# LQR control for a system
describing the interaction between a floating solid and the surrounding fluid

Marius Tucsnak  Institut de Mathématiques de Bordeaux UMR 5251, Université de
Bordeaux/Bordeaux INP/CNRS, 351 Cours de la Libération, 33 405 TALENCE, France
Zhuo Xu  Institut de Mathématiques de Bordeaux UMR 5251, Université de
Bordeaux/Bordeaux INP/CNRS, 351 Cours de la Libération, 33 405 TALENCE, France

(Dedicated to professor for Hélène Frankowska for her 70th anniversary)

###### Abstract

This paper studies an infinite time horizon LQR optimal control problem for a
system describing, within a linear approximation, the vertical oscillations of
a floating solid, coupled to the motion of the free boundary fluid on which it
floats. The fluid flow is described by a viscous version of the linearized
Saint-Venant equations (shallow water regime). The major difficulty we are
facing is that the domain occupied by the fluid is unbounded so that the
system is not exponentially stable. This fact firstly raises challenges in
proving the wellposedness, requiring the combined use of analytic semigroup
theory and of an interpolation technique. The main contribution of the paper
is that we show that, in spite of the lack of exponential stabilizability, we
can define a wellposed LQR problem for which a Riccati based approach to
design feedback controls can be implemented.

Key words: floating solid; shallow water equations; analytic semigroup; strong
stability; Riccati equation.

AMS subject classification: 93C20, 93B52, 35Q35, 49J21.

###### Contents

  1. 1 Introduction
  2. 2 Some background on LQR problems
  3. 3 Operator form of the governing equations and semigroup generator
  4. 4 Semigroup generation and well-posedness
  5. 5 The optimal control problem
    1. 5.1 Feedback design
    2. 5.2 Proof of Theorem 1.2
  6. 6 Appendix
    1. 6.1 Appendix A
    2. 6.2 Appendix B

##  1 Introduction

In this paper we consider a coupled PDEs-ODEs system describing the motion of
a floating solid in a free surface viscous fluid which fills the remaining
part of the space. The considered system can be seen as a simplified model for
a point absorber, one on the most popular models of wave energy converters,
see Bocchi [3], Falnes and Johannes [6], Haak, Maity, Takahashi and Tucsnak
[7]. More generally, problems describing the interaction of floating or
immersed bodies with the fluid surrounding them have been studied, in
particular, in John [8], [9], Lannes [12], Maity, Martín, Takahashi and
Tucsnak [14], Maity and Tucsnak [15].

In the present work, the fluid flow is supposed to be viscous and to obey the
shallow water regime. The corresponding system (without inputs) has been
considered in [14] in the case when the fluid-solid system is confined in a
bounded container. The novelty we bring in this work is, besides introducing
an input function, which we consider the situation in which the fluid is not
bounded (of obvious interest when we aim at describing a point absorber
situated in the ocean) and that we discuss an optimal control problem, with an
input acting on the solid. One of the main difficulties which have to be
tackled when solving this problem is that the semigroup corresponding to the
system with no inputs is not exponentially stable. This type of control is of
interest for wave energy converters in order to synchronize the oscillations
of the rigid floating object and of the incoming waves, see Korde and Ringwood
[10].

More precisely, we are concerned with an infinite horizon LQR optimal control
problem for the system obtained by linearizing the equations which describe
the interaction of a floating solid with the fluid on which it floats. This
fluid is assumed to be infinite in the horizontal direction. This fact
contrasts with the situation described in the previous paper [14], where the
fluid was supposed to be in a bounded container. The extension of this model
to an unbounded fluid was first considered in Vergara-Hermosilla, Matignon and
Tucsnak [19], where input-output stability questions have been investigated.

To be more specific, we suppose that the floating solid has vertical walls and
it undergoes only vertical motions, see Figure 1. In this case the projection
of the solid on the fluid bottom (supposed to be horizontal) does not depend
on time. We denote by $\mathcal{I}=[-a,a]$, with $a>0$, the projection on the
fluid bottom of the solid domain and we set
$\mathcal{E}=\mathbb{R}\verb|\|[-a,a]$. We designate by $\mu>0$ the viscosity
coefficient of the fluid. The solid is supposed to have unit mass and it is
constrained to move only in the vertical direction. We denote by $q(t,x)$ the
flux at instant $t$ through the section $x$, by $h(t,x)$ the height of the
free surface of the fluid at the point of abscissa $x$ at instant $t$, whereas
$H(t)$ stands for the distance from the bottom of rigid body to the sea bottom
at instant $t$. Moreover, $p(t,x)$, designs the pressure under the rigid body
(which can be seen as a Lagrange multiplier, see [14, Section 2]. The input
signal $u$ designs the variation with respect to time of the vertical force
exerted by the controller on the rigid body. With the above notation, the
considered system can be written, see [14], [19], in the form:

$\displaystyle\frac{\partial h}{\partial t}+\frac{\partial q}{\partial x}=0$
$\displaystyle($ $\displaystyle t>0,\;x\in\mathbb{R}),$ (1.1)
$\displaystyle\frac{\partial q}{\partial t}+\frac{\partial h}{\partial
x}-\mu\frac{\partial^{2}q}{\partial x^{2}}=0$ $\displaystyle($ $\displaystyle
t>0,\;x\in\mathcal{E}),$ (1.2) $\displaystyle H(t)=h(t,x)$ $\displaystyle($
$\displaystyle t>0,\;x\in\mathcal{I}),$ (1.3) $\displaystyle p(t,x)=0$
$\displaystyle($ $\displaystyle t>0,\;x\in\mathcal{E}),$ (1.4) $\displaystyle
h\left(t,-a^{-}\right)-\mu\frac{\partial q}{\partial
x}(t,-a^{-})=p\left(t,-a^{+}\right)+H(t)-\mu\frac{\partial q}{\partial
x}\left(t,-a^{+}\right)$ $\displaystyle($ $\displaystyle t>0),$ (1.5)
$\displaystyle h\left(t,a^{+}\right)-\mu\frac{\partial q}{\partial
x}\left(t,a^{+}\right)=p\left(t,a^{-}\right)+H(t)-\mu\frac{\partial
q}{\partial x}(t,a^{-})$ $\displaystyle($ $\displaystyle t>0),$ (1.6)
$\displaystyle\ddot{H}(t)=\int_{-a}^{a}p(t,x){\rm d}x+u(t)$ $\displaystyle($
$\displaystyle t>0),$ (1.7) $\displaystyle\frac{\partial q}{\partial
t}+\frac{\partial p}{\partial x}=0$ $\displaystyle($ $\displaystyle
t>0,\;x\in\mathcal{I}),$ (1.8) $\displaystyle
q(t,-a^{-})=q\left(t,-a^{+}\right)=q(t,-a),\quad
q\left(t,a^{+}\right)=q(t,a^{-})=q(t,a)$ $\displaystyle($ $\displaystyle
t>0),$ (1.9) $\displaystyle\lim_{|x|\rightarrow\infty}q(t,x)=0$
$\displaystyle($ $\displaystyle t>0),$ (1.10) $\displaystyle
H(0)=H_{0},\qquad\dot{H}(0)=G_{0},$ (1.11) $\displaystyle h(0,x)=h_{0}(x)$
$\displaystyle($ $\displaystyle x\in\mathcal{E}),$ (1.12) $\displaystyle
q(0,x)=q_{0}(x)$ $\displaystyle($ $\displaystyle x\in\mathbb{R}),$ (1.13)

where

$f\left(a^{+}\right)=\lim_{x\rightarrow a\atop x>a}f(x),\qquad
f(a^{-})=\lim_{x\rightarrow a\atop x<a}f(x).$

Our first main result concerns the well-posedness of above system.

######  Theorem 1.1.

Let $T>0$. Then for every $\left[\begin{gathered}H_{0}\ G_{0}\ h_{0}\
q_{0}\end{gathered}\right]^{\top}\in\mathbb{R}^{2}\times
H^{1}(\mathcal{E})\times H^{1}(\mathbb{R})$ satisfying the compatibility
condition

$q_{0}(x)=-G_{0}x+\frac{q_{0}(a)+q_{0}(-a)}{2}\qquad\quad(x\in\mathcal{I}),$
(1.14)

and every $u\in L^{2}(0,\infty)$, there exists a unique solution
$\left[\begin{gathered}H\ h\ q\ p\end{gathered}\right]^{\top}$ of (1.1)-(1.13)
satisfying

$\displaystyle H\in H^{2}(0,T),\quad h\in
H^{1}\left((0,T);\;H^{1}(\mathcal{E})\right),\quad p\in
L^{2}\left((0,T);\;\mathcal{P}_{2}(\mathcal{I})\right),$ (1.15) $\displaystyle
q\in H^{1}\left((0,T);\;L^{2}(\mathbb{R})\right)\cap
C\left([0,T];\;H^{1}(\mathbb{R})\right),\quad q\in
H^{1}\left((0,T);\;\mathcal{P}_{1}(\mathcal{I})\right),$ (1.16) $\displaystyle
q\in L^{2}\left((0,T);\;H^{2}(\mathcal{E})\right).$ (1.17)

where $\mathcal{P}_{1}(\mathcal{I})$ and $\mathcal{P}_{2}(\mathcal{I})$ stand
for the spaces formed by first and second degree polynomials on $\mathcal{I}$,
respectively. Moreover, this solution depends continuously on the control
function and on the initial data.

Figure 1: The body floating in an unbounded fluid

We next consider the optimal control problem associated to the system
(1.1)-(1.13) which consists in minimizing the cost functional
$J:L^{2}(0,\infty)\to\mathbb{R}\cup\\{+\infty\\}$ defined by

$\displaystyle
J(u)=\int_{0}^{\infty}\left(\left|u(t)\right|^{2}+\left|\dot{H}(t)\right|^{2}\right)\,{\rm
d}t\qquad\qquad\left(u\in L^{2}(0,\infty)\right).$ (1.18)

At this stage, our second main result can be stated as follows:

######  Theorem 1.2.

For every initial data satisfying the assumptions in Theorem 1.1, there exists
a unique $\bar{u}\in L^{2}(0,\infty)$ such that $J(\bar{u})<J(u)$ for every
$u\in L^{2}(0,\infty)$, $u\neq\bar{u}$. Moreover, $\bar{u}$ can be written in
feedback form.

A more detailed version of the above result, including details on the feedback
design, will be given, after introducing the necessary notation, in Theorem
5.7.

As for wellposedness, the main difficulty in the proof of Theorem 1.2 is that
system (1.1)-(1.13) is not exponentially stable and even not exponentially
stabilizable. This fact does not allow the application of the standard theory
of infinite horizon LQR problems. Nevertheless, we show below that our system
is strongly stable and output stabilizable, which entitles us to use the
theory developed by Curtain and Oostveen [4] for this type of systems.

The remaining part of this work is organized as follows. In Section 2 we
provide some necessary background on LQR problems for infinite dimensional
systems. In Sections 3 and 4 we give the proof of Theorem 1.1. Finally,
Section 5 is devoted to the proof of Theorem 1.2.

##  2 Some background on LQR problems

As shown below in Section 5, equations (1.1)-(1.13) do not define an
exponentially stabilizable system, so that the standard way to prove that the
system is optimizable and to develop a Riccati type theory (see, for instance,
Curtain and Zwart [5, Section 6.2]) does not apply. We will instead establish
directly the optimizability by ad-hoc arguments and use the LQR theory for
strongly stable systems developed in Curtain and Oostveen [4], see also
Bensoussan, Prato, Delfour and Mitter [2] or Oostveen, Curtain and Ito [16].
For the reader’s convenience, in this section we provide, following the
references above, the necessary background on infinite horizon LQR problems,
with focus on systems which are strongly stable.

Let $X$, $U$ and $Y$ be Hilbert spaces. We consider the operators
$A:\mathcal{D}(A)\to X$, which generates a $C^{0}$ semigroup $\mathbb{T}$ on
$X$, $B\in\mathcal{L}(U,X)$ and $C\in\mathcal{L}(X,Y)$. We consider the linear
system $\Sigma(A,B,C)$, of state space $X$, input space $U$ and output space
$Y$, described by the equations

$\displaystyle\dot{z}(t)=Az(t)+Bu(t)$ $\displaystyle($ $\displaystyle t>0),$
(2.1) $\displaystyle z(0)=z_{0},$ (2.2) $\displaystyle y(t)=Cz(t)$
$\displaystyle($ $\displaystyle t>0),$ (2.3)

with $z_{0}$ arbitrarily chosen in $X$. The infinite horizon LQR problem for
(2.1)-(2.3), consists in minimizing the cost functional

$J(u)=\int_{0}^{\infty}\left(\left\|Cz(t)\right\|^{2}_{Y}+\left\|u(t)\right\|^{2}_{U}\right)\,{\rm
d}t\qquad\qquad(u\in L^{2}\left((0,\infty);\;U)\right),$ (2.4)

over $L^{2}((0,\infty);\;U)$. The above problem makes sense for every
$z_{0}\in X$ if and only if the system is optimizable, i.e., if for every
$z_{0}\in X$ there exists $\overline{u}\in L^{2}\left((0,\infty);\;U\right)$
such that $J(\overline{u})<\infty$. According to a classical result, see for
instance, [5, Theorem 6.2.4], under the appropriate assumptions (including
optimizability) the considered LQR problem admits a unique solution $u^{*}\in
L^{2}((0,\infty);\;U)$ and there exists a self-adjoint, nonnegative operator
$\Pi\in\mathcal{L}(X)$ such that $J(u^{*})=\left\langle z_{0},\Pi
z_{0}\right\rangle_{X}$. Under supplementary assumptions (which imply, in
particular, that $(A,B)$ is exponentially stabilizable), the operator $\Pi$
can be derived from the solution of an algebraic Riccati equation (see [5,
Theorem 6.2.7]). As already mentioned, exponential stabilizability does not
hold for the system we consider in the forthcoming sections. Therefore, in
order to apply a Riccati type theory, we use the approach developed in [2],
[4] and [16], for systems which are only strongly stable. To state the result,
we need the following definition.

######  Definition 2.1.

For every $t>0$, the output map $\Psi_{t}:$ ${X}\to L^{2}((0,\infty);Y)$ is
defined by

$(\Psi_{t}z_{0})(\sigma)=\begin{cases}C\mathbb{T}(\sigma)z_{0}&\qquad\qquad(\sigma\in(0,t],\
z_{0}\in X),\\\ 0&\qquad\qquad(\sigma>t,\ z_{0}\in X).\end{cases}$

The system (2.1)-(2.3) is called output stable if there exists
$\Psi\in\mathcal{L}\left({X},\;L^{2}((0,\,\infty);\;Y)\right)$ such that

$\lim_{t\to\infty}\|\Psi_{t}-\Psi\|_{\mathcal{L}\left(X,\;L^{2}((0,\,\infty);\;Y)\right)}=0.$

######  Remark 2.2.

According to the terminology introduced in Tucsnak and Weiss [18, Section
4.6], if operators $A$ and $C$ are as in the above definition, we say that $C$
is an infinite-time admissible observation operator for $\mathbb{T}$ and we
design $\Psi$ as the extended output map.

An essential tool in proving our second main result is the proposition below,
for which we refer again to [4].

######  Proposition 2.3.

Suppose that there exists an operator $F\in\mathcal{L}(X,U)$ such that the
operator $A+BF$ generates a strongly stable semigroup $\mathbb{T}^{F}$ on $X$
and with the system $\Sigma\left(A+BF,\,0,\,\begin{bmatrix}C\\\
F\end{bmatrix}\right)$ (with state space $X$, input space $U$ and output space
$Y\times U$) output stable. Then there exist a unique optimal control
$\bar{u}\in L^{2}((0,\infty);\;U)$ minimizing (2.4). Moreover $\bar{u}$ is
given by $\bar{u}(t)=-B^{*}P_{0}z(t)$, where $P_{0}\in\mathcal{L}(X)$ is the
minimal self-adjoint nonnegative solution to the algebraic Riccati equation

${A}^{*}Pz+P{A}z-P{B}{B}^{*}Pz+{C}^{*}{C}z=0\qquad\qquad(z\in\mathcal{D}(A)).$

Finally, the corresponding minimal cost functional is given by

$J(\bar{u},\,z_{0})=\left<P_{0}z_{0},\,z_{0}\right>_{X},$

where $\left<\cdot,\,\cdot\right>_{X}$ is the inner product in the Hilbert
space ${X}$.

######  Remark 2.4.

The sense which $P$ satisfies the Riccati equation above is described in [2,
Part IV 3.1, p.398 and Part V, p.481], where it is shown, in particular, that
$Pz\in\mathcal{D}(A^{*})$ for every $z\in\mathcal{D}(A)$.

##  3 Operator form of the governing equations and semigroup generator

In this section, we show that system (1.1)-(1.13) can be written in the
standard form

$\displaystyle\dot{z}(t)=\mathcal{A}z(t)+\mathcal{B}u(t)\qquad(t>0),$ (3.1)
$\displaystyle z(0)=z_{0},$ (3.2)

where $\mathcal{A}$ is the generator of an analytic semigroup on a Hilbert
space $\mathcal{X}$ and $\mathcal{B}\in\mathcal{L}(\mathcal{U},\mathcal{X})$,
where $\mathcal{U}$ is another Hilbert space.

To write system (1.1)-(1.13) in the form (3.1)-(3.2) we follow the procedure
proposed in [14], where the corresponding system, with the fluid confined in a
bounded container, has been investigated.

We begin by setting

$q_{-}(t)=q(t,-a),\qquad q_{+}(t)=q(t,a)\qquad\qquad(t>0).$ (3.3)

Next we remark that the methodology developed in [14] to eliminate the
pressure term $p$ can be applied with no major modification to our case. In
this way we obtain the following system, equivalent to the original governing
equations (1.1)-(1.13):

$\displaystyle\dot{H}(t)=-\frac{q_{+}(t)-q_{-}(t)}{2a}$ $\displaystyle($
$\displaystyle t>0),$ (3.4) $\displaystyle\frac{\partial h}{\partial
t}=-\frac{\partial q}{\partial x}$ $\displaystyle($ $\displaystyle
t>0,\;x\in\mathcal{E}),$ (3.5) $\displaystyle\frac{\partial q}{\partial
t}=-\frac{\partial h}{\partial x}+\mu\frac{\partial^{2}q}{\partial x^{2}}$
$\displaystyle($ $\displaystyle t>0,\;x\in\mathcal{E}),$ (3.6)
$\displaystyle\left[\begin{gathered}\dot{q}_{-}(t)\\\
\dot{q}_{+}(t)\end{gathered}\right]=2aM\left[\begin{gathered}u(t)\\\
-u(t)\end{gathered}\right]$ (3.11)
$\displaystyle+4a^{2}M\left[\begin{gathered}\mu\left(\frac{q_{+}(t)-q_{-}(t)}{2a}\right)+\left(h(t,-a^{-})-H(t)\right)-\mu\frac{\partial
q}{\partial x}(t,-a^{-})\\\
-\mu\left(\frac{q_{+}(t)-q_{-}(t)}{2a}\right)-\left(h(t,a^{+})-H(t)\right)+\mu\frac{\partial
q}{\partial x}\left(t,a^{+}\right)\end{gathered}\right]$ $\displaystyle($
$\displaystyle t>0),$ (3.14)
$\displaystyle\lim_{|x|\rightarrow\infty}q(t,x)=0$ $\displaystyle($
$\displaystyle t>0),$ (3.15) $\displaystyle H(0)=H_{0},$ (3.16) $\displaystyle
h(0,x)=h_{0}(x)$ $\displaystyle($ $\displaystyle x\in\mathcal{E}),$ (3.17)
$\displaystyle q(0,x)=q_{0}(x)$ $\displaystyle($ $\displaystyle
x\in\mathcal{E}),$ (3.18) $\displaystyle q_{-}(0)=q_{-,0},\qquad
q_{+}(0)=q_{+,0},$ (3.19)

where the matrix $M$ is defined by

$\displaystyle
M=\frac{1}{8a^{3}\left(1+\frac{2a^{3}}{3}\right)}\begin{bmatrix}1+\frac{8a^{3}}{3}&&1-\frac{4a^{3}}{3}\\\
1-\frac{4a^{3}}{3}&&1+\frac{8a^{3}}{3}\end{bmatrix}.$ (3.20)

At this stage we introduce more notation, which will be used in all the
remaining part of this paper. Firstly, we define the spaces

$\displaystyle\mathcal{X}=\left\\{[H\,\;h\,\;q\,\;q_{-}\,\;q_{+}]^{\top}\in\mathbb{C}\times
H^{1}(\mathcal{E})\times
L^{2}(\mathcal{E})\times\mathbb{C}\times\mathbb{C}\right\\},$ (3.21)
$\displaystyle\mathcal{Z}=\left\\{[H\,\;h\,\;q\,\;q_{-}\,\;q_{+}]^{\top}\in\mathbb{C}\times
H^{1}(\mathcal{E})\times
H^{2}(\mathcal{E})\times\mathbb{C}\times\mathbb{C}\;|\;q(-a)=q_{-},\,q(a)=q_{+}\right\\},$
(3.22)
$\displaystyle\mathcal{W}=\left\\{[H\,\;h\,\;q\,\;q_{-}\,\;q_{+}]^{\top}\in\mathbb{C}\times
H^{1}(\mathcal{E})\times
H^{1}(\mathcal{E})\times\mathbb{C}\times\mathbb{C}\;|\;q(-a)=q_{-},\,q(a)=q_{+}\right\\},$
(3.23) $\displaystyle\mathcal{U}=\mathbb{C}.$ (3.24)

Moreover, denoting by $z=[H\,\;h\,\;q\,\;q_{-}\,\;q_{+}]^{\top}$ a generic
element of $\mathcal{X}$, we also introduce the operators $\mathcal{A}:$
$\mathcal{D}(\mathcal{A})\rightarrow{\mathcal{X}}$ and
$\mathcal{B}\in\mathcal{L}(\mathcal{U},{\mathcal{X}})$ defined by

$\mathcal{D}(\mathcal{A})=\mathcal{Z},$ (3.25)
$\displaystyle\mathcal{A}z=\left[\begin{gathered}-\frac{q_{+}-q_{-}}{2a}\\\
-\frac{{\rm d}q}{{\rm d}x}\\\ -\frac{{\rm d}h}{{\rm d}x}+\mu\frac{{\rm
d}^{2}q}{{\rm d}x^{2}}\\\ R_{1}z\\\
R_{2}z\end{gathered}\right],\qquad\qquad\mathcal{B}u=\left[\begin{gathered}0\\\
0\\\ 0\\\ \frac{au}{(1+\frac{2a^{3}}{3})}\\\
-\frac{au}{(1+\frac{2a^{3}}{3})}\end{gathered}\right]\qquad(z\in\mathcal{D}(\mathcal{A}),\
u\in\mathcal{U}),$ (3.36)

where

$\displaystyle\left[\begin{gathered}R_{1}z\\\
R_{2}z\end{gathered}\right]=4a^{2}M\left[\begin{gathered}\mu\left(\frac{q_{+}-q_{-}}{2a}\right)+\left(h(-a^{-})-H\right)-\mu\frac{{\rm
d}q}{{\rm d}x}(-a^{-})\\\
-\mu\left(\frac{q_{+}-q_{-}}{2a}\right)-\left(h(a^{+})-H\right)+\mu\frac{{\rm
d}q}{{\rm d}x}\left(a^{+}\right)\end{gathered}\right].$ (3.41)

It is easy to check that $\mathcal{B}\in\mathcal{L}(\mathcal{U},\mathcal{X})$.

With the above notation, following step by step the procedure in [14], we
have:

######  Proposition 3.1.

Let $T>0$ and let $\left[\begin{gathered}H_{0}\ G_{0}\ h_{0}\
q_{0}\end{gathered}\right]^{\top}$ satisfies the assumptions in Theorem 1.1.
Let $\left[\begin{gathered}H\ h\ q\ p\end{gathered}\right]^{\top}$ be a
solution of (1.1)-(1.13) satisfying (1.15)-(1.17). Then
$z=[H\,\;h\,\;q\,\;q_{-}\,\;q_{+}]^{\top}$, with $q_{+}$ and $q_{-}$ defined
in (3.3), lies in

$C([0,T];\;\mathcal{W})\cap L^{2}((0,T);\;\mathcal{Z})\cap
H^{1}((0,T);\;\mathcal{X}),$ (3.42)

and it satisfies

$\displaystyle\dot{z}(t)=\mathcal{A}z(t)+\mathcal{B}u(t)\qquad\qquad(t>0),$
(3.43) $\displaystyle z(0)=z_{0}.$ (3.44)

Conversely, assume that $z$ satisfies (3.42)-(3.44). Then there exists an
extensions of $q$ (still denoted by $q$) and $p$ such that
$\left[\begin{gathered}H\ h\ q\ p\end{gathered}\right]^{\top}$ is a solution
of (1.1)-(1.13) satisfying (1.15)-(1.17).

In this section and in the following one we prove that operator $\mathcal{A}$
generates an analytic semigroup on $\mathcal{X}$ and we give the first
consequences of this result on the wellposedness of system (3.4)-(3.19). The
remaining part of this section is devoted to the study of the spectrum and of
the resolvents of the operator $\mathcal{A}$ defined above.

As usual, we denote by $\mathcal{\sigma}(\mathcal{A})$ and
$\mathcal{\rho}(\mathcal{A})$ the spectrum and the resolvent set of operator
$\mathcal{A}$, respectively. We also set

$\mathbb{C}_{-}=\left\\{{\lambda\in\mathbb{C}\;\,|\;\,{\rm
Re}\,\lambda<0}\right\\},\qquad\mathbb{R}_{-}=\mathbb{C}_{-}\cap\mathbb{R}.$
(3.45)

In order to state the main theorem of this section, we need four technical
lemmas. Although based on elementary facts, the proofs of these four lemmas
require a certain computational effort, so that they are postponed to the
appendix in subsection 6.1. The first one of these lemmas is:

######  Lemma 3.2.

Assume that $\mu>0$ and $\lambda\in\mathbb{C}\verb|\|\\{0\\}$,
$\lambda\neq-\frac{1}{\mu}$. Then
$\frac{\lambda^{2}}{1+\mu\lambda}\in\mathbb{R}_{-}$ if and only if

$\lambda\in\left(-\infty,-\frac{1}{\mu}\right)\quad{\rm
or}\quad\left|\lambda+\frac{1}{\mu}\right|=\frac{1}{\mu}\ .$ (3.46)

The second auxiliary result to be proved in the Appendix A is:

######  Lemma 3.3.

Let $a>0$ and $\mu>0$. We assume that $\lambda\in\mathbb{C}$ satisfies

$\lambda\not\in\left(-\infty,-\frac{1}{\mu}\right]\quad{\rm
and}\quad\left|\lambda+\frac{1}{\mu}\right|\neq\frac{1}{\mu}\ .$ (3.47)

Let

$\omega_{\lambda}:=\sqrt{\frac{\lambda^{2}}{1+\mu\lambda}}\,,$ (3.48)

where $\sqrt{\cdot}$ stands for the principal determination of the square root
function. (Note that this definition is correct due to Lemma 3.2.)

Let $\mathcal{M}_{\lambda}$ be defined by

$\displaystyle\mathcal{M}_{\lambda}=\begin{bmatrix}\lambda\left(1+\frac{8a^{3}}{3}\right)+2a\left(\mu+\frac{1}{\lambda}\right)+\frac{4a^{2}\lambda}{\omega_{\lambda}}&\
&-\lambda\left(1-\frac{4a^{3}}{3}\right)-2a\left(\mu+\frac{1}{\lambda}\right)\\\
-\lambda\left(1-\frac{4a^{3}}{3}\right)-2a\left(\mu+\frac{1}{\lambda}\right)&\
&\lambda\left(1+\frac{8a^{3}}{3}\right)+2a\left(\mu+\frac{1}{\lambda}\right)+\frac{4a^{2}\lambda}{\omega_{\lambda}}\end{bmatrix}.$
(3.49)

Then, there exists a set $\mathcal{S}\subset\mathbb{C}_{-}$, having at most
four elements, such that for every $\lambda$ satisfying (3.47) and not lying
in $\mathcal{S}$ we have that the matrix $\mathcal{M}_{\lambda}$ is
invertible.

An important role in proving our resolvent estimates is played by the
following result:

######  Lemma 3.4.

Let $\omega\in\mathbb{C}$, ${\rm Re}\,\omega>0$ and $a>0$. We define the
following operators:

$\displaystyle\left(D_{-}(\omega)\gamma_{-}\right)(x)=\gamma_{-}\exp[\omega(a+x)]$
$\displaystyle($ $\displaystyle\gamma_{-}\in\mathbb{C},\quad x\leqslant-a),$
(3.50)
$\displaystyle\left(D_{+}(\omega)\gamma_{+}\right)(x)=\gamma_{+}\exp[\omega(a-x)]$
$\displaystyle($ $\displaystyle\gamma_{+}\in\mathbb{C},\quad x\geqslant a),$
(3.51) $\displaystyle\left(R_{-}(\omega)\varphi\right)(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2\omega}\left(\int_{x}^{-a}\exp(-\omega\xi)\varphi(\xi){\rm
d}\xi\right)\exp(\omega
x)+\frac{1}{2\omega}\left(\int_{-\infty}^{x}\exp(\omega\xi)\varphi(\xi){\rm
d}\xi\right)\exp(-\omega x)$
$\displaystyle-\frac{1}{2\omega}\left(\int_{-\infty}^{-a}\exp(\omega\xi)\varphi(\xi){\rm
d}\xi\right)\exp(2\omega a)\exp(\omega x)\qquad\quad\left(\varphi\in
L^{2}(-\infty,-a),\;x\leqslant-a\right),$ (3.52)
$\displaystyle\left(R_{+}(\omega)\varphi\right)(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2\omega}\left(\int_{a}^{x}\exp(\omega\xi)\varphi(\xi){\rm
d}\xi\right)\exp(-\omega
x)+\frac{1}{2\omega}\left(\int_{x}^{\infty}\exp(-\omega\xi)\varphi(\xi){\rm
d}\xi\right)\exp(\omega x)$
$\displaystyle-\frac{1}{2\omega}\left(\int_{a}^{\infty}\exp(-\omega\xi)\varphi(\xi){\rm
d}\xi\right)\exp(2\omega a)\exp(-\omega x)\qquad\quad\left(\varphi\in
L^{2}(a,\infty),\;x\geqslant a\right).$ (3.53)

Then we have

$\displaystyle
D_{-}(\omega)\in\mathcal{L}\left(\mathbb{C},\;L^{2}(-\infty,-a)\right),$
$\displaystyle D$
${}_{+}(\omega)\in\mathcal{L}\left(\mathbb{C},\;L^{2}(a,\infty)\right),$
$\displaystyle R_{-}(\omega)\in\mathcal{L}\left(L^{2}(-\infty,-a)\right),$
$\displaystyle R$ ${}_{+}(\omega)\in\mathcal{L}\left(L^{2}(a,\infty)\right),$

with the norm estimates

$\displaystyle\|D_{-}(\omega)\|_{\mathcal{L}\left(\mathbb{C},\;L^{2}(-\infty,-a)\right)}\leqslant\left(\frac{1}{2{\rm
Re}\,\omega}\right)^{\frac{1}{2}},\quad$ $\displaystyle\|$ $\displaystyle
D_{+}(\omega)\|_{\mathcal{L}\left(\mathbb{C},\;L^{2}(a,\infty)\right)}\leqslant\left(\frac{1}{2{\rm
Re}\,\omega}\right)^{\frac{1}{2}},$ (3.54)
$\displaystyle\|R_{-}(\omega)\|_{\mathcal{L}(L^{2}(-\infty,-a))}\leqslant\frac{3}{2|\omega|{\rm
Re}\,\omega},\quad$ $\displaystyle\|$ $\displaystyle
R_{+}(\omega)\|_{\mathcal{L}(L^{2}(a,\infty))}\leqslant\frac{3}{2|\omega|{\rm
Re}\,\omega}.$ (3.55)

Finally, the structure of the resolvent of our semigroup generator will be
show to be a consequence of the following lemma:

######  Lemma 3.5.

Let $a>0$ and $\mu>0$ and assume that $\lambda\in\mathbb{C}$ satisfies
condition (3.47). Then for every $\varphi\in L^{2}(\mathcal{E})$ and every
$\gamma_{-}$, $\gamma_{+}\in\mathbb{C}$, there exists a unique $q_{\lambda}\in
H^{2}(\mathcal{E})$ such that

$\displaystyle\left\\{\begin{aligned} -&\frac{{\rm d}^{2}q_{\lambda}}{{\rm
d}x^{2}}+\omega^{2}_{\lambda}q_{\lambda}=\varphi(x)\qquad\qquad(x\in\mathcal{E}),\\\
&q_{\lambda}(-a)=\gamma_{-},\quad
q_{\lambda}(a)=\gamma_{+},\end{aligned}\right.$ (3.56)

where $\omega_{\lambda}$ is defined in (3.48). Moreover, we have

$\displaystyle q_{\lambda}(x)=\left\\{\begin{aligned}
&[D_{-}(\omega_{\lambda})\gamma_{-}](x)+[R_{-}(\omega_{\lambda})\varphi](x)&(&x\leqslant-a),\\\
\\\
&[D_{+}(\omega_{\lambda})\gamma_{+}](x)+[R_{+}(\omega_{\lambda})\varphi](x)&(&x\geqslant
a),\end{aligned}\right.$ (3.57)

where the operators $D_{+}(\omega_{\lambda})$, $D_{-}(\omega_{\lambda})$,
$R_{+}(\omega_{\lambda})$ and $R_{-}(\omega_{\lambda})$ have been defined in
Lemma 3.4.

The main result in this section is:

######  Theorem 3.6.

For every $a>0$ and $\mu>0$, there exists a set
$\mathcal{S}\subset\mathbb{C}_{-}$, having at most four elements, such that
the spectrum of the operator $\mathcal{A}$ defined in (3.21)-(3.41) satisfies

$\sigma(\mathcal{A})\subset E,$ (3.58)

where $E$ is defined by

$E=\\{0\\}\cup\mathcal{S}\cup\left(-\infty,-\frac{1}{\mu}\right]\cup\left\\{\lambda\in\mathbb{C}_{-}\quad\left|\quad\left|\lambda+\frac{1}{\mu}\right|=\frac{1}{\mu}\right\\}\right..$

Moreover, for every $\lambda\not\in E$,
$\mathcal{F}=[f_{1}\;\,f_{2}\;\,f_{3}\;\,f_{4}\;\,f_{5}]^{\top}\in\mathcal{X}$
we denote

$\displaystyle(\lambda\mathbb{I}-\mathcal{\mathcal{A}})^{-1}\mathcal{F}=z_{\lambda}=[H_{\lambda}\;\,h_{\lambda}\;\,q_{\lambda}\;\,q_{-,\lambda}\;\,q_{+,\lambda}]^{\top}\in\mathcal{D}(\mathcal{A}).$
(3.59)

Then we have

$\displaystyle\left[\begin{gathered}q_{-,\lambda}\\\
q_{+,\lambda}\end{gathered}\right]$ (3.62) $\displaystyle=$
$\displaystyle\mathcal{M}^{-1}_{\lambda}\left(M^{-1}\left[\begin{gathered}f_{4}\\\
f_{5}\end{gathered}\right]+\frac{4a^{2}}{\lambda}\left[\begin{gathered}f_{2}(-a^{-})-f_{1}\\\
f_{1}-f_{2}\left(a^{+}\right)\end{gathered}\right]+\frac{4a^{2}\lambda}{\omega^{2}_{\lambda}}\left[\begin{gathered}-\int_{-\infty}^{0}\exp(\omega_{\lambda}\xi)\varphi_{\lambda}(\xi-a){\rm
d}\xi\\\
\int_{0}^{\infty}\exp(-\omega_{\lambda}\xi)\varphi_{\lambda}(\xi+a){\rm
d}\xi\end{gathered}\right]\right),$ (3.69)

where

$\varphi_{\lambda}(x)=\frac{\lambda}{1+\mu\lambda}f_{3}(x)-\frac{1}{1+\mu\lambda}\frac{{\rm
d}f_{2}}{{\rm d}x}(x)\qquad\qquad(x\in\mathcal{E}),$

and $\omega_{\lambda}$, $\mathcal{M}_{\lambda}$ have been defined in (3.48)
and (3.49), respectively. In addition, we have

$\displaystyle q_{\lambda}(x)=\left\\{\begin{aligned}
&[D_{-}(\omega_{\lambda})q_{-,\lambda}](x)+[R_{-}(\omega_{\lambda})\varphi_{\lambda}](x)\quad\;\,&(&x\leqslant-a),\\\
&[D_{+}(\omega_{\lambda})q_{+,\lambda}](x)+[R_{+}(\omega_{\lambda})\varphi_{\lambda}](x)&(&x\geqslant
a),\end{aligned}\right.$ (3.70) $\displaystyle
h_{\lambda}(x)=\frac{1}{\lambda}\left(f_{2}(x)-\frac{{\rm
d}q_{\lambda}(x)}{{\rm
d}x}\right)\qquad\qquad\qquad\qquad\;\;\,(x\in\mathcal{E}),$ (3.71)
$\displaystyle
H_{\lambda}=\frac{1}{\lambda}\left(f_{1}-\frac{q_{+,\lambda}-q_{-,\lambda}}{2a}\right),$
(3.72)

where the operators $D_{-}(\omega_{\lambda})$, $D_{+}(\omega_{\lambda})$,
$R_{-}(\omega_{\lambda})$ and $R_{+}(\omega_{\lambda})$ have been defined in
Lemma 3.4.

###### Proof.

We firstly remark that the resolvent equation (3.59) can be written as:

$\displaystyle\lambda
H_{\lambda}+\frac{q_{+,\lambda}-q_{-,\lambda}}{2a}=f_{1},$ (3.73)
$\displaystyle\lambda h_{\lambda}(x)+\frac{{\rm d}q_{\lambda}}{{\rm
d}x}(x)=f_{2}(x)\qquad\qquad\qquad\quad\;(x\in\mathcal{E}),$ (3.74)
$\displaystyle\lambda q_{\lambda}(x)+\frac{{\rm d}h_{\lambda}}{{\rm
d}x}(x)-\mu\frac{{\rm d}^{2}q_{\lambda}}{{\rm
d}x^{2}}(x)=f_{3}(x)\qquad(x\in\mathcal{E}),$ (3.75) $\displaystyle
q_{\lambda}(-a)=q_{-,\lambda},\qquad\quad q_{\lambda}(a)=q_{+,\lambda},$
(3.76) $\displaystyle\lambda\left[\begin{gathered}q_{-,\lambda}\\\
q_{+,\lambda}\end{gathered}\right]-4a^{2}M\left[\begin{gathered}\mu\left(\frac{q_{+,\lambda}-q_{-,\lambda}}{2a}\right)+\left(h_{\lambda}(-a^{-})-H_{\lambda}\right)-\mu\frac{{\rm
d}q_{\lambda}}{{\rm d}x}(-a^{-})\\\
-\mu\left(\frac{q_{+,\lambda}-q_{-,\lambda}}{2a}\right)-\left(h_{\lambda}\left(a^{+}\right)-H_{\lambda}\right)+\mu\frac{{\rm
d}q_{\lambda}}{{\rm
d}x}\left(a^{+}\right)\end{gathered}\right]=\left[\begin{gathered}f_{4}\\\
f_{5}\end{gathered}\right],$ (3.83)

where the matrix $M$ has been defined in (3.20). From equations (3.74)-(3.76),
it follows that

$\displaystyle\left\\{\begin{aligned} -&\frac{{\rm d}^{2}q_{\lambda}}{{\rm
d}x^{2}}(x)+\omega^{2}_{\lambda}q_{\lambda}(x)=\frac{\lambda}{1+\mu\lambda}f_{3}(x)-\frac{1}{1+\mu\lambda}\frac{{\rm
d}f_{2}}{{\rm d}x}(x)\quad(x\in\mathcal{E}),\\\
&q_{\lambda}(-a)=q_{-,\lambda},\quad q_{\lambda}(a)=q_{+,\lambda},\\\
\end{aligned}\right.$ (3.84)

This implies, using Lemma 3.5, that (3.70) holds true. Formulas (3.71) and
(3.72) are then obtained as direct consequences of (3.73) and (3.75).

We still have to prove (3.62). To this aim, we use Lemma 3.5 in order to
reduce equations (3.83) to a linear algebraic system of unknowns
$q_{-,\lambda}$ and $q_{+,\lambda}$. We note that from the first equation in
(3.70), it follows that for every $x\geqslant a$, we have

$\displaystyle\frac{{\rm d}q_{\lambda}}{{\rm d}x}(x)=$
$\displaystyle-\frac{1}{2}\left(\int_{a}^{x}\exp(\omega_{\lambda}\xi)\varphi_{\lambda}(\xi){\rm
d}\xi\right)\exp(-\omega_{\lambda}x)+\frac{1}{2}\left(\int_{x}^{\infty}\exp(-\omega_{\lambda}\xi)\varphi_{\lambda}(\xi){\rm
d}\xi\right)\exp(\omega_{\lambda}x)$
$\displaystyle-\omega_{\lambda}q_{+,\lambda}\exp(\omega_{\lambda}(a-x))+\frac{1}{2}\left(\int_{a}^{\infty}\exp(-\omega_{\lambda}\xi)\varphi_{\lambda}(\xi){\rm
d}\xi\right)\exp(2\omega_{\lambda}a)\exp(-\omega_{\lambda}x).$ (3.85)

Taking the limit when $x\to a^{+}$ in the above equation, it follows that

$\frac{{\rm d}q_{\lambda}}{{\rm
d}x}\left(a^{+}\right)=\left(\int_{a}^{\infty}\exp(-\omega_{\lambda}\xi)\varphi_{\lambda}(\xi){\rm
d}\xi\right)\exp(\omega_{\lambda}a)-\omega_{\lambda}q_{+,\lambda}.$ (3.86)

In a similar manner, we have

$\frac{{\rm d}q_{\lambda}}{{\rm
d}x}(-a^{-})=-\left(\int_{-\infty}^{-a}\exp(\omega_{\lambda}\xi)\varphi_{\lambda}(\xi){\rm
d}\xi\right)\exp(\omega_{\lambda}a)+\omega_{\lambda}q_{-,\lambda}.$ (3.87)

Inserting the above two formulas in (3.83) and using the fact that

$\frac{1+\mu\lambda}{\lambda}=\frac{\lambda}{\omega^{2}_{\lambda}},$

we obtain that

$\displaystyle\lambda M^{-1}\left[\begin{gathered}q_{-,\lambda}\\\
q_{+,\lambda}\end{gathered}\right]-4a^{2}\left[\begin{gathered}-\left(\frac{\mu}{2a}+\frac{\lambda}{\omega_{\lambda}}+\frac{1}{2a\lambda}\right)q_{-,\lambda}+\left(\frac{\mu}{2a}+\frac{1}{2a\lambda}\right)q_{+,\lambda}\\\
\left(\frac{\mu}{2a}+\frac{1}{2a\lambda}\right)q_{-,\lambda}-\left(\frac{\mu}{2a}+\frac{\lambda}{\omega_{\lambda}}+\frac{1}{2a\lambda}\right)q_{+,\lambda}\end{gathered}\right]$
(3.92) $\displaystyle=$ $\displaystyle M^{-1}\left[\begin{gathered}f_{4}\\\
f_{5}\end{gathered}\right]+\frac{4a^{2}}{\lambda}\left[\begin{gathered}f_{2}(-a^{-})-f_{1}\\\
f_{1}-f_{2}\left(a^{+}\right)\end{gathered}\right]+\frac{4a^{2}\lambda}{\omega^{2}_{\lambda}}\left[\begin{gathered}-\int_{-\infty}^{0}\exp(\omega_{\lambda}\xi)\varphi_{\lambda}(\xi-a){\rm
d}\xi\\\
\int_{0}^{\infty}\exp(-\omega_{\lambda}\xi)\varphi_{\lambda}(\xi+a){\rm
d}\xi\end{gathered}\right].$ (3.99)

The above equation can be rewritten

$\mathcal{M}_{\lambda}\left[\begin{gathered}q_{-,\lambda}\\\
q_{+,\lambda}\end{gathered}\right]=M^{-1}\left[\begin{gathered}f_{4}\\\
f_{5}\end{gathered}\right]+\frac{4a^{2}}{\lambda}\left[\begin{gathered}f_{2}(-a^{-})-f_{1}\\\
f_{1}-f_{2}\left(a^{+}\right)\end{gathered}\right]+\frac{4a^{2}\lambda}{\omega^{2}_{\lambda}}\left[\begin{gathered}-\int_{-\infty}^{0}\exp(\omega_{\lambda}\xi)\varphi_{\lambda}(\xi-a){\rm
d}\xi\\\
\int_{0}^{\infty}\exp(-\omega_{\lambda}\xi)\varphi_{\lambda}(\xi+a){\rm
d}\xi\end{gathered}\right],$

where the matrices $M$ and $\mathcal{M}_{\lambda}$ have been defined in(3.20),
(3.49), respectively. By combining the above formula and Lemma 3.3, we obtain
(3.62) and thus the conclusion of Theorem 3.6. ∎

##  4 Semigroup generation and well-posedness

In this section we continue to use the notation introduced in the previous
one, in particular for the space $\mathcal{X}$ defined in (3.21) and for the
operator $\mathcal{A}$ defined in (3.25)-(3.41). We prove below that
$\mathcal{A}$ generates an analytic semigroup on $\mathcal{X}$ and we give the
proof of Theorem 1.1.

To proof of the fact that $\mathcal{A}$ generates an analytic semigroup on
$\mathcal{X}$ is partially based on two technical results given below.
Although based on elementary facts, the proofs of these two lemmas are
provided, for completeness, in the Appendix on subsection 6.2. The first of
these lemmas is:

######  Lemma 4.1.

Let $\mu>0$ and $\theta\in[0,\frac{\pi}{2})$. Then

$\frac{\lambda^{2}}{1+\mu\lambda}\not\in\mathbb{R}_{-}\qquad\qquad\left(\lambda\in\Sigma_{\theta},\;|\lambda|\geqslant\frac{4}{\mu(1-\sin(\theta))}\right),$
(4.1)

and

${\rm
Re}\,\left(\sqrt{\frac{\lambda^{2}}{1+\mu\lambda}}\right)\geqslant\frac{1}{4}\sqrt{\frac{|\lambda|(1-\sin(\theta))}{\mu}}\qquad\qquad\left(\lambda\in\Sigma_{\theta},\;|\lambda|\geqslant\frac{4}{\mu(1-\sin(\theta))}\right),$
(4.2)

where

$\Sigma_{\theta}=\left\\{{\lambda=\rho\exp(i\phi)\in\mathbb{C}\;\,\Big{|}\;\,\arg\phi\in\left(-\frac{\pi}{2}-\theta,\;\frac{\pi}{2}+\theta\right)}\right\\}.$
(4.3)

The second technical result, which is also proved in the Appendix on
subsection 6.2 is:

######  Lemma 4.2.

Let $a,\,\mu>0$ and $\theta\in\left[0,\frac{\pi}{2}\right)$. Then there exists
a constant $R(a,\,\mu,\,\theta)$, such that for every

$\lambda\in\Sigma_{\theta},\quad|\lambda|\geqslant R(a,\,\mu,\,\theta),$ (4.4)

the matrix $\mathcal{M}_{\lambda}$ defined in (3.49) is invertible and there
is a constant $K(\theta,\,\mu)>0$, such that

$\left\|\lambda\mathcal{M}^{-1}_{\lambda}\right\|\leqslant
K(\theta,\mu)\qquad\left(\lambda\in\Sigma_{\theta},\;|\lambda|\geqslant
R(a,\,\mu,\,\theta)\right).$ (4.5)

An important ingredient of the proof of the fact that $\mathcal{A}$ generates
an analytic semigroup on $\mathcal{X}$ is the following result:

######  Proposition 4.3.

With the notation in Theorem 3.6, for every $\lambda\in\Sigma_{\theta}$,
$|\lambda|\geqslant\frac{4}{\mu(1-\sin\theta)}$ we have

$\displaystyle\left\|q_{\lambda}\right\|_{L^{2}(\mathcal{E})}\leqslant\tilde{c}\left(\frac{1}{|\omega_{\lambda}|{\rm
Re}\,\omega_{\lambda}}\|\varphi_{\lambda}\|_{L^{2}(\mathcal{E})}+\frac{1}{({\rm
Re}\,\omega_{\lambda})^{\frac{1}{2}}}\left\|\left[\begin{gathered}q_{-,\lambda}\\\
q_{+,\lambda}\end{gathered}\right]\right\|_{\mathbb{C}^{2}}\right),$ (4.8)

and

$\displaystyle\left\|\frac{{\rm d}q_{\lambda}}{{\rm
d}x}\right\|_{L^{2}(\mathcal{E})}\leqslant\tilde{c}\left(\frac{1}{{\rm
Re}\,\omega_{\lambda}}\|\varphi_{\lambda}\|_{L^{2}(\mathcal{E})}+\left\|\omega_{\lambda}\left[\begin{gathered}q_{-,\lambda}\\\
q_{+,\lambda}\end{gathered}\right]\right\|_{\mathbb{C}^{2}}\right),$ (4.11)

where $\tilde{c}>0$ is a constant depending only on $a$, $\theta$ and $\mu$.

###### Proof.

Firstly, by combining (3.70) and Lemma 3.4, we have

$\left\|q_{\lambda}\right\|_{L^{2}(\mathcal{E})}\leqslant\tilde{c}\left(\frac{1}{|\omega_{\lambda}|{\rm
Re}\,\omega_{\lambda}}\|\varphi_{\lambda}\|_{L^{2}(\mathcal{E})}+\frac{1}{({\rm
Re}\,\omega_{\lambda})^{\frac{1}{2}}}\left\|\left[\begin{gathered}q_{-,\lambda}\\\
q_{+,\lambda}\end{gathered}\right]\right\|_{\mathbb{C}^{2}}\right)\qquad\qquad(\lambda\in\sigma(\mathcal{A})).$

Similarly, we obtain

$\left\|\frac{{\rm d}q_{\lambda}}{{\rm
d}x}\right\|_{L^{2}(\mathcal{E})}\leqslant\tilde{c}\left(\frac{1}{{\rm
Re}\,\omega_{\lambda}}\|\varphi_{\lambda}\|_{L^{2}(\mathcal{E})}+\left\|\frac{\omega_{\lambda}}{({\rm
Re}\,\omega_{\lambda})^{\frac{1}{2}}}\left[\begin{gathered}q_{-,\lambda}\\\
q_{+,\lambda}\end{gathered}\right]\right\|_{\mathbb{C}^{2}}\right),$

for $\lambda\in\Sigma_{\theta}$,
$|\lambda|\geqslant\frac{4}{\mu(1-\sin\theta)}$. The last estimate and the
fact that

$\frac{1}{({\rm Re}\,\omega_{\lambda})^{\frac{1}{2}}}\geqslant 1.$

imply the conclusion (4.11). ∎

We are now in position to prove that $\mathcal{A}$ is a sectorial operator on
$\mathcal{X}$.

######  Theorem 4.4.

Let $a,\;\mu>0$ and $\theta\in\left[0,\frac{\pi}{2}\right)$, then there exists
positive constants $c=c(a,\,\theta,\,\mu)$ and $R(a,\,\mu,\,\theta)$ such that

$\left\|\lambda(\lambda\mathbb{I}-\mathcal{A})^{-1}\right\|_{\mathcal{L}(\mathcal{X})}\leqslant
c\qquad\qquad\left(\lambda\in\Sigma_{\theta},\;|\lambda|\geqslant
R(a,\,\mu,\,\theta)\right),$ (4.12)

so that the operator $\mathcal{A}$ generates an analytic semigroup on
$\mathcal{X}$.

###### Proof.

From Theorem 3.6, it follows that if $R(a,\,\mu,\,\theta)$ is large enough
then

$\left\\{\lambda\in\Sigma_{\theta},\;|\lambda|\geqslant
R(a,\,\mu,\,\theta)\right\\}\subset\rho(\mathcal{A}).$

Using again Theorem 3.6, it follows that for $\lambda$ satisfying (4.4) and
every
$\mathcal{F}=[f_{1}\,\;f_{2}\,\;f_{3}\,\;f_{4}\,\;f_{5}]^{\top}\in\mathcal{X}$,
we can set

$(\lambda\mathbb{I}-\mathcal{A})^{-1}\mathcal{F}=z_{\lambda}=\left[H_{\lambda}\,\;h_{\lambda}\,\;q_{\lambda}\,\;q_{-,\lambda}\,\;q_{+,\lambda}\right]^{\top},$

with $H_{\lambda}$, $h_{\lambda}$, $q_{\lambda}$, $q_{-,\lambda}$ and
$q_{+,\lambda}$ satisfying (3.62)-(3.72). Combining this with (4.5), we get

$\displaystyle\left\|\lambda\left[\begin{gathered}q_{-,\lambda}\\\
q_{+,\lambda}\end{gathered}\right]\right\|_{\mathbb{C}^{2}}\leqslant$
$\displaystyle\tilde{c}\left(\left|\int_{-\infty}^{0}\exp(\omega_{\lambda}\xi)\varphi_{\lambda}(\xi-a){\rm
d}\xi\right|+\left|\int_{0}^{\infty}\exp(-\omega_{\lambda}\xi)\varphi_{\lambda}(\xi+a){\rm
d}\xi\right|\right)$ (4.15)
$\displaystyle+\tilde{c}\left(|f_{1}|+\|f_{2}\|_{H^{1}(\mathcal{E})}+|f_{4}|+|f_{5}|\right)\qquad\qquad\left(\lambda\in\Sigma_{\theta},\;|\lambda|\geqslant
R(a,\,\mu,\,\theta)\right),$

where

$\varphi_{\lambda}(x)=\frac{\lambda}{1+\mu\lambda}f_{3}(x)-\frac{1}{1+\mu\lambda}\frac{{\rm
d}f_{2}(x)}{{\rm d}x}\qquad\qquad(x\in\mathcal{E}),$

and $\tilde{c}=\tilde{c}(\theta,\;\mu\;,a)$. Noting that

$\|\varphi_{\lambda}\|_{L^{2}(\mathcal{E})}\leqslant\tilde{c}\left(\|f_{3}\|_{L^{2}(\mathcal{E})}+\frac{1}{|\lambda|}\|f_{2}\|_{H^{1}(\mathcal{E})}\right)\qquad\qquad\left(\lambda\in\Sigma_{\theta},\;|\lambda|\geqslant
R(a,\,\mu,\,\theta)\right),$

the above inequalities, combined with Hölder’s inequality, imply that

$\displaystyle\left\|\lambda\left[\begin{gathered}q_{-,\lambda}\\\
q_{+,\lambda}\end{gathered}\right]\right\|_{\mathbb{C}^{2}}$ (4.18)
$\displaystyle\leqslant$
$\displaystyle\tilde{c}\left(|f_{1}|+\|f_{2}\|_{H^{1}(\mathcal{E})}+\|f_{3}\|_{L^{2}(\mathcal{E})}+|f_{4}|+|f_{5}|\right)\qquad\qquad\left(\lambda\in\Sigma_{\theta},\;|\lambda|\geqslant
R(a,\,\mu,\,\theta)\right).$ (4.19)

On the other hand, combining Lemma 3.4, Lemma 4.1 and (3.70), we have

$\displaystyle\|\lambda q_{\lambda}\|_{L^{2}(\mathcal{E})}\leqslant$
$\displaystyle\tilde{c}\left\|\lambda\left[\begin{gathered}q_{-,\lambda}\\\
q_{+,\lambda}\end{gathered}\right]\right\|_{\mathbb{C}^{2}}$ (4.22)
$\displaystyle+\frac{|\lambda|}{2|\omega_{\lambda}|{\rm
Re}\,\omega_{\lambda}}\left(\|f_{3}\|_{L^{2}(\mathcal{E})}+\frac{1}{|\lambda|}\|f_{2}\|_{H^{1}(\mathcal{E})}\right)\qquad\qquad\left(\lambda\in\Sigma_{\theta},\;|\lambda|\geqslant
R(a,\,\mu,\,\theta)\right).$ (4.23)

On the other hand, since

${\rm Re}\,\sqrt{z}=\frac{1}{\sqrt{2}}\sqrt{|z|+{\rm
Re}\,z}\qquad\qquad(z\in\mathbb{C}\verb|\|\mathbb{R}_{-}),$ (4.24)

for $|\lambda|$ large enough, we have

$\frac{|\lambda|}{2|\omega_{\lambda}|{\rm Re}\,\omega_{\lambda}}\leqslant c,$
(4.25)

where $c$ is a constant depending only on $a$, $\mu$ and $\theta$.

Estimates (4.22) and (4.25), combined with (4.2) and (4.5) yield that

$\displaystyle\|\lambda q_{\lambda}\|_{L^{2}(\mathcal{E})}$
$\displaystyle\leqslant$
$\displaystyle\tilde{c}\left(|f_{1}|+\|f_{2}\|_{H^{1}(\mathcal{E})}+\|f_{3}\|_{L^{2}(\mathcal{E})}+|f_{4}|+|f_{5}|\right)\qquad\qquad\left(\lambda\in\Sigma_{\theta},\;|\lambda|\geqslant
R(a,\,\mu,\,\theta)\right).$ (4.26)

Using Theorem 3.6 and Remark 4.3, we derive:

$\displaystyle\left\|\lambda h_{\lambda}\right\|_{H^{1}(\mathcal{E})}$
$\displaystyle=$ $\displaystyle\left\|f_{2}-\frac{{\rm d}q_{\lambda}}{{\rm
d}x}\right\|_{H^{1}(\mathcal{E})}$ $\displaystyle\leqslant$
$\displaystyle\|f_{2}\|_{H^{1}(\mathcal{E})}+\left\|\frac{{\rm
d}q_{\lambda}}{{\rm d}x}\right\|_{L^{2}(\mathcal{E})}+\left\|\frac{{\rm
d}^{2}q_{\lambda}}{{\rm d}x^{2}}\right\|_{L^{2}(\mathcal{E})}$
$\displaystyle\leqslant$
$\displaystyle\tilde{c}\left(|f_{1}|+\|f_{2}\|_{H^{1}(\mathcal{E})}+\|f_{3}\|_{L^{2}(\mathcal{E})}+|f_{4}|+|f_{5}|\right)\qquad\qquad\left(\lambda\in\Sigma_{\theta},\;|\lambda|\geqslant
R(a,\,\mu,\,\theta)\right),$ (4.27)

and

$\displaystyle\|\lambda H_{\lambda}\|_{\mathbb{C}}$ $\displaystyle=$
$\displaystyle\left\|f_{1}-\frac{q_{+,\lambda}-q_{-,\lambda}}{2a}\right\|_{\mathbb{C}}$
$\displaystyle\leqslant$
$\displaystyle\tilde{c}\left(|f_{1}|+\|f_{2}\|_{H^{1}(\mathcal{E})}+\|f_{3}\|_{L^{2}(\mathcal{E})}+|f_{4}|+|f_{5}|\right)\qquad\qquad\left(\lambda\in\Sigma_{\theta},\;|\lambda|\geqslant
R(a,\,\mu,\,\theta)\right).$ (4.28)

Putting together (4.19), (4.26), (4.27) and (4.28) yield the conclusion
(4.12). ∎

We are now in a position to prove our first main result.

###### Proof of Theorem 1.1.

The proof is based on Theorem 4.4 and on an interpolation argument. Following
Lunardi [13, Chapter 1], given two Banach spaces $X,\ Y$, where $Y\subset X$
with continuous embedding, we denote by $(X,\,Y)_{\frac{1}{2},2}$ the real
interpolation space of order $\frac{1}{2}$ between $Y$ and $X$.

Recall the operators $\mathcal{A}$ and $\mathcal{B}$ defined in (3.25) and
(3.36). It is not difficult to check that the real interpolation space
$\left(\mathcal{X},\,\mathcal{D}(\mathcal{A})\right)_{\frac{1}{2},2}$
coincides with the space $\mathcal{W}$ defined in (3.23). Moreover, we know
from Theorem 4.4 that $\mathcal{A}$ generates an analytic semigroup on
$\mathcal{X}$ and we also know that
$\mathcal{B}\in\mathcal{L}(\mathcal{U},\mathcal{X})$. According to a classical
result on analytic semigroups (see, for instance, [2, Theorem 2.1, p.207]),
for every
$z_{0}\in\left(\mathcal{X},\mathcal{D}(\mathcal{A})\right)_{\frac{1}{2},2}=\mathcal{W}$,
$T>0$ and $\mathcal{B}u\in L^{2}((0,T);\;\mathcal{X})$, the system (3.1),
(3.2) admits a unique solution

$z\in C([0,T];\;\mathcal{W})\cap L^{2}((0,T);\;\mathcal{Z})\cap
H^{1}((0,T);\;\mathcal{X}).$

The desired conclusion follows now by applying Proposition 3.1. ∎

##  5 The optimal control problem

We give below the proof of our main result in Theorem 1.2 and we describe the
construction of the feedback control by applying Proposition 2.3. To apply
this result we have to prove the existence of a feedback operator $F$ and of
an observation operator $C$ such that our system satisfies all the assumptions
of Proposition 2.3. The definition of $C$ is easy to find, whereas the
construction of $F$ is slightly more involved. Therefore, in the first
subsection below we derive some energy estimates which will suggest the choice
of this feedback operator.

###  5.1 Feedback design

We begin by deriving the following energy estimate:

######  Proposition 5.1.

Let $\left[\begin{gathered}H\ h\ q\ p\end{gathered}\right]^{\top}$ be the
solution of system (1.1)-(1.13) whose existence and uniqueness has been proved
in Theorem 1.1 and let

$\displaystyle
E(t)=\int_{\mathbb{R}}\frac{1}{2}\left[q^{2}(t,x)+h^{2}(t,x)\right]{\rm
d}x+\frac{1}{2}\dot{H}^{2}(t)\qquad\qquad(t>0).$ (5.1)

Then we have

$\frac{\rm d}{{\rm d}t}E(t)=-\mu\int_{\mathbb{R}}\left(\frac{\partial
q}{\partial x}\right)^{2}(t,x){\rm d}x+u(t)\dot{H}(t)\qquad\qquad(t>0).$ (5.2)

###### Proof.

From (5.1), it follows that

$\displaystyle\frac{\rm d}{{\rm
d}t}E(t)=\int_{\mathbb{R}}\left[q(t,x)\frac{\partial q}{\partial
t}(t,x)+h(t,x)\frac{\partial h}{\partial t}(t,x)\right]{\rm
d}x+\dot{H}(t)\ddot{H}(t)\qquad\qquad(t>0\ a.e.).$

Combining the above formula with (1.1), (1.2), (1.3) and (1.4), we obtain

$\displaystyle\frac{\rm d}{{\rm d}t}E(t)=$
$\displaystyle-\int_{\mathcal{E}}\frac{\partial(hq)}{\partial x}(t,x){\rm
d}x+\mu\int_{\mathcal{E}}q(t,x)\frac{\partial^{2}q}{\partial x^{2}}(t,x){\rm
d}x$ $\displaystyle+\mu\int_{\mathcal{I}}q(t,x)\frac{\partial^{2}q}{\partial
x^{2}}(t,x){\rm d}x-H(t)\int_{\mathcal{I}}\frac{\partial q}{\partial
x}(t,x){\rm d}x$ $\displaystyle-\int_{\mathcal{I}}q(t,x)\frac{\partial
p}{\partial x}(t,x){\rm
d}x+\dot{H}(t)\ddot{H}(t)\qquad\qquad\qquad\quad\qquad(t>0\ a.e.).$

Integrating by parts and combining (1.5), (1.6), (1.8) and (1.10), we obtain

$\frac{\rm d}{{\rm d}t}E(t)=-\mu\int_{\mathbb{R}}\left(\frac{\partial
q}{\partial x}\right)^{2}(t,x){\rm d}x-\dot{H}(t)\int_{\mathcal{I}}p(t,x){\rm
d}x+\dot{H}(t)\ddot{H}(t)\qquad\qquad(t>0\ a.e.).$

Using (1.7) and above equality, it follows that

$\frac{\rm d}{\rm dt}E(t)=-\mu\int_{\mathbb{R}}\left(\frac{\partial
q}{\partial x}\right)^{2}(t,x){\rm d}x+u(t)\dot{H}(t)\qquad\qquad(t>0),$

which concludes the proof of this lemma. ∎

The above lemma suggests that $\tilde{u}_{\alpha}(t)=-\alpha\dot{H}(t)$, with
$\alpha\geqslant 0$, is a feedback control for which the energy of the system
is not increasing. Moreover, if $\alpha>0$ it follows from (5.2) that
$J(\tilde{u}_{\alpha})$ is finite. In order to express this feedback law in
terms of $z$ introduced in Section 3, we note that from (3.4) this feedback
can be rewritten

$\tilde{u}_{\alpha}(t)=\alpha\left(\frac{q_{+}(t)-q_{-}(t)}{2a}\right)\qquad\qquad(t>0,\;\alpha\geqslant
0).$

In the next subsection we show how the above formula allows us to construct a
feedback operator $F$ and operators $A$, $B$ and $C$ such that all the
assumptions in Proposition 2.3 are satisfied.

###  5.2 Proof of Theorem 1.2

In this subsection we show that equations (1.1)-(1.10), together with the
output law $y(t)=\dot{H}(t)$ for $t>0$ define a system satisfying all the
assumptions in Proposition 2.3. To this aim, we define the following spaces
and operators:

  * •

The state space is $X=\mathcal{W}$, where $\mathcal{W}$ has been introduced in
(3.23);

  * •

$A$ is the part in $\mathcal{W}$ of the operator $\mathcal{A}$ introduced in
(3.25)-(3.41);

  * •

$Y=\mathbb{C}$ and $C\in\mathcal{L}(X,Y)$ is defined by

$Cz=-\frac{q_{+}-q_{-}}{2a}\qquad\qquad\left(z=[H\,\;h\,\;q\,\;q_{-}\,\;q_{+}]^{\top}\in
X\right).$ (5.3)

  * •

$U=\mathbb{C}$ and $B=\mathcal{B}\in\mathcal{L}(U,X)$, where $\mathcal{B}$ has
been defined in (3.36);

  * •

$F\in\mathcal{L}(X,U)$ is defined by

$Fz=\frac{q_{+}-q_{-}}{2a}\qquad\qquad(z\in X).$ (5.4)

######  Remark 5.2.

The motivation behind of the above choice of the feedback operator has been
explained above, in the comments following the proof of Proposition 5.1.

######  Proposition 5.3.

With the above notation, the operators $A$ and $A_{F}=A+BF$ generate analytic
semigroups on $X$.

###### Proof.

As already mentioned in the Proof of Theorem 1.1, $X=\mathcal{W}$ coincides
with the real interpolation space
$\left(\mathcal{X},\mathcal{D}(\mathcal{A})\right)_{\frac{1}{2},2}$, where
$\mathcal{A}$ has been introduced in (3.25)-(3.41). On the other hand, we have
seen in Theorem 4.4 that $\mathcal{A}$ is a sectorial operator on the space
$\mathcal{X}$ defined in (3.21). The two above facts, combined with [13,
Theorem 6.5, p.165] imply that $A$ is the part of $\mathcal{A}$ in $X$ and $A$
is sectorial on $X$, so that $A$ generates an analytic semigroup on $X$. This,
the fact that $BF\in\mathcal{L}(X)$ and Corollary 2.2 in Pazy [17, p.81] imply
that $A_{F}$ generates an analytic semigroup on $X$. ∎

The result below provides some useful spectral properties of the operator
$A_{F}$ introduced above.

######  Proposition 5.4.

With the notation in Proposition 5.3, there exists a set
$\mathcal{S}\subset\mathbb{C}_{-}$ (recall (3.45) for the definition of
$\mathbb{C}_{-}$), having at most four elements, such that the spectrum of
$A_{F}$ satisfies

$\sigma\left(A_{F}\right)\subset\\{0\\}\cup\mathcal{S}\cup\left(-\infty,-\frac{1}{\mu}\right]\cup\left\\{\lambda\in\mathbb{C}_{-}\quad\left|\quad\left|\lambda+\frac{1}{\mu}\right|=\frac{1}{\mu}\right\\}\right..$
(5.5)

###### Proof.

The proof can be completed following line by line the Theorem 3.6. The only
difference is that we need to replace the matrix $\mathcal{M}_{\lambda}$
defined in (3.49) by

$\displaystyle\tilde{\mathcal{M}}_{\lambda}=\begin{bmatrix}\lambda\left(1+\frac{8a^{3}}{3}\right)+2a\left(\mu+\frac{1}{2a}+\frac{1}{\lambda}\right)+\frac{4a^{2}\lambda}{\omega_{\lambda}}&\
&-\lambda\left(1-\frac{4a^{3}}{3}\right)-2a\left(\mu+\frac{1}{\lambda}\right)\\\
-\lambda\left(1-\frac{4a^{3}}{3}\right)-2a\left(\mu+\frac{1}{\lambda}\right)&\
&\lambda\left(1+\frac{8a^{3}}{3}\right)+2a\left(\mu+\frac{1}{2a}+\frac{1}{\lambda}\right)+\frac{4a^{2}\lambda}{\omega_{\lambda}}\end{bmatrix},$

where $\omega_{\lambda}$ is defined in (3.48). ∎

We also need the result below, which is easy to check, so we skip the proof.

######  Proposition 5.5.

With the notation in Proposition 5.3, we have that $0$ is not an eigenvalue of
$A_{F}$. Moreover, for every $G\in\mathcal{L}(X,\mathbb{C})$, we have that
$0\in\sigma(A+BG)$, which implies that system $\Sigma(A,B,C)$ is not
exponentially stabilizable.

The result below show that the system
$\Sigma\left(A+BF,\,0,\,\begin{bmatrix}C\\\ F\end{bmatrix}\right)$ satisfies
one of the main assumptions in Proposition 2.3.

######  Proposition 5.6.

With the notation in Proposition 5.3, the semigroup $\mathbb{T}^{F}$ generated
by $A_{F}$ is strongly stable in $X$.

###### Proof.

We first remark that, since $\mathbb{T}^{F}$ is analytic, its growth bound
$\omega_{0}\left(\mathbb{T}^{F}\right)$ is given ([2, Proposition 2.9,
p.120-121]) by

$\omega_{0}\left(\mathbb{T}^{F}\right)=\sup_{\lambda\in\sigma(A_{F})}{\rm
Re}\,\lambda.$

This fact, combined with Proposition 5.4 and 5.5, implies that
$\mathbb{T}^{F}$ is a bounded semigroup.

On the other hand, from Proposition 5.4 and 5.5, it follows that $A_{F}$ has
no eigenvalues on the imaginary axis and that $\sigma\left(A_{F}\right)\cap
i\mathbb{R}=\\{0\\}$. Using the above properties and applying the Arendt-Batty
theorem (see Arendt and Batty [1]), we obtain the announced conclusion. ∎

We are now in a position to prove the main result of this section, which can
be seen as a reinforced version of Theorem 1.2, already announced in Section
1. Recall that the operators $A,\ B$ and $C$ appearing in the theorem below
are those introduced of the beginning of this subsection.

######  Theorem 5.7.

For every initial data satisfying the assumptions in Theorem 1.1, there exists
a unique optimal control $\bar{u}\in L^{2}(0,\infty)$ minimizing the cost
functional $J(u)$ defined in (1.18). Moreover, $\bar{u}$ can be expressed in
the feedback form $\bar{u}(t)=-B^{*}P_{0}z(t)$, where $P_{0}\in\mathcal{L}(X)$
is the minimal self-adjoint nonnegative solution to the algebraic Riccati
equation

$A^{*}Pz+PAz-PBB^{*}Pz+C^{*}Cz=0\qquad\quad(z\in\mathcal{D}(A)).$

Finally, the corresponding minimal cost functional (1.18) is given by

$J(\bar{u},\;z_{0})=\left<P_{0}z_{0},\;z_{0}\right>_{X}.$

###### Proof.

As stated in Proposition 3.1, system (1.1)-(1.13) is equivalent to system
(3.4)-(3.19). Hence, finding an optimal control for the latter system implies
that the optimal control is also suitable for the original system.

To apply Proposition 2.3 we first remark that, according to Proposition 5.6,
the semigroup generated by $A_{F}=A+BF$ is strongly stable. Moreover, from
Proposition 5.1 it follows that

$\int_{0}^{\infty}\left|\dot{H}(t)\right|^{2}{\rm d}t\leqslant
2\|z_{0}\|_{X}^{2}.$

Combining the last estimate with (3.4) and with the definitions (5.3), (5.4)
of the operators $C$ and $F$, it follows that the system
$\Sigma\left(A+BF,\,0,\,\begin{bmatrix}C\\\ F\end{bmatrix}\right)$ is output
stable. We can thus apply Proposition 2.3 to obtain the announced conclusions.
∎

##  6 Appendix

###  6.1 Appendix A

This paragraph is devoted to the proof of Lemmas 3.2-3.5.

###### Proof of Lemma 3.2.

Since

$\frac{\lambda^{2}}{1+\mu\lambda}=\frac{1}{\mu^{2}}\frac{(\mu\lambda)^{2}}{\mu\lambda+1},$

it clearly suffices to prove (4.1) and (4.2) for $\mu=1$.

We take $\lambda\neq-1$ and write $\lambda=\rho\exp({i\phi})$ with $\rho>0$.
Since

$\displaystyle\frac{\lambda^{2}}{1+\lambda}=\frac{\rho^{2}\left(\cos(2\phi)+\rho\cos(\phi)+i(\sin(2\phi)+\rho\sin(\phi)\right)}{|1+\lambda|^{2}},$

it follows that $\frac{\lambda^{2}}{1+\lambda}\in\mathbb{R}_{-}$ if and only
if

$\cos(2\phi)+\rho\cos(\phi)<0\quad{\rm and}\quad\sin(2\phi)+\rho\sin(\phi)=0.$

The above condition implies that

$\phi=\pi\quad{\rm and}\quad\rho>1\quad{\rm or}\quad 2\cos(\phi)+\rho=0.$
(6.1)

The first condition in (6.1) writes $\lambda\in(-\infty,-1)$. If the second
condition in (6.1) holds then

$\displaystyle|1+\lambda|^{2}$ $\displaystyle=$
$\displaystyle\left(1+\rho\cos(\phi)+i\rho\sin(\phi)\right)\left(1+\rho\cos(\phi)-i\rho\sin(\phi)\right)$
$\displaystyle=$ $\displaystyle\rho^{2}+2\rho\cos(\phi)+1$ $\displaystyle=$
$\displaystyle\rho(2\cos(\phi)+\rho)+1$ $\displaystyle=$ $\displaystyle 1.$

So that the second condition is equivalent to $|\lambda+1|=1$.

We have thus proved the conclusion for $\mu=1$ and, consequently, for any
positive $\mu$. ∎

###### Proof of Lemma 3.3.

We first note that

$\det\mathcal{M}_{\lambda}=4a^{2}\lambda\left(a+\frac{1}{\omega_{\lambda}}\right)\times\left[\lambda\left(2+\frac{4a^{3}}{3}\right)+4a\left(\mu+\frac{1}{\lambda}\right)+\frac{4a^{2}\lambda}{\omega_{\lambda}}\right].$

In order to obtain the conclusion of Lemma 3.3, it suffices to check that the
equation

$4a^{2}\lambda\left[a+\frac{1}{\omega_{\lambda}}\right]\times\left[\lambda\left(2+\frac{4a^{3}}{3}\right)+4a\left(\mu+\frac{1}{\lambda}\right)+\frac{4a^{2}\lambda}{\omega_{\lambda}}\right]=0,$
(6.2)

admits at most four solutions satisfying (3.47) and lying in $\mathbb{C}_{-}$.
To this aim, we begin by remarking that for any $\lambda$ satisfying (3.47) we
have

${\rm Re}\,\left(a+\frac{1}{\omega_{\lambda}}\right)\geqslant a\,.$

Consequently, the complex number $\lambda$ satisfying (3.47) is a solution of
$\det\mathcal{M}_{\lambda}=0$, if and only if

$\lambda\left(2+\frac{4a^{3}}{3}\right)+4a\left(\mu+\frac{1}{\lambda}\right)+\frac{4a^{2}\lambda}{\omega_{\lambda}}=0.$
(6.3)

Let now $\lambda\in\mathbb{C}^{*}$ with ${\rm Re}\,\lambda\geqslant 0$. Then
$\lambda=\rho\exp({i\phi})$ with $|\phi|\leqslant\frac{\pi}{2}$ and $\rho>0$.
Then there exists an angle $\psi$ with $|\psi|\leqslant\frac{\pi}{2}$ such
that

$1+\mu\lambda=1+\mu\rho\cos(\phi)+i\mu\rho\sin(\phi)=|1+\mu\lambda|\exp({i\psi}).$

Moreover, it is easy to see that $\psi\geqslant 0$ if and only if
$\phi\geqslant 0$, thus
$\left|\phi-\frac{\psi}{2}\right|\leqslant\frac{\pi}{2}$. Combining this
estimate with the fact, following from the definition (3.48) of
$\omega_{\lambda}$, that

$\omega_{\lambda}=\frac{\rho}{|1+\mu\lambda|^{\frac{1}{2}}}\exp\left({i\left(\phi-\frac{\psi}{2}\right)}\right),$

it follows that

${\rm Re}\,\left(\frac{\omega_{\lambda}}{\lambda}\right)\geqslant
0\qquad\qquad(\lambda\in\mathbb{C}^{*},\ \ {\rm Re}\,\lambda\geqslant 0).$

The last inequality implies that for every $\lambda\in\mathbb{C}^{*}$ with
${\rm Re}\,\lambda\geqslant 0$ we have

${\rm
Re}\left(\lambda\left(2+\frac{4a^{3}}{3}\right)+4a\left(\mu+\frac{1}{\lambda}\right)+\frac{4a^{2}\lambda}{\omega_{\lambda}}\right)\geqslant
4a\mu>0.$

We have thus shown that any the solution of (6.3) either vanishes or it lies
in the left half plane.
Moreover, if $\lambda$ satisfying (3.47) is a solution of
$\det\mathcal{M}_{\lambda}=0$, then we have

$\left[\lambda^{2}\left(2+\frac{4a^{3}}{3}\right)+4a\left(\mu\lambda+1\right)\right]^{2}=16a^{4}\lambda^{2}(1+\mu\lambda),$

so that $\det\mathcal{M}_{\lambda}$ can vanish for at most four values of
$\lambda$ satisfying (3.47). ∎

###### Proof of Lemma 3.4.

Firstly, by the definition of $D_{+}(\omega)$ (3.51), for every
$\gamma_{+}\in\mathbb{C}$, we have

$\int_{a}^{\infty}|(D_{+}(\omega)\gamma_{+})(x)|^{2}{\rm
d}x\leqslant\exp(2({\rm
Re}\,\omega)a)|\gamma_{+}|^{2}\int_{a}^{\infty}\exp\left(-2({\rm
Re}\,\omega)x\right){\rm d}x=\frac{|\gamma_{+}|^{2}}{2{\rm Re}\,\,\omega},$

which implies the second equality in (3.54).

On the other hand, to estimate the norm of operator $R_{+}(\omega)$, defined
in (3.53), we remark that

$\displaystyle\left|\frac{1}{2\omega}\left(\int_{a}^{x}\exp(\omega\xi)\varphi(\xi){\rm
d}\xi\right)\exp(-\omega x)\right|$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{2|\omega|}\left(\int_{a}^{x}\exp\left(\left({\rm
Re}\,\omega\right)\xi\right)|\varphi(\xi)|{\rm d}\xi\right)\exp\left(-({\rm
Re}\,\omega)x\right)$ $\displaystyle\leqslant$
$\displaystyle\frac{\|\varphi\|_{L^{\infty}(a,\infty)}}{2|\omega|}\left(\int_{a}^{x}\exp(({\rm
Re}\,\omega)\xi){\rm d}\xi\right)\exp(-({\rm Re}\,\omega)x)$ $\displaystyle=$
$\displaystyle\frac{\|\varphi\|_{L^{\infty}(a,\infty)}}{2|\omega|}\frac{\exp(({\rm
Re}\,\omega)x)-\exp(({\rm Re}\,\omega)a)}{{\rm Re}\,\omega}\exp(-({\rm
Re}\,\omega)x)$ $\displaystyle\leqslant$
$\displaystyle\frac{\|\varphi\|_{L^{\infty}(a,\infty)}}{2|\omega|{\rm
Re}\,\omega}\qquad\qquad\quad(\varphi\in
L^{\infty}(a,\infty),\;x\in(a,\infty)).$ (6.4)

With a similar calculation for the other two terms of $R_{+}(\omega)$, we
obtain

$\left|\frac{1}{2\omega}\left(\int_{x}^{\infty}\exp(-\omega\xi)\varphi(\xi){\rm
d}\xi\right)\exp(\omega
x)\right|\leqslant\frac{\|\varphi\|_{L^{\infty}(a,\infty)}}{2|\omega|{\rm
Re}\,\omega},$ (6.5)

and

$\left|\frac{1}{2\omega}\left(\int_{a}^{\infty}\exp(-\omega\xi)\varphi(\xi){\rm
d}\xi\right)\exp(2\omega a)\exp(-\omega
x)\right|\leqslant\frac{\|\varphi\|_{L^{\infty}(a,\infty)}}{2|\omega|{\rm
Re}\,\omega},$ (6.6)

where

$\varphi\in L^{\infty}(a,\infty)\qquad{\rm and}\qquad x\in(a,\infty).$

Combining the defintion (3.53) of $R_{+}(\omega)$ with the (6.1)-(6.6), it
follows that

$\left\|R_{+}(\omega)\right\|_{\mathcal{L}(L^{\infty}(a,\infty))}\leqslant\frac{3}{2|\omega|{\rm
Re}\,\omega}\qquad\qquad({\rm Re}\,\omega>0,\;\varphi\in
L^{\infty}(a,\infty)).$ (6.7)

We next estimate the norm in $\mathcal{L}(L^{1}(a,\infty))$ of the operator
$R_{+}(\omega)$ defined in (3.53). For the first term in the right hand side
of (3.53) we can use Tonelli’s theorem to get

$\displaystyle\int_{a}^{\infty}\left|\frac{1}{2\omega}\left(\int_{a}^{x}\exp(\omega\xi)\varphi(\xi){\rm
d}\xi\right)\exp(-\omega x)\right|{\rm d}x$ $\displaystyle\leqslant$
$\displaystyle\frac{1}{2|\omega|}\int_{a}^{\infty}\left(\int_{a}^{x}\exp(({\rm
Re}\,\omega)\xi)|\varphi(\xi)|{\rm d}\xi\right)\exp(-({\rm Re}\,\omega)x){\rm
d}x$ $\displaystyle=$
$\displaystyle\frac{1}{2|\omega|}\int_{a}^{\infty}\left(\int_{\xi}^{\infty}\exp(-({\rm
Re}\,\omega)x){\rm d}x\right)\exp(({\rm Re}\,\omega)\xi)|\varphi(\xi)|{\rm
d}\xi$ $\displaystyle\leqslant$
$\displaystyle\frac{\|\varphi\|_{L^{1}(a,\infty)}}{2|\omega|{\rm
Re}\,\omega}\qquad\qquad\qquad(\varphi\in L^{1}(a,\infty)).$ (6.8)

In a fully similar manner it can be checked that, for every $\varphi\in
L^{1}(a,\infty)$, the other terms in the right hand side of (3.53) satisfy the
estimates,

$\int_{a}^{\infty}\left|\frac{1}{2\omega}\left(\int_{x}^{\infty}\exp(-\omega\xi)\varphi(\xi){\rm
d}\xi\right)\exp(\omega x)\right|{\rm
d}x\leqslant\frac{\|\varphi\|_{L^{1}(a,\infty)}}{2|\omega|{\rm Re}\,\omega},$
(6.9)

and

$\int_{a}^{\infty}\left|\frac{1}{2\omega}\left(\int_{a}^{\infty}\exp(-\omega\xi)\varphi(\xi){\rm
d}\xi\right)\exp(2\omega a)\exp(-\omega x)\right|{\rm
d}x\leqslant\frac{\|\varphi\|_{L^{1}(a,\infty)}}{2|\omega|{\rm Re}\,\omega}.$
(6.10)

Combining the definition (3.53) of $R_{+}(\omega)$ with formulas (6.1)-(6.10),
we obtain that

$\|R_{+}(\omega)\|_{\mathcal{L}(L^{1}(a,\infty))}\leqslant\frac{3}{2|\omega|{\rm
Re}\,\omega}\qquad\qquad\qquad({\rm Re}\,\omega>0).$ (6.11)

Using the (6.7) and (6.11), combined with the Riesz-Thorin theorem (see, for
instance, [11, p.318]), we find the second inequality in (3.55) also holds.

The first inequalities in (3.54) and (3.55) can be proved in a fully similar
manner, so we skip the corresponding details. ∎

###### Proof of Lemma 3.5.

After a standard caculation, we find that the general solution of equation
(3.56) is

$\displaystyle q_{\lambda}(x)=$
$\displaystyle\frac{1}{2\omega_{\lambda}}\left(\int_{a}^{x}\exp(\omega_{\lambda}\xi)\varphi(\xi){\rm
d}\xi\right)\exp(-\omega_{\lambda}x)-\frac{1}{2\omega_{\lambda}}\left(\int_{a}^{x}\exp(-\omega_{\lambda}\xi)\varphi(\xi){\rm
d}\xi\right)\exp(\omega_{\lambda}x)$
$\displaystyle+K_{1}\exp(-\omega_{\lambda}x)+K_{2}\exp(\omega_{\lambda}x)\qquad\qquad\qquad\qquad(x\geqslant
a),$

where $K_{1}$ and $K_{2}$ are complex constants.
Combined Lemma 3.2 and 3.3, we get ${\rm Re}\,\omega_{\lambda}>0$, where
$\omega_{\lambda}$ is defined by (3.48). In order to have $q_{\lambda}\in
L^{2}(\mathcal{E})$ we necessarily choose

$K_{2}=\frac{1}{2\omega_{\lambda}}\left(\int_{a}^{\infty}\exp(-\omega_{\lambda}\xi)\varphi(\xi){\rm
d}\xi\right),$

so that

$\displaystyle q_{\lambda}(x)=$
$\displaystyle\frac{1}{2\omega_{\lambda}}\left(\int_{a}^{x}\exp(\omega_{\lambda}\xi)\varphi(\xi){\rm
d}\xi\right)\exp(-\omega_{\lambda}x)+\frac{1}{2\omega_{\lambda}}\left(\int_{x}^{\infty}\exp(-\omega_{\lambda}\xi)\varphi(\xi){\rm
d}\xi\right)\exp(\omega_{\lambda}x)$
$\displaystyle+K_{1}\exp(-\omega_{\lambda}x)\qquad\qquad\qquad\qquad(x\geqslant
a).$

Using next the boundary condition in (3.56) we obtain that

$\gamma_{+}=\frac{1}{2\omega_{\lambda}}\left(\int_{a}^{\infty}\exp(-\omega_{\lambda}\xi)\varphi(\xi){\rm
d}\xi\right)\exp(\omega_{\lambda}a)+K_{1}\exp(-\omega_{\lambda}a),$

which yields

$K_{1}=\gamma_{+}\exp(\omega_{\lambda}a)-\frac{1}{2\omega_{\lambda}}\left(\int_{a}^{\infty}\exp(-\omega_{\lambda}\xi)\varphi(\xi){\rm
d}\xi\right)\exp(2\omega_{\lambda}a).$ (6.12)

With the above obtained values of $K_{1}$ and $K_{2}$ we see that if
$q_{\lambda}\in H^{2}(\mathcal{E})$ satisfies (3.56) then $q_{\lambda}$
satisfies the second equation in (3.57). By a fully similar calculation we can
check that if $q_{\lambda}\in H^{2}(\mathcal{E})$ satisfies (3.56) then
$q_{\lambda}$ satisfies the first equation in (3.57). Thus the unique possible
solution $q_{\lambda}\in H^{2}(\mathcal{E})$ of (3.56) is given by (3.57).

On the other hand, we know from Lemma 3.4 (respectively from the above
calculations) that $q_{\lambda}$ defined by (3.57) lies in
$L^{2}(\mathcal{E})$ (respectively that it satisfies (3.56)). This implies
that indeed $q_{\lambda}$ defined by (3.57) lies in $H^{2}(\mathcal{E})$ and
it is the unique solution in this space of (3.56). ∎

###  6.2 Appendix B

This subsection is devoted to the proofs of the lemmas in Section 4.

###### Proof of Lemma 4.1.

By obvious homogeneity reasons (see the proof of Lemma 3.2), it suffices to
prove the conclusion of the present lemma for $\mu=1$. In this case we have

$|1+\lambda|^{2}=1+\rho^{2}+2\rho\cos(\phi),$

so that

$\displaystyle\frac{\lambda^{2}}{1+\lambda}$
$\displaystyle=\frac{\left(\rho^{2}\cos(2\phi)+i\rho\sin(2\phi)\right)\left(1+\rho\cos(\phi)-i\rho\sin(\phi)\right)}{1+\rho^{2}+2\rho\cos(\phi}$
$\displaystyle=\frac{\rho^{2}\cos(2\phi)+\rho^{3}\cos(\phi)+i(\rho^{2}\sin(2\phi)+\rho^{3}\sin(\phi))}{1+\rho^{2}+2\rho\cos(\phi)}\qquad\qquad\quad(\lambda\in\Sigma_{\theta}).$
(6.13)

Moreover, we have that

$\rho^{2}\sin(2\phi)+\rho^{3}\sin(\phi)=\rho^{2}\sin(\phi)(2\cos(\phi)+\rho)\neq
0,$

where

$\phi\in\left(-\frac{\pi}{2}-\theta,\;\frac{\pi}{2}+\theta\right)\verb|\|\\{0\\}\qquad{\rm
and}\qquad\rho\geqslant\frac{4}{1-\sin(\theta)}.$

The above three formulas imply that the imaginary part of
$\frac{\lambda^{2}}{1+\lambda}$, with $\lambda=\rho\exp(i\phi)$ and
$\rho\geqslant\frac{4}{1-\sin(\theta)}$ vanishes iff $\phi=0$. On the other
hand, formula (6.13) clearly yields that ${\rm
Re}\,\frac{\lambda^{2}}{1+\lambda}>0$. We have thus proved that (4.1) holds.

We still prove that formula (4.2) holds for $\mu=1$. To this aim, we first
recall the classical formula

${\rm Re}\,\sqrt{z}=\frac{1}{\sqrt{2}}\sqrt{|z|+{\rm
Re}\,z}\qquad\qquad\quad(z\in\mathbb{C}\verb|\|\mathbb{R}_{-}).$ (6.14)

We next apply the above formula for $z=\frac{\lambda^{2}}{1+\lambda}$ and
$\lambda=\rho\exp(i\phi)$, with

$\rho\geqslant\frac{4}{1-\sin(\theta)}\qquad{\rm
and}\qquad\phi\in\left(-\frac{\pi}{2}-\theta,\,\frac{\pi}{2}+\theta\right).$
(6.15)

Combining (6.14), (6.15) and (6.13), we obtain that

$\displaystyle 2\left[{\rm
Re}\,\left(\sqrt{\frac{\lambda^{2}}{1+\lambda}}\right)\right]^{2}$
$\displaystyle=\frac{\rho^{2}\sqrt{1+\rho^{2}+2\rho\cos(\phi)}}{1+\rho^{2}+2\rho\cos(\phi)}+\frac{\rho^{2}\cos(2\phi)+\rho^{3}\cos(\phi)}{1+\rho^{2}+2\rho\cos(\phi)}$
$\displaystyle=\frac{\rho^{2}}{1+\rho^{2}+2\rho\cos(\phi)}\left(\sqrt{1+\rho^{2}+2\rho\cos(\phi)}+\cos(2\phi)+\rho\cos(\phi)\right),$
(6.16)

for every $\lambda=\rho\exp(i\phi)$ with $\rho$ and $\phi$ satisfying (6.15).
Moreover, since $\rho>1$, we have

$\displaystyle\sqrt{1+\rho^{2}+2\rho\cos(\phi)}+\cos(2\phi)+\rho\cos(\phi)$
$\displaystyle\geqslant$ $\displaystyle\rho-1+\cos(2\phi)+\rho\cos(\phi)$
$\displaystyle\geqslant$ $\displaystyle\rho(1-\sin(\theta))-2$
$\displaystyle\geqslant$ $\displaystyle\frac{1}{2}\rho(1-\sin(\theta)).$

Using the above equation and (6.2), for every $\lambda=\rho\exp(i\phi)$
satisfying (6.15), it follows that

$2\left[{\rm
Re}\,\left(\sqrt{\frac{\lambda^{2}}{1+\lambda}}\right)\right]^{2}\geqslant\frac{\rho^{3}(1-\sin(\theta))}{2(1+\rho^{2}+2\rho\cos(\phi))}\geqslant\frac{\rho^{3}(1-\sin(\theta))}{2(1+\rho)^{2}}\geqslant\frac{\rho(1-\sin(\theta))}{8},$

which ends the proof of Lemma 4.1. ∎

###### Proof of Lemma 4.2.

From Lemma 3.3, we find for every $\lambda$ satisfying (4.4), the matrix
$\mathcal{M}_{\lambda}$ defined in (3.49) is invertible. Moreover, from the
definition (3.49) of $\mathcal{M}_{\lambda}$ , we have

$\lambda^{-1}\mathcal{M}_{\lambda}=\left[\begin{gathered}1+\frac{8a^{3}}{3}\qquad\quad-1+\frac{4a^{3}}{3}\\\
-1+\frac{4a^{3}}{3}\qquad\quad
1+\frac{8a^{3}}{3}\end{gathered}\right]+2a\left(\frac{\mu}{\lambda}+\frac{1}{\lambda^{2}}\right)\left[\begin{gathered}1\quad-1\\\
-1\quad
1\end{gathered}\right]+\frac{4a^{2}}{\omega_{\lambda}}\left[\begin{gathered}1\quad
0\\\ 0\quad 1\end{gathered}\right].$

The above formula can be rewritte,

$\lambda^{-1}\mathcal{M}_{\lambda}=\left[\begin{gathered}1+\frac{8a^{3}}{3}\qquad-1+\frac{4a^{3}}{3}\\\
-1+\frac{4a^{3}}{3}\qquad
1+\frac{8a^{3}}{3}\end{gathered}\right]+\mathcal{P}_{\lambda},$

where

$\mathcal{P}_{\lambda}=2a\left(\frac{\mu}{\lambda}+\frac{1}{\lambda^{2}}\right)\left[\begin{gathered}1\quad-1\\\
-1\quad
1\end{gathered}\right]+\frac{4a^{2}}{\omega_{\lambda}}\left[\begin{gathered}1\quad
0\\\ 0\quad 1\end{gathered}\right].$

Using next Lemma 4.1 we see that there exists a constant
$\tilde{K}(\theta,\,\mu)>0$ such that

$\|\mathcal{P}_{\lambda}\|\leqslant\frac{\tilde{K}(\theta,\,\mu)}{\sqrt{|\lambda|}}\qquad\qquad\qquad\left(\lambda\in\Sigma_{\theta},\;|\lambda|\geqslant
R(a,\,\mu,\,\theta)\right).$

Moreover, after a simple calculation, we see that the inverse of the matrix
$M$ defined in (3.20) is

$\displaystyle
M^{-1}=\left[\begin{gathered}1+\frac{8a^{3}}{3}\qquad\quad-\left(1-\frac{4a^{3}}{3}\right)\\\
-\left(1-\frac{4a^{3}}{3}\right)\qquad\quad
1+\frac{8a^{3}}{3}\end{gathered}\right],$

Consequently,

$\displaystyle\|\lambda\mathcal{M}^{-1}_{\lambda}\|$ $\displaystyle=$
$\displaystyle\left\|(M^{-1}+\mathcal{P}_{\lambda})^{-1}\right\|$
$\displaystyle\leqslant$
$\displaystyle\left\|M^{-1}\right\|\left\|\left(\mathbb{I}+\frac{\mathcal{P}_{\lambda}}{M^{-1}}\right)\right\|$
$\displaystyle\leqslant$
$\displaystyle\left\|M^{-1}\right\|\left\|\frac{1}{1-\left\|\frac{\mathcal{P}_{\lambda}}{\left\|M^{-1}\right\|}\right\|}\right\|\qquad\qquad\left(\lambda\in\Sigma_{\theta},\;|\lambda|\geqslant
R(a,\,\mu,\,\theta)\right).$ (6.17)

The above formula implies the inequality (4.5). ∎

Acknowledgments. Funded by the European Union (Horizon Europe MSCA project
Modconflex, grant number 101073558).

The authors are grateful to Debayan Maity, Jorge San Martín and Takéo
Takahashi for numerous suggestions, in particular those involving the detailed
estimates from Section 6.

## References

  * [1] W. Arendt and C. J. Batty, Tauberian theorems and stability of one-parameter semigroups, Transactions of the American Mathematical Society, 306 (1988), pp. 837–852.
  * [2] A. Bensoussan, G. D. Prato, M. C. Delfour, and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Representation and Control of Infinite Dimensional Systems, 2007.
  * [3] E. Bocchi, Floating structures in shallow water: local well-posedness in the axisymmetric case, SIAM Journal on Mathematical Analysis, 52 (2020).
  * [4] R. F. Curtain and J. C. Oostveen, Necessary and sufficient conditions for strong stability of distributed parameter systems, Systems & Control Letters, 37 (1999), pp. 11–18.
  * [5] R. F. Curtain and H. Zwart, An introduction to infinite-dimensional linear systems theory, vol. 21, New York, 1995.
  * [6] Falnes and Johannes, Ocean waves and oscillating systems (linear interactions including wave-energy extraction) interaction between oscillations and waves, 10.1017/CBO9780511754630 (2002), pp. 43–57.
  * [7] B. H. Haak, D. Maity, T. Takahashi, and M. Tucsnak, Mathematical analysis of the motion of a rigid body in a compressible navier-stokes-fourier fluid, Mathematische Nachrichten, 292 (2017).
  * [8] F. John, On the motion of floating bodies. i, Communications on Pure and Applied Mathematics, 2 (1949), pp. 13–57.
  * [9] , On the motion of floating bodies ii. simple harmonic motions, Communications on pure and applied mathematics, 3 (1950), pp. 45–101.
  * [10] U. A. Korde and J. Ringwood, Hydrodynamic control of wave energy devices, Cambridge University Press, 2016.
  * [11] S. G. Krantz, A panorama of harmonic analysis, vol. 27, American Mathematical Soc., 2019.
  * [12] D. Lannes, On the dynamics of floating structures, Annals of PDE, 3 (2017), p. 11.
  * [13] A. Lunardi, Interpolation theory, vol. 16, Springer, 2018.
  * [14] D. Maity, J. S. Martín, T. Takahashi, and M. Tucsnak, Analysis of a simplified model of rigid structure floating in a viscous fluid, Journal of Nonlinear Science, (2019), pp. 1–46.
  * [15] D. Maity and M. Tucsnak, A maximal regularity approach to the analysis of some particulate flows, 2017.
  * [16] J. C. Oostveen, R. F. Curtain, and K. Ito, An approximation theory for strongly stabilizing solutions to the operator $lq$ riccati equation, SIAM Journal on Control and Optimization, 38 (2000), pp. 1909–1937.
  * [17] A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44, Springer Science & Business Media, 2012.
  * [18] M. Tucsnak and G. Weiss, Observation and control for operator semigroups, Springer Science & Business Media, 2009.
  * [19] G. Vergara-Hermosilla, D. Matignon, and M. Tucsnak, Well-posedness and input-output stability for a system modelling rigid structures floating in a viscous fluid, IFAC-PapersOnLine, 53 (2020), pp. 7491–7496.