Semistability of Nonlinear Impulsive Systems with Delays

Abstract

This paper is concerned with the stability analysis and semistability theorems for delay impulsive systems having a continuum of equilibria. We relate stability and semistability to the classical concepts of system storage functions to impulsive systems providing a generalized hybrid system energy interpretation in terms of storage energy. We show a set of Lyapunov-based sufficient conditions for establishing these stability properties. These make it possible to deduce properties of the Lyapunov functional and thus lead to sufficient conditions for stability and semistability. Our proposed results are evaluated using an illustrative example to show their effectiveness.

1. Introduction

Due to their numerous applications in various fields of sciences and engineering, impulsive differential systems have become a large and growing interdisciplinary area of research. In recent years, the issues of stability in impulsive differential equations with time delays have attracted increasing interest in both theoretical research and practical applications [1–9], while difficulties and challenges remain in the area of impulsive differential equations [10], especially those involving time delays [11]. Various mathematical models in the study of biology, population dynamics, ecology and epidemic, and so forth can be expressed by impulsive delay differential equations. These processes and phenomena, for which the adequate mathematical models are impulsive delay differential equations, are characterized by the fact that there is sudden change of their state and that the processes under consideration depend on their prehistory at each moment of time. In the transmission of the impulse information, input delays are often encountered. Control and synchronization of chaotic systems are considered in [12, 13]. By utilizing impulsive feedback control, all the solutions of the Lorenz chaotic system will converge to an equilibrium point. The application of networked control systems is considered in [14–17], while in [14], when analyzing the asymptotic stability for discrete-time neural networks, the activation functions are not required to be differentiable or strictly monotonic. The existence of the equilibrium point is first proved under mild conditions. By constructing a new Lyapnuov-Krasovskii functional, a linear matrix inequality (LMI) approach is developed to establish sufficient conditions for the discrete-time neural networks to be globally asymptotically stable. In [18], Razumikhin-type theorems are established which guarantee ISS/iISS for delayed impulsive systems with external input affecting both the continuous dynamics and the discrete dynamics. It is shown that when the delayed continuous dynamics are ISS/iISS but the discrete dynamics governing the impulses are not, the ISS/iISS property of the impulsive system can be retained if the length of the impulsive interval is large enough. Conversely, when the delayed continuous dynamics are not ISS/iISS but the discrete dynamics governing the impulses are, the impulsive system can achieve ISS/iISS. In [19, 20], the authors consider linear time invariant uncertain sampled-data systems in which there are two sources of uncertainty: the values of the process parameters can be unknown while satisfying a polytopic condition and the sampling intervals can be uncertain and variable. They model such systems as linear impulsive systems and they apply their theorem to the analysis and state-feedback stabilization. They find a positive constant which determines an upper bound on the sampling intervals for which the stability of the closed loop is guaranteed. Population growth and biological systems are considered in [21, 22]. Stochastic systems are considered in [23–25], and so forth. However, the corresponding theory for impulsive systems with time delays having a continuum of equilibria has been relatively less developed.

The purpose of this paper is to study the stability and semistability properties for nonlinear delayed impulsive systems with continuum of equilibria. Examples of such systems include mechanical systems having rigid-body modes and isospectral matrix dynamical systems [26]. Such systems also arise in chemical kinetics, compartmental modeling, and adaptive control. Since every neighborhood of a nonisolated equilibrium contains another equilibrium, a nonisolated equilibrium cannot be asymptotically stable. Thus asymptotic stability is not the appropriate notion of stability for systems having a continuum of equilibria. Two notions that are of particular relevance to such systems are convergence and semistability. Convergence is the property whereby every solution converges to a limit point that may depend on the initial condition. Semistability is the additional requirement that all solutions converge to limit points that are Lyapunov stable. More precisely, an equilibrium is semistable if it is Lyapunov stable, and every trajectory starting in a neighborhood of the equilibrium converges to a (possibly different) Lyapunov stable equilibrium. It can be seen that, for an equilibrium, asymptotic stability implies semistability, while semistability implies Lyapunov stability. We will employ the method of Lyapunov function for the study of stability and semistability of impulsive systems with time delays. Several stability criteria are established. A set of Lyapunov-based sufficient conditions is provided for stability criteria, then we extend the notion of stability to develop the concept of semistability for delay impulsive systems. Finally, an example illustrates the effectiveness of our approach.

2. Preliminaries

Let ℕ denote the set of positive integer numbers. Let PC_𝑡 denote the set of piecewise right continuous functions 𝜙∶[𝑡−𝑟,𝑡]→ℝ^𝑛 with the norm defined by ‖𝜙‖^𝑡_𝑟=sup_{−𝑟≤𝑠≤0}‖𝜙(𝑡+𝑠)‖. For simplicity, define ‖𝜙‖_𝑟=‖𝜙‖⁰_𝑟, for 𝜙∈PC₀. For given 𝑟>0, if 𝑥∈PC([𝑡₀−𝑟,+∞),ℝ^𝑛), then for each 𝑡≥𝑡₀, we define 𝑥_𝑡,𝑥_𝑡−∈PC₀ by 𝑥_𝑡(𝑠)=𝑥(𝑡+𝑠)(−𝑟≤𝑠≤0) and 𝑥_𝑡−(𝑠)=𝑥(𝑡+𝑠)(−𝑟≤𝑠<0), respectively. A function 𝛼∶ℝ⁺→ℝ⁺ is of class 𝒦, if 𝛼 is continuous, strictly increasing, and 𝛼(0)=0. For a given scalar 𝜌≥0, let ℬ(𝜌)={𝑥∈ℝ^𝑛;‖𝑥‖≤𝜌}.

Let Ω∈ℝ^𝑛 be an open set and 𝔅(𝜌)⊂Ω for some 𝜌>0. Given functionals 𝑓∶ℝ⁺×PC([−𝑟,0],Ω)→ℝ^𝑛,𝑔∶ℝ⁺×Ω→ℝ^𝑛, satisfying 𝑓(𝑡,0)=0,𝑔(0,0)=0. Considering the following nonlinear time-delay impulsive system Σ_𝑡 described by the state equation 𝑥̇(𝑡)=𝑓𝑡,𝑥_𝑡,𝑡>𝑡₀,𝑡≠𝑡_𝑘𝑥𝑡,𝑘∈ℕ,₊=𝑔(𝑡,𝑥(𝑡)),𝑡=𝑡_𝑘𝑥𝑡,𝑘∈ℕ,₀+𝜃=𝜙(𝜃),𝜃∈[−𝑟,0], (2.1)(2.2)(2.3) where 𝑥(𝑡)∈ℝ^𝑛 is the system state, 𝑥̇(𝑡) denotes the right-hand derivative of 𝑥(𝑡), 𝑥(𝑡⁺) and 𝑥(𝑡⁻) denote the limit from the right and the limit from the left at point 𝑡, respectively. 𝑡₀ is the initial time. Here we assume that the solutions of system Σ_𝑡 are right continuous, that is, 𝑥(𝑡⁺)=𝑥(𝑡). {𝑡_𝑘},𝑘∈ℕ is a strictly increasing sequence of impulse times in (𝑡₀,∞) where lim_𝑘→∞𝑡_𝑘=∞.

Definition 2.1. The function 𝑓∶ℝ×PC→ℝ^𝑛 is said to be composite-PC, if for each 𝑡₀∈ℝ and 𝛼>0, 𝑥∈PC([𝑡₀−𝑟,𝑡₀+𝛼],ℝ^𝑛) and 𝑥 is continuous at each 𝑡≠𝑡_𝑘 in [𝑡₀,𝑡₀+𝛼], then the composite function ℎ(𝑥)=𝑓(𝑡,𝑥_𝑡)∈PC([𝑡₀−𝑟,𝑡₀+𝛼],ℝ^𝑛).

Definition 2.2. The function 𝑓∶ℝ×PC→ℝ^𝑛 is said to be quasi-bounded, if for each 𝑡₀∈ℝ⁺, 𝛼>0, and for each compact set 𝐹∈ℝ^𝑛, there exists some 𝑀>0, such that ‖𝑓(𝑡,𝜓)‖≤𝑀 for all (𝑡,𝜓)∈[𝑡₀,𝑡₀+𝛼]×PC([−𝑟,0],𝐹).

Definition 2.3. The function 𝑥∶[𝑡₀−𝑟,𝑡₀+𝛼]→ℝ^𝑛 with 𝛼>0 is said to be a solution of Σ_𝑡 if(i)𝑥 is continuous at each 𝑡≠𝑡_𝑘 in (𝑡₀,𝑡₀+𝛼];(ii)the derivative of 𝑥 exists and is continuous at all but at most a finite number of points 𝑡 in 𝑡∈[𝑡₀,𝑡₀+𝛼);(iii)the right-hand derivative of 𝑥 exists and satisfies (2.1) in 𝑡∈[𝑡₀,𝑡₀+𝛼], while for each 𝑡_𝑘∈[𝑡₀,𝑡₀+𝛼], (2.2) holds;(iv)Equation (2.3) holds, that is, 𝑥(𝑡₀+𝜃)=𝜙(𝜃),𝜃∈[−𝑟,0].
We denote by 𝑥(𝑡,𝑡₀,𝜙)(or 𝑥(𝑡), if in not confusing) the solution of Σ_𝑡. 𝑥(𝑡) is said to be a solution defined on [𝑡₀−𝑟,∞) if all above conditions hold for any 𝛼>0.
We make the following assumptions on system Σ_𝑡.(A1)𝑓(𝑡,𝜓) is composite-PC, quasi-bounded and locally Lipschitzian in 𝜓.(A2) For each fixed 𝑡∈ℝ⁺, 𝑓(𝑡,𝜓) is a continuous function of 𝜓 on PC([−𝜏,0],ℝ^𝑛).
Under the assumptions above, it was shown in [11] that for any 𝜙∈PC([−𝑟,0],ℝ^𝑛), system Σ_𝑡 admits a solution 𝑥(𝑡,𝑡₀,𝜙) that exists in a maximal interval [𝑡₀−𝑟,𝑡₀+𝑏)(0<𝑏≤+∞) and the zero solution of the system exists.

Definition 2.4. An equilibrium point of Σ_𝑡 is a point 𝑥_𝑒∈PC([𝑡₀−𝑟,𝑡₀+𝛼],ℝ^𝑛) satisfying 𝑥(𝑡,𝑡₀,𝜙)=𝑥_𝑒 for all 𝑡≥0 where 𝑥(𝑡,𝑡₀,𝜙) is the solution of Σ_𝑡. Let ℰ denote the set of equilibrium points of Σ_𝑡.

Definition 2.5. Consider the delay impulsive system Σ_𝑡.(i)An equilibrium point 𝑥(𝑡)≡𝑥_𝑒 of Σ_𝑡 is Lyapunov stable if for any 𝜀>0 there exists 𝛿(𝜀,𝑡₀), such that ‖𝜙−𝜙_𝑒‖_𝑟<𝛿 implies ‖𝑥(𝑡)−𝑥_𝑒‖<𝜀 for all 𝑡≥𝑡₀, where 𝜙_𝑒 is the initial function for 𝑥_𝑒. An equilibrium point 𝑥 is uniformly Lyapunov stable, if, in addition, the number 𝛿 is independent of 𝑡₀.(ii)An equilibrium point 𝑥 of Σ_𝑡 is semistable if it is Lyapunov stable and there exists an open subset of Ω containing 𝑥 such that for all initial conditions in Ω the trajectory of Σ_𝑡 converges to a Lyapunov stable equilibrium point, that is, lim_𝑡→∞𝑥(𝑡,𝑡₀,𝜙)=𝑦,𝜙∈Ω, where 𝑦 is a Lyapunov stable equilibrium point.(iii)System Σ_𝑡 is said to be uniformly asymptotically stable in the sense of Lyapunov with respect to the zero solution, if it is uniformly stable and lim_𝑡→∞‖𝑥(𝑡)‖=0.

Definition 2.6. The function 𝑉∶[𝑡₀,+∞)×PC([−𝜏,0],ℬ(𝜌))→ℝ⁺ is said to belong to the class 𝒱₀ if(i)𝑉 is continuous in each of the sets [𝑡_𝑘−1,𝑡_𝑘)×PC([−𝜏,0],ℬ(𝜌)) and for each 𝑘∈ℕ,lim_{(𝑡,𝑦)→(𝑡}⁻_𝑘,𝑥)𝑉(𝑡,𝑦)=𝑉(𝑡⁻_𝑘,𝑥) exists;(ii)𝑉(𝑡,𝑥) is locally Lipschitzian in 𝑥∈PC([−𝜏,0],ℬ(𝜌)), and for all 𝑡≥𝑡₀, 𝑉(𝑡,0)≡0.

Definition 2.7. Let 𝑉∈𝒱₀. For any (𝑡,𝜓)∈[𝑡₀,+∞)×PC([−𝜏,0],ℬ(𝜌)), the upper right-hand derivative of 𝑉 with respect to system Σ_𝑡 is defined by 𝐷⁺𝑉𝑡,𝜓(0)∶=limsup_ℎ→0+1ℎ𝑉𝑡+ℎ,𝜓(0)+ℎ𝑓𝑡,𝜓−𝑉𝑡,𝜓(0). (2.4)

3. Main Results

In the following, we will establish several sufficient conditions for Lyapunov stability and semistability for impulsive differential system Σ_𝑡 with time delays.

Theorem 3.1. System Σ_𝑡 is uniformly stable, and the zero solution of Σ_𝑡 is asymptotically stable if there exists a Lyapunov function 𝑉∈𝒱₀ which satisfies the following.(i)∃𝑎,𝑏∈𝒦 such that 𝑎(‖𝑥‖)≤𝑉(𝑡,𝑥)≤𝑏(‖𝑥‖). (3.1)(ii)For any 𝑡∈[𝑡₀,+∞), 𝑡≠𝑡_𝑘 and 𝜓∈PC([−𝑟,0],ℝ^𝑛), there exists 𝑐>0, such that 𝐷⁺𝑉𝑡,𝜓(0)≤−𝑐𝑉𝑡,𝜓(0). (3.2)(iii)There exist a 𝜇(0<𝜇<1) and a subsequence {𝑡_𝑘𝑗} of the impulsive moments {𝑡_𝑘} such that 𝑉𝑡_𝑘𝑗+1,𝑥_𝑟≤𝜇𝑉𝑡_𝑘𝑗,𝑥_𝑟. (3.3)(iv)For any 𝑚,0≤𝑚≤𝑘₀, forall𝑥_𝑒∈ℰ there exists a function 𝛼∈𝒦, such that 𝑉𝑡_𝑚,𝑥−𝑥_𝑒𝑟≤𝛼𝜙−𝜙_𝑒𝑟. (3.4)(v)For any 𝑚∈ℕ,𝑘_𝑗≤𝑚<𝑘_𝑗+1,𝑗=0,1,2,…, there exists a function 𝛽∈𝒦 such that 𝑉𝑡_𝑚,𝑥−𝑥_𝑒𝑟≤𝛽𝑉𝑡_𝑘𝑗,𝑥−𝑥_𝑒𝑟. (3.5)

Proof. Let 𝑥_𝑒 be an equilibrium point of the system Σ_𝑡. We first prove that 𝑥_𝑒 is uniformly stable, that is, for forall𝜀>0, there exists 𝛿=𝛿(𝜀)>0 such that ‖𝜙−𝜙_𝑒‖_𝑟<𝛿 implies ‖𝑥(𝑡)−𝑥_𝑒‖<𝜀 for all 𝑡≥𝑡₀.
For all𝜀>0, let 0<𝛿<𝜀 such that 𝑎(𝜀)>max𝛼(𝛿),𝛽(𝛼(𝛿)). (3.6) For any ‖𝜙−𝜙_𝑒‖_𝑟<𝛿, by condition (3.4), we get 𝑉𝑡_𝑚,𝑥−𝑥_𝑒𝑟≤𝛼𝜙−𝜙_𝑒𝑟≤𝛼(𝛿),0≤𝑚≤𝑘₀. (3.7)
By (3.3), it is clear that ‖𝑉(𝑡,𝑥)‖_𝑟 is nonincreasing along the subsequence {𝑡_𝑘𝑗}, so we have 𝑉𝑡_𝑘𝑗,𝑥−𝑥_𝑒𝑟≤𝑉𝑡_𝑘0,𝑥−𝑥_𝑒𝑟≤𝛼(𝛿),𝑗=0,1,2,…. (3.8)
For any 𝑚,𝑘_𝑗≤𝑚<𝑘_𝑗+1,𝑗=0,1,2,…, by (3.5), we get 𝑉𝑡_𝑚,𝑥−𝑥_𝑒𝑟≤𝛽(𝛼(𝛿)). (3.9) Combining (3.7), (3.8), and (3.9), we conclude that 𝑉𝑡_𝑘,𝑥−𝑥_𝑒𝑟<𝑎(𝜀),𝑘=1,2,…. (3.10) By condition (3.2), for any 𝑡∈[𝑡_𝑘,𝑡_𝑘+1),𝑘=0,1,2…, we have 𝑉𝑡,𝑥−𝑥_𝑒≤𝑉𝑡_𝑘,𝑥−𝑥_𝑒<𝑎(𝜀), (3.11) and then, by (3.10), for any 𝑡≥𝑡₀ we derive that 𝑉(𝑡,𝑥−𝑥_𝑒)<𝑎(𝜀). Hence, by (3.1) we obtain that 𝑎(‖𝑥−𝑥_𝑒‖)≤𝑉(𝑡,𝑥−𝑥_𝑒)<𝑎(𝜀). Since 𝑎∈𝒦, we get 𝑥(𝑡)−𝑥_𝑒<𝜀,𝑡≥𝑡₀, (3.12) which implies that system Σ_𝑡 is uniformly Lyapunov stable.
Next, we will prove that the zero solution of Σ_𝑡 is asymptotically stable.
Since system Σ_𝑡 is uniformly stable, from (3.1), there must exist a real number 𝑀>0 such that ‖𝑉(𝑡,𝑥)‖_𝑟≤𝑀,𝑡≥𝑡₀. Hence, there exists a 𝑣≥0 such that limsup_𝑡→∞‖𝑉(𝑡,𝑥)‖_𝑟=𝑣≤𝑀. (3.13)
In the following, we will show that 𝑣=0. Without loss of generality, we can suppose that there exists a sequence {𝑡_𝑛}⊂[𝑡₀,∞),𝑛=1,2,…, such that lim_𝑛→∞𝑉𝑡_𝑛,𝑥_𝑟=limsup_𝑛→∞‖𝑉(𝑡,𝑥)‖_𝑟=𝑣. (3.14) From (3.3) we get 𝑉𝑡_𝑘𝑗,𝑥_𝑟<𝜇^𝑗𝑣𝑡_𝑘0,𝑥_𝑟. (3.15) Since 0<𝜇<1, we obtain lim_𝑗→∞𝑉𝑡_𝑘𝑗,𝑥_𝑟=0. (3.16) If the sequence {𝑡_𝑛}⊂[𝑡₀,∞),𝑛=1,2,… is the same as the sequence {𝑡_𝑘𝑗},𝑗=0,1,2,…, then it is obvious that 𝑣=0. If 0≤𝑛<𝑘₀, it follows from the assumptions above that (3.16) holds. Otherwise, we assume that 𝑛≥𝑘₀; there exists a 𝑗∈ℕ such that 𝑘_𝑗≤𝑛<𝑘_𝑗+1. Then from condition (3.5) we get 𝑉𝑡_𝑛,𝑥_𝑟≤𝛽𝑉𝑡_𝑘𝑗,𝑥_𝑟. (3.17) So lim_𝑛→∞𝑉𝑡_𝑛,𝑥_𝑟≤lim_𝑗→∞𝛽𝑉𝑡_𝑘𝑗,𝑥_𝑟=0, (3.18) which implies 𝑣=0.
Hence, we derive that lim_𝑡→∞‖𝑉(𝑡,𝑥)‖=0. Finally, by (3.1), we have lim_𝑡→∞‖𝑥(𝑡)‖=0 which implies that the zero solution of the system Σ_𝑡 is asymptotically stable. The proof is completed.

Next, we present a sufficient condition for semistability for system Σ_𝑡.

Let 𝔏₁∶={𝑓∶[0,∞)→ℝ;𝑓ismeasurableand∫^∞₀|𝑓(𝑡)|𝑑𝑡<∞}.

Theorem 3.2. Consider the system Σ_𝑡; assume that there exists nonnegative-definite continuous function 𝑊∶ℝ×ℝ^𝑛→ℝ such that 𝐷⁺𝑉𝑡,𝜓(0)≤−𝑊𝑡,𝜓(0). (3.19) Let 𝑊⁻¹(0)∶={𝑥∣𝑊(𝑡,𝑥)≡0,𝑓𝑜𝑟𝑎𝑙𝑙𝑡≥𝑡₀}. If every equilibrium point of system Σ_𝑡 is Lyapunov stable, then every point in 𝑊⁻¹(0) is semistable.

Proof. Define 𝜑(𝑡)𝑊(0)∶=𝑡,𝜓,𝑡≠𝑡_𝑘,𝑘∈ℕ,0,𝑡=𝑡_𝑘,𝑘∈ℕ. (3.20) It follows from (3.19) and (3.3) that ^𝑡₀𝜑(𝑠)𝑑𝑠≤𝑉𝑥𝑡₁−𝑉(𝑥(𝑡))≤𝑉𝑥𝑡₁. (3.21)
Since 𝜑(⋅) is nonnegative, it follows that 𝜑(⋅)∈𝔏₁. Next, we show that 𝜑(𝑡)→0 as 𝑡→∞.
If it is not true, then there exists 𝜀>0 and an infinite sequence of times 𝜏₁,𝜏₂,… such that |𝜑(𝜏_𝑖)|≥𝜀. By definition of 𝜑(⋅) we have 𝜏_𝑖,𝑖=1,2… that does not belong to the set of impulsive times {𝑡_𝑘}.
Note that from (3.19), it follows from Proposition 3.1 of [26] that 𝑥(𝑡) is bounded for all 𝑡≥0. Hence, it follows from the Lipschitz continuity of 𝑓(⋅) that 𝑥̇(𝑡) is bounded for all 𝑡≥0; thus, 𝜑(⋅) is uniformly continuous on[𝑡₀,+∞)⧵{𝑡_𝑛}. So, there exists 𝛿>0 such that every 𝜏_𝑖 is contained in some interval of 𝐼_𝑖,𝜏_𝑖∈𝐼_𝑖 of length 𝛿 on which 𝜑(𝑡)≥𝜀/2,𝑡∈𝐼_𝑖. This contradicts 𝜑(⋅)∈𝔏₁. Hence 𝜑(𝑡)→0 as 𝑡→∞. It follows that 𝑊(𝑡,𝜓(0))→0 as 𝑡→∞. Since 𝑥(𝑡) is bounded, we get 𝑥(𝑡)→𝑊⁻¹(0) (as 𝑡→∞).
Next, let 𝑥_𝑒∈𝑊⁻¹(0). For every open neighborhood 𝑈 and 𝑥₀∈𝑈, 𝑥(𝑡)→𝑊⁻¹(0)(as𝑡→∞), it follows from Proposition 5.1 of [26] that there exists 𝑦∈𝑊⁻¹(0) such that lim_𝑡→∞𝑥(𝑡)=𝑦. Since every point in ℰis Lyapunov stable, and hence 𝑦 is a Lyapunov stable equilibrium of Σ_𝑡, it follows that 𝑥_𝑒 is semistable. Finally, since 𝑥_𝑒∈𝑊⁻¹(0) is arbitrary, this implies every point in 𝑊⁻¹(0) is semistable. The proof is completed.

4. Numerical Example

In this section, we give an example about compartmental systems to illustrate the effectiveness of the proposed method. Compartmental systems involve dynamical models that are characterized by conservation laws (e.g., mass and energy) capturing the exchange of material between coupled macroscopic subsystems known as compartments. Each compartment is assumed to be kinetically homogeneous, that is, any material entering the compartment is instantaneously mixed with the material of the compartment.

Example 4.1. Consider the nonlinear two-compartment time-delay impulsive systems given by 𝑥̇₁(𝑡)=−𝑥₁(𝑡)+𝑥₂(𝑡)1−sin𝑥₁(𝑡−𝑟)+𝑥³₂(𝑡)−𝑥³₁(𝑡),𝑡≠𝑡_𝑘𝑥̇,𝑘∈ℕ,₂(𝑡)=−𝑥₂(𝑡)+𝑥₁(𝑡)1+sin𝑥₁(𝑡−𝑟)+𝑥³₁(𝑡)−𝑥³₂(𝑡),𝑡≠𝑡_𝑘𝑥,𝑘∈ℕ,₁𝑡⁺=0.8𝑥₁(𝑡),𝑡=𝑡_𝑘𝑥,𝑘∈ℕ,₂𝑡⁺=0.9𝑥₂(𝑡),𝑡=𝑡_𝑘𝑥𝑡,𝑘∈ℕ,₀+𝜃=𝜙(𝜃)=[],cos𝜃sin𝜃,𝜃∈−0.2,0 (4.1) where 𝑟≥0. Let Lyapunov function 𝑉(𝜓(0))=(1/2)𝜓²₁(0)+(1/2)𝜓²₂(0), then for any 𝜓∈PC([−𝑟,0],ℬ(𝜌)) we have 𝐷⁺𝑉𝜓(01)=−2𝑥²₁+𝑥²₂−12𝑥₁−𝑥₂²𝑥²₁+𝑥₁𝑥₂+𝑥²₂1≤−2𝑥²₁+𝑥²₂=−𝑉𝜓(0). (4.2) Let 𝑐=1, 𝑎(‖𝑥‖)=𝑏(‖𝑥‖)=(1/2)‖𝑥‖, 𝜇=0.9, and 𝛽(‖𝑥‖)=‖𝑥‖, then the conditions of Theorem 3.1 are satisfied, which means the equilibrium points of the system are Lyapunov stable, and 𝐷⁺𝑉𝜓(0)1=−2𝑥₁−𝑥₂²31+4𝑥²₁+12𝑥₁+𝑥₂²1≤−2𝑥₁−𝑥₂². (4.3) Let 𝑊(𝑥₁,𝑥₂)=(1/2)(𝑥₁−𝑥₂)² then we derive that 𝐷⁺𝑉(𝜓(0))≤−𝑊(𝜓(0)); it follows from Theorem 3.2 that every point in 𝑊⁻¹(0) is semistable.
The simulation result is depicted in Figure 1, where the length of the impulsive intervals is 𝑇=0.3 second and the time delay 𝑟=0.1 second.

Figure 1 

State trajectory: semistable.

Acknowledgments

The authors would like to thank the Editor and the anonymous reviewers for their valuable comments and suggestions that improved the overall quality of this paper. This work was supported by Grants of NSFC-60874006.