# Finite-Time Lyapunov Functions and Impulsive Control Design

Authors: Huijuan Li; Qingxia Ma
Journal: Complexity
 (2020)

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## Abstract

In this paper, we introduce finite-time Lyapunov functions for impulsive systems. The relaxed sufficient conditions for asymptotic stability of an equilibrium of an impulsive system are given via finite-time Lyapunov functions. A converse finite-time Lyapunov theorem for controlling the impulsive system is proposed. Three examples are presented to show how to analyze the stability of an equilibrium of the considered impulsive system via finite-time Lyapunov functions. Furthermore, according to the results, we design an impulsive controller for a chaotic system modified from the Lorenz system.


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## Body

## 1. Introduction

Impulsive systems have been investigated by researchers, since impulsive systems can describe many practical problems from fields such as engineering, finance, chemistry, and biology (see [1, 2]). In references [1, 3–8], the authors studied the stability of equilibria of impulsive systems. Lyapunov functions are widely used to analyze stability problems of dynamical systems because it is not necessary to compute an analytic solution of the considered system. Many researchers try to relax the conditions imposed on Lyapunov functions. In [1, 2, 9], Lyapunov functions are allowed to be nonincreasing at the impulses. In [10], the conditions imposed on Lyapunov functions are more relaxed. Lyapunov functions could be nonincreasing at some of resetting times. The authors in [6] investigated the stability of hybrid systems by nonmonotonic Lyapunov functions which are not monotonically decreasing along the considered system trajectories. In [11, 12], Lyapunov functions for the considered impulsive system may increase during the continuous part of the trajectory of the state. In [5], Lyapunov functions are monotonically decreasing along the continuous part of the trajectory of the considered system and could increase at the resetting times. Based on the attained results, the authors designed a H∞ controller for the considered problem. In [13], utilizing results from [2], the author got some sufficient conditions for impulsive control for a class of systems. In [14, 15], the authors studied chaotic communication systems. Then, they designed impulsive control for the considered systems. In [7], stochastic switched systems with impulses and time delay were investigated. Based on vector Lyapunov functions, input-to-state stability of the considered system was discussed. In [16], the authors investigated switched systems with time delay. Lyapunov functions can be nonincreasing at the switching times. According to the results, adaptive control was designed for the considered system. In [17], the authors presented an overview of the research investigations on impulsive control systems. Using Lyapunov functions with relaxed constraints, the authors in [18] discussed sufficient and necessary conditions for asymptotic stability of an equilibrium of a discrete-time homogeneous dynamical system. The results were extended to discrete-time systems in [19, 20]. A converse Lyapunov theorem was proposed for continuous-time systems via Lyapunov functions with relaxed conditions in [21]. In [22], two ways were designed for the computation of Lyapunov functions with relaxed conditions for continuous-time systems. In [23], the authors discussed input-to-state stability for continuous-time systems via input-to-state stable (ISS) Lyapunov functions with relaxed conditions. In [24], the authors proposed relaxed sufficient conditions for asymptotic stability of an equilibrium of time varying impulsive systems via indefinite Lyapunov functions and designed an impulsive controller for a chaotic system.In this paper, we will analyze the stability of equilibria of time invariant impulsive systems by Lyapunov functions with relaxed constraints, later named finite-time Lyapunov functions, and then design impulsive control for a chaotic system adapted from the Lorenz system. The ideas of the paper are inspired by the results discussed above. We obtained some novel results. Finite-time Lyapunov functions can increase during some continuous part of the trajectory of the considered system and have positive jumps at some impulses (see Theorem1, example). It is worthy to point out that a converse finite-time Lyapunov theorem (see Theorem 2) is proposed. Based on Theorem 2, a finite-time Lyapunov function can be constructed for the considered impulsive system. Moreover, we design an impulse controller to get a chaotic system stabilized based on Theorem 1 and Corollary 1.This paper is organized as follows. In Section2, we introduce notations and basic definitions. Finite-time Lyapunov functions for impulsive systems are introduced. The main problem studied in this paper is described. In Section 3, the main results are discussed. We first study how to prove the origin of system (1) is asymptotic stable by finite-time Lyapunov functions. Then, a converse finite-time Lyapunov theorem for impulsive systems is obtained, that is, if the origin of system (1) is asymptotically stable and Condition 1 holds, then there exists a finite-time Lyapunov function for system (1). In Section 4, we show the efficiency of our main results via three examples. Especially, Example 3 shows that finite-time Lyapunov functions can increase along some continuous portion of the trajectory of system (31) and increase at some resetting times. Furthermore, according to our main results, we design impulsive control for a chaotic system in Section 5. We present simulation results of the chaotic system with the designed controller. Some concluding remarks are discussed in Section 6.


## 2. Notations and Preliminaries

The real numbers, the nonnegative real numbers, and the nonnegative integers are denoted byℝ, ℝ+, and ℤ+, respectively. The Euclidean norm of the real vector x∈ℝn is denoted by x. For x=x1,…,xnT∈ℝn, we denote 1−norm for x by x1=∑i=1,…,nxi. The open ball of radius r around z in the norm of ⋅ is defined by Bz,r=x∈ℝn|x−z<r. For a set Ω⊂ℝn, the boundary and the interior of Ω are denoted by ∂Ω and int Ω, respectively.It is well known that comparison functions are widely used in stability analysis. Comparison functions are described as follows. If a continuous functionα:ℝ+⟼ℝ+ satisfies α0=0 and αs>0 for all s>0, then we say it is positive definite. A positive definite function is of classK if it is strictly increasing and of classK∞ if it belongs to the class K and unbounded. We say a continuous function γ:ℝ+⟼ℝ+ belongs to classℒ if γr is strictly decreasing to 0 as r⟶∞. A continuous function β:ℝ+×ℝ+⟼ℝ+ is said to be of classKℒ if it is of class K∞ in the first argument and of class ℒ in the second argument.In this paper, we are going to study the stability property of the following system with impulses described by(1)x˙t=fxt,t≠tk,k∈1,2,3,…,,xtk=gxtk−,t=tk,k∈1,2,3,…,,where 0<t1<t2<⋯<tk<⋯ are resetting times in 0,∞ and limk⟶∞tk=∞. The functions f,g: ℝn⟼ℝn are locally Lipschitz continuous and satisfy the conditions f0=0, g0=0. It is evident that the origin is an equilibrium of system (1). Suppose that a sequence of impulse times tk is given; the solution of system (1) corresponding to an initial condition x0=x0 is denoted by xt,x0. The limits of xt from left and right are denoted by xt− and xt+, respectively. It is easy to see that the solution xt of system (1) is right continuous, that is, it is continuous in 0,t1, tk,tk+1, and the following conditions hold.(2)xtk−=limε⟶0+xtk−ε,xtk=xtk+=limε⟶0+xtk+ε.For a constantT>0 and a sequence of impulse times tk, we define a positively T−invariant set for system (1).Definition 1.
Given a constantT>0 and a sequence of impulse times tk, a compact set Ω⊂ℝn is called a positively T−invariant set for system (1) if for all xt,x0∈Ω, it satisfies xt+T,x0∈Ω for t∈ℝ+.Remark 1.
In Definition1, if T=0 holds, then we call the set Ω a positively invariant set for system (1).
The following definition describes asymptotic stability of system (1) we are interested in.Definition 2.
For system (1), we suppose that a sequence of impulse times tk is given. The origin of system (1) is asymptotically stable in a compact set Ω⊂ℝn if there exists a function β∈Kℒ such that for every initial condition x0∈Ω, it holds that(3)xt,x0≤βx0,t,t≥0.
In the literature, sufficient conditions for asymptotic stability of system (1) were obtained via Lyapunov functions described by the following definition.Definition 3.
Given a sequence of impulse timestk for system (1). A continuous function V:ℝn⟼ℝ+ is said to be a Lyapunov function for system (1) on a compact set Ω⊂ℝn, if there exist functions α1,α2∈K∞, a positive definite function α:ℝ+⟼ℝ+, and a continuous function θ:ℝ+⟼0,1⊂ℝ+ such that(i)
(C1)(4)α1x≤Vx≤α2x,∀x∈ℝn.(ii)
(C2)(5)∇Vxt,x0fxt≤−αVxt,x0,for,x0∈Ωt≠tk,k∈1,2,3,…,.(ii)
(C3)(6)Vgxt,x0≤θVxt−,x0,forx0∈Ω,t=tk,k∈1,2,3,…,.
We are going to relax the conditions imposed on Lyapunov functions in Definition3 and then demonstrate how to prove asymptotic stability of system (1) via finite-time Lyapunov functions defined by the following definition.Definition 4.
Consider system (1) with a given impulsive time sequence tk. A continuous function V:ℝn⟼ℝ+ is said to be a finite-time Lyapunov function for system (1) on a compact set Ω⊂ℝn, if there exist a positive constant T, functions α1,α2∈K∞, and a function ρ∈K with ρ<id such that(7)α1x≤Vx≤α2x,∀x∈ℝn,(8)VxT+t,x0≤ρVxt,x0,fort≥0,x0∈Ω.Remark 2.
(i)
From constraint (8), we have that the function V defined in Definition 4 may increase during some continuous portion of the trajectory and at some impulses. It is clear that the conditions imposed on finite-time Lyapunov functions are more relaxed than those on Lyapunov functions defined by Definition 3.(ii)
In order to verify condition (8), we have to construct finite-time Lyapunov functions. However, it is not easy to give a formulation of a finite-time Lyapunov function. Under certain conditions, Theorem 2 will explain how to construct a finite-time Lyapunov function. When we check if the condition (8) holds, it is necessary to calculate the solution xT+t,x0. For easy computation, the Euler method will be used to calculate the solution xT+t,x0 for examples considered in Section 4.
The following impulsive integral inequality of Gronwall type will be used in deducing inequalities in the proofs of our main results.Lemma 1.
Lett1,t2,…,tk,…, be a strictly increasing sequence of impulse times in 0,∞ and limk⟶∞tk=∞, the function m:ℝ+⟼ℝ a continuous function for t≠tk and right continuous at t=tkk=1,2,…,, and the function p:ℝ+⟼ℝ+ be a continuous function. Moreover, we assume that(9)mt≤A+∫0tpsmsds+∑0<tk≤tλkmtk,t≥0,k>0,k∈ℤ+,where λk≥0 and A are constants. Then it holds that(10)mt≤A∏0<tk≤t1+λke∫0tpsds,t≥0.Proof.
The proof is similar to the proof of Theorem 16.1 in [25].


## 3. Main Results

In this section, we first demonstrate how to prove asymptotic stability of system (1) via finite-time Lyapunov functions defined by Definition 4. Then, we propose a converse finite-time Lyapunov theorem for system (1).Theorem 1.
Consider system (1) with a given impulsive time sequence tk. Let T be a positive constant, and a compact set Ω⊂ℝn with 0∈intΩ be a positively T−invariant set for system (1). If there exists a finite-time Lyapunov function V:ℝn⟼ℝ with T for system (1) on Ω, then the origin of system (1) is asymptotically stable in Ω over the given impulse sequence. Furthermore, an estimate of the domain of attraction of the origin of system (1) is given by Da=x∈Ω|Vx≤minx∈∂ΩVx.Proof.
According to the conditions, there exist functionsα1,α2∈K∞ and a function ρ∈K with ρ<id such that inequalities (7) and (8) from Definition 4 hold. For any t>0, t∈ℝ+, there exists a integer N>0 such that t=NT+j,j∈0,T. Utilizing (8) recursively, we obtain that(11)Vxt,x0=VxNT+j,x0=VxN−1T+T+j,x0≤ρVxN−1T+j,x0⋯≤ρNVxj,x0≤ρNα2xj,x0,where ρN represents the N−times composition of ρ and α2 comes from (7). Let k∈ℤ+ denote the number of impulsive time ti∈0,ji∈ℤ+,i>0, and(12)σ=0,k=0,1,k≥1.
The solutionxt,x0 at time t=j is given by(13)xj,x0=x0+∑d=1kgxtd−+∫0jfxsds,fork≥1,x0+∫0jfxsds,fork=0.
Using (12), xj,x0 is rewritten as(14)xj,x0=x0+σ∑d=1kgxtd−+∫0jfxsds,for anyj≥0.
Then, we have(15)xj,x0−x0≤σ∑d=1kgxtd−−gx0+gx0+∫0jfxs−fx0+fx0ds≤σ∑d=1kgxtd−−gx0+σ∑d=1kgx0+∫0jfxs−fx0ds+∫0jfx0ds.
Using the Lipschitz conditions forf,g, and Lemma 1, we get that(16)xj,x0−x0≤σ∑d=1kLgxtd−−x0+σkgx0+∫0jLfxs−x0ds+∫0jfx0ds≤C1+σLgkeLfj,where Lf,Lg are Lipschitz constants for the functions f,g, respectively, and C=σk∑d=1kgx0+∫0jfx0ds. Therefore, it holds that(17)xj,x0≤C1+σLgkeLfj+x0≤C1+σLgkeLfT+x0≔HTx0.
Based on the assumptions on the functionsf,g, it is obtained that HTx0 is continuous with respect to x0. Moreover, it is clear that HT0=0, and HTs>0 for s>0. Thus, HTx0 is a positive definite function with respect to x0. It is easy to see that there exists a function γ∈K∞ such that HTx0≤γx0. Then, it is satisfied that xj,x0≤γx0 for all 0≤j<T.
The idea of the proof of the existence of the functionρ1 is inspired by Theorem 2.1 in [21] and Lemma 12 in [22]. Because ρ is positive definite, without loss of generality, we can assume that ρ is invertible, that is, ρ is a one-to-one and onto function. According to Theorem 3.16 in [26] and the above discussion, it holds that ρ−1 is continuous and ρ−10=ρ−1ρ0=0. Thus, for t=NT+j, we obtain that(18)Vxt,x0≤ρt/T∘ρ−1∘α2∘γx0.
From the above analysis, the functionρ−1 is positive definite. Then, there exists a function ρ1∈K∞ satisfying ρ1s≥ρ−1s for s≥0. Thus, it holds that(19)Vxt,x0≤ρt/T∘ρ1∘α2∘γx0.
Since the conditionα1∈K∞ is satisfied, then the function α1 is a one-to-one and onto function. Therefore, the function α1−1 exists and α1−1∈K∞ holds. Furthermore, utilizing (7), we have that(20)xt,x0≤α1−1∘ρt/T∘ρ1∘α2∘γx0.
Letβx0,t=α1−1∘ρt/T∘ρ1∘α2∘γx0. Since α1−1,ρ1,α2,γ are K∞ functions, for fixed t, the function β increases as the argument x0 increases. Because ρ<id holds, for fixed x0, the function β decreases as the argument t increases. Therefore, we obtain that β∈Kℒ is a K∞ function in the argument x0 and a ℒ function in the argument t. Then, it holds that(21)xt,x0≤βx0,t.
Therefore, the origin of system (1) is asymptotically stable in Ω over the given impulse sequence. Moreover, an estimate of the domain of attraction of the origin of system (1) is obtained by Da=x∈Ω|Vx≤minx∈∂ΩVx.Remark 3.
To make surexT,x0∈Ω for all x0∈Ω, we have to ensure that the set Ω is a positively T−invariant set for system (1).
In the following, a converse finite-time Lyapunov theorem is investigated. To obtain the desired result, it is necessary to require the following condition.Condition 1.
Consider system (1) with a given impulsive time sequence tk. There exists a Kℒ function β which satisfies (2) for system (1) and the inequality(22)βs,T<s,for some T>0 and s>0.Theorem 2.
Consider system (1) with a given impulsive time sequence tk. If the origin of system (1) is asymptotically stable in an invariant set Ω⊂ℝn over the given impulse sequence and Condition 1 holds, then for any function η∈K∞, the function V:ℝn⟼ℝ+ with(23)Vx=ηx,forx∈ℝnsatisfies inequalities (7) and (8) with T from Condition 1.Proof.
Becauseη∈K∞, it is evident that there exist functions α1,α2∈K∞ such that inequality (7) holds. Since Condition 1 is satisfied, then there exists a positive constant T such that inequality (22) holds. Thus, we have that(24)Vxt+T,x0=ηxt+T,x0=ηxT,xt,x0≤ηβxt,x0,T=η°βη−1Vxt,x0,T.
Letρ=η°βη−1Vxt,x0,T. It is obvious that ρ is a positive definite function of the variable Vxt,x0. Based on Condition 1, it is obtained that ρs<η∘η−1s=s holds. Thus, V satisfies inequality (8) with T from Condition 1.Remark 4.
Theorem2 provides a way to construct finite-time Lyapunov functions for systems. However, for the considered system, it is not easy to check if Vx=ηx satisfies inequality (8).


## 4. Examples

In this section, three examples are presented to illustrate how to analyze stability of impulsive systems with finite-time Lyapunov functions. Based on the definition of finite-time Lyapunov function (see Definition4), in order to check if a continuous function V:ℝn⟼ℝ+ is a finite-time Lyapunov function for system (1), it is necessary to calculate VxT,x0 in (8) from Definition 4. For a constant 0<T<+∞, we compute the value of xT,x0 of system (1) with respect to an initial condition x0 by the Euler method with the time step denoted by ht. In order to simplify the computation, for the following examples, 1-norm ⋅1 is utilized.


### 4.1. Example 1

Consider the following one-dimensional system described by(25)x˙t=−xt,t≠tk,xt=2xtk−,t=tk=3k4,where x∈ℝ, k>0,k∈ℤ+.LetV1x=x be a finite-time Lyapunov function candidate for system (25). To show there exists a constant T>0 such that V1 satisfies (8) from Definition 4, we calculate VxT,x0 with T=3/4,ht=1/4.(26)x14,x0=34x0,x12,x0=916x0,xt1−,x0=x34−,x0=2764x0,x34,x0=2732x0,x1,x0≤81128x0,x54,x0≤243512x0,xt2−,x0=x32−,x0=7292048x0,x32,x0=7291024x0.From the above calculation, the following inequalities hold.(27)Vx34,x0≤2732Vx0,x0∈ℝ,Vx32+t,xt,x0≤2732Vxt,x0,t∈ℝ+,x0∈ℝ.Then, it is obvious thatV1 is a finite-time Lyapunov function for system (25) in ℝ. Based on Theorem 1, we have that the origin of system (25) is asymptotically stable in ℝ over the given impulse sequence (see Figure 1). Figure 1 clearly shows that V1x=x is not a Lyapunov function for system (25).Figure 1
The trajectory of the state of system (25) corresponding to the initial condition x0=64 and the resetting time tk=0.75k,k=1,2,…,.


### 4.2. Example 2

In this section, we consider the following one-dimensional system described by(28)x˙t=xt,t≠tk,xt=12xtk−,t=tk=k2,where x∈ℝ, k>0,k∈ℤ+.LetV2x=x be a finite-time Lyapunov function candidate for system (28). To check if V2 satisfies condition (8) from Definition 4 with T=1/2, we have to calculate with ht=1/4.(29)x14,x0=54x0,xt1−,x0=x12−,x0=2516x0,x12,x0=2532x0,x34,x0=125128x0,xt2−,x0=x1−,x0=625512x0,x1,x0=6251024x0.It is evident that the following inequality is satisfied.(30)V2x12+t,x0≤V2xt,x0,t∈ℝ+,x0∈ℝ.According to the above analysis, we conclude thatV2 is a finite-time Lyapunov function for system (28) in ℝ. Moreover, utilizing Theorem 1, it is attained that the origin of system (28) is asymptotically stable in ℝ over the given impulse sequence. Figure 2 demonstrates that V2 is not a Lyapunov function for system (28).Figure 2
The trajectory of the state of system (28) corresponding to the initial condition x0=64 and the resetting time tk=0.5k,k=1,2,…,.


### 4.3. Example 3

We consider the following two-dimensional system described by(31)x˙1t=−0.2x2t,t≠tk,x˙2t=x1t−x2t,t≠tk,x1t=x1tk−+18sinx1tk−+x2tk−x2tk−,t=tk=5k,x2t=x2tk−+18cosx1tk−+x2tk−x1tk−,t=tk=5k,where x=x1,x2T∈ℝ2, k>0, k∈ℤ+.For system (31), we choose V3x=x1 as a finite-time Lyapunov function candidate. Now, we show there exists a constant T such that inequality (8) from Definition 4 is satisfied. The following calculation is done with T=5, ht=1. For the calculation, we utilize the following notations: x10=x10,x20=x20,x11=x1ht,x0,x21=x2ht,x0,x12=x12ht,x0,x22=x22ht,x0,…,x1k=x1kht,x0,x2k=x2kht,x0.(32)x11=x10−0.2x20,x21=x10,…,x14=0.44x10−0.12x20,x24=0.6x10−0.16x20,x15−,x0=0.32x10−0.088x20,x25−,x0=0.44x10−0.12x20,x15+x25≤98⋅0.76x01=0.855x01.Thus, we have that(33)V3x5,x0≤0.855V3x0,V3x5+t,x0≤0.855V3xt,x0,t∈ℝ+,x0∈ℝ2.From the above calculation, it is obvious thatV3 is a finite-time Lyapunov function for system (31) in ℝ2. Based on Theorem 1, it is obtained that the origin of system (31) is asymptotically stable in ℝ2 over the given impulse sequence (see Figures 3 and 4). Figure 5 demonstrates that the function V3 is not a Lyapunov function for system (31).Figure 3
The trajectory of the statex1 of system (31) corresponding to the initial condition x0=−6,,2T and the resetting time tk=5k,k=1,2,…,.Figure 4
The trajectory of the statex2 of system (31) corresponding to the initial condition x0=−6,,2T and the resetting time tk=5k,k=1,2,…,.Figure 5
The trajectory of the finite-time Lyapunov functionV3 for system (31) with the initial condition x0=−6,2T and the resetting time tk=5k,k=1,2,…,.


## 4.1. Example 1

Consider the following one-dimensional system described by(25)x˙t=−xt,t≠tk,xt=2xtk−,t=tk=3k4,where x∈ℝ, k>0,k∈ℤ+.LetV1x=x be a finite-time Lyapunov function candidate for system (25). To show there exists a constant T>0 such that V1 satisfies (8) from Definition 4, we calculate VxT,x0 with T=3/4,ht=1/4.(26)x14,x0=34x0,x12,x0=916x0,xt1−,x0=x34−,x0=2764x0,x34,x0=2732x0,x1,x0≤81128x0,x54,x0≤243512x0,xt2−,x0=x32−,x0=7292048x0,x32,x0=7291024x0.From the above calculation, the following inequalities hold.(27)Vx34,x0≤2732Vx0,x0∈ℝ,Vx32+t,xt,x0≤2732Vxt,x0,t∈ℝ+,x0∈ℝ.Then, it is obvious thatV1 is a finite-time Lyapunov function for system (25) in ℝ. Based on Theorem 1, we have that the origin of system (25) is asymptotically stable in ℝ over the given impulse sequence (see Figure 1). Figure 1 clearly shows that V1x=x is not a Lyapunov function for system (25).Figure 1
The trajectory of the state of system (25) corresponding to the initial condition x0=64 and the resetting time tk=0.75k,k=1,2,…,.


## 4.2. Example 2

In this section, we consider the following one-dimensional system described by(28)x˙t=xt,t≠tk,xt=12xtk−,t=tk=k2,where x∈ℝ, k>0,k∈ℤ+.LetV2x=x be a finite-time Lyapunov function candidate for system (28). To check if V2 satisfies condition (8) from Definition 4 with T=1/2, we have to calculate with ht=1/4.(29)x14,x0=54x0,xt1−,x0=x12−,x0=2516x0,x12,x0=2532x0,x34,x0=125128x0,xt2−,x0=x1−,x0=625512x0,x1,x0=6251024x0.It is evident that the following inequality is satisfied.(30)V2x12+t,x0≤V2xt,x0,t∈ℝ+,x0∈ℝ.According to the above analysis, we conclude thatV2 is a finite-time Lyapunov function for system (28) in ℝ. Moreover, utilizing Theorem 1, it is attained that the origin of system (28) is asymptotically stable in ℝ over the given impulse sequence. Figure 2 demonstrates that V2 is not a Lyapunov function for system (28).Figure 2
The trajectory of the state of system (28) corresponding to the initial condition x0=64 and the resetting time tk=0.5k,k=1,2,…,.


## 4.3. Example 3

We consider the following two-dimensional system described by(31)x˙1t=−0.2x2t,t≠tk,x˙2t=x1t−x2t,t≠tk,x1t=x1tk−+18sinx1tk−+x2tk−x2tk−,t=tk=5k,x2t=x2tk−+18cosx1tk−+x2tk−x1tk−,t=tk=5k,where x=x1,x2T∈ℝ2, k>0, k∈ℤ+.For system (31), we choose V3x=x1 as a finite-time Lyapunov function candidate. Now, we show there exists a constant T such that inequality (8) from Definition 4 is satisfied. The following calculation is done with T=5, ht=1. For the calculation, we utilize the following notations: x10=x10,x20=x20,x11=x1ht,x0,x21=x2ht,x0,x12=x12ht,x0,x22=x22ht,x0,…,x1k=x1kht,x0,x2k=x2kht,x0.(32)x11=x10−0.2x20,x21=x10,…,x14=0.44x10−0.12x20,x24=0.6x10−0.16x20,x15−,x0=0.32x10−0.088x20,x25−,x0=0.44x10−0.12x20,x15+x25≤98⋅0.76x01=0.855x01.Thus, we have that(33)V3x5,x0≤0.855V3x0,V3x5+t,x0≤0.855V3xt,x0,t∈ℝ+,x0∈ℝ2.From the above calculation, it is obvious thatV3 is a finite-time Lyapunov function for system (31) in ℝ2. Based on Theorem 1, it is obtained that the origin of system (31) is asymptotically stable in ℝ2 over the given impulse sequence (see Figures 3 and 4). Figure 5 demonstrates that the function V3 is not a Lyapunov function for system (31).Figure 3
The trajectory of the statex1 of system (31) corresponding to the initial condition x0=−6,,2T and the resetting time tk=5k,k=1,2,…,.Figure 4
The trajectory of the statex2 of system (31) corresponding to the initial condition x0=−6,,2T and the resetting time tk=5k,k=1,2,…,.Figure 5
The trajectory of the finite-time Lyapunov functionV3 for system (31) with the initial condition x0=−6,2T and the resetting time tk=5k,k=1,2,…,.


## 5. Impulsive Control of a Chaotic System

In this section, based on Theorem1, impulsive control is designed for a chaotic system described by(34)x˙=Ax+γx,where x=x1,x2,x3T∈ℝ3, and(35)A=−aa0c−1000−b,γx=x2x3−x1x3x1x2,in which a,b,c∈ℝ are positive constants. The constants a and b are called the Prandtl number and Rayleigh number, respectively. In [27], the authors studied system (34) adapted from the Lorenz system.In order to ensure the origin is an asymptotic stable equilibrium, we design an impulsive controller for system (34) described by(36)x˙t=Axt+γxt,fort≠tk,xt=xtk+=Dxtk−,fort=tk,where tkk=1,2,…,, 0<t1<t2<⋯<tk<tk+1<⋯ (tk⟶∞ as k⟶∞) are impulses, h=t1=tk+1−tkk=1,2,…,, and(37)Dx=d11x1d12x2d13x3d21x1d22x2d23x3d31x1d32x2d33x3,in which dij:ℝ⟼ℝ are continuous. We assume that there exist constants d¯ij∈ℝ+i,j=1,2,3 satisfying dijs≤d¯ijs for s∈ℝ.For convenience, the following notations are introduced:(38)d1=d¯111−ah+d¯12ch+d¯12hR+d¯13hR+d¯22ch+d¯211−ah+d¯22hR+d¯23hR+d¯311−ah+d¯32ch+d¯32hR+d¯33hR,d2=d¯11ah+d¯11hR+d¯121−h+d¯13hR+d¯221−h+d¯21hR+d¯21ah+d¯23hR+d¯31ah+d¯31hR+d¯321−h+d¯33hR,d3=d¯11Rh+d¯12hR+d¯131−bh+d¯22Rh+d¯21hR+d¯231−bh+d¯31Rh+d¯32hR+d¯331−bh,where R from Corollary 1 is a positive constant. It is clear that dii=1,2,3 are positive constants.Corollary 1.
LetR>0 be a constant. Consider system (36) on a bounded set D⊂B0,R⊂ℝ3. Let M=maxi=1,2,3ci. If M<1 is satisfied, then the function V:ℝ3⟼ℝ defined by Vx=x1 is a finite-time Lyapunov function for system (36) with T=h. Therefore, the origin of system (36) is asymptotically stable in D over the given impulse sequence.Proof.
Under the conditions, we proveVx=x1 is a finite-time Lyapunov function for system (36). It is obvious that there exist functions α1,α2∈K∞ such that(39)α1x≤Vx≤α2x.
Now, we have to calculateVxh,x0 for system (36) with x0∈D by the Euler method with the step size h. Let x0=x10,x20,x30T.(40)x1h−,x0=x10+ax20−x10+x20x30h,x2h−,x0=x20+cx10−x20−x10x30h,x3h−,x0=x30+x10x20−bx30h,xh,x0=Dxh−.
By calculation, it is obtained thatVxh,x0≤Mx01=MVx0. Thus, the function V satisfies inequality (8) from Definition 4. By Theorem 1, we obtain that the origin of system (36) is asymptotically stable in Ω over the given impulse sequence.Remark 5.
In the proof, we calculatexh,x0 by the Euler method with the step size h. The reason for letting the step size being h is that it is easy to estimate the value of xh−,x01.


### 5.1. Simulation Results

In this section, system (34) is considered as an example with the coefficients a=35, b=8/3, c=25, and x0=3,4,5T. Figure 6 shows that a chaotic attractor exists for system (34) with the given conditions and is similar to that for the Lorenz system.Figure 6
The trajectory of the states of system (34) with the coefficients a=35, b=8/3, and c=25 and the initial condition x0=3,4,5T.An impulsive controller is designed as follows:tk=0.01kk=1,2,…,, dij=0i,j=1,2,3,i≠j, dii=0.6i=1,2,3. We consider system (36) with the given coefficients on D=B0,20⊂ℝ3. The constraints of Corollary 1 are satisfied with R=20,h=0.01, M=0.84. Then, Vx=|x|1 is a finite-time Lyapunov function for system (36) with the given coefficients. Hence, the origin of system (36) with the given coefficients is asymptotic stable on D (see Figures 7–9). Figure 10 clearly shows that V is not a Lyapunov function for system (36).Figure 7
The trajectory of the statex1 of system (36) with the coefficients a=35, b=8/3, c=25, tk=0.01kk=1,2,…,, dij=0i,j=1,2,3,i≠j, and dii=0.6i=1,2,3 and the initial condition x0=3,4,5T.Figure 8
The trajectory of the statex2 of system (36) with the coefficients a=35, b=8/3, c=25, tk=0.01kk=1,2,…,, dij=0i,j=1,2,3,i≠j, and dii=0.6i−1,2,3 and the initial condition x0=3,4,5T.Figure 9
The trajectory of the statex3 of system (36) with the coefficients a=35, b=8/3, c=25, tk=0.01kk=1,2,…,, dij=0i,j=1,2,3,i≠j, and dii=0.6i=1,2,3 and the initial condition x0=3,4,5T.Figure 10
The trajectory of the finite-time Lyapunov functionV of system (36) with the coefficients a=35, b=8/3, c=25, tk=0.01kk=1,2,…,, dij=0i,j=1,2,3,i≠j, and dii=0.6i=1,2,3 and the initial condition x0=3,4,5T.Remark 6.
From Figures7–9, we obtain that the simulation results are similar to that of [24], since the designed impulsive control is similar to each other. However, from examples of Section 4 and this chaotic system, we can conclude that construction of finite-time Lyapunov functions or design of impulsive control based on Theorem 1 is easier than the method proposed in [24]. The reason for this point is that we do not have to calculate the derivative of Lyapunov function along the continuous part of the trajectory of the considered system.


## 5.1. Simulation Results

In this section, system (34) is considered as an example with the coefficients a=35, b=8/3, c=25, and x0=3,4,5T. Figure 6 shows that a chaotic attractor exists for system (34) with the given conditions and is similar to that for the Lorenz system.Figure 6
The trajectory of the states of system (34) with the coefficients a=35, b=8/3, and c=25 and the initial condition x0=3,4,5T.An impulsive controller is designed as follows:tk=0.01kk=1,2,…,, dij=0i,j=1,2,3,i≠j, dii=0.6i=1,2,3. We consider system (36) with the given coefficients on D=B0,20⊂ℝ3. The constraints of Corollary 1 are satisfied with R=20,h=0.01, M=0.84. Then, Vx=|x|1 is a finite-time Lyapunov function for system (36) with the given coefficients. Hence, the origin of system (36) with the given coefficients is asymptotic stable on D (see Figures 7–9). Figure 10 clearly shows that V is not a Lyapunov function for system (36).Figure 7
The trajectory of the statex1 of system (36) with the coefficients a=35, b=8/3, c=25, tk=0.01kk=1,2,…,, dij=0i,j=1,2,3,i≠j, and dii=0.6i=1,2,3 and the initial condition x0=3,4,5T.Figure 8
The trajectory of the statex2 of system (36) with the coefficients a=35, b=8/3, c=25, tk=0.01kk=1,2,…,, dij=0i,j=1,2,3,i≠j, and dii=0.6i−1,2,3 and the initial condition x0=3,4,5T.Figure 9
The trajectory of the statex3 of system (36) with the coefficients a=35, b=8/3, c=25, tk=0.01kk=1,2,…,, dij=0i,j=1,2,3,i≠j, and dii=0.6i=1,2,3 and the initial condition x0=3,4,5T.Figure 10
The trajectory of the finite-time Lyapunov functionV of system (36) with the coefficients a=35, b=8/3, c=25, tk=0.01kk=1,2,…,, dij=0i,j=1,2,3,i≠j, and dii=0.6i=1,2,3 and the initial condition x0=3,4,5T.Remark 6.
From Figures7–9, we obtain that the simulation results are similar to that of [24], since the designed impulsive control is similar to each other. However, from examples of Section 4 and this chaotic system, we can conclude that construction of finite-time Lyapunov functions or design of impulsive control based on Theorem 1 is easier than the method proposed in [24]. The reason for this point is that we do not have to calculate the derivative of Lyapunov function along the continuous part of the trajectory of the considered system.


## 6. Conclusion

In this paper, we introduced the definition of finite-time Lyapunov function for impulsive systems. It was proved that if there exists a finite-time Lyapunov function for system (1), then the origin of system (1) is asymptotically stable (Theorem 1). It is worthy to point out that the conditions imposed on finite-time Lyapunov functions for system (1) are more relaxed than those on Lyapunov function which decrease along the continuous part of the trajectory or have negative jumps at all impulses. Finite-time Lyapunov functions can increase during some continuous part of the trajectory of the considered system and have positive jumps at some resetting times. This point was demonstrated by example. A converse finite-time Lyapunov theorem (Theorem 2) was proposed. Three examples were presented to illustrate how to analyze stability of the origin of an impulsive system via finite-time Lyapunov functions. According to our main results, impulsive control was designed to ensure the origin of the considered chaotic system is asymptotically stable. Some simulation results of the chaotic system with impulsive control were presented to show how to design an impulsive controller for the chaotic system by finite-time Lyapunov functions.



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