# On Boundedness and Attractiveness of Nonlinear Switched Delay Systems

Authors: Yi Zhang; Yuyun Zhao; Honglei Xu; Hongting Shi; Kok Lay Teo
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

This paper is concerned with the boundedness and attractiveness of nonlinear switched delay systems whose subsystems have different equilibria. Some sufficient conditions which can guarantee the system’s boundedness are obtained. In addition, we work out the region where the solution will remain and furthermore the relationship between the initial function and the bounded region. Based on the new concept of attractor with switching laws, we show that the nonlinear switched delay system is attractive and then obtain the attractive region.


---

## Body

## 1. Introduction

A switched system is a collection of finite continuous variable systems (called subsystems) along with a discrete event governing the “switching” among them (called the switching law). In reality, there are many switched systems that occur naturally or by design, such as those in the fields of control, communication, computer, and signal processes. Indeed, these systems are always suited to describe practical dynamical behaviors with hybrid nature in engineering and technology [1, 2]. In the last decades, they have attracted considerable attention among control theorists, computer scientists, and practicing engineers in the study of switched systems and switching control design and hence rich theoretical results have been obtained (see [3–20] and the references therein).Up to now, most of the studies on switched systems require that all subsystems share a common equilibrium. However, in many real world problems, the assumption that all subsystems share a common equilibrium may not hold and thus the assumption may limit the applicability of stability results. Recently, it is pointed out in [21, 22] that, when subsystems have different equilibria or no equilibrium, a switched system can still exhibit interesting behaviors under appropriate switching laws. Such behaviors are similar to those of a conventional bounded or stable system near an equilibrium point. In this paper, we introduce some boundedness and attractiveness notions to define such behaviors for switched delay systems. Such notions are extensions of the traditional boundedness and attractiveness concepts in [15, 16].In this paper, we focus on the boundedness and attractiveness problems for a simple yet important class of nonlinear switched delay systems, which includes three main contributions. First, we propose the concept of attractor with the switching law and the notation of a functionφ∈Cτ in the neighbourhood of a subset Ω∈Rn which is different from the conventional concept of the distance between two subsets of Rn. Second, we propose sufficient conditions which guarantee that the switched delay system is bounded or attractive. Third, we explicitly construct a switching law and the region in which the solutions remain (boundedness) and explicitly construct the attractor for the switched delay systems.


## 2. Problem Statement

Consider the nonlinear switched delay system(1)x˙=Aγx+fγ(x(t-τ)),γ∈𝕀≜{1,20,…,N},
where x∈Rn, fγ∈C(Rn,Rn), and Aγ∈Rn×n, τ≥0. It is assumed that(H1)
there existα>0,M≥1 such that ∥eAγt∥≤Me-αt;(H2)
there existsL, 0<L<α/M, such that ∥fγ(x)-fγ(y)∥≤L∥x-y∥,x,y∈ℝn, γ∈𝕀;(H3)
Aγx+fγ(x)=0 has only one solution xγ∈ℝn and xγ≠xl for γ≠l, γ,l∈𝕀. Let Γ={xγ(t;0,xj),γ,j∈𝕀,t∈R+}, where
(2)xγ(t;0,xj)=eAγtxj+∫0t‍eAγ(t-u)fγ(xγ(u-τ))du
and denote that D~=max{∥xγ-xj∥,γ≠j,γ,j∈𝕀}, Cτ≜C([-τ,0],Rn), and for φ∈Cτ, define
(3)∥φ∥τ=supθ∈[-τ,0]∥φ(θ)∥.
Let xγ(t;t0,φ)≜xγ(t) be the solution of the initial value problem
(4)x˙γ(t)=Aγxγ+fγ(xγ(t-τ)),xγ(θ)=φ(θ),θ∈[-τ,0].
If φ≡xj especially, denote that xγ(t;t0,xj)≜x~γj(t),j∈𝕀, and let x(t;t0,φ,σ)≜x(t) be the solution of (1) with the switching law σ as
(5)x˙(t)=Aγx+fγ(x(t-τ)),γ∈𝕀,x(θ)=φ(θ),θ∈[-τ,0],
where xt≜x(t+θ),θ∈[-τ,0]. Since all the subsystems are autonomous, we can take t0=0 without loss of generality. Let σ=((t1,i1),…,(tk,ik),…) be the switching law of the switched system (1), where ik∈𝕀 and 0=t0<t1<⋯<tk<⋯; that is, under the switching law σ, the subsystem ik is active during the time interval [tk-1,tk). Suppose that ik≠ik+1 holds, which means that a subsystem will be definitely switched to a different subsystem when the switching triggers.


## 3. Boundedness

In this section, we will study the boundedness problem of the switched delay system (1).Definition 1.
Suppose thatδ>0, Ω⊂Rn, and then a neighborhood of Ω in Cτ is defined as
(6)Bδ(Ω)={φ∣φ∈Cτ,supθ∈[-τ,0]ρ(φ(θ),Ω)≤δ},
where the distance ρ between x∈Rn and Ω is defined as
(7)ρ(x,Ω)=inf{∥x-y∥∣y∈Ω}.Definition 2.
A nonnegative numberμ is called the dwell time of the switching law
(8)σ=((t1,i1),…,(tk,ik),…),
if
(9)μ=inf{|tk+1-tk|,k=1,2,…}
holds.Lemma 3.
Letμ be the dwell time of the switching law σ=((t1,i1),…,(tk,ik),…), and δ>0 is a constant. If μ≥max{τ,(ln(M(δ+D~)/δ))/(α-ML)}, then φ∈Bδ(xi0) implies that xtk∈Bδ(xik), where
(10)xtk=xtk(θ)=x(tk+θ,0,φ,σ),θ∈[-τ,0].Proof.
Suppose thatφ∈Bδ(xi0) and the switching law is defined as σ=((t1,i1),…,(tk,ik),…). Then, we will use induction to prove the result. When m=1, t∈[0,t1], we will prove that
(11)supθ∈[-τ,0]∥x(t1+θ,0,φ,σ)-xi1∥≤δ.
In fact,
(12)∥x(t,0,φ,σ)-xi1∥=∥xi1(t)-xi1∥=∥eAi1tφ(0)+∫0t‍eAi1(t-u)fi1(xi1(u-τ))dud-eAi1txi1-∫0t‍eAi1(t-u)fi1(xi1)du∥≤∥eAi1t∥∥φ(0)-xi1∥+∫0t‍∥eAi1(t-u)∥∥fi1(xi1(u-τ))-fi1(xi1)∥du≤Me-αt∥φ(0)-xi1∥+ML∫0t‍e-α(t-u)∥xi1(u-τ)-xi1∥du.
Then,
(13)eαt∥xi1(t)-xi1∥≤M∥φ(0)-xi1∥+ML∫0t‍eαu∥xi1(u-τ)-xi1∥du≤M∥φ(0)-xi1∥+ML∫0t‍eαu∥xi1(u-τ)-xi1∥du≤M[∥φ(0)-xi0∥+∥xi0-xi1∥]+ML∫0t‍eαu∥xi1(u-τ)-xi1∥du≤M[δ+D~]+ML∫0t‍eαu∥xi1u-xi1∥τdu.
Furthermore, since the right hand side of the inequality is increasing function, we have
(14)eαt∥xi1t-xi1∥τ≤M[δ+D~]+ML∫0t‍eαu∥xi1u-xi1∥τdu,
and then, by applying the Bellman-Gronwall inequality to (14), we have
(15)eαt∥xi1t-xi1∥τ≤M[δ+D~]eMLt,
and thus
(16)∥xi1t-xi1∥τ≤M[δ+D~]e-(α-ML)t.
Let t=t1; we get
(17)∥xt1-xi1∥τ=∥xi1t1-xi1∥τ≤M[δ+D~]e-(α-ML)t1≤δ,
and (11) is proved. Suppose that, when m=k, the result is right; that is, xtk∈Bδ(xik), and then we will prove that xtk+1∈Bδ(xik+1); that is,
(18)supθ∈[-τ,0]∥x(tk+1+θ,0,φ,σ)-xik+1∥≤δ.
In fact, when t∈[tk,tk+1], x(t,0,φ,σ)=xik+1(t-tk,0,xtk), and thus
(19)∥x(t,0,φ,σ)-xik+1∥=∥xik+1(t-tk,0,xtk)-xik+1∥=∥∫0t-tk‍eAik+1(t-tk)xtk(0)d+∫0t-tk‍eAik+1(t-tk-u)fik+1(xik+1(u-τ))dud-eAik+1(t-tk)xik+1d-∫0t-tk‍eAik+1(t-tk-u)fik+1(xik+1)du∥≤∥eAik+1(t-tk)∥∥xtk(0)-xik+1∥+∫0t-tk‍∥eAik+1(t-tk-u)∥dddd×∥fik+1(xik+1(u-τ))-fik+1(xik+1)∥du≤Me-α(t-tk)∥xtk(0)-xik+1∥+ML∫0t-tk‍e-α(t-tk-u)∥xik+1(u-τ)-xik+1∥du,
and then we have
(20)eα(t-tk)∥xik+1(t-tk,0,φ)-xik+1∥≤M∥xtk(0)-xik+1∥+ML∫0t-tk‍eαu∥xik+1(u-τ)-xik+1∥du≤M[∥xtk(0)-xik∥+∥xik-xik+1∥]+ML∫0t-tk‍eαu∥xik+1(u-τ)-xik+1∥du≤M[δ+D~]+ML∫0t-tk‍eαu∥xik+1u-xik+1∥τdu.
Since the right hand side of the inequality is increasing function, we have
(21)eα(t-tk)∥xik+1(t-tk)-xik+1∥τ≤M[δ+D~]+ML∫0t-tk‍eαu∥xik+1u-xik+1∥τdu.
Applying the Bellman-Gronwall inequality to (21) yields
(22)∥xik+1(t-tk)-xik+1∥τ=∥xt-xik+1∥τ≤M[δ+D~]e-(α-ML)(t-tk).
Let t=tk+1, and then we get
(23)∥xtk+1-xik+1∥τ≤M[δ+D~]e-(α-ML)(tk+1-tk)≤δ,
and (18) is proved. The proof is completed.Remark 4.
Lemma3 shows that if the initial function φ is in the δ neighborhood of xi0, then the solution x(t,0,φ,σ) with the switching law σ whose dwell time satisfies μ≥max{τ,(ln(M(δ+D~)/δ))/(α-ML)} has the following property: at the kth switching point, the function x(tk+θ,0,φ,σ)(θ∈[-τ,0]) is in the δ neighborhood of xik.Theorem 5.
Letμ be the dwell time of the switching law σ=((t1,i1),…,(tk,ik),…) and δ>0 is a constant. If μ≥max{τ,ln(M(δ+D~)/δ)/(α-ML)}, then φ∈Bδ(xi0) implies that the solution x(t,0,φ,σ) of (1) satisfies x(t)∈BMδ(Γ).Proof.
We use mathematical induction to prove the result. Whent∈[0,t1],
(24)ρ(x(t,0,φ,σ),Γ)≤∥x(t,0,φ,σ)-x(t,0,xi0,σ)∥=∥xi1(t,0,φ,σ)-x~i1(t,0,xi0)∥=∥xi1(t)-x~i1(t)∥=∥eAi1tφ(0)+∫0t‍eAi1(t-u)fi1(xi1(u-τ))dud-eAi1txi0-∫0t‍eAi1(t-u)fi1(x~i1(u-τ))du∥≤∥eAi1t(φ(0)-xi0)∥+∥∫0t‍eAi1(t-u)(fi1(xi1(u-τ))-fi1(x~i1(u-τ)))du∥≤Me-αt∥φ(0)-xi0∥+LM∫0t‍e-α(t-u)∥xi1(u-τ)-x~i1(u-τ)∥du≤e-αt[Mδ+LM∫0t‍eαu∥xi1u-x~i1u∥τdu],
which implies that
(25)eαt∥xi1t-x~i1t∥τ≤Mδ+LM∫0t‍eαu∥xi1u-x~i1u∥τdu,∥xi1t-x~i1t∥τ≤Mδe-(α-LM)t≤Mδ.
Thus, for t∈[0,t1] and φ∈Bδ,
(26)ρ(x(t,0,φ,σ),Γ)≤Mδ.
Suppose that, when t∈[tk-1,tk], (26) holds. Now we prove that, when t∈[tk,tk+1],
(27)ρ(x(t,0,φ,σ),Γ)≤Mδ.
In fact, by Lemma 3, we know that xtk∈Bδ(xik) and
(28)ρ(x(t,0,φ,σ),Γ)≤∥x(t,0,φ,σ)-x~ik+1ik(t-tk,0,xik)∥=∥xik+1(t-tk,0,xtk,σ)-x~ik+1ik(t-tk,0,xik)∥=∥xik+1(t-tk)-x~ik+1ik(t-tk)∥=∥∫0t-tk‍eAik+1(t-tk)xtk(0)d+∫0t-tk‍eAik+1(t-tk-u)fik+1(xik+1(u-τ))dud-eAik+1(t-tk)xikd-∫0t-tk‍eAik+1(t-tk-u)fik+1(x~ik+1ik(u-τ))du∥≤Me-α(t-tk)∥xtk(0)-xik∥+LM∫0t-tk‍e-α(t-tk-u)∥xik+1(u-τ)-x~ik+1ik(u-τ)∥du≤e-α(t-tk)[Mδ+LM∫0t-tk‍eαu∥xik+1u-x~ik+1uik∥τdu],
which implies that
(29)∥xik+1(t-tk)-x~ik+1ik(t-tk)∥τ≤Mδe-(α-LM)(t-tk)≤Mδ
and thus
(30)ρ(x(t,0,φ,σ),Γ)≤∥xik+1(t-tk)-x~ik+1ik(t-tk)∥τ≤Mδ.
Thus, for all t≥0, we have
(31)ρ(x(t,0,φ,σ),Γ)≤Mδ.
The proof is completed.Remark 6.
(i) The result in Theorem5 shows that if the initial function φ is in the δ neighborhood of xi0, then the solution x(t,0,φ,σ) with the switching law whose dwell time satisfies μ≥max{τ,(ln(M(δ+D~)/δ))/(α-ML)} is in the Mδ neighborhood of the set Γ.
(ii) Theorem5 not only shows the boundedness of the switched system but also gives the region in which the solution will remain.By the same procedure of Lemma3 and Theorem 5, we can study the boundedness of the solution of the following nonlinear switched multidelays system:
(32)x˙=Aγx+∑j=1P‍fγj(x,x(t-τγj)),γ∈𝕀≜{1,2,…,N},
where x∈Rn, fγj∈C(Rn×Rn,Rn), Aγ∈Rn×n, and 0≤τγj≤τ.Assume that(H1)’
there existα>0,M≥1 such that ∥eAγt∥≤Me-αt;(H2)’
there existsL, 0<L<α/2PM, such that ∥fγj(x1,x2)-fγj(y1,y2)∥≤L[∥x1-y1∥+∥x2-y2∥],x1,x2,y1,y2∈Rn, γ∈𝕀, and j=1,2,…,P;(H3)’
Aγx+∑j=1P‍fγj(x,x)=0 has only one solution xγ∈Rn and xj≠xl for j≠l, j,l∈𝕀. Let Γ′={x~ij(t;0,xj),i,j∈𝕀,t∈R+}. Similarly, we can obtain the following.Lemma 7.
Letμ be the dwell time of the switching law σ=((t1,i1),…,(tk,ik),…), and δ>0 is a constant. If μ≥max{τ,(ln(M(δ+D~)/δ))/(α-2PML)}, then φ∈Bδ(xi0) implies that the solution x(t,0,φ,σ) of (32) satisfies xtk∈Bδ(xik).Theorem 8.
Letμ be the dwell time of the switching law σ=((t1,i1),…,(tk,ik),…), and δ>0 is a constant. If μ≥max{τ,(ln(M(δ+D~)/δ))/(α-2PML)}, then φ∈Bδ(xi0) implies that the solution x(t,0,φ,σ) of (32) satisfies x(t)∈BMδ(Γ′) for all t∈R+.


## 4. Attractiveness

In this section, we first give the concept of an attractor of the switched system with switching law, and then we study the existence of the attractor of the nonlinear switched delay system (1). We will establish sufficient conditions which guarantee the existence of the attractor and furthermore we can find out the switching law and the attractor of the switching delay system (1).Definition 9.
A setΩ⊂Rn is called an attractor of the switched system (1) with a switching law σ, if there exists a δ>0 such that, for any ε>0, there exists a T>0, such that, for any φ∈Bδ(Ω)∩Cτ, t≥T implies ρ(x(t,0,φ,σ),Ω)≤ε.Remark 10.
According to Definition9, Ω is an attractor of the switched system (1) with a switching law σ if and only if there exists δ>0 such that φ∈Bδ(Ω) implies
(33)limt→∞x(t,0,φ,σ)=0.
In this section, we will prove that Γ is the attractor of system (1). For convenience, we first show that
(34)φ∈Bδ(Γ)⟹limt→∞x(t,0,φ,σ)=0
is equivalent to
(35)φ∈Bδ(xi)⟹limt→∞x(t,0,φ,σ)=0.Lemma 11.
For any givenε>0, λ>0, there exists T(ε,λ)=ln(MΔ/λε)/(α-ML)>0 such that if φ∈Bε(Γ), φ∈Cτ, then the solution xi(t,0,φ) of the ith subsystem of (1) satisfies xit∈Bλε(xi),t≥T, where Δ≜sup{∥x-y∥,x,y∈Bε(Γ)}., If φ∈Bε(xj) and φ∈Cτ especially, then one can take T(ε,λ)=ln(M(ε+D~)/λε)/(α-ML)>0 such that when t≥T implies the solution of the ith subsystem of (1) xi(t,0,φ) satisfies xit∈Bλε(xi).Proof.
Letxi(t,0,φ)≜xi(t) be the solution of the ith subsystem of (1) and φ∈Bε(Γ), and then
(36)∥xi(t)-xi∥=∥eAitφ(0)+∫0t‍eAi(t-u)fi(xi(u-τ))dud-eAitxi-∫0t‍eAi(t-u)fi(xi)du∥≤∥eAit(φ(0)-xi)∥+∥∫0t‍eAi(t-u)[fi(xi(u-τ))-fi(xi)]du∥≤Me-αt∥φ(0)-xi∥+∫0t‍MLe-α(t-u)∥xi(u-τ)-xi∥du,
which implies that
(37)eαt∥xit-xi∥τ≤M∥φ-xi∥τ+ML∫0t‍eαu∥xiu-xi∥τdu
and thus
(38)∥xit-xi∥τ≤MΔe-(α-ML)t.
So, when t>T=ln(MΔ/λε)/(α-ML), we have
(39)∥xit-xi∥τ≤λεandthusxit∈Bλε(xi).
Ifφ∈Bε(xj) specially, it follows from (37), that
(40)eαt∥xit-xi∥τ≤M∥φ-xi∥τ+ML∫0t‍eαu∥xiu-xi∥τdu≤M[∥φ-xj∥τ+∥xj-xi∥]+ML∫0t‍eαu∥xiu-xi∥τdu≤M[ε+D~]+ML∫0t‍eαu∥xiu-xi∥τdu,
which implies that
(41)∥xit-xi∥τ≤M[ε+D~]e-(α-ML)t
and thus, when t≥T=ln(M(ε+D~)/λε)/(α-ML), we have ∥xit-xi∥τ≤λε; that is, xit∈Bλε(xi). This completes the proof.Theorem 12.
Suppose that the dwell timeμ of the switching law σ={(t1,i1),(t2,i2),…} satisfies t1≥ln(MΔ/λε)/(α-ML). If μ≥max{τ,ln(MΔ/λε)/(α-ML)}, then the set Γ is an attractor of the switched delay system (1).Proof.
From Lemma11, we can see that we only need to prove that, for any φ∈Cτ, φ∈B1(∪i=1Nxi), the solution x(t,0,φ,σ) satisfies limt→∞ρ(x(t,0,φ,σ,),Γ)=0. In fact, we take ε=1 and λ=1/2 in Lemma 11, and then φ∈Cτ and φ∈B1(xi0)(i0∈𝕀) implies that xtk∈B1/2k(xik) if μ≥ln[2M(1+D~)]/(α-ML), k=1,2,… . On the other hand, for t∈[0,t1], we have
(42)ρ(x(t,0,φ,σ),Γ)=ρ(xi1(t,0,φ),Γ)≜ρ(xi1(t),Γ)=inf{∥xi1(t)-y∥,y∈Γ}≤∥xi1(t)-xi1(t,0,xi0)∥=∥xi1(t)-x~i1i0(t)∥=∥eAi1tφ(0)+∫0t‍eAi1(t-u)fi1(xi1(u-τ))dud-eAi1txi0-∫0t‍eAi1(t-u)fi1(x~i1i0(u-τ))du∥≤∥eAi1t(φ(0)-xi0)∥+∥∫0t‍eAi1(t-u)[fi1(xi1(u-τ))-fi1(x~i1i0(u-τ))]du∥≤Me-αt∥φ(0)-xi0∥+∫0t‍MLe-α(t-u)∥xi1(u-τ)-x~i1i0(u-τ)∥du,
which implies that
(43)eαt∥xi1(t)-x~i1i0(t)∥≤M∥φ-xi0∥τ+∫0t‍MLeαu∥xi1u-x~i1ui0∥τdu.
Since the right hand side of the above inequality is an increasing function, we have
(44)eαt∥xi1t-x~i1ti0∥τ≤M∥φ-xi0∥τ+∫0t‍MLeαu∥xi1u-x~i1ui0∥τdu,∥xi1t-x~i1ti0∥τ≤M∥φ-xi0∥τe-(α-ML)t,
and thus
(45)ρ(x(t,0,φ,σ),Γ)≤∥xi1(t)-x~i1(t)∥≤M,
since φ∈B1(xi0). When t∈[t1,t2], since xt1∈B1/2(xi1), we have
(46)ρ(x(t,0,φ,σ),Γ)=ρ(xi2(t-t1,0,xt1),Γ)≜ρ(xi2(t-t1),Γ)=inf{∥xi2(t-t1)-y∥,y∈Γ}≤∥xi2(t-t1)-xi2(t-t1,0,xi1)∥=∥xi2(t-t1)-x~i2i1(t-t1)∥=∥∫0t-t1‍eAi2(t-t1)xt1(0)d+∫0t-t1‍eAi2(t-t1-u)fi2(xi2(u-τ))dud-eAi2txi1-∫0t-t1‍eAi2(t-t1-u)fi2(x~i2i1(u-τ))du∥≤∥eAi2(t-t1)(xt1(0)-xi1)∥+∥∫0t-t1‍eAi2(t-t1-u)ssssddssss×[fi2(xi2(u-τ))-fi2(x~i2i1(u-τ))]du∫0t-t1‍∥≤Me-α(t-t1)∥xt1(0)-xi1∥+∫0t-t1‍MLe-α(t-t1-u)∥xi2(u-τ)-x~i2i1(u-τ)∥du,
which implies that
(47)eα(t-t1)∥xi2t-x~i2ti1∥τ≤M∥xt1-xi1∥τ+∫0t-t1‍MLeαu∥xi2u-x~i2ui1∥τdu,∥xi2t-x~i2ti1∥τ≤M∥xt1-xi1∥τe-(α-ML)(t-t1),
and thus
(48)ρ(x(t,0,φ,σ),Γ)≤∥xi2t-x~i2ti1∥τ≤12M,
since xt1∈B1/2(xi1). Suppose that, for t∈[tk-1,tk],
(49)ρ(x(t,0,φ,σ),Γ)≤12k-1M.
When t∈[tk,tk+1], since xtk∈B1/2k(xik), (take ε=1/2k in Lemma 11), we have
(50)ρ(x(t,0,φ,σ),Γ)=ρ(xik+1(t-tk,0,xtk),Γ)=ρ(xik+1(t-tk),Γ)=inf{∥xik+1(t-tk)-y∥,y∈Γ}≤∥xik+1(t-tk)-xik+1(t-tk,0,xik)∥≜∥xik+1(t-tk)-x~ik+1ik(t-tk)∥=∥∫0t-tk‍eAik+1(t-tk)xtk(0)d+∫0t-tk‍eAik+1(t-tk-u)fik+1(xik+1(u-τ))dud-eAik+1(t-tk)xikd-∫0t-tk‍eAik+1(t-tk-u)fik+1(x~ik+1ik(u-τ))du∥≤Me-α(t-tk)∥xtk(0)-xik∥+ML∫0t-tk‍e-α(t-tk-u)∥xik+1(u-τ)-x~ik+1ik(u-τ)∥du
which implies that
(51)∥xik+1t-x~ik+1tik∥τ≤M∥xtk-xik∥τe-(α-ML)(t-t1)≤12kM
and thus
(52)ρ(x(t,0,φ,σ),Γ)≤12kM,t∈[tk,tk+1].
Therefore, we have
(53)limt→∞ρ(x(t,0,φ,σ),Γ)=0,
which means that Γ is the attractor of system (1) with a switching law σ.
Similarly, we can study the existence of the attractor of the switched delay system (32) with the assumptions (H1)’–(H3)’ and obtain the following results.Lemma 13.
For any givenε>0, λ>0, there exists
(54)T(ε,λ)=max{τ,ln(M(δ+Δ)/δ)α-2PML},
such that if φ∈Bε(Γ′) and φ∈Cτ, then the solution xi(t,0,φ) of the ith subsystem of (32) satisfies xit∈Bλε(xi),t≥T. If φ∈Bε(xj) especially, then one can take T(ε,λ)=ln(M(ε+D~)/λε)/(α-2PML)>0 such that when t≥T implies the solution xi(t,0,φ) of the ith subsystem of (32) satisfies xit∈Bλε(xi),t≥T.Theorem 14.
Suppose that the dwell timeμ of the switching law σ={(t1,i1),(t2,i2),…} satisfies t1≥ln(MΔ/λε)/(α-2PML); if μ≥max{τ,ln(MΔ/λε)/(α-2PML)}, then the set Γ′ is an attractor of the switched delay system (32).



---

*Source: 713847-2013-11-19.xml*