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

Abstract

We consider the exponential stabilization for Timoshenko beam with distributed delay in the boundary control. Suppose that the controller outputs are of the form 𝛼₁𝑢₁(𝑡)+𝛽₁𝑢₁(𝑡−𝜏)+∫⁰_−𝜏𝑔₁(𝜂)𝑢₁(𝑡+𝜂)𝑑𝜂 and 𝛼₂𝑢₂(𝑡)+𝛽₂𝑢₂(𝑡−𝜏)+∫⁰_−𝜏𝑔₂(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 𝛼_𝑖>𝛽_𝑖>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 𝛼_𝑖,𝛽_𝑖,𝑖=1,2.

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∫⁰_−∞𝑑𝛼(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 space𝐿²_loc(ℝ), then the memory controller will take the form ⁰_−𝜏(𝑠)𝑢()(𝑡)()+𝑑𝛼𝑡+𝑠=𝛼𝑢+𝛽𝑢𝑡−𝜏_0−𝜏𝑔(𝑠)𝑢(𝑡+𝑠)𝑑𝑠. (1)

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∫⁰_−𝜏|𝑔(𝜂)|𝑑𝜂≠0, Nicaise and Pignotti in [10] discussed a high dimensional wave equation. Under the condition that𝛼>∫⁰_−𝜏|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 Section 2, 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: 𝜌𝑤_𝑡𝑡(𝑥,𝑡)−𝐾𝑤_𝑥𝑥−𝜑_𝑥()()𝐼𝑥,𝑡=0,𝑥∈0,1,𝑡>0,_𝜌𝜑_𝑡𝑡(𝑥,𝑡)−𝐸𝐼𝜑_𝑥𝑥(𝑥,𝑡)−𝐾𝑤_𝑥()()𝑤()()𝐾𝑤−𝜑𝑥,𝑡=0,𝑥∈0,1,𝑡>0,0,𝑡=𝜑0,𝑡=0,𝑡>0,_𝑥−𝜑(1,𝑡)=𝑣₁(𝑡),𝐸𝐼𝜑_𝑥(1,𝑡)=𝑣₂(𝑡),𝑤(𝑥,0)=𝑤₀(𝑥),𝑤_𝑡(𝑥,0)=𝑤₁(𝑥),𝜑(𝑥,0)=𝜑₀(𝑥),𝜑_𝑡(𝑥,0)=𝜑₁(𝑥), (2)

where𝑣₁(𝑡)and𝑣₂(𝑡)are the control force and torque from the controllers, respectively. If the controllers have no memory, namely, 𝑢_𝑗(𝑡)=𝑣_𝑗(𝑡),𝑗=1,2, where𝑢_𝑗(𝑡)are controller inputs, this model had been studied in [14]. If the controllers have memory, then the Timoshenko beam became 𝜌𝑤_𝑡𝑡(𝑥,𝑡)−𝐾𝑤_𝑥𝑥−𝜑_𝑥()()𝐼𝑥,𝑡=0,𝑥∈0,1,𝑡>0,_𝜌𝜑_𝑡𝑡(𝑥,𝑡)−𝐸𝐼𝜑_𝑥𝑥(𝑥,𝑡)−𝐾𝑤_𝑥()()𝑤()()𝐾𝑤−𝜑𝑥,𝑡=0,𝑥∈0,1,𝑡>0,0,𝑡=𝜑0,𝑡=0,𝑡>0,_𝑥−𝜑(1,𝑡)=𝛼₁𝑢₁(𝑡)+𝛽₁𝑢₁()+𝑡−𝜏_0−𝜏𝑔₁𝜂𝑢₁𝑡+𝜂𝑑𝜂,𝐸𝐼𝜑_𝑥(1,𝑡)=𝛼₂𝑢₂(𝑡)+𝛽₂𝑢₂()+𝑡−𝜏_0−𝜏𝑔₂𝜂𝑢₂𝑤()𝑡+𝜂𝑑𝜂,𝑥,0=𝑤₀(𝑥),𝑤_𝑡(𝑥,0)=𝑤₁(𝑥),𝜑(𝑥,0)=𝜑₀(𝑥),𝜑_𝑡(𝑥,0)=𝜑₁(𝑥),𝑢₁(𝜃)=𝑓₁(𝜃),𝑢₂(𝜃)=𝑓₂(𝜃),𝜃∈(−𝜏,0), (3)

where𝜏is the delay time,𝛼_𝑖,𝛽_𝑖∈ℝ(𝑖=1,2)are the controller parameters, and𝑔_𝑖(𝜂)∈𝐿²[−𝜏,0],𝑗=1,2, and𝑓_𝑖(𝜃),𝜃∈(−𝜏,0)(𝑖=1,2)are bounded measurable functions that are memory values of controllers. When𝑔_𝑗≡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),𝑤_𝑡(𝑥,𝑡),𝜑_𝑡(𝑥,𝑡))is measurable. We introduce an auxiliary system as follows: 𝑤_𝑠()=𝑧()()(),𝜑𝑥,𝑠,𝑡𝑥,𝑠,𝑡,𝑥∈0,1,𝑠∈0,𝜏_𝑠()=𝜓()()(),𝑧𝑥,𝑠,𝑡𝑥,𝑠,𝑡,𝑥∈0,1,𝑠∈0,𝜏_𝑠()=𝐾𝑥,𝑠,𝑡𝜌𝑤_𝑥𝑥(𝑥,𝑠,𝑡)−𝜑_𝑥(),𝜓𝑥,𝑠,𝑡_𝑠(𝑥,𝑠,𝑡)=𝐸𝐼𝐼_𝜌𝜑_𝑥𝑥()+𝐾𝑥,𝑠,𝑡𝐼_𝜌𝑤_𝑥()−𝜑(),𝑤()=𝜑()(),𝐾𝑤𝑥,𝑠,𝑡𝑥,𝑠,𝑡0,𝑠,𝑡0,𝑠,𝑡=0,𝑠∈0,𝜏_𝑥−𝜑(1,𝑠,𝑡)=𝛽₁𝑢₁()+𝑡+𝑠−𝜏_{−𝑠−𝜏}𝑔₁𝜂𝑢₁(),𝜑𝑡+𝑠+𝜂𝑑𝜂,𝑠∈0,𝜏𝐸𝐼_𝑥(1,𝑠,𝑡)=𝛽₂𝑢₂()+𝑡+𝑠−𝜏_{−𝑠−𝜏}𝑔₂𝜂𝑢₂()𝑤()(),𝑧()𝑡+𝑠+𝜂𝑑𝜂,𝑠∈0,𝜏,𝑡>0,𝑥,0,𝑡=𝑤𝑥,𝑡𝑥,0,𝑡=𝑤_𝑡(),(),𝜑()(),𝜓𝑥,𝑡𝑥∈0,1𝑥,0,𝑡=𝜑𝑥,𝑡_𝑡(𝑥,0,𝑡)=𝜑_𝑡(),()𝑥,𝑡𝑥∈0,1,𝑡>0. (4)

Equation (4) is a partial state predictor.

Denote the state of (4) at the moment𝑠=𝜏by 𝑝₁(𝑥,𝑡),𝑝₂(𝑥,𝑡),𝑞₁(𝑥,𝑡),𝑞₂()=𝑤(),𝜑(),𝑧(),𝜓().𝑥,𝑡𝑥,𝜏,𝑡𝑥,𝜏,𝑡𝑥,𝜏,𝑡𝑥,𝜏,𝑡 (5)

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: 𝑝_1,𝑡(𝑥,𝑡)=𝑞₁(𝑥,𝑡)+𝑎₁(𝑥)𝑢₁(𝑡)+𝑎₂(𝑥)𝑢₂(𝑡),()𝑝𝑥∈0,1,𝑡>0,_2,𝑡(𝑥,𝑡)=𝑞₂(𝑥,𝑡)+𝑎₃(𝑥)𝑢₁(𝑡)+𝑎₄(𝑥)𝑢₂(𝑡),()𝑞𝑥∈0,1,𝑡>0,_1,𝑡()=𝐾𝑥,𝑡𝜌𝑝_1,𝑥𝑥−𝑝_2,𝑥(𝑥,𝑡)+𝑏₁(𝑥)𝑢₁(𝑡)+𝑏₂(𝑥)𝑢₂(𝑡),𝑞_2,𝑡(𝑥,𝑡)=𝐸𝐼𝐼_𝜌𝑝_2,𝑥𝑥()+𝐾𝑥,𝑡𝐼_𝜌𝑝_1,𝑥−𝑝₂(𝑥,𝑡)+𝑏₃(𝑥)𝑢₁(𝑡)+𝑏₄(𝑥)𝑢₂(𝑡),𝑝₁(0,𝑡)=𝑝₂(0,𝑡)=𝑞₁(0,𝑡)=𝑞₂()𝐾𝑝0,𝑡=0,𝑡>0,_1,𝑥−𝑝₂(1,𝑡)=𝛽₁𝑢₁(𝑡),𝑡>0,𝐸𝐼𝑝_2,𝑥(1,𝑡)=𝛽₂𝑢₂(𝑡)𝑝,𝑡>0,₁(𝑥,0)=𝐸₁𝑤₀,𝜑₀,𝑤₁,𝜑₁(𝑥)−⁰_−𝜏𝑎₁(𝑥,𝑠)𝑓₁(𝑠)−𝑑𝑠_0−𝜏𝑎₂(𝑥,𝑠)𝑓₂(𝑠)𝑝𝑑𝑠,₂(𝑥,0)=𝐸₂𝑤₀,𝜑₀,𝑤₁,𝜑₁(𝑥)−⁰_−𝜏𝑎₃(𝑥,𝑠)𝑓₁(𝑠)𝑑𝑠−⁰_−𝜏𝑎₄(𝑥,𝑠)𝑓₂(𝑠)𝑞𝑑𝑠,₁(𝑥,0)=𝐸₃𝑤₀,𝜑₀,𝑤₁,𝜑₁(𝑥)+⁰_−𝜏𝑏₁(𝑥,𝑠)𝑓₁(𝑠)+𝑑𝑠_0−𝜏𝑏₂(𝑥,𝑠)𝑓₂(𝑠)𝑞𝑑𝑠,₂(𝑥,0)=𝐸₄𝑤₀,𝜑₀,𝑤₁,𝜑₁(𝑥)+⁰_−𝜏𝑏₃(𝑥,𝑠)𝑓₁(𝑠)+𝑑𝑠_0−𝜏𝑏₄(𝑥,𝑠)𝑓₂(𝑠)𝑑𝑠, (6)

where𝑎_𝑖(𝑥,𝑟),𝑏_𝑖(𝑥,𝑟),𝑎_𝑖(𝑥),𝑏_𝑖(𝑥)(𝑖=1,2,3,4)are measurable function and𝐸_𝑖(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) 𝐸(𝑡)=12𝑝₁,𝑝₂²_{𝑉1𝐾(0,1)×𝑉1𝐸𝐼(0,1)}+12(𝑞₁,𝑞₂)²_{𝐿2𝜌(0,1)×𝐿2𝐼𝜌(0,1)}=12¹₀𝐾𝑝_1,𝑥(𝑥,𝑡)−𝑝₂(𝑥,𝑡)²+1𝑑𝑥2¹₀𝐸𝐼𝑝_2,𝑥(𝑥,𝑡)²+1𝑑𝑥2¹₀𝜌𝑞₁(𝑥,𝑡)²+1𝑑𝑥2¹₀𝐼_𝜌𝑞₂(𝑥,𝑡)²𝑑𝑥. (7)

A direct calculation gives 𝑑𝐸(𝑡)𝑑𝑡=𝑢₁(𝑡)𝛽₁𝑞₁(1,𝑡)+¹₀𝐾𝑝_1,𝑥−𝑝₂𝑎^′₁(𝑥)−𝑎₃(𝑥+)𝑑𝑥₁₀𝐸𝐼𝑝_2,𝑥(𝑥,𝑡)𝑎^′₃(𝑥)𝑑𝑥+¹₀𝜌𝑞₁(𝑥,𝑡)𝑏₁(𝑥)+𝑑𝑥₁₀𝐼_𝜌𝑞₂(𝑥,𝑡)𝑏₃(𝑥)𝑑𝑥+𝑢₂(𝑡)𝛽₂𝑞₂(1,𝑡)+¹₀𝐾𝑝_1,𝑥−𝑝₂×𝑎^′₂(𝑥)−𝑎₄(𝑥)𝑑𝑥+¹₀𝐸𝐼𝑝_2,𝑥(𝑥,𝑡)𝑎^′₄(𝑥)+𝑑𝑥₁₀𝜌𝑞₁(𝑥,𝑡)𝑏₂(𝑥)𝑑𝑥+¹₀𝐼_𝜌𝑞₂(𝑥,𝑡)𝑏₄(𝑥)𝑑𝑥. (8)

Set 𝑈₁𝑝₁,𝑝₂,𝑞₁,𝑞₂=𝛽₁𝑞₁()+1,𝑡₁₀𝐾𝑝_1,𝑥(𝑥,𝑡)−𝑝₂(𝑥,𝑡)𝑎^′₁(𝑥)−𝑎₃(𝑥)+𝑑𝑥₁₀𝜌𝑞₁(𝑥,𝑡)𝑏₁(𝑥)𝑑𝑥+¹₀𝐸𝐼𝑝_2,𝑥(𝑥,𝑡)𝑎^′₃(𝑥)+𝑑𝑥₁₀𝐼_𝜌𝑞₂(𝑥,𝑡)𝑏₃(𝑥)𝑈𝑑𝑥;₂𝑝₁,𝑝₂,𝑞₁,𝑞₂=𝛽₂𝑞₂()+1,𝑡₁₀𝐾𝑝_1,𝑥(𝑥,𝑡)−𝑝₂(𝑥,𝑡)𝑎^′₂(𝑥)−𝑎₄(𝑥)+𝑑𝑥₁₀𝜌𝑞₁(𝑥,𝑡)𝑏₂(𝑥)𝑑𝑥+¹₀𝐸𝐼𝑝_2,𝑥(𝑥,𝑡)𝑎^′₄(𝑥)+𝑑𝑥₁₀𝐼_𝜌𝑞₂(𝑥,𝑡)𝑏₄(𝑥)𝑑𝑥. (9)

We take the feedback control law as 𝑢₁(𝑡)=−𝑈₁𝑝₁,𝑝₂,𝑞₁,𝑞₂,𝑢₂(𝑡)=−𝑈₂𝑝₁,𝑝₂,𝑞₁,𝑞₂. (10)

Then, the closed loop system associated with (6) is 𝑝_1,𝑡(𝑥,𝑡)=𝑞₁(𝑥,𝑡)−𝑎₁(𝑥)𝑈₁𝑝₁,𝑝₂,𝑞₁,𝑞₂−𝑎₂(𝑥)𝑈₂𝑝₁,𝑝₂,𝑞₁,𝑞₂,()𝑝𝑥∈0,1,𝑡>0,_2,𝑡(𝑥,𝑡)=𝑞₂(𝑥,𝑡)−𝑎₃(𝑥)𝑈₁𝑝₁,𝑝₂,𝑞₁,𝑞₂−𝑎₄(𝑥)𝑈₂𝑝₁,𝑝₂,𝑞₁,𝑞₂,()𝑞𝑥∈0,1,𝑡>0,_1,𝑡()=𝐾𝑥,𝑡𝜌𝑝_1,𝑥𝑥−𝑝_2,𝑥(𝑥,𝑡)−𝑏₁(𝑥)𝑈₁×𝑝₁,𝑝₂,𝑞₁,𝑞₂−𝑏₂(𝑥)𝑈₂𝑝₁,𝑝₂,𝑞₁,𝑞₂,𝑞_2,𝑡(𝑥,𝑡)=𝐸𝐼𝐼_𝜌𝑝_2,𝑥𝑥()+𝐾𝑥,𝑡𝐼_𝜌𝑝_1,𝑥−𝑝₂(𝑥,𝑡)−𝑏₃(𝑥)𝑈₁𝑝₁,𝑝₂,𝑞₁,𝑞₂−𝑏₄(𝑥)𝑈₂𝑝₁,𝑝₂,𝑞₁,𝑞₂,𝑝₁(0,𝑡)=𝑝₂(0,𝑡)=𝑞₁(0,𝑡)=𝑞₂()𝐾𝑝0,𝑡=0,𝑡>0,_1,𝑥−𝑝₂(1,𝑡)=−𝛽₁𝑈₁𝑝₁,𝑝₂,𝑞₁,𝑞₂,𝑡>0,𝐸𝐼𝑝_2,𝑥𝑥(1,𝑡)=−𝛽₂𝑈₂𝑝₁,𝑝₂,𝑞₁,𝑞₂𝑝,𝑡>0,₁(𝑥,0)=𝐸₁𝑤₀,𝜑₀,𝑤₁,𝜑₁(𝑥)−⁰_−𝜏𝑎₁(𝑥,𝑠)𝑓₁(𝑠)−𝑑𝑠_0−𝜏𝑎₂(𝑥,𝑠)𝑓₁(𝑠)𝑝𝑑𝑠,₂(𝑥,0)=𝐸₂𝑤₀,𝜑₀,𝑤₁,𝜑₁(𝑥)−⁰_−𝜏𝑎₃(𝑥,𝑠)𝑓₁(𝑠)−𝑑𝑠_0−𝜏𝑎₄(𝑥,𝑠)𝑓₁(𝑠)𝑞𝑑𝑠,₁(𝑥,0)=𝐸₃𝑤₀,𝜑₀,𝑤₁,𝜑₁(𝑥)+⁰_−𝜏𝑏₁(𝑥,𝑠)𝑓₁(𝑠)+𝑑𝑠_0−𝜏𝑏₂(𝑥,𝑠)𝑓₁(𝑠)𝑞𝑑𝑠,₂(𝑥,0)=𝐸₄𝑤₀,𝜑₀,𝑤₁,𝜑₁(𝑥)+⁰_−𝜏𝑏₃(𝑥,𝑠)𝑓₁(𝑠)+𝑑𝑠_0−𝜏𝑏₄(𝑥,𝑠)𝑓₁(𝑠)𝑑𝑠. (11)

We estimate the error of the system (3) with control (10) and the system (11).

Let(𝑤(𝑥,𝑡),𝜑(𝑥,𝑡),𝑤_𝑡(𝑥,𝑡),𝜑_𝑡(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))and𝑊_𝑡(𝑥,𝑡)=(𝑤_𝑡(𝑥,𝑡),𝜑_𝑡(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),𝑊_𝑡(𝑥,𝑡)), we consider the error both solutions in the energy space ‖𝑃(⋅,𝑡)−𝑊(⋅,𝑡+𝜏)‖²_{𝑉1𝐾(0,1)×𝑉1𝐸𝐼(0,1)}+𝑄(⋅,𝑡)−𝑊_𝑡(⋅,𝑡+𝜏)²_{𝐿2𝜌(0,1)×𝐿2𝐼𝜌(0,1)}. (12)

In this paper, we will prove the following results.

Theorem 1. Let(𝑊(𝑥,𝑡),𝑊_𝑡(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/𝐼_𝜌. Let𝜇_𝑛,𝑛∈ℕ be the eigenvalues of the free system (the system (2) without controls). Set 𝜉_𝑛⁽¹⁾=⁰_−𝜏𝑔₁𝜂𝑒^{√−𝑖𝜇𝑛(𝜏+𝜂)}𝜉𝑑𝜂,_𝑛(2)=⁰_−𝜏𝑔₂𝜂𝑒^{√−𝑖𝜇𝑛(𝜏+𝜂)}𝑑𝜂. (13) Then the following assertions are true:(1)when inf_𝑛𝛽₁𝜌+𝛼₁𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽¹⁾>0,inf_𝑛𝛽₂𝐼_𝜌+𝛼₂𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽²⁾>0, (14)the system (11) is exponentially stable;(2)if for all𝑛∈ℕ, 𝛽₁𝜌+𝛼₁𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽¹⁾𝛽>0,₂𝐼_𝜌+𝛼₂𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽²⁾>0 (15)but inf_𝑛𝛽₁𝜌+𝛼₁𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽¹⁾=0,inf_𝑛𝛽₂𝐼_𝜌+𝛼₂𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽²⁾=0, (16) then the system (11) is asymptotically stable.

In the following sections, we will prove our results. In Section 3, we will determine functions𝑎_𝑖(𝑥),𝑏_𝑖(𝑥),𝑎_𝑖(𝑥,𝑠),𝑏_𝑖(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 functions𝑎_𝑖(𝑥),𝑏_𝑖(𝑥),𝑎_𝑖(𝑥,𝑠),𝑏_𝑖(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 in𝐿²_𝜌(0,1)×𝐿²_𝐼𝜌(0,1)as follows: ℒ=−𝐾𝑤,𝜑𝜌𝑤^′′(𝑥)−𝜑^′(𝑥),−𝐸𝐼𝐼_𝜌𝜑^′′(𝑥)−𝐾𝐼_𝜌𝑤^′(𝑥)−𝜑(𝑥)^𝑇, (17) with domain 𝒟(ℒ)=𝑤(𝑥),𝜑(𝑥)∈𝐻²(0,1)×𝐻²()∣𝑤(0)(0)𝐾𝑤0,1=𝜑=0,_′(1)−𝜑(1)=0,𝐸𝐼𝜑^′(1).=0 (18) Thenℒis a positive define operator with compact resolvent in𝐿²_𝜌(0,1)×𝐿²_𝐼𝜌(0,1); its eigenvalues are 0<𝜇₁<𝜇₂<⋯<𝜇_𝑛<⋯ (19) and the eigenfunctionsΦ_𝑛(𝑥)=(𝑤_𝑛(𝑥),𝜑_𝑛(𝑥))^𝑇corresponding to𝜇_𝑛are real functions and form a normalized orthogonal basis for𝐿²_𝜌(0,1)×𝐿²_𝐼𝜌(0,1).

Lemma 4 (see [13]). LetΦ_𝑛(𝑥)=(𝑤_𝑛(𝑥),𝜑_𝑛(𝑥))be the normalized eigenfunction corresponding to the eigenvalue𝜇_𝑛ofℒ. Then it holds that ¹₀𝐾𝜔^′_𝑛(𝑥)−𝜑_𝑛(𝑥)²𝑑𝑥+¹₀𝐸𝐼𝜑^′_𝑛(𝑥)²𝑑𝑥=𝜇_𝑛,0<inf_𝑛𝜌𝑤_𝑛(1)²+𝐼_𝜌𝜑_𝑛(1)²≤sup_𝑛𝜌𝑤_𝑛(1)²+𝐼_𝜌𝜑_𝑛(1)²<∞. (20)

Now let us return to (3). We write the equation in (3) into the vector form 𝑤_𝑡𝑡()𝜑𝑥,𝑡_𝑡𝑡()−𝐾𝑥,𝑡𝜌𝜕_𝑥𝑥−𝐾𝜌𝜕_𝑥𝐾𝐼_𝜌𝜕_𝑥𝐸𝐼𝐼_𝜌𝜕_𝑥𝑥−𝐾𝐼_𝜌𝑤()𝜑()𝑥,𝑡𝑥,𝑡=0 (21)

and the boundary conditions are^𝑤(0,𝑡)_𝜑(0,𝑡)=0, and 𝐾𝜕_𝑥−𝐾0𝐸𝐼𝜕_𝑥𝑤(𝑥,𝑡)𝜑(𝑥,𝑡)_𝑥=1=𝛼₁00𝛼₂𝑢₁(𝑡)𝑢₂(𝑡)+𝛽₁00𝛽₂𝑢₁()𝑢𝑡−𝜏₂(𝑡−𝜏)+⁰_−𝜏𝑔₁𝜂00𝑔₂𝜂𝑢₁𝑢𝑡+𝜂₂𝑡+𝜂𝑑𝜂. (22)

The initial datum are 𝑤()𝜑()=𝑥,0𝑤𝑥,0⁰(𝑥)𝜑₀(𝑥),𝑤_𝑡()𝜑𝑥,0_𝑡()=𝑤𝑥,0¹(𝑥)𝜑₁(𝑥). (23)

Set𝑊(𝑥,𝑡)=(𝑤(𝑥,𝑡),𝜑(𝑥,𝑡))^𝑇and𝑈(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡))^𝑇. Define2×2matrices Λ₁=𝛼₁00𝛼₂,Λ₂=𝛽₁00𝛽₂,Λ₃𝜂=𝑔₁𝜂00𝑔₂𝜂 (24)

and define an operator𝐵fromℝ²to𝐻⁻¹(0,1)×𝐻⁻¹(0,1), where𝐻⁻¹(0,1)=(𝑉¹_𝜔(0,1))^∗is dual space, 𝛿()0()𝐵=𝑥−10𝛿𝑥−1, (25)

and define an operatorΓ_𝑁from𝐻²(0,1)×𝐻²(0,1)toℝ²by Γ_𝑁𝐾𝑤𝑊=^′(1)−𝜑(1)𝐸𝐼𝜑^′(1), (26)

where𝑊(𝑥)=(𝑤(𝑥),𝜑(𝑥))^𝑇.

With help of these notations, we can rewrite (3) into 𝑊_𝑡𝑡()()Λ𝑥,𝑡+ℒ𝑊𝑥,𝑡=𝐵₁𝑈(𝑡)+Λ₂𝑈(𝑡−𝜏)+⁰_−𝜏Λ₃𝜂𝑈,𝑊()𝑡+𝜂𝑑𝜂𝑡>0,0,𝑡=0,Γ_𝑁𝑊()𝑊()1,𝑡=0,𝑥,0=𝑊₀(𝑥)=𝑤₀(𝑥),𝜑₀(𝑥)^𝑇,𝑊_𝑡(𝑥,0)=𝑊₁(𝑥)=𝑤₁(𝑥),𝜑₁(𝑥)^𝑇, (27)

and (4) into 𝑊_𝑠()=𝑉(),𝑉𝑥,𝑠,𝑡𝑥,𝑠,𝑡_𝑠(𝑥,𝑠,𝑡)+ℒ𝑊(𝑥,𝑠,𝑡)=𝐵Λ₂𝑈(𝑡+𝑠−𝜏)+𝐵^−𝑠_−𝜏Λ₃𝜂𝑈𝑊()𝑡+𝑠+𝜂𝑑𝜂,0,𝑠,𝑡=0,Γ_𝑁𝑊()𝑊()(),𝑉()1,𝑠,𝑡=0,𝑥,0,𝑡=𝑊𝑥,𝑡𝑥,0,𝑡=𝑊_𝑡(𝑥,𝑡), (28)

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 on𝐿²_𝜌(0,1)×𝐿²_𝐼𝜌(0,1)by Cos(𝑡ℒ)𝐹=^∞_𝑛=1cos√𝜇_𝑛𝑡𝐹,Φ_{𝑛𝐿2𝜌×𝐿2𝐼𝜌}Φ_𝑛,Sin(𝑡ℒ)𝐹=^∞_𝑛=1sin√𝜇_𝑛𝑡√𝜇_𝑛𝐹,Φ_{𝑛𝐿2𝜌×𝐿2𝐼𝜌}Φ_𝑛. (29)

Clearly, the following equalities hold, for any𝑡∈ℝ, Sin(𝑡ℒ)=^𝑡₀()𝑑Cos𝑡ℒ𝑑𝑡,((().𝑑𝑡Cos𝑡ℒ))=−ℒSin𝑡ℒ (30)

It is easy to know that the vector-valued function 𝑊(𝑥,𝑡)=Cos(𝑡ℒ)𝑊₀+Sin(𝑡ℒ)𝑊₁+^𝑡₀Sin((𝑡−𝑠)ℒ)𝐵Λ₁𝑈(𝑠)+Λ₂𝑈(+𝑠−𝜏)𝑑𝑠_𝑡0Sin((𝑡−𝑠)ℒ)𝐵⁰_−𝜏Λ₃𝜂𝑈𝑠+𝜂𝑑𝜂𝑑𝑠 (31)

is differentiable with respect to𝑡and 𝑊_𝑡(𝑥,𝑡)=−ℒSin(𝑡ℒ)𝑊₀+Cos(𝑡ℒ)𝑊₁+^𝑡₀Cos((𝑡−𝑠)ℒ)𝐵Λ₁𝑈(𝑠)+Λ₂𝑈(+𝑠−𝜏)𝑑𝑠_𝑡0Cos((𝑡−𝑠)ℒ)𝐵⁰_−𝜏Λ₃𝜂𝑈𝑠+𝜂𝑑𝜂𝑑𝑠. (32)

Further,𝑊(𝑥,𝑡)satisfies (27).

Similarly, we know the vector-valued function 𝑊()()𝑊()()𝑊𝑥,𝑠,𝑡=Cos𝑠ℒ⋅,𝑡+Sin𝑠ℒ_𝑡()+⋅,𝑡_𝑠0Sin((𝑠−𝑟)ℒ)𝐵Λ₂𝑈()+𝑡+𝑟−𝜏𝑑𝑟_𝑠0Sin((𝑠−𝑟)ℒ)𝐵^−𝑟_−𝜏Λ₃𝜂𝑈𝑉()()𝑊()()𝑊𝑡+𝑟+𝜂𝑑𝜂𝑑𝑟,𝑥,𝑠,𝑡=−ℒSin𝑠ℒ⋅,𝑡+Cos𝑠ℒ_𝑡(⋅,𝑡)+^𝑠₀Cos((𝑠−𝑟)ℒ)𝐵Λ₂𝑈()+𝑡+𝑟−𝜏𝑑𝑟_𝑠0Cos((𝑠−𝑟)ℒ)𝐵^−𝑟_−𝜏Λ₃𝜂𝑈𝑡+𝑟+𝜂𝑑𝜂𝑑𝑟. (33)(34)

satisfy (28).

Set 𝑃(𝑥,𝑡)=𝑊(𝑥,𝜏,𝑡),𝑄(𝑥,𝑡)=𝑉(𝑥,𝜏,𝑡). (35)

Then we have 𝑃()𝑄()=()()()()𝑊()𝑊𝑥,𝑡𝑥,𝑡Cos𝜏ℒSin𝜏ℒ−ℒSin𝜏ℒCos𝜏ℒ𝑥,𝑡_𝑡()+𝑥,𝑡_𝜏0)ℒ))ℒ)Sin((𝜏−𝑟Cos((𝜏−𝑟𝐵Λ₂𝑈()+𝑡+𝑟−𝜏𝑑𝑟_𝜏0)ℒ))ℒ)Sin((𝜏−𝑟Cos((𝜏−𝑟×𝐵^−𝑟_−𝜏Λ₃𝜂𝑈𝑡+𝑟+𝜂𝑑𝜂𝑑𝑟. (36)

Thus, 𝑃_𝑡()𝑄𝑥,𝑡_𝑡()=()()()()𝑊𝑥,𝑡Cos𝜏ℒSin𝜏ℒ−ℒSin𝜏ℒCos𝜏ℒ^𝑡()𝑊𝑥,𝑡_𝑡𝑡()+𝑥,𝑡_𝜏0)ℒ))ℒ)Cos((𝜏−𝑟−ℒSin((𝜏−𝑟×𝐵Λ₂𝑈()+0𝑡+𝑟−𝜏𝑑𝑟𝐵Λ₂𝑈(𝑡)−()()Sin𝜏ℒCos𝜏ℒ𝐵Λ₂𝑈()−()()𝐵𝑡−𝜏Sin𝜏ℒCos𝜏ℒ^0−𝜏Λ₃𝜂𝑈𝑡+𝜂𝑑𝜂+^𝜏₀)ℒ))ℒ)𝐵Cos((𝜏−𝑟−ℒSin((𝜏−𝑟×^−𝑟_−𝜏Λ₃𝜂𝑈+𝑡+𝑟+𝜂𝑑𝜂𝑑𝑟_0−𝜏ℒℒSin𝜏+𝜂Cos𝜏+𝜂𝐵Λ₃𝜂𝑈(𝑡)𝑑𝜂. (37)

Note that 𝑊_𝑡()𝑊𝑥,𝑡_𝑡𝑡()=𝑊()𝑊𝑥,𝑡0𝐼−ℒ0𝑥,𝑡_𝑡()+0𝐵Λ𝑥,𝑡₁𝑈(𝑡)+Λ₂𝑈(𝑡−𝜏)+∫⁰_−𝜏Λ₃𝜂𝑈.𝑡+𝜂𝑑𝜂 (38)

So it holds that𝑃_𝑡()𝑄𝑥,𝑡_𝑡()=𝑃()𝑄()+()𝑥,𝑡0𝐼−ℒ0𝑥,𝑡𝑥,𝑡Sin𝜏ℒ𝐵Λ₁𝑈(𝑡)+∫⁰_−𝜏Sin𝜏+𝜂ℒ𝐵Λ₃𝜂𝑈(𝑡)()𝑑𝜂Cos𝜏ℒ𝐵Λ₁𝑈(𝑡)+∫⁰_−𝜏Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑈(𝑡)𝑑𝜂+𝐵Λ₂𝑈(𝑡). (39)

Therefore, we have equations 𝑃_𝑡(𝑥,𝑡)=𝑄(𝑥,𝑡)+Sin(𝜏ℒ)𝐵Λ₁𝑈(𝑡)+⁰_−𝜏Sin𝜏+𝜂ℒ𝐵Λ₃𝜂𝑈(𝑡)𝑄𝑑𝜂,_𝑡(𝑥,𝑡)=−ℒ𝑃(𝑥,𝑡)+Cos(𝜏ℒ)𝐵Λ₁𝑈(𝑡)+⁰_−𝜏Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑈(𝑡)𝑃()()𝑑𝜂,0,𝑡=𝑄0,𝑡=0,Γ_𝑁𝑃(⋅,𝑡)=Λ₂𝑈(𝑡) (40)

and initial conditions 𝑃(𝑥,0)=Cos(𝜏ℒ)𝑊₀+Sin(𝜏ℒ)𝑊₁−⁰_−𝜏Sin(𝑠ℒ)𝐵Λ₂𝑓(𝑠)+𝑑𝑠_{0−𝜏𝑠−𝜏}Sin𝜏−𝑠+𝜂ℒ×𝐵Λ₃𝜂𝑓(𝑠)𝑄()()𝑊𝑑𝜂𝑑𝑠,𝑥,0=−ℒSin𝜏ℒ₀+Cos(𝜏ℒ)𝑊₁+⁰_−𝜏Cos(𝑠ℒ)𝐵Λ₂𝑓(𝑠)+𝑑𝑠_{0−𝜏𝑠−𝜏}Cos𝜏−𝑠+𝜂ℒ𝐵Λ₃𝜂𝑓(𝑠)𝑑𝜂𝑑𝑠, (41)(42)

where𝑓(𝑠)=(𝑓₁(𝑠),𝑓₂(𝑠))^𝑇.

Since all entries of𝐵are meaningful as linear functional on𝐻¹(0,1), so for any𝑍=(𝑧₁,𝑧₂)∈ℝ²andΦ_𝑛(𝑥)∈𝐻¹(0,1)×𝐻¹(0,1), 𝐵𝑍,Φ_{𝑛𝐿2𝜌(0,1)×𝐿2𝐼𝜌(0,1)}=𝑧₁¹₀𝜌𝛿(𝑥−1)𝑤_𝑛(𝑥)𝑑𝑥+𝑧₂¹₀𝐼_𝜌𝛿(𝑥−1)𝜑_𝑛(𝑥)𝑑𝑥=𝜌𝑧₁𝑤_𝑛(1)+𝐼_𝜌𝑧₂𝜑_𝑛(1)=𝑧₁,𝑧₂𝜌𝑤_𝑛(1),𝜑_𝑛(1)^𝑇. (43)

Therefore, we have the following results.

Theorem 5. Let{𝜇_𝑛;𝑛∈ℕ} be the list of all eigenvalues of ℒ. Then the functions that appear in (6) are 𝑎₁(𝑥)=𝜌^∞_𝑛=1𝑤_𝑛(1)𝑤_𝑛(𝑥)×𝛼₁sin𝜏√𝜇_𝑛√𝜇_𝑛+⁰_−𝜏𝑔₁𝜂sin𝜏+𝜂√𝜇_𝑛√𝜇_𝑛,𝑎𝑑𝜂₂(𝑥)=𝐼_𝜌^∞_𝑛=1𝜑_𝑛(1)𝑤_𝑛(𝑥)×𝛼₂sin𝜏√𝜇_𝑛√𝜇_𝑛+⁰_−𝜏𝑔₂𝜂sin𝜏+𝜂√𝜇_𝑛√𝜇_𝑛,𝑎𝑑𝜂₃(𝑥)=𝜌^∞_𝑛=1𝑤_𝑛(1)𝜑_𝑛(𝑥)×𝛼₁sin𝜏√𝜇_𝑛√𝜇_𝑛+⁰_−𝜏𝑔₁𝜂sin𝜏+𝜂√𝜇_𝑛√𝜇_𝑛,𝑎𝑑𝜂₄(𝑥)=𝐼_𝜌^∞_𝑛=1𝜑_𝑛(1)𝜑_𝑛(𝑥)×𝛼₂sin𝜏√𝜇_𝑛√𝜇_𝑛+⁰_−𝜏𝑔₂𝜂sin𝜏+𝜂√𝜇_𝑛√𝜇_𝑛,𝑏𝑑𝜂₁(𝑥)=𝜌^∞_𝑛=1𝑤_𝑛(1)𝑤_𝑛(𝑥)×𝛼₁cos𝜏√𝜇_𝑛+⁰_−𝜏𝑔₁𝜂cos𝜏+𝜂√𝜇_𝑛,𝑏𝑑𝜂₂(𝑥)=𝐼_𝜌^∞_𝑛=1𝜑_𝑛(1)𝑤_𝑛(𝑥)×𝛼₂cos𝜏√𝜇_𝑛+⁰_−𝜏𝑔₂𝜂cos𝜏+𝜂√𝜇_𝑛,𝑏𝑑𝜂₃(𝑥)=𝜌^∞_𝑛=1𝑤_𝑛(1)𝜑_𝑛(𝑥)×𝛼₁cos𝜏√𝜇_𝑛+⁰_−𝜏𝑔₁𝜂cos𝜏+𝜂√𝜇_𝑛,𝑏𝑑𝜂₄(𝑥)=𝐼_𝜌^∞_𝑛=1𝜑_𝑛(1)𝜑_𝑛(𝑥)×𝛼₂cos𝜏√𝜇_𝑛+⁰_−𝜏𝑔₂𝜂cos𝜏+𝜂√𝜇_𝑛,𝑎𝑑𝜂₁(𝑥,𝑠)=𝜌^∞_𝑛=1𝑤_𝑛(1)𝑤_𝑛(𝑥)×𝛽₁sin𝑠√𝜇_𝑛√𝜇_𝑛−^𝑠_−𝜏𝑔₁𝜂sin𝜏−𝑠+𝜂√𝜇_𝑛√𝜇_𝑛,𝑎𝑑𝜂₂(𝑥,𝑠)=𝐼_𝜌^∞_𝑛=1𝜑_𝑛(1)𝑤_𝑛(𝑥)×𝛽₂sin𝑠√𝜇_𝑛√𝜇_𝑛−^𝑠_−𝜏𝑔₂𝜂sin𝜏−𝑠+𝜂√𝜇_𝑛√𝜇_𝑛,𝑎𝑑𝜂₃(𝑥,𝑠)=𝜌^∞_𝑛=1𝑤_𝑛(1)𝜑_𝑛(𝑥)×𝛽₁sin𝑠√𝜇_𝑛√𝜇_𝑛−^𝑠_−𝜏𝑔₁𝜂sin𝜏−𝑠+𝜂√𝜇_𝑛√𝜇_𝑛,𝑎𝑑𝜂₄(𝑥,𝑠)=𝐼_𝜌^∞_𝑛=1𝜑_𝑛(1)𝜑_𝑛(𝑥)×𝛽₂sin𝑠√𝜇_𝑛√𝜇_𝑛−^𝑠_−𝜏𝑔₂𝜂sin𝜏−𝑠+𝜂√𝜇_𝑛√𝜇_𝑛,𝑏𝑑𝜂₁(𝑥,𝑠)=𝜌^∞_𝑛=1𝑤_𝑛(1)𝑤_𝑛(𝑥)×𝛽₁cos𝑠√𝜇_𝑛+^𝑠_−𝜏𝑔₁𝜂cos𝜏−𝑠+𝜂√𝜇_𝑛,𝑏𝑑𝜂₂(𝑥,𝑠)=𝐼_𝜌^∞_𝑛=1𝜑_𝑛(1)𝑤_𝑛(𝑥)×𝛽₂cos𝑠√𝜇_𝑛+^𝑠_−𝜏𝑔₂𝜂cos𝜏−𝑠+𝜂√𝜇_𝑛,𝑏𝑑𝜂₃(𝑥,𝑠)=𝜌^∞_𝑛=1𝑤_𝑛(1)𝜑_𝑛(𝑥)×𝛽₁cos𝑠√𝜇_𝑛+^𝑠_−𝜏𝑔₁𝜂cos𝜏−𝑠+𝜂√𝜇_𝑛,𝑏𝑑𝜂₄(𝑥,𝑠)=𝐼_𝜌^∞_𝑛=1𝜑_𝑛(1)𝜑_𝑛(𝑥)×𝛽₂cos𝑠√𝜇_𝑛+^𝑠_−𝜏𝑔₂𝜂cos𝜏−𝑠+𝜂√𝜇_𝑛𝑑𝜂, (44)

and the linear operators are 𝐸₁𝜔₀,𝜑₀,𝜔₁,𝜑₁(𝑥)=^∞_𝑛=1cos𝜏√𝜇_𝑛𝑊₀,Φ_𝑛+sin𝜏√𝜇_𝑛√𝜇_𝑛𝑊₁,Φ_𝑛𝑤_𝑛(𝑥),𝐸₂𝜔₀,𝜑₀,𝜔₁,𝜑₁(𝑥)=^∞_𝑛=1cos𝜏√𝜇_𝑛𝑊₀,Φ_𝑛+sin𝜏√𝜇_𝑛√𝜇_𝑛𝑊₁,Φ_𝑛𝜑_𝑛(𝑥),𝐸₃𝜔₀,𝜑₀,𝜔₁,𝜑₁(𝑥)=^∞_𝑛=1−√𝜇_𝑛sin𝜏√𝜇_𝑛𝑊₀,Φ_𝑛+cos𝜏√𝜇_𝑛𝑊₁,Φ_𝑛𝑤_𝑛(𝑥),𝐸₄𝜔₀,𝜑₀,𝜔₁,𝜑₁(𝑥)=^∞_𝑛=1−√𝜇_𝑛sin𝜏√𝜇_𝑛𝑊₀,Φ_𝑛+cos𝜏√𝜇_𝑛𝑊₁,Φ_𝑛𝜑_𝑛(𝑥), (45)

4. The Proof of Theorem 1

In this section, we will prove Theorem 1. Here we mainly estimate the error: ‖𝑃(⋅,𝑡)−𝑊(⋅,𝑡+𝜏)‖²_{𝑉1𝐾(0,1)×𝑉1𝐸𝐼(0,1)}+𝑄(⋅,𝑡)−𝑊_𝑡(⋅,𝑡+𝜏)²_{𝐿2𝜌(0,1)×𝐿2𝐼𝜌(0,1)}. (46)

According to the calculation in Section 3, we have 𝑃(𝑥,𝑡)−𝑊(𝑥,𝑡+𝜏)=−^𝜏₀Sin((𝜏−𝑟)ℒ)𝐵Λ₁𝑈()−𝑡+𝑟𝑑𝑟_𝜏0Sin((𝜏−𝑟)ℒ)𝐵⁰_−𝜏Λ₃𝜂𝑈𝑄()𝑡+𝑟+𝜂𝑑𝜂𝑑𝑟,𝑥,𝑡−𝑊_𝑡(𝑥,𝑡+𝜏)=−^𝜏₀Cos((𝜏−𝑟)ℒ)𝐵Λ₁𝑈()−𝑡+𝑟𝑑𝑟_𝜏0Cos((𝜏−𝑟)ℒ)𝐵⁰_−𝜏Λ₃𝜂𝑈𝑡+𝑟+𝜂𝑑𝜂𝑑𝑟. (47)(48)

So, ‖𝑃(⋅,𝑡)−𝑊(⋅,𝑡+𝜏)‖²_{𝑉1𝐾(0,1)×𝑉1𝐸𝐼(0,1)}+𝑄(⋅,𝑡)−𝑊_𝑡(⋅,𝑡+𝜏)²_{𝐿2𝜌(0,1)×𝐿2𝐼𝜌(0,1)}≤4𝛼₁𝜌^2∞_𝑛=1𝜔_𝑛(1)²×^𝜏₀sin√𝜇_𝑛(𝜏−𝑟)𝑢₁(𝑟+𝑡)𝑑𝑟²+4𝛼₂𝐼_𝜌^2∞_𝑛=1𝜑_𝑛(1)²×^𝜏₀sin√𝜇_𝑛(𝜏−𝑟)𝑢₂(𝑟+𝑡)𝑑𝑟²+4𝜌^2∞_𝑛=1𝜔_𝑛(1)²×^𝜏₀sin√𝜇_𝑛(𝜏−𝑟)⁰_−𝜏𝑔₁𝜂𝑢₁𝑟+𝑡+𝜂𝑑𝜂𝑑𝑟²+4𝐼²_𝜌^∞_𝑛=1𝜑_𝑛(1)²×^𝜏₀sin√𝜇_𝑛(𝜏−𝑟)⁰_−𝜏𝑔₂𝜂𝑢₂𝑟+𝑡+𝜂𝑑𝜂𝑑𝑟²+4𝛼₁𝜌^2∞_𝑛=1𝜔_𝑛(1)²×^𝜏₀cos√𝜇_𝑛(𝜏−𝑟)𝑢₁(𝑟+𝑡)𝑑𝑟²+4𝛼₂𝐼_𝜌^2∞_𝑛=1𝜑_𝑛(1)²×^𝜏₀cos√𝜇_𝑛(𝜏−𝑟)𝑢₂(𝑟+𝑡)𝑑𝑟²+4𝜌^2∞_𝑛=1𝜔_𝑛(1)²×^𝜏₀cos√𝜇_𝑛(𝜏−𝑟)⁰_−𝜏𝑔₁𝜂𝑢₁𝑟+𝑡+𝜂𝑑𝜂𝑑𝑟²+4𝐼²_𝜌^∞_𝑛=1𝜑_𝑛(1)²×^𝜏₀cos√𝜇_𝑛(𝜏−𝑟)⁰_−𝜏𝑔₂𝜂𝑢₂𝑟+𝑡+𝜂𝑑𝜂𝑑𝑟². (49)

Note that{cos√𝜇_𝑛𝑡,sin√𝜇_𝑛𝑡;𝑛∈ℕ}is a Riesz basis sequence for𝐿²[0,𝜏]. Thus, there exist positive constants𝑀_𝑖(𝑖=1,2,3,4)such that ‖𝑃(⋅,𝑡)−𝑊(⋅,𝑡+𝜏)‖²_{𝑉1𝐾(0,1)×𝑉1𝐸𝐼(0,1)}+𝑄(⋅,𝑡)−𝑊_𝑡(⋅,𝑡+𝜏)²_{𝐿2𝜌(0,1)×𝐿2𝐼𝜌(0,1)}≤𝑀₁^2𝜏₀𝑢₁(𝑟+𝑡)²𝑑𝑟+𝑀₂^2𝜏₀𝑢₂(𝑟+𝑡)²𝑑𝑟+𝑀₃^2𝜏₀⁰_−𝜏𝑔₁𝜂𝑢₁𝑡+𝑟+𝜂𝑑𝜂²𝑑𝑟+𝑀₄^2𝜏₀⁰_−𝜏𝑔₂𝜂𝑢₂𝑡+𝑟+𝜂𝑑𝜂²𝑑𝑟≤𝑀₁^2𝜏₀𝑢₁(𝑟+𝑡)²𝑑𝑟+𝑀₂^2𝜏₀𝑢₂(𝑟+𝑡)²𝑑𝑟+𝑀₃²𝜏²⁰_−𝜏𝑔₁𝜂²𝑑𝜂^𝑡+𝜏_𝑡−𝜏𝑢₁(𝑠)²𝑑𝑠+𝑀₄²𝜏²⁰_−𝜏𝑔₂𝜂²𝑑𝜂^𝑡+𝜏_𝑡−𝜏𝑢₂(𝑠)²𝑑𝑠. (50)

Let(𝑃(𝑥,𝑡),𝑄(𝑥,𝑡)) be the solution to (11), and 𝐸(𝑡)be its energy functional; then we have 𝐸(𝑡)=12‖(𝑃(𝑥,𝑡),𝑄(𝑥,𝑡))‖²_ℋ,𝑑𝐸(𝑡)𝑑𝑡=−𝑈²₁(𝑃,𝑄)(𝑡)−𝑈²₂(𝑃,𝑄)(𝑡). (51)

Therefore, we have ^𝜏₀‖𝑈(𝑡+𝑟)‖²_ℝ2=𝑑𝑟_{𝑡𝑡+𝜏}𝑈²₁(𝑃,𝑄)(𝑟)+𝑈²₂(𝑟)(𝑡)(),𝑃,𝑄)(𝑑𝑟=𝐸−𝐸𝑡+𝜏_𝜏0‖𝑈(𝑡−𝑟)‖²_ℝ2𝑑𝑟=𝐸(𝑡−𝜏)−𝐸(𝑡). (52)

So, we can get ‖𝑃(⋅,𝑡)−𝑊(⋅,𝑡+𝜏)‖²_{𝑉1𝐾(0,1)×𝑉1𝐸𝐼(0,1)}+𝑄(⋅,𝑡)−𝑊_𝑡(⋅,𝑡+𝜏)²_{𝐿2𝜌(0,1)×𝐿2𝐼𝜌(0,1)}≤max_𝑖=1,2𝑀_𝑖²[𝐸(𝑡)−𝐸(𝑡+𝜏)]+max_𝑗=3,4𝑀_𝑗²𝜏²⁰_−𝜏𝑔₁𝜂²+𝑔₂𝜂²×[𝐸()(.𝑑𝜂𝑡−𝜏−𝐸𝑡+𝜏)] (53)

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: ‖𝑃(⋅,𝑡)−𝑊(⋅,𝑡+𝜏)‖²_{𝑉1𝐾(0,1)×𝑉1𝐸𝐼(0,1)}+𝑄(⋅,𝑡)−𝑊_𝑡(⋅,𝑡+𝜏)²_{𝐿2𝜌()0,1×𝐿2𝐼𝜌()0,1}≤𝑀𝑒^{−𝜀(𝑡−𝜏)}, (54)

where𝑀is a positive constant. So(𝑊(𝑥,𝑡),𝑊_𝑡(𝑥,𝑡))also decays exponentially.

5. The Proof of Theorem 2

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, 𝑃_𝑡(𝑥,𝑡)=𝑄(𝑥,𝑡)+Sin(𝜏ℒ)𝐵Λ₁𝑈(𝑡)+⁰_−𝜏Sin𝜏+𝜂ℒ𝐵Λ₃𝜂𝑈(𝑡)𝑄𝑑𝜂,_𝑡(𝑥,𝑡)=−ℒ𝑃(𝑥,𝑡)+Cos(𝜏ℒ)𝐵Λ₁𝑈(𝑡)+⁰_−𝜏Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑈(𝑡)Γ𝑑𝜂,_𝑁𝑃(⋅,𝑡)=Λ₂𝑈(𝑡),𝑃()()𝑃()0,𝑡=𝑄0,𝑡=0,𝑥,0=𝑃₀(𝑥),𝑄(𝑥,0)=𝑄₀(𝑥). (55)

The observation system corresponding to (55) is 𝑊_𝑡()()()𝑉𝑥,𝑡=𝑉𝑥,𝑡,𝑥∈0,1,𝑡>0,_𝑡()()()𝑊()𝑥,𝑡=−ℒ𝑊𝑥,𝑡,𝑥∈0,1,𝑡>0,𝑥,0=𝑊₀,𝑉(𝑥,0)=𝑉₀,𝑦₁(𝑡)=𝛽₁𝑧(1,𝑡)+¹₀𝐾𝑤_𝑥()()×𝑎𝑥,𝑡−𝜑𝑥,𝑡_′1(𝑥)−𝑎₃(𝑥+)𝑑𝑥₁₀𝐸𝐼𝜑_𝑥(𝑥,𝑡)𝑎^′₃(𝑥)+𝑑𝑥₁₀𝜌𝑧(𝑥,𝑡)𝑏₁(𝑥)𝑑𝑥+¹₀𝐼_𝜌𝜓(𝑥,𝑡)𝑏₃(𝑥)𝑦𝑑𝑥;₂(𝑡)=𝛽₂𝜓(1,𝑡)+¹₀𝐾𝑤_𝑥()()×𝑎𝑥,𝑡−𝜑𝑥,𝑡_′2(𝑥)−𝑎₄(𝑥)+𝑑𝑥₁₀𝐸𝐼𝜑_𝑥(𝑥,𝑡)𝑎^′₄(𝑥)+𝑑𝑥₁₀𝜌𝑧(𝑥,𝑡)𝑏₂(𝑥)𝑑𝑥+¹₀𝐼_𝜌𝜓(𝑥,𝑡)𝑏₄(𝑥)𝑑𝑥, (56)

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 𝑦₁(𝑡)=𝛽₁𝑧(1,𝑡)+𝑊(⋅,𝑡),𝑎₁,𝑎₃^𝑇_{𝑉1𝑘×𝑉1𝐸𝐼}+𝑉(⋅,𝑡),𝑏₁,𝑏₃^𝑇_{𝐿2𝜌×𝐿2𝐼𝜌},𝑦₂(𝑡)=𝛽₁𝜓(1,𝑡)+𝑊(⋅,𝑡),𝑎₂,𝑎₄^𝑇_{𝑉1𝑘×𝑉1𝐸𝐼}+𝑉(⋅,𝑡),𝑏₂,𝑏₄^𝑇_{𝐿2𝜌×𝐿2𝐼𝜌}. (57)

Since 𝑎₁(𝑥)𝑎₂(𝑥)𝑎₃(𝑥)𝑎₄(𝑥)=Sin(𝜏ℒ)𝐵Λ₁+⁰_−𝜏Sin𝜏+𝜂ℒ𝐵Λ₃𝜂𝑏𝑑𝜂,₁(𝑥)𝑏₂(𝑥)𝑏₃(𝑥)𝑏₄(𝑥)=Cos(𝜏ℒ)𝐵Λ₁+⁰_−𝜏Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑌(𝑡)()𝑑𝜂,=𝐶𝑊,𝑉=Λ₂𝑉()+Λ1,𝑡₁ℒ^1/2Sin(𝜏ℒ)𝐵+ℒ^1/20_−𝜏Sin((𝜏+𝜂)ℒ)𝐵Λ₃(𝜂)𝑑𝜂^∗ℒ^1/2𝑊(𝑡)+Λ₁()𝐵+Cos𝜏ℒ_0−𝜏Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑑𝜂^∗𝑉(𝑡). (58)

Taking the Laplace transform for above equation leads to, for any ℜ𝜆>0, 𝜆𝑃(𝑥)=𝑄(𝑥)+Sin(𝜏ℒ)𝐵Λ₁𝑈(𝜆)+⁰_−𝜏Sin𝜏+𝜂ℒ𝐵Λ₃𝜂𝑈(𝜆)(),(𝑥)(𝑥)()𝑑𝜂,𝑥∈0,1𝜆𝑄=−ℒ𝑃+Cos𝜏ℒ𝐵Λ₁𝑈(𝜆)+⁰_−𝜏Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑈(𝜆)Γ𝑑𝜂,_𝑁𝑃(⋅,𝑡)=Λ₂𝑈(𝜆),𝑃()()𝑌(𝜆)0,𝑡=𝑄0,𝑡=0,=Λ₂𝑄(1)+Λ₁ℒ^1/2Sin(𝜏ℒ)𝐵+ℒ^1/20_−𝜏Sin((𝜏+𝜂)ℒ)𝐵Λ₃(𝜂)𝑑𝜂^∗ℒ^1/2𝑃(⋅)+Λ₁()𝐵+Cos𝜏ℒ_0−𝜏Cos((𝜏+𝜂)ℒ)𝐵Λ₃(𝜂)𝑑𝜂^∗𝑄(⋅). (59)

We have the following results by solving (59): 𝜆²𝑝(𝑥)=𝜆()+ℒSin𝜏ℒ𝐵Λ₁+⁰_−𝜏Sin𝜏+𝜂ℒ𝐵Λ₃𝜂()𝑑𝜂+Cos𝜏ℒ𝐵Λ₁+⁰_−𝜏Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑑𝜂+𝐵Λ₂𝑈(𝜆),𝑌(𝜆)=𝜆Sin(𝜏ℒ)𝐵Λ₁+⁰_−𝜏Sin𝜏+𝜂ℒ𝐵Λ₃𝜂()𝑑𝜂+Cos𝜏ℒ𝐵Λ₁+⁰_−𝜏Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑑𝜂+𝐵Λ₂^∗𝑃. (60)

So we can get 𝑌(𝜆)=𝜆sin(𝜏ℒ)𝐵Λ₁+⁰_−𝜏Sin𝜏+𝜂ℒ𝐵Λ₃𝜂()𝑑𝜂+Cos𝜏ℒ𝐵Λ₁+⁰_−𝜏Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑑𝜂+𝐵Λ₂^∗𝜆²+ℒ⁻¹×𝜆Sin(𝜏ℒ)𝐵Λ₁+⁰_−𝜏Sin𝜏+𝜂ℒ𝐵Λ₃𝜂()𝑑𝜂+Cos𝜏ℒ𝐵Λ₁+⁰_−𝜏Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑑𝜂+𝐵Λ₂𝑈(𝜆), (61)

and hence the transform matrix is 𝐻(𝜆)=𝜆Sin(𝜏ℒ)𝐵Λ₁+⁰_−𝜏Sin𝜏+𝜂ℒ𝐵Λ₃𝜂()𝑑𝜂+Cos𝜏ℒ𝐵Λ₁+⁰_−𝜏Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑑𝜂+𝐵Λ₂^∗𝜆²+ℒ⁻¹×𝜆Sin(𝜏ℒ)𝐵Λ₁+⁰_−𝜏Sin𝜏+𝜂ℒ𝐵Λ₃𝜂()𝑑𝜂+Cos𝜏ℒ𝐵Λ₁+⁰_−𝜏Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑑𝜂+𝐵Λ₂. (62)

For any𝑍=(𝑧₁,𝑧₂)∈ℂ², we can get (𝐻(𝜆)𝑍,𝑍)_ℂ²=^∞_𝑛=11𝜆²+𝜇_𝑛×𝜆Sin(𝜏ℒ)𝐵Λ₁+⁰_−𝜏Sin𝜏+𝜂ℒ𝐵Λ₃𝜂()𝑑𝜂+Cos𝜏ℒ𝐵Λ₁+⁰_−𝜏Cos𝜏+𝜂ℒ×𝐵Λ₃𝜂𝑑𝜂+𝐵Λ₂𝑍,Φ_{𝑛𝐿2𝜌×𝐿2𝐼𝜌}². (63)

We can easily get 𝐵Λ₂𝑍,Φ_{𝑛𝐿2𝜌×𝐿2𝐼𝜌}=𝛽₁𝑧₁𝜌𝑤_𝑛(1)+𝛽₂𝑧₂𝐼_𝜌𝜑_𝑛(1),Sin(𝜏ℒ)𝐵Λ₁𝑍,Φ_{𝑛𝐿2𝜌×𝐿2𝐼𝜌}=Sin𝜏√𝜇_𝑛√𝜇_𝑛𝛼₁𝑧₁𝜌𝑤_𝑛(1)+𝛼₂𝑧₂𝐼_𝜌𝜑_𝑛(1),Cos(𝜏ℒ)𝐵Λ₁𝑍,Φ_{𝑛𝐿2𝜌×𝐿2𝐼𝜌}=Cos𝜏√𝜇_𝑛𝛼₁𝑧₁𝜌𝑤_𝑛(1)+𝛼₂𝑧₂𝐼_𝜌𝜑_𝑛(1)_0−𝜏,Sin(𝜏+𝜂)ℒ𝐵Λ₃(𝜂)𝑑𝜂𝑍,Φ_{𝑛𝐿2𝜌×𝐿2𝐼𝜌}=⁰_−𝜏Sin𝜏+𝜂√𝜇_𝑛√𝜇_𝑛×𝑔₁𝜂𝑧₁𝜌𝑤_𝑛(1)+𝑔₂𝜂𝑧₂𝐼_𝜌𝜑_𝑛(1)𝑑𝜂,_0−𝜏Cos(𝜏+𝜂)ℒ𝐵Λ₃(𝜂)𝑑𝜂𝑍,Φ_{𝑛𝐿2𝜌×𝐿2𝐼𝜌}=⁰_−𝜏Cos𝜏+𝜂√𝜇_𝑛×𝑔₁𝜂𝑧₁𝜌𝑤_𝑛(1)+𝑔₂𝜂𝑧₂𝐼_𝜌𝜑_𝑛(1)𝑑𝜂. (64)

Thus, we have 𝜆Sin(𝜏ℒ)𝐵Λ₁+⁰_−𝜏Sin𝜏+𝜂ℒ𝐵Λ₃𝜂()𝑑𝜂+Cos𝜏ℒ𝐵Λ₁+⁰_−𝜏Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑑𝜂+𝐵Λ₂𝑍,Φ_{𝑛𝐿2𝜌×𝐿2𝐼𝜌}²|𝜆|≤𝐹²𝜇_𝑛+2𝜌𝜔_𝑛(1)²+𝐼_𝜌𝜑_𝑛(1)²‖𝑍‖²_ℂ2, (65)

where𝐹is a positive constant dependent on𝛼_𝑖,𝛽_𝑖,𝑔_𝑖(𝜂),𝑖=1,2. Therefore, we have the following result: ‖𝐻(𝜆)‖≤𝐹^∞_𝑛=11|𝜆|²+𝜇_𝑛|𝜆|²𝜇_𝑛×𝜌𝜔+2_𝑛(1)²+𝐼_𝜌𝜑_𝑛(1)². (66)

From Lemma 4, we have sup_{ℜ𝜆>𝛿>0}‖𝐻(𝜆)‖<∞. (67)

Hence the system (6) is𝐿²_locwell posed (see, [15]).

Next, we consider the exact observability of the system (6).

Lemma 6 (see [16]). Let ℋ 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)={𝜇_𝑛;𝑛∈ℕ};(2)the spectra of ℒ satisfy the separable condition inf_𝑛≠𝑚√𝜇_𝑛−√𝜇_𝑚=𝛿>0; (68)(3)the corresponding eigenvectors {Φ_𝑛;𝑛∈ℕ} with ‖Φ_𝑛‖_ℋ=1 form a normalized orthogonal basis for ℋ.
Let 𝕐 be a Hilbert space. Assume that 𝐶∶𝐷(ℒ)→𝕐 is an admissible observation operator for ℒ. Then the following system: 𝑍_𝑡𝑡+ℒ𝑍(𝑡)=0,𝑍(0)=𝑍₀,𝑍_𝑡(0)=𝑍₁,𝑌(𝑡)=𝐶𝑍,𝑍_𝑡 (69) is exactly observable in finite time in the energy space 𝐷(ℒ^1/2)×ℋ if and only if inf_𝑛∈ℕ𝐶Φ_𝑛𝑖√𝜇_𝑛,Φ_𝑛𝕐>0. (70)

Now we apply Lemma 6 to the system (55). We can easily know that the condition (68) is fulfilled when𝑐₁=𝐾/𝜌≠𝑐₂=𝐸𝐼/𝐼_𝜌 (see Remark 2.1 in [8]).

ForΦ_𝑛(𝑥)=(𝑤_𝑛(𝑥),𝜑_𝑛(𝑥))^𝑇, we have 𝐶Φ_𝑛𝑖√𝜇_𝑛,Φ_𝑛=Λ₂Φ_𝑛(1)+Λ₁ℒ^1/2Sin(𝜏ℒ)𝐵+ℒ^1/20_−𝜏Sin((𝜏+𝜂)ℒ)𝐵Λ₃(𝜂)𝑑𝜂^∗ℒ^1/2Φ_𝑛𝑖√𝜇_𝑛+Λ₁()𝐵+Cos𝜏ℒ_0−𝜏Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑑𝜂^∗Φ_𝑛. (71)

For any𝑍=(𝑧₁,𝑧₂)∈ℂ², we have Λ₁ℒ^1/2Sin(𝜏ℒ)𝐵^∗ℒ^1/2Φ_𝑛𝑖√𝜇_𝑛,𝑍_ℂ2=ℒ^1/2Φ_𝑛𝑖√𝜇_𝑛,ℒ^1/2Sin(𝜏ℒ)𝐵Λ₁𝑍_{𝐿2𝜌×𝐿2𝐼𝜌}=−𝑖√𝜇_𝑛Φ_𝑛,Sin(𝜏ℒ)𝐵Λ₁𝑍_{𝐿2𝜌×𝐿2𝐼𝜌}=−𝑖sin√𝜇_𝑛𝜏𝛼₁𝑧₁𝜌𝜔_𝑛(1)+𝛼₂𝑧₂𝐼_𝜌𝜑_𝑛(1). (72)

Similarity, we have ℒ^1/20_−𝜏Sin𝜏+𝜂ℒ𝐵Λ₃𝜂𝑑𝜂^∗ℒ^1/2Φ_𝑛𝑖√𝜇_𝑛,𝑍_ℂ2=−𝑖⁰_−𝜏sin𝜏+𝜂√𝜇_𝑛×𝑔₁𝜂𝑧₁𝜌𝜔_𝑛(1)+𝑔₂𝜂𝑧₂𝐼_𝜌𝜑_𝑛(1Λ)𝑑𝜂,₁(Cos(𝜏ℒ)𝐵)^∗Φ_𝑛,𝑍_ℂ2=cos√𝜇_𝑛𝜏𝛼₁𝑧₁𝜌𝜔_𝑛(1)+𝛼₂𝑧₂𝐼_𝜌𝜑_𝑛(1)_0−𝜏,Cos𝜏+𝜂ℒ𝐵Λ₃𝜂𝑑𝜂^∗Φ_𝑛,𝑍_ℂ2=⁰_−𝜏cos𝜏+𝜂√𝜇_𝑛×𝑔₁𝜂𝑧₁𝜌𝜔_𝑛(1)+𝑔₂𝜂𝑧₂𝐼_𝜌𝜑_𝑛(1)𝑑𝜂. (73)

Thus it holds that 𝐶Φ_𝑛𝑖√𝜇_𝑛,Φ_𝑛=𝛽₁𝜌+𝛼₁𝑒^{√−𝑖𝜏𝜇𝑛}+∫⁰_−𝜏𝑔₁𝜂𝑒^{√−𝑖((𝜏+𝜂)𝜇𝑛)}𝑑𝜂𝜌𝑤_𝑛(1)𝛽₂𝐼_𝜌+𝛼₂𝑒^{√−𝑖𝜏𝜇𝑛}+∫⁰_−𝜏𝑔₂𝜂𝑒^{√−𝑖((𝜏+𝜂)𝜇𝑛)}𝑑𝜂𝐼_𝜌𝜑_𝑛(1). (74)

Set 𝜉_𝑛⁽¹⁾=⁰_−𝜏𝑔₁𝜂𝑒^{√−𝑖𝜇𝑛(𝜏+𝜂)}𝜉𝑑𝜂,_𝑛(2)=⁰_−𝜏𝑔₂𝜂𝑒^{√−𝑖𝜇𝑛(𝜏+𝜂)}𝑑𝜂. (75)

Then 𝐶Φ_𝑛𝑖√𝜇_𝑛,Φ_𝑛²_ℂ2=𝛽₁𝜌+𝛼₁𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽¹⁾²𝜌𝑤_𝑛(1)²+𝛽₂𝐼_𝜌+𝛼₂𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽²⁾²𝐼_𝜌𝜑_𝑛(1)². (76)

Obviously, when 𝐴₁=inf_𝑛𝛽₁𝜌+𝛼₁𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽¹⁾𝐴>0,₂=inf_𝑛𝛽₂𝐼_𝜌+𝛼₂𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽²⁾>0, (77)

we have 𝐶Φ_𝑛𝑖√𝜇_𝑛,Φ_𝑛²_ℂ2≥𝐴²₁𝜌𝑤_𝑛(1)²+𝐴²₂𝐼_𝜌𝜑_𝑛(1)²≥min𝐴₁,𝐴₂𝜌𝑤_𝑛(1)²+𝐼_𝜌𝜑_𝑛(1)²; (78)

using Lemma 4, inf_𝑛𝐶Φ_𝑛𝑖√𝜇_𝑛,Φ_𝑛²_ℂ2>0. (79)

According to Lemma 6, the system (56) is exactly observable in finite time, and hence the closed-loop system (11) is exponentially stable.

If for all𝑛∈ℕ, 𝛽₁𝜌+𝛼₁𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽¹⁾𝛽>0,₂𝐼_𝜌+𝛼₂𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽²⁾>0, (80)

we can see that in this case, there is no eigenvalue of system (11) on the imaginary axis. Moreover, if the conditions inf_𝑛𝛽₁𝜌+𝛼₁𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽¹⁾=0,inf_𝑛𝛽₂𝐼_𝜌+𝛼₂𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽²⁾=0 (81)

hold, 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/𝐼_𝜌is 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, 𝐶Φ_𝑛𝑖√𝜇_𝑛,Φ_𝑛²_ℂ2=𝛽₁𝜌+𝛼₁𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽¹⁾²𝜌𝑤_𝑛(1)²+𝛽₂𝐼_𝜌+𝛼₂𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽²⁾²|𝐼_𝜌𝜑_𝑛(1)|²; (82)

one only needs to request inf_𝑛𝐶Φ_𝑛𝑖√𝜇_𝑛,Φ_𝑛²_ℂ2>0, (83)

so, the conditions inf_𝑛𝛽₁𝜌+𝛼₁𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽¹⁾>0,inf_𝑛𝛽₂𝐼_𝜌+𝛼₂𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽²⁾>0, (84)

are sufficient, but not necessary. Since 𝜉_𝑛⁽¹⁾=⁰_−𝜏𝑔₁𝜂𝑒^{𝑖√𝜇𝑛(𝜏−𝜂)}𝜉𝑑𝜂,_𝑛(2)=⁰_−𝜏𝑔₂𝜂𝑒^{𝑖√𝜇𝑛(𝜏−𝜂)}𝑑𝜂, (85)

so lim_𝑛→∞𝜉_𝑛^(𝑗)=0,𝑗=1,2. Therefore, when 𝛽₁𝜌+𝛼₁𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽¹⁾𝛽>0,₂𝐼_𝜌+𝛼₂𝑒^{√−𝑖𝜏𝜇𝑛}+𝜉_𝑛⁽²⁾>0 (86)

that means that there is no eigenvalue on the imaginary axis, and inf_𝑛𝛽₁𝜌+𝛼₁𝑒^{√−𝑖𝜏𝜇𝑛}>0,inf_𝑛𝛽₂𝐼_𝜌+𝛼₂𝑒^{√−𝑖𝜏𝜇𝑛}>0, (87)

we have inf_𝑛𝐶Φ_𝑛𝑖√𝜇_𝑛,Φ_𝑛²_ℂ2>0. (88)

Clearly, when𝛽₁≠𝜌𝛼₁and𝛽₂≠𝐼_𝜌𝛼₂, we also have inf_𝑛𝐶Φ_𝑛𝑖√𝜇_𝑛,Φ_𝑛²_ℂ2>0. (89)

Therefore, the conditions in Theorem 2 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.

Acknowledgment

This research was supported by the National Science Natural Foundation in China (NSFC-61174080).