\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 75, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/75\hfil 
Stabilization of Euler-Bernoulli beam equations]
{Stabilization of Euler-Bernoulli beam equations with variable
 coefficients under delayed boundary output feedback}

\author[K.-Y. Yang, J.-J. Li, J. Zhang \hfil EJDE-2015/75\hfilneg]
{Kun-Yi Yang, Jing-Jing Li, Jie Zhang}

\address{Kun-Yi Yang (corresponding author)\newline
College of Science, North China University of Technology,
Beijing 100144, China}
\email{kyy@amss.ac.cn}

\address{Jing-Jing Li \newline
College of Science, North China University of Technology,
Beijing 100144, China}
\email{ 350572566@qq.com}

\address{Jie Zhang \newline
College of Science, North China University of Technology,
Beijing 100144, China}
\email{jzhang26@ncut.edu.cn}

\thanks{Submitted December 30, 2014. Published March 24, 2015.}
\subjclass[2000]{35J10, 93C20, 93C25}
\keywords{Euler-Bernoulli beam equation; variable coefficients; 
 time delay; \hfill\break\indent observer; feedback control; exponential stability}

\begin{abstract}
 In this article, we study the stabilization of an Euler-Bernoulli
 beam equation with variable coefficients where boundary observation
 is subject to a time delay. To resolve the mathematical complexity
 of variable coefficients, we design an observer-predictor based on
 the well-posed open-loop system: the state of system is estimated
 with available observation and then predicted without observation.
 We show that the closed-loop system is stable exponentially under 
 estimated state feedback by a numerical simulation illustrating 
 our results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks

\section{Introduction}

The phenomenon of time delay is commonly observed in modern engineering
and scientific research \cite{Datko1986, Datko1988, Datko1991,Datko1993,Fia,
Fridman,Logemann,Krstic2008}.
Much attention has been devoted to the stability of control systems with
time delay. Nevertheless, even a small delay may break the system's
stability \cite{Datko1986,Datko1988,Datko1991,Datko1993,Fia,Gumowski}.
It is indicated in \cite{Flem} that for distributed parameter control
systems, time delay in observation and control can cause complications.
Stimulated by the work in \cite{Guo3}, we solve the stabilization problem
with delayed observation and boundary control, for the one-dimensional
Euler-Bernoulli beam equation \cite{Yang1}.

In this article, we focus on the boundary stabilization of an Euler-Bernoulli
beam equation with variable coefficients where boundary observation contains
a fixed time delay. This is a generalization of the similar work such as
\cite{Yang1} for the beam equation with constant coefficients.
It is obvious that variable coefficients present more mathematical challenges,
making the stabilization problems of the system much more complicated since
it is difficult to construct the Lyapunov functions and estimate the eigenvalues
and eigenfunctions by asymptotic analysis.

Consider the following nonuniform Euler-Bernoulli beam equation with linear
boundary feedback control:

\begin{equation}\label{101}
 \begin{gathered}
\rho(x)w_{tt}(x,t)+(EI(x)w_{xx}(x,t))_{xx}=0, \quad 0<x<1,\; t>0,\\
w(0,t)=w_x(0,t)=w_{xx}(1,t)=0, \quad t \geq 0,\\
(EI(x)w_{xx})_x(1,t)=u(t),\quad t \geq 0,\\
y(t)=w_t(1,t-\tau),\quad t>\tau,\\
w(x,0)=w_0(x),w_t(x,0)=w_1(x), \quad 0 \leq x \leq 1,
\end{gathered}
\end{equation}
where $x$ stands for the position and $t$ the time, $w$ is the state,
$u$ is the boundary controller input,
$(w_0,w_1)^T$ is the initial value, $\tau>0$ is a known constant time
delay, and $y$ is the delayed observation(or output)
 which suffers from a given time delay $\tau$.
$EI(x)(>0) \in C^2[0,1]$ is the flexural rigidity of the beam, and
$\rho(x)(>0) \in C[0,1]$ is the mass density at $x$.

The system above is considered in the energy state space
\[
\mathcal{H} = H_E^2(0,1)\times L^2(0,1), \quad
H_E^2(0,1)=\{f\in H^2(0,1):f(0)=f'(0)=0\}.
\]
The energy of the system is
 $$
 E_0(t)= \frac{1}{2}\int^1_0[EI(x)w^2_{xx}(x,t)+\rho(x)w^2_{t}(x,t)]dx.
 $$

As noted in \cite{Datko1988} (where $EI(x)=\rho(x)=1$), even a small
amount of time delay in the stabilizing boundary output feedback schemes
 destabilizes the system. Therefore, it is important to design stabilizing
controllers that are robust to time delay for systems described in \eqref{101}.

The next section shows the well-posedness of the considered open-loop system.
In section 3, we design the observer and predictor for the system.
The asymptotic stability of the closed-loop system under the estimated
 state feedback control is then studied in
section 4. Section 5 illustrates the simulation results and concludes the paper.

\section{Well-posedness of the open-loop system}

We introduce a new variable $z(x,t)=w_t(1,t-x\tau)$. Then the system
\eqref{101} becomes
\begin{equation}\label{102}
\begin{gathered}
\rho(x)w_{tt}(x,t)+(EI(x)w_{xx}(x,t))_{xx}=0, \quad 0<x<1,\; t>0,\\
w(0,t)=w_x(0,t)=w_{xx}(1,t)=0,\quad t \geq 0,\\
(EI(x)w_{xx})_x(1,t)=u(t),\quad t \geq 0,\\
\tau z_t(x,t)+z_x(x,t)=0, \quad 0<x<1,t\geq0,\\
z(0,t)=w_t(1,t),\quad t\geq0,\\
w(x,0)=w_0(x),w_t(x,0)=w_1(x), \quad 0 \leq x \leq 1,\\
z(x,0)=z_0(x),\quad 0 \leq x \leq 1,\\
y(t)=z(1,t),\quad t \geq \tau,\\
\end{gathered}
\end{equation}
where $z_0$ is the initial value of the variable $z$.

We consider the system \eqref{102} in the energy state space
$\mathbb{H}=\mathcal{H} \times L^2(0,1)$, with the state
variable $(w(\cdot,t),w_t(\cdot,t),z(\cdot,t))^T$
for which the inner product induced norm is defined as following:
\begin{align*}
E_1(t)&=\frac{1}{2}\|(w(\cdot,t),w_t(\cdot,t),z(\cdot,t))^T\|^2_{\mathbb{H}}\\
&= \frac{1}{2}\int^1_0[EI(x)w^2_{xx}(x,t)+\rho(x)w^2_{t}(x,t)+z^2(x,t)]dx.
\end{align*}
The input space and the output space are the same $U=Y=\mathbb{C}$.

\begin{theorem}\label{thm2.1}
System \eqref{102} is well-posed: For any
$(w_0,w_1,z_0)^T\in \mathbb{H}$ and  $u \in L_{\rm loc}^2(0, \infty)$, 
there exists a unique solution of \eqref{102} such that
$(w(\cdot,t),w_t(\cdot,t),z(\cdot,t))^T$ belongs to $C(0, \infty; \mathbb{H})$;
and for any $T>0$, there exist a constant $ C_T>0$ such that
\begin{align*}
&\|(w(\cdot,T),w_t(\cdot,t),z(\cdot,T))^T\|_{\mathbb{H}}^2+\int_0^T|y(t)|^2dt\\
&\leq C_T\Big[\|(w_0,w_1,z_0)^T\|_{\mathbb{H}}^2+\int_0^T|u(t)|^2dt\Big].
\end{align*}
\end{theorem}

\begin{proof}
Firstly, we represent the system
\begin{equation}
\begin{gathered}
\rho(x)w_{tt}(x,t)+(EI(x)w_{xx}(x,t))_{xx}=0,\quad 0<x<1,\;t\geq0,\\
w(0,t)=w_x(0,t)=w_{xx}(1,t)=0,\quad t\geq0,\\
(EI(x)w_{xx})_x(1,t)=u(t),\quad t\geq0,\\
y_{w}(t)=w_t(1,t),\quad t\geq0,
\end{gathered}
\end{equation}
as a second-order system in $\mathcal{H}$,
\begin{equation}\label{103}
\begin{gathered}
w_{tt}(\cdot,t)+Aw(\cdot,t)+Bu(t)=0,\quad 0<x<1,\;t\geq0,\\
y_{\omega}(t)=B^{*}w_t(\cdot,t),\quad t\geq0,
\end{gathered}
\end{equation}
where $A$ is a self-adjoint operator in $\mathcal{H}$£ and $B$ is
the input operator:
\begin{equation}\label{104}
 \begin{gathered}
Af=\frac{1}{\rho(x)}(EI(x)f'')'', \\
\forall f \in D(A)=\{f\in H^4(0,1)\cap H_E^2(0,1) : f''(1)=(EI f'')'(1)=0\},\\
B=\delta(x-1).
\end{gathered}
\end{equation}
Here $\delta(\cdot)$ denote the Dirac distribution.
It was shown in \cite{Guoluo}
that system \eqref{103} and \eqref{104} is well-posed in the
sense of Salamon \cite{Curtain}:
for any $u \in L_{\rm loc}^2(0, \infty)$ and $(w_0,w_1)^T\in \mathcal{H}$,
 there exists a unique solution
$(w(\cdot,t),w_t(\cdot,t))^T\in C(0,\infty;\mathcal{H})$ to \eqref{103} and for
any $T>0$, there exists a
constant $D_T>0$ such that
\begin{equation}\label{105}
\begin{aligned}
&\|(w(\cdot,T),w_t(\cdot,T))^T\|_{\mathcal{H}}^2+\int_0^T|y_w(t)|^2dt\\
&\leq D_T\Big[\|(w_0,w_1)^T\|_{\mathcal{H}}^2+\int_0^T|u(t)|^2dt\Big].
\end{aligned}
\end{equation}
Then the following inequality can be shown similarly as those
 in \cite{GuoYang}:
\begin{align*}
&\|(w(\cdot,T),w_t(\cdot,T),z(\cdot,T))^T\|_{\mathcal{H}}^2+\int_0^T | y(t) |^2 dt\\
&\leq C_T\left[\|(w_0,w_1,z_0)^T\|_{\mathcal{H}}^2+\int_0^T|u(t)|^2dt\right],
\end{align*}
for a constant $C_T>0$. The details are omitted.
\end{proof}


Theorem \ref{thm2.1} illustrates that, for any initial value
in the state space, the output belongs to $L^2_{\rm loc}(\tau,\infty)$
as long as the input $u$ belongs to $L^2_{\rm loc}(0,\infty)$. This fact
is particularly necessary to the solvability of observer shown in
the next section (\cite{Guoluo,Guo3}).


\section{Observer and predictor design}

For any fixed time delay $\tau>0$, and when $t>\tau$, we propose a two-step method
 to estimate the state of \eqref{101} by designing the observer
and predictor systems.
\smallskip

\noindent\textbf{Step 1.}
 From the known observation signal $\{y(s+\tau): s\in [0,t-\tau], t>\tau \}$,
we construct an observer system to estimate the state
$\{w(x,s): s\in [0,t-\tau], t>\tau \}$ which satisfies
\begin{equation}\label{108}
\begin{gathered}
\rho(x)w_{ss}(x,s)+(EI(x)w_{xx}(x,s))_{xx}=0,\quad
 0<x<1,\; 0<s<t-\tau,\; t>\tau,\\
w(0,s)=w_x(0,s)=w_{xx}(1,s)=0,\quad 0 \leq s \leq t-\tau,\; t>\tau,\\
(EI(x)w_{xx})_x(1,s)=u(s), \quad 0 \leq s \leq t-\tau, \; t>\tau,\\
y(s+\tau)=w_s(1,s), \quad 0 \leq s \leq t-\tau,\; t>\tau.
\end{gathered}
\end{equation}
Then a Luenberger observer naturally can be constructed for
the system \eqref{108},
\begin{equation}\label{109}
 \begin{gathered}
\rho(x)\widehat{w}_{ss}(x,s)+(EI(x)\widehat{w}_{xx}(x,s))_{xx}=0,\quad
 0<x<1, \; 0<s<t-\tau, \; t>\tau,\\
\widehat{w}(0,s)=\widehat{w}_x(0,s)=\widehat{w}_{xx}(1,s)=0, \quad
 0 \leq s \leq t-\tau,\; t>\tau,\\
(EI(x)\widehat{w}_{xx})_x(1,s)=u(s)+k_1[\widehat{w}_s(1,s)-y(s+\tau)], \quad
0 \leq s \leq t-\tau, \;t>\tau, k_1>0, \\
\widehat{w}(x,0)=\widehat{w}_0(x),\widehat{w}_s(x,0)=\widehat{w}_1(x), \quad
 0 \leq x \leq 1,
\end{gathered}
\end{equation}
where $(\widehat{w}_0,\widehat{w}_1)^T$ is an arbitrary assigned
initial state of the observer.

For \eqref{109} to be an observer for \eqref{108}, we have
to show its convergence. To do this, we set
\begin{equation}\label{10901}
\varepsilon(x,s)=\widehat {w}(x,s)-w(x,s), \quad
 0 \leq s \leq t-\tau, t>\tau.
 \end{equation}
Then by \eqref{108} and \eqref{109}, $\varepsilon$ satisfies
\begin{equation}\label{110}
 \begin{gathered}
\rho(x)\varepsilon_{ss}(x,s)+(EI(x)\varepsilon_{xx}(x,s))_{xx}=0, \quad
 0<x<1,\; 0<s<t-\tau,\; t>\tau,\\
\varepsilon(0,s)=\varepsilon_x(0,s)=\varepsilon_{xx}(1,s)=0, \quad
 0 \leq s \leq t-\tau,\; t>\tau,\\
(EI(x)\varepsilon_{xx})_x(1,s)=k_1\varepsilon_s(1,s),\quad
0 \leq s \leq t-\tau,\; t>\tau,\; k_1>0, \\
\varepsilon(x,0)=\widehat{w}_0(x)-w_0(x),\quad 0 \leq x \leq 1,\\
\varepsilon_s(x,0)=\widehat{w}_1(x)-w_1(x),\quad 0 \leq x \leq 1.
\end{gathered}
\end{equation}
The system above can be written as
\begin{equation}\label{10503}
\frac{d}{ds} \begin{pmatrix}
 \varepsilon(\cdot, s) \\
 \varepsilon_s(\cdot, s) 
 \end{pmatrix}
=\mathbb{B}  \begin{pmatrix}
 \varepsilon(\cdot, s) \\
 \varepsilon_s(\cdot, s) 
 \end{pmatrix},
\end{equation}
where
\begin{equation}\label{10501}
 \begin{gathered}
\mathbb{B}(f,g)^T = (g, -\frac{1}{\rho(x)}(EI(x)f''(x))'')^T,\\
\begin{aligned}
D(\mathbb{B})=\big\{&(f,g)\in (H^4(0,1) \cap H_E^2(0,1))
\times H^2_E(0,1) :\\
 &f''(1)=0,(EIf'')'(1)=k_1g(1)\big\},
\end{aligned}
\end{gathered}
\end{equation}
and $\mathbb{B}$ generates an exponentially stable $C_0$-semigroup on
$\mathcal{H}$ satisfying:
\begin{equation}
\label{111}
\| e^{\mathbb{B}s}\| \leq Me^{-\omega s}, \quad \forall s \ge 0,
\end{equation}
for some positive constants $M,\omega$. Hence, for any 
$(w_0,w_1)^T\in \mathcal{H}$ and $(\widehat{w}_0,\widehat{w}_1)^T\in \mathcal{H}$, there
exists a unique solution to \eqref{110} such that
\begin{equation}\label{10301}
\|(\varepsilon(\cdot,s),\varepsilon_s(\cdot,s))^T\|_{\mathcal{H}}
\leq Me^{-\omega s}\|(\widehat{w}_0-w_0,\widehat{w}_1-w_1)^T\|_{\mathcal{H}},
\end{equation}
for all $s \in[0,t-\tau]$ and all $t>\tau$.
\smallskip


\noindent\textbf{Step 2.}
 Predict $\{(w(x,s),w_s(x,s))^T, s\in (t-\tau,t],
t>\tau\}$ by 
\[
\{(\widehat{w}(x,s),\widehat{w}_s(x,s))^T, s\in
[0,t-\tau], t>\tau\}.
\]
This is done by solving \eqref{101} with estimated initial value
$(\widehat{w}(x,t-\tau),\widehat{w}_s(x,t-\tau))^T$ obtained from
\eqref{109}:
\begin{equation}
 \begin{gathered}
\rho(x)\widehat{w}_{ss}(x,s,t)+(EI(x)\widehat{w}_{xx}(x,s,t))_{xx}=0,\quad
 0<x<1, \; t-\tau<s<t,\;  t>\tau,\\
\widehat{w}(0,s,t)=\widehat{w}_x(0,s,t)=\widehat{w}_{xx}(1,s,t)=0, \quad
 t-\tau \leq s \leq t,\; t>\tau,\\
(EI(x)\hat{w}_{xx})_x(1,s,t)=u(s),\quad t-\tau \leq s \leq t,\;
 t>\tau,\\
\widehat{w}(x,t-\tau,t)=\widehat{w}(x,t-\tau),\widehat{w}_s(x,t-\tau,t)
=\widehat{w}_s(x,t-\tau),\\
0 \leq x \leq 1,\; t-\tau \leq s \leq t,\; t>\tau.
\end{gathered}
\end{equation}
We finally get the estimated state variable by
\begin{equation}
\label{10601} \widetilde{w}(x,t) 
= \widehat{w}(x,t,t), \quad \forall t>\tau,
\end{equation}
which is assured by Theorem \ref{thm3.1} below.

\begin{theorem}\label{thm3.1}
For all $t>\tau$, we have
\begin{equation} \label{112}
\| (w(\cdot,t)-\tilde{w_t}(\cdot,t),w_t(\cdot,t)-\tilde{w_t}(\cdot,t))^T\|_{\mathcal{H}}
\leq  Me^{-\omega(t-\tau)}\|(\widehat{w}_0-w_0,\widehat{w}_1-w_1)^T\|_{\mathcal{H}},
\end{equation}
where $(\widehat{w}_0,\widehat{w}_1)^T$ is the initial state of
observer \eqref{109}, $(w_0,w_1)^T$ is the initial state of original
system \eqref{101}, $M,\omega$ are constants in \eqref{111}.
\end{theorem}

\begin{proof}
Let \begin{equation}\label{10701}
\varepsilon(x,s,t)=\widehat{w}(x,s,t)-w(x,s),\quad t-\tau \leq s \leq
t,\; t>\tau.
\end{equation}
Then $\varepsilon(x,s,t)$ satisfies
\begin{equation}
\begin{gathered}
\rho(x)\varepsilon_{ss}(x,s,t)+(EI(x)\varepsilon_{xx}(x,s,t))_{xx}=0, \\
0<x<1,\;t-\tau<s<t,\; t>\tau;\\
\varepsilon(0,s,t)=\varepsilon_x(0,s,t)=\varepsilon_{xx}(1,s,t)
=(EI(x)\varepsilon_{xx})_x(1,s,t)=0, \\ 
t-\tau \leq s \leq t,\; t>\tau;\\
\varepsilon(x,t-\tau,t)=\varepsilon(x,t-\tau),\varepsilon_s(x,t-\tau,t)
=\varepsilon_s(x,t-\tau),\\
 0 \leq x \leq 1,\; t-\tau \leq s \leq t, \;t>\tau;
\end{gathered}
\end{equation}
which is a conservative system
\begin{equation}\label{10702}
\|(\varepsilon(\cdot,s,t),\varepsilon_s(\cdot,s,t))^T\|_{\mathcal{H}}
= \|(\varepsilon(\cdot,t-\tau),\varepsilon_s(\cdot,t-\tau))^T\|_{\mathcal{H}}.
\end{equation}
Collecting \eqref{10301}, \eqref{10601} and \eqref{10702} gives
\eqref{112}. 
\end{proof}


\section{Stabilization by the estimated state feedback}

Since the feedback $u(t)=k_2\tilde{w_t}(1,t)=k_2\widehat w_s(1,t,t)$ $(k_2>0)$
stabilizes exponentially the system \eqref{101}, and we have the
estimation $\tilde{w_t}(1,t)$ of $w_t(1,t)$, it is natural to design
the estimated state feedback control law of the following:
\begin{equation} \label{113} 
u^*(t)= \begin{cases}
k_2\tilde{w_t}(1,t)=k_2\widehat w_s(1,t,t),\quad t>\tau,\; k_2>0, \\
0,\quad t\in[0,\tau].
\end{cases}
\end{equation}
The closed-loop system becomes a system of partial differential
equations \eqref{114}-\eqref{116} via applying the control law
above:
\begin{equation}\label{114}
\begin{gathered}
\rho(x)w_{tt}(x,t)+(EI(x)w_{xx}(x,t))_{xx}=0,\quad 0<x<1,\;t>0,\\
w(0,t)=w_x(0,t)=w_{xx}(1,t)=0,\quad t \geq 0,\\
(EI(x)w_{xx})_{x}(1,t)=u^*(t),\quad t \geq 0,\\
w(x,0)=w_0(x), \quad w_t(x,0)=w_1(x),\quad 0 \leq x \leq 1,
\end{gathered}
\end{equation}
and
% \label{115}
\begin{gather*}
\rho(x)\widehat{w}_{ss}(x,s)+(EI(x)\widehat{w}_{xx}(x,s))_{xx}=0,\quad
 0<x<1,\; 0<s<t-\tau,\; t>\tau,\\
\widehat{w}(0,s)=\widehat{w}_x(0,s)=\widehat{w}_{xx}(1,s)=0,\quad
 0 \leq s \leq t-\tau,\; t>\tau,\\
(EI(x)\widehat{w}_{xx})_x(1,s)=u^*(s)+k_1[\widehat{w}_s(1,s)-w_s(1,s)],\quad
0 \leq s \leq t-\tau,\; t>\tau,\; k_1>0,\\
\widehat{w}(x,0)=\widehat{w}_0(x), \quad 
\widehat{w}_s(x,0)=\widehat{w}_1(x),\; 0 \leq x \leq 1,
\end{gather*}
and
\begin{equation} \label{116}
\begin{gathered}
\rho(x)\widehat{w}_{ss}(x,s,t)+(EI(x)\widehat{w}_{xx}(x,s,t))_{xx}=0,
\quad 0<x<1,\; t-\tau<s<t,\; t>\tau,\\
\widehat{w}(0,s,t)=\widehat{w}_x(0,s,t)=\widehat{w}_{xx}(1,s,t)=0,\quad
t-\tau \leq s \leq t,\; t>\tau,\\
(EI(x)\widehat{w}_{xx})_x(1,s,t)=u^*(s),\;t-\tau \leq s \leq t,\quad t>\tau,\\
\widehat{w}(x,t-\tau,t)=\widehat{w}(x,t-\tau),\widehat{w}_s(x,t-\tau,t)
=\widehat{w}_s(x,t-\tau),\quad 0 \leq x \leq 1,\; t>\tau.
\end{gathered}
\end{equation}

We consider the closed-loop system \eqref{114}-\eqref{116} in the
state space $X=\mathcal{H}^3$. Obviously the system
\eqref{114}-\eqref{116} is equivalent to the system
\eqref{117}-\eqref{119} for $t>\tau$:
\begin{equation}\label{117}
\begin{gathered}
\rho(x)w_{tt}(x,t)+(EI(x)w_{xx}(x,t))_{xx}=0,\quad 0<x<1,\; t>\tau,\\
w(0,t)=w_x(0,t)=w_{xx}(1,t)=0,\quad t>\tau,\\
(EI(x)w_{xx})_{x}(1,t)=k_2[w_t(1,t)+\varepsilon_s(1,t,t)],\quad t>\tau,\; k_2>0, \\
w(x,0)=w_0(x),w_t(x,0)=w_1(x),\quad 0 \leq x \leq 1,
\end{gathered}
\end{equation}
and
\begin{equation}\label{118}
\begin{gathered}
\rho(x)\varepsilon_{ss}(x,s)+(EI(x)\varepsilon_{xx}(x,s))_{xx}=0,
\quad 0<x<1,\; 0<s<t-\tau,\; t>\tau,\\
\varepsilon(0,s)=\varepsilon_x(0,s)
=\varepsilon_{xx}(1,s)=0,\quad  0 \leq s \leq t-\tau,\; t>\tau,\\
(EI(x)\varepsilon_{xx})_x(1,s)=k_1\varepsilon_s(1,s),\quad
 0 \leq s \leq t-\tau,\; t>\tau,\; k_1>0,\\
\varepsilon(x,0)=\widehat{w}_0(x)-w_0(x),\quad
\varepsilon_s(x,0)=\widehat{w}_1(x)-w_1(x),\quad 0 \leq x \leq 1,
\end{gathered}
\end{equation}
and
\begin{equation}\label{119}
\begin{gathered}
\rho(x)\varepsilon_{ss}(x,s,t)+(EI(x)\varepsilon_{xx}(x,s,t))_{xx}=0,\\
 0<x<1, \; t-\tau<s<t,\; t>\tau;\\
\varepsilon(0,s,t)=\varepsilon_x(0,s,t)=\varepsilon_{xx}(1,s,t)
=(EI(x)\varepsilon_{xx})_x(1,s,t)=0,\\ t-\tau \leq s \leq t,\; t>\tau;\\
\varepsilon(x,t-\tau,t)=\varepsilon(x,t-\tau,t),\quad
\varepsilon_s(x,t-\tau,t)=\varepsilon_s(x,t-\tau),\\
 0 \leq x \leq 1,\; t>\tau;
\end{gathered}
\end{equation}
where $\varepsilon (x,s)$ and $\varepsilon(x,s,t)$ are given by
\eqref{10901} and \eqref{10701} respectively.

\begin{theorem}\label{thm4.1}
Let $t>\tau$, for any $(w_0,w_1)^T \in \mathcal{H}$,
$(\widehat{w}_0,\widehat{w}_1)^T \in \mathcal{H}$, there exists a unique solution
of systems \eqref{117}-\eqref{119} such that
$(w(\cdot,t), w_t(\cdot,t))^T \in \mathcal{C}(\tau, \infty; \mathcal{H})$,
$(\varepsilon(\cdot,s),\varepsilon_s(\cdot,s))^T \in \mathcal{C}(0,t-\tau;\mathcal{H})$,
$(\varepsilon(\cdot,s,t), \varepsilon_s(\cdot,s,t))^T 
\in \mathcal{C}([t-\tau, t]\times[\tau,\infty);\mathcal{H})$
for any $(\widehat{w}_0-w_0, \widehat{w}_1-w_1)^T \in D(\mathbb B)$,
 where $\mathbb B$ is defined by \eqref{10501}, system \eqref{117}
decays exponentially in the sense that
\begin{equation}\label{131}
\begin{aligned}
&\|(w(\cdot,t), w_t(\cdot,t))^T \|_{\mathcal{H}} \\
&\leq M_0 e^{-\omega_0(t-\tau)} \| (w_0, w_1)^T \|_{\mathcal{H}}\\
&\quad + \frac{L_0 C MM_0 e^{\omega_0 \tau}}{\sqrt{2\omega}} 
\Big(e^{-\frac{\omega_0 t}{2}} + e^{\omega \tau} 
\cdot e^{-\frac{\omega t}{2}}\Big) 
\|\mathbb B \big(\varepsilon(\cdot,0), \varepsilon_s(\cdot,0)\big)^T\|_{\mathcal{H}}.
\end{aligned}
\end{equation}
\end{theorem}


\begin{proof}
For any $(w_0,w_1)^T \in \mathcal{H}$,
$(\widehat{w}_0, \widehat{w}_1)^T\in \mathcal{H}$, since $\mathbb B$ defined by
\eqref{10501} generates an exponentially stable $C_0-$semigroup on $\mathcal{H}$,
there is a unique solution 
$(\varepsilon(\cdot,s), \varepsilon_s(\cdot,s))^T \in \mathcal{C}(0,t-\tau; \mathcal{H})$
to \eqref{118} such that \eqref{10301} holds.

Now, for any given time $t>\tau$, write \eqref{119} as
\begin{equation}\label{11901}
\frac{d}{ds} \begin{pmatrix}
 \varepsilon(\cdot,s,t) \\
 \varepsilon_s(\cdot,s,t)
 \end{pmatrix}
=\mathbb{A} \begin{pmatrix}
 \varepsilon(\cdot,s,t) \\
 \varepsilon_s(\cdot,s,t)
 \end{pmatrix},
\end{equation}
where $\mathbb{A}$ is defined by
\begin{equation}\label{11902}
\begin{gathered}
\mathbb{A}(f,g)^T=(g, -\frac{1}{\rho(x)}(EI(x)f'')'')^T, \\
 D(\mathbb{A})=\{(f,g)^T\in(H^4(0,1)\cap H^2_E(0,1))\times H^2_E(0,1) :
 f''(1)= (EI f'')'(1)=0 \}.
\end{gathered}
\end{equation}


Then $\mathbb{A}$ is skew-adjoint in $\mathcal{H}$ and hence generates
a conservative $C_0$-semigroup on $\mathcal{H}$. For any
$(\varepsilon(\cdot, t-\tau), 
\varepsilon_s(\cdot,t-\tau))^T \in \mathcal{H}$
that is determined by \eqref{118}, there exists a unique
solution to \eqref{119} such that
\begin{equation}\label{11903}
\|(\varepsilon(\cdot,s,t),\varepsilon_s(\cdot,s,t))^T \|_{\mathcal{H}}
= \|(\varepsilon(\cdot,t-\tau),\varepsilon_s(\cdot,t-\tau))^T \|_{\mathcal{H}},
\end{equation}
for all $s \in [t-\tau,t]$.
So, $(\varepsilon(\cdot,s,t),\varepsilon_s(\cdot,s,t)) \in
\mathcal{C}([t-\tau,t]\times [\tau,\infty);{\mathcal{H}})$. Moreover, since
$\mathbb{A}$ is skew-adjoint with compact resolvent, the solution of
\eqref{119} can be, in terms of $s$, represented as
\begin{equation}\label{11904}
 \begin{pmatrix}
 \varepsilon(x,s,t) \\
 \varepsilon_s(x,s,t) 
 \end{pmatrix}
=\sum_{n=0}^{\infty} a_n(t)e^{\lambda_ns}
 \begin{pmatrix}
 \frac{1}{\lambda_n}\phi_n(x) \\
 \phi_n(x) 
 \end{pmatrix}
+\sum_{n=0}^{\infty} b_n(t)e^{-\lambda_ns}
 \begin{pmatrix}
 -\frac{1}{\lambda_n}\phi_n(x) \\
 \phi_n(x)
 \end{pmatrix}
\end{equation}
where $(\pm \frac{1}{\lambda}\phi(x), \phi(x))$ is a sequence of all 
$\omega$-linearly
independent approximated normalized orthogonal eigenfunctions of
$\mathbb{A}$ corresponding to eigenvalues $\pm \lambda$ satisfies:
\begin{equation}\label{120}
 \begin{gathered}
  \phi^{(4)}(x)+\frac{2EI'(x)}{EI(x)}\phi'''(x)+\frac{EI''(x)}{EI(x)}\phi''(x)
+\lambda^2\frac{\rho(x)}{EI(x)}\phi(x)=0,\\
\phi(0)=\phi'(0)=\phi''(1)=\phi'''(1)=0.
\end{gathered}
\end{equation}

Set $h=\int_0^1(\frac{\rho(\tau)}{EI(\tau)})^{1/4}d\tau$ and 
$\lambda_n= \beta_n^2/ h^2$, then from the reference \cite{Guo2},
 when $n$ is large enough the solutions of the equations above can be represented 
as
 \begin{equation}\label{g129}
 \begin{gathered}
 \beta_n=\frac{1}{\sqrt{2}}(n+\frac{1}{2})\pi(1+i)
+ \mathcal{O}\big(\frac{1}{n}\big),\\
\begin{aligned}
 \phi_n(x) &= e^{-\frac{1}{4}\int_0^z a(\tau) d\tau}\sqrt{2}(i-1)
\Big[\sin\Big((n+\frac{1}{2})\pi z\Big) 
- \cos\Big((n+\frac{1}{2})\pi z\Big)\\
&\quad +e^{-(n+\frac{1}{2})\pi z} 
   +(-1)^{n}e^{-(n+\frac{1}{2})\pi (1-z)}\Big]
+ \mathcal{O}\big(\frac{1}{n}\big),
\end{aligned}\\
\begin{aligned}
\beta^{-2}_n\phi''_n(x)
&= \frac{1}{h^2}\Big(\frac{\rho(x)}{EI(x)}\Big)^{1/2}
e^{-\frac{1}{4}\int_0^za(\tau)d\tau}\sqrt{2}(1+i) 
\Big[ \cos\Big((n+\frac{1}{2})\pi z\Big) \\
&\quad - \sin\Big((n+\frac{1}{2})\pi z\Big)
  +e^{-(n+\frac{1}{2})\pi z}+(-1)^n e^{-(n+\frac{1}{2})\pi(1-z)}\Big] \\
&\quad + \mathcal{O}\big(\frac{1}{n}\big).
\end{aligned}
\end{gathered}
\end{equation}
From \eqref{11904},
\begin{equation}\label{13201}
 \varepsilon_s(1,t,t)={\sum_{n=0}^{\infty}}[a_n(t)e^{\lambda_n t}+b_n(t)
e^{-\lambda_n t}]\phi_n(1).
\end{equation}
For \eqref{119} we have
\begin{align*} %\label{132}
&l_n a_n(t) e^{\lambda_n (t-\tau)} \\
& =  \Big\langle 
 \begin{pmatrix}
 \varepsilon(\cdot,t-\tau) \\
 \varepsilon_s(\cdot,t-\tau) 
 \end{pmatrix},
 \begin{pmatrix}
 \frac{1}{\lambda_n}\phi_n(\cdot) \\
 \phi_n(\cdot)
 \end{pmatrix}
\Big\rangle_{\mathcal{H}} \\
& = \frac{1}{\lambda_n} \Big\langle 
 \begin{pmatrix}
 \varepsilon(\cdot,t-\tau) \\
 \varepsilon_s(\cdot,t-\tau) \\
 \end{pmatrix}, 
\mathbb A
 \begin{pmatrix}
 \frac{1}{\lambda_n}\phi_n(\cdot) \\
 \phi_n(\cdot)\\
 \end{pmatrix}
\Big\rangle_{\mathcal{H}}\\
 & =   \frac{1}{\lambda_n} \Big\langle 
 \begin{pmatrix}
 \varepsilon(\cdot,t-\tau) \\
 \varepsilon_s(\cdot,t-\tau) 
 \end{pmatrix},
 \begin{pmatrix}
 \phi_n(\cdot) \\
 -\frac{1}{\lambda_n\rho(\cdot)}(EI(\cdot)\phi''_n(\cdot))''\\
 \end{pmatrix}
\Big\rangle_{\mathcal{H}}\\
& =   \frac{1}{\lambda_n} \Big[\int^1_0 EI(x)\varepsilon_{xx}(x,t-\tau)\phi''_n(x)dx
-\frac{1}{\lambda_n}\int^1_0\varepsilon_s(x,t-\tau)(EI(x)\phi''_n(x))''dx\Big]\\
& =   \frac{1}{\lambda_n} \Big[ -\! \int_0^1 (EI(x)
\varepsilon_{xx}(x,t-\tau))_{x} \phi'_n(x)dx + \frac{1}{\lambda_n}
\int_0^1 \varepsilon_{sx}(x,t-\tau) (EI(x)\phi''_n(x))' dx \Big] \\
& =   \frac{1}{\lambda_n} \Big[ -(EI(x)\varepsilon_{xx})_{x}(1,t-\tau)\phi_n(1) 
+ \int_0^1 (EI(x)\varepsilon_{xx}(x,t-\tau))_{xx}\phi_n(x)dx \\
& \quad - \frac{1}{\lambda_n} \int_0^1 \varepsilon_{sxx}(x,t-\tau) EI(x)
\phi''_n(x) dx\Big]\\
& =   \frac{1}{\lambda_n} \Big[ -k_1\varepsilon_s(1,t-\tau)\phi_n(1) + \int_0^1
(EI(x)\varepsilon_{xx}(x,t-\tau))_{xx}\phi_n(x)dx\\
& \quad - \frac{1}{\lambda_n} \int_0^1 \varepsilon_{sxx}(x,t-\tau) EI(x)
\phi''_n(x) dx\Big]\\
& =   \frac{1}{\lambda_n} \Big\{ -k_1\varepsilon_s(1,t-\tau)\phi_n(1) 
+ \int_0^1 (EI(x)\varepsilon_{xx}(x,t-\tau))_{xx} 
 \Big[ e^{-\frac{1}{4}\int_0^z a(\tau) d\tau}\sqrt{2}(i-1) \\
& \quad\times\Big(\sin\big((n+\frac{1}{2})\pi z\big) 
- \cos\big((n+\frac{1}{2})\pi z\big)+e^{-(n+\frac{1}{2})\pi z} 
+(-1)^{n}e^{-(n+\frac{1}{2})\pi (1-z)}\Big) \Big] dx \\
&\quad - \int_0^1 \varepsilon_{sxx}(x,t-\tau) \sqrt{EI(x)} 
 \sqrt{\rho(x)} e^{-\frac{1}{4}\int_0^za(\tau)d\tau}\sqrt{2}(1+i) 
\Big[ \cos\big((n+\frac{1}{2})\pi z\big) \\
&\quad - \sin\big((n+\frac{1}{2})\pi z\big)
 +e^{-(n+\frac{1}{2})\pi z}+(-1)^n e^{-(n+\frac{1}{2})\pi(1-z)}\Big]dx 
+ \mathcal{O}\big(\frac{1}{n}\big) \Big\}.
\end{align*}
By the expression of $\phi_n(x)$, there exists a constant $c_0>0$ such that
\begin{equation}\label{13206}
| \phi_n(1) | \leq c_0.
\end{equation}
Notice that
 \begin{equation} \label{13206b}
\begin{aligned}  
| \varepsilon_s(1,t-\tau) | 
&=  | \int_0^1 \varepsilon_{sx}(x,t-\tau) dx | 
=| \int_0^1 \int_0^x \varepsilon_{sxx} (y,t-\tau) dy dx | \\
&\leq \int_0^1 \Big[ \int_0^x \varepsilon^2_{sxx} (y,t-\tau) dy\Big]^{1/2} dx 
\leq \Big[\int_0^1 \varepsilon_{sxx}^2(x,t-\tau) dx\Big]^{1/2} \\
&\leq \frac{1}{m}\Big[\int_0^1 EI(x)\varepsilon_{sxx}^2(x,t-\tau) dx\Big]^{1/2}
\end{aligned}
\end{equation}
where $m = min_{(0\leq x \leq 1)} \{EI(x)\}$.
Then
 \begin{equation} \label{13202}
\begin{aligned}
| l_n a_n(t) | 
&\leq \frac{1}{| \lambda_n |}
\Big\{ c_0 k_1 | \varepsilon_s(1,t-\tau)| 
+ 8 \Big[\int_0^1 \rho(x) (EI(x)\varepsilon_{xx}(x,t-\tau))_{xx}^2 dx \Big]^{1/2} \\
&\quad\times \Big(\int_0^1 \frac{1}{\rho(x)} dx \Big)^{1/2} 
   + 8 \Big[\int_0^1 EI(x) \varepsilon_{sxx}^2(x,t-\tau) dx\Big]^{1/2} 
 \int_0^1 \rho(x) dx \Big\}\\
&\leq \frac{1}{| \lambda_n |} \Big[ \frac{c_0 k_1}{m} 
+ 8 \Big(\int_0^1 \frac{1}{\rho(x)} dx\Big)^{1/2} 
+ 8 \int_0^1 \rho(x) dx\Big] \\
&\quad\times \| \mathbb B(\varepsilon(\cdot,t-\tau), 
\varepsilon_s(\cdot,t-\tau))^T \|_{\mathcal{H}}.
\end{aligned}
\end{equation}
Similarly,
\begin{align*}  % \label{137}
&l_n b_n(t) e^{-\lambda_n (t-\tau)} \\
& =  \Big\langle 
 \begin{pmatrix}
 \varepsilon(\cdot,t-\tau) \\
 \varepsilon_s(\cdot,t-\tau) 
 \end{pmatrix},
 \begin{pmatrix}
 -\frac{1}{\lambda_n}\phi_n(\cdot) \\
 \phi_n(\cdot)
 \end{pmatrix}
 \Big\rangle_{\mathcal{H}} \\
 & =   \frac{1}{\lambda_n} \Big\langle 
 \begin{pmatrix}
 \varepsilon(\cdot,t-\tau) \\
 \varepsilon_s(\cdot,t-\tau) 
 \end{pmatrix}, \mathbb A
  \begin{pmatrix}
 -\frac{1}{\lambda_n}\phi_n(\cdot) \\
 \phi_n(\cdot)
 \end{pmatrix}
 \Big\rangle_{\mathcal{H}} \\
 & =   \frac{1}{\lambda_n} \Big\langle 
 \begin{pmatrix}
 \varepsilon(\cdot,t-\tau) \\
 \varepsilon_s(\cdot,t-\tau) 
 \end{pmatrix},
 \begin{pmatrix}
 \phi_n(\cdot) \\
 \frac{1}{\lambda_n \rho(\cdot)}(EI(\cdot)\phi''_n(\cdot))''\\
 \end{pmatrix}
\Big\rangle_{\mathcal{H}}\\
& =   \frac{1}{\lambda_n}
 \Big[ \int^1_0EI(x)\varepsilon_{xx}(x,t-\tau)\phi''_n(x)dx
+\frac{1}{\lambda_n}\int^1_0\varepsilon_s(x,t-\tau)(EI(x)\phi''_n(x))''dx\Big]\\
 & =   \frac{1}{\lambda_n}\Big[ - \int_0^1 (EI(x)\varepsilon_{xx}(x,t-\tau)) 
d \phi_n(x) +\frac{1}{\lambda_n} \int_0^1 \varepsilon_{sxx}(x,t-\tau)
EI(x)\phi''_n(x) dx \Big] \\
& =   \frac{1}{\lambda_n} \Big[-(EI(x) \varepsilon_{xx})_x(1,t-\tau)\phi_n(1) 
+ \int_0^1 (EI(x)\varepsilon_{xx}(x,t-\tau))_{xx}\phi_n(x)dx  \\
 &\quad +\frac{1}{\lambda_n} \int_0^1 \varepsilon_{sxx}(x,t-\tau) EI(x) 
\phi''_n(x) dx \Big],\\
& =   \frac{1}{\lambda_n} \Big\{ -k_1\varepsilon_s(1,t-\tau)\phi_n(1) 
+ \int_0^1 (EI(x)\varepsilon_{xx}(x,t-\tau))_{xx} 
\Big[ e^{-\frac{1}{4}\int_0^z a(\tau) d\tau}\sqrt{2}(i-1)\ \\
&\quad\times \Big(\sin\big((n+\frac{1}{2})\pi z\big) - 
\cos\big((n+\frac{1}{2})\pi z\big)+e^{-(n+\frac{1}{2})\pi z} 
+(-1)^{n}e^{-(n+\frac{1}{2})\pi (1-z)}\Big) \Big] dx \\
& \quad + \int_0^1 \varepsilon_{sxx}(x,t-\tau) \sqrt{EI(x)} 
 \sqrt{\rho(x)} e^{-\frac{1}{4}\int_0^za(\tau)d\tau}\sqrt{2}(1+i) 
 \Big[ \cos\big((n+\frac{1}{2})\pi z\big)  \\
&\quad - \sin\big((n+\frac{1}{2})\pi z\big)
 +e^{-(n+\frac{1}{2})\pi z}+(-1)^n e^{-(n+\frac{1}{2})\pi(1-z)}\Big]dx
 + \mathcal{O}\big(\frac{1}{n}\big) \Big\}.
\end{align*}
Then
 \begin{equation} \label{137021}
\begin{aligned}
| l_n b_n(t) |
& \leq \frac{1}{ | \lambda_n |}
\Big\{ c_0 k_1 | \varepsilon_s(1,t-\tau)| 
 + 8 \Big[\int_0^1 \rho(x) (EI(x)\varepsilon_{xx}(x,t-\tau))_{xx}^2 dx \Big]^{1/2}\\
&\quad \times \Big(\int_0^1 \frac{1}{\rho(x)} dx \Big)^{1/2} 
   + 8 \Big[\int_0^1 EI(x) \varepsilon_{sxx}^2(x,t-\tau) dx\Big]^{1/2} 
 \int_0^1 \rho(x) dx \Big\}\\
&\leq \frac{1}{ | \lambda_n |} \Big[ \frac{c_0 k_1}{m} 
+ 8 \Big(\int_0^1 \frac{1}{\rho(x)} dx\Big)^{1/2} + 8 \int_0^1 \rho(x) dx\Big] \\
&\quad\times \| \mathbb B(\varepsilon(\cdot,t-\tau), \varepsilon_s(\cdot,t-\tau))^T \|_{\mathcal{H}}.
\end{aligned}
\end{equation}

Collecting \eqref{13201}, \eqref{13202}, \eqref{137021}, and the expression 
of $\lambda_n$ gives 
\begin{equation} \label{13702}
 | \varepsilon_s(1,t,t) | \leq C \| \mathbb B(\varepsilon(\cdot,t-\tau), 
\varepsilon_s(\cdot,t-\tau))^T \|_{\mathcal{H}}
\end{equation}
for some constant $C>0$ independent of $t$.
Now by \eqref{10301} and $C_0-$semigroup theory, we have
\begin{equation} \label{13703}
 \| \mathbb B(\varepsilon(\cdot,t-\tau), \varepsilon_s(\cdot,t-\tau)) \|_{\mathcal{H}}
\leq M e^{-\omega(t-\tau)} \| \mathbb B(\varepsilon(\cdot,0),
 \varepsilon_s(\cdot,0))^T \|_{\mathcal{H}}
\end{equation} 
for any $t\in [\tau, +\infty)$, where $M, \omega$ are given by \eqref{111}.
 We finally get
\begin{equation} \label{13704}
 | \varepsilon_s(1,t,t) | \leq CM e^{-\omega(t-\tau)} 
\| \mathbb B(\varepsilon(\cdot,0), \varepsilon_s(\cdot,0))^T \|_{\mathcal{H}}.
\end{equation}
Furthermore, the equation \eqref{117} can be written as
\begin{equation}\label{11701}
\frac{d}{dt} 
 \begin{pmatrix}
 w(\cdot, t) \\
 w_t(\cdot, t) 
 \end{pmatrix}
  = \mathcal{A}_0 
 \begin{pmatrix}
 w(\cdot, t) \\
 w_t(\cdot, t) 
 \end{pmatrix}
  + \mathcal{B}_0 \varepsilon_s(1,t,t)
\end{equation}
where
\begin{equation}\label{11702}
\begin{gathered} 
\mathcal{A}_0 (f,g)^T = (g,
-\frac{1}{\rho(x)}(EI(x)f'')'')^T,\\
\begin{aligned}
\forall (f,g)^T \in D(\mathcal{A}_0) 
= \big\{& (f,g)^T \in (H^4(0,1)\cap H^2_E(0,1))
 \times H_E^2(0,1) :\\
&f''(1)=0, (EIf'')'(1) = k_2 g(1)\big\},
\end{aligned}\\
\mathcal{B}_0=
 \begin{pmatrix}
 0 \\
 \delta(x-1) 
 \end{pmatrix}.
\end{gathered}
\end{equation}
A direct computation shows that
\begin{equation}\label{11704}
 \mathcal{B}_0 \mathcal{A}_0^{-1} (f,g)^T = f(1), \quad  \forall (f,g)^T \in \mathcal{H},
\end{equation}
which means ${\mathcal{B}}_0 {\mathcal{A}_0}^{-1}$ is bounded.

For the energy $E_0(t)$ of the system \eqref{117}, simple computations 
tells us that
\begin{equation}\label{11707}
\dot{E}_0(t) = -k_2 w_t^2(1,t),
\end{equation}
which shows that
\begin{equation}\label{11708}
 k_2 \int_0^T | w_t(1,t) |^2 dt \leq E_0(0),
\end{equation} 
for any $T>0$. 
This inequality together with \eqref{11704} illustrates that
${\mathcal{B}}_0$ is admissible for $e^{\mathcal{A}_0 t}$. Therefore, there exists a unique 
solution to \eqref{11701} such that 
$(w(\cdot,t), w_t(\cdot,t))^T \in \mathcal{C}(\tau,\infty; \mathcal{H})$.
Since $\mathcal{A}_0$ generates an exponentially stable $C_0$-semigroup, 
it follows from \cite[Proposition 2.5]{Weiss} and \eqref{13704} that
\begin{align*}
 \| \int_{\tau}^{t/2}e^{\mathcal{A}_0
(t/2-s)}\mathcal{B}_0\varepsilon_s(1,s,s)ds\|_\mathcal{H}
&\leq L_0 \|\varepsilon_s(1,\cdot,\cdot)\|_{L^2(\tau,t/2)} \\
&\leq \frac{L_0CM}{\sqrt{2\omega}} 
 \| \mathbb B(\varepsilon(\cdot,0), \varepsilon_s(\cdot,0))^T \|_{\mathcal{H}},
\end{align*}
and
\begin{align*}
\|\int_{t/2}^t e^{\mathcal{A}_0 (t-s)}\mathcal{B}_0\varepsilon_s(1,s,s)ds\|_\mathcal{H}
&\leq \|\int_{0}^te^{\mathcal{A}_0 (t-s)}\mathcal{B}_0(0\mathop{\Diamond}\limits_{t/2}
\varepsilon_s(1,s,s))ds\|_\mathcal{H} \\
&\leq L_0\|\varepsilon_s(1,\cdot,\cdot)\|_{L^2(t/2, t)} \\ 
&\leq \frac{L_0 C M e^{\omega \tau}e^{-\frac{\omega t}{2}}}{\sqrt{2\omega}} \|
 \mathbb B(\varepsilon(\cdot,0), \varepsilon_s(\cdot,0))^T \|_{\mathcal{H}},
\quad \forall t\ge 0,
\end{align*}
for some constant $L_0>0$ that is independent of
$\varepsilon_s(1,t,t)$, and
$$
(u\mathop{\Diamond}\limits_{\tau}v)(t)
=\begin{cases}
u(t), & 0\leq t\leq\tau,\\
v(t), & t>\tau.
\end{cases}
$$
On the other hand, the solutions of the systems \eqref{11701} can be 
represented as
\begin{equation}\label{11709}
\begin{aligned}
&(w(\cdot,t), w_t(\cdot,t))^T \\
&= e^{\mathcal{A}_0(t-\tau)} (w(\cdot,\tau), w_t(\cdot,\tau))^T 
 + \int_{\tau}^t e^{\mathcal{A}_0(t-s-\tau)} \mathcal{B}_0 \varepsilon_s(1,s,s)ds\\
&= e^{\mathcal{A}_0 (t-\tau)} (w(\cdot,\tau),w_t(\cdot,\tau))^T
  + e^{\mathcal{A}_0(t/2-\tau)} \int_{\tau}^{t/2} e^{\mathcal{A}_0(t/2-s)} 
\mathcal{B}_0 \varepsilon_s(1,s,s) ds\\
&\quad + e^{-\mathcal{A}_0 \tau} \int_{t/2}^t e^{\mathcal{A}_0(t-s)} \mathcal{B}_0 \varepsilon_s(1,s,s)ds.
\end{aligned}
\end{equation}
Since $\mathcal{A}_0$ generates an exponentially stable $C_0$-semigroup, there exists 
two positive constants $M_0, \omega_0$ such that 
$\|e^{{\mathcal{A}}_0t}\|\leq M_0e^{-\omega_0 t}$, which together with \eqref{11709} 
and the conservative property of the system \eqref{114} for $u^*(t)=0$ lead to
\begin{align*} %\label{131}
&\|(w(\cdot,t), w_t(\cdot,t))^T \|_{\mathcal{H}}\\
&\leq M_0 e^{-\omega_0(t-\tau)} \| (w(\cdot,\tau), w_t(\cdot,\tau))^T \|_{\mathcal{H}} \\
&\quad + \frac{L_0 C MM_0 e^{\omega_0 \tau}}{\sqrt{2\omega}} 
\big(e^{-\frac{\omega_0 t}{2}} + e^{\omega \tau} 
 e^{-\frac{\omega t}{2}}\big) 
\|\mathbb B (\varepsilon(\cdot,0), \varepsilon_s(\cdot,0))^T\|_{\mathcal{H}}\\
&= M_0 e^{-\omega_0(t-\tau)} \| (w(\cdot,0), w_t(\cdot,0))^T \|_{\mathcal{H}} \\
&\quad + \frac{L_0 C MM_0 e^{\omega_0 \tau}}{\sqrt{2\omega}} 
\big(e^{-\frac{\omega_0 t}{2}} + e^{\omega \tau} \cdot e^{-\frac{\omega t}{2}}\big)
 \|\mathbb B \big(\varepsilon(\cdot,0), \varepsilon_s(\cdot,0)\big)^T\|_{\mathcal{H}}.
\end{align*}
\end{proof}

\section{Simulation results}


\begin{figure}[ht]
\begin{center}
\includegraphics[width=10cm]{fig1a} % submitted_beam_variable_state.jpg}
\includegraphics[width=10cm]{fig1b} %submitted_beam_variable_velocity.jpg}
\end{center}
\caption{Displacement $w(x,t)$ (top), 
and velocity $w_t(x,t)$} (bottom) of the solution\label{fig1}
\end{figure}

In this section, using the finite difference method we present the numerical 
simulation for the closed-loop system \eqref{117}-\eqref{119}.
Here we choose the space grid size $N=30$, time step $dt=0.0003$ 
and time span $[0, 40]$. Parameters and coefficients respectively 
are chosen to be $\tau=k_1=k_2=1, \rho(x)=1+0.2\sin(x), EI(x)=1+0.2\cos(x)$. 
For the initial values:
\begin{gather*}
w_0(x)=x^2,\quad w_1(x)=1, \\
\varepsilon(x,0)=x^2,\quad 
\varepsilon_s(x,0)=1,
\quad  \forall x \in [0,1],
\end{gather*}
the displacement $w(x,t)$ and velocity $w_t(x,t)$ are plotted in
Figure \ref{fig1}.
It shows clearly that the system is very stable with small displacement 
under time-variable coefficients. This simple simulation illustrates 
that the observer-predictor based scheme is useful to make the unstable 
system exponentially stable for the Euler-Bernoulli beam equation 
with variable coefficients.


\subsection*{Acknowledgments}
This work was supported by the National Natural Science
Foundation of China under Grant 61203058, and the training
program for outstanding young teachers of North College University of Technology.

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\end{document}