\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 216, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/216\hfil Well-posedness and exact controllability]
{Well-posedness and exact controllability of a fourth order Schr\"{o}dinger
equation with variable coefficients and Neumann boundary control and collocated
observation}

\author[R. Wen, S. Chai \hfil EJDE-2016/216\hfilneg]
{Ruili Wen, Shugen Chai}

\address{Ruili Wen \newline
School of Mathematical Sciences,
Shanxi University, Taiyuan 030006, China}
\email{wenruili@sxu.edu.cn}

\address{Shugen Chai (corresponding author) \newline
School of Mathematical Sciences,
Shanxi University, Taiyuan 030006, China}
\email{sgchai@sxu.edu.cn}

\thanks{Submitted May 25, 2016. Published August 12, 2016.}
\subjclass[2010]{93C20, 35L35, 35B37}
\keywords{Fourth order Schr\"{o}dinger equation; variable coefficients; 
\hfill\break\indent well-posedness; exact controllability; boundary control;
boundary observation}

\begin{abstract}
 We consider an open-loop system of a fourth order Schr\"{o}dinger equation
 with variable coefficients and Neumann boundary control and collocated
 observation. Using the multiplier method on Riemannian manifold we show that
 that the system is well-posed in the sense of Salamon.
 This implies that the exponential stability of the closed-loop system under
 the direct proportional output feedback control and the exact controllability
 of open-loop system are equivalent. So in order to conclude feedback
 stabilization from well-posedness, we study the exact controllability under
 a uniqueness assumption by presenting the observability inequality for the
 dual system. In addition, we show that the system is regular in the sense
 of Weiss, and that the feedthrough operator is zero.

\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction and statement of main results}

There is a wide class of infinite-dimensional linear systems introduced by Salamon
and Weiss in the 1980s \cite{Salamon1,Salamon2,Weiss1,Weiss2},
which cover many control systems described by
partial differential equations (PDEs) with the
actuators and sensors supported at isolated points, subregions, or
on boundaries of the spatial regions. This class of infinite-dimensional systems,
although the input and output operators are allowed to be unbounded, may
possess many properties that make them similar in many ways to
finite-dimensional ones, such as representation, transfer function, internal
model based tracking and disturbance rejection, stabilizing
controller parametrization, and quadratic optimal control \cite{5C}.
As of now, many multi-dimensional PDEs have been verified to be well-posed and
regular; see
\cite{1A,2AT,3BGSW,4C,5C,CGweaklycoupled,CSIAM,GSSchrodinger,
GStransplate,GSplate,GuoZhang,Staffans2012,Zwart}
and the references therein.

The fourth order Schr\"{o}dinger equation arises in many scientific fields
such as quantum mechanics, plasma physics, nonlinear optics and so on.
In quantum mechanics, the solution $\varphi(x,t)$ of system \eqref{Eq4.1}
denotes the probability amplitude function,
and the conservation of the norms validates the Born's statistical
interpretation of $\varphi(x,t)$.
Further more, $\int_{\Omega}|\varphi(x,t)|^2dx$ represents the probability
of finding the particle in domain $\Omega$ at the time $t$ and the conservation
law provides the particle which will not disappear in $\Omega$.
Here we consider the control problem of a fourth order Schr\"{o}dinger
equation with Neumann boundary conditions. On the one hand, we generalize
the well-posedness for fourth order Schr\"{o}dinger equation
with Neumann boundary control and collocated observation \cite{WenSIAM}
to the variable coefficients case, and on the other hand,
establish the exact controllability of this system.
The system that we are concerned with in this paper is described by
the PDEs
\begin{equation}\label{Se1}
\begin{gathered}
 \mathrm{i}w_{t}(x,t)+P^2w(x,t)=0, \quad x\in \Omega, t>0, \\
 w(x,t)=0, \quad x\in \partial\Omega,\; t \geqslant 0, \\
 \frac{\partial w(x,t)}{\partial\nu_{\mathcal{A}}}=0, \quad
 x\in\Gamma_1,\; t\geqslant 0,\\
 \frac{\partial w(x,t)}{\partial\nu_{\mathcal{A}}}=u(x,t), \quad
 x\in \Gamma_0, \; t \geqslant 0, \\
 y(x,t)=-\mathrm{i}\mathcal{A}(A^{-1}w(x,t)), \quad x\in \Gamma_0,\;
 t\geqslant 0,
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^{n}(n\geqslant2)$ is an open bounded region with
$C^{3}$-boundary $\partial\Omega=\Gamma
=\overline{\Gamma_0}\cap\overline{\Gamma_1}$ and assume that
$\Gamma_0\ (\text{int}(\Gamma_0)\neq\emptyset)$ and $\Gamma_1$
are relatively open in $\partial\Omega$ and $\Gamma_0\cap\Gamma_1=\emptyset$.
The operators $\mathcal{A}$ and $A$ are defined in \eqref{Eq1.3}
and \eqref{Eq1.5} later respectively, and $P$ is a second-order partial
differential operator
$$
P=\sum_{i,j=1}^{n}\frac{\partial}{\partial
x_{i}}\Big(a_{ij}(x)\frac{\partial}{\partial x_{j}}\Big),
$$
which, for some constants $a,b>0$, satisfies
\begin{equation}\label{Eq1.2}
\begin{gathered}
 a\sum_{i=1}^{n}|\xi_{i}|^2\leqslant
\sum_{i,j=1}^{n}a_{ij}(x)\xi_{i}\overline{\xi}_{j}\leqslant
b\sum_{i=1}^{n}|\xi_{i}|^2,\quad \forall x\in\overline{\Omega},\;
\xi=(\xi_1,\xi_2,\dots, \xi_n)\in \mathbb{C}^{n}, \\
a_{ij}(x)=a_{ji}(x)\in
C^{\infty}(\mathbb{R}^n),\quad \forall i,j=1,2,\dots,n.
\end{gathered}
\end{equation}
We define the operator $\mathcal{A}$ as
\begin{equation}\label{Eq1.3}
\mathcal{A}f=Pf, \quad D(\mathcal{A})=H^2(\Omega)\cap H_0^{1}(\Omega),
\end{equation}
and
\begin{equation}\label{Eq1.4}
\begin{gathered}
 \nu_{\mathcal{A}}
=\Big(\sum_{k=1}^{n}\nu_{k}a_{k1}(x),\sum_{k=1}^{n}\nu_{k}a_{k2}(x),
 \dots,\sum_{k=1}^{n}\nu_{k}a_{kn}(x)\Big),\\
\frac{\partial}{\partial\nu_{\mathcal{A}}}
=\sum_{i,j=1}^{n}a_{ij}(x)\nu_{j}\frac{\partial}{\partial x_{i}},
\end{gathered}
\end{equation}
where $\nu=(\nu_1,\nu_2,\dots,\nu_n)$ is the unit normal
vector of $\partial\Omega$ pointing outward of $\Omega$,
$u$ and $y$ are the boundary control and the boundary observation of
system \eqref{Se1}.

Now, let $A$ be the positive self-adjoint operator in $L^2(\Omega)$ defined by
\begin{equation}\label{Eq1.5}
Af=P^2f, \quad D(A)=H^{4}(\Omega)\cap H_0^2(\Omega).
\end{equation}
Just as in \cite{GZwaveV}, one can show that
\begin{equation}\label{Eq1.6}
A^{1/2}=-\mathcal{A}.
\end{equation}

Let $H=H^{-2}(\Omega)$ and $U=Y=L^2(\Gamma_0)$, where $H^{-2}(\Omega)$
is the dual of $H_0^2(\Omega)$
with respect to the pivot space $L^2(\Omega)$.
The following Theorem \ref{Th1.1} shows that system \eqref{Se1} is
well-posed with the state space $H$ and the
 input and output space $U=Y=L^2(\Gamma_0)$ \cite{GuoZhang}.

\begin{theorem}\label{Th1.1}
System \eqref{Se1} is well-posed. More precisely, for any $T>0$,
initial value $w_0\in H$, and the control input $u\in
L^2(0,T;U)$, there exists a unique solution $w\in C(0,T;H)$ to
 \eqref{Se1} such that
\begin{equation}\label{Se2}
\|w(\cdot,T)\|_{H}^2+\| y\|_{L^2(0,T;U)}^2
 \leqslant C_{T}\big[\|w_0\|_{H}^2+\|u\|_{L^2(0,T;U)}^2\big],
\end{equation}
 where $C_T>0$ is used to represent the constant that depends only on
$\Omega, \Gamma_0,$ and $T$,
 although it may have different values in different contexts.
\end{theorem}

It is proved in \cite[Theorem 5.8]{GCbook} (see also \cite[Theorem 5.2]{WenSIAM})
 that if the abstract system \eqref{Eq2.19} introduced later is well-posed,
it must be regular in the sense of Weiss with the zero feedthrough operator.
The following result is hence a consequence of Theorem \ref{Th1.1}.

\begin{corollary}\label{Co1.1}
 System \eqref{Se1} is regular and the feedthrough operator is zero.
\end{corollary}

Theorem \ref{Th1.1} implies that the open-loop system \eqref{Se1} is well-posed
in the sense of Salamom with the state space $H$ and the same input and
output space $U=Y$.
From this result and \cite[Theorems 6.7 and 6.8]{GCbook}
(see also \cite[Theorems 5.3 and 5.4]{WenSIAM})
on the first order abstract system formulation (see also \cite{GuoLuo}
for the second order abstract system),
 we know that system \eqref{Se1} is exactly controllable
on some interval $[0,T]$ $(T>0)$ if and only if its corresponding
closed loop systems under the output proportional feedback $u=-ky,
k>0$ is exponentially stable. So, based on this argument, to get the
feedback stabilization of system \eqref{Se1} from the
well-posedness, we need to discuss the exact controllability of
the open loop system \eqref{Se1}. We show that under the assumptions
(H1) and (H2) stated below, system \eqref{Se1}
is exactly controllable on some interval $[0,T]$, $T>0$.

It should be emphasized that due to the variable coefficients,
the classical multipliers method in Euclidean space seems invalid \cite{WenMMAS}
to prove Theorem \ref{Th1.1} and \ref{Th1.2}, some computations on
the Riemannian manifold are needed.

By the ellipticity condition \eqref{Eq1.2}, we denote the coefficients
 matrix and its inverse by $A(x)$ and $G(x)$, respectively,
and the determinant of $G(x)$ by $\rho(x)$,
\begin{equation}\label{Eq1.8}
\begin{gathered}
A(x)=[a_{ij}(x)]_{n\times n}, \quad
 G(x)=[g_{ij}(x)]_{n\times n}=[a_{ij}(x)]_{n\times n}^{-1}=A(x)^{-1}, \\
 \rho(x)=\det [g_{ij}(x)]_{n\times n},\quad \forall x\in \mathbb{R}^{n}.
\end{gathered}
\end{equation}
Let $\mathbb{R}^n$ be the usual Euclidean space. For each
$x=(x_1,x_2,\dots,x_n)\in \mathbb{R}^n$, define the inner
product and norm over the tangent space $\mathbb{R}_x^n$ of $\mathbb{R}^{n}$
by
\begin{equation}\label{Eq1.9}
\begin{gathered}
 g(X,Y):=\langle
X,Y\rangle_g=\sum_{i,j=1}^{n}g_{ij}\alpha_{i}\beta_{j},\\
 |X|_g:=\langle X,X\rangle_g^{1/2}, \quad
 \forall X=\sum_{i=1}^{n}\alpha_{i}\frac{\partial}{\partial x_{i}},\;
Y=\sum_{i=1}^{n}\beta_{i}\frac{\partial}{\partial x_{i}}\in
\mathbb{R}_x^n.
\end{gathered}
\end{equation}
Then $(\mathbb{R}^n,g)$ becomes a Riemannian manifold with
Riemannian metric $g$ \cite{Yao1,Yao2}. Denote by $D$ the
Levi-Civita connection with respect to $g$ and let $N$ be a smooth
vector field on $(\mathbb{R}^n,g)$. Then for each $x\in \mathbb{R}^n$,
the covariant differential $DN$ of $N$ determines a
bilinear form on $\mathbb{R}_x^n\times \mathbb{R}_x^n$:
\begin{equation}\label{Eq1.10}
DN(X,Y)=\langle D_{Y}N,X\rangle_g,\quad \forall X,Y\in
\mathbb{R}_x^n,
\end{equation}
where $D_{Y}N$ stands for the covariant derivative of the vector
field $N$ with respect to $Y$.

In this article we use the following assumptions:
\begin{itemize}
\item[(H1)] There exists a vector field $N$ on ($\mathbb{R}^{n},g$) such that
\begin{equation}\label{Eq1.11}
DN(X,X)=b(x)|X|_g^2, \; \forall \; X\in\mathbb{R}_{x}^{n}, \; x\in\Omega,
\end{equation}
where $b(x)$ is a function defined on $\Omega$ so that
\begin{equation}\label{Eq1.12}
b_0=\inf_{x\in\Omega}b(x)>0.
\end{equation}

\item[(H2)] (Uniqueness assumption]) The problem
 \begin{equation}\label{Eq1.13}
\begin{gathered}
 \mathcal{A}^2v=\zeta v, \quad x\in \Omega, \\
 v=\frac{\partial v}{\partial\nu_{\mathcal{A}}}=0, \quad x\in \Gamma, \\
 \mathcal{A}v=0, \quad x\in \Gamma_0,
\end{gathered}
\end{equation}
possesses a unique zero solution, where $\zeta$ is an arbitrary complex
 number and $\Gamma_0$
is relatively open in $\Gamma$ and satisfies
\begin{equation}\label{Eq1.14}
\Gamma_0=\{x\in \Gamma|N(x)\cdot\nu>0\}.
\end{equation}
\end{itemize}

For the variable case, several corollaries were presented to show how to
verify Assumption (H1) by means of the Riemannian geometry method in \cite{Yao1}.
In fact, when $a_{ij}(x)=\delta_{ij}$, then for some given $x_0$,
 the radial field $N=x-x_0$ satisfies Assumption (H1) with $b(x)\equiv1$.
As for Assumption (H2), it is a valid fact
(\cite[Theorem 4.2]{Liu} and \cite[Theorem 1.3]{GStransplate}),
but it is not verified, as was indicated in \cite{Yao2},
the problem is not a Cauchy problem, and hence many uniqueness theorems
cannot be applied.
We propose it as an unsolved problem here.

\begin{theorem}\label{Th1.2}
Under Assumptions {\rm (H1), (H2)}, system \eqref{Se1} is exactly controllable
on some $[0,T],T>0$. That is, given initial data $w_0\in H$ and any time $T>0$,
there exists a boundary control $u\in L^2(0,T;L^2(\Gamma_0))$
such that the unique solution $w\in C(0,T;H)$ of \eqref{Se1} satisfies $w(T)=0$.
\end{theorem}

The following result is a direct consequence of Theorems \ref{Th1.1} and
\ref{Th1.2}.

\begin{corollary}\label{Co1.2}
Suppose \eqref{Eq1.14} holds. Then system \eqref{Se1} is exponentially
stable under the proportional output feedback $u=-ky$ for any $k>0$.
\end{corollary}

This article is organized as follows. In Section 2, we
formulate system \eqref{Se1} into a collocated
abstract first-order system. Some basic knowledge on Riemannian
geometry is stated. Sections 3 and 4 are devoted to the proofs of
Theorems 1.1 and 1.2, respectively.


\section{Collocated formulation and preliminary results}

In this section, we introduce some notations and facts in Riemannian
geometry that we need in the following sections.
For any $\varphi\in
C^2(\mathbb{R}^n)$ and
$ N=\sum_{i=1}^{n}h_{i}(x)\frac{\partial}{\partial x_{i}}$, denote
\begin{equation}\label{Eq2.1}
\begin{gathered}
\operatorname{div}_0(N)=\sum_{i=1}^{n}\frac{\partial
h_{i}(x)}{\partial x_{i}},\quad
D\varphi=\nabla_g\varphi=\sum_{i,j=1}^{n}\frac{\partial\varphi}{\partial
x_{j}}a_{ij}(x)\frac{\partial}{\partial x_{i}},\\
\operatorname{div}_g(N)=\sum_{i=1}^{n}\frac{1}{\sqrt{\rho(x)}}
\frac{\partial }{\partial x_{i}}(\sqrt{\rho(x)}h_{i}(x)),
\\
\Delta_g\varphi=\sum_{i,j=1}^{n}\frac{1}{\sqrt{\rho(x)}}\frac{\partial}{\partial
x_{i}}\Big(\sqrt{\rho(x)}a_{ij}(x)\frac{\partial\varphi}{\partial
x_{j}}\Big)=P\varphi-(Dp)\varphi,\\
 p=\frac{1}{2}\ln (\det [a_{ij}(x)]),
\end{gathered}
\end{equation}
where $\operatorname{div}_0$ is the divergence operator in Euclidean space
$\mathbb{R}^n$, and $\nabla _g,{\rm div_g}$ and $\Delta_g$ are
the gradient operator, the divergence operator and the
Beltrami-Laplace operator in $(\mathbb{R}^n,g)$ respectively.

 Let $\mu=\frac{\nu_{\mathcal{A}}}{|\nu_{\mathcal{A}}|_g}$ be
the unit outward-pointing normal to $\partial\Omega$ in terms of the
Riemannian metric $g$. The following Lemma \cite[p. 128,138]{Taylor} provides
some useful identities.

\begin{lemma} \label{lem2.1}
 Let $\varphi,\psi\in C^2(\overline{\Omega})$
and let $N$ be a vector field on $(\mathbb{R}^n,g)$. Then we have:
(1) Divergence formulae and theorems
\begin{gather*}
\operatorname{div}_0(\varphi N)=\varphi\operatorname{div}_0(N)+N(\varphi),\quad
\operatorname{div}_g(\varphi N)=\varphi\operatorname{div}_g(N)+N(\varphi), \\
\int_{\Omega}\operatorname{div}_0(N)dx=\int_{\Gamma}N\cdot\nu d\Gamma,\quad
 \int_{\Omega}\operatorname{div}_g(N)dx
=\int_{\Gamma}\langle N,\mu\rangle_g d\Gamma.
\end{gather*}
(2) Green$'$s formulae
\begin{gather*}
\int_{\Omega}\psi P\varphi dx
=\int_{\Gamma}\psi\frac{\partial\varphi}{\partial\nu_{\mathcal{A}}}
d\Gamma-\int_{\Omega}\langle\nabla_g\varphi,\nabla_g\psi\rangle_g dx,\\
\int_{\Omega}\psi\Delta_g\varphi
dx=\int_{\Gamma}\psi\frac{\partial\varphi}{\partial\mu}d\Gamma
-\int_{\Omega}\langle\nabla_g\varphi,\nabla_g\psi\rangle_gdx.
\end{gather*}
\end{lemma}

\begin{lemma} \label{lem2.2}
We denote by $T^2(\mathbb{R}_x^n)$ the set of all covariant tensors
of order 2 on $\mathbb{R}_x^n$. Then $T^2(\mathbb{R}_x^n)$ is an
inner product space of dimension $n^2$ with the inner product
\begin{equation}\label{Eq2.2}
\langle F,G\rangle_{T^2(\mathbb{R}_x^n)}
=\sum_{i,j=1}^{n}F(e_{i},e_{j})G(e_{i},e_{j}),
\quad \forall F,G\in T^2(\mathbb{R}_x^n),
\end{equation}
where $\{e_1,e_2,\dots,e_n\}$ is an arbitrarily chosen orthonormal
basis for $(\mathbb{R}_x^n,g)$.

Let $\mathcal{X}(\mathbb{R}^n)$ be the set of all vector fields on
$\mathbb{R}^n$. We denote by
$\Delta:\mathcal{X}(\mathbb{R}^n)\to\mathcal{X}(\mathbb{R}^n)$
the Hodge-Laplace operator. Then \cite[(2.2.7),(2.2.14)]{Yao2}:
\begin{equation}\label{Eq2.3}
\begin{gathered}
\Delta_g(N(\varphi))=(\Delta N)(\varphi)+2\langle DN,
D^2\varphi\rangle_{T^2(\mathbb{R}_x^n)}
 +N(\Delta_g\varphi)+\operatorname{Ric}(N,D\varphi),\\
 N(\Delta_g\varphi)=N(\mathcal{A}\varphi)-D^2p(N,D\varphi)-D^2\varphi(N,Dp),\quad
 \forall \varphi\in C^2(\mathbb{R}^n),
\end{gathered}
\end{equation}
where $\operatorname{Ric}(\cdot,\cdot)$ is the Ricci curvature tensor of the
Riemannian metric $g$, $D^2\varphi$ and $D^2p$ are the Hessian
of $\varphi$ and $p$, respectively, in
terms of the Riemannian metric $g$.

For a fixed $x\in \mathbb{R}^{n}$. Let $E_1,E_2,\dots,E_n$ be a frame
field normal at $x$ on $(\mathbb{R}^{n},g)$,
which means that $\langle E_{i},E_{j}\rangle=\delta_{ij}$ in some neighborhood
of $x$ and $(D_{E_{i}}E_{j})(x)=0$
for $1\leqslant i,j\leqslant n$. Set $ N=\sum_{i=1}^{n}\gamma_{i}E_{i}$, then
$ N(\varphi)=\sum_{i=1}^{n}\gamma_{i}E_{i}(\varphi)$, where $E_{i}(\varphi)$
is the covariant derivative of $\varphi$
with respect to $E_{i}$ under the Riemannian metric $g$. Then
\begin{equation}\label{Eq2.4}
\begin{aligned}
\langle Dp,D(N(\varphi))\rangle_g
&=E_{i}(p)E_{i}(N(\varphi))\\
&=E_{i}(p)[E_{i}(\gamma_{j})E_{j}(\varphi)+\gamma_{j}E_{i}E_{j}(\varphi)]\\
&=DN(D\varphi, Dp)+D^2\varphi(N,Dp).
\end{aligned}
\end{equation}
From \eqref{Eq2.3} and \eqref{Eq2.4}, we obtain
\begin{equation}\label{Eq2.5}
\begin{aligned}
\mathcal{A}(N(\varphi))&=(\Delta_g\varphi+Dp)(N(\varphi)) \\
&=\Delta_g(N(\varphi))+\langle Dp,D(N(\varphi))\rangle_g\\
 &=(\Delta N)(\varphi)+2\langle DN,D^2\varphi\rangle_{T^2(\mathbb{R}_{x}^{n})}
 +N(\mathcal{A}\varphi) -D^2p(N,D\varphi) \\
&\quad +\operatorname{Ric}(N,D\varphi)+DN(D\varphi,Dp).
\end{aligned}
\end{equation}
\end{lemma}

\begin{lemma}[{see \cite[Lemma 4.1]{GZplateV}}] \label{lem2.3}
Let $\psi$ be a smooth function on $\overline{\Omega}$ and satisfy
$\psi|_{\Gamma}=0$. Then there exists a continuous function $q(x)$ on
$\Gamma$ which is independent of $\psi$ such that
\begin{equation}\label{Eq2.6}
\Delta_g\psi(x)=\frac{\partial^2\psi}{\partial\mu^2}
+q(x)\frac{\partial\psi(x)}{\partial\mu},\quad \forall x\in\Gamma.
\end{equation}
Moreover, if $\psi$ satisfies
$\frac{\partial\psi}{\partial\nu_{\mathcal{A}}}|_{\Gamma}=0$, then
\begin{equation}\label{Eq2.7}
N(\psi)|_{\Gamma}=0\quad\text{on $\overline{\Omega}$ for any vector field $N$}.
\end{equation}
So,
\begin{equation}\label{Eq2.8}
\mathcal{A}(\psi)=\Delta_g\psi+(Df)(\psi)
=\Delta_g\psi=\frac{\partial^2\psi}{\partial\mu^2}
 =\frac{1}{|\nu_{\mathcal{A}}|_g^2}
\frac{\partial^2\psi}{\partial\nu_{\mathcal{A}}^2}\quad\text{on }\Gamma,
\end{equation}
and
\begin{equation}\label{Eq2.9}
\begin{aligned}
 \frac{\partial N(\psi)}{\partial\nu_{\mathcal{A}}}
& =N\Big(\frac{\partial\psi}{\partial\nu_{\mathcal{A}}}\Big)
=\langle N,\frac{\nu_{\mathcal{A}}}{|\nu_{\mathcal{A}}|_g}\rangle_g
\frac{\nu_{\mathcal{A}}}{|\nu_{\mathcal{A}}|_g}
\Big(\frac{\partial\varphi}{\partial\nu_{\mathcal{A}}}\Big)\\
& =N\cdot\nu\frac{1}{|\nu_{\mathcal{A}}|_g^2}
\frac{\partial^2\psi}{\partial\nu_{\mathcal{A}}^2}
=\mathcal{A}\psi N\cdot\nu \quad \text{on } \Sigma.
\end{aligned}
\end{equation}
\end{lemma}

\begin{lemma} \label{lem2.4}
 Let $\varphi$ be a complex function defined on
$\overline{\Omega}$ with suitable regularity. Then there exist some
constants $C$, possibly depending on $g,N$ and $\Omega$, such that:
(1)
\begin{gather*}
 \sup_{x\in\overline{\Omega}}|N|_g\leqslant C,\quad
 \sup_{x\in\overline{\Omega}}|DN|_g\leqslant C,\quad
\sup_{x\in\overline{\Omega}}|\operatorname{div}_g(N)|\leqslant C, \\
\sup_{x\in\overline{\Omega}}|Dp|_g\leqslant C,\quad
\sup_{x\in\overline{\Omega}}|\nabla_g(\operatorname{div}_gN)|_g\leqslant C,
\end{gather*}
(2)
\begin{gather*}
|N(\varphi)|\leqslant C|\nabla_g\varphi|_g,\quad
 |Dp(\varphi)|\leqslant C|\nabla_g\varphi|_g,\quad
 |DN(\nabla_g\varphi,\nabla_g\overline{\varphi})|\leqslant
 C|\nabla_g\varphi|_g^2, \\
 |\langle\nabla_g\varphi,\nabla_g(\operatorname{div}_gN)\rangle_g|\leqslant
 C|\nabla_g\varphi|_g,\quad
 |(\Delta N)\varphi|_g\leqslant C|\Delta N|_g|\nabla_g\varphi|_g\leqslant
 C|\nabla_g\varphi|_g, \\
|\langle DN,D^2\varphi\rangle_{T^2(\mathbb{R}_x^n)}|
\leqslant C|DN|_g|D^2\varphi|_g\leqslant
 C|D^2\varphi|_g, \\
|D^2p(N,D\varphi)|\leqslant|D^2p|_g|N|_g|D\varphi|_g
\leqslant C|D\varphi|_g,\\
|D^2\varphi(N,Dp)|\leqslant|D^2\varphi|_g|N|_g|Dp|_g
\leqslant C|D^2\varphi|_g, \\
|\operatorname{Ric}(N,D\varphi)|\leqslant|
\operatorname{Ric}|_g|N|_g|D\varphi|_g\leqslant C|D\varphi|_g,
\end{gather*}
where $p(x)=\frac{1}{2}\ln (\det [a_{ij}(x)])$.

(3)
\[
 \int_{\Omega}|\varphi|^2dx
\leqslant C\|\varphi\|_{H^2(\Omega)}^2,\quad
\int_{\Omega}|D\varphi|_g^2dx
\leqslant C\|\varphi\|_{H^2(\Omega)}^2,\quad
\int_{\Omega}|D^2\varphi|_g^2dx
\leqslant C\|\varphi\|_{H^2(\Omega)}^2.
\]
\end{lemma}

 Now we cast the system \eqref{Se1} into an abstract
first-order system in the state space
$H=H^{-2}(\Omega)$ and control and output spaces $U=Y$.

Let $A_1$ be the positive self -adjoint operator in $H$ induced
by the bilinear form $a(\cdot,\cdot)$
defined by
$$
\langle A_1f,g\rangle_{H^{-2}(\Omega)\times H_0^2(\Omega)}
=a(f,g)
=\int_{\Omega}Af\overline{Ag}dx, \quad
 \forall f,g\in H_0^2(\Omega).
$$
By  the Lax-Milgram theorem, $A_1$ is a canonical isomorphism
from $D(A_1)=H_0^2(\Omega)$
onto $H$. It is easy to show that $A_1f=Af$ whenever
$f\in H^{4}(\Omega)\cap H_0^2(\Omega)$
and that $A_1^{-1}g=A^{-1}g$ for any $g\in L^2(\Omega)$.
Hence $A_1$ is an extension of $A$ to the space $H_0^2(\Omega)$.

It is well known that $D(A_1^{1/2})=L^2(\Omega)$ and $A_1^{1/2}$ is a
canonical isomorphism from $L^2(\Omega)$ onto $H$ (see \cite{GSplate}).
 Define the map $\gamma\in\mathcal{L}(L^2(\Gamma_0),H^{3/2}(\Omega))$
\cite[p. 189]{LM} so that $\gamma u=\phi$ if and only if
\begin{equation}\label{Eq2.10}
\begin{gathered}
P^2\phi(x)=0,\quad x\in\Omega,\\
 \phi(x)\big|_{\Gamma}=0, \quad
 \frac{\partial\phi(x)}{\partial\nu_{\mathcal{A}}}\big|_{\Gamma_1}=0,\quad
 \frac{\partial\phi(x)}{\partial\nu_{\mathcal{A}}}\big|_{\Gamma_0}=u(x).
\end{gathered}
\end{equation}
In terms of the Dirichlet map, we can write \eqref{Se1} as
\begin{equation}\label{Eq2.11}
\mathrm{i}\dot{w}+A_1(w-\gamma u)=0.
\end{equation}
It is clear that $D(A_1)$ is dense in $H$, so is $D(A_1^{1/2})$.
We identify $H$ with its dual $H'$. Then the following
Gelfand-triple of continuous and dense inclusions hold:
\begin{equation}\label{Eq2.12}
D(A_1^{1/2})\hookrightarrow
H=H'\hookrightarrow D(A_1^{1/2})'.
\end{equation}
Define an extension $\widetilde{A}\in
{\mathcal{L}}(D(A_1^{1/2}),D(A_1^{1/2})')$
of $A_1$ by
\begin{equation}\label{Eq2.13}
 \langle \widetilde{A}f,g\rangle_{D(A_1^{1/2})', D(A_1^{1/2})}
= \langle A_1^{1/2}f,A_1^{1/2}g\rangle_{H},\quad
 \forall f,g\in D(A_1^{1/2}).
\end{equation}
Hence \eqref{Eq2.11} can be written in $D(A_1^{1/2})'$ as
\begin{equation}\label{Eq2.14}
\dot{w}=\mathrm{i}\widetilde{A}w +Bu,
\end{equation}
where $B\in \mathcal{L}(U,D(A_1^{1/2})')$ is given by
\begin{equation}\label{Eq2.15}
 Bu=-\mathrm{i}\widetilde{A}\gamma u, \quad \forall u\in U.
\end{equation}
Define $B^{\ast}\in {\mathcal{L}}(D(A_1^{1/2}),U)$ by
\begin{equation}\label{Eq2.16}
\langle B^{\ast}f,u\rangle_{U}
=\langle f,Bu\rangle_{D(A_1^{1/2}),
D(A_1^{1/2})'}, \quad \forall
f\in D(A_1^{1/2})=H_0^{1}(\Omega),\;u\in U.
\end{equation}
Then for any $f\in D(A_1^{1/2})$ and $u\in C^{\infty}_0(\Gamma)$, we have
\begin{equation}\label{Eq2.17}
\begin{aligned}
\langle Bu,f\rangle_{D(A_1^{1/2})',D(A_1^{1/2})}
&=\langle\widetilde{A}^{-1}Bu,\widetilde{A}f\rangle_{D(A_1^{1/2}),
 D(A_1^{1/2})'}\\
& = \langle
 A_1^{1/2}\widetilde{A}^{-1}Bu,A_1^{1/2}f\rangle_{H} \\
&=\langle A_1^{-1}A_1^{1/2}\widetilde{A}^{-1}Bu,A_1^{-1}A_1^{1/2}f\rangle
 _{H_0^2(\Omega)}\\
& = \langle A_1^{-1/2}(-\mathrm{i}\gamma u),A_1^{-1/2}f\rangle_{H_0^2(\Omega)} \\
&=\langle -\mathrm{i}\gamma u,f\rangle_{L^2(\Omega)}\\
&= \langle-\mathrm{i}\gamma u,AA^{-1}f\rangle_{L^2(\Omega)}
 =\langle u,-\mathrm{i}\mathcal{A}(A^{-1}f)\rangle_{L^2(\Gamma_0)}.
 \end{aligned}
\end{equation}
Since $C^{\infty}_0(\Gamma_0)$ is dense in $L^2(\Gamma_0)$, we
obtain
\begin{equation}\label{Eq2.18}
B^{\ast}f=-\mathrm{i}\mathcal{A}(A^{-1}f),\quad \forall
 f\in D(A_1^{1/2})=L^2(\Omega).
\end{equation}
Thus, we have formulated the open loop system \eqref{Se1} into an abstract
 first-order form in $H$:
\begin{equation}\label{Eq2.19}
\begin{gathered}
\dot{w}=\mathrm{i}\widetilde{A}w+Bu,\\
y=B^{\ast}w.
\end{gathered}
\end{equation}
where $\widetilde{A}$, $B$ and $B^{\ast}$ are defined by
\eqref{Eq2.13}, \eqref{Eq2.15} and \eqref{Eq2.18}, respectively.


\section{Proof of Theorem \ref{Th1.1}}

We need the following Lemma which comes from \cite[Theorem A.1]{ChaiJDE}.

\begin{lemma}\label{Le3.1}
If there exist constants $T>0$, $C_{T}>0$ such that the input and
output of system \eqref{Se1} satisfy
\begin{equation}\label{Eq3.1}
\int ^{T}_0\|y(t)\|^2_{U}dt \leqslant C_{T}\int
^{T}_0\|u(t)\|^2_{U}dt, \quad \forall u\in L^2(0,T;L^2(\Gamma_0)),
\end{equation}
with $w(\cdot,0)\equiv 0$, then system \eqref{Se1} is well-posed.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{Th1.1}]
Introduce the transformation $z=A_1^{-1}w\in H_0^2(\Omega)$.
Then $z$ satisfies
\begin{equation}\label{Eq3.2}
\begin{gathered}
 z_{t}(x,t)-\mathrm{i}\mathcal{A}^2z(x,t)
=-\mathrm{i}(\gamma u(\cdot,t))(x,t), \quad (x,t)\in \Omega\times(0,T]=: Q, \\
 z(x,0)=z_0(x), \quad x\in \Omega, \\
 z(x,t)=\frac{\partial z(x,t)}{\partial\nu_{\mathcal{A}}}=0, \quad
(x,t)\in \partial\Omega\times [0,T]=:\Sigma,
 \end{gathered}
\end{equation}
and from \eqref{Eq2.18}, the output of system \eqref{Se1} is changed
into the form
\begin{equation}\label{Eq3.3}
y(x,t)=B^{\ast}w(x,t)
=B^{\ast}A_1A_1^{-1}w(x,t)
=B^{\ast}A_1z(x,t)
=-\mathrm{i}\mathcal{A}z(x,t)\quad x\in\Gamma_0,\; t>0.
\end{equation}
Therefore, by Lemma \ref{Le3.1}, Theorem \ref{Th1.1} amounts to saying
for some (and hence for all) $T>0$,
that the solution to system \eqref{Eq3.2} with zero initial data
satisfies
\begin{equation}\label{Eq3.4}
 \int_0^{T}\int_{\Gamma_0}|\mathcal{A}z(x,t)|^2d\Gamma dt
\leqslant C_{T}\int_0^{T}\int_{\Gamma_0}|u(x,t)|^2d\Gamma dt.
\end{equation}
We proceed with the proof in three steps.
\smallskip

\noindent\textbf{Step 1.} (Energy identity)
Since $\partial\Omega$ is of class $C^{3}$, it follows from
\cite[Lemma 4.1]{GZwaveV}
that there exists a $C^2$ vector field $N$ on $\overline{\Omega}$ such that
\begin{equation}\label{Eq3.5}
 N(x)=\mu(x),\; x\in\Gamma;\quad |N(x)|\leqslant 1,\; x\in\Omega.
\end{equation}
Now, multiply both sides of the first equation in \eqref{Eq3.2}
 by $N(\overline{z})$ and integrate over
$Q$ to obtain
\begin{equation}\label{Eq3.6}
 \int_{Q}z_{t}N(\overline{z})dQ- \mathrm{i}\int_{Q}\mathcal{A}^2zN(\overline{z})dQ
 =-\mathrm{i}\int_{Q}\gamma uN(\overline{z})dQ.
\end{equation}
Compute the first term on the left-hand side of \eqref{Eq3.6} to yield
\begin{equation}\label{Eq3.7}
\begin{aligned}
&\int_{Q}z_{t}N(\overline{z})dQ\\
& =\int_{\Omega}zN(\overline{z})dx\Big|_0^{T}-\int_{Q}zN(\overline{z}_{t})dQ \\
& =\Big(\int_{\Omega}\operatorname{div}_g(|z|^2N)dx-\int_{\Omega}\overline{z}N(z)dx
 -\int_{\Omega}|z|^2\operatorname{div}_g(N)dx\Big)\Big|_0^{T}\\
& \quad -\Big(\int_{Q}\operatorname{div}_g(z\overline{z}_{t}N)dQ
 -\int_{Q}\overline{z}_{t}N(z)dQ-\int_{Q}z\overline{z}_{t}
 \operatorname{div}_g(N)dQ\Big),
\end{aligned}
\end{equation}
and hence
\begin{equation}\label{Eq3.8}
\begin{aligned}
 2\mathrm{i}\operatorname{Im}
\int_{Q}z_{t}N(\overline{z})dQ
 &= \int_{Q}z\overline{z}_{t}\operatorname{div}_g(N)dQ
 -\Big(\int_{\Omega}\overline{z}N(z)dx
 +\int_{\Omega}|z|^2\operatorname{div}_g(N)dx\Big)\Big|_0^{T}\\
 &= \int_{Q}\mathrm{i}\overline{\gamma u}z\operatorname{div}_g(N)dQ
 -\int_{Q}\mathrm{i}\mathcal{A}^2\overline{z}z\operatorname{div}_g(N)dQ\\
 &\quad -\Big(\int_{\Omega}\overline{z}N(z)dx+\int_{\Omega}|z|^2
\operatorname{div}_g(N)dx\Big)\Big|_0^{T}.
\end{aligned}
\end{equation}
Straightforward computations yield
\begin{equation}\label{Eq3.9}
\begin{aligned}
&\int_{Q}\mathcal{A}^2\overline{z}z\operatorname{div}_g(N)dQ\\
&=\int_{Q}|\mathcal{A}z|^2\operatorname{div}_g(N)dQ
 +\int_{Q}z\mathcal{A}(\operatorname{div}_g(N))\mathcal{A}\overline{z}dQ
 +2\int_{Q}\mathcal{A}\overline{z}\langle\nabla_gz,\nabla_g
 \operatorname{div}_g(N)\rangle_gdQ,
\end{aligned}
\end{equation}
where we used the fact
$\mathcal{A}(\varphi\psi)=\psi\mathcal{A}\varphi+\varphi\mathcal{A}\psi
+2\langle\nabla_g\varphi,\nabla_g\psi\rangle_g$.
Substituting \eqref{Eq3.9} in \eqref{Eq3.8} yileds
\begin{equation}\label{Eq3.10}
\begin{aligned}
 \operatorname{Im}\int_{Q}z_{t}N(\overline{z})dQ
&=\frac{1}{2}\int_{Q}\overline{\gamma u}z\operatorname{div}_g(N)dQ
 -\frac{1}{2}\int_{Q}|\mathcal{A}z|^2\operatorname{div}_g(N)dQ\\
&\quad -\frac{1}{2}\int_{Q}z\mathcal{A}(\operatorname{div}_g(N))
 \mathcal{A}\overline{z}dQ
 -\int_{Q}\mathcal{A}\overline{z}\langle\nabla_gz,
 \nabla_g\operatorname{div}_g(N)\rangle_gdQ\\
&\quad +\frac{\mathrm{i}}{2}\Big(\int_{\Omega}\overline{z}N(z)dx
 +\int_{\Omega}|z|^2\operatorname{div}_g(N)dx\Big)\Big|_0^{T}.
\end{aligned}
\end{equation}

Next we compute the second term on the left-hand side of \eqref{Eq3.6} to yield
\begin{equation}\label{Eq3.11}
\begin{aligned}
&\operatorname{Im}\mathrm{i}
\int_{Q}\mathcal{A}^2zN(\overline{z})dQ\\
&=\operatorname{Re}\int_{Q}\mathcal{A}^2zN(\overline{z})dQ
=\operatorname{Re}\int_{Q}\Delta_g(\mathcal{A}z)N(\overline{z})dQ
 +\operatorname{Re}\int_{Q}(Dp)(\mathcal{A}z)N(\overline{z})dQ\\
&= -\operatorname{Re}\int_{\Sigma}\frac{\partial(N(\overline{z}))}{\partial\mu}\mathcal{A}zd\Sigma
 +\operatorname{Re}\int_{Q}\Delta_g(N(\overline{z}))\mathcal{A}zdQ \\
&\quad +\operatorname{Re}\int_{Q}(Dp)(\mathcal{A}z)N(\overline{z})dQ\\
&=-\operatorname{Re}\int_{\Sigma}|\mathcal{A}z|^2d\Sigma
 +\operatorname{Re}\int_{\Sigma}\mathcal{A}z(Dp)(\overline{z})d\Sigma
 +\operatorname{Re}\int_{Q}(\Delta N)(\overline{z})\mathcal{A}zdQ\\
&+2\operatorname{Re}\int_{Q}\mathcal{A}z\langle DN,D^2\overline{z}\rangle_{T^2(\mathbb{R}_{x}^{n})}dQ
 +\operatorname{Re}\int_{Q}N(\mathcal{A}\overline{z})\mathcal{A}zdQ\\
&\quad -\operatorname{Re}\int_{Q}D^2p(N,D\overline{z})\mathcal{A}zdQ
 -\operatorname{Re}\int_{Q}D^2\overline{z}(N,Dp)\mathcal{A}zdQ \\
&\quad +\operatorname{Re}\int_{Q}\operatorname{Ric}(N,D\overline{z})\mathcal{A}zdQ
 +\operatorname{Re}\int_{Q}(Dp)(\mathcal{A}z)N(\overline{z})dQ,
\end{aligned}
\end{equation}
where we have used \eqref{Eq2.3} and the fact that
$$
\overline{z}\big|_{\Gamma}=\frac{\partial\overline{z}}{\partial\mu}\Big|_{\Gamma}=0
\Rightarrow \frac{\partial^2\overline{z}}{\partial\mu^2}\Big|_{\Gamma}
=\Delta_g\overline{z}\big|_{\Gamma},
$$
while
\begin{gather}\label{Eq3.12}
\operatorname{Re}\int_{Q}N(\mathcal{A}\overline{z})\mathcal{A}zdQ
 =\frac{1}{2}\int_{\Sigma}|\mathcal{A}z|^2d\Sigma-\frac{1}{2}
\int_{Q}|\mathcal{A}z|^2\operatorname{div}_g(N)dQ, \\
\label{Eq3.13}
\begin{aligned}
&\operatorname{Re}\int_{Q}Dp(\mathcal{A}z)N(\overline{z})dQ \\
&=-\operatorname{Re}\int_{Q}\mathcal{A}zDp(N(\overline{z}))dQ
 -\operatorname{Re}\int_{Q}N(\overline{z})\mathcal{A}z\operatorname{div}_g(Dp)dQ.
\end{aligned}
\end{gather}
Combining \eqref{Eq3.6}, \eqref{Eq3.10}, \eqref{Eq3.11}, \eqref{Eq3.12} and
\eqref{Eq3.13} to obtain
\begin{equation}\label{Eq3.14}
\begin{aligned}
& \frac{1}{2}\int_{\Sigma}|\mathcal{A}z|^2d\Sigma\\
&= \frac{1}{2}\int_{Q}z\mathcal{A}(\operatorname{div}_g(N))\mathcal{A}\overline{z}dQ
 +\int_{Q}\mathcal{A}\overline{z}\langle\nabla_gz,
 \nabla_g(\operatorname{div}_g(N))\rangle_gdQ \\
&\quad +\operatorname{Re}\int_{Q}(\Delta N)(\overline{z})\mathcal{A}zdQ
 +2\operatorname{Re}\int_{Q}\mathcal{A}z\langle DN,
 D^2\overline{z}\rangle_{T^2(\mathbb{R}_{x}^{n})}dQ \\
&\quad -\operatorname{Re}\int_{Q}D^2p(N,D\overline{z})\mathcal{A}zdQ
 -\operatorname{Re}\int_{Q}D^2\overline{z}(N,Dp)\mathcal{A}zdQ\\
&\quad + R_1 + R_2 + b_{0,T},
\end{aligned}
\end{equation}
where
\begin{gather*}
\begin{aligned}
R_1&=\operatorname{Re}\int_{Q}\operatorname{Ric}(N,D\overline{z})\mathcal{A}zdQ
 -\operatorname{Re}\int_{Q}(\mathcal{A}z)Dp(N(\overline{z}))dQ\\
&\quad -\operatorname{Re}\int_{Q}N(\overline{z})
\mathcal{A}z\operatorname{div}_g(Dp)dQ,
\end{aligned}\\
R_2=-\frac{1}{2}\int_{Q}\overline{\gamma u}z\operatorname{div}_g(N)dQ
 -\operatorname{Re}\int_{Q}\gamma uN(\overline{z})dQ, \\
b_{0,T}= -\frac{\mathrm{i}}{2}\Big(\int_{\Omega}\overline{z}N(z)dx
+\int_{\Omega}|z|^2\operatorname{div}_g(N)dx\Big)\Big|_0^{T}
\end{gather*}
\smallskip

\noindent\textbf{Step 2.} (Estimation of $R_1$).
Let $\gamma u=0$ in the first identity of \eqref{Eq3.2} and note that
$z=A_1^{-1}w\in H_0^2(\Omega)$.
We know that the solution to \eqref{Eq3.2} is associated with a
$C_0$-group on the space $H_0^2(\Omega)$. That is to say, for any
$z_0\in H_0^2(\Omega)$, there exists a unique solution
$z\in H_0^2(\Omega)$ to \eqref{Eq3.2}, which
depends continuously on $z_0$. This fact together with
\eqref{Eq3.14} implies that
\begin{equation}\label{Eq3.15}
 \frac{1}{2}\int_{\Sigma}|\mathcal{A}z|^2d\Sigma
\leqslant C_{T}\|z_0\|_{H_0^2(\Omega)}.
\end{equation}
This shows that the operator $B^{\ast}$ is admissible, and so is
$B$ \cite{4C}.
In other words,
\begin{equation}\label{Eq3.16}
u\mapsto w \text{ is continuous from } L^2(\Sigma) \text{ to }
C(0,T;H^{-2}(\Omega)).
\end{equation}
Moreover, by \eqref{Eq3.16},
\begin{equation}\label{Eq3.17}
z=A_1^{-1}w\in H_0^2(\Omega)
 \text{ depends continuously on } u\in L^2(0,T;L^2(\Gamma_0)).
\end{equation}
Therefore,
\begin{equation}\label{Eq3.18}
R_1\leqslant C_{T}\|u\|_{L^2(0,T;L^2(\Gamma_0))}^2, \quad \forall u\in
L^2(0,T;L^2(\Gamma_0)).
\end{equation}
where we have used Lemma 2.4.
\smallskip

\noindent\textbf{Step 3.} (Estimation of $R_2$ and $b_{0,T}$). This can be
easily obtained from the representations of $R_2$ and $b_{0,T}$ in
\eqref{Eq3.14} and \eqref{Eq3.17} that
\begin{equation}\label{Eq3.19}
R_2+b_{0,T}\leqslant C_{T}\|u\|_{L^2(0,T;L^2(\Gamma_0))}^2, \;\forall
u\in L^2(0,T;L^2(\Gamma_0)).
\end{equation}

Finally, it follows from \eqref{Eq3.14}, \eqref{Eq3.18}, and
\eqref{Eq3.19} that \eqref{Eq3.4} holds. The proof of Theorem \ref{Th1.1} is
complete.
\end{proof}

\section{Proof of Theorem \ref{Th1.2}}

In this section, we show the exact controllability by means of the Hilbert
Uniqueness Method (HUM) \cite{Lions1988}.
Since by Theorem \ref{Th1.1}, system \eqref{Se1} is well-posed which is
cast into the abstract first-order formulation \eqref{Eq2.19}
and $(\mathrm{i}\widetilde{A})^{\ast}=-\mathrm{i}\widetilde{A}$ in
$H^{-2}(\Omega)$, it follows that $\dot{w}=\mathrm{i}\widetilde{A}w+Bu$
is exactly controllable if and only if
$\dot{w}=\mathrm{i}\widetilde{A}w, y=B^{\ast}w$ is exactly observable.
More precisely, the exact controllability of system \eqref{Se1} is equivalent
to the exact observability of the dual system of system \eqref{Se1}
as follows:
\begin{equation}\label{Eq4.1}
\begin{gathered}
 \mathrm{i}\varphi_{t}+P^2\varphi=0 \quad \text{in } \Omega\times(0,T]=: Q, \\
 \varphi=0,\quad \frac{\partial\varphi}{\partial\nu_{\mathcal{A}}}=0 \quad \text{on }
 \partial\Omega\times [0,T]=: \Sigma, \\
 \varphi(x,0)=\varphi^{0}(x) \quad \text{in } \Omega, \\
 \end{gathered}
\end{equation}
with the output $y=-\mathrm{i}\mathcal{A}\varphi$. That is to say, the
``observability inequality'' holds for system \eqref{Eq4.1} in
the sense of (cf. \eqref{Eq3.2} and \eqref{Eq3.4}):
\begin{equation}\label{Eq4.2}
 \int_0^{T}\int_{\Gamma_0}\big|\mathcal{A}\varphi\big|^2d\Gamma dt \geqslant
C_{T}\|\varphi^{0}\|^2_{H_0^2(\Omega)}, \quad \forall
\varphi^{0}\in H_0^2(\Omega),
\end{equation}
for some (and hence for all) positive $T>0$.

To prove \eqref{Eq4.2}, we let $A$ be defined by \eqref{Eq1.5} and
let $\varphi$ be a solution to \eqref{Eq4.1}.
 Then $\mathrm{i}A$ generates a strongly continuous unitary group on the space
$H_0^2(\Omega)$ and hence
\begin{equation}\label{Eq4.3}
\begin{aligned}
\|\varphi(t)\|_{H_0^2(\Omega)}
&=\|A^{1/2}\varphi(t)\|_{L^2(\Omega)}
 =\|e^{{\rm i}At}\varphi^{0}\|_{H_0^2(\Omega)}\\
&=\|\varphi^{0}\|_{H_0^2(\Omega)}
 =\|A^{1/2}\varphi^{0}\|_{L^2(\Omega)}.
 \end{aligned}
\end{equation}

In this Section, let $N$ be an arbitrary vector field on $(\mathbb{R}^{n},g)$.
Assume that $\varphi$ solves problem \eqref{Eq4.1}.
Multiply the both sides of the first equation in \eqref{Eq4.1}
by $N(\overline{\varphi})$ and integrate on $Q$ to obtain
\begin{equation}\label{Eq4.4}
 \int_{Q}\varphi_{t}N(\overline{\varphi})dQ
-\mathrm{i}\int_{Q}\mathcal{A}^2\varphi N(\overline{\varphi})dQ=0.
\end{equation}

Now use the same process as the computation from \eqref{Eq3.7} to \eqref{Eq3.10}
to obtain
\begin{equation}\label{Eq4.5}
\begin{aligned}
 \operatorname{Im}\int_{Q}\varphi_{t}N(\overline{\varphi})dQ
&= -\frac{1}{2}\int_{Q}|\mathcal{A}\varphi|^2\operatorname{div}_0(N)dQ
 -\frac{1}{2}\int_{Q}\varphi\mathcal{A}(\operatorname{div}_0(N))
\mathcal{A}\overline{\varphi}dQ \\
&\quad -\int_{Q}\mathcal{A}\overline{\varphi}\langle\nabla_g\varphi,
 \nabla_g\operatorname{div}_0(N)\rangle_gdQ \\
&\quad +\frac{\mathrm{i}}{2}\Big(\int_{\Omega}
\overline{\varphi}N(\varphi)dx+\int_{\Omega}|\varphi|^2
 \operatorname{div}_0(N)dx\Big)\Big|_0^{T}.
\end{aligned}
\end{equation}

Next we compute the second term on the left hand side of \eqref{Eq4.4}
\begin{equation}\label{Eq4.6}
\begin{aligned}
& \operatorname{Im}\mathrm{i}\int_{Q}\mathcal{A}^2\varphi N(\overline{\varphi})dQ\\
&= \operatorname{Re}\int_{Q}\mathcal{A}^2\varphi N(\overline{\varphi})dQ \\
&= -\operatorname{Re}\int_{\Sigma}\frac{\partial(N(\overline{\varphi}))}
 {\partial\nu_{\mathcal{A}}}\mathcal{A}(\overline{\varphi})d\Sigma
 +\operatorname{Re}\int_{Q}\mathcal{A}(N(\overline{\varphi}))
 \mathcal{A}\varphi dQ\\
&= -\frac{1}{2}\int_{\Sigma}|\mathcal{A}\varphi|^2N\cdot\nu d\Sigma
 +\operatorname{Re}\int_{Q}\mathcal{A}\varphi[(\Delta N)(\overline{\varphi})
 +2\langle DN,D^2\overline{\varphi}\rangle_{T^2(\mathbb{R}_{x}^{n})} \\
&\quad +\operatorname{Ric}(N,D\overline{\varphi})
 -D^2p(N,D\overline{\varphi})+DN(D\overline{\varphi},Dp)]dQ \\
 &\quad -\frac{1}{2}\operatorname{Re}
\int_{Q}|\mathcal{A}\varphi|^2\operatorname{div}_0(N)dQ,
\end{aligned}
\end{equation}
where we have used \eqref{Eq2.5}, \eqref{Eq2.7} and \eqref{Eq2.9}.

To obtain the observability inequality, we define
$T\in T^2(\mathbb{R}_{x}^{n})$ for any $x\in\overline{\Omega}$
as follows:
\begin{equation}\label{Eq4.7}
T(X,Y)=DN(X,Y)+DN(Y,X),\quad \forall X,Y \in\mathbb{R}_{x}^{n}.
\end{equation}
It is clear that $T(\cdot,\cdot)$ is symmetric, and from \eqref{Eq1.11}, we have
\begin{equation}\label{Eq4.8}
DN(X,Y)+DN(Y,X)=2b(x)\langle X,Y\rangle_g,\quad
\forall X,Y\in\mathbb{R}_{x}^{n},\; x\in\overline{\Omega}.
\end{equation}
Fix $x\in\overline{\Omega}$, and let $\{e_{i}\}_{i=1}^{n}$ be an orthonormal
 basis of $(\mathbb{R}_{x}^{n},g)$.
By \eqref{Eq4.8}, we have
\begin{equation}\label{Eq4.9}
\begin{aligned}
 \langle DN,D^2\varphi\rangle_{T^2(\mathbb{R}_{x}^{n})}
&=\sum_{i,j=1}^{n}DN(e_{i},e_{j})D^2\varphi(e_{i},e_{j}) \\
&=b(x)\Delta_g\varphi=b(x)(\mathcal{A}\varphi-Dp(\varphi)).
\end{aligned}
\end{equation}
Combining \eqref{Eq4.4}, \eqref{Eq4.5}, \eqref{Eq4.6} and \eqref{Eq4.9} we obtain
\begin{equation}\label{Eq4.10}
\frac{1}{2}\int_{\Sigma}|\mathcal{A}\varphi|^2N\cdot\nu d\Sigma\\
 = M_1+ M_2+M_3+M_4
\end{equation}
where
\begin{gather*}
M_1= 2\int_{Q}b(x)|\mathcal{A}\varphi|^2dQ, \\
\begin{aligned}
M_2&= \Big[\frac{1}{2}\int_{Q}\varphi\mathcal{A}(\operatorname{div}_0(N))
 \mathcal{A}\overline{\varphi}dQ
 +\int_{Q}\mathcal{A}\overline{\varphi}\langle\nabla_g\varphi,\nabla_g
 \operatorname{div}_0(N)\rangle_gdQ \\
&\quad +\operatorname{Re}\int_{Q}\mathcal{A}\varphi[(\Delta N)(\overline{\varphi})+\operatorname{Ric}(N,D\overline{\varphi})
 -D^2p(N,D\overline{\varphi})+DN(D\overline{\varphi},Dp)]dQ \\
&\quad -2\operatorname{Re}\int_{Q}b(x)\mathcal{A}\varphi Dp
 (\overline{\varphi})dQ\Big],
\end{aligned}\\
M_3= -\frac{\mathrm{i}}{2}\Big(\int_{\Omega}\overline{\varphi}
 N(\varphi)dx\Big|_0^{T},\\
M_4=-\frac{\mathrm{i}}{2}\Big(\int_{\Omega}|\varphi|^2\operatorname{div}_0(N)dx\Big)
\Big|_0^{T}
\end{gather*}

Now define the energy function for \eqref{Eq4.1} as
\begin{equation}\label{Eq4.11}
 E(t)=E(\varphi,t)=\frac{1}{2}\int_{\Omega}|\mathcal{A}\varphi|^2dx.
\end{equation}
Then $E(t)= E(0)$ for all $t>0$. Set
\begin{equation}\label{Eq4.12}
 L(t)=\int_{\Omega}(|\varphi|^2+|\nabla_g\varphi|_g^2)dx
\end{equation}
be the lower order terms in composition of $E(t)$.

\begin{lemma} \label{lem4.1}
 Suppose that {\rm (H2)} holds. Let $\varphi$ is the solution of \eqref{Eq4.1}
with $\mathcal{A}\varphi=0$
on $\Sigma_0$. Then $\varphi\equiv 0$ in $Q$.
\end{lemma}

\begin{proof}
Let
$$
J=\{\varphi\in X= C(0,T;H_0^2(\Omega)); \varphi \text{ is the solution of
\eqref{Eq4.1} with } \mathcal{A}\varphi|_{\Sigma_0}=0 \}.
$$
We shall prove $J=0$. First, note that for any given initial data
$\varphi^{0}\in H_0^2(\Omega)$,
Equation \eqref{Eq4.1} admits a unique weak solution
\begin{equation}\label{Eq4.13}
 \varphi(t)\in C(0,T;H_0^2(\Omega)).
\end{equation}
From this and \eqref{Eq4.24} below, we have
\begin{equation}\label{Eq4.14}
E(0)\leqslant C(\|\mathcal{A}\varphi\|_{L^2(\Sigma_0)}^2
+\|\varphi\|_{L^{\infty}(0,T;H_0^{1}(\Omega))}^2),\quad
 \forall \varphi\in X \text{solves } \eqref{Eq4.1}.
\end{equation}
Now, we show that there exists a constant $C>0$ such that for any
 $\varphi\in X$ satisfying
\begin{equation}\label{Eq4.15}
\|\varphi\|_{L^{\infty}(0,T;H_0^{1}(\Omega))}^2
\leqslant C(\|\mathcal{A}\varphi\|_{L^2(\Sigma_0)}^2
+\|\varphi\|_{L^{\infty}(0,T;L^2(\Omega))}^2).
\end{equation}
Actually, if \eqref{Eq4.15} does not hold, then there exists a solution
sequence $\{\varphi_n\}\in X$ to \eqref{Eq4.1} satisfying
\begin{gather}\label{Eq4.16}
\|\mathcal{A}\varphi_n\|_{L^2(\Sigma_0)}^2+\|\varphi_n\|_{L^{\infty}
(0,T;L^2(\Omega))}^2\to 0,\quad\text{as } n\to\infty, \\
\label{Eq4.17}
\|\varphi_n\|_{L^{\infty}(0,T;H_0^{1}(\Omega))}^2=1.
\end{gather}
It is easy to learn from \eqref{Eq4.13} and \eqref{Eq4.14} that
$\{\varphi_n\}$ is bounded in $X$ and hence is relatively compact in
$L^{\infty}(0,T;H_0^{1}(\Omega))$.
Without loss of generality, we extract a subsequence $\{\varphi_n\}$
and assume it converges strongly to $\varphi\in L^{\infty}(0,T;H_0^{1}(\Omega))$,
by \eqref{Eq4.17}, and satisfies
\begin{equation}\label{Eq4.18}
\|\varphi\|_{L^{\infty}(0,T;H_0^{1}(\Omega))}^2=1.
\end{equation}
However, \eqref{Eq4.16} implies $\varphi\equiv 0$ in $Q$, which contradicts
\eqref{Eq4.18}. So \eqref{Eq4.15} holds.

From \eqref{Eq4.14} and \eqref{Eq4.15}, we have
\begin{equation}\label{Eq4.19}
E(0)\leqslant C(\|\mathcal{A}\varphi\|_{L^2(\Sigma_0)}^2
+\|\varphi\|_{L^2(0,T;L^2(\Omega))}^2),\quad \forall \varphi\in X
 \text{that solves \eqref{Eq4.1}}.
\end{equation}
Then \eqref{Eq4.19} still holds for $\varphi\in L^{\infty}(0,T;L^2(\Omega))$
satisfying \eqref{Eq4.1} by a denseness argument.
Thus, we have proved that
$\varphi\in J$ implies that $\psi=\dot{\varphi}$ satisfies \eqref{Eq4.1}
 with $\mathcal{A}\psi|_{\Sigma_0}=0$ and
$\psi\in L^{\infty}(0,T;L^2(\Omega))$. This together with \eqref{Eq4.19} gets
\begin{equation}\label{Eq4.20}
\psi(0)\in H_0^2(\Omega).
\end{equation}

At last, because of \eqref{Eq4.13}, $\psi\in X$, it follows from
\eqref{Eq4.19} that the map
$\frac{\partial}{\partial t}:\varphi\to\dot{\varphi}$ is continuous from
$J$ to $J$ and the injection of $\{\varphi\in J;\dot{\varphi}\in J\}$ is compact.
Therefore, $J$ is a finite dimensional space.
There must be an $\eta\in\mathbb{C}$ and $\varphi\in J\backslash\{0\}$ such
that $\dot{\varphi}=\eta\varphi$, which implies
\begin{equation}\label{Eq4.21}
\varphi(x,t)=e^{\eta t}\varphi(x,0).
\end{equation}
Substitute \eqref{Eq4.21} into \eqref{Eq4.1} to obtain \eqref{Eq1.13}
with $v(x)=\varphi(x,0)$ and $\zeta=-\mathrm{i}\eta$.
By (H2), we obtain $\varphi(x,t)\equiv 0$, hence $J=\{0\}$.
\end{proof}

Next, we evaluate the terms on the right-hand side of \eqref{Eq4.10}.
\begin{equation}\label{Eq4.22}
\begin{gathered}
 M_1=2\operatorname{Re}\int_{Q}b(x)|\mathcal{A}\varphi|^2dQ\geqslant 4b_0TE(0),\\
 |M_2|\leqslant C_1\varepsilon TE(0)+\frac{C_2}{\varepsilon}\int_0^{T}L(t)dt,\\
 |M_3|=\Big|-\frac{\mathrm{i}}{2}\int_{\Omega}
\overline{\varphi}N(\varphi)dx\Big|\leqslant
 \varepsilon E(0)+\frac{1}{16\varepsilon}L(t),\\
\begin{aligned}
 |M_4|&=\Big|-\frac{\mathrm{i}}{2}\int_{\Omega}|\varphi|^2
 \operatorname{div}_g(N)dx\Big| \\
&=\Big|-\frac{\mathrm{i}}{2}\int_{\Omega}\varphi\overline{\varphi}
 \operatorname{div}_g(N)dx\Big|\\
&\leqslant C_3\varepsilon E(0)+\frac{C_4}{\varepsilon}L(t),
\end{aligned}
\end{gathered}
\end{equation}
where we have used Lemma 2.4 in the evaluation of $M_2$, and \eqref{Eq4.3}
in the evaluations of $M_3$ and $M_4$, respectively. So
\begin{equation}\label{Eq4.23}
\begin{aligned}
&\frac{1}{2}\int_{\Sigma_0}|\mathcal{A}\varphi|^2N\cdot\nu d\Sigma\\
&\geqslant\frac{1}{2}\int_{\Sigma}|\mathcal{A}\varphi|^2N\cdot\nu d\Sigma\\
&\geqslant 4b_0\big[T-\frac{C_1T+2+2C_3}{4b_0}\varepsilon\big]E(0) \\
&\quad -\frac{C_2}{\varepsilon}\int_0^{T}L(t)dt
 -\frac{1+16C_4}{16\varepsilon}L(T)
 -\frac{1+16C_4}{16\varepsilon}L(0).
\end{aligned}
\end{equation}
Setting $\varepsilon>0$ small enough, we obtain
\begin{equation}\label{Eq4.24}
E(0)\leqslant C_{T}\int_{\Sigma_0}|\mathcal{A}\varphi|^2d\Sigma
+C\Big(\int_0^{T}L(t)dt+L(T)+L(0)\Big).
\end{equation}

Next, use the standard compact uniqueness argument to absorb the lower-order
terms in \eqref{Eq4.24}.
That is to say, we want to show that there exists a constant $C>0$ such that
\begin{equation}\label{Eq4.25}
 \|\varphi\|_{L^{\infty}(0,T;H_0^{1}(\Omega))}^2
\leqslant C\int_{\Sigma_0}|\mathcal{A}\varphi|^2d\Sigma,
\end{equation}
for the solution $\varphi$ of \eqref{Eq4.1}. We will assume \eqref{Eq4.25}
is not true to obtain a contradiction.
To this purpose, let $\{\varphi_n\}$ be the solution sequence of \eqref{Eq4.1}
such that
\begin{gather}\label{Eq4.26}
 \int_{\Sigma_0}|\mathcal{A}\varphi_n|^2d\Sigma\to 0,\quad n\to\infty, \\
\label{Eq4.27}
 \|\varphi_n\|_{L^{\infty}(0,T;H_0^{1}(\Omega))}^2=1.
\end{gather}
Then it follows from \eqref{Eq4.24} that $\{\varphi_n\}$ is a bounded sequence
in $C(0,T;H_0^2(\Omega))$,
and so relatively compact in $L^{\infty}(0,T;H_0^{1}(\Omega))$ because
of the injection
$$
C(0,T;H_0^2(\Omega))\to L^{\infty}(0,T;H_0^{1}(\Omega))
$$
is compact. Without loss of generality, we extract a subsequence
$\{\varphi_n\}$ and assume that $\{\varphi_n\}$ converges strongly to
$\varphi\in L^{\infty}(0,T;H_0^{1}(\Omega))$.
From \eqref{Eq4.27},
\begin{equation}\label{Eq4.28}
 \|\varphi\|_{L^{\infty}(0,T;H_0^{1}(\Omega))}^2=1.
\end{equation}
Furthermore, $\{\varphi_n\}$ converges to $\varphi$ in
$L^{\infty}(0,T;H_0^2(\Omega))$ in weak star topology.
Therefore, $\varphi$ is a solution to \eqref{Eq4.1} with
\begin{equation}\label{Eq4.29}
 \varphi\in C(0,T;H_0^2(\Omega)).
\end{equation}
By \eqref{Eq3.15}, we know that
\[
 \frac{1}{2}\int_{\Sigma_0}|\mathcal{A}\varphi|^2d\Sigma
\leqslant C_{T}\|\varphi_0\|_{H_0^2(\Omega)}.
\]
From this fact and \eqref{Eq4.26} to have
\begin{equation}\label{Eq4.30}
 \mathcal{A}\varphi=0\quad \text{on } \Sigma_0.
\end{equation}
Finally, by Lemma 4.1, we have
\begin{equation}\label{Eq4.31}
 \varphi=0\quad \text{in}\ Q,
\end{equation}
contradicting \eqref{Eq4.28}. So the proof of Theorem \ref{Th1.2} is complete.

\subsection*{Acknowledgments}
This work was supported by the National Natural Science Foundation of China
for the Youth (No.61503230), and by the National Natural Science Foundation
of China for the Youth (No. 61403239).

\begin{thebibliography}{99}

\bibitem{1A} K. Ammari;
Dirichlet boundary stabilization of the wave equation,
 {\it Asymptotic Anal.}, 30 (2002), 117-130.

\bibitem{2AT} K. Ammari, M. Tucsnak;
 Stabilization of second-order evolution equations by a class of
 unbounded feedbacks, {\it ESAIM Control Optim. Calc.Var.},
 6 (2001), 361-386.

\bibitem{3BGSW} C. I. Byrnes, D. S. Gilliam, V. I. Shubov, G. Weiss;
 Regular linear systems governed by a boundary controlled heat
 equation, {\it J. Dyn. Control Syst.}, 8 (2002), 341-370.

\bibitem{4C} R. F. Curtain;
 The Salamon-Weiss class of well-posed infinite dimensional linear systems:
 A survey, {\it IMA J. Math. Control Inform.}, 14 (1997), 207-223.

\bibitem{5C} R. F. Curtain;
 Linear operator inequalities for strongly stable weakly
 regular linear systems, {\it Math. Control Signals Syst.},
14 (2001), 299-337.

\bibitem{ChaiJDE} S. G. Chai, B. Z. Guo;
 Well-posedness and regularity of Naghdi's shell equation under
 boundary control, {\it J. Differential Equations}, 249 (2010), 3174-3214.

\bibitem{CGweaklycoupled} S. G. Chai, B. Z. Guo;
 Well-posedness and regularity of weakly coupled
 wave-plate equation with boundary control and observation,
{\it J. Dyn. Control Syst.}, 15 (2003), 331-358.

\bibitem{CSIAM} S. G. Chai, B. Z. Guo;
 Feedthrough operator for linear elasticity system
 with boundary control and observation,
{\it SIAM J. Control Optim.}, 48 (2010), 3708-3734.

\bibitem{GCbook} B. Z. Guo, S. G. Chai;
 {\it Infinite-Dimensional Linear Dystem Control Theory}, Science
 Press, Beijing, 2012 (in Chinese).

\bibitem{GuoLuo} B. Z. Guo, Y. H. Luo;
 Controllability and Stability of a seconder order hyperbolic system
 with collocated sensor/actuator,
{\it Systems Control Lett.}, 46 (2002), 45-65.

\bibitem{GSSchrodinger} B. Z. Guo, Z. C. Shao;
 Regularity of a Schr\"{o}dinger equation with Dirichlet
control and collocated observation, {\it Systems Control Lett.} 54 (2005),
1135-1142.

\bibitem{GStransplate} B. Z. Guo, Z. C. Shao;
 On well-posedness, regularity and exact controllability for problems
 of transmission of plate equation with variable coefficients, {\it
 Quart. Appl. Math.}, 65 (2007), 705-736.

\bibitem{GSplate} B. Z. Guo, Z. C. Shao;
 Regularity of an Euler-Bernoulli equation with Neumann control and collocated
 observation, {\it J. Dyn. Control Syst.}, 12(2006), 405-418.

\bibitem{GuoZhang} B. Z. Guo, X. Zhang;
 The Regularity of wave equation with partial Dirichlet control and observation,
 {\it SIAM J. Control Optim.}, 44 (2005), 1598-1613.

\bibitem{GZwaveV} B. Z. Guo, Z. X. Zhang;
On the well-posedness and regularity of wave equations with variable
coefficients and partial boundary Dirichlet control and collocated observation,
 {\it ESAIM Control Optim. Calc. Var.}, 13 (2007), 776-792.

\bibitem{GZplateV} B. Z. Guo, Z. X. Zhang;
 Well-posedness and regularity for an Euler-Bernoulli plate with variable
coefficients and boundary control and observation,
{\it Math. Control Signals Syst.}, 19 (2007), 337-360.

\bibitem{Lions1988} J. L. Lions;
 Exact contrllability, Stabilization and perturbations for distributed systems,
 {\it SIAM Rev}., 30 (1988), 1-68.

\bibitem{LM} J. L. Lions, E. Magenes;
 {\it Non-Homogeneous Boundary Value Problems and Application}, Vol. I,
 II, Springer-Verlag, Birlin, 1972.

\bibitem{Liu} W. Liu, G. H. Williams;
 Exact controllability for problems of transmission of the plate equation
with lower order terms, {\it Quart. Appl. Math.}, 58 (2000), 37-68.

\bibitem{Salamon1} D. Salamon;
 Infinite dimensional systems with unbounded control and observation:
 a functional analytic approach, {\it Trans. Amer. Math. Soc.},
300 (1987), 383-481.

\bibitem{Salamon2} D. Salamon;
 Realization theory in Hilbert space, {\it Math. Systems Theory,}
 21 (1989), 147-164.

\bibitem{Staffans2012} O. J. Staffans, G. Weiss;
 A physically motivated class of scattering passive linear systems,
 {\it SIAM J. Control Optim.}, 50 (2012), 3083-3112.

\bibitem{Taylor} M. E. Taylor;
 {\it Partial Diffirential Equations I: Basic Theory,}
Springer-Verlag, New York, 1996.

\bibitem{Weiss1} G. Weiss;
 Admissible observation operators for linear semigroups,
{\it Israel J. Math.}, 65 (1989), no. 1, 17-43.

\bibitem{WenSIAM} R. L. Wen, S. G. Chai, B. Z. Guo;
 Well-posedness and exact controllability of fourth order Schr\"{o}dinger
 equation with boundary control snd collocated onservation,
{\it SIAM J. Control Optim.}, 52 (2014), 365-396.

\bibitem{WenMMAS} R. L. Wen, S. G. Chai, B. Z. Guo;
 Well-posedness and regularity of Euler-Bernoulli equation with variable
coefficient and Dirichlet boundary control and collocated observation,
 {\it Math. Meth. Appl. Sci.}, 37 (2014), 2889-2905.

\bibitem{Weiss2} G. Weiss;
 Admissibility of unbounded control operators, {\it SIAM J. Control Optim.},
27 (1989), 527-545.

\bibitem{Yao1} P. F. Yao;
 On the observatility inequalities for exact controllability of wave equations
with variable coefficients, \emph{SIAM Journal on Control and Optimization}
37 (1999), 1568-1599.

\bibitem{Yao2} P. F. Yao;
 Observatility inequalities for the Euler-Bernoulli plate with variable coefficients,
 \emph{Contemporary. Mathematics}, Amer. Math. Soc., Providence,
RI 268 (2000), 383-406.

\bibitem{Zwart} H. Zwart;
Transfer functions for infinite-dimensional systems,
{\it Systems Control Lett.}, 52 (2004), 247-255.

\end{thebibliography}

\end{document}