# Exponential Stabilization for Timoshenko Beam with Distributed Delay in the Boundary Control

Authors: Xiu Fang Liu; Gen Qi Xu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider the exponential stabilization for Timoshenko beam with distributed delay in the boundary control. Suppose that the controller outputs are of the formα1u1(t)+β1u1(t-τ)+∫-τ0‍g1(η)u1(t+η)dη and α2u2(t)+β2u2(t-τ)+∫-τ0‍g2(η)u2(t+η)dη; where u1(t) and u2(t) are the inputs of boundary controllers. In the past, most stabilization results for wave equations and Euler-Bernoulli beam with delay are required αi>βi>0,i=1,2. In the present paper, we will give the exponential stabilization about Timoshenko beam with distributed delay and demand to satisfy the lesser conditions for αi,βi,i=1,2.


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## Body

## 1. Introduction

Since the extensive applications of Timoshenko beam in high-Tech, the stabilization problem has been a hot topic in the mathematical control theory and engineering; for instance, see [1–5] and the references therein. In many literature, the control delay problem has been neglected. Due to extensive applications of the system with delay, more and more scholars devoted to study the stabilization of the system with controller delay. It is well known that time delay caused by controller memory usually takes the form∫-∞0dα(s)u(t+s), whereα(s)is a bounded variation function (or matrix-valued function) andu(t)is the control input. If the control is in the spaceLloc2(ℝ), then the memory controller will take the form
(1)∫-τ0‍dα(s)u(t+s)=αu(t)+βu(t-τ)+∫-τ0‍g(s)u(t+s)ds.Based on this reason, Xu et al. (see [6]) studied firstly stabilization of the 1-d wave systems with delay of the formαu(t)+βu(t-τ). They proved that the system with control delay is exponential stable ifα>β>0and unstable ifβ>α. Nicaise and Pignotti in [7] studied the stability and instability of the wave equation with delay in boundary and internal distributed delay. Nicaise and Valein in [8] extend the 1-d wave equation to the networks of 1-d wave equations. Shang et al. in [9] studied Euler-Bernoulli beam and showed thatβ>0is not necessary, but the conditionα>|β|is necessary. For the case of distributed delay, that is,β>0and∫-τ0‍|g(η)|dη≠0, Nicaise and Pignotti in [10] discussed a high dimensional wave equation. Under the condition thatα>∫-τ0‍|g(η)|dη≠0, they proved the velocity feedback control law also stabilizes exponentially the system.From above we see thatα,β, andg(η)are determined by the controller. We cannot determine whether or notα>β>0includingα>|β|in practice. Under the assumption of state being measurable, Shang and Xu in [11] designed a dynamic feedback controller for cantilever Euler-Bernoulli beam that stabilizes exponentially the system for any real|α|≠|β|. Recently Han and Xu in [12] extended this result to the case of output being measurable; they showed that a state observer can realize the state reconstruction from the output of the system. Xu and Wang in [13] discussed the Timoshenko beam with boundary control delay, and they also stabilized the system by a dynamic feedback controller. Note that the difference between [11, 13], one is a system of single input and single output, the other is a system of 2 inputs and 2 outputs. Such discussion will lead us to extend the method to a general system of multiinput and multioutput. So far, however, there is no result for anyα,β, andg(η)about Timoshenko beams. In this paper, we still consider Timoshenko beam with boundary control distributed delay. We will seek for a dynamic feedback control law that exponentially stabilizes the Timoshenko beam with distributed delay under certain conditions.The rest is organized as follows. In Section2, we will describe the design process of controllers, including predict system and generation of signal, and then state the main results of this paper. In Section 3, we will give the representation of the transform system. In Section 4, we will prove our first result on the stabilization of the original system. In Section 5, we will prove the second result on the exponential stabilization of the induced system. In Section 6, we conclude the paper.


## 2. Design of Controllers and Main Results

Letw(x,t)be the displacement andφ(x,t)the rotation angle of the beam. The motion of a cantilever beam is governed by the following partial differential equations:
(2)ρwtt(x,t)-K(wxx-φx)(x,t)=0,x∈(0,1),t>0,Iρφtt(x,t)-EIφxx(x,t)-K(wx-φ)(x,t)=0,x∈(0,1),t>0,w(0,t)=φ(0,t)=0,t>0,K(wx-φ)(1,t)=v1(t),EIφx(1,t)=v2(t),w(x,0)=w0(x),wt(x,0)=w1(x),φ(x,0)=φ0(x),φt(x,0)=φ1(x),wherev1(t)andv2(t)are the control force and torque from the controllers, respectively. If the controllers have no memory, namely, uj(t)=vj(t),j=1,2, whereuj(t)are controller inputs, this model had been studied in [14]. If the controllers have memory, then the Timoshenko beam became
(3)ρwtt(x,t)-K(wxx-φx)(x,t)=0,x∈(0,1),t>0,Iρφtt(x,t)-EIφxx(x,t)-K(wx-φ)(x,t)=0,x∈(0,1),t>0,w(0,t)=φ(0,t)=0,t>0,K(wx-φ)(1,t)=α1u1(t)+β1u1(t-τ)+∫-τ0‍g1(η)u1(t+η)dη,EIφx(1,t)=α2u2(t)+β2u2(t-τ)+∫-τ0‍g2(η)u2(t+η)dη,w(x,0)=w0(x),wt(x,0)=w1(x),φ(x,0)=φ0(x),φt(x,0)=φ1(x),u1(θ)=f1(θ),u2(θ)=f2(θ),θ∈(-τ,0),whereτis the delay time,αi,βi∈ℝ(i=1,2)are the controller parameters, andgi(η)∈L2[-τ,0],j=1,2, andfi(θ),θ∈(-τ,0)(i=1,2)are bounded measurable functions that are memory values of controllers. Whengj≡0,j=1,2, (3) is just the model in [13].We suppose that the state of (3) is measurable; that is,(w(x,t),φ(x,t),wt(x,t),φt(x,t))is measurable. We introduce an auxiliary system as follows:
(4)w^s(x,s,t)=z^(x,s,t),x∈(0,1),s∈(0,τ),φ^s(x,s,t)=ψ^(x,s,t),x∈(0,1),s∈(0,τ),z^s(x,s,t)=Kρ(w^xx(x,s,t)-φ^x(x,s,t)),ψ^s(x,s,t)=EIIρφ^xx(x,s,t)+KIρ(w^x(x,s,t)-φ^(x,s,t)),w^(0,s,t)=φ^(0,s,t)=0,s∈(0,τ),K(w^x-φ^)(1,s,t)=β1u1(t+s-τ)+∫-τ-s‍g1(η)u1(t+s+η)dη,s∈(0,τ),EIφ^x(1,s,t)=β2u2(t+s-τ)+∫-τ-s‍g2(η)u2(t+s+η)dη,s∈(0,τ),t>0,w^(x,0,t)=w(x,t),z^(x,0,t)=wt(x,t),x∈(0,1),φ^(x,0,t)=φ(x,t),ψ^t(x,0,t)=φt(x,t),x∈(0,1),t>0.Equation (4) is a partial state predictor.Denote the state of (4) at the moments=τby
(5)(p1(x,t),p2(x,t),q1(x,t),q2(x,t))=(w^(x,τ,t),φ^(x,τ,t),z^(x,τ,t),ψ^(x,τ,t)).Using (3) we can verify that the functions group (p1(x,t),p2(x,t),q1(x,t),q2(x,t))satisfy the following partial differential equations:
(6)p1,t(x,t)=q1(x,t)+a1(x)u1(t)+a2(x)u2(t),x∈(0,1),t>0,p2,t(x,t)=q2(x,t)+a3(x)u1(t)+a4(x)u2(t),x∈(0,1),t>0,q1,t(x,t)=Kρ(p1,xx-p2,x)(x,t)+b1(x)u1(t)+b2(x)u2(t),q2,t(x,t)=EIIρp2,xx(x,t)+KIρ(p1,x-p2)(x,t)+b3(x)u1(t)+b4(x)u2(t),p1(0,t)=p2(0,t)=q1(0,t)=q2(0,t)=0,t>0,K(p1,x-p2)(1,t)=β1u1(t),t>0,EIp2,x(1,t)=β2u2(t),t>0,p1(x,0)=E1(w0,φ0,w1,φ1)(x)-∫-τ0‍a1(x,s)f1(s)ds-∫-τ0‍a2(x,s)f2(s)ds,p2(x,0)=E2(w0,φ0,w1,φ1)(x)-∫-τ0‍a3(x,s)f1(s)ds-∫-τ0‍a4(x,s)f2(s)ds,q1(x,0)=E3(w0,φ0,w1,φ1)(x)+∫-τ0‍b1(x,s)f1(s)ds+∫-τ0‍b2(x,s)f2(s)ds,q2(x,0)=E4(w0,φ0,w1,φ1)(x)+∫-τ0‍b3(x,s)f1(s)ds+∫-τ0‍b4(x,s)f2(s)ds,whereai(x,r),bi(x,r),ai(x),bi(x)(i=1,2,3,4)are measurable function andEi(i=1,2,3,4)are bounded linear operators on[H1[0,1]×L2[0,1]]2; they are determined later.Equation (6) is a system without delay, but the controls appear in the system interior and boundary. First we consider the stabilization problem of (6). Let us consider the energy functional of (6)
(7)E(t)=12∥(p1,p2)∥VK1(0,1)×VEI1(0,1)2+12∥(q1,q2)∥Lρ2(0,1)×LIρ2(0,1)2=12∫01K|p1,x(x,t)-p2(x,t)|2dx+12∫01EI|p2,x(x,t)|2dx+12∫01ρ|q1(x,t)|2dx+12∫01‍Iρ|q2(x,t)|2dx.A direct calculation gives(8)dE(t)dt=u1(t)[+∫01‍Iρq2(x,t)b3(x)dx]β1q1(1,t)+∫01‍K(p1,x-p2)[a1′(x)-a3(x)]dx+∫01‍EIp2,x(x,t)a3′(x)dx+∫01‍ρq1(x,t)b1(x)dx+∫01‍Iρq2(x,t)b3(x)dx]+u2(t)[+∫01‍Iρq2(x,t)b4(x)dx]β2q2(1,t)+∫01K(p1,x-p2)×[a2′(x)-a4(x)]dx+∫01EIp2,x(x,t)a4′(x)dx+∫01ρq1(x,t)b2(x)dx+∫01‍Iρq2(x,t)b4(x)dx].Set(9)U1(p1,p2,q1,q2)=β1q1(1,t)+∫01‍K(p1,x(x,t)-p2(x,t))(a1′(x)-a3(x))dx+∫01‍ρq1(x,t)b1(x)dx+∫01‍EIp2,x(x,t)a3′(x)dx+∫01‍Iρq2(x,t)b3(x)dx;U2(p1,p2,q1,q2)=β2q2(1,t)+∫01‍K(p1,x(x,t)-p2(x,t))(a2′(x)-a4(x))dx+∫01‍ρq1(x,t)b2(x)dx+∫01‍EIp2,x(x,t)a4′(x)dx+∫01‍Iρq2(x,t)b4(x)dx.We take the feedback control law as(10)u1(t)=-U1(p1,p2,q1,q2),u2(t)=-U2(p1,p2,q1,q2).Then, the closed loop system associated with (6) is
(11)p1,t(x,t)=q1(x,t)-a1(x)U1(p1,p2,q1,q2)-a2(x)U2(p1,p2,q1,q2),x∈(0,1),t>0,p2,t(x,t)=q2(x,t)-a3(x)U1(p1,p2,q1,q2)-a4(x)U2(p1,p2,q1,q2),x∈(0,1),t>0,q1,t(x,t)=Kρ(p1,xx-p2,x)(x,t)-b1(x)U1×(p1,p2,q1,q2)-b2(x)U2(p1,p2,q1,q2),q2,t(x,t)=EIIρp2,xx(x,t)+KIρ(p1,x-p2)(x,t)-b3(x)U1(p1,p2,q1,q2)-b4(x)U2(p1,p2,q1,q2),p1(0,t)=p2(0,t)=q1(0,t)=q2(0,t)=0,t>0,K(p1,x-p2)(1,t)=-β1U1(p1,p2,q1,q2),t>0,EIp2,xx(1,t)=-β2U2(p1,p2,q1,q2),t>0,p1(x,0)=E1(w0,φ0,w1,φ1)(x)-∫-τ0‍a1(x,s)f1(s)ds-∫-τ0‍a2(x,s)f1(s)ds,p2(x,0)=E2(w0,φ0,w1,φ1)(x)-∫-τ0‍a3(x,s)f1(s)ds-∫-τ0‍a4(x,s)f1(s)ds,q1(x,0)=E3(w0,φ0,w1,φ1)(x)+∫-τ0‍b1(x,s)f1(s)ds+∫-τ0‍b2(x,s)f1(s)ds,q2(x,0)=E4(w0,φ0,w1,φ1)(x)+∫-τ0‍b3(x,s)f1(s)ds+∫-τ0‍b4(x,s)f1(s)ds.We estimate the error of the system (3) with control (10) and the system (11).Let(w(x,t),φ(x,t),wt(x,t),φt(x,t))be the solution to (3) with control signals (10) and let function group(p1(x,t),p2(x,t),q1(x,t),q2(x,t))be the solution to (11). SetW(x,t)=(w(x,t),φ(x,t))andWt(x,t)=(wt(x,t),φt(x,t)), and setP(x,t)=(p1(x,t),p2(x,t))andQ(x,t)=(q1(x,t),q2(x,t)).To discuss the stability(W(x,t),Wt(x,t)), we consider the error both solutions in the energy space
(12)∥P(·,t)-W(·,t+τ)∥VK1(0,1)×VEI1(0,1)2+∥Q(·,t)-Wt(·,t+τ)∥Lρ2(0,1)×LIρ2(0,1)2.In this paper, we will prove the following results.Theorem 1.
Let(W(x,t),Wt(x,t))be the solution to (3) with controls (10) and let (P(x,t),Q(x,t))be the solution to the closed-loop system (11). If the system (11) is asymptotically (exponentially) stable, then the system (3) also is asymptotically (exponentially) stable.Theorem 2.
Suppose thatK/ρ≠EI/Iρ. Letμn,n∈ℕ be the eigenvalues of the free system (the system (2) without controls). Set
(13)ξn(1)=∫-τ0‍g1(η)e-iμn(τ+η)dη,ξn(2)=∫-τ0‍g2(η)e-iμn(τ+η)dη.
Then the following assertions are true:(1)
when(14)infn|β1ρ+α1e-iτμn+ξn(1)|>0,infn|β2Iρ+α2e-iτμn+ξn(2)|>0,the system (11) is exponentially stable;(2)
if for alln∈ℕ,
(15)|β1ρ+α1e-iτμn+ξn(1)|>0,|β2Iρ+α2e-iτμn+ξn(2)|>0but
(16)infn|β1ρ+α1e-iτμn+ξn(1)|=0,infn|β2Iρ+α2e-iτμn+ξn(2)|=0,
then the system (11) is asymptotically stable.In the following sections, we will prove our results. In Section3, we will determine functionsai(x),bi(x),ai(x,s),bi(x,s)(i=1,2,3,4). In Section 4, we will prove Theorem 1. In Section 5, we pay our attention to the proof of Theorem 2.


## 3. Representation of the System (6)

In this section, we will obtain the expressions for the functionsai(x),bi(x),ai(x,s),bi(x,s)(i=1,2,3,4)appearing in system (6) using (3) and (4).We begin with introducing two useful lemmas.Lemma 3 (see [13]).
Define the differential operator inLρ2(0,1)×LIρ2(0,1)as follows:
(17)ℒ(w,φ)=(-Kρ(w′′(x)-φ′(x)),-EIIρφ′′(x)-KIρ(w′(x)-φ(x)))T,
with domain
(18)𝒟(ℒ)={(w(x),φ(x))∈H2(0,1)×H2(0,1)∣w(0)=φ(0)=0,K(w′(1)-φ(1))=0,EIφ′(1)=0}.
Thenℒis a positive define operator with compact resolvent inLρ2(0,1)×LIρ2(0,1); its eigenvalues are
(19)0<μ1<μ2<⋯<μn<⋯
and the eigenfunctionsΦn(x)=(wn(x),φn(x))Tcorresponding toμnare real functions and form a normalized orthogonal basis forLρ2(0,1)×LIρ2(0,1).Lemma 4 (see [13]).
LetΦn(x)=(wn(x),φn(x))be the normalized eigenfunction corresponding to the eigenvalueμnofℒ. Then it holds that
(20)∫01K|ωn′(x)-φn(x)|2dx+∫01EI|φn′(x)|2dx=μn,0<infn{ρ|wn(1)|2+|Iρφn(1)|2}≤supn{ρ|wn(1)|2+|Iρφn(1)|2}<∞.Now let us return to (3). We write the equation in (3) into the vector form
(21)(wtt(x,t)φtt(x,t))-(Kρ∂xx-Kρ∂xKIρ∂xEIIρ∂xx-KIρ)(w(x,t)φ(x,t))=0and the boundary conditions are(w(0,t)φ(0,t))=0, and
(22)(K∂x-K0EI∂x)(w(x,t)φ(x,t))x=1=(α100α2)(u1(t)u2(t))+(β100β2)(u1(t-τ)u2(t-τ))+∫-τ0(g1(η)00g2(η))(u1(t+η)u2(t+η))dη.The initial datum are(23)(w(x,0)φ(x,0))=(w0(x)φ0(x)),(wt(x,0)φt(x,0))=(w1(x)φ1(x)).SetW(x,t)=(w(x,t),φ(x,t))TandU(t)=(u1(t),u2(t))T. Define2×2matrices
(24)Λ1=(α100α2),Λ2=(β100β2),Λ3(η)=(g1(η)00g2(η))and define an operatorBfromℝ2toH-1(0,1)×H-1(0,1), whereH-1(0,1)=(Vω1(0,1))*is dual space,
(25)B=(δ(x-1)00δ(x-1)),and define an operatorΓNfromH2(0,1)×H2(0,1)toℝ2by
(26)ΓNW=(K(w′(1)-φ(1))EIφ′(1)),whereW(x)=(w(x),φ(x))T.With help of these notations, we can rewrite (3) into
(27)Wtt(x,t)+ℒW(x,t)=B(Λ1U(t)+Λ2U(t-τ)+∫-τ0‍Λ3(η)U(t+η)dη),t>0,W(0,t)=0,ΓNW(1,t)=0,W(x,0)=W0(x)=(w0(x),φ0(x))T,Wt(x,0)=W1(x)=(w1(x),φ1(x))T,and (4) into
(28)W^s(x,s,t)=V^(x,s,t),V^s(x,s,t)+ℒW^(x,s,t)=BΛ2U(t+s-τ)+B∫-τ-s‍Λ3(η)U(t+s+η)dη,W^(0,s,t)=0,ΓNW^(1,s,t)=0,W^(x,0,t)=W(x,t),V^(x,0,t)=Wt(x,t),whereW^(x,s,t)=(w^(x,s,t),φ^(x,s,t))T,V^(x,s,t)=(z^(x,s,t),ψ^(x,s,t))T.We define two families of the bounded linear operators onLρ2(0,1)×LIρ2(0,1)by
(29)Cos(tℒ)F=∑n=1∞‍cosμnt(F,Φn)Lρ2×LIρ2Φn,Sin(tℒ)F=∑n=1∞sinμntμn(F,Φn)Lρ2×LIρ2Φn.Clearly, the following equalities hold, for anyt∈ℝ,
(30)Sin(tℒ)=∫0t‍Cos(tℒ)dt,ddt(Cos(tℒ))=-ℒSin(tℒ).It is easy to know that the vector-valued function(31)W(x,t)=Cos(tℒ)W0+Sin(tℒ)W1+∫0tSin((t-s)ℒ)B[Λ1U(s)+Λ2U(s-τ)]ds+∫0tSin((t-s)ℒ)B∫-τ0Λ3(η)U(s+η)dηdsis differentiable with respect totand
(32)Wt(x,t)=-ℒSin(tℒ)W0+Cos(tℒ)W1+∫0tCos((t-s)ℒ)B[Λ1U(s)+Λ2U(s-τ)]ds+∫0tCos((t-s)ℒ)B∫-τ0‍Λ3(η)U(s+η)dηds.Further,W(x,t)satisfies (27).Similarly, we know the vector-valued function(33)W^(x,s,t)=Cos(sℒ)W(·,t)+Sin(sℒ)Wt(·,t)+∫0sSin((s-r)ℒ)BΛ2U(t+r-τ)dr+∫0sSin((s-r)ℒ)B∫-τ-rΛ3(η)U(t+r+η)dηdr,(34)V^(x,s,t)=-ℒSin(sℒ)W(·,t)+Cos(sℒ)Wt(·,t)+∫0sCos((s-r)ℒ)BΛ2U(t+r-τ)dr+∫0sCos((s-r)ℒ)B∫-τ-rΛ3(η)U(t+r+η)dηdr.satisfy (28).Set(35)P(x,t)=W^(x,τ,t),Q(x,t)=V^(x,τ,t).Then we have(36)(P(x,t)Q(x,t))=(Cos(τℒ)Sin(τℒ)-ℒSin(τℒ)Cos(τℒ))(W(x,t)Wt(x,t))+∫0τ(Sin((τ-r)ℒ)Cos((τ-r)ℒ))BΛ2U(t+r-τ)dr+∫0τ(Sin((τ-r)ℒ)Cos((τ-r)ℒ))×B∫-τ-r‍Λ3(η)U(t+r+η)dηdr.Thus,(37)(Pt(x,t)Qt(x,t))=(Cos(τℒ)Sin(τℒ)-ℒSin(τℒ)Cos(τℒ))(Wt(x,t)Wtt(x,t))+∫0τ(Cos((τ-r)ℒ)-ℒSin((τ-r)ℒ))×BΛ2U(t+r-τ)dr+(0BΛ2U(t))-(Sin(τℒ)Cos(τℒ))BΛ2U(t-τ)-(Sin(τℒ)Cos(τℒ))B∫-τ0‍Λ3(η)U(t+η)dη+∫0τ(Cos((τ-r)ℒ)-ℒSin((τ-r)ℒ))B×∫-τ-rΛ3(η)U(t+r+η)dηdr+∫-τ0(Sin((τ+η)ℒ)Cos((τ+η)ℒ))BΛ3(η)U(t)dη.Note that(38)(Wt(x,t)Wtt(x,t))=(0I-ℒ0)(W(x,t)Wt(x,t))+(0B(Λ1U(t)+Λ2U(t-τ)+∫-τ0Λ3(η)U(t+η)dη)).So it holds that(39)(Pt(x,t)Qt(x,t))=(0I-ℒ0)(P(x,t)Q(x,t))+(Sin(τℒ)BΛ1U(t)+∫-τ0Sin((τ+η)ℒ)BΛ3(η)U(t)dηCos(τℒ)BΛ1U(t)+∫-τ0Cos((τ+η)ℒ)BΛ3(η)U(t)dη+BΛ2U(t)).Therefore, we have equations(40)Pt(x,t)=Q(x,t)+Sin(τℒ)BΛ1U(t)+∫-τ0‍Sin((τ+η)ℒ)BΛ3(η)U(t)dη,Qt(x,t)=-ℒP(x,t)+Cos(τℒ)BΛ1U(t)+∫-τ0‍Cos((τ+η)ℒ)BΛ3(η)U(t)dη,P(0,t)=Q(0,t)=0,ΓNP(·,t)=Λ2U(t)and initial conditions(41)P(x,0)=Cos(τℒ)W0+Sin(τℒ)W1-∫-τ0Sin(sℒ)BΛ2f(s)ds+∫-τ0∫-τsSin((τ-s+η)ℒ)×BΛ3(η)f(s)dηds,(42)Q(x,0)=-ℒSin(τℒ)W0+Cos(τℒ)W1+∫-τ0Cos(sℒ)BΛ2f(s)ds+∫-τ0∫-τsCos((τ-s+η)ℒ)BΛ3(η)f(s)dηds,wheref(s)=(f1(s),f2(s))T.Since all entries ofBare meaningful as linear functional onH1(0,1), so for anyZ=(z1,z2)∈ℝ2andΦn(x)∈H1(0,1)×H1(0,1),
(43)(BZ,Φn)Lρ2(0,1)×LIρ2(0,1)=z1∫01ρδ(x-1)wn(x)dx+z2∫01Iρδ(x-1)φn(x)dx=ρz1wn(1)+Iρz2φn(1)=[z1,z2][ρwn(1),φn(1)]T.Therefore, we have the following results.Theorem 5.
Let{μn;n∈ℕ} be the list of all eigenvalues of ℒ. Then the functions that appear in (6) are
(44)a1(x)=ρ∑n=1∞wn(1)wn(x)×(α1sinτμnμn+∫-τ0‍g1(η)sin(τ+η)μnμndη),a2(x)=Iρ∑n=1∞‍φn(1)wn(x)×(α2sinτμnμn+∫-τ0‍g2(η)sin(τ+η)μnμndη),a3(x)=ρ∑n=1∞‍wn(1)φn(x)×(α1sinτμnμn+∫-τ0‍g1(η)sin(τ+η)μnμndη),a4(x)=Iρ∑n=1∞‍φn(1)φn(x)×(α2sinτμnμn+∫-τ0‍g2(η)sin(τ+η)μnμndη),b1(x)=ρ∑n=1∞‍wn(1)wn(x)×(+∫-τ0‍g1(η)cos(τ+η)μndη)α1cosτμn+∫-τ0‍g1(η)cos(τ+η)μndη),b2(x)=Iρ∑n=1∞‍φn(1)wn(x)×(+∫-τ0‍g2(η)cos(τ+η)μndη)α2cosτμn+∫-τ0‍g2(η)cos(τ+η)μndη),b3(x)=ρ∑n=1∞‍wn(1)φn(x)×(+∫-τ0‍g1(η)cos(τ+η)μndη)α1cosτμn+∫-τ0‍g1(η)cos(τ+η)μndη),b4(x)=Iρ∑n=1∞‍φn(1)φn(x)×(+∫-τ0‍g2(η)cos(τ+η)μndη)α2cosτμn+∫-τ0‍g2(η)cos(τ+η)μndη),a1(x,s)=ρ∑n=1∞‍wn(1)wn(x)×(β1sinsμnμn-∫-τs‍g1(η)sin(τ-s+η)μnμndη),a2(x,s)=Iρ∑n=1∞‍φn(1)wn(x)×(β2sinsμnμn-∫-τs‍g2(η)sin(τ-s+η)μnμndη),a3(x,s)=ρ∑n=1∞‍wn(1)φn(x)×(β1sinsμnμn-∫-τs‍g1(η)sin(τ-s+η)μnμndη),a4(x,s)=Iρ∑n=1∞‍φn(1)φn(x)×(β2sinsμnμn-∫-τs‍g2(η)sin(τ-s+η)μnμndη),b1(x,s)=ρ∑n=1∞‍wn(1)wn(x)×(+∫-τs‍g1(η)cos(τ-s+η)μndη)β1cossμn+∫-τs‍g1(η)cos(τ-s+η)μndη),b2(x,s)=Iρ∑n=1∞‍φn(1)wn(x)×(+∫-τs‍g2(η)cos(τ-s+η)μndη)β2cossμn+∫-τs‍g2(η)cos(τ-s+η)μndη),b3(x,s)=ρ∑n=1∞‍wn(1)φn(x)×(+∫-τs‍g2(η)cos(τ-s+η)μndη)β1cossμn+∫-τs‍g1(η)cos(τ-s+η)μndη),b4(x,s)=Iρ∑n=1∞‍φn(1)φn(x)×(+∫-τs‍g2(η)cos(τ-s+η)μndη)β2cossμn+∫-τs‍g2(η)cos(τ-s+η)μndη),and the linear operators are(45)E1(ω0,φ0,ω1,φ1)(x)=∑n=1∞‍[cosτμn(W0,Φn)+sinτμnμn(W1,Φn)]wn(x),E2(ω0,φ0,ω1,φ1)(x)=∑n=1∞‍[+sinτμnμn(W1,Φn)]cosτμn(W0,Φn)+sinτμnμn(W1,Φn)]φn(x),E3(ω0,φ0,ω1,φ1)(x)=∑n=1∞‍[-μnsinτμn(W0,Φn)+cosτμn(W1,Φn)]wn(x),E4(ω0,φ0,ω1,φ1)(x)=∑n=1∞‍[-μnsinτμn(W0,Φn)+cosτμn(W1,Φn)]φn(x),


## 4. The Proof of Theorem1

In this section, we will prove Theorem1. Here we mainly estimate the error:
(46)∥P(·,t)-W(·,t+τ)∥VK1(0,1)×VEI1(0,1)2+∥Q(·,t)-Wt(·,t+τ)∥Lρ2(0,1)×LIρ2(0,1)2.According to the calculation in Section3, we have
(47)P(x,t)-W(x,t+τ)=-∫0τ‍Sin((τ-r)ℒ)BΛ1U(t+r)dr-∫0τ‍Sin((τ-r)ℒ)B∫-τ0‍Λ3(η)U(t+r+η)dηdr,(48)Q(x,t)-Wt(x,t+τ)=-∫0τ‍Cos((τ-r)ℒ)BΛ1U(t+r)dr-∫0τ‍Cos((τ-r)ℒ)B∫-τ0‍Λ3(η)U(t+r+η)dηdr.So,(49)∥P(·,t)-W(·,t+τ)∥VK1(0,1)×VEI1(0,1)2+∥Q(·,t)-Wt(·,t+τ)∥Lρ2(0,1)×LIρ2(0,1)2≤4(α1ρ)2∑n=1∞|ωn(1)|2×|∫0τsinμn(τ-r)u1(r+t)dr|2+4(α2Iρ)2∑n=1∞|φn(1)|2×|∫0τsinμn(τ-r)u2(r+t)dr|2+4ρ2∑n=1∞|ωn(1)|2×|∫0τsinμn(τ-r)(∫-τ0‍g1(η)u1(r+t+η)dη)dr|2+4Iρ2∑n=1∞|φn(1)|2×|∫0τsinμn(τ-r)(∫-τ0‍g2(η)u2(r+t+η)dη)dr|2+4(α1ρ)2∑n=1∞|ωn(1)|2×|∫0τcosμn(τ-r)u1(r+t)dr|2+4(α2Iρ)2∑n=1∞|φn(1)|2×|∫0τcosμn(τ-r)u2(r+t)dr|2+4ρ2∑n=1∞|ωn(1)|2×|∫0τcosμn(τ-r)(∫-τ0g1(η)u1(r+t+η)dη)dr|2+4Iρ2∑n=1∞|φn(1)|2×|∫0τcosμn(τ-r)(∫-τ0g2(η)u2(r+t+η)dη)dr|2.Note that{cosμnt,sinμnt;n∈ℕ}is a Riesz basis sequence forL2[0,τ]. Thus, there exist positive constantsMi(i=1,2,3,4)such that
(50)∥P(·,t)-W(·,t+τ)∥VK1(0,1)×VEI1(0,1)2+∥Q(·,t)-Wt(·,t+τ)∥Lρ2(0,1)×LIρ2(0,1)2≤M12∫0τ|u1(r+t)|2dr+M22∫0τ|u2(r+t)|2dr+M32∫0τ|∫-τ0‍g1(η)u1(t+r+η)dη|2dr+M42∫0τ|∫-τ0‍g2(η)u2(t+r+η)dη|2dr≤M12∫0τ|u1(r+t)|2dr+M22∫0τ|u2(r+t)|2dr+M32τ2∫-τ0|g1(η)|2dη∫t-τt+τ|u1(s)|2ds+M42τ2∫-τ0|g2(η)|2dη∫t-τt+τ|u2(s)|2ds.Let(P(x,t),Q(x,t)) be the solution to (11), and E(t)be its energy functional; then we have
(51)E(t)=12∥(P(x,t),Q(x,t))∥ℋ2,dE(t)dt=-U12(P,Q)(t)-U22(P,Q)(t).Therefore, we have(52)∫0τ‍∥U(t+r)∥ℝ22dr=∫tt+τ(U12(P,Q)(r)+U22(P,Q)(r))dr=E(t)-E(t+τ),∫0τ∥U(t-r)∥ℝ22dr=E(t-τ)-E(t).So, we can get(53)∥P(·,t)-W(·,t+τ)∥VK1(0,1)×VEI1(0,1)2+∥Q(·,t)-Wt(·,t+τ)∥Lρ2(0,1)×LIρ2(0,1)2≤maxi=1,2{Mi2}[E(t)-E(t+τ)]+maxj=3,4{Mj2}τ2∫-τ0‍(|g1(η)|2+|g2(η)|2)dη×[E(t-τ)-E(t+τ)].If(P(x,t),Q(x,t))is exponential stable, there exists a positive constantε>0such thatE(t)≤E(0)e-εt.We can obtain the following result from above:
(54)∥P(·,t)-W(·,t+τ)∥VK1(0,1)×VEI1(0,1)2+∥Q(·,t)-Wt(·,t+τ)∥Lρ2(0,1)×LIρ2(0,1)2≤Me-ε(t-τ),whereMis a positive constant. So(W(x,t),Wt(x,t))also decays exponentially.


## 5. The Proof of Theorem2

In this section, we will discuss the stability of system (11). At first we considerL2well posed of the system (6). For the sake of simplicity, we use the vector form of (6); that is,
(55)Pt(x,t)=Q(x,t)+Sin(τℒ)BΛ1U(t)+∫-τ0Sin((τ+η)ℒ)BΛ3(η)U(t)dη,Qt(x,t)=-ℒP(x,t)+Cos(τℒ)BΛ1U(t)+∫-τ0Cos((τ+η)ℒ)BΛ3(η)U(t)dη,ΓNP(·,t)=Λ2U(t),P(0,t)=Q(0,t)=0,P(x,0)=P0(x),Q(x,0)=Q0(x).The observation system corresponding to (55) is
(56)Wt(x,t)=V(x,t),x∈(0,1),t>0,Vt(x,t)=-ℒW(x,t),x∈(0,1),t>0,W(x,0)=W0,V(x,0)=V0,y1(t)=β1z(1,t)+∫01K(wx(x,t)-φ(x,t))×(a1′(x)-a3(x))dx+∫01EIφx(x,t)a3′(x)dx+∫01ρz(x,t)b1(x)dx+∫01Iρψ(x,t)b3(x)dx;y2(t)=β2ψ(1,t)+∫01K(wx(x,t)-φ(x,t))×(a2′(x)-a4(x))dx+∫01‍EIφx(x,t)a4′(x)dx+∫01ρz(x,t)b2(x)dx+∫01Iρψ(x,t)b4(x)dx,whereW(x,t)=(w(x,t),φ(x,t))TandV(x,t)=(z(x,t),ψ(x,t))T.We can write the observation as(57)y1(t)=β1z(1,t)+〈W(·,t),(a1,a3)T〉Vk1×VEI1+〈V(·,t),(b1,b3)T〉Lρ2×LIρ2,y2(t)=β1ψ(1,t)+〈W(·,t),(a2,a4)T〉Vk1×VEI1+〈V(·,t),(b2,b4)T〉Lρ2×LIρ2.Since(58)(a1(x)a2(x)a3(x)a4(x))=Sin(τℒ)BΛ1+∫-τ0Sin((τ+η)ℒ)BΛ3(η)dη,(b1(x)b2(x)b3(x)b4(x))=Cos(τℒ)BΛ1+∫-τ0‍Cos((τ+η)ℒ)BΛ3(η)dη,Y(t)=C(W,V)=Λ2V(1,t)+[ℒ1/2∫-τ0sin((τ+η)ℒ)BΛ3(η)dη]Λ1ℒ1/2Sin(τℒ)B+ℒ1/2∫-τ0Sin((τ+η)ℒ)BΛ3(η)dη]*ℒ1/2W(t)+[+∫-τ0cos((τ+η)ℒ)BΛ3(η)dη]Λ1Cos(τℒ)B+∫-τ0Cos((τ+η)ℒ)BΛ3(η)dη]*V(t).Taking the Laplace transform for above equation leads to, for anyℜλ>0,
(59)λP(x)=Q(x)+Sin(τℒ)BΛ1U(λ)+∫-τ0Sin((τ+η)ℒ)BΛ3(η)U(λ)dη,x∈(0,1),λQ(x)=-ℒP(x)+Cos(τℒ)BΛ1U(λ)+∫-τ0Cos((τ+η)ℒ)BΛ3(η)U(λ)dη,ΓNP(·,t)=Λ2U(λ),P(0,t)=Q(0,t)=0,Y(λ)=Λ2Q(1)+[∫-τ0sin((τ+η)ℒ)BΛ3(η)dη]Λ1ℒ1/2Sin(τℒ)B+ℒ1/2∫-τ0Sin((τ+η)ℒ)BΛ3(η)dη]*ℒ1/2P(·)+[∫-τ0cos((τ+η)ℒ)BΛ3(η)dη]Λ1Cos(τℒ)B+∫-τ0Cos((τ+η)ℒ)BΛ3(η)dη]*Q(·).We have the following results by solving (59):
(60)(λ2+ℒ)p(x)=[+BΛ2[λ(sin(τℒ)BΛ1+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)]λ(+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)Sin(τℒ)BΛ1+∫-τ0Sin((τ+η)ℒ)BΛ3(η)dη)+Cos(τℒ)BΛ1+∫-τ0Cos((τ+η)ℒ)BΛ3(η)dη+BΛ2[λ(sin(τℒ)BΛ1+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)]U(λ),Y(λ)=[λ(+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)Sin(τℒ)BΛ1+∫-τ0Sin((τ+η)ℒ)BΛ3(η)dη)+Cos(τℒ)BΛ1+∫-τ0Cos((τ+η)ℒ)BΛ3(η)dη+BΛ2[λ(sin(τℒ)BΛ1+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)]*P.So we can get(61)Y(λ)=[λ(+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)sin(τℒ)BΛ1+∫-τ0Sin((τ+η)ℒ)BΛ3(η)dη)+Cos(τℒ)BΛ1+∫-τ0Cos((τ+η)ℒ)BΛ3(η)dη+BΛ2[λ(sin(τℒ)BΛ1+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)]*(λ2+ℒ)-1×[λ(+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)Sin(τℒ)BΛ1+∫-τ0Sin((τ+η)ℒ)BΛ3(η)dη)+Cos(τℒ)BΛ1+∫-τ0Cos((τ+η)ℒ)BΛ3(η)dη+BΛ2[λ(sin(τℒ)BΛ1+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)]U(λ),and hence the transform matrix is(62)H(λ)=[λ(+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)Sin(τℒ)BΛ1+∫-τ0Sin((τ+η)ℒ)BΛ3(η)dη)+Cos(τℒ)BΛ1+∫-τ0Cos((τ+η)ℒ)BΛ3(η)dη+BΛ2[λ(sin(τℒ)BΛ1+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)]*(λ2+ℒ)-1×[λ(+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)Sin(τℒ)BΛ1+∫-τ0Sin((τ+η)ℒ)BΛ3(η)dη)+Cos(τℒ)BΛ1+∫-τ0Cos((τ+η)ℒ)BΛ3(η)dη+BΛ2[λ(sin(τℒ)BΛ1+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)].For anyZ=(z1,z2)∈ℂ2, we can get
(63)(H(λ)Z,Z)ℂ2=∑n=1∞1λ2+μn×|BΛ3(η)dη+BΛ2[λ(sin(τℒ)BΛ1+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)]Z,Φn)Lρ2×LIρ2|([λ(+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)Sin(τℒ)BΛ1+∫-τ0Sin((τ+η)ℒ)BΛ3(η)dη)+Cos(τℒ)BΛ1+∫-τ0Cos((τ+η)ℒ)×BΛ3(η)dη+BΛ2[λ(sin(τℒ)BΛ1+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)]Z,Φn)Lρ2×LIρ2|2.We can easily get(64)(BΛ2Z,Φn)Lρ2×LIρ2=β1z1ρwn(1)+β2z2Iρφn(1),(Sin(τℒ)BΛ1Z,Φn)Lρ2×LIρ2=Sin(τμn)μn[α1z1ρwn(1)+α2z2Iρφn(1)],(Cos(τℒ)BΛ1Z,Φn)Lρ2×LIρ2=Cos(τμn)[α1z1ρwn(1)+α2z2Iρφn(1)],(∫-τ0‍Sin(τ+η)ℒBΛ3(η)dηZ,Φn)Lρ2×LIρ2=∫-τ0‍Sin((τ+η)μn)μn×[g1(η)z1ρwn(1)+g2(η)z2Iρφn(1)]dη,(∫-τ0‍Cos(τ+η)ℒBΛ3(η)dηZ,Φn)Lρ2×LIρ2=∫-τ0‍Cos((τ+η)μn)×[g1(η)z1ρwn(1)+g2(η)z2Iρφn(1)]dη.Thus, we have(65)|BΛ2[λ(Sin(τℒ)BΛ1+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)]Z,Φn)Lρ2×LIρ2|([λ(Sin(τℒ)BΛ1+∫-τ0Sin((τ+η)ℒ)BΛ3(η)dη)+Cos(τℒ)BΛ1+∫-τ0Cos((τ+η)ℒ)BΛ3(η)dη+BΛ2[λ(Sin(τℒ)BΛ1+∫-τ0sin((τ+η)ℒ)BΛ3(η)dη)]Z,Φn)Lρ2×LIρ2|2≤F||λ|2μn+2|[ρ|ωn(1)|2+Iρ|φn(1)|2]∥Z∥ℂ22,whereFis a positive constant dependent onαi,βi,gi(η),i=1,2. Therefore, we have the following result:
(66)∥H(λ)∥≤F∑n=1∞1|λ|2+μn||λ|2μn+2|×[ρ|ωn(1)|2+Iρ|φn(1)|2].From Lemma4, we have
(67)supℜλ>δ>0∥H(λ)∥<∞.Hence the system (6) isLloc2well posed (see, [15]).Next, we consider the exact observability of the system (6).Lemma 6 (see [16]).
Letℋ be a separable Hilbert space, and let ℒ be a unbounded positive definite operator. Assume that ℒ satisfies the following conditions:(1)
ℒ has compact resolvent and its spectrum isσ(ℒ)={μn;n∈ℕ};(2)
the spectra ofℒ satisfy the separable condition
(68)infn≠m|μn-μm|=δ>0;(3)
the corresponding eigenvectors{Φn;n∈ℕ} with ∥Φn∥ℋ=1 form a normalized orthogonal basis for ℋ.
Let𝕐 be a Hilbert space. Assume that C:D(ℒ)→𝕐 is an admissible observation operator for ℒ. Then the following system:
(69)Ztt+ℒZ(t)=0,Z(0)=Z0,Zt(0)=Z1,Y(t)=C(Z,Zt)
is exactly observable in finite time in the energy space D(ℒ1/2)×ℋ if and only if
(70)infn∈ℕ∥C(Φniμn,Φn)∥𝕐>0.Now we apply Lemma6 to the system (55). We can easily know that the condition (68) is fulfilled whenc1=K/ρ≠c2=EI/Iρ (see Remark 2.1 in [8]).ForΦn(x)=(wn(x),φn(x))T, we have
(71)C(Φniμn,Φn)=Λ2Φn(1)+[ℒ1/2∫-τ0sin((τ+η)ℒ)BΛ3(η)dη]Λ1ℒ1/2Sin(τℒ)B+ℒ1/2∫-τ0Sin((τ+η)ℒ)BΛ3(η)dη]*ℒ1/2Φniμn+[∫-τ0cos((τ+η)ℒ)BΛ3(η)dη]Λ1Cos(τℒ)B+∫-τ0Cos((τ+η)ℒ)BΛ3(η)dη]*Φn.For anyZ=(z1,z2)∈ℂ2, we have
(72)(Λ1(ℒ1/2Sin(τℒ)B)*ℒ1/2Φniμn,Z)ℂ2=(ℒ1/2Φniμn,ℒ1/2Sin(τℒ)BΛ1Z)Lρ2×LIρ2=-iμn(Φn,Sin(τℒ)BΛ1Z)Lρ2×LIρ2=-isin(μnτ)[α1z1ρωn(1)+α2z2Iρφn(1)]¯.Similarity, we have(73)((ℒ1/2∫-τ0‍Sin((τ+η)ℒ)BΛ3(η)dη)*ℒ1/2Φniμn,Z)ℂ2=-i∫-τ0‍sin((τ+η)μn)×[g1(η)z1ρωn(1)+g2(η)z2Iρφn(1)]¯dη,(Λ1(Cos(τℒ)B)*Φn,Z)ℂ2=cos(μnτ)[α1z1ρωn(1)+α2z2Iρφn(1)]¯,((∫-τ0Cos((τ+η)ℒ)BΛ3(η)dη)*Φn,Z)ℂ2=∫-τ0cos((τ+η)μn)×[g1(η)z1ρωn(1)+g2(η)z2Iρφn(1)]¯dη.Thus it holds that(74)C(Φniμn,Φn)=((β1ρ+α1e-iτμn+∫-τ0g1(η)e-i((τ+η)μn)dη)ρwn(1)(β2Iρ+α2e-iτμn+∫-τ0g2(η)e-i((τ+η)μn)dη)Iρφn(1)).Set(75)ξn(1)=∫-τ0g1(η)e-iμn(τ+η)dη,ξn(2)=∫-τ0g2(η)e-iμn(τ+η)dη.Then(76)∥C(Φniμn,Φn)∥ℂ22=|β1ρ+α1e-iτμn+ξn(1)|2|ρwn(1)|2+|β2Iρ+α2e-iτμn+ξn(2)|2|Iρφn(1)|2.Obviously, when(77)A1=infn|β1ρ+α1e-iτμn+ξn(1)|>0,A2=infn|β2Iρ+α2e-iτμn+ξn(2)|>0,we have(78)∥C(Φniμn,Φn)∥ℂ22≥A12|ρwn(1)|2+A22|Iρφn(1)|2≥min{A1,A2}(|ρwn(1)|2+|Iρφn(1)|2);using Lemma4,
(79)infn∥C(Φniμn,Φn)∥ℂ22>0.According to Lemma6, the system (56) is exactly observable in finite time, and hence the closed-loop system (11) is exponentially stable.If for alln∈ℕ,
(80)|β1ρ+α1e-iτμn+ξn(1)|>0,|β2Iρ+α2e-iτμn+ξn(2)|>0,we can see that in this case, there is no eigenvalue of system (11) on the imaginary axis. Moreover, if the conditions
(81)infn|β1ρ+α1e-iτμn+ξn(1)|=0,infn|β2Iρ+α2e-iτμn+ξn(2)|=0hold, then the imaginary axis is an asymptote of the eigenvalues of the system (11). Therefore, the stability theorem [17] asserts then that the system (11) is asymptotically stable. Therefore, we get the result of the Theorem 2.


## 6. Conclusion

In this paper, we designed a new controller for a Timoshenko beam with distributed delay in the boundary that stabilizes exponentially the system. In the design process of new controllers, there are main steps: (1) to translate the delay system into a system without delay; (2) for the undelay system, we used the collocated feedback law to obtain the control signals; (3) using the obtained control signals, act on the delay system. This control strategy can be regarded as extension form of [15]. In the stability analysis, the key trick is to use the exact observability of the dual system in finite time to obtain the exponential stability of the closed-loop system.In the proof of main result, the conditionK/ρ≠EI/Iρis used to ensure the separability of the spectrum (see, the condition (2)) in Lemma 6). In the statement of our result (Theorem 2), the conditions are stronger than the practice; in fact,
(82)∥C(Φniμn,Φn)∥ℂ22=|β1ρ+α1e-iτμn+ξn(1)|2|ρwn(1)|2+|β2Iρ+α2e-iτμn+ξn(2)|2|Iρφn(1)|2;one only needs to request(83)infn∥C(Φniμn,Φn)∥ℂ22>0,so, the conditions(84)infn|β1ρ+α1e-iτμn+ξn(1)|>0,infn|β2Iρ+α2e-iτμn+ξn(2)|>0,are sufficient, but not necessary. Since(85)ξn(1)=∫-τ0‍g1(η)eiμn(τ-η)dη,ξn(2)=∫-τ0‍g2(η)eiμn(τ-η)dη,solimn→∞ξn(j)=0,j=1,2. Therefore, when
(86)|β1ρ+α1e-iτμn+ξn(1)|>0,|β2Iρ+α2e-iτμn+ξn(2)|>0that means that there is no eigenvalue on the imaginary axis, and(87)infn|β1ρ+α1e-iτμn|>0,infn|β2Iρ+α2e-iτμn|>0,we have(88)infn∥C(Φniμn,Φn)∥ℂ22>0.Clearly, whenβ1≠ρα1andβ2≠Iρα2, we also have
(89)infn∥C(Φniμn,Φn)∥ℂ22>0.Therefore, the conditions in Theorem2 are easily verified.The control method proposed in this paper can be used to the system of output availed system by using the Luenberger observer. Also we have noted that the method is only fitting the continuous model; for the model of data-driven system (e.g., see, [18]), it might fail. So we need to study the corresponding control strategy for the data-driven system.



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*Source: 726794-2013-12-16.xml*