# Output Feedback Exponential Stabilization for  a 1-d Wave PDE with Dynamic
Boundary ††thanks: This work was supported by the Natural Science Foundation
of Shaanxi Province 2018JM1051, 2014JQ1017.  ††thanks: 2020 Mathematics
Subject Classification. 37L15; 93D15; 93B51; 93B52.

Zhan-Dong Mei School of Mathematics and Statistics, Xi’an Jiaotong University,
Xi’an 710049, China. Email<EMAIL_ADDRESS>
###### Abstract

We study the output feedback exponential stabilization for a 1-d wave PDE with
dynamic boundary. With only one measurement, we construct an infinite-
dimensional state observer to trace the state and design an estimated state
based controller to exponentially stabilize the original system. This is an
essentially important improvement for the existence literature [Ö. Morgül,
B.P. Rao and F. Conrad, IEEE Transactions on Automatic Control, 39(10) (1994),
2140-2145] where two measurements including the high order angular velocity
feedback were adopted. When a control matched nonlinear internal uncertainty
and external disturbance are taken into consideration, we construct an
infinite-dimensional extended state observer (ESO) to estimate the total
disturbance and state simultaneously. By compensating the total disturbance,
an estimated state based controller is designed to exponentially stabilize the
original system while making the closed-loop system bounded. Riesz basis
approach is crucial to the verifications of the exponential stabilities of two
coupled systems of the closed-loop systems. Some numerical simulations are
presented to illustrate the effectiveness.

Key words: Exponential stabilization, disturbance, extended state observer,
Riesz basis, tip mass.

##  1 Introduction

In this paper, we are concerned with the output feedback exponential
stabilization of a 1-d wave PDE with dynamic boundary described as follows

$\left\\{\begin{array}[]{l}u_{tt}(x,t)=u_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
u(0,t)=0,u_{x}(1,t)+mu_{tt}(1,t)=U(t)+F(t),t\geq 0,\\\
y(t)=\\{u_{x}(0,t),u(1,t)\\},\;\;t\geq 0,\end{array}\right.$ (1)

where $x$ and $t$ denote the independent spatial and time variables,
respectively, $u(x,t)$ denotes the cable displacement at $x$ for time $t$,
$m>0$ is the tip mass at the actuator end, $U(t)$ is the boundary control (or
input), $F(t)=f(u(\cdot,t),u_{t}(\cdot,t))+d(t)$, $f:H^{1}(0,1)\times
L^{2}(0,1)\rightarrow\mathds{R}$ is the internal uncertainty, $d(t)$ is the
external disturbance, $y(t)$ is the observation (output). System (1) can be
used to describe flexible cable with tip mass [10, 18, 32].

When there is no disturbance ($F(t)\equiv 0$), the velocity feedback control
law $U(t)=-\alpha u_{t}(1,t)$ $(\alpha>0)$ is sufficient for the strong
stabilization but not sufficient for the exponential stabilization of the
system (1), see [27]. By virtue of energy multiplier approach, Morgül et al.
[32] proved that the output feedback control law $U(t)=-\alpha
u_{t}(1,t)-au_{xt}(1,t)(\alpha,a>0)$ exponentially stabilize the right system.
Here the angular velocity $u_{xt}(1,t)$ was adopted, which is quite different
from the case $m=0$ where velocity feedback is sufficient for the exponential
stability [35]. Moreover, in the special case $\alpha=m/a\neq 1$, Morgül et
al. [32] verified the Riesz basis generation of the closed-loop system, and
thereby the spectrum-determined growth condition is satisfied. In [10], Guo
and Xu showed by virtue of essential spectral analysis that the spectrum-
determined growth condition always holds for the closed-loop system under
output feedback control law $U(t)=-\alpha
u_{t}(1,t)-au_{xt}(1,t)(\alpha,a>0)$. However, the Riesz basis generation for
the general case is still unsolved. This will be presented in Lemma 3.

Observe that the aforemention literatures used static collocated feedback
where the actuators and sensors are located in the same end $x=1$. The reason
why energy multiplier method is successfully used for the stability analysis
is that such output feedback makes the system dissipative. However, it has
known for a long time, the performance of closed-loop system under collocated
output feedback may be not so good [2]. For the non-collocated setting,
sensors can be placed according to the performance requirement, thereby such
design approach for specific systems has been widely used [2, 29, 38]. Since
the non-collocated closed-loop system is usually non-dissipative, the well-
posedness as well as stability analysis is most difficult. Dynamic boundary
feedback approach may be a better choice to overcome the difficulty stemmed
from non-collocated feedback. In [12], with one measurement $u_{x}(0,t),$ an
infinite-dimensional observer-based feedback technique has been employed to
construct a stabilizing boundary feedback controller for a wave equation (1)
with $m=0$ and $F(t)=0$. Such method has been applied to solve non-collocated
boundary control designs of Euler-Bernoulli beam equation [13]. For (1) with
tip mass ($m\neq 0$), non-collocated control is still a long standing unsolved
problem. The reason is that tip mass makes the system much more complex; such
system is indeed a hybrid system consists of a PDE and an ODE. One of our task
in this paper is to construct a non-collocated output feedback stabilizing
controller for system (1) without disturbance. Compared to the existence
references [10, 32], our main contributions for the case $F(t)\equiv 0$ are:
1) non-collocated feedback is used, 2) only one measurement is employed, 3)
the Riesz basis generations for closed-loop systems in [10, 32] have been
generalized to the case $m\neq a$, 4) Riesz basis approach is used to prove
the exponential stability for a coupled system which is equivalent to the
closed-loop system obtained from our feedback control law, where the
verification of exponential stabilization as well as Riesz basis generation
for coupled equation is most changeable [12, 13, 16].

Uncertainties and disturbances widely exist in various practical engineering
control systems. When the disturbance is taking into consideration ($F(t)\neq
0$), the vibration cable model (1) is quite different from the previous work,
because it is governed by a nonhomogeneous hyperbolic PDE. Even a small
disturbance can damage the stabilizing output feedback design for system
without disturbance. Thus, anti-disturbance problem is most changeable: in
order to stabilize the system with disturbance, the controller should be
redesigned. Many engineers and mathematicians focus on developing various
approaches to deal with uncertainty and disturbance in control system problem.
In [31] and [33], in order to reject the disturbance under certain conditions,
Morgül proposed a kind of dynamic boundary controller for elastic beam and
string, respectively. Guo et al. [17] and Ge et al. [9] used adaptive control
method to stabilize the wave equation and beam equation with disturbance,
respectively. Sliding-mode control (SMC) approach was adopted to reject the
disturbance for wave equations [14], beam equation [15, 21], and the cascade
of ODE-wave systems [28]. Backstepping approach developed by Krstic and
Smyshlyaev [22, 23, 24, 25, 26, 37] is also helpful for reject disturbance of
PDEs [1, 4, 5, 6, 39] and output regulation for coupled linear wave-ODE
systems (similar to our setup) [7]. It should be noted that the active
disturbance rejection control (ADRC) proposed by Han [19] is another powerful
approach to reject disturbance. The core idea of ADRC is
estamaite/cancellation in real time and the key step of control strategy is
the construction of an extended state observer (ESO) for the estimating of the
state and uncertainty. In the earlier works for PDEs by ADRC like [15, 20,
36], they dealt with the disturbance by ODEs reduced from the associated PDEs
through some special test functions. Therefore, the ESO is of finite-
dimension; slow variation, high gains and boundedness of the derivation of the
disturbance were used in the ESO. In [8], Feng and Guo developed a new
infinite-dimensional ESO to relax such restricts of the conventional ESO for a
class of anti-stable wave equations with external disturbance. Later on, the
same method was used to study the stabilization of other wave equations [30,
41, 43] and Euler-Bernoulli beams [42, 44].

In section 3, we shall clarify that the estimated state based output feedback
control law (28) for system (1) with $F(t)\equiv 0$ is not robust to the total
disturbance. For system (1) with $f(w,w_{t})=0$ and $d(t)$ being bounded, He
and Ge [18] adopted adaptive control approach to compensate the uncertainty.
In the case $f(w,w_{t})=0$ and $m=a\alpha$, by constructing a finite-
dimensional ESO, Xie and Xu [40] employed active disturbance rejection control
approach to cope with the disturbance. However, they excluded the internal
uncertainty as well as the general case $m\neq a\alpha$. Moreover, as in [32],
the author in [40] used two measurements including the high order angular
velocity feedabck $w_{xt}(1,t)$. In this paper, with two low order
measurements $w_{x}(0,t)$ and $w(1,t)$, we shall design an infinite-
dimensional ESO to estimate the original state and total disturbance online.
By compensating the total disturbance, an estimated state based output
feedback control law is constructed in order to exponentially stabilize the
original system. Moreover, all the other states of the closed-loop system are
verified to be bounded. The main contribution for the system (1) with
disturbance ($F(t)\neq 0$) lies in that, 1) the internal uncertainty is
considered, 2) the high order angular velocity feedback $w_{xt}(1,t)$ is
avoided, 3) we consider the case $m\neq a,\alpha>0$ while [40] just solved the
special case $\alpha,a>0$ and $m=a\alpha$, 4) our control strategy can be also
applied to simplify the existence references [30, 42, 43].

Consider system (1) in the energy state Hilbert space
$\mathbf{H}_{1}=H_{E}^{1}(0,1)\times L^{2}(0,1)\times\mathds{C},\
H_{E}^{1}(0,1)=\\{f|f\in H^{1}(0,1),f(0)=0\\}$, with the following norm
$\|(f,g,$
$\eta)\|^{2}_{\mathbf{H}_{1}}=\int_{0}^{1}[|f^{\prime}(x)|^{2}+|g(x)|^{2}]dx+\frac{1}{m}|\eta|^{2},(f,g,\eta)\in\mathbf{H}_{1}.$
Define the operator
$\mathbf{A}_{1}:D(\mathbf{A}_{1})(\subset\mathbf{H}_{1})\rightarrow\mathbf{H}_{1}$
by $\mathbf{A}_{1}(f,g,\eta)=(g,f^{\prime\prime},-f^{\prime}(1)),$
$\forall\;(f,g)\in
D(\mathbf{A}_{1})=\\{(f,g,\eta)\in\mathbf{H}_{1}|(g,f^{\prime\prime},-f^{\prime}(1))\in\mathbf{H}_{1},\eta=mg(1)\\}.$
Obviously, $\mathbf{A}_{1}$ is skew-adjoint and generates a unity group.
System (1) is written abstractly by

$\displaystyle\frac{d}{dt}\left(u(\cdot,t),u_{t}(\cdot,t),mu_{t}(1,t)\right)$
$\displaystyle=\mathbf{A}_{1}\left(u(\cdot,t),u_{t}(\cdot,t),mu_{t}(1,t)\right)+\mathbf{B}_{1}[U(t)+F(t)],$

where $\mathbf{B}_{1}=(0,0,1)^{T}$ is a bounded linear operator on
$\mathbf{H}_{1}$. The following proposition can be derived directly from the
proof of [42, Proposition 1.1].

###### Proposition 1

Assume that $f:H^{1}(0,1)\times L^{2}(0,1)\rightarrow\mathds{R}$ is continuous
and satisfies global Lipschitz condition in $H^{1}(0,1)\times L^{2}(0,1)$.
Then, for any $(u(\cdot,0),u_{t}(\cdot,0),mu_{t}(1,$ $0))\in\mathbf{H}_{1}$,
$u,d\in L^{2}_{loc}(0,\infty)$, there exists a unique global solution (mild
solution) to (1) such that $(u(\cdot,t),u_{t}(\cdot,t),$ $mu_{t}(1,t))\in
C(0,\infty;\mathbf{H}_{1})$.

We proceed as follows. In section 2, for system (1) without disturbance, we
design a Luenberger state observer and an estimated state based stabilizing
control law. We design in section 3 an infinite-dimensional ESO for system (1)
to estimated total disturbance and state in real time. An estimated total
disturbance and estimated state based stabilizing control law is then
designed. Moreover, it is proved that an important couple subsystem including
the original equation of the closed-loop is exponential stability and the
closed-loop system is bounded. In section 4, some numerical simulations are
presented.

##  2 Stabilization in the absence of disturbance

We rewrite (1) with $F(t)\equiv 0$ as follows

$\left\\{\begin{array}[]{l}u_{tt}(x,t)=u_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
u(0,t)=0,u_{x}(1,t)+mu_{tt}(1,t)=U(t),\;t\geq 0,\\\ y(t)=u_{x}(0,t),\;\;t\geq
0.\end{array}\right.$ (2)

Our aim in this section is to design a stabilizing control law for system (2)
by virtue of only one noncollocated measurement $u_{x}(0,t)$. To this end, we
first construct an infinite-dimensional Luenberger state observer as follows

$\left\\{\begin{array}[]{l}\widehat{u}_{tt}(x,t)=\widehat{u}_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
\widehat{u}_{x}(0,t)=\gamma\widehat{u}_{t}(0,t)+\beta\widehat{u}(0,t)+u_{x}(0,t),\\\
\widehat{u}_{x}(1,t)+m\widehat{u}_{tt}(1,t)=U(t),\;\;t\geq
0,\end{array}\right.$ (3)

where $\beta,\gamma>0$. Set $\widetilde{u}(x,t)=\widehat{u}(x,t)-u(x,t)$ to
get

$\left\\{\begin{array}[]{l}\widetilde{u}_{tt}(x,t)=\widetilde{u}_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
\widetilde{u}_{x}(0,t)=\gamma\widetilde{u}_{t}(0,t)+\beta\widetilde{u}(0,t),\;\;t\geq
0,\\\ \widetilde{u}_{x}(1,t)+m\widetilde{u}_{tt}(1,t)=0,\;\;t\geq
0.\end{array}\right.$ (4)

Denote $\mathbf{H}_{2}=H^{1}(0,1)\times L^{2}(0,1)\times\mathds{C}$, with norm
$\|(f,g,\eta)\|^{2}_{\mathbf{H}_{2}}=\int_{0}^{1}|f^{\prime}(x)|^{2}+|g(x)|^{2}+\beta|f(0)|^{2}+|\eta|^{2}/m,$
$(f,g,\eta)\in\mathbf{H}_{2}.$ Then we can write (4) abstractly by

$\displaystyle\frac{d}{dt}(\widetilde{u}(\cdot,t),\widetilde{u}_{t}(\cdot,t),m\widetilde{u}_{t}(1,t))$
$\displaystyle=\mathbf{A}_{2}(\widetilde{u}(\cdot,t),\widetilde{u}_{t}(\cdot,t),m\widetilde{u}_{t}(1,t)).$
(5)

Here
$\mathbf{A}_{2}:D(\mathbf{A}_{2})(\subset\mathbf{H}_{2})\rightarrow\mathbf{H}_{2}$
is defined by $\mathbf{A}_{2}(f,$
$g,\eta)=(g,f^{\prime\prime},-f^{\prime}(1)),\;\forall\;(f,g,\eta)\in
D(\mathbf{A}_{2})=\\{(f,g,$
$\eta)\in\mathbf{H}_{2}|(g,f^{\prime\prime},-f^{\prime}(1))\in\mathbf{H}_{2},f^{\prime}(0)=\gamma
g(0)+\beta f(0),$ $\eta=mg(1))\\}.$ By [34], $\mathbf{A}_{2}$ generates an
exponentially stable $C_{0}$-semigroup, that is, $\|e^{\mathbf{A}_{2}t}\|\leq
M_{\mathbf{A}_{2}}e^{\omega_{\mathbf{A}_{2}}t},\;t\geq 0$, where
$M_{\mathbf{A}_{2}}$ and $\omega_{\mathbf{A}_{2}}$ are two positive constants.
The following Lemma 2 presents that the Riesz basis property of
$\mathbf{A}_{2}$ also holds.

###### Lemma 2

Assume that $\gamma\neq 1$. Then, there exist a sequence of generalized
eigenfunctions of operator $(\mathbf{A}_{2},D(\mathbf{A}_{2}))$ that forms
Riesz basis for $\mathbf{H}_{2}$; the eigenvalues
$\\{\lambda_{n}\\}_{n=-\infty}^{+\infty}$ have asymptotic expression
$\lambda_{n}=\frac{1}{2}{\rm ln}\frac{|\gamma-1|}{\gamma+1}+n_{\beta}\pi
i+O(|n|^{-1}),$ where $n_{\beta}$ $=n$ if $\gamma>1$ and $n_{\beta}=n+1/2$ if
$0<\gamma<1$; the eigenfunction is given by $(f_{n},\lambda_{n}f_{n},$
$m\lambda_{n}f(1))$ with
$f_{n}(x)=[(1+\gamma)\lambda_{n}+\beta]e^{\lambda_{n}x}+[(1-\gamma)\lambda_{n}-\beta]e^{-\lambda_{n}x}$
and $F_{n}(x)$ $=(f_{n}^{{}^{\prime}}(x),\lambda_{n}f_{n}(x),\beta
f_{n}(0),-\lambda_{n}^{-1}f^{\prime}_{n}(1))/\lambda_{n}^{2}=\big{(}(1+\gamma)$
$\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}e^{in_{\beta}x}-(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_{\beta}x},(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}$
$e^{in_{\beta}x}+(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_{\beta}x},0,0\big{)}+O(n^{-1}).$

Proof. It follows from [34] that operator $(\mathbf{A}_{2},D(\mathbf{A}_{2}))$
generates an exponentially stable $C_{0}$-semigroup, and it is a densely
defined and discrete operator on $\mathbf{H}_{2}$. Therefore, for any
$\lambda\in\sigma(\mathbf{A}_{2})=\sigma_{P}(\mathbf{A}_{2})$, ${\rm
Re}\lambda<0$. We divide the proof into several steps.

Step 1: We claim that for any $\lambda\in\sigma(\mathbf{A}_{2})$, there
corresponds one eigenfunction $(f,g,\eta)$ given by

$\displaystyle\left\\{\begin{array}[]{ll}f(x)=[(1+\gamma)\lambda+\beta]e^{\lambda
x}+[(1-\gamma)\lambda-\beta]e^{-\lambda x},\\\ g=\lambda f,\\\
\eta=-\frac{f^{\prime}(1)}{\lambda}\end{array}\right.$ (9)

and $\lambda$ satisfies the characteristic equation

$\displaystyle
e^{2\lambda}[(1+\gamma)\lambda+\beta](1+m\lambda)=[(1-\gamma)\lambda-\beta](1-m\lambda).$
(10)

This implies that each eigenvalue of $\mathbf{A}_{2}$ is geometrically simple.

Indeed, $f^{\prime\prime}(x)-\lambda^{2}f(x)=0$ implies that $f$ is of the
form $f(x)=Ee^{\lambda x}+Fe^{-\lambda x}$. Use the boundary condition
$f^{\prime}(0)=(\gamma\lambda+\beta)f(0)$ to derive
$E=(1+\gamma)\lambda+\beta,F=(1-\gamma)\lambda-\beta$. We can then derive
$\eta=mg(1)=m\lambda f(1)=-\frac{f^{\prime}(1)}{\lambda}$ and (10) by virtue
of the boundary condition $f^{\prime}(1)+m\lambda^{2}f(1)=0$.

Step 2: We show that the eigenvalues $\\{\lambda_{n}\\}_{n=-\infty}^{+\infty}$
have asymptotic expression

$\lambda_{n}=\frac{1}{2}{\rm ln}\frac{|\gamma-1|}{\gamma+1}+n_{\beta}\pi
i+O(|n|^{-1}),$ (11)

where

$\displaystyle n_{\beta}=\left\\{\begin{array}[]{ll}n,\gamma>1\\\
n+\frac{1}{2},0<\gamma<1;\end{array}\right.$ (14)

the corresponding eigenfunction $(f_{n},\lambda_{n}f_{n},$ $m\lambda_{n}f(1))$
of $\mathbf{A}_{2}$ can be chosen such that

$\displaystyle F_{n}(x)=$
$\displaystyle\left(\begin{array}[]{c}(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}e^{in_{\beta}x}-(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_{\beta}x}\\\
(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}e^{in_{\beta}x}+(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_{\beta}x}\\\
0\\\ 0\\\ \end{array}\right)$ (19) $\displaystyle+O(n^{-1}),$ (20)

where

$F_{n}(x)=\frac{1}{\lambda_{n}^{2}}\begin{pmatrix}f_{n}^{{}^{\prime}}(x)\\\
\lambda_{n}f_{n}(x)\\\ \beta f_{n}(0)\\\ -\lambda_{n}^{-1}f^{\prime}_{n}(1)\\\
\end{pmatrix}^{T}.$ (21)

Now we give the proof. By (10), it follows that

$\displaystyle
e^{2\lambda}=\frac{m(\gamma-1)\lambda^{2}+(m\beta-\gamma+1)\lambda-\beta}{m(\gamma+1)\lambda^{2}+(m\beta+\gamma+1)\lambda+\beta}$
$\displaystyle=\frac{\gamma-1}{\gamma+1}+O(|\lambda|^{-1}).$ (22)

The asymptotic expression (11) is derived directly by Routhe’s theorem. Then
we obtain

$\begin{split}&\lambda_{n}\frac{f_{n}(x)}{\lambda^{2}_{n}}=\frac{(1+\gamma)\lambda_{n}+\beta}{\lambda_{n}}e^{\lambda_{n}x}+\frac{(1-\gamma)\lambda_{n}-\beta}{\lambda_{n}}e^{-\lambda_{n}x}\\\
&=(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}e^{in_{\beta}x}+(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_{\beta}x}\\\
&+O(|n|^{-1}),\end{split}$ (23)
$\begin{split}&\frac{f^{\prime}_{n}(x)}{\lambda^{2}_{n}}=\frac{(1+\gamma)\lambda_{n}+\beta}{\lambda_{n}}e^{\lambda_{n}x}-\frac{(1-\gamma)\lambda_{n}-\beta}{\lambda_{n}}e^{-\lambda_{n}x}\\\
=&(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}e^{in_{\beta}x}-(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_{\beta}x}\\\
&+O(|n|^{-1}),\end{split}$ (24) $\begin{split}&\frac{\beta
f_{n}(0)}{\lambda^{2}_{n}}=\frac{(1+\gamma)\lambda_{n}+\beta}{\lambda^{2}_{n}}+\frac{(1-\gamma)\lambda_{n}-\beta}{\lambda^{2}_{n}}=O(|n|^{-1}),\end{split}$
(25)
$\begin{split}&\frac{-\lambda^{-1}_{n}f^{\prime}_{n}(1)}{\lambda^{2}_{n}}=-\frac{(1+\gamma)\lambda_{n}+\beta}{\lambda^{2}_{n}}e^{\lambda_{n}}+\frac{(1-\gamma)\lambda_{n}-\beta}{\lambda^{2}_{n}}e^{-\lambda_{n}}\\\
&=O(|n|^{-1}),\end{split}$ (26)

The combination of (23), (24), (25) and (26) indicates that the asymptotic
expression (2) of $F_{n}(x)$ holds.

Step 3: We show that there exist a sequence of generalized eigenfunctions of
$\mathbf{A}_{2}$ which forms Riesz basis for $\mathbf{H}_{2}$.

Now we give the proof. Denote

$P(x)=\left(\begin{array}[]{cc}(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}&-(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}\\\
(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}&(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}\\\
\end{array}\right)$

for $\lambda>1$,

$P(x)=\left(\begin{array}[]{cc}(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}i&(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}i\\\
(1+\gamma)\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}i&-(1-\gamma)\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}i\\\
\end{array}\right)$

for $\lambda<1$. It is easily seen that the operator $\mathbf{T}$ defined by
$\mathbf{T}w=P(x)w$ for any $w\in\left(L^{2}(0,1)\right)^{2}$, is a bounded
and invertible linear operator on $\left(L^{2}(0,1)\right)^{2}$. Combine this
with the fact that $\\{(e^{in\pi x},e^{-in\pi x})\\}_{n=-\infty}^{+\infty}$
forms Riesz basis for $\left(L^{2}(0,1)\right)^{2}$, to obtain that
$\\{P(x)(e^{in\pi x},e^{-in\pi x})\\}_{n=-\infty}^{+\infty}$ also forms Riesz
basis for $\left(L^{2}(0,1)\right)^{2}$. Thereby,
$\\{(0,0,1,0)\\}\bigcup\\{(0,0,0,1)\\}\bigcup$ $\\{(P(x)(e^{in\pi x},e^{-in\pi
x}),0,$ $0)$ $\\}_{n=-\infty}^{+\infty}$ forms Riesz basis for
$\left(L^{2}(0,1)\right)^{2}\times\mathds{C}^{2}$.

From (2), we derive

$\displaystyle F_{n}(x)=(P(x)(e^{in\pi x},e^{-in\pi x}),0,0)+O(n^{-1}),$ (27)

which implies that,  there exists enough big positive integer $N$ such that

$\displaystyle\sum_{|n|>N}\|F_{n}-(P(x)(e^{in\pi x},e^{-in\pi
x}),0,0)\|_{\left(L(0,1)\right)^{2}\times\mathds{C}^{2}}^{2}$
$\displaystyle=\sum_{|n|>N}O(|n|^{-2})<\infty.$

Since $\\{(0,0,1,0)\\}\bigcup\\{(0,0,0,1)\\}\bigcup\\{(P(x)(e^{in\pi
x},e^{-in\pi x}),0,0)\\}_{n=-\infty}^{+\infty}$ forms Riesz basis for
$\big{(}L^{2}(0,$ $1)\big{)}^{2}\times\mathds{C}$, it follows by [11, Lemma 1]
that there exists a $M\geq N$ such that
$\\{(0,0,1,0)\\}\bigcup\\{(0,0,0,1)\\}$ $\bigcup\\{(P(x)(e^{in\pi x},e^{-in\pi
x}),0,0)\\}_{n=-M}^{M}\bigcup\\{F_{n}\\}_{|n|>M}$ forms Riesz basis for
$\left(L^{2}(0,1)\right)^{2}\times\mathds{C}^{2}$. We define an isometric
isomorphism
$\mathbb{T}_{1}:\mathbf{H}_{2}\rightarrow\left(L^{2}(0,1)\right)^{2}\times\mathds{C}^{2}$
by
$\mathbb{T}_{1}(f,g,\eta)=(f^{\prime},g,\sqrt{\beta}f(0),\eta),\;\forall\;(f,g,\eta)\in\mathbf{H}_{2}.$
Then there exist $\\{W_{n}(x),V_{n}(x),0\\}_{n=-M}^{M}\subset\mathbf{H}_{2}$
such that $\\{(0,0,1)\\}\bigcup\\{(W_{n}(x),V_{n}(x),0)\\}_{n=-M}^{M}$
$\bigcup\\{(\lambda^{-2}_{n}f_{n}(x),\lambda^{-1}_{n}f_{n}(x),-\lambda_{n}^{-3}f^{\prime}_{n}(1))\\}_{|n|>M}$
forms Riesz basis for $\mathbf{H}_{2}$. Since
$\\{(\lambda^{-2}_{n}f_{n}(x),\lambda^{-1}_{n}f_{n}(x),$
$-\lambda_{n}^{-3}f^{\prime}_{n}(1))\\}_{n=-\infty}^{\infty}$ is a sequence of
generalized eigenvectors of $\mathbf{A}_{2}$, and by [34], $\mathbf{A}_{2}$ is
a densely defined and discrete operator, it follows from [11, Theorem 1] that
there exists generalized eigenfunctions of $\mathbf{A}_{2}$ that forms Riesz
basis for $\mathbf{H}_{2}$. The proof is therefore completed.

Since the state feedback $U(t)=-\alpha u_{t}-au_{xt}(1,t)$ makes the system
(2) exponentially stable [10, 32], Lemma 2 allows us to design the following
control law

$\displaystyle U(t)=-\alpha\widehat{u}_{t}(1,t)-a\widehat{u}_{xt}(1,t).$ (28)

The closed-loop system is then described by

$\displaystyle\left\\{\begin{array}[]{l}u_{tt}(x,t)=u_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
u(0,t)=0,u_{x}(1,t)+mu_{tt}(1,t)=-\alpha\widehat{u}_{t}(1,t)\\\
-a\widehat{u}_{xt}(1,t),\;\;t\geq 0,\\\
\widehat{u}_{tt}(x,t)=\widehat{u}_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
{\color[rgb]{0,0,1}\widehat{u}_{x}(0,t)=\gamma\widehat{u}_{t}(0,t)+\beta\widehat{u}(0,t)+u_{x}(0,t)},\;\;t\geq
0,\\\
\widehat{u}_{x}(1,t)+m\widehat{u}_{tt}(1,t)=-\alpha\widehat{u}_{t}(1,t)-a\widehat{u}_{xt}(1,t),\end{array}\right.$
(35)

which is equivalent to

$\displaystyle\left\\{\begin{array}[]{l}u_{tt}(x,t)=u_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
u(0,t)=0,u_{x}(1,t)+mu_{tt}(1,t)=-\alpha u_{t}(1,t)\\\
-au_{xt}(1,t)-\alpha\widetilde{u}_{t}(1,t)-a\widetilde{u}_{xt}(1,t),\;\;t\geq
0,\\\ \widetilde{u}_{tt}(x,t)=\widetilde{u}_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
\widetilde{u}_{x}(0,t)=\gamma\widetilde{u}_{t}(0,t)+\beta\widetilde{u}(0,t),\;\;t\geq
0,\\\ \widetilde{u}_{x}(1,t)+m\widetilde{u}_{tt}=0,\end{array}\right.$ (42)

Consider system (42) in Hilbert state space $\mathcal{H}=\mathbf{H}$
$\times\mathbf{H}_{2}$, where $\mathbf{H}=H_{E}^{1}(0,1)\times
L^{2}(0,1)\times\mathds{C}$, $H_{E}^{1}(0,1)=\\{f$ $\in H^{1}(0,1):f(0)=0\\}$.
The norm of $\mathbf{H}$ is given by
$\|(f,g,\eta)\|^{2}_{\mathbf{H}}=\int_{0}^{1}[|f^{\prime}(x)|^{2}+|g(x)|^{2}]dx+\frac{1}{m+\alpha
a}|\eta|^{2},$ $(f,g)\in\mathbf{H}.$ Define the operator
$\mathbf{A}:D(\mathbf{A})(\subset\mathbf{H})\rightarrow\mathbf{H}$ by
$\mathbf{A}(f,g,\eta)=(g,f^{\prime\prime},-f^{\prime}(1)-\alpha
g(1)),\;\forall\;(f,g)\in
D(\mathbf{A})=\\{(f,g)\in\mathbf{H}|f^{\prime\prime}(1)=0,\eta=mg(1)+af^{\prime}(1)\\}.$
By [10, 32], it follows that $\mathbf{A}$ generates an exponentially stable
$C_{0}$-semigroup, and there exist a sequence of generalized eigenfunctions of
$\mathbf{A}$ that forms Riesz basis for $\mathbf{H}$ provided $\alpha=m/a\neq
1$. In the following Lemma 3, we generalize the result to the case $a\neq
m,\alpha>0$.

###### Lemma 3

Suppose that $a\neq m$. Then, there exist a sequence of generalized
eigenfunctions of operator $(\mathbf{A},D(\mathbf{A}))$ that forms Riesz basis
for $\mathbf{H}$; the characteristic equation of $(\mathbf{A},D(\mathbf{A}))$
is $e^{2\lambda}[1+\alpha+(a+m)\lambda]+(1-\alpha)+a-m=0$.

Proof. We divide the proof into several steps.

Step 1. We claim that operator $(\mathbf{A},D(\mathbf{A}))$ a densely defined
and discrete operator on $\mathbf{H}$; for any
$\lambda\in\sigma(\mathbf{A})=\sigma_{P}(\mathbf{A})$, ${\rm Re}\lambda<0$.

Indeed, since by [32],$(\mathbf{A},D(\mathbf{A}))$ generates an exponentially
stable $C_{0}$-semigroup, it is densely defined and has bounded inverse. Let
$(f,g,\eta)\in\mathbf{H}$. Since by Sobolev imbedding theory $H^{2}(0,1)\times
H^{1}(0,1)\times\mathds{C}$ is compactly imbedding in $H^{1}(0,1)\times
L^{2}(0,1)\times\mathds{C}$, $\mathbf{A}^{-1}$ is compact. Moreover, for any
$\lambda\in\sigma(\mathbf{A})=\sigma_{P}(\mathbf{A})$, ${\rm Re}\lambda<0$.

Step 2. We claim that for any $\lambda\in\sigma(\mathbf{A})$, there
corresponds one eigenfunction $(f,g,\eta)$ given by

$\displaystyle\left\\{\begin{array}[]{ll}f(x)=e^{\lambda x}-e^{-\lambda x},\\\
g=\lambda f,\\\ \eta=-\frac{f^{\prime}(1)+\alpha\lambda
f(1)}{\lambda},\end{array}\right.$ (46)

and $\lambda$ satisfies the characteristic equation

$\displaystyle e^{2\lambda}[1+\alpha+(a+m)\lambda]+(1-\alpha)+(a-m)=0.$ (47)

This implies that each eigenvalue of $\mathbf{A}$ is geometrically simple.

Step 3: We show that the eigenvalues $\\{\lambda_{n}\\}_{n=-\infty}^{+\infty}$
have asymptotic expression

$\lambda_{n}=\frac{1}{2}{\rm ln}\frac{|m-a|}{m+a}+n_{a}\pi i+O(|n|^{-1}),$
(48)

where $n_{a}=n$ if $m>a$ and $n_{a}=n+\frac{1}{2}$ if $0<m<a$, the
corresponding eigenfunction $(f_{n},\lambda_{n}f_{n},$
$m\lambda_{n}f(1)+af^{\prime}_{n}(1))$ of $\mathbf{A}_{2}$ can be chosen such
that

$\displaystyle
F_{n}(x)=\left(\begin{array}[]{c}\left|\frac{m-a}{m+a}\right|^{x/2}e^{in_{a}x}+\left|\frac{m+a}{m-a}\right|^{x/2}e^{-in_{a}x}\\\
\left|\frac{m-a}{m+a}\right|^{x/2}e^{in_{a}x}-\left|\frac{m+a}{m-a}\right|^{x/2}e^{-in_{a}x}\\\
0\\\ \end{array}\right)$ (52) $\displaystyle+O(n^{-1}),$ (53)

where

$F_{n}(x)=\frac{1}{\lambda_{n}}\begin{pmatrix}f_{n}^{{}^{\prime}}(x)\\\
\lambda_{n}f_{n}(x)\\\
-\lambda_{n}^{-1}[f^{\prime}_{n}(1)+\alpha\lambda_{n}f_{n}(1)]\\\
\end{pmatrix}^{T}.$ (54)

Now we give the proof. By (47), it follows that

$\displaystyle
e^{2\lambda}=-\frac{1-\alpha+(a-m)\lambda}{1+\alpha+(a+m)\lambda}=\frac{m-a}{m+a}+O(|\lambda|^{-1}).$
(55)

The asymptotic expression (48) is derived directly by Routhe’s theorem.

Step 4: We show that there exist a sequence of generalized eigenfunctions of
$\mathbf{A}$ which forms Riesz basis for $\mathbf{H}$.

Now we give the proof. Denote

$\displaystyle
Q(x)=\left\\{\begin{array}[]{ll}\left(\begin{array}[]{cc}\left|\frac{m-a}{m+a}\right|^{x/2}&\left|\frac{m+a}{m-a}\right|^{x/2}\\\
\left|\frac{m-a}{m+a}\right|^{x/2}&-\left|\frac{m+a}{m-a}\right|^{x/2}\\\
\end{array}\right),m>a,\\\
\left(\begin{array}[]{cc}\left|\frac{m-a}{m+a}\right|^{x/2}i&-\left|\frac{m+a}{m-a}\right|^{x/2}i\\\
\left|\frac{m-a}{m+a}\right|^{x/2}i&\left|\frac{m+a}{m-a}\right|^{x/2}i\\\
\end{array}\right),m<a.\end{array}\right.$

It is easily seen that the operator $\mathbf{T}$ defined by
$\mathbf{T}w=Q(x)w$ for any $w\in\left(L^{2}(0,1)\right)^{2}$, is a bounded
and invertible operator on $\left(L^{2}(0,1)\right)^{2}$. Combine this with
the fact that $\\{(e^{in\pi x},e^{-in\pi x})\\}_{n=-\infty}^{+\infty}$ forms
Riesz basis for $\left(L^{2}(0,1)\right)^{2}$, to obtain that
$\\{Q(x)(e^{in\pi x},e^{-in\pi x})\\}_{n=-\infty}^{+\infty}$ also forms Riesz
basis for $\left(L^{2}(0,1)\right)^{2}$. Thereby,
$\\{(0,0,1)\\}\bigcup\\{(Q(x)(e^{in\pi x},e^{-in\pi x}),$
$0)\\}_{n=-\infty}^{+\infty}$ forms Riesz basis for
$\left(L^{2}(0,1)\right)^{2}\times\mathds{C}$.

By (2), it follows that

$\displaystyle F_{n}(x)=(Q(x)(e^{in\pi x},e^{-in\pi x}),0)+O(n^{-1}),$ (56)

which implies that, there exists enough big positive integer $N$ such that

$\displaystyle\sum_{|n|>N}\|F_{n}(x)-(Q(x)(e^{in\pi x},e^{-in\pi
x}),0)\|_{\left(L(0,1)\right)^{2}\times\mathds{C}^{2}}^{2}=\sum_{|n|>N}O(|n|^{-2})<\infty.$

Since $\\{(0,0,1)\\}\bigcup\\{(Q(x)(e^{in\pi x},e^{-in\pi
x}),0)\\}_{n=-\infty}^{+\infty}$ forms Riesz basis for
$\left(L^{2}(0,1)\right)^{2}\times\mathds{C}$, it follows by [11, Lemma 1]
that there exists a $M\geq N$ such that $\\{(0,0,1)\\}\bigcup\\{(Q(x)(e^{in\pi
x},e^{-in\pi x}),0)\\}_{n=-M}^{M}\bigcup$ $\\{F_{n}\\}_{|n|>M}$ forms Riesz
basis for $\left(L^{2}(0,1)\right)^{2}\times\mathds{C}$. We define an
isometric isomorphism
$\mathbb{T}_{1}:\mathbf{H}\rightarrow\left(L^{2}(0,1)\right)^{2}\times\mathds{C}$
by
$\mathbb{T}_{1}(f,g,\eta)=(f^{\prime},g,\eta),\;\forall\;(f,g,\eta)\in\mathbf{H}.$
Then there exist $\\{W_{n}(x),V_{n}(x),0\\}_{n=-M}^{M}\subset\mathbf{H}$ such
that
$\\{(0,0,1)\\}\bigcup\\{(W_{n}(x),V_{n}(x),0)\\}_{n=-M}^{M}\bigcup\\{(\lambda^{-1}_{n}f_{n}(x),f_{n}(x),$
$-\lambda_{n}^{-2}[f^{\prime}_{n}(1)+\alpha\lambda_{n}f_{n}(1)])$
$\\}_{|n|>M}$ forms Riesz basis for $\mathbf{H}$. Since
$\\{(\lambda^{-1}_{n}(x),f_{n}(x),$
$-\lambda_{n}^{-2}[f^{\prime}_{n}(1)+\alpha\lambda_{n}f_{n}(1)])\\}_{n=-\infty}^{\infty}$
is a sequence of generalized eigenvectors of $\mathbf{A}$, and $\mathbf{A}$ is
a densely defined and discrete operator. By [11, Theorem 1], it follows that
there exists generalized eigenfunctions of $\mathbf{A}$ that forms Riesz basis
for $\mathbf{H}$. This completes the proof.

Define operator
$\mathcal{A}:D(\mathcal{A})(\subset\mathcal{H})\rightarrow\mathcal{H}$ by
$\mathcal{A}(f,g,\eta,\phi,\psi,$
$h)=(g,f^{\prime\prime},-f^{\prime}(1)-\alpha
g(1)-\alpha\psi(1),\mathbf{A}_{2}(\phi,\psi,h)),$
$D(\mathcal{A})=\\{(f,g,\eta,\phi,\psi,h)\in(H_{E}^{1}(0,1)\bigcap$
$H^{2}(0,1))\times H_{E}^{1}(0,1)\times\mathds{C}\times
D(\mathbf{A}_{2}),\eta=af^{\prime}(1)+mg(1)+a\phi^{\prime}(1),h=m\psi(1)\\}.$
Then the system (42) is abstractly described by

$\frac{d}{dt}Z(t)=\mathcal{A}Z(t),$

where
$Z(t)=\big{(}u(\cdot,t),u_{t}(\cdot,t),au_{x}(1,t)+mu_{t}(1,t)+a\widetilde{u}_{x}(1,t),\widetilde{u}(\cdot,t),$
$\widetilde{u}_{t}(\cdot,t),m\widetilde{u}_{t}(1,t)\big{)}.$ We shall use
Riesz basis approach to verify semigroup generation and exponential stability.
However, it is difficult to verify the Riesz basis generation of couple wave
equations [12, 16, 30], not to mention wave equations with tip mass. We shall
adopt Bari’s theorem to prove Riesz basis generation of operator $\mathcal{A}$
by finding out complicated relations between sequences of generalized
eigenfunctions.

###### Theorem 4

Assume that $m\neq a$, $m\neq a\gamma$ and $\gamma\neq 1$. Then, the operator
$\mathcal{A}$ generates an exponentially stable $C_{0}$-semigroup on
$\mathcal{H}$.

Proof. It is easy to show that $\mathcal{A}^{-1}$ exists and is compact on
$\mathcal{H}$, thereby the spectrum of $\mathcal{A}$ consists of eigenvalues.
Now we show that
$\sigma(\mathcal{A})=\sigma(\mathbf{A})\bigcup\sigma(\mathbf{A}_{2})$. By [12]
and Lemma 10, $\mathbf{A}$ and $\mathbf{A}_{2}$ are also discrete operators.
Hence
$\sigma(\mathcal{A})\supseteq\sigma(\mathbf{A})\bigcup\sigma(\mathbf{A}_{2}).$
Let $\lambda\in\sigma(\mathcal{A})$ and $(f,g,\eta,\phi,\psi,h)$ be the
corresponding eigenfunction. If $(\phi,\psi,h)\neq 0$,
$\lambda\in\sigma(\mathbf{A}_{2})$; if $(\phi,\psi,h)=0$, we have that
$(f,g,\eta)\neq 0$, $(f,g,\eta)\in D(\mathbf{A})$, and
$\lambda(f,g,\eta)=\mathbf{A}(f,g,\eta)$, which implies that
$\lambda\in\sigma(\mathbf{A})$. Hence
$\sigma(\mathcal{A})\subseteq\sigma(\mathbf{A})\bigcup\sigma(\mathbf{A}_{2}).$
Therefore
$\sigma(\mathcal{A})=\sigma(\mathbf{A})\bigcup\sigma(\mathbf{A}_{2})$.

Next, we shall show that the generalized eigenfunction of $\mathcal{A}$ forms
a Riesz basis for $\mathcal{H}$. Let $\\{\mu_{n}\\}_{n=-\infty}^{\infty}$ and
$\\{\lambda_{n}\\}_{n=-\infty}^{\infty}$ be respectively the eigenvalues of
$\mathbf{A}$ and $\mathbf{A}_{2}$. Let
$\\{(\mu_{n}^{-1}f_{n},f_{n},-\mu_{n}^{-2}(f^{\prime}(1)$
$+\alpha\mu_{n}f_{n}(1)))\\}_{n=-\infty}^{\infty}$ and
$\\{(\lambda^{-2}_{n}\phi_{n},\lambda_{n}^{-1}\phi_{n},\lambda^{-3}_{n}\phi^{\prime}_{n}(1)\\}_{n=-\infty}^{\infty}$
be the generalized eigenfunctions corresponding to
$\\{\mu_{n}\\}_{n=-\infty}^{\infty}$ and
$\\{\lambda_{n}\\}_{n=-\infty}^{\infty}$ such that they form Riesz basises for
$\mathbf{H}$ and $\mathbf{H}_{2}$, respectively. Hence, the sequence
$\\{(\mu_{n}^{-1}f_{n},f_{n},-\mu_{n}^{-2}(f^{\prime}(1)+\alpha\mu_{n}f_{n}(1)),0,0,0)\\}_{n=-\infty}^{\infty}\bigcup$
$\\{(0,0,0,\lambda^{-2}_{n}\phi_{n},\lambda_{n}^{-1}\phi_{n},$
$\lambda^{-3}_{n}\phi^{\prime}_{n}(1))\\}_{n=-\infty}^{\infty}$ forms a Riesz
basis for $\mathcal{H}$, which is equivalent to that
$\\{(\mu_{n}^{-1}f^{\prime}_{n},f_{n},-\mu_{n}^{-2}$
$(f^{\prime}(1)+\alpha\mu_{n}f_{n}(1)),0,0,0,0)\\}_{n=-\infty}^{\infty}\bigcup\\{(0,0,0,\lambda^{-2}_{n}\phi^{\prime}_{n},$
$\lambda_{n}^{-1}\phi_{n},\lambda_{n}^{-2}\beta\phi_{n}(0),\lambda^{-3}_{n}\phi^{\prime}_{n}(1)$
$\\}_{n=-\infty}^{\infty}$ forms a Riesz basis for
$\left(L^{2}(0,1)\right)^{2}\times\mathds{C}\times\left(L^{2}(0,1)\right)^{2}\times\mathds{C}^{2}$.

Let $\lambda\in\sigma(\mathcal{A})$ and
$\big{(}\lambda^{-2}f,\lambda^{-1}f,-\lambda^{-3}(f^{\prime}(1)+\alpha\lambda
f(1)+\alpha\lambda\phi(1)),$
$\lambda^{-2}\phi,\lambda^{-1}\phi,\lambda^{-3}\phi^{\prime}(1)\big{)}$ be the
corresponding eigenfunction. If $\phi=0$, then
$\big{(}\lambda^{-2}f,\lambda^{-1}f,-\lambda^{-3}(f^{\prime}(1)+\alpha\lambda
f(1))\big{)}\neq 0$ and $\lambda\in\sigma(\mathbf{A})$. Hence the eigenvalues
$\\{\mu_{n}\\}_{n=1}^{\infty}$ corresponds the eigenfunctions
$\\{\big{(}\mu_{n}^{-2}f_{n},\mu_{n}^{-1}f_{n},$
$-\mu_{n}^{-3}(f_{n}^{\prime}(1)+\alpha\mu_{n}f_{n}(1)),0,0,0\big{)}\\}_{n=-\infty}^{\infty}$.

If $\phi\neq 0$, then $\lambda\in\sigma(\mathbf{A}_{2})$. The eigenvalues
$\\{\lambda_{n}\\}_{n=-\infty}^{\infty}$ corresponds the eigenfunction
$\big{(}\lambda_{n}^{-2}\phi_{n},$
$\lambda_{n}^{-1}\phi_{n},\lambda_{n}^{-3}\phi_{n}^{\prime}(1)\big{)}$ of
$\mathbf{A}_{2}$. By Lemma 2, it follows that

$\begin{split}\phi_{n}(x)=[(1+\gamma)\lambda_{n}+\beta]e^{\lambda_{n}x}+[(1-\gamma)\lambda_{n}-\beta]e^{-\lambda_{n}x}.\end{split}$
(57)

Let
$\big{(}\lambda_{n}^{-2}f_{1n},\lambda_{n}^{-1}f_{1n},-\lambda_{n}^{-3}(f_{1n}^{\prime}(1)+\alpha\lambda_{n}f_{1n}(1)+\alpha\lambda_{n}\phi_{n}(1)),\lambda_{n}^{-2}\phi_{n},$
$\lambda_{n}^{-1}\phi_{n},\lambda_{n}^{-3}\phi_{n}^{\prime}(1)\big{)}\in\mathcal{H}$
be the eigenfunction of $\mathcal{A}$ corresponding to eigenvalues
$\\{\lambda_{n}\\}_{n=-\infty}^{\infty}$. Then, $f_{1n}$ satisfies
$f^{\prime\prime}_{1n}(x)=\lambda_{n}^{2}f_{1n}(x),$
$f_{1n}(0)=0,(1+a\lambda_{n})f^{\prime}_{1n}(1)+(m\lambda^{2}_{n}+\alpha\lambda_{n})f_{1n}=-\alpha\lambda_{n}\phi_{n}(1)-a\lambda_{n}\phi^{\prime}_{n}(1).$
The solution of the ode $f^{\prime\prime}_{1n}(x)=\lambda^{2}_{n}f_{1n}(x)$
with boundary conditions $f_{1n}(0)=0$ is of the form

$\displaystyle f_{1n}(x)=E(e^{\lambda_{n}x}-e^{-\lambda_{n}x}).$ (58)

Combine (57) and
$(1+a\lambda_{n})f^{\prime}_{1n}(1)+(m\lambda^{2}_{n}+\alpha\lambda_{n})f_{1n}=-\alpha\lambda_{n}\phi_{n}(1)-a\lambda_{n}\phi^{\prime}_{n}(1)$
to derive
$\frac{E}{\lambda_{n}}=-\frac{\alpha\phi_{n}(1)+a\phi^{\prime}_{n}(1)}{E_{11}e^{\lambda_{n}}+E_{12}e^{-\lambda_{n}}}$
$=-\frac{E_{21}e^{\lambda_{n}}+E_{22}e^{-\lambda_{n}}}{E_{11}e^{\lambda_{n}}+E_{12}e^{-\lambda_{n}}},$
where
$E_{11}=\lambda_{n}[1+\alpha+(a+m)\lambda_{n}],E_{12}=\lambda_{n}[1-\alpha+(a-m)\lambda_{n}],E_{21}=(\alpha$
$+a\lambda_{n})[(1+\gamma)\lambda_{n}+\beta],E_{22}=(\alpha-a\lambda_{n})[(1-\gamma)\lambda_{n}-\beta].$
Use the characteristic equation in Lemma 3 to get
$\frac{E}{\lambda_{n}}=-\frac{E_{21}F_{1}+E_{22}F_{2}}{E_{11}F_{1}+E_{12}F_{2}}=\frac{a(\gamma^{2}-1)}{\gamma
a-m}+O(|n^{-1}|),$ where
$F_{1}=[(1-\gamma)\lambda_{n}-\beta](1-m\lambda_{n})$,
$F_{2}=[(1+\gamma)\lambda_{n}+\beta](1+m\lambda_{n})$. This, together with
(58) indicates

$\displaystyle\lambda_{n}^{-2}f^{\prime}_{1n}(x)=\lambda_{n}^{-1}E[e^{\lambda_{n}x}+e^{-\lambda_{n}x}]=\frac{a(\gamma^{2}-1)}{\gamma
a-m}\cdot$
$\displaystyle\bigg{[}\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}e^{in_{\beta}x}+\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_{\beta}x}\bigg{]}+O(|n|^{-1}),$
$\displaystyle\lambda_{n}^{-1}f_{1n}(x)=\lambda_{n}^{-1}E[e^{\lambda_{n}x}-e^{-\lambda_{n}x}]=\frac{a(\gamma^{2}-1)}{\gamma
a-m}\cdot$
$\displaystyle\left[\left|\frac{\gamma-1}{\gamma+1}\right|^{x/2}e^{in_{\beta}x}-\left|\frac{\gamma+1}{\gamma-1}\right|^{x/2}e^{-in_{\beta}x}\right]+O(|n|^{-1}).$

Denote $W=\left(\begin{array}[]{cc}I_{3}&V\\\ 0&I_{3}\\\ \end{array}\right),$
with $V=\frac{2a}{m-a\gamma}\left(\begin{array}[]{ccc}-\gamma&1&0\\\
1&-\gamma&0\\\ 0&0&0\\\ \end{array}\right).$ Obviously, $W$ has bounded
inverse. Combine this with Lemma 2 to derive

$\displaystyle\big{(}\mu_{n}^{-2}f_{n},\mu_{n}^{-1}f_{n},-\mu_{n}^{-3}(f_{n}^{\prime}(1)+\alpha\mu_{n}f_{n}(1)),0,0,0\big{)}^{T}=$
$\displaystyle
W\big{(}\mu_{n}^{-2}f_{n},\mu_{n}^{-1}f_{n},-\mu_{n}^{-3}(f_{n}^{\prime}(1)+\alpha\mu_{n}f_{n}(1)),0,0,0\big{)}^{T},$
(59)
$\displaystyle\big{(}\lambda_{n}^{-2}f_{1n},\lambda_{n}^{-1}f_{1n},-\lambda_{n}^{-3}(f_{1n}^{\prime}(1)+\alpha\lambda_{n}f_{1n}(1)$
$\displaystyle+\alpha\lambda_{n}\phi_{n}(1)),\lambda_{n}^{-2}\phi_{n},\lambda_{n}^{-1}\phi_{n},\lambda_{n}^{-3}\phi_{n}^{\prime}(1)\big{)}^{T}=W\big{(}0,0,0,$
$\displaystyle\lambda_{n}^{-2}\phi_{n},\lambda_{n}^{-1}\phi_{n},\lambda_{n}^{-3}\phi_{n}^{\prime}(1)\big{)}^{T}+O(|n|^{-1}).$
(60)

Then, by Bari’s theorem and [11, Theorem 1], the sequence
$\\{\big{(}\mu_{n}^{-2}f^{\prime}_{n},\mu_{n}^{-1}f_{n},$
$-\mu_{n}^{-3}(f_{n}^{\prime}(1)+\alpha\mu_{n}f_{n}(1)),$
$0,0,0,0\big{)}\\}_{n=-\infty}^{\infty}\bigcup\\{\big{(}\lambda_{n}^{-2}f^{\prime}_{1n},$
$\lambda_{n}^{-1}f_{1n},-\lambda_{n}^{-3}(f_{1n}^{\prime}(1)+\alpha\lambda_{n}f_{1n}(1)+\alpha\lambda_{n}\phi_{n}(1)),\lambda_{n}^{-2}\phi^{\prime}_{n},$
$\lambda_{n}^{-1}\phi_{n},\lambda_{n}^{-2}\beta\phi_{n}(0),$
$\lambda_{n}^{-3}\phi_{n}^{\prime}(1)\big{)}\\}_{n=-\infty}^{\infty}$ forms
Riesz basis for the Hilbert space
$\big{(}L^{2}(0,1)\big{)}^{2}\times\mathds{C}\times\left(L^{2}(0,1)\right)^{2}\times\mathds{C}^{2}$,
which is equivalent to that the sequence
$\\{\big{(}\mu_{n}^{-2}f_{n},\mu_{n}^{-1}f_{n},$
$-\mu_{n}^{-3}(f_{n}^{\prime}(1)+\alpha\mu_{n}f_{n}(1)),0,0,0\big{)}\\}_{n=-\infty}^{\infty}$
$\bigcup\\{\big{(}\lambda_{n}^{-2}f_{1n},$
$\lambda_{n}^{-1}f_{1n},-\lambda_{n}^{-3}(f_{1n}^{\prime}(1)+\alpha\lambda_{n}f_{1n}(1)+\alpha\lambda_{n}\phi_{n}(1)),\lambda_{n}^{-2}\phi_{n},$
$\lambda_{n}^{-1}\phi_{n},\lambda_{n}^{-3}\phi_{n}^{\prime}(1)\big{)}\\}_{n=-\infty}^{\infty}$
forms Riesz basis for $\mathcal{H}$.

The semigroup generation and spectrum-determined growth condition of
$\mathcal{A}$ are derived directly from the Riesz basis property. Since both
$\mathbf{A}$ and $\mathbf{A}_{2}$ are generator of exponentially stable
$C_{0}$-semigroups, $e^{\mathcal{A}t}$ is exponentially stable.

###### Remark 5

We have to mention that, although it forms Riesz basis for $\mathcal{H}$,
$\\{(\mu_{n}^{-1}f_{n},f_{n},-\mu_{n}^{-2}(f^{\prime}(1)+\alpha\mu_{n}f_{n}(1)),$
$0,0,0)\\}_{n=-\infty}^{\infty}\bigcup\\{(0,0,0,\lambda^{-2}_{n}\phi_{n},\lambda_{n}^{-1}\phi_{n},\lambda^{-3}_{n}\phi^{\prime}_{n}(1)\\}_{n=-\infty}^{\infty}$
are not all the generalized eigenfunctions of $\mathcal{A}$. This is the
reason why we choose to compute the generalized eigenfunctions of
$\mathcal{A}$ and find out the relations (59) and (60), which are the key
steps for the proof.

###### Theorem 6

Given initial value $(u(\cdot,0),u_{t}(\cdot,0),mu_{t}(1,$
$0)+a\widehat{u}_{x}(1,0),\widehat{u}(\cdot,0),\widehat{u}_{t}(\cdot,0),m\widehat{u}_{t}(1,0)+a\widehat{u}_{x}(1,0))\in\mathcal{H}_{1}$,
there exists a unique solution to system (35) such that
$(u(\cdot,t),u_{t}(\cdot,t),mu_{t}(1,t)+a\widehat{u}_{x}(1,t),\widehat{u}(\cdot,t),\widehat{u}_{t}(\cdot,t),m\widehat{u}_{t}(1,$
$t)+a\widehat{u}_{x}(1,t))\in C(0,\infty;\mathcal{H}_{1})$ satisfying
$\int_{0}^{1}(|u_{t}(x,t)|^{2}+|u_{x}(x,t)|^{2}+|\widehat{u}_{t}(x,t)|^{2}+|\widehat{u}_{x}(x,t)|^{2})dx+|mu_{t}(1,t)+a\widehat{u}_{x}(1,t)|^{2}/m+{\color[rgb]{0,0,1}|m\widehat{u}_{t}(1,0)+a\widehat{u}_{x}(1,0)|^{2}/(m+a\alpha)}+\beta|\widehat{u}(0,t)|^{2}$
$\leq M_{1}e^{-\gamma_{1}t},$ where $M_{1}$ and $\gamma_{1}$ are two positive
constants.

Proof. Fix $(u(\cdot,0),u_{t}(\cdot,0),mu_{t}(1,0)+a\widehat{u}_{x}(1,0),$
$\widehat{u}(\cdot,0),\widehat{u}_{t}(\cdot,0),m\widehat{u}_{t}(1,0)+a\widehat{u}_{x}(1,0))\in\mathbf{H}_{1}\times\mathbf{H}_{3}$.
Then we have
$(u(\cdot,0),u_{t}(\cdot,0),mu_{t}(1,0)+a[u_{x}(1,0)+\widetilde{u}_{x}(1,0)],\widetilde{u}(\cdot,0),\widetilde{u}_{t}(\cdot,0),m\widetilde{u}_{x}(1,0))\in\mathbf{H}\times\mathbf{H}_{2}$.
By Theorem 4, it follows that
$(u(\cdot,t),u_{t}(\cdot,t),mu_{t}(1,t)+a[u_{x}(1,t)+\widetilde{u}_{x}(1,t)],\widetilde{u}(\cdot,t),\widetilde{u}_{t}(\cdot,t),m\widetilde{u}_{x}(1,t))=e^{\mathcal{A}t}(u(\cdot,$
$0),u_{t}(\cdot,0),mu_{t}(1,0)+a[u_{x}(1,0)+\widetilde{u}_{x}(1,0)],\widetilde{u}(\cdot,0),$
$\widetilde{u}_{t}(\cdot,0),m\widetilde{u}_{x}(1,0))\in
C(0,\infty;\mathbf{H}\times\mathbf{H}_{2})$ and

$\displaystyle\int_{0}^{1}(|u_{t}(x,t)|^{2}+|u_{x}(x,t)|^{2}+|\widehat{u}_{t}(x,t)|^{2}+|\widehat{u}_{x}(x,t)|^{2})dx$
$\displaystyle+\frac{1}{m}|mu_{t}(1,0)+a\widehat{u}_{x}(1,t)|^{2}+\frac{1}{m+a\alpha}|m\widehat{u}_{t}(1,0)$
$\displaystyle+a\widehat{u}_{x}(1,t)|^{2}+\beta|\widehat{u}(0,t)|^{2}$
$\displaystyle\leq\int_{0}^{1}(|u_{t}(x,t)|^{2}+|u_{x}(x,t)|^{2}+2|u_{t}(x,t)|^{2}$
$\displaystyle+2|u_{x}(x,t)|^{2}+2|\widetilde{u}_{t}(x,t)|^{2}+2|\widetilde{u}_{x}(x,t)|^{2})dx$
$\displaystyle+\frac{1}{m}|mu_{t}(1,t)+a[u_{x}(1,t)+\widetilde{u}_{x}(1,t)]|^{2}$
$\displaystyle+\frac{2}{m+a\alpha}|mu_{t}(1,t)+a[u_{x}(1,t)+\widetilde{u}_{x}(1,t)]|^{2}$
$\displaystyle+\frac{2}{m+a\alpha}|m\widetilde{u}_{t}(1,t)|^{2}+\beta|\widetilde{u}(0,t)|^{2}$
$\displaystyle\leq
3\left[1+\frac{m+a\alpha}{m}\right]\|(u(\cdot,t),u_{t}(\cdot,t),mu_{t}(1,t)$
$\displaystyle+a[u_{x}(1,t)+\widetilde{u}_{x}(1,t)],\widetilde{u}(\cdot,t),\widetilde{u}_{t}(\cdot,t),m\widetilde{u}_{x}(1,t))\|^{2}$
$\displaystyle=3\left[1+\frac{m+a\alpha}{m}\right]\|e^{\mathcal{A}t}(u(\cdot,0),u_{t}(\cdot,0),mu_{t}(1,0)$
$\displaystyle+a[u_{x}(1,0)+\widetilde{u}_{x}(1,0)],\widetilde{u}(\cdot,0)\|^{2}$
$\displaystyle\leq M_{1}e^{-\gamma_{1}t},$

where
$M_{1}=3M^{2}_{\mathcal{A}}\left[1+\frac{m+a\alpha}{m}\right]\|(u(\cdot,0),u_{t}(\cdot,0),mu_{t}(1,$
$0)+a[u_{x}(1,0)+\widetilde{u}_{x}(1,0)],\widetilde{u}(\cdot,0)\|^{2}$,
$\gamma_{1}=2{\color[rgb]{0,0,1}\omega_{\mathcal{A}}}$. This completes the
proof.

###### Remark 7

Theorem 6 presents not only the exponential stability of
$(u(\cdot),u_{t}(\cdot),\widehat{u}(\cdot,t),\widehat{u}_{t}(\cdot,t))$ but
also the exponential stability of
$\eta(t)=mu_{t}(1,t)+a[u_{x}(1,t)+\widetilde{u}_{x}(1,t)]$ and
$\psi(t)=\widehat{u}_{t}(1,t)+a[u_{x}(1,t)+\widetilde{u}_{x}(1,t)]$, because
$\eta(t)$ and $\psi(t)$ are the states of the boundary ode dynamic parts of
the closed-loop system. This is quite different from the case in [12] where
$m=0$. By Theorem 6, with only one non-collocated measurement $u_{x}(0,t)$ we
can design the estimated state based controller (28) to exponentially
stabilize the vibration cable with tip mass (2). Therefore, we improve the
results in references [10, 32], where two collocated measurements $u_{t}(1,t)$
and $u_{xt}(1,t)$ with $u_{xt}(1,t)$ being angular velocity were used.

###### Remark 8

In [12], Guo and Xu verified the exponential stability of closed-loop system
(35) with $m=0$. By virtue of Riesz basis approach, the authors used 2.5 pages
of two columns to verify the semigroup generation and exponential stability of
system [12, (3.1)]. When $m\neq 0$, we transfer the closed-loop system to an
equivalent system (42) which contains the original system and error systems.
Although we also used Riesz basis approach and our case is more complicated
($m\neq 0$), our proof is much shorter than that in [12] because (42) is
easier to be dealt with: it contains an independent subsystem related to
$\widetilde{u}$. Our method can be used to simplify the verification of
exponential stability in [12].

##  3 Stabilization in presence of internal uncertainty and external
disturbance

When we consider disturbance $F(t)\equiv F$, the boundary condition
$u_{x}(1,t)+mu_{tt}(1,t)=-\alpha\widehat{u}_{t}(1,t)-a\widehat{u}_{xt}(1,t)$
of the closed-loop system (35) is changed to
$u_{x}(1,t)+mu_{tt}(1,t)=-\alpha\widehat{u}_{t}(1,t)-a\widehat{u}_{xt}(1,t)+F$.
One can see that $u(x,t)=Fx,\widehat{u}(x,t)=-F/\beta$ is a solution of the
closed-loop system, which is not stable. Therefore, when there is disturbance,
the stabilizing control law should be redesigned. Motivated by [8], for system
(2) with $F(t)\neq 0$, we propose in this section an infinite-dimensional ESO
without high gain described as follow

$\left\\{\begin{array}[]{l}v_{tt}(x,t)=v_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
v_{x}(0,t)=\gamma v_{t}(0,t)+\beta v(0,t)+u_{x}(0,t),\\\
v_{x}(1,t)+mv_{tt}(1,t)=U(t),\;\;t\geq 0,\\\
q_{tt}(x,t)=q_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\ q_{x}(0,t)=\gamma
q_{t}(0,t)+\beta q(0,t),\;\;t\geq 0,\\\ q(1,t)=v(1,t)-u(1,t),\;\;t\geq
0.\end{array}\right.$ (61)

Since it depends only on the input $u(t)$ and output $u_{x}(0,t),u(1,t)$,
system (61) is completely known. Although there exists disturbance, the high
order angular velocity $u_{xt}(1,t)$ is not used.

Set $\widehat{v}(x,t)=v(x,t)-u(x,t)$ to derive the error equation with unknown
input as follows

$\left\\{\begin{array}[]{l}\widehat{v}_{tt}(x,t)=\widehat{v}_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
\widehat{v}_{x}(0,t)=\gamma\widehat{v}_{t}(0,t)+\beta\widehat{v}(0,t),\;\;t\geq
0,\\\ \widehat{v}_{x}(1,t)+m\widehat{v}_{tt}(1,t)=-F(t),\;\;t\geq
0\end{array}\right.$ (62)

which is written abstractly by

$\displaystyle\frac{d}{dt}{\color[rgb]{0,0,1}(\widehat{v}(\cdot,t),\widehat{v}_{t}(\cdot,t),m\widehat{v}_{t}(1,t))}=\mathbf{A}_{2}{\color[rgb]{0,0,1}(\widehat{v}(\cdot,t),\widehat{v}_{t}(\cdot,t),m\widehat{v}_{t}(1,t))}-\mathbf{B}_{2}F(t),$
(63)

where $\mathbf{B}_{2}=(0,0,1)^{T}$ is a bounded linear operator. Since
$\mathbf{A}_{2}$ is a generator of an exponentially stable $C_{0}$-semigroup,
the role of the $v$-part of the system (61) is to transfer the total
disturbance $F(t)$ into an exponentially stable system that is relatively
easier to be dealt with.

Similar to the proof of [42, Lemma A.1 and A.2], we derive the following
lemma.

###### Lemma 9

Assume that $d\in L^{\infty}(0,\infty)$ (or $d\in L^{2}(0,\infty))$,
$f:H^{1}(0,1)\times L^{2}(0,1)\rightarrow\mathds{R}$ is continuous and system
(1) admits a unique bounded solution
${\color[rgb]{0,0,1}(u(\cdot,t),u_{t}(\cdot,t))}\in
C(0,\infty;H^{1}(0,1)\times L^{2}(0,1))$. Then for any initial value
${\color[rgb]{0,0,1}(\widehat{v}(\cdot,0),\widehat{v}_{t}(\cdot,0),m\widehat{v}_{t}(1,0))}\in\mathbf{H}_{2}$,
there exists a unique solution to system (62) such that
$(\widehat{v}(\cdot,t),\widehat{v}_{t}(\cdot,t),m\widehat{v}_{t}(1,t))\in
C(0,\infty;\mathbf{H}_{2})$ and
$\|(\widehat{v}(\cdot,t),\widehat{v}_{t}(\cdot,t),m\widehat{v}_{t}(1,t))\|_{\mathbf{H}_{2}}<+\infty$.
Moreover, if $\lim_{t\rightarrow\infty}f(w(\cdot,t),w_{t}(\cdot,t))=0$ and
$d\in L^{2}(0,\infty)$, then
$\lim_{t\rightarrow\infty}\|(\widehat{v}(\cdot,t),\widehat{v}_{t}(\cdot,t),m\widehat{v}_{t}(1,$
$t))\|_{\mathbf{H}_{2}}=0$.

Set $\widehat{q}(x,t)=q(x,t)-\widehat{v}(x,t)$ to get

$\left\\{\begin{array}[]{l}\widehat{q}_{tt}(x,t)=\widehat{q}_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
{\color[rgb]{0,0,1}\widehat{q}_{x}(0,t)=\gamma\widehat{q}_{t}(0,t)+\beta\widehat{q}(0,t)},\widehat{q}(1,t)=0.\end{array}\right.$
(64)

Consider system (64) in the space $\mathbb{H}=H^{1}_{e}(0,1)\times
L^{2}(0,1)$, $H^{1}_{e}(0,1)=\\{f\in H^{1}(0,1)|f(1)=0\\}$ with inner product
induced norm
$\|(f,g)\|^{2}_{\mathbb{H}}=\int_{0}^{1}[|f^{\prime}(x)|^{2}+|g(x)|^{2}]dx$.
Define $\mathbb{A}(f,g)=(g,f^{\prime\prime}),\forall(f,g)\in
D(\mathbb{A})=\\{(f,g)\in(H_{e}^{1}\bigcap$ $H^{2}(0,1))\times
H^{1}_{e}(0,1)|f^{\prime}(0)=\gamma g(0)+\beta f(0)\\}.$ Then, system (64) can
be written abstractly as

$\displaystyle\frac{d}{dt}(\widehat{q}(\cdot,t),\widehat{q}_{t}(\cdot,t))=\mathbb{A}(\widehat{q}(\cdot,t),\widehat{q}_{t}(\cdot,t)).$

By [3], $\mathbb{A}$ is generates an exponentially stable $C_{0}$-semigroup.
The following Lemma 10 tells us that the system (64) possess Riesz basis
property.

###### Lemma 10

Assume that $\alpha\neq 1$. Then, $\mathbb{A}$ is a discrete operator; there
exist a sequence of generalized eigenfunctions of $\mathbb{A}$ which forms
Riesz basis for $\mathbb{H}$; there exist a family of eigenvalue
$\\{\lambda_{n}\\}_{n=-\infty}^{\infty}$ of operator $\mathbb{A}_{2}$
asymptotically expressed by
$\lambda_{n}=\frac{1}{2}\ln\bigg{|}\frac{\gamma-1}{\gamma+1}\bigg{|}+n_{\gamma}\pi
i+O(|n|^{-1}),$ where $n_{\gamma}=n-1/2$ if $0<\gamma<1$ and $n_{\gamma}=n$ if
$\gamma>1$; the corresponding eigenfunctions is given by
$(\lambda_{n}^{-1}f_{n},f_{n})$ with $f_{n}(x)=\sinh\lambda_{n}(x-1)$ and
$F_{n}(x):=\big{(}\lambda_{n}^{-1}f_{n}^{\prime}(x),f_{n}(x)\big{)}=\big{(}\big{|}\frac{\gamma-1}{\gamma+1}\big{|}^{(x-1)/2}e^{n_{\gamma}\pi
i(x-1)}+\big{|}\frac{\alpha+1}{\gamma-1}\big{|}^{(x-1)/2}e^{-n_{\gamma}\pi
i(x-1)},\big{|}\frac{\gamma-1}{\gamma+1}\big{|}^{(x-1)/2}e^{n_{\gamma}\pi
i(x-1)}-\big{|}\frac{\alpha+1}{\gamma-1}\big{|}^{(x-1)/2}$ $e^{-n_{\gamma}\pi
i(x-1)}\big{)}+O(|n|^{-1}).$

The following lemma is due to Feng and Guo [8].

###### Lemma 11

Assume that $(\widehat{q}(\cdot,0),\widehat{z}_{t}(\cdot,0))\in
D(\mathbb{A})$. The solution of (64) satisfies $|\widehat{q}_{x}(1,t)|\leq
Me^{-\mu t}$ for some constant $M,\;\mu>0$.

###### Remark 12

On one hand, since by (64) we obtain $\widehat{q}(1,t)=0$, it follows from
Lemma 11 that $F(t)=$
$-\widehat{v}_{x}(1,t)-m\widehat{v}_{tt}(1,t)=-q_{x}(1,t)-mq_{tt}(1,t)+{\color[rgb]{0,0,1}\widehat{q}_{x}(1,t)}+m\widehat{q}_{tt}(1,t)=-q_{x}(1,t)-mq_{tt}(1,t)+{\color[rgb]{0,0,1}\widehat{q}_{x}(1,t)}\approx-
q_{x}(1,t)-mq_{tt}(1,t)$ provided
$(\widehat{q}(\cdot,0),\widehat{q}_{t}(\cdot,0))\in D(\mathbb{A})$. This
indicates that $-q_{x}(1,t)-mq_{tt}(1,t)$ is an estimate of total disturbance
$F(t)$. On the other hand, since (64) decays exponentially, we obtain
$u(\cdot,t)=v(\cdot,t)-q(\cdot,t)+\widehat{q}(\cdot,t)\approx
v(\cdot,t)-q(\cdot,t)$. This implies that $v(\cdot,t)-q(\cdot,t)$ is an
estimate of $w(\cdot,t)$. The combination of the two hands tells us that the
system (61) estimates both the total disturbance and original state. This is
the reason why we call the system (61) infinite-dimensional ESO for (1). We
shall show in the rest of the present section that the disturbance estimator
(61) is enough to exponentially stabilize system (1) for general initial state
$(\widehat{q}(\cdot,0),\widehat{q}_{t}(\cdot,0))\in\mathbb{H}$.

Since by [32] the state feedback $u(t)=-\alpha u_{t}(1,t)-au_{xt}(1,t)$
exponentially stabilizes system (2), it is natural to design an estimated
state-based controller

$\displaystyle U(t)=q_{x}(1,t)+mq_{tt}(1,t)-\alpha[v_{t}(1,t)-q_{t}(1,t)]$
$\displaystyle-a[v_{xt}(1,t)-q_{xt}(1,t)],$ (65)

where $q_{x}(1,t)+mq_{tt}(1,t)$ is used to cancel the total disturbance
$F(t)$, $v_{t}(1,t)-q_{t}(1,t)$ and $v_{xt}(1,t)-q_{xt}(1,t)$ are respectively
applied to estimate $u_{t}(1,t)$ and $u_{xt}(1,t)$.

###### Remark 13

In [30, 42, 43], infinite-dimensional ESOs were used to estimate the total
disturbance only; the authors design additional ESO-based Luenberger state
observers in order to derive the estimations of original states. In our
control law (3), we directly estimated state from the infinite-dimensional
ESO. Our control strategy is more concise and energy-saving, and it can help
one simplify the design of references [30, 42, 43].

The closed-loop system of (1) under the controller (3) is

$\left\\{\begin{array}[]{l}u_{tt}(x,t)=u_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
u(0,t)=0,u_{x}(1,t)+mu_{tt}(1,t)=q_{x}(1,t)+mq_{tt}(1,t)\\\
-\alpha[v_{t}(1,t)-q_{t}(1,t)]-a[v_{xt}(1,t)-q_{xt}(1,t)]+F(t),\\\
v_{tt}(x,t)=v_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\ v_{x}(0,t)=\gamma
v_{t}(0,t)+\beta v(0,t)+u_{x}(0,t),\;\;t\geq 0,\\\
v_{x}(1,t)+mv_{tt}(1,t)=q_{x}(1,t)+mq_{tt}(1,t)\\\
-\alpha[v_{t}(1,t)-q_{t}(1,t)]-a[v_{xt}(1,t)-q_{xt}(1,t)],\;\;t\geq 0,\\\
q_{tt}(x,t)=q_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\ q_{x}(0,t)=\gamma
q_{t}(0,t)+\beta q(0,t),\;\;t\geq 0,\\\ q(1,t)=v(1,t)-u(1,t),\;\;t\geq
0,\end{array}\right.$ (66)

which is equivalent to

$\left\\{\begin{array}[]{l}u_{tt}(x,t)=u_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
u(0,t)=0,u_{x}(1,t)+mu_{tt}(1,t)=-\alpha u_{t}(1,t)\\\
-au_{xt}(1,t)+a\widehat{q}_{xt}(1,t)+\widehat{q}_{x}(1,t),\;\;t\geq 0,\\\
\widehat{v}_{tt}(x,t)=\widehat{v}_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
\widehat{v}_{x}(0,t)=\gamma\widehat{v}_{t}(0,t)+\beta\widehat{v}(0,t),\;\;t\geq
0,\\\ \widehat{v}_{x}(1,t)+m\widehat{v}_{tt}(1,t)=-F(t),\;\;t\geq 0,\\\
\widehat{q}_{tt}(x,t)=\widehat{q}_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
\widehat{q}_{x}(0,t)=\gamma\widehat{q}_{t}(0,t)+\beta\widehat{q}(0,t),\widehat{q}(1,t)=0.\end{array}\right.$
(67)

Our target in the rest of this section is to show that the state
$(u(\cdot,t),u_{t}(\cdot,t))$ of the closed-loop system (66) is exponentially
stable while guaranteeing the boundedness of the other variables. Since it
does not involve the total disturbance $F(t)$, we choose to firstly consider
the $(w,\widehat{z})$-part of (67) described by

$\left\\{\begin{array}[]{l}u_{tt}(x,t)=u_{xx}(x,t),\;\;x\in(0,1),\;t>0,\\\
u(0,t)=0,u_{x}(1,t)+mu_{tt}(1,t)=-au_{xt}(1,t)\\\ -\alpha
u_{t}(1,t)+a\widehat{q}_{xt}(1,t)+\widehat{q}_{x}(1,t),\;\;t\geq 0,\\\
\widehat{q}_{tt}(x,t)=\widehat{q}_{xx}(x,t),\;\;x\in(0,1),\\\
\widehat{q}_{x}(0,t)=\gamma\widehat{q}_{t}(0,t)+\beta\widehat{q}(0,t),\widehat{q}_{x}(1,t)=0.\end{array}\right.$
(68)

We consider system (68) in Hilbert state space $\mathcal{X}_{1}=\mathbf{H}$
$\times\mathbb{H}$. Define the operator
$\mathcal{A}_{1}:D(\mathcal{A}_{1})\subset\mathcal{X}_{1}\rightarrow\mathcal{X}_{1}$
by $\mathcal{A}_{1}(f,g,\eta,$
$\phi,\psi)=(g,f^{\prime\prime},{\color[rgb]{0,0,1}-f^{\prime}(1)-\alpha
g(1)+\phi^{\prime}(1)},\psi,\phi^{\prime\prime}),$ $(f,g,\phi,\psi)\in
D(\mathcal{A}_{1})=\\{(f,g,\eta,\phi,\psi)\in(H^{1}_{E}(0,1)\bigcap$
$H^{2}(0,1))\times H^{1}_{E}(0,1)\times
D(\mathbb{A})|{\color[rgb]{0,0,1}\eta=af^{\prime}(1)+mg(1)-a\phi^{\prime}(1)}\\}$.
System (68) is abstractly described by

$\frac{d}{dt}X(t)=\mathcal{A}_{1}X(t),$

where $X(t)=(u(\cdot,t),u_{t}(\cdot,t),au_{x}(1,t)$
$+mu_{t}(1,t)-a\widehat{q}_{x}(1,t),\widehat{q}(\cdot,t),\widehat{q}_{t}(\cdot,t)).$
One can easily verify that the operator $\mathcal{A}_{1}$ is not dissipative,
we shall adopt Riesz basis approach to verify the stability.

###### Theorem 14

Assume that $\gamma\neq 1$, $m\neq a$ and $m\neq a\gamma$. Then, system (68)
is governed by an exponentially stable $C_{0}$-semigroup.

Proof. By the same procedure as the proof of Theorem 4, we can obtain that the
operator $\mathcal{A}_{1}$ has bounded and compact inverse on
$\mathcal{X}_{1}$ and
$\sigma(\mathcal{A}_{1})=\sigma(\mathbf{A})\bigcup\sigma(\mathbb{A})$. Next,
we shall show that the generalized eigenfunction of $\mathcal{A}_{1}$ forms
Riesz basis for $\mathcal{X}_{1}$. Let
$\\{\lambda_{n}\\}_{n=-\infty}^{\infty}$ and
$\\{\lambda_{1n}\\}_{n=-\infty}^{\infty}$ be the eigenvalues of $\mathbb{A}$
and $\mathbf{A}$, respectively. Let
$\\{(\lambda_{n}^{-1}\phi_{n},\phi_{n})\\}_{n=-\infty}^{\infty}$ and
$\\{(\lambda_{1n}^{-1}f_{n},f_{n},af_{n}^{\prime}(1)+m\lambda_{1n}f_{n}(1))\\}_{n=-\infty}^{\infty}$
be the generalized eigenfunctions corresponding to
$\\{\lambda_{n}\\}_{n=1}^{\infty}$ and
$\\{\lambda_{1n}\\}_{n=-\infty}^{\infty}$ that form Riesz basises for
$\mathbb{H}$ and $\mathbf{H}$, respectively. As a result,
$\\{(0,0,0,\lambda_{n}^{-1}\phi_{n},\phi_{n})\\}_{n=-\infty}^{\infty}\bigcup\\{(\lambda_{1n}^{-1}f_{n},f_{n},$
$af_{n}^{\prime}(1)+m\lambda_{1n}f_{n}(1),0,0)\\}_{n=-\infty}^{\infty}$ forms
a Riesz basis for $\mathcal{X}_{1}$, which is equivalent to that
$\\{(0,0,0,\lambda_{n}^{-1}\phi^{\prime}_{n},$
$\phi_{n})\\}_{n=-\infty}^{\infty}\bigcup\\{(\lambda_{1n}^{-1}$
$f^{\prime}_{n},f_{n},af_{n}^{\prime}(1)+m\lambda_{1n}f_{n}(1),0,0)\\}_{n=-\infty}^{\infty}$
forms a Riesz basis for
$(L^{2}(0,1))^{2}\times\mathds{C}\times(L^{2}(0,1))^{2}$.

Let $\lambda\in\sigma(\mathcal{A})$ and
$(\lambda^{-1}f,f,af^{\prime}(1)+m\lambda f(1),\lambda^{-1}\phi,\phi)$ be the
corresponding eigenfunction. If $\phi=0$, then
$(\lambda^{-1}f,f,af^{\prime}(1)+m\lambda f(1))\neq 0$ and
$\lambda\in\sigma(\mathbf{A})$. Hence in this case the eigenvalues
$\\{\lambda_{1n}\\}_{n=1}^{\infty}$ corresponds the eigenfunctions
$\\{(\lambda_{1n}^{-1}f_{n},f_{n},af^{\prime}_{n}(1)+m\lambda_{1n}f_{n}(1),0,0)\\}_{n=1}^{\infty}$.

If $\phi\neq 0$, then $\lambda\in\sigma(\mathbb{A})$. The eigenvalues
$\\{\lambda_{n}\\}_{n=1}^{\infty}$ corresponds the eigenfunction
$(\lambda_{n}^{-1}\phi_{n},\phi_{n})$ $\\}_{n=1}^{\infty}$ of $\mathbb{A}$,
where $\phi_{n}=\sinh\lambda_{n}(x-1).$ Denote by
$(\lambda_{n}^{-1}f_{1n},f_{1n},af^{\prime}_{1n}(1)-a\phi^{\prime}_{n}(1)+m\lambda_{n}f_{1n}(1),$
$\lambda_{n}^{-1}\phi_{n},\phi_{n})$ the eigenfunction of $\mathcal{A}_{1}$
corresponding to $\lambda_{n}$. Then we have
$f^{\prime\prime}_{1n}(x)=\lambda_{n}f_{1n}(x),f_{1n}(0)=0,(1+a\lambda_{n})f^{\prime}_{1n}(1)+(m\lambda^{2}_{n}+\alpha\lambda_{n})f_{1n}=(a\lambda_{n}+1)\phi^{\prime}_{n}(1)$
whose solution is of the form

$\displaystyle f_{1n}(x)=F\sinh\lambda_{n}(x-1)+G\cosh\lambda_{n}(x-1).$

Combine (57) and the boundary conditions to derive
$F=[(1+a\lambda_{n})\cosh\lambda_{n}]/h_{n},G=[(1+a\lambda_{n})\sinh\lambda_{n}]/h_{n},$
where
$h_{n}=(1+a\lambda_{n})\cosh\lambda_{n}+(m\lambda_{n}+\alpha)\sinh\lambda_{n}.$
Use the fact that
$\lambda_{n}\cosh\lambda_{n}=-(\gamma\lambda_{n}+\beta)\sinh\lambda_{n}$ to
get $F=\frac{a\gamma}{a\gamma-m}+O(|n|^{-1})$ and
$G=-\frac{a}{a\gamma-m}+O(|n|^{-1}).$ Denote
$Q=\left(\begin{array}[]{cc}I_{3}&J\\\ 0&I_{2}\\\ \end{array}\right)$ with
$J=\frac{a}{a\gamma-m}\left(\begin{array}[]{ccccc}\gamma&-1&0\\\
-1&\gamma&0\\\ 0&0&0\\\ \end{array}\right)^{T}.$ Then $Q$ is a bounded linear
operator and it has bounded inverse. Moreover, we obtain the following
relations

$\displaystyle(\lambda_{1n}^{-1}f^{\prime}_{n},f_{n},af^{\prime}_{n}(1)+m\lambda_{1n}f_{n}(1),0,0)^{T}$
$\displaystyle=Q(\lambda_{1n}^{-1}f^{\prime}_{n},f_{n},af^{\prime}_{n}(1)+m\lambda_{1n}f_{n}(1),0,0)^{T},$
(69)
$\displaystyle(\lambda_{n}^{-1}f^{\prime}_{1n},f_{1n},af^{\prime}_{1n}(1)-a\phi^{\prime}_{n}(1)+m\lambda_{n}f_{1n}(1),\lambda_{n}^{-1}\phi^{\prime}_{n},$
$\displaystyle\phi_{n})=Q(0,0,0,\lambda_{n}^{-1}\phi^{\prime}_{n},\phi_{n})+O(n^{-1}).$
(70)

Then, by Bari’s theorem the sequence
$\\{(\lambda_{1n}^{-1}f^{\prime}_{n},f_{n},$
$af^{\prime}_{n}(1)+m\lambda_{1n}f_{n}(1),0,0)\\}_{n=-\infty}^{\infty}\bigcup\\{(\lambda_{n}^{-1}f^{\prime}_{1n},f_{1n},$
$af^{\prime}_{1n}(1)-a\phi^{\prime}_{n}(1)+m\lambda_{n}f_{1n}(1),\lambda_{n}^{-1}\phi^{\prime}_{n},\phi_{n})\\}_{n=-\infty}^{\infty}$
forms Riesz basis for
$\left(L^{2}(0,1)\right)^{2}\times\mathds{C}\times\left(L^{2}(0,1)\right)^{2}$,
which is equivalent to that
$\\{(\lambda_{1n}^{-1}f_{n},f_{n},af^{\prime}_{n}(1)+m\lambda_{1n}f_{n}(1),0,0)\\}_{n=-\infty}^{\infty}\bigcup$
$\\{(\lambda_{n}^{-1}f^{\prime}_{1n},f_{1n},af_{1n}(1)-a\phi^{\prime}_{n}(1)+m\lambda_{n}f_{1n}(1),\lambda_{n}^{-1}\phi_{n},\phi_{n})$
$\\}_{n=-\infty}^{\infty}$ forms Riesz basis for $\mathcal{X}_{1}$.

The semigroup generation and spectrum-determined growth condition of
$\mathcal{A}_{1}$ are directly derived by the Riesz basis property. Since
$\mathbf{A}$ and $\mathbb{A}$ generate exponentially stable $C_{0}$-semigroups
and spectrum-determined growth condition holds, $e^{\mathcal{A}_{1}t}$ is
exponentially stable.

###### Remark 15

In the proof of Theorem 14, although it forms a Riesz basis for
$\mathcal{H}_{1}$,  not all the elements of the sequence
$\\{(0,0,0,\lambda_{n}^{-1}\phi_{n},$
$\phi_{n})\\}_{n=-\infty}^{\infty}\bigcup\\{(\lambda_{1n}^{-1}f_{n},f_{n},af_{n}^{\prime}(1)+m\lambda_{1n}f_{n}(1),0,0)\\}_{n=-\infty}^{\infty}$
are generalized eigenfunctions of $\mathcal{A}$. In order to overcome this
difficulty, our key step is to find out the relations (69) and (70).

We consider the closed-loop system (66) in Hilbert state space
$\mathfrak{X}=\mathbf{H}_{1}\times\mathbf{H}_{2}\times\mathbb{H}_{1}$, where
$\mathbb{H}_{1}=H^{1}(0,1)\times L^{1}(0,1)$ with norm
$\|(f,g)\|^{2}_{\mathbb{H}_{1}}=\int_{0}^{1}[|f^{\prime}(x)|^{2}+|g(x)|^{2}]dx+\beta|f(0)|^{2}$.
The state of the closed-loop system (66) is
$(u(\cdot,t),u_{t}(\cdot,t),mu_{t}(1,t)-mq_{t}(1,t)+a(v_{x}(1,t)-q_{x}(1,t)),v(\cdot,t),v_{t}(\cdot,t),mv_{t}(1,t)-mq_{t}(1,t)+a[v_{x}(1,t)-q_{x}(1,t)],q(\cdot,t),q_{t}(\cdot,t))$.

###### Theorem 16

Suppose that $d\in L^{\infty}(0,\infty)$ (or $d\in L^{2}(0,\infty)$) and
$f:H^{1}(0,1)\times L^{2}(0,1)\rightarrow\mathds{R}$ is continuous. For any
initial value
$(u(\cdot,0),u_{t}(\cdot,0),mu_{t}(1,0)-mq_{t}(1,0)+a(v_{x}(1,0)-q_{x}(1,0)),v(\cdot,0),v_{t}(\cdot,$
$0),mv_{t}(1,0)-mq_{t}(1,0)+a[v_{x}(1,0)-q_{x}(1,0)],q(\cdot,0),q_{t}(\cdot,0))\in\mathfrak{X}$
with $q(1,0)=v(1,0)-u(1,0)$, then there exists a unique solution to system
(66) such that $(u(\cdot,t),u_{t}(\cdot,t),mu_{t}(1,t)$
$-mq_{t}(1,t)+a(v_{x}(1,t)-q_{x}(1,t)),v(\cdot,t),v_{t}(\cdot,t),mv_{t}(1,t)-mq_{t}(1,t)+a[v_{x}(1,t)-q_{x}(1,t)],q(\cdot,t),q_{t}(\cdot,t))\in
C(0,\infty;$ $\mathfrak{X})$ satisfy $q(1,t)=v(1,t)-u(1,t)$,

$\displaystyle\int_{0}^{1}[|u_{t}(x,t)|^{2}+|u_{x}(x,t)|^{2}]{\color[rgb]{0,0,1}dx}+\frac{1}{m}|mu_{t}(1,t)$
$\displaystyle+a[v_{x}(1,t)-q_{x}(1,t)]|^{2}\leq M_{2}e^{-\gamma_{2}t},$ (71)
$\displaystyle\sup_{t\geq
0}\int_{0}^{1}[|u_{t}(x,t)|^{2}+|u_{x}(x,t)|^{2}]{\color[rgb]{0,0,1}dx}+\frac{1}{m}|mu_{t}(1,t)$
$\displaystyle-mq_{t}(1,t)+a[v_{x}(1,t)-q_{x}(1,t)]|^{2}<\infty,$ (72)
$\displaystyle\sup_{t\geq
0}\bigg{[}\int_{0}^{1}[|v_{t}(x,t)|^{2}+|v_{x}(x,t)|^{2}+|q_{t}(x,t)|^{2}+|q_{x}(x,$
$\displaystyle
t)|^{2}]dx+\beta(|v(0,t)|^{2}+|q(0,t)|^{2})+|mv_{t}(1,t)-mq_{t}(1,$
$\displaystyle t)+a[v_{x}(1,t)-q_{x}(1,t)]|^{2})\bigg{]}<\infty,$ (73)

where $M_{2}$ and $\gamma_{2}$ are two positive constants.

If $f(0,0)=0$ and $d\in L^{2}(0,\infty)$, then
$\lim_{t\rightarrow+\infty}\int_{0}^{1}(|u_{t}(x,$
$t)|^{2}+|u_{x}(x,t)|^{2}+\frac{1}{m}|mu_{t}(1,t)-$
$mq_{t}(1,t)+a(v_{x}(1,t)-q_{x}(1,t))|^{2}=0,\lim_{t\rightarrow+\infty}\big{[}\int_{0}^{1}(|v_{t}(x,$
$t)|^{2}+|v_{x}(x,t)|^{2}+|q_{t}(x,t)|^{2}+|q_{x}(x,t)|^{2})dx+\beta(v(0,t)|^{2}$
$+|q(0,t)|^{2})+|mv_{t}(1,t)-mq_{t}(1,t)+a[v_{x}(1,t)-q_{x}(1,t)|^{2})\big{]}$
$=0.$

Proof. Given initial value $(u(\cdot,0),u_{t}(\cdot,0),mu_{t}(1,0)$
$-mq_{t}(1,0)+a(v_{x}(1,0)-q_{x}(1,0)),v(\cdot,0),v_{t}(\cdot,0),$
$mv_{t}(1,0)-mq_{t}(1,0)+a[v_{x}(1,0)-q_{x}(1,0)],q(\cdot,0),q_{t}(\cdot,0))\in\mathbf{H}_{1}\times\mathbf{H}_{2}\times\mathbb{H}_{1}$
with $q(1,0)=v(1,0)-u(1,0)$, we have
$(u(\cdot,0),u_{t}(\cdot,0),mu_{t}(1,0)+a[u_{t}(1,0)-\widehat{q}_{t}(1,0)],\widehat{q}(1,0),$
$\widehat{q}_{t}(1,0))\in\mathcal{X}_{1}$ and
$\big{(}\widehat{v}(\cdot,0),\widehat{v}_{t}(\cdot,$
$0),m\widehat{v}_{t}(1,0)\big{)}\in\mathbf{H}_{2}$. By Theorem 14, we obtain
$(u(\cdot,t),u_{t}(\cdot,t),mu_{t}(1,t)+a[u_{t}(1,t)-\widehat{q}_{t}(1,t)],\widehat{q}(1,t),$
$\widehat{q}_{t}(1,t))\in C(0,\infty;\mathcal{X}_{1}),q(1,t)$ $=v(1,t)-u(1,t)$
and

$\displaystyle\int_{0}^{1}(|u_{t}(x,t)|^{2}+|u_{x}(x,t)|^{2})dx+|mu_{t}(1,t)+a(v_{x}(1,t)$
$\displaystyle-
q_{x}(1,t))|^{2}=\int_{0}^{1}(|u_{t}(x,t)|^{2}+|u_{x}(x,t)|^{2})dx$
$\displaystyle+|mu_{t}(1,t)+a(u_{x}(1,t)-\widehat{q}_{x}(1,t))|^{2}$
$\displaystyle\leq
M^{2}_{\mathcal{A}_{1}}e^{-2\omega_{\mathcal{A}_{1}}t}\|(u(\cdot,0),u_{t}(\cdot,0),mu_{t}(1,0)$
$\displaystyle+a(u_{x}(1,0)-\widehat{q}_{x}(1,0)),\widehat{q}(\cdot,0),\widehat{q}_{t}(\cdot,0))\|^{2}_{\mathcal{X}_{1}}=M^{2}e^{-\gamma_{2}t},$

where $\gamma_{2}=2\omega_{\mathcal{A}_{1}}$ and
$M_{2}=M^{2}_{\mathcal{A}_{1}}\|(u(\cdot,0),u_{t}(\cdot,0),mu_{t}(1,$
$0)+a(u_{x}(1,0)-\widehat{q}_{x}(1,0)),\widehat{q}(\cdot,0),\widehat{q}_{t}(\cdot,0))\|^{2}_{\mathcal{X}_{1}}$
$=M^{2}_{\mathcal{A}_{1}}\|(u(\cdot,$
$0),u_{t}(\cdot,0),mu_{t}(1,0)+a(v_{x}(1,0)-q_{x}(1,0)),q(\cdot,0)-v(\cdot,0)+u(\cdot,0),q_{t}(\cdot,0)-v_{t}(\cdot,0)+u_{t}(\cdot,0))\|^{2}_{\mathcal{X}_{1}}.$

Therefore, we derive the continuity and exponential stability of
$(u(\cdot,t),u_{t}(\cdot,t))$ on $H^{1}(0,1)\times L^{2}(0,1)$. This, together
with the continuity of $f$ indicates that $f(w(\cdot,t),w_{t}(\cdot,t))\in
L^{\infty}(0,\infty)$. It follows from Lemma 9 that
$\big{(}\widehat{v}(\cdot,t),\widehat{v}_{t}(\cdot,t),m\widehat{v}_{t}(1,t)\big{)}\in
C(0,\infty;\mathbf{H}_{2})$ and $\sup_{t\geq
0}\|\big{(}\widehat{v}(\cdot,t),\widehat{v}_{t}(\cdot,t),m\widehat{v}_{t}(1,$
$t)\big{)}\|_{\mathbf{H}_{2}}<+\infty$. Hence,
$mu_{t}(1,t)-mq_{t}(1,t)+a(v_{x}(1,t)-q_{x}(1,t))=mu_{t}(1,t)$
$+a[u_{t}(1,t)-q_{t}(1,t)]-m\widehat{v}_{t}(1,t)\in C(0,\infty,\mathds{C}),$
$mv_{t}(1,t)$
$-mq_{t}(1,t)+a(v_{x}(1,t)-q_{x}(1,t))=mu_{t}(1,t)+a(v_{x}(1,t)-q_{x}(1,t))\in
C(0,\infty,\mathds{C}),$ $\sup_{t\geq 0}|mu_{t}(1,t)-$
$mq_{t}(1,t)+a(v_{x}(1,t)-q_{x}(1,t))|\leq\sup_{t\geq
0}|mu_{t}(1,t)+a[u_{t}(1,t)-\widehat{q}_{t}(1,t)]|$ $+\sup_{t\geq
0}|m\widehat{v}_{t}(1,t)|<\infty,$ $\sup_{t\geq 0}|m$
$v_{t}(1,t)-mq_{t}(1,t)+a(v_{x}(1,t)-q_{x}(1,t))|=\sup_{t\geq 0}|mu_{t}(1,$
$t)+a(v_{x}(1,t)-q_{x}(1,t))|<\infty,$

$\displaystyle\int_{0}^{1}[|v_{t}(x,t)|^{2}+|v_{x}(x,t)|^{2}]dx+\beta|v(0,t)|^{2}\leq
2\int_{0}^{1}[|\widehat{v}_{t}(x,$ $\displaystyle
t)|^{2}+|\widehat{v}_{x}(x,t)|^{2}]dx+\beta|\widehat{v}(0,t)|^{2}+2\int_{0}^{1}[|u_{t}(x,t)|^{2}$
$\displaystyle+|u_{x}(x,t)|^{2}]dx\leq 2\sup_{t\geq
0}\bigg{[}\|\big{(}\widehat{v}(\cdot,t),\widehat{v}_{t}(\cdot,t),m\widehat{v}_{t}(1,t)\big{)}\|^{2}_{\mathbf{H}_{2}}$
$\displaystyle+2\|(u(\cdot,t),u_{t}(\cdot,t),mu_{t}(1,t)+a(u_{x}(1,t)-\widehat{q}_{x}(1,t)),$
$\displaystyle\widehat{q}(\cdot,t),\widehat{q}_{t}(\cdot,t))\|^{2}_{\mathcal{X}_{1}}\bigg{]}<\infty,$
$\displaystyle\int_{0}^{1}[|q_{t}(x,t)|^{2}+|q_{x}(x,t)|^{2}]dx+\beta|q(0,t)|^{2}\leq
2\int_{0}^{1}[|\widehat{q}_{t}(x,$ $\displaystyle
t)|^{2}+|\widehat{q}_{x}(x,t)|^{2}]dx+2\beta|\widehat{q}(0,t)|^{2}+2\int_{0}^{1}[|\widehat{v}_{t}(x,t)|^{2}$
$\displaystyle+|\widehat{v}_{xx}(x,t)|^{2}]dx+2\beta|\widehat{v}(0,t)|^{2}\leq
2\int_{0}^{1}[|\widehat{q}_{t}(x,t)|^{2}$
$\displaystyle+|\widehat{q}_{x}(x,t)|^{2}]dx+2\beta\int_{0}^{1}|\widehat{q}_{x}(x,t)|^{2}dx+2\int_{0}^{1}[|\widehat{v}_{t}(x,t)|^{2}$
$\displaystyle+|\widehat{v}_{xx}(x,t)|^{2}]dx+2\beta|\widehat{v}(0,t)|^{2}\leq
2(1$
$\displaystyle+\beta)M^{2}_{\mathbb{A}}e^{-2\omega_{\mathbb{A}}}\|(\widehat{q}(\cdot,t),\widehat{q}_{t}(\cdot,t))\|^{2}_{\mathbb{H}}+2\int_{0}^{1}[|\widehat{v}_{t}(x,t)|^{2}$
$\displaystyle+|\widehat{v}_{xx}(x,t)|^{2}]dx+2\beta|\widehat{v}(0,t)|^{2}<\infty,$

where the fact
$|\widehat{q}(0,t)|^{2}=\big{|}\widehat{q}(1,t)-\int_{0}^{1}\widehat{q}_{x}(x,t)dx\big{|}^{2}\leq\int_{0}^{1}|\widehat{q}_{x}(x,t)|^{2}dx$
is used. Then (72) is derived, system (66) has a unique solution and
$(u(\cdot,t),u_{t}(\cdot,t),mu_{t}(1,t)-mq_{t}(1,t)+a(v_{x}(1,t)-q_{x}(1,t)),v(\cdot,t),v_{t}(\cdot,t),mv_{t}(1,t)-mq_{t}(1,t)+a[v_{x}(1,t)-q_{x}(1,t)],q(\cdot,t),q_{t}(\cdot,t))\in
C(0,\infty;$ $\mathfrak{X})$ and (73) is derived.

If $f(0,0)=0$, we derive
$\lim_{t\rightarrow\infty}f(w(\cdot,t),w_{t}(\cdot,t))=0$ by the continuity of
$f$ and the exponential stability of $(w(\cdot,t),w_{t}(\cdot,t))$. Moreover,
we use the assumption $d\in L^{2}(0,\infty)$ and [42, Lemma A.1] to get
$\lim_{t\rightarrow\infty}\|(\widehat{v}(\cdot,t),\widehat{v}_{t}(\cdot,t),m\widehat{v}_{t}(1,t))\|_{\mathbb{H}_{1}}=0.$
Then, the two limits are verified. This completes the proof.

###### Remark 17

The subsystem consisting of $(u,\widehat{q})$-part of (68) is vital to prove
Theorem 16. Indeed, by virtue of semigroup theorem, we can derive the
continuity and exponential stability of the solution of (1) without the global
Lipsichitz condition. Then we obtain that $(u(\cdot,t),u_{t}(\cdot,t))$ is
continuous and bounded on $H^{1}_{E}(0,1)\times L^{2}(0,1)$. By Lemma 9, the
existence and boundedness of
$(\widehat{v}(\cdot,t),\widehat{v}_{t}(\cdot,t),m\widehat{v}_{t}(1,t))$ are
verified by viewing $f(w(\cdot,t),w_{t}(\cdot,t))+d(t)$ as the boundary input,
provided $f$ is continuous. Furthermore, we not only derive the exponential
stability of $(u(\cdot,t),u_{t}(\cdot,t))$ but also verify the exponential
stability of $\eta(t)=mu_{t}(1,t)+a(v_{t}(1,t)-q_{t}(1,t))$, because
$mu_{t}(1,t)+a(v_{t}(1,t)-q_{t}(1,t))$ is the state of boundary dynamic of the
exponentially stable system (67). Theorem 16 also tells us that the other
states (including the state of the boundary dynamic) are bounded.

###### Remark 18

In Theorem 16, compared to the results in [40], the differences and
improvements mainly lie in that, 1) the internal uncertainty is taken into
consideration, while [40] just studies the case of $f(w,w_{t})=0$, 2) only
“low order” measurements $w(1,t)$ and $w_{x}(0,t)$ are adopted, while [40]
used the velocity $w_{t}(1,t)$ as well as the high order angular velocity
$w_{xt}(1,t)$, 3) we consider arbitrary $m\neq a,m\neq a\gamma,\gamma\neq 1$
while [40] just solved the special case $m=a\alpha$.

##  4 Numerical simulation

We present in this section some numerical simulations for the closed-loop
system (66). We use the finite difference scheme and the numerical results are
programmed in Matlab. The time step and the space step are taken as $1/200$
and $1/100$, respectively. We take the nonlinear internal uncertainty
$f(u(\cdot,t),u_{t}(\cdot,t))=\sin(u(1,t))$ and the external disturbance
$d(t)=\cos(2t)$. Let $m=5$. The parameters and the initial values are chosen
as
$\alpha=a=2,\beta=\gamma=1.5,u(x,0)=x^{3}-3x^{2},u_{t}(x,0)=0,v(x,0)=-2x^{3},v_{t}(x,0)=0,q(x,0)=q_{t}(x,0)=0.$

Fig.1, Fig.3 and Fig.5 show the displacements $u(x,t)$, $v(x,t)$ and $q(x,t)$
of the closed-loop system (66), respectively; while Fig.2, Fig 4 and Fig 6
respectively present the velocities $u_{t}(x,t)$, $v_{t}(x,t)$ and
$q_{t}(x,t)$. Fig.7 and Fig.8 display the boundary sates
$\eta(t)=mu_{t}(1,t)+a[v_{x}(1,t)-q_{x}(1,t)]$ and
$\psi(t)=mu_{t}(1,t)-mq_{t}(1,t)+a[v_{x}(1,t)-q_{x}(1,t)]$. One can see from
the simulations that $(u(\cdot,t),u_{t}(\cdot,t))$ and $\eta(t)$ decays
rapidly, while $(v(\cdot,t),v_{t}(\cdot,t))$, $(q(\cdot,t),q_{t}(\cdot,t))$
and $\psi(t)$ are bounded.

Figure 1: The state $u(x,t)$.

Figure 2: The state $u_{t}(x,t)$.

Figure 3: The state $v(x,t)$.

Figure 4: The state $v_{t}(x,t)$.

Figure 5: The state $q(x,t)$.

Figure 6: The state $q_{t}(x,t)$.

Figure 7: The state $\eta(x,t)$

Figure 8: The state $\psi(x,t)$

##  5 Concluding remarks

In this paper, the output feedback exponential stabilization for a 1-d wave
PDE with boundary and with or without disturbance is investigated. When there
is no disturbance, with only one non-collocated measurement $w_{x}(0,t)$, we
design a Luenberger state observer and an estimated state based stabilizing
controller. This improves the existence references [10, 32] where the authors
used two collocated measurements $w_{t}(1,t)$ and $w_{xt}(1,t)$. By modifying
the proof, we can simplify the proof of the exponential stability of the
closed-loop system [12, (3.1)] where $m=0$, because our coupled system (42)
contains an independent subsystem. When the boundary internal uncertainty and
external disturbance are considered, we construct an infinite-dimensional ESO
to estimate the original state and total disturbance online. Then, the
estimated state and estimated total disturbance allows us to design a
stabilizing controller while guaranteeing the boundedness of the closed-loop
system. Riesz basis approach is the main tool for the proofs of the
exponential stabilities of two coupled systems.

## References

  * [1] J. Auriol, M.D. Meglio, Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs, Automatica, 115 (2020), 108896.
  * [2] R.H. Cannon, E. Schmitz, Initial experiments on the end-point control of a flexible one-link robot, Int. J. Robot. Res., 3 (1984), 62-75.
  * [3] G. Chen, A note on the boundary stabilization of the wave equation, SIAM J. Control Optim., 19 (1981), 106-113.
  * [4] J. Deutscher, Finite-time output regulation for linear $2\times 2$ hyperbolic systems using backstepping, Automatica, 75 (2017), 54-62.
  * [5] J. Deutscher, J. Gabriel, Periodic output regulation for general linear heterodirectional hyperbolic systems, Automatica, 103 (2019), 208-216.
  * [6] J. Deutscher, N. Gehring, Output Feedback Control of Coupled Linear Parabolic ODE-PDE-ODE Systems, IEEE Trans. Automat. Control, DOI 10.1109/TAC.2020.3030763.
  * [7] J. Deutscher, J. Gabriel, A backstepping approach to output regulation for coupled linear wave-ODE systems, Automatica, 123 (2020).
  * [8] H. Feng, B.Z. Guo, A new active disturbance rejection control to output feedback stabilization for a one-dimensional anti-stable wave equation with disturbance, IEEE Trans. Automat. Control, 62(8) (2017), 3774-3787.
  * [9] S.S. Ge, S. Zhang and W. He, Vibration control of an Euler-Bernoulli beam under unknown spatiotemporally varying disturbance, Int. J. Control, 84 (2011), 947-960.
  * [10] B.Z. Guo, C.Z. Xu, On the spectrum-determined growth condition of a vibration cable with a tip mass, IEEE Trans. Autom. Control, 45(1) (2000), 89-93.
  * [11] B.Z. Guo, R. Yu, The Riesz basis property of discrete operators and application to a Euler-Bernoulli beam equation with boundary linear feedback control, IMA J. Math. Control I., 18(2) (2001), 241-251.
  * [12] B.Z. Guo, C.Z. Xu, The stabilization of a one-dimensional wave equation by boundary feedback with non-collocated observation, IEEE Trans. Automat. Control., 52 (2007), 371-377.
  * [13] B.Z. Guo, J.M. Wang, K.Y. Yang, Dynamic stabilization of an Euler-Bernoulli beam under boundary control and non-collocated observation, Syst. Control Lett., 57 (2008), 740-749.
  * [14] B.Z. Guo, F.F. Jin, Sliding mode and active disturbance rejection control to the stabilization of anti-stable one-dimensional wave equation subject to boundary input disturbance, IEEE Trans. Automat. Control, 58 (2013), 1269-1274.
  * [15] B.Z. Guo, F.F. Jin, The active disturbance rejection and sliding mode control approach to the stabilization of Euler-Bernoulli beam equation with boundary input disturbance, Automatica, 49 (2013), 2911-2918.
  * [16] B.Z. Guo, J.M. Wang, Control of wave and beam PDEs: the Riesz basis approach, Springer, 2019.
  * [17] W. Guo, B.Z. Guo, Adaptive output feedback stabilization for one dimensional wave equation with corrupted observation by harmonic disturbance, SIAM J. Control Optim., 51 (2013), 1679-1706.
  * [18] W. He, S.S. Ge, Robust adaptive boundary control of a vibrating string under unknown time-varying disturbance, IEEE Trans. Control Syst. Technol., 20 (2012), 48-58.
  * [19] J.Q. Han, From PID to active disturbance rejection control, IEEE Trans. Ind. Electron., 56 (2009), 900-906.
  * [20] F.F. Jin, B.Z. Guo, Lyapunov approach to output feedback stabilization for Euler-Bernoulli beam equation with boundary input disturbance, Automatica, 52 (2015), 95-102.
  * [21] D. Karagiannis, V. Radisavljevic-Gajic, Sliding mode boundary control for vibration suppression in a pinned-pinned Euler-Bernoulli beam with disturbances, J. Vib. Control, 24(6) (2018), 1109-1122.
  * [22] M. Krstic, A. Balogh and A. Smyshlyaev, Backstepping boundary controller and observer for the undamped shear beam, 17th International Symposium on Mathematical Theory of Networks and Systems, 2006.
  * [23] M. Krstic, A. Smyshlyaev, Boundary control of PDEs: A course on backstepping designs, Vol. 16. Siam., 2008.
  * [24] M. Krstic, B.Z. Guo, A. Balogh and A. Smyshlyaev, Control of a tip-force destabilized shear beam by non-collocated observer-based boundary feedback, SIAM J Control Optim., 47(2008), 553-574.
  * [25] M. Krstic, Compensating actuator and sensor dynamics governed by diffusion PDEs, Syst. Control Lett., 58(5) (2009), 372-377.
  * [26] M. Krstic and A. Smyshlyaev, Backstepping boundary control for first order hyperbolic PDEs and application to systems with actuator and sensor delays, Syst. Control Lett., 57(2008), 750-758.
  * [27] E.B. Lee, Y.C. You, Stabilization of a hybrid (string/point mass) system. In: Proceedings of the 15th International Conference on Systems Engineering. Dayton; 1987. p. 109-112.
  * [28] J.J. Liu J, J.M. Wang, Boundary stabilization of a cascade of ODE-wave systems subject to boundary control matched disturbance, Int. J. Robust Nonlinear Control, 27 (2017), 252-280.
  * [29] L.Y. Liu, K. Yuan, Noncollocated passivity-based control of a single-link flexible manipulator, Robotica, 21 (2003), 117-135.
  * [30] Z.D. Mei, H.C. Zhou, Output Feedback Exponential Stabilization for a One-Dimensional Wave Equation with Control Matched Nonlinear Disturbance, IEEE Trans. Automat. Control, 66(5) (2021), 2273-2280.
  * [31] Ö. Morgül,Stabilization and disturbance rejection for the wave equation. In: Proceedings of the 33rd IEEE Conference on Decision and Control (CDC). Orlando; 1994. p. 1776-1777.
  * [32] Ö. Morgül, B. Rao, and F. Conrad, On the stabilization of a cable with a tip mass, IEEE Trans. Automat. Control, 39 (1994), 2140-2145.
  * [33] Ö. Morgül, Stabilization and disturbance rejection for the beam equation, IEEE Trans. Autom. Control, 46 (2001), 1913-1918.
  * [34] B.P. Rao, Decay estimates of solutions for a hybrid system of flexible structures, Euro. J. Appl. Math., 4 (1993), 303-319.
  * [35] P. Rideau, Contrôle d’un assemblage de poutres flexibles par des capteurs-actionneurs ponctuels: Étude du spectre du système, Thèse de l’École Nationale Supérieure des Mines de Paris. Sophia-Antipolis, France, Nov. 1985.
  * [36] L.L. Su, J.M. Wang, Stabilisation of an anti-stable joint string with boundary disturbance, Int. J. Control, 93(5) (2020), 1027-1038.
  * [37] A.Smyshlyaev, M. Krstic, Backstepping observers for a class of parabolic PDEs, Syst. Control Lett., 54 (2005), 613-625.
  * [38] Z.Q. Tian, H. Feng, Stabilization of one-dimensional wave equation by non-collocated boundary feedback, Eur. J. Control, 32 (2016), 39-42.
  * [39] J. Wang, S.X. Tang, M. Krstic, Adaptive output-feedback control of torsional vibration in off-shore rotary oil drilling systems, Automatica, 111 (2020), 108640.
  * [40] Y.R. Xie, G.Q. Xu, Stabilization of a wave equation with a tip mass based on disturbance observer of time-varying gain, J. Dyn. Control. Syst., 23 (2017):667-677
  * [41] H.C. Zhou, B.Z. Guo, Unknown input observer design and output feedback stabilization for multi-dimensional wave equation with boundary control matched uncertainty, J. Differ. Equations, 263 (2017), 2213-2246.
  * [42] H.C. Zhou, H. Feng, Disturbance estimator based output feedback exponential stabilization for Euler-Bernoulli beam equation with boundary control, Automatica, 91 (2018), 79-88.
  * [43] H.C. Zhou, G. Weiss, Output feedback exponential stabilization for one-dimensional unstable wave equations with boundary control matched disturbance, SIAM J. Control Optim., 56 (2018), 4098-4129.
  * [44] H.C. Zhou and H. Feng, Stabilization for Euler-Bernoulli beam equation with boundary moment control and disturbance via a new disturbance estimator, J. Dyn. Control. Syst., 27 (2021), 247-259.