# Semistability of Nonlinear Impulsive Systems with Delays

Authors: Xiaowu Mu; Yongliang Gao
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 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

Letℕ denote the set of positive integer numbers. Let PCt denote the set of piecewise right continuous functions ϕ:[t-r,t]→ℝn with the norm defined by ∥ϕ∥rt=sup-r≤s≤0∥ϕ(t+s)∥. For simplicity, define ∥ϕ∥r=∥ϕ∥r0, for ϕ∈PC0. For given r>0, if x∈PC([t0-r,+∞),ℝn), then for each t≥t0, we define xt,xt-∈PC0 by xt(s)=x(t+s)(-r≤s≤0) and xt-(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∈ℝn;∥x∥≤ρ}.LetΩ∈ℝn be an open set and 𝔅(ρ)⊂Ω for some ρ>0. Given functionals f:ℝ+×PC([-r,0],Ω)→ℝn,g:ℝ+×Ω→ℝn, satisfying f(t,0)=0,g(0,0)=0. Considering the following nonlinear time-delay impulsive system Σt described by the state equation
(2.1)x˙(t)=f(t,xt),t>t0,t≠tk,k∈ℕ,(2.2)x(t+)=g(t,x(t)),t=tk,k∈ℕ,(2.3)x(t0+θ)=ϕ(θ),θ∈[-r,0],
where x(t)∈ℝn is the system state, x˙(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 Σt are right continuous, that is, x(t+)=x(t). {tk},k∈ℕ is a strictly increasing sequence of impulse times in (t0,∞) where limk→∞tk=∞.Definition 2.1.
The functionf:ℝ×PC→ℝn is said to be composite-PC, if for each t0∈ℝ and α>0, x∈PC([t0-r,t0+α],ℝn) and x is continuous at each t≠tk in [t0,t0+α], then the composite function h(x)=f(t,xt)∈PC([t0-r,t0+α],ℝn).Definition 2.2.
The functionf:ℝ×PC→ℝn is said to be quasi-bounded, if for each t0∈ℝ+, α>0, and for each compact set F∈ℝn, 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 functionx:[t0-r,t0+α]→ℝn with α>0 is said to be a solution of Σt if(i)
x is continuous at each t≠tk in (t0,t0+α];(ii)
the derivative ofx 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 ofx exists and satisfies (2.1) in t∈[t0,t0+α], while for each tk∈[t0,t0+α], (2.2) holds;(iv)
Equation (2.3) holds, that is, x(t0+θ)=ϕ(θ),θ∈[-r,0].
We denote byx(t,t0,ϕ)(or x(t), if in not confusing) the solution of Σt. 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 systemΣt.(A1)
f(t,ψ) is composite-PC, quasi-bounded and locally Lipschitzian in ψ.(A2)
For each fixedt∈ℝ+, f(t,ψ) is a continuous function of ψ on PC([-τ,0],ℝn).
Under the assumptions above, it was shown in [11] that for any ϕ∈PC([-r,0],ℝn), system Σt 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 ofΣt is a point xe∈PC([t0-r,t0+α],ℝn) satisfying x(t,t0,ϕ)=xe for all t≥0 where x(t,t0,ϕ) is the solution of Σt. Let ℰ denote the set of equilibrium points of Σt.Definition 2.5.
Consider the delay impulsive systemΣt.(i)
An equilibrium pointx(t)≡xe of Σt is Lyapunov stable if for any ε>0 there exists δ(ε,t0), such that ∥ϕ-ϕe∥r<δ implies ∥x(t)-xe∥<ε for all t≥t0, where ϕe is the initial function for xe. An equilibrium point x is uniformly Lyapunov stable, if, in addition, the number δ is independent of t0.(ii)
An equilibrium pointx of Σt 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 Σt converges to a Lyapunov stable equilibrium point, that is, limt→∞x(t,t0,ϕ)=y,ϕ∈Ω, where y is a Lyapunov stable equilibrium point.(iii)
SystemΣt is said to be uniformly asymptotically stable in the sense of Lyapunov with respect to the zero solution, if it is uniformly stable and limt→∞∥x(t)∥=0.Definition 2.6.
The functionV:[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 [tk-1,tk)×PC([-τ,0],ℬ(ρ)) and for each k∈ℕ,lim(t,y)→(tk-,x)V(t,y)=V(tk-,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.
LetV∈𝒱0. For any (t,ψ)∈[t0,+∞)×PC([-τ,0],ℬ(ρ)), the upper right-hand derivative of V with respect to system Σt is defined by
(2.4)D+V(t,ψ(0)):=limsuph→0+1h{V(t+h,ψ(0)+hf(t,ψ))-V(t,ψ(0))}.


## 3. Main Results

In the following, we will establish several sufficient conditions for Lyapunov stability and semistability for impulsive differential systemΣt with time delays.Theorem 3.1.
SystemΣt is uniformly stable, and the zero solution of Σt is asymptotically stable if there exists a Lyapunov function V∈𝒱0 which satisfies the following.(i)
∃a,b∈𝒦 such that
(3.1)a(∥x∥)≤V(t,x)≤b(∥x∥).(ii)
For anyt∈[t0,+∞), t≠tk and ψ∈PC([-r,0],ℝn), there exists c>0, such that
(3.2)D+V(t,ψ(0))≤-cV(t,ψ(0)).(iii)
There exist aμ(0<μ<1) and a subsequence {tkj} of the impulsive moments {tk} such that
(3.3)∥V(tkj+1,x)∥r≤μ∥V(tkj,x)∥r.(iv)
For anym,0≤m≤k0, forallxe∈ℰ there exists a function α∈𝒦, such that
(3.4)∥V(tm,x-xe)∥r≤α(∥ϕ-ϕe∥r).(v)
For anym∈ℕ,kj≤m<kj+1,j=0,1,2,…, there exists a function β∈𝒦 such that
(3.5)∥V(tm,x-xe)∥r≤β∥V(tkj,x-xe)∥r.Proof.
Letxe be an equilibrium point of the system Σt. We first prove that xe is uniformly stable, that is, for forallε>0, there exists δ=δ(ε)>0 such that ∥ϕ-ϕe∥r<δ implies ∥x(t)-xe∥<ε for all t≥t0.
Forallε>0, let 0<δ<ε such that
(3.6)a(ε)>max{α(δ),β(α(δ))}.
For any ∥ϕ-ϕe∥r<δ, by condition (3.4), we get
(3.7)∥V(tm,x-xe)∥r≤α(∥ϕ-ϕe∥r)≤α(δ),0≤m≤k0.
By (3.3), it is clear that ∥V(t,x)∥r is nonincreasing along the subsequence {tkj}, so we have
(3.8)∥V(tkj,x-xe)∥r≤∥V(tk0,x-xe)∥r≤α(δ),j=0,1,2,….
For anym,kj≤m<kj+1,j=0,1,2,…, by (3.5), we get
(3.9)∥V(tm,x-xe)∥r≤β(α(δ)).
Combining (3.7), (3.8), and (3.9), we conclude that
(3.10)∥V(tk,x-xe)∥r<a(ε),k=1,2,….
By condition (3.2), for any t∈[tk,tk+1),k=0,1,2…, we have
(3.11)V(t,x-xe)≤V(tk,x-xe)<a(ε),
and then, by (3.10), for any t≥t0 we derive that V(t,x-xe)<a(ε). Hence, by (3.1) we obtain that a(∥x-xe∥)≤V(t,x-xe)<a(ε). Since a∈𝒦, we get
(3.12)∥x(t)-xe∥<ε,t≥t0,
which implies that system Σt is uniformly Lyapunov stable.
Next, we will prove that the zero solution ofΣt is asymptotically stable.
Since systemΣt 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
(3.13)limsupt→∞∥V(t,x)∥r=v≤M.
In the following, we will show thatv=0. Without loss of generality, we can suppose that there exists a sequence {tn}⊂[t0,∞),n=1,2,…, such that
(3.14)limn→∞∥V(tn,x)∥r=limsupn→∞∥V(t,x)∥r=v.
From (3.3) we get
(3.15)∥V(tkj,x)∥r<μj∥v(tk0,x)∥r.
Since 0<μ<1, we obtain
(3.16)limj→∞∥V(tkj,x)∥r=0.
If the sequence {tn}⊂[t0,∞),n=1,2,… is the same as the sequence {tkj},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 kj≤n<kj+1. Then from condition (3.5) we get
(3.17)∥V(tn,x)∥r≤β(∥V(tkj,x)∥r).
So
(3.18)limn→∞∥V(tn,x)∥r≤limj→∞β(∥V(tkj,x)∥r)=0,
which implies v=0.
Hence, we derive thatlimt→∞∥V(t,x)∥=0. Finally, by (3.1), we have limt→∞∥x(t)∥=0 which implies that the zero solution of the system Σt is asymptotically stable. The proof is completed.Next, we present a sufficient condition for semistability for systemΣt.Let𝔏1:={f:[0,∞)→ℝ;fismeasurableand∫0∞‍|f(t)|dt<∞}.Theorem 3.2.
Consider the systemΣt; assume that there exists nonnegative-definite continuous function W:ℝ×ℝn→ℝ such that
(3.19)D+V(t,ψ(0))≤-W(t,ψ(0)).
Let W-1(0):={x∣W(t,x)≡0,forallt≥t0}. If every equilibrium point of system Σt is Lyapunov stable, then every point in W-1(0) is semistable.Proof.
Define(3.20)φ(t):={W(t,ψ(0)),t≠tk,k∈ℕ,0,t=tk,k∈ℕ.
It follows from (3.19) and (3.3) that
(3.21)∫0t‍φ(s)ds≤V(x(t1))-V(x(t))≤V(x(t1)).
Sinceφ(·) is nonnegative, it follows that φ(·)∈𝔏1. Next, we show that φ(t)→0 as t→∞.
If it is not true, then there existsε>0 and an infinite sequence of times τ1,τ2,… such that |φ(τi)|≥ε. By definition of φ(·) we have τi,i=1,2…  that does not belong to the set of impulsive times {tk}.
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 x˙(t) is bounded for all t≥0; thus, φ(·) is uniformly continuous on[t0,+∞)∖{tn}. So, there exists δ>0 such that every τi is contained in some interval of Ii,τi∈Ii of length δ on which φ(t)≥ε/2,t∈Ii. 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, letxe∈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 limt→∞x(t)=y. Since every point in ℰis Lyapunov stable, and hence y is a Lyapunov stable equilibrium of Σt, it follows that xe is semistable. Finally, since xe∈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(4.1)x˙1(t)=-x1(t)+x2(t){1-sin(x1(t-r))}+x23(t)-x13(t),t≠tk,k∈ℕ,x˙2(t)=-x2(t)+x1(t){1+sin(x1(t-r))}+x13(t)-x23(t),t≠tk,k∈ℕ,x1(t+)=0.8x1(t),t=tk,k∈ℕ,x2(t+)=0.9x2(t),t=tk,k∈ℕ,x(t0+θ)=ϕ(θ)=(cosθsinθ),θ∈[-0.2,0],
where r≥0. Let Lyapunov function V(ψ(0))=(1/2)ψ12(0)+(1/2)ψ22(0), then for any ψ∈PC([-r,0],ℬ(ρ)) we have
(4.2)D+V(ψ(0))=-12(x12+x22)-12(x1-x2)2(x12+x1x2+x22)≤-12(x12+x22)=-V(ψ(0)).
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
(4.3)D+V(ψ(0))=-12(x1-x2)2[1+34x12+(12x1+x2)2]≤-12(x1-x2)2.
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 Figure1, 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.



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*Source: 260236-2012-10-14.xml*