# Positive and negative exact boundary controllability results for the linear
Biharmonic Schrödinger equation

Kaïs Ammari Hedi Bouzidi LR Analysis and Control of PDEs, LR22ES03, Department
of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5019
Monastir, Tunisia<EMAIL_ADDRESS>Analysis and Control of PDEs,
LR22ES03, Department of Mathematics, Faculty of Sciences of Monastir,
University of Monastir, 5019 Monastir, Tunisia<EMAIL_ADDRESS>
###### Abstract

In this paper, we study the exact boundary controllability of the linear
Biharmonic Schrödinger equation $\displaystyle
i\partial_{t}y=-\partial_{x}^{4}y+\gamma\partial_{x}^{2}y$ on a bounded domain
with hinged boundary conditions and boundary control acts on the second
spatial derivative at the left endpoint, where the parameter
$\displaystyle\gamma<0$. We prove that this system is exactly controllable in
time $\displaystyle T>0$, if and only if, the parameter $\displaystyle\gamma$
does not belong to a critical countable set of negative real numbers. The
analysis in this work is based on spectral analysis together with the
nonharmonic Fourier series method.

2010 Mathematics Subject Classification. 35P05, 35G05, 81Q10, 81Q93, 93C15,
93D15.
Key words and phrases. Optical fibers, laser beams, Biharmonic Schrödinger,
boundary control.

##  1 Introduction and main result

In this paper, we study the boundary controllability of the linear fourth-
order Schrödinger equation on the bounded interval
$\displaystyle\left(0,\ell\right)$, where $\displaystyle\ell>0$. More
precisely, we consider the following control system

$\begin{cases}i\partial_{t}y(t,x)=-\partial_{x}^{4}y(t,x)+\gamma\partial_{x}^{2}y(t,x),&(t,x)\in(0,T)\times(0,\ell),\\\
y(t,0)=y(t,\ell)=\partial_{x}^{2}y(t,\ell)=0,\leavevmode\nobreak\
\partial_{x}^{2}y(t,0)=f(t),&t\in(0,T),\\\
y(0,x)=y_{0}(x),&x\in(0,\ell),\end{cases}$ (1.1)

where the parameter $\displaystyle\gamma<0$, $\displaystyle f$ is a control
that acts on the left end $\displaystyle x=0$, and the function $\displaystyle
y_{0}$ is the initial condition. For system (1.1), the appropriate control
notion to study is the exact controllability, which is defined as follows:
System (1.1) is said to be exactly controllable at time $\displaystyle T>0$
if, given any initial state $\displaystyle y_{0}$, there exists a control
$\displaystyle f$ such that the corresponding solution $\displaystyle
y=y(t,x)$ satisfies $\displaystyle y(T,.)=0$.
The fourth-order Schrödinger equation has been modeled by Karpman [8], and
Karpman $\displaystyle\&$ Shagalov [9] in order to describe the propagation of
intense laser beams in a bulk medium with Kerr nonlinearity when small fourth-
order dispersion is taken into an account. Nowadays, equation (1.1) is often
called the biharmonic Schrödinger equation and has various applications in
several areas of physics, such as nonlinear optics, plasma physics and
nonuniform dielectric media, see [5, 7, 10, 11].
Notice that, in the case where $\displaystyle\partial_{x}^{4}y\equiv 0$,
Equation (1.1) collapses to the standard second order Schrödinger equation. In
this direction, the exact controllability of the second order Schrödinger
equation has been extensively investigated, see for instance [1, 3, 4]. In the
case where $\displaystyle\gamma=0$, Zheng and Zhongcheng [15] proved that the
biharmonic Schrödinger equation (1.1) with clamped boundary conditions and
Neumann boundary control is exactly controllable at time $\displaystyle T>0$.
Later on, Wen et al. [14] established well-posedness and exact controllability
results for a system described by the fourth order Schrödinger equation in
(1.1) for $\displaystyle\gamma=0,$ on a bounded domain of
$\displaystyle\mathbb{R}^{n}\leavevmode\nobreak\ (n\geq 2)$ with boundary
control and collocated observation. Along similar lines, the same authors in
[13], extended these results to the case of a linear fourth-order multi-
dimensional Schrödinger equation with hinged boundary control and collocated
observation. Recently, in the case where $\displaystyle\gamma=1,$ Capistrano-
Filho and Cavalcante [6] proved global stabilization and exact controllability
results for the biharmonic Schrödinger equation (1.1) on a periodic domain
$\displaystyle\mathbb{T}$ with internal control supported on an arbitrary sub-
domain of $\displaystyle\mathbb{T}$. Whereas, in the case where
$\displaystyle\gamma(x)\geq 0$, the authors [2] showed that the fourth-order
Schrödinger equation (1.1) with clamped boundary conditions and boundary
control acting on the first spatial derivative at one endpoint is exactly
controllable at time $\displaystyle T>0$.
As we know, the exact controllability of System (1.1) with
$\displaystyle\gamma<0$ has not been studied in the literature yet. As we will
see, that the parameter $\displaystyle\gamma$ plays a key role in
understanding the dynamics of System (1.1). Indeed, we first prove that the
eigenvalues $\displaystyle\left({\lambda_{n}}\right)_{n\in\mathbb{N}^{*}}$
associated to Problem (1.1) with $\displaystyle f(t)\equiv 0,$ are bounded
from below by the quantity $\displaystyle-\frac{\gamma^{2}}{4}$. Furthermore,
we show that the finite number of negative eigenvalues
$\displaystyle\left({\lambda_{n}}\right)_{n\in\mathbb{N}^{*}}$ are
algebraically simple, if and only if, the parameter $\displaystyle\gamma$ does
not belong to the following critical countable set of negative real numbers:

${{\Gamma^{*}}=\left\\{-\,\frac{\pi^{2}}{\ell^{2}}\left(p^{2}+q^{2}\right)\leavevmode\nobreak\
:\leavevmode\nobreak\ p,q\in\mathbb{N}^{*},\leavevmode\nobreak\ 1\leq
p<q\right\\}.}$ (1.2)

Therefore, using the nonharmonic Fourier series method, we establish positive
and negative exact controllability results for System (1.1). Namely, our main
result is the following:

######  Theorem 1.1.

Given $\displaystyle T>0$ and $\displaystyle y_{0}\in
H^{-1}\left(0,\ell\right)$. Assume that
$\displaystyle\gamma\not\in\Gamma^{*}$. Then, there exists a control
$\displaystyle f\in L^{2}(0,T)$ such that the solution $\displaystyle y$ of
the control problem (1.1) satisfies

$y(T,x)=0,\leavevmode\nobreak\ \leavevmode\nobreak\ x\in\left[0,\ell\right].$

Moreover, if $\displaystyle\gamma\in\Gamma^{*},$ then System (1.1) is not
exactly controllable at time $\displaystyle T>0$.

The remainder of this paper is organized as follows: In Section $\displaystyle
2$, we investigate the main properties of all the eigenvalues
$\displaystyle\left({\lambda_{n}}\right)_{n\in\mathbb{N}^{*}}$ associated to
the control system (1.1). In Section $\displaystyle 3$, we prove the
observability estimate for uncontrolled system (1.1) with $\displaystyle
f(t)\equiv 0$. Finally, in Section $\displaystyle 4$, we give the proof of our
main exact controllability result.

##  2 Spectral analysis

We consider the following spectral problem which arises by applying separation
of variables to system (1.1) with $\displaystyle f(t)\equiv 0$,

$\displaystyle\displaystyle\begin{cases}\phi^{\prime\prime\prime\prime}-\gamma\phi^{\prime\prime}=\lambda\phi,\leavevmode\nobreak\
\leavevmode\nobreak\ x\in(0,\ell),\\\
\phi(0)=\phi^{\prime\prime}(0)=\phi(\ell)=\phi^{\prime\prime}(\ell)=0.\end{cases}$
(2.1)

Let $\displaystyle L^{2}(0,\ell)$ be the Lebesgue space of all functions
$\displaystyle y$ defined on $\displaystyle(0,\ell),$ being equipped with the
norm

$\displaystyle\|y\|_{L^{2}(0,\ell)}=\left(\int_{0}^{\ell}|y(x)|^{2}dx\right)^{\frac{1}{2}},\leavevmode\nobreak\
\forall\leavevmode\nobreak\ y\in L^{2}(0,\ell).$

We consider the Sobolev space $\displaystyle H^{2}(0,\ell)\cap
H^{1}_{0}(0,\ell)$ endowed with the norm

$\displaystyle\|y\|_{H^{2}\cap
H^{1}_{0}(0,\ell)}=\|{y}^{\prime\prime}\|_{L^{2}(0,\ell)},\leavevmode\nobreak\
\forall\leavevmode\nobreak\ y\in H^{2}(0,\ell)\cap H^{1}_{0}(0,\ell).$

We introduce the operator $\displaystyle\mathcal{A}$ defined in $\displaystyle
L^{2}(0,\ell)$ by setting:

$\displaystyle\mathcal{A}y=\phi^{\prime\prime\prime\prime}-\gamma\phi^{\prime\prime},$

on the domain

$\displaystyle\mathcal{D}\left(\mathcal{A}\right):=\left\\{\phi\in
H^{4}(0,\ell)\leavevmode\nobreak\ :\leavevmode\nobreak\
\phi(0)=\phi^{\prime\prime}(0)=\phi(\ell)=\phi^{\prime\prime}(\ell)=0\right\\},$

which is dense in $\displaystyle L^{2}(0,\ell)$.

######  Proposition 2.1.

The linear operator $\displaystyle\mathcal{A}$ is positive and self-adjoint
such that $\displaystyle\mathcal{A}^{-1}$ is compact. Moreover, the spectrum
of $\displaystyle\mathcal{A}$ is discrete and consists of a sequence of real
eigenvalues $\displaystyle(\lambda_{n})_{n\in\mathbb{N}^{*}}$ tending to
$\displaystyle+\infty$:

$\displaystyle-\frac{\gamma^{2}}{4}\leq\lambda_{1}\leq\lambda_{2}\leq\ldots\leq\lambda_{n}\leq\ldots\underset{n\rightarrow+\infty}{\to}+\infty.$

The corresponding eigenfunctions
$\displaystyle(\Phi_{n})_{n\in\mathbb{N}^{*}}$ can be chosen to form an
orthonormal basis of $\displaystyle L^{2}(0,\ell)$.

###### Proof.

Let $\displaystyle y\in\mathcal{D}\left(\mathcal{A}\right)$, then using an
integration by parts, we have

$\displaystyle\displaystyle\langle\mathcal{A}\phi,\phi\rangle_{L^{2}(0,\ell)}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{\ell}\left(\phi^{\prime\prime\prime\prime}(x)-\gamma\phi^{\prime\prime}(x)\right){\phi}(x)dx$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{\ell}\left(\phi^{\prime\prime}(x)\right)^{2}+\gamma\left(\phi^{\prime}(x)\right)^{2}dx.$

Since
$\displaystyle\displaystyle\int_{0}^{\ell}\left({\phi^{\prime\prime}(x)}+\frac{\gamma\phi(x)}{2}\right)^{2}dx\geq
0,$ then

$\displaystyle\langle\mathcal{A}\phi,\phi\rangle_{L^{2}(0,\ell)}\geq-\frac{\gamma^{2}}{4}\|\phi\|^{2}_{L^{2}(0,\ell)}.$

This implies that the quadratic form is bounded from below by
$\displaystyle-\frac{\gamma^{2}}{4}$, and then, the linear operator
$\displaystyle\mathcal{A}$ is symmetric. Furthermore, it is easy to show that
$\displaystyle Ran(\mathcal{A}-iId)=L^{2}(0,\ell)$, and this means that
$\displaystyle\mathcal{A}$ is self-adjoint. Since, the space $\displaystyle
H^{2}(0,\ell)\cap H^{1}_{0}(0,\ell)$ is continuously and compactly embedded in
the space $\displaystyle L^{2}(0,\ell)$, then $\displaystyle\mathcal{A}^{-1}$
is compact in $\displaystyle L^{2}(0,\ell)$. ∎

######  Theorem 2.2.

One has:
(a) All the positives eigenvalues of Problem (2.1) are algebraically simple
and satisfy

$\lambda_{n}=\frac{n^{2}\pi^{2}}{\ell^{2}}\left(\frac{n^{2}\pi^{2}}{\ell^{2}}+\gamma\right)\hbox{
for }\leavevmode\nobreak\ n\geq
n_{0}:=\left[\ell{\pi}^{-1}{\sqrt{|\gamma|}}\right]+1,$ (2.2)

where $\displaystyle\left[\bullet\right]$ denotes the integer part of
$\displaystyle\bullet$.
(b) $\displaystyle\lambda=0$ is a simple eigenvalue of Problem (2.1) if and
only if

$\gamma=-\frac{p^{2}\pi^{2}}{\ell^{2}},\leavevmode\nobreak\
p\in\mathbb{N}^{*}.$ (2.3)

(c) All the negatives eigenvalues of Problem (2.1) satisfy

$\lambda_{n}=\frac{n^{2}\pi^{2}}{\ell^{2}}\left(\frac{n^{2}\pi^{2}}{\ell^{2}}+\gamma\right)\hbox{
for }n\leq\left[\ell{\pi}^{-1}{\sqrt{|\gamma|}}\right]-1.$ (2.4)

Moreover, they are algebraically simple if and only if
$\displaystyle\gamma\not\in\Gamma^{*}$.

###### Proof.

(a) Since $\displaystyle\lambda>0$, any non-trivial solution
$\displaystyle\phi(x,\lambda)$ of the fourth-order linear differential
equation

$\phi^{\prime\prime\prime\prime}-\gamma\phi^{\prime\prime}-\lambda\phi=0,\leavevmode\nobreak\
\leavevmode\nobreak\ x\in(0,\ell),$ (2.5)

may be expressed as follows:

$\displaystyle\phi(x,\lambda)=C_{1}\sin(x\eta(\lambda))+C_{2}\cos(x\eta(\lambda))+C_{3}\sinh(x\bar{\eta}(\lambda))+C_{4}\cosh(x\bar{\eta}(\lambda)),$

for some constants $\displaystyle C_{j},\leavevmode\nobreak\ j=1,2,3,4,$ where

$\displaystyle\eta(\lambda)=\sqrt{\frac{\sqrt{\gamma^{2}+4\lambda}-\gamma}{2}}\hbox{
and }\bar{\eta}(\lambda)=\sqrt{\frac{\sqrt{\gamma^{2}+4\lambda}+\gamma}{2}}.$

Using the fact that $\displaystyle\bar{\eta}(\lambda)>0,$ and the boundary
conditions in (2.1), we deduce that the positive eigenvalues of (2.1) are
solutions of the characteristic equation
$\displaystyle\sin(\ell\eta(\lambda))=0.$ Consequently, all positives
eigenvalues of Problem (2.1) satisfy

$\displaystyle\lambda_{n}=\frac{n^{2}\pi^{2}}{\ell^{2}}\left(\frac{n^{2}\pi^{2}}{\ell^{2}}+\gamma\right),\leavevmode\nobreak\
n>\frac{\ell\sqrt{|\gamma|}}{\pi},$

and this proves relation (2.2). The corresponding eigenfunctions
$\displaystyle\phi(x,\lambda_{n}),$ up to a multiplicative constant, have the
form

$\phi(x,\lambda_{n})=\sin(x\eta(\lambda_{n})),\leavevmode\nobreak\
n>\frac{\ell\sqrt{|\gamma|}}{\pi},$ (2.6)

and then, each eigenvalue $\displaystyle\lambda_{n}$ of Problem (2.1) is
simple.
(b) In this case, any non-trivial solution $\displaystyle\phi(x,\lambda)$ of
Equation (2.5) may be written as the following,

$\displaystyle\phi(x,\lambda)=C_{1}+C_{2}x+C_{3}\sin\left(\sqrt{|\gamma|}x\right)+C_{4}\cos\left(\sqrt{|\gamma|}x\right),$

for some constants $\displaystyle C_{j},\leavevmode\nobreak\ j=1,2,3,4.$ Using
the boundary conditions at $\displaystyle x=0$, one gets $\displaystyle
C_{1}=C_{4}=0$. If (2.3) is not satisfied, then, from the boundary conditions
at $\displaystyle x=\ell$, one has $\displaystyle C_{2}=C_{3}=0$. This implies
that $\displaystyle\phi(x,\lambda)\equiv 0,$ which is a contradiction.
Conversely, if (2.3) holds, then, $\displaystyle C_{2}=0$. Thus, the
eigenfunction $\displaystyle\phi(x,\lambda)$ associated to the eigenvalue
$\displaystyle\lambda=0$, up to a multiplicative constant, has the form

$\phi(x,\lambda)=\sin\left(\sqrt{|\gamma|}x\right).$ (2.7)

Therefore, $\displaystyle\lambda=0$ is a simple eigenvalue of Problem (2.1).
(c) Since $\displaystyle\lambda<0$, any non-trivial solution
$\displaystyle\phi(x,\lambda)$ of Equation (2.5) may be expressed as follows:

$\displaystyle\phi(x,\lambda)=C_{1}\sin(x\xi(\lambda))+C_{2}\cos(x\xi(\lambda))+C_{3}\sin(x\bar{\xi}(\lambda))+C_{4}\cos(x\bar{\xi}(\lambda)),$

for some constants $\displaystyle C_{j},\leavevmode\nobreak\ j=1,2,3,4,$ where

$\xi(\lambda)=\sqrt{\frac{\sqrt{\gamma^{2}+4\lambda}-\gamma}{2}}\hbox{ and
}\bar{\xi}(\lambda)=\sqrt{\frac{-\sqrt{\gamma^{2}+4\lambda}-\gamma}{2}}.$
(2.8)

Using the boundary conditions at $\displaystyle x=0$, one has

$\phi(x,\lambda)=C_{1}\sin(x\xi(\lambda))+C_{3}\sin(x\bar{\xi}(\lambda)).$
(2.9)

Thus, by the boundary conditions at $\displaystyle x=\ell$,
$\displaystyle\lambda$ is an eigenvalue of Problem (2.1) if and only if the
function

$\sin(\ell\xi(\lambda))\sin(\ell\bar{\xi}(\lambda))=0.$ (2.10)

Therefore, relation (2.4) is proved. Since,

$\displaystyle\xi(\lambda)\not=\bar{\xi}(\lambda)\hbox{ and
}\left(\xi(\lambda)\right)^{2}+\left(\bar{\xi}(\lambda)\right)^{2}=|\gamma|,$

then, by relation (2.10), the following cases must be examined :
(i) $\displaystyle\dfrac{\xi(\lambda)}{\bar{\xi}(\lambda)}=\frac{p}{q}$, where
$\displaystyle p,q\in\mathbb{N}^{*}$ and $\displaystyle p\not=q$. Or
equivalently, $\displaystyle\gamma\in\Gamma^{*}$. Then, by relation (2.9) and
the boundary conditions at $\displaystyle x=\ell$, the eigenfunctions
$\displaystyle\phi(x,\lambda_{n})$ of Problem (2.1), have the form

$\phi(x,\lambda_{n})=C_{1}\sin(x\xi(\lambda_{n}))+C_{3}\sin(x\bar{\xi}(\lambda_{n})),$
(2.11)

where $\displaystyle|C_{1}|+|C_{3}|\not=0.$ Thus, each eigenvalue
$\displaystyle\lambda_{n}$ of Problem (2.1) is of multiplicity two.
(ii) $\displaystyle\dfrac{\xi(\lambda)}{\bar{\xi}(\lambda)}\not=\frac{p}{q}$,
where $\displaystyle p,q\in\mathbb{N}^{*}$. Or equivalently,
$\displaystyle\gamma\not\in\Gamma^{*}$. Then, by relation (2.9) and the
boundary conditions at $\displaystyle x=\ell$, the eigenfunctions
$\displaystyle\phi(x,\lambda_{n})$ of Problem (2.1), up to a multiplicative
constant, have the form

$\phi(x,\lambda_{n})=\sin(x\eta(\lambda_{n})),\leavevmode\nobreak\
n<\frac{\ell\sqrt{|\gamma|}}{\pi}.$ (2.12)

Therefore, each eigenvalue $\displaystyle\lambda_{n}$ of Problem (2.1) is a
simple. The proof is then complete. ∎

As a consequence of Theorem 2.2, we have the following result.

######  Corollary 2.1.

One has:
(a) Let $\displaystyle(\lambda_{n})_{n\geq n_{0}}$ be the increasing sequence
of positive eigenvalues of the spectral problem (2.1) given by relation (2.2).
Then, the following uniform gap condition holds

${\lambda_{n+1}}-{\lambda_{n}}>C,\leavevmode\nobreak\ \hbox{as }n\to\infty,$
(2.13)

for some positive constant $\displaystyle C>0$.
(b) Let $\displaystyle(\Phi_{n})_{n\geq 1}$ be the sequence of normalized
eigenfunctions of the spectral problem (2.1) so that
$\displaystyle\lim_{n\to\infty}\|{\Phi}_{n}\|_{L^{2}_{\rho}(0,\ell)}=1$. Then,
one has

$\dfrac{\left|{\Phi_{n}^{\prime}(0)}\right|}{\sqrt[4]{\lambda_{n}}}=\sqrt{\frac{2}{\ell}},\leavevmode\nobreak\
\hbox{as }n\to\infty.$ (2.14)

Now, we give a characterization of some fractional powers of the linear
operator $\displaystyle\mathcal{A}$ which will be useful in the next section.
According to Lemma 2.1, the operator $\displaystyle\mathcal{A}$ is positive
and self-adjoint, and hence it generates a scale of interpolation spaces
$\displaystyle\mathcal{H}_{\theta}$, $\displaystyle\theta\in\mathbb{R}$. For
$\displaystyle\theta\geq 0$, the space $\displaystyle\mathcal{H}_{\theta}$
coincides with $\displaystyle\mathcal{D}(\mathcal{A}^{\theta})$ and is
equipped with the norm
$\displaystyle\|u\|_{\theta}^{2}=\langle\mathcal{A}^{\theta}u,\mathcal{A}^{\theta}u\rangle_{L^{2}(0,\ell)}$,
and for $\displaystyle\theta<0$ it is defined as the completion of
$\displaystyle L^{2}(0,\ell)$ with respect to this norm. Furthermore, we have
the following spectral representation of space
$\displaystyle\mathcal{H}_{\theta}$,

$\mathcal{H}_{\theta}=\left\\{u(x)=\sum\limits_{n\in\mathbb{N}^{*}}c_{n}\Phi_{n}(x)\leavevmode\nobreak\
:\leavevmode\nobreak\
\|u\|_{\theta}^{2}=\sum\limits_{n\in\mathbb{N}^{*}}|\lambda_{n}|^{2\theta}|c_{n}|^{2}<\infty\right\\},$
(2.15)

where $\displaystyle\theta\in\mathbb{R}$, and the eigenfunctions
$\displaystyle\left(\Phi_{n}\right)_{n\in\mathbb{N}^{*}}$ are defined in Lemma
2.1. In particular,

$\mathcal{H}_{0}=L^{2}(0,\ell)\hbox{ and
}\mathcal{H}_{1/4}=H^{1}_{0}(0,\ell).$ (2.16)

##  3 Observability of the biharmonic Schrödinger equation

In this section, we prove some observability results for the uncontrolled
system,

${\begin{cases}i\partial_{t}z(t,x)=-\partial_{x}^{4}z(t,x)+\gamma\partial_{x}^{2}z(t,x),&(t,x)\in(0,T)\times(0,\ell),\\\
z(t,0)=\partial_{x}^{2}z(t,0)=z(t,\ell)=\partial_{x}^{2}z(t,\ell)=0,&t\in(0,T),\\\
z(0,x)=z_{0},&x\in(0,\ell).\end{cases}}$ (3.1)

We start by mentioning the well-posedness of System (3.1). Obviously, the
solutions of Problem (3.1) can be written in Fourier series as

${z}(t,x):=\sum\limits_{n\in\mathbb{N}^{*}}c_{n}e^{i\lambda_{n}t}{\Phi}_{n}(x),$
(3.2)

where the Fourier coefficients are given by

$\displaystyle
c_{n}:=\int_{0}^{\ell}z_{0}(x)\bar{\Phi}_{n}(x)\rho(x)dx,\leavevmode\nobreak\
n\in\mathbb{N}^{*},$

and $\displaystyle\left(c_{n}\right)\in\ell^{2}\left(\mathbb{N}^{*}\right)$.
Let us denote by $\displaystyle\mathcal{E}_{\theta}(t)$ the energy associated
to the space $\displaystyle\mathcal{H}_{\theta}$, then

$\displaystyle\displaystyle\mathcal{E}_{\theta}(t)$
$\displaystyle\displaystyle=\|z\|_{\theta}^{2}=\sum\limits_{n\in\mathbb{N}^{*}}|{\lambda}_{n}|^{2\theta}|c_{n}e^{i\lambda_{n}t}|^{2}$
$\displaystyle\displaystyle=\sum\limits_{n\in\mathbb{N}^{*}}|{\lambda}_{n}|^{2\theta}|c_{n}|^{2}=\mathcal{E}_{\theta}(0),$

which establishes the conservation of energy along time. Consequently, one
has:

######  Lemma 3.1.

Let $\displaystyle\theta\in\mathbb{R}$ and $\displaystyle
z_{0}\in\mathcal{H}_{\theta}$. Then Problem (3.1) has a unique solution
$\displaystyle{z}\in C([0,T],\mathcal{H}_{\theta})$. Moreover, the energy
$\displaystyle\mathcal{E}_{\theta}(t)$ of System (3.1) is conserved along the
time.

######  Proposition 3.2.

Assume that $\displaystyle\gamma\not\in\Gamma^{*}$, where
$\displaystyle\Gamma^{*}$ is given by relation (1.2). Let $\displaystyle T>0$
and $\displaystyle z_{0}\in H^{1}_{0}(0,\ell)$, then

$C_{T}^{-1}\|z_{0}\|_{H^{1}_{0}(0,\ell)}^{2}\leq\int_{0}^{T}|\partial_{x}z(t,0)|^{2}dt\leq
C_{T}\|z_{0}\|_{H^{1}_{0}(0,\ell)}^{2},$ (3.3)

for some positive constant $\displaystyle C_{T}>0$, depending on
$\displaystyle T,$ where $\displaystyle z$ is the solution of the adjoint
system (3.1).

###### Proof.

From (3.2), we have

$\int_{0}^{T}|\partial_{x}z(t,0)|^{2}dt=\int_{0}^{T}\Big{|}\sum\limits_{n\in\mathbb{N}^{*}}c_{n}e^{i{\lambda}_{n}t}\Phi_{n}^{\prime}\left(0\right)\Big{|}^{2}dt,\leavevmode\nobreak\
\forall T>0.$ (3.4)

Since $\displaystyle\gamma\in\Gamma^{*}$, then by Theorem 2.2 and the gap
condition (2.13), Beurling’s Lemma (e.g., [2]) states that, for any
$\displaystyle T>2\pi D^{+}\left(\lambda_{n}\right)$, the family
$\displaystyle\left(e^{i\lambda_{n}t}\right)_{n\in\mathbb{N}^{*}}$ forms a
Riesz basis of $\displaystyle L^{2}(0,T)$, where $\displaystyle
D^{+}\left(\lambda_{n}\right):=\displaystyle\lim_{r\to\infty}\frac{n^{+}\left(r,\lambda_{n}\right)}{r}$
is the Beurling upper density of the sequence
$\displaystyle(\lambda_{n})_{n\in\mathbb{N}^{*}},$ and $\displaystyle
n^{+}\left(r,\lambda_{n}\right)$ denotes the maximum number of terms of the
sequence $\displaystyle(\lambda_{n})_{n\in\mathbb{N}^{*}}$ contained in an
interval of length $\displaystyle r$. Therefore, from relation (3.4), we
deduce that for every $\displaystyle T>2\pi D^{+}\left(\lambda_{n}\right),$
there exists a positive constant $\displaystyle C_{T}>0$, depending on
$\displaystyle T$, such that

$C_{T}^{-1}\sum\limits_{n\in\mathbb{N}^{*}}\left|c_{n}\Phi^{\prime}_{n}(0)\right|^{2}\leq\int_{0}^{T}|\partial_{x}z(t,0)|^{2}dt\leq
C_{T}\sum\limits_{n\in\mathbb{N}^{*}}\left|c_{n}\Phi^{\prime}_{n}(0)\right|^{2}.$
(3.5)

By relations (2.2) and (2.6), we find that $\displaystyle
D^{+}\left(\lambda_{n}\right)=0.$ It is easy to check from relations
(2.6)-(2.7) and (2.12), that the eigenfunctions
$\displaystyle\left(\Phi_{n}\right)_{n\in\mathbb{N}^{*}}$ satisfy

$\Phi_{n}^{\prime}(\ell)\not=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\forall n\in\mathbb{N}^{*}.$ (3.6)

Moreover, using the second statement of Corollary 2.1, one has

$\displaystyle
C^{-1}\sqrt{\lambda_{n}}\leq\left|\Phi^{\prime}_{n}(0)\right|^{2}\leq
C\sqrt{\lambda_{n}},\leavevmode\nobreak\ \hbox{ as }n\to\infty,$

for some positive constant $\displaystyle C>0$. Therefore, by relation (3.5),
for any $\displaystyle T>0$, one gets

$C_{T}^{-1}\sum\limits_{n\in\mathbb{N}^{*}}\sqrt{\left|\lambda_{n}\right|}\left|c_{n}\right|^{2}\leq\int_{0}^{T}|\partial_{x}z(t,0)|^{2}dt\leq
C_{T}\sum\limits_{n\in\mathbb{N}^{*}}\sqrt{\left|\lambda_{n}\right|}\left|c_{n}\right|^{2},$
(3.7)

for some new constant $\displaystyle C_{T}>0$. Thus, by relations (2.15) and
(2.16), we get (3.3). ∎

##  4 Proof of Theorem 1.1

In this section, we prove our main controllability result, which is given by
Theorem 1.1. Let us first mention that the control problem (1.1) is well posed
in the sense of ”transposition”, see [12]. More precisely, we have:

######  Proposition 4.3.

Let $\displaystyle T>0$, and $\displaystyle f\in L^{2}(0,T)$. Then for any
$\displaystyle y_{0}\in H^{-1}(0,\ell)$, there exists a unique weak solution
of System (1.1) in the sense of transposition, satisfying

$y\in C\left([0,T];H^{-1}(0,\ell)\right).$ (4.1)

Moreover, there exists a constant $\displaystyle C(T)>0$ such that

$\|y\|_{L^{\infty}\left([0,T];H^{-1}(0,\ell)\right)}\leq
C(T)\left(\|y_{0}\|_{H^{-1}(0,\ell)}+\|f\|_{L^{2}(0,T)}\right).$ (4.2)

For the proof of this Proposition, the reader is referred to [2, 15]. In that
papers, the result is proven for the linear fourth-order Schrödinger equation
with clamped boundary conditions and boundary control acts on the first
spatial derivative at one endpoint, but the proof can easily be adapted to our
setting. We are now ready to prove Theorem 1.1.

###### Proof of Theorem 1.1.

Since, System (1.1) is linear and reversible in time, then, by applying the
Lions’ HUM [12], the control problem is reduced to the obtention of suitable
observability inequalities for the adjoint system (3.1), that is,

$C_{T}^{-1}\|z_{0}\|_{H^{1}_{0}(0,\ell)}^{2}\leq\int_{0}^{T}|\partial_{x}z(t,0)|^{2}dt\leq
C_{T}\|z_{0}\|_{H^{1}_{0}(0,\ell)}^{2}.$ (4.3)

for some positive constant $\displaystyle C_{T}>0$, depending on
$\displaystyle T,$ where $\displaystyle z$ is the solution of the adjoint
system (3.1). Let $\displaystyle\gamma\not\in\Gamma^{*},$ where
$\displaystyle\Gamma^{*}$ is given by relation (1.2). As a direct application
of Proposition 3.2, both inequalities in (4.3) hold for all $\displaystyle
T>0.$ This implies that System (1.1) is exactly controllable in time
$\displaystyle T>0$ for any initial data $\displaystyle y_{0}\in
H^{-1}(0,\ell)$. Now, let $\displaystyle\gamma\in\Gamma^{*}$ and let
$\displaystyle z$ be the solution of the adjoint system (3.1) with initial
data

$\displaystyle
z_{0}:=\Phi_{n}(x)=\sqrt{\frac{\ell}{2}}{\phi(x,\lambda_{n})},\leavevmode\nobreak\
\hbox{ for some }n\leq n_{0}=\left[\ell{\pi}^{-1}{\sqrt{|\gamma|}}\right]-1,$

where the eigenfunctions $\displaystyle\phi(x,\lambda_{n})$ are given by
(2.11). By Theorem 2.2, the eigenvalues
$\displaystyle\left(\lambda_{n}\right)_{n\leq n_{0}}$ of Problem (2.1) are of
multiplicity two, and this implies that,
$\displaystyle\partial_{\lambda}\phi(x,\lambda)_{|_{\lambda_{n}}}=0,$ for each
fixed $\displaystyle x\in\left(0,\ell\right)$. Hence, by relations (3.2),
(2.10) and the orthogonality properties of the eigenfunctions, we deduce that
the solution $\displaystyle z$ of the adjoint system (3.1) has the form

$\displaystyle{z}(t,x)=\sqrt{\frac{\ell}{2}}{e^{i\lambda_{n}t}}\left(\bar{\xi}(\lambda_{n})\sin(x\xi(\lambda_{n}))-\xi(\lambda_{n})\sin(x\bar{\xi}(\lambda_{n}))\right),\hbox{
for some }n\leq n_{0},$

where $\displaystyle\xi(\lambda)$ and $\displaystyle\bar{\xi}(\lambda)$ are
given by (2.8). Therefore, $\displaystyle\partial_{x}{z}(t,0)=0$.
Consequently, the right hand side of the first inequality in (4.3) is zero,
while, the left hand side is not zero. Thus, the first inequality in (4.3)
cannot be valid. The proof is then complete. ∎

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