞)𝑐₁𝑀‖𝜙‖^𝑝<𝑐₁𝑀‖𝜙‖^𝑝. Now, we assume that (3.9) is not true, set 𝑡^∗=inf{𝑡∈[𝑡_𝑚,𝑡_𝑚+1)∶𝑊(𝑡)⩾𝑐₁𝑀‖𝜙‖^𝑝}. Hence 𝑡^∗∈(𝑡_𝑚,𝑡_𝑚+1) and 𝑊(𝑡^∗)=𝑐₁𝑀‖𝜙‖^𝑝. If 𝑊(𝑡)>(1/𝑞)𝑐₁𝑀‖𝜙‖^𝑝 for 𝑡∈[𝑡_𝑚,𝑡^∗], set 𝑡=𝑡_𝑚, otherwise, set 𝑡=sup{𝑡∈[𝑡_𝑚,𝑡^∗)∶𝑊(𝑡)⩽(1/𝑞)𝑐₁𝑀‖𝜙‖^𝑝}. Thus, we have 𝑞𝑊(𝑡)⩾𝑐₁𝑀‖𝜙‖^𝑝⩾𝑊(𝑡+𝜃) for 𝑡∈[𝑡,𝑡^∗]. Hence, by (3.15), 𝐷⁺𝑊(𝑡)⩽0 for 𝑡∈[𝑡,𝑡^∗]. Then 𝑊(𝑡^∗)⩽𝑊(𝑡)<𝑐₁𝑀‖𝜙‖^𝑝, which yields a contradiction. Therefore, (3.9) holds. The rest is the same as in the proof of Theorem 3.1.

Remark 3.3. By Theorems 3.1 and 3.2, we can design impulsive control {𝐼_𝑘(𝑥(𝑡⁻_𝑘)),𝑘∈ℕ} for the following FDEs 𝑥^′(𝑡)=𝑓(𝑡,𝑥_𝑡),𝑡⩾𝑡₀,𝑥_𝑡0(𝑠)=𝜙(𝑠),−𝜏⩽𝑠⩽0,𝜙∈𝐶,(3.17) such that the system can be impulsively stabilized to its trivial solution. In Theorem 3.1, the constant 𝑐 may be chosen as a positive number. In the stability theory of FDEs, the condition 𝐷⁺𝑉(𝑡,𝜑(0))⩽𝑐𝑉(𝑡,𝜑(0)) allows the derivative of the Lyapunov function to be positive which may not even guarantee the stability of functional differential system (see, e.g., [4, 15]). However, as we can see from Theorem 3.1, impulses play an important role in making a functional differential system globally exponentially stable even if it may be unstable itself.

Remark 3.4. It is important to emphasize that, in contrast with some existing exponential stability results for IFDEs in the literature [15, 16], Theorems 3.1 and 3.2 are also valid for any finite delay. Therefore, our new results are more practically applicable than those in the literature, since the restrictive condition that the supper bound of time delay is less than the length of all the impulsive intervals is actually removed here.

4. Applications and Examples

Now, we will apply the general Razumikhin-type theorems established in Section 3 to deal with the global exponential stability of impulsive delay differential equations (IDDEs).

Consider a delay system of the form

𝑥^′(𝑡)=𝐹(𝑡,𝑥(𝑡),𝑥(𝑡−𝛿₁(𝑡)),…,𝑥(𝑡−𝛿_𝑚(𝑡))),𝑡≠𝑡_𝑘,𝑡⩾0,Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘)=𝐼_𝑘(𝑥(𝑡⁻_𝑘)),𝑡=𝑡_𝑘,𝑘∈ℕ,𝑥(𝑠)=𝜙(𝑠),−𝜏⩽𝑠⩽0,(4.1) where 𝜙∈𝑃𝐶, 𝛿_𝑖∶ℝ⁺→[0,𝜏],1⩽𝑖⩽𝑚, are all continuous, and

𝐹∶ℝ⁺×ℝ^𝑛×ℝ^𝑛×𝑚→ℝ^𝑛(4.2) is continuous. We also assume that (4.1) has a global solution which is again denoted by 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝜙), 𝐹(𝑡,0,…,0)=0 and 𝐼_𝑘(0)=0,𝑘∈ℕ.

Theorem 4.1. Let 𝜌=sup_𝑘∈ℕ{𝑡_𝑘−𝑡_𝑘−1}<∞, 𝜆₀ be a real number, and 𝜆₁,…,𝜆_𝑚,𝑐₁,𝑐₂,𝑝,𝛾 be all positive numbers, 0<𝛾<1. Suppose that there exists a function 𝑉∈𝜈₁ such that(i)𝑐₁|𝑥|^𝑝⩽𝑉(𝑡,𝑥)⩽𝑐₂|𝑥|^𝑝 for all 𝑡⩾𝑡₀⩾−𝜏, and 𝑥∈ℝ^𝑛;(ii)𝑉_𝑡(𝑡,𝑥)+𝑉_𝑥(𝑡,𝑥)𝐹(𝑡,𝑥,𝑦)⩽𝜆₀𝑉(𝑡,𝑥)+^𝑚_𝑖=1𝜆_𝑖𝑉(𝑡−𝛿_𝑖(𝑡),𝑦_𝑖)(4.3) for all 𝑡∈[𝑡_𝑘−1,𝑡_𝑘)(𝑘∈ℕ),𝑥∈ℝ^𝑛,𝑦=(𝑦₁,…,𝑦_𝑚)∈ℝ^𝑛×𝑚;(iii)𝑉(𝑡_𝑘,𝑥+𝐼_𝑘(𝑥))⩽𝛾𝑉(𝑡⁻_𝑘,𝑥), 𝑘∈ℕ, 𝑥∈ℝ^𝑛.If 𝜆₀+∑^𝑚_𝑖=1𝜆_𝑖/𝛾+ln𝛾/𝜌<0, then the trivial solution of (4.1) is globally exponentially stable.

Proof. For 𝑡∈[𝑡_𝑘−1,𝑡_𝑘),(𝑘∈ℕ),𝜙∈𝑃𝐶, let 𝑓(𝑡,𝜙)=𝐹(𝑡,𝜙(0),𝜙(−𝛿₁(𝑡)),…,𝜙(−𝛿_𝑚(𝑡))).(4.4) Then system (4.1) becomes system (2.2), and 𝐷⁺𝑉(𝑡,𝜙(0)) becomes 𝐷⁺𝑉(𝑡,𝜙(0))=𝑉_𝑡(𝑡,𝜙(0))+𝑉_𝑥(𝑡,𝜙(0))𝐹(𝑡,𝜙(0),𝜙(−𝛿₁(𝑡)),…,𝜙(−𝛿_𝑚(𝑡))).(4.5) If 𝜆₀+∑^𝑚_𝑖=1𝜆_𝑖/𝛾+ln𝛾/𝜌<0, then there exists a constant 𝑞>1/𝛾, such that 𝜆₀+𝑞^𝑚_𝑖=1𝜆_𝑖+ln𝛾𝜌<0.(4.6) So, if 𝑡∈[𝑡_𝑘−1,𝑡_𝑘)(𝑘∈ℕ) and 𝑞𝑉(𝑡,𝜑(0))⩾𝑉(𝑡+𝜃,𝜑(𝜃)) for all 𝜃∈[−𝜏,0], then 𝐷⁺𝑉(𝑡,𝜑(0))⩽𝜆₀𝑉(𝑡,𝜑(0))+^𝑚_𝑖=1𝜆_𝑖𝑉(𝑡−𝛿_𝑖(𝑡),𝜑(−𝛿_𝑖(𝑡)))⩽(𝜆₀+𝑞^𝑚_𝑖=1𝜆_𝑖)𝑉(𝑡,𝜑(0))=𝑐𝑉(𝑡,𝜑(0)),(4.7) where 𝑐=𝜆₀+𝑞∑^𝑚_𝑖=1𝜆_𝑖<−ln𝛾/𝜌. By Theorem 3.1, the trivial solution of (4.1) is globally exponentially stable.

Corollary 4.2. Let 𝜌=sup_𝑘∈ℕ{𝑡_𝑘−𝑡_𝑘−1}<∞. Assume that there exist scalar numbers 𝛼_𝑖⩾0(1⩽𝑖⩽𝑚), 0<𝛼<1 and 𝜂 such that 𝑥^𝑇𝐹(𝑡,𝑥,0)⩽𝜂|𝑥|²,(4.8)|𝐹(𝑡,𝑥,𝑦)−𝐹(𝑡,𝑥,0)|⩽^𝑚_𝑖=1𝛼_𝑖|𝑦_𝑖|,(4.9)|𝑥+𝐼_𝑘(𝑥)|⩽𝛼|𝑥|(4.10) for all 𝑡⩾𝑡₀,𝑥∈ℝ^𝑛, 𝑦=(𝑦₁,…,𝑦_𝑚)∈ℝ^𝑛×𝑚, 𝑘∈ℕ.
If ∑𝜂+^𝑚𝑖=1𝛼_𝑖𝛼+ln𝛼𝜌<0,(4.11) then the trivial solution of (4.1) is globally exponentially stable.

Proof. Let 𝑉(𝑡,𝑥)=𝑥^𝑇𝑥=|𝑥|², then we can easily see that condition (i) of Theorem 4.1 holds.
For 𝑥∈ℝ^𝑛 and 𝑦=(𝑦₁,…,𝑦_𝑚)∈ℝ^𝑛×𝑚, from (4.8) and (4.9), we can calculate that 𝑉_𝑡(𝑡,𝑥)+𝑉_𝑥(𝑡,𝑥)𝐹(𝑡,𝑥,𝑦)=2𝑥^𝑇𝐹(𝑡,𝑥,𝑦)=2𝑥^𝑇𝐹(𝑡,𝑥,0)+2𝑥^𝑇[𝐹(𝑡,𝑥,𝑦)−𝐹(𝑡,𝑥,0)]⩽2𝜂|𝑥|²+2^𝑚_𝑖=1𝛼_𝑖|𝑥|⋅|𝑦_𝑖|.(4.12) Let 𝛾=𝛼², by the inequality 𝑎𝑏⩽(𝑎²+𝑏²)/2, we have |𝑥|⋅|𝑦_𝑖|=(𝛾^−1/4|𝑥|)(𝛾^1/4|𝑦_𝑖𝛾|)⩽^−1/2|𝑥|²+𝛾^1/2|𝑦_𝑖|²2.(4.13) Substituting (4.13) into (4.12), we obtain 𝑉_𝑡(𝑡,𝑥)+𝑉_𝑥∑(𝑡,𝑥)𝐹(𝑡,𝑥,𝑦)⩽(2𝜂+^𝑚𝑖=1𝛼_𝑖𝛼)|𝑥|²+𝛾^𝑚1/2_𝑖=1𝛼_𝑖|𝑦_𝑖|²=𝜆₀|𝑥|²+^𝑚_𝑖=1𝜆_𝑖|𝑦_𝑖|²,(4.14) where 𝜆₀=2𝜂+∑^𝑚_𝑖=1𝛼_𝑖/𝛼, 𝜆_𝑖=𝛾^1/2𝛼_𝑖, 𝑖=1,…,𝑚.
From (4.11), we have 𝜆₀+∑^𝑚_𝑖=1𝜆_𝑖𝛾+ln𝛾𝜌<0.(4.15) The conclusion follows from Theorem 4.1 immediately and the proof is completed.

Remark 4.3. Let 𝜚=inf_𝑘∈𝑁{𝑡_𝑘−𝑡_𝑘−1}>0. If 𝛾⩾1, by Theorem 3.2, we can also give some other results on global exponential stability for (4.1). For the details, we omit them here.

Example 4.4. Consider a scalar nonlinear impulsive delay differential equation 𝑥^′(𝑡)=𝐹(𝑡,𝑥(𝑡),𝑥(𝑡−𝛿(𝑡))),𝑡≠𝑡_𝑘,Δ𝑥(𝑡_𝑘)=𝑐_𝑘𝑥(𝑡⁻_𝑘),𝑡=𝑡_𝑘,𝑘∈ℕ,𝑥(𝑠)=𝜙(𝑠),𝑠∈[−𝜏,0](4.16) on 𝑡⩾0, where 𝛿∶𝑅⁺→[0,𝜏] is a continuous function, 𝑐_𝑘∈ℝ, and 1𝐹(𝑡,𝑥,𝑦)=𝑏𝑥−𝑥10³−𝑦cos𝑡,(4.17) with 𝑥=𝑥(𝑡),𝑦=𝑥(𝑡−𝛿(𝑡)), 𝑏>0.
From (4.8)-(4.9), we can see 𝜂=𝑏, 𝛼₁=1. By Corollary 4.2, if there exists a scalar number 0<𝛼<1, such that |1+𝑐_𝑘1|⩽𝛼,𝑏+𝛼+ln𝛼𝜌<0,(4.18) then the trivial solution of (4.16) is global exponentially stable.

Example 4.5. Consider the following linear impulsive delay system: 𝑥^′(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝛿(𝑡)),𝑡≠𝑡_𝑘,𝑡⩾0,Δ𝑥(𝑡_𝑘)=𝐶_𝑘𝑥(𝑡⁻_𝑘),𝑡=𝑡_𝑘,𝑘∈ℕ,𝑥(𝑠)=𝜙(𝑠),−𝜏⩽𝑠⩽0,𝜙∈𝑃𝐶,(4.19) where 𝐴=0.10.2−0.10.20.150.300.240.1,𝐵=−0.120.0300.12−0.20.0500.14−0.1,𝐶_𝑘=−0.5000−0.8000−0.4,(4.20)𝛿∶ℝ⁺→[0,𝜏] is a continuous function. Since 𝜆_min(𝐴)=−0.2315 and 𝜆_max(𝐴)=0.4388, we can not use the results in [5, 14] to determine the exponential stability. From Corollary 4.2, we can choose 𝜂=𝜆_max((𝐴+𝐴^𝑇)/2)=0.4409, 𝛼₁=‖𝐵‖=0.2905, 𝛼=‖𝐼+𝐶_𝑘‖=0.6, by (4.11), we obtain that if sup_𝑘∈ℕ{𝑡_𝑘−𝑡_𝑘−1}<−𝛼ln𝛼/(𝛼₁+𝛼𝜆₁)=0.5521, then trivial solution of (4.19) is globally exponentially stable.

Remark 4.6. For (4.19), the authors in [15] chose 𝛿(𝑡)=(1/40)(1+𝑒^−𝑡), 𝜏=0.05, sup_𝑘∈ℕ{𝑡_𝑘−𝑡_𝑘−1}⩽0.2 such that 𝜏⩽𝑡_𝑘−𝑡_𝑘−1⩽0.2. However, if 𝜏>sup_𝑘∈ℕ{𝑡_𝑘−𝑡_𝑘−1}, the results in [15, 16] fail to determine the global exponential stability.

5. Conclusion

In this paper, some new Razumikhin-type theorems on global exponential stability for IFDEs are obtained by employing Lyapunov-Razumikhin technique. Some applications to IDDEs are also given. It should be mentioned that our results may allow us to develop an effective impulsive control strategy to stabilize an underlying delay dynamical system even if it may be unstable in practice, which is particularly meaningful for applications in engineering and technology. Two examples are also given to demonstrate the effectiveness of the theoretical results.

Acknowledgments

This work was supported by the Natural Science Foundation of Guangdong Province (8351009001000002) and the National Natural Science Foundation of China (10971232).