# Controllability and Stabilizability of the linearized compressible Navier-
Stokes system with Maxwell’s law

Sakil Ahamed∗, and Subrata Majumdar†

###### Abstract.

In this paper, we study the control properties of the linearized compressible
Navier-Stokes system with Maxwell’s law around a constant steady state
$(\rho_{s},u_{s},0),\rho_{s}>0,u_{s}>0$ in the interval $(0,2\pi)$ with
periodic boundary data. We explore the exact controllability of the coupled
system by means of a localized interior control acting in any of the equations
when time is large enough. We also study the boundary exact controllability of
the linearized system using a single control force when the time is
sufficiently large. In both cases, we prove the exact controllability of the
system in the space $L^{2}(0,2\pi)\times L^{2}(0,2\pi)\times L^{2}(0,2\pi)$.
We establish the exact controllability results by proving an observability
inequality with the help of an Ingham-type inequality. Next, we prove the
small time lack of controllability of the concerned system for the case of
localized interior control.

Further, using a Gramian-based approach demonstrated by Urquiza, we prove the
exponential stabilizability of the corresponding closed-loop system with an
arbitrary prescribed decay rate using boundary feedback control law.

###### Key words and phrases:

Linearized compressible Navier-Stokes equation with Maxwell’s law, Exact
controllability, Boundary control, distributed control, Ingham inequality,
Observability inequality, Feedback stabilization

###### 2020 Mathematics Subject Classification:

35Q35,35Q30, 93B05, 93B07, 93D15

∗Department of Mathematics & Statistics, Indian Institute of Technology
Kanpur, Kanpur, UP- 208016, India; e-mail<EMAIL_ADDRESS>
† Department of Mathematics, Indian Institute of Technology Bombay, Powai,
Mumbai- 400076, India; e-mail<EMAIL_ADDRESS>
∗Corresponding author: Sakil Ahamed

###### Contents

  1. 1 Introduction and Main results
    1. 1.1 Setting of the problem
    2. 1.2 Interior controllability
    3. 1.3 Boundary controllability
    4. 1.4 Stabilizability
    5. 1.5 Organization of the paper
  2. 2 Linearized operator
    1. 2.1 Spectral analysis of the linearized operator
  3. 3 Interior controllability
    1. 3.1 Proof of Theorem 1.2:
  4. 4 Boundary controllability
    1. 4.1 Control in density
      1. 4.1.1 Well-posedness
      2. 4.1.2 Exact Controllability
      3. 4.1.3 Proof of Theorem 1.7
    2. 4.2 Control in velocity
    3. 4.3 Control in stress
  5. 5 Small time lack of controllability
  6. 6 Rapid exponential stabilization
    1. 6.1 Urquiza’s method
    2. 6.2 Control in density
    3. 6.3 Feedback control law and proof of Theorem 1.9
  7. 7 Construction of the biorthogonal family and Ingham-type inequality

##  1\. Introduction and Main results

###  1.1. Setting of the problem

The control and stabilization of fluid flow have been extensively studied due
to their numerous practical applications such as weather prediction, water
flow in a pipe, designing aircraft and cars. Many researchers have been
interested in the subject of the controllability of fluid flows, more so for
incompressible flow (see [2], [19], [33], [34], [37]) than for compressible
flow (see [11], [12]). In our research, we focus on investigating the exact
controllability and stabilizability of the linearized compressible Navier-
Stokes system with Maxwell’s law.

Let us consider the one-dimensional compressible Navier-Stokes system in the
domain $(0,2\pi)$:

$\left.\begin{aligned}
&\partial_{t}\hat{\rho}+\partial_{x}(\hat{\rho}\hat{u})=0\qquad\qquad\qquad\qquad\text{
in }(0,T)\times(0,2\pi),\\\
&\partial_{t}\left({\hat{\rho}\hat{u}}\right)+\partial_{x}\left({\hat{\rho}\hat{u}^{2}}\right)+\partial_{x}p=\partial_{x}\hat{S}\qquad\text{in
}(0,T)\times(0,2\pi),\end{aligned}\right\\}$ (1.1)

where, $\hat{\rho}$, $\hat{u}$, $p$ and $\hat{S}$ represent the density,
velocity, pressure and stress tensor of the fluid, respectively. Equation
(1.1)1 is the consequence of conservation of mass and equation (1.1)2 is the
consequence of conservation of momentum. We assume that the pressure $p$
satisfies the following constitutive law:

$p(\hat{\rho})=a\hat{\rho}^{\gamma},\>\quad\gamma\geq 1,$ (1.2)

where $a$ is a positive constant. While, the stress tensor $\hat{S}$ is
assumed to satisfy the Maxwell’s law:

$\kappa\partial_{t}\hat{S}+\hat{S}=\mu\partial_{x}\hat{u},\text{ in
}(0,T)\times(0,2\pi).$ (1.3)

Here $\mu$ and $\kappa$ are positive constants, with $\mu$ representing the
fluid viscosity and $\kappa$ denoting the relaxation time that characterizes
the time delay in the response of the stress tensor to the velocity gradient.
For more detailed information, refer to [22] and references therein. Relation
$\eqref{max law}$ is first proposed by Maxwell, in order to describe the
relation of stress tensor and velocity gradient for a non-simple fluid.

In this paper, we study the control aspects of the one-dimensional
compressible Navier-Stokes system with Maxwell’s law and with periodic
boundary conditions linearized around a constant steady state
$\left(\rho_{s},u_{s},0\right),\rho_{s}>0,u_{s}>0$ of (1.1). More precisely we
consider the following system:

$\left.\begin{aligned}
&\partial_{t}\rho+u_{s}\partial_{x}\rho+\rho_{s}\partial_{x}u=\mathbbm{1}_{\mathcal{O}_{1}}f_{1},\qquad&&\text{in
}(0,T)\times(0,2\pi),\\\
&\partial_{t}u+u_{s}\partial_{x}u+a\gamma{\rho_{s}}^{\gamma-2}\partial_{x}\rho-\frac{1}{\rho_{s}}\partial_{x}S=\mathbbm{1}_{\mathcal{O}_{2}}f_{2},\qquad&&\text{in
}(0,T)\times(0,2\pi),\\\
&\partial_{t}S+\frac{1}{\kappa}S-\frac{\mu}{\kappa}\partial_{x}u=\mathbbm{1}_{\mathcal{O}_{3}}f_{3},\qquad&&\text{in
}(0,T)\times(0,2\pi),\\\
&\rho(t,0)=\rho(t,2\pi),\>u(t,0)=u(t,2\pi),\>S(t,0)=S(t,2\pi),\qquad&&t\in(0,T),\\\
&\rho(0,x)=\rho_{0}(x),\quad u(0,x)=u_{0}(x),\quad
S(0,x)=S_{0}(x),\qquad&&x\in(0,2\pi),\end{aligned}\right\\}$ (1.4)

where $\mathbbm{1_{\mathcal{O}_{j}}}$ is the characteristic function of an
open set ${\mathcal{O}}_{j}\subset(0,2\pi),$ $j=1,2,3$ and $f_{i}$, $i=1,2,3$
are the distributed controls.

To establish the controllability results, we have to consider the following
restriction on the constant steady state $(\rho_{s},u_{s},0)$:

$\left(a\gamma\rho_{s}^{\gamma-1}-u_{s}^{2}\right)\geq 0.$ (1.5)

######  Definition 1.1.

The system (1.4) is exactly controllable in $(L^{2}(0,2\pi))^{3}$ at time
$T>0$, if for any initial condition
$(\rho_{0},u_{0},S_{0})^{\top}\in(L^{2}(0,2\pi))^{3}$ and any other
$(\rho_{1},u_{1},S_{1})^{\top}\in(L^{2}(0,2\pi))^{3}$, there exist controls
$f_{i}\in L^{2}\left(0,T;L^{2}(\mathcal{O}_{i})\right),i=1,2,3,$ such that the
corresponding solution $(\rho,u,S)^{\top}$ of (1.4) satisfies

$(\rho,u,S)^{\top}(T,x)=(\rho_{1},u_{1},S_{1})^{\top}(x),\text{ for all
}x\in(0,2\pi).$ (1.6)

Let us first mention controllability results for some fluid models related to
our system. The existence of the solution of the compressible Navier-Stokes
system with Maxwell’s law, along with the blow-up results has been studied,
for example, in [21, 22, 41] and references therein. If $\kappa=0$, then the
Maxwell’s law (1.3) turns into the Newtonian law
$\hat{S}=\mu\partial_{x}\hat{u}$ and the equation (1.1) becomes Navier-Stokes
system of a viscous, compressible, isothermal barotropic fluid (density is
function of pressure only), in a bounded domain $(0,2\pi)$. The compressible
Navier-Stokes system linearized around a constant trajectory
$(\rho_{s},u_{s})$ for $\rho_{s}>0,u_{s}>0$, yields a coupled transport-
parabolic system with constant coefficients. The controllability of such
systems with constant coefficients in one dimension has been extensively
studied in the literature. In [9, 10], the authors studied this system in the
domain $(0,2\pi)$ with periodic boundary conditions and localized interior
control acting only in the parabolic equation. In [9], using the moment
method, the authors proved the null controllability in
$H^{s+1}_{\textnormal{m}}(0,2\pi)\times H^{s}(0,2\pi),$ $s>6.5,$ at time
$T>\frac{2\pi}{|u_{s}|},$ where $H^{s}_{\textnormal{m}}(0,2\pi)$ is the space
of periodic Sobolev space with mean zero. This result was improved in [10] by
showing that the null controllability holds for any initial data in
$H^{1}_{\textnormal{m}}(0,2\pi)\times L^{2}(0,2\pi).$ Moreover, the authors
also proved that, the system in consideration is not null controllable in
$H^{s}_{\textnormal{m}}(0,2\pi)\times L^{2}(0,2\pi),$ $0\leqslant s<1,$ at any
time $T>0$ by $L^{2}$-control acting in the parabolic equation. Thus
$H^{1}_{\textnormal{m}}(0,2\pi)\times L^{2}(0,2\pi)$ is the largest space in
which the system is null controllable by a $L^{2}$-parabolic control.

Recently, in [3], the above results have been extended to more general coupled
transport-parabolic systems with constant coefficients. The authors considered
coupling of several transport and parabolic equations in one-dimensional torus
$\mathbb{T}$ and studied the null controllability with localized interior
control in optimal time. Moreover, an algebraic necessary and sufficient
condition on the coupling term was obtained when controls act only on the
parabolic or transport components. For the extension of these results to the
general coupling matrix, one can refer to the work [24].

The local null controllability of the nonlinear compressible Navier-Stokes
system around a trajectory with non-zero velocity at large time $T>0$ has been
obtained in [16, 15, 17, 31, 30, 32].

In [1], the authors considered the one-dimensional compressible Navier-Stokes
equations with Maxwell’s law linearized around a constant steady state
$(\rho_{s},0,0),\rho_{s}>0$ with Dirichlet boundary conditions and with
interior controls in the interval $(0,\pi)$. They have proved that the system
is not null controllable at any time using localized controls in density and
stress equations and even everywhere control in the velocity equation.
Moreover, they have shown that the system is null controllable at any time
$T>0$ in the space $(L^{2}(0,\pi))^{3}$, if the control acts everywhere in the
density or stress equation. Approximate controllability at large time using
localized controls has also been studied.

In the following subsection, we state our main results regarding the interior
controllability of the system (1.4)

###  1.2. Interior controllability

At first, we consider the case when $f_{2}=0=f_{3}$ in (1.4). Performing
integration by parts and using the boundary conditions, from the system (1.4),
we deduce

$\frac{d}{dt}\left(\int_{0}^{2\pi}u(t,x)\>{\rm
d}x\right)=0,\quad\frac{d}{dt}\left(\int_{0}^{2\pi}S(t,x)e^{\frac{t}{\kappa}}\>{\rm
d}x\right)=0,$

and therefore,

$\int_{0}^{2\pi}u(T,x)\>{\rm d}x=\int_{0}^{2\pi}u_{0}(x)\>{\rm
d}x,\quad\int_{0}^{2\pi}S(T,x)e^{\frac{T}{\kappa}}\>{\rm
d}x=\int_{0}^{2\pi}S_{0}(x)\>{\rm d}x.$

Thus if the system (1.4) with $f_{2}=0=f_{3}$ is exactly controllable at time
$T>0$ then necessarily

$\int_{0}^{2\pi}u_{0}(x)\>{\rm d}x=0=\int_{0}^{2\pi}S_{0}(x)\>{\rm d}x.$ (1.7)

Let us define the space

$L^{2}_{m}(0,2\pi)=\left\\{f\in L^{2}(0,2\pi):\int_{0}^{2\pi}f(x)\,{\rm
d}x=0\right\\}.$

######  Theorem 1.2.

Let $f_{2}=0=f_{3}$ in (1.4) and $\mathcal{O}_{1}\subseteq(0,2\pi)$. Let us
assume (1.5). Then there exists a $T_{0}>0$ such that the system (1.4) is
exactly controllable in $L^{2}(0,2\pi)\times L^{2}_{m}(0,2\pi)\times
L_{m}^{2}(0,2\pi)$ at time $T>T_{0}$, by a interior control $f_{1}\in
L^{2}\left(0,T;L^{2}(\mathcal{O}_{1})\right)$ for the density.

######  Remark 1.3.

In fact, if the control is used in one of the equations, then the following
exact controllability results can be obtained:

  1. (1)

Let $f_{1}=0=f_{3}$ in (1.4) and $\mathcal{O}_{2}\subseteq(0,2\pi)$. Let us
assume (1.5). Then the system (1.4) is exactly controllable in
$L_{m}^{2}(0,2\pi)\times L^{2}(0,2\pi)\times L_{m}^{2}(0,2\pi)$ at time
$T>T_{0}$, by a interior control $f_{2}\in
L^{2}\left(0,T;L^{2}(\mathcal{O}_{2})\right)$ for the velocity.

  2. (2)

Let $f_{1}=0=f_{2}$ in (1.4) and $\mathcal{O}_{3}\subseteq(0,2\pi)$. Let us
assume (1.5). Then the system (1.4) is exactly controllable in
$L_{m}^{2}(0,2\pi)\times L_{m}^{2}(0,2\pi)\times L^{2}(0,2\pi)$ at time
$T>T_{0}$, by a interior control $f_{3}\in
L^{2}\left(0,T;L^{2}(\mathcal{O}_{3})\right)$ for the stress.

######  Remark 1.4.

Note that the characteristics equations associated to our system (1.4) can be
written as:

$x+\beta_{1}t=c_{1},\quad x+\beta_{2}t=c_{2},\quad x+\beta_{3}t=c_{3},$

where $c_{1},c_{2},c_{3}$ are constants and $\beta_{1},\beta_{2},\beta_{3}$
are the roots (distinct and nonzero) of the equation

$r^{3}+2u_{s}r^{2}+\left(u_{s}^{2}-b\rho_{s}-\frac{\mu}{\kappa\rho_{s}}\right)r-\frac{\mu
u_{s}}{\kappa\rho_{s}}=0.$ (1.8)

In the above theorem, the waiting time $T_{0}$ is of the form

$T_{0}={2\pi}{\left(\frac{1}{|\beta_{1}|}+\frac{1}{|\beta_{2}|}+\frac{1}{|\beta_{3}|}\right)}.$

This result is a consequence of the particular construction on the
biorthogonal family in Theorem 7.8. In Theorem 1.2, the exact controllability
result is proved for any time $T>T_{0}$. We can not claim that $T_{0}$ is the
minimal time to have the exact controllability of the system. Determine the
minimal time $T_{min}>0$, such that the system is exactly controllable at
$T\geq T_{min}$ and the system is not exactly controllable at $T<T_{min}$ is a
challenging open problem. In particular, notice that the minimal control time
should depend on the support of the localized control.

######  Remark 1.5.

Assumption (1.5) may not be a necessary condition for the controllability of
the system (1.4). In this paper, the proof of the controllability results
relies on the spectrum of the linear operator associated with the system. We
have used an Ingham-type inequality to prove these results. To use this
inequality, we need a uniform spectral gap(see 7.1) and condition (1.5)
ensures the same. Thus assuming condition (1.5) is a limitation of our method.

Next, we will study the controllability properties of the linearized
compressible Navier–Stokes system with Maxwell’s law when the control acts
only in the boundary.

###  1.3. Boundary controllability

Let us consider the following system:

$\left.\begin{aligned}
&\partial_{t}\rho+u_{s}\partial_{x}\rho+\rho_{s}\partial_{x}u=0,\qquad&&\text{in
}(0,T)\times(0,2\pi),\\\
&\partial_{t}u+u_{s}\partial_{x}u+a\gamma{\rho_{s}}^{\gamma-2}\partial_{x}\rho-\frac{1}{\rho_{s}}\partial_{x}S=0,\qquad&&\text{in
}(0,T)\times(0,2\pi),\\\
&\partial_{t}S+\frac{1}{\kappa}S-\frac{\mu}{\kappa}\partial_{x}u=0,\qquad&&\text{in
}(0,T)\times(0,2\pi),\\\ &\rho(0,x)=\rho_{0}(x),\quad u(0,x)=u_{0}(x),\quad
S(0,x)=S_{0}(x),\qquad&&x\in(0,2\pi),\end{aligned}\right\\}$ (1.9)

along with one of the following boundary conditions in the time interval
$(0,T):$

$\displaystyle\textit{Control in density:
}\rho(t,0)=\rho(t,2\pi)+q(t),\>u(t,0)=u(t,2\pi),\>S(t,0)=S(t,2\pi),$ (1.10)
$\displaystyle\textit{Control in velocity:
}\rho(t,0)=\rho(t,2\pi),\>u(t,0)=u(t,2\pi)+r(t),\>S(t,0)=S(t,2\pi),$ (1.11)
$\displaystyle\textit{Control in stress:
}\rho(t,0)=\rho(t,2\pi),\>u(t,0)=u(t,2\pi),\>S(t,0)=S(t,2\pi)+p(t),$ (1.12)

where $q,r,p$, are the controls. We recall the definition of boundary exact
controllability of the system (1.9)-(1.10).

######  Definition 1.6.

The system (1.9)-(1.10) is boundary exactly controllable in
$(L^{2}(0,2\pi))^{3}$ at time $T>0$, if for any initial condition
$(\rho_{0},u_{0},S_{0})^{\top}\in(L^{2}(0,2\pi))^{3}$ and any other
$(\rho_{1},u_{1},S_{1})^{\top}\in(L^{2}(0,2\pi))^{3}$, there exists control
$q\in L^{2}\left(0,T\right)$ such that the corresponding solution
$(\rho,u,S)^{\top}$ of (1.9)-(1.10) satisfies

$(\rho,u,S)^{\top}(T,x)=(\rho_{1},u_{1},S_{1})^{\top}(x),\text{ for all
}x\in(0,2\pi).$

We now mention some boundary controllability results of the linearized
compressible Navier-Stokes system. In [9], the authors studied null
controllability of the concerned system in
$H^{s+1}_{\textnormal{m}}(0,2\pi)\times H^{s}(0,2\pi),$ $s>4.5,$ using
boundary control acting at the parabolic component at time
$T>\frac{2\pi}{|u_{s}|}$ with periodic boundary data. Recently, in the work
[27], the above result has been improved. The author established the boundary
null controllability of the system in the space $L^{2}(0,2\pi)\times
L^{2}(0,2\pi)$ using density control and in the space $H^{1}_{m}(0,2\pi)\times
L^{2}(0,2\pi)$ using velocity control in time $T>\frac{2\pi}{|u_{s}|}$. This
paper extends these two results for the linearized compressible Navier-Stokes
equation with non-barotropic fluids. In the context of Dirichlet boundary, the
paper [4] dealt with the boundary null controllability of this system with a
density control in the space $H^{s}(0,1)\times L^{2}(0,1),s>\frac{1}{2}$ at
time $T>1$. The authors have also explored the approximate controllability of
the system in the space $L^{2}(0,1)\times L^{2}(0,1)$ at time $T>1$. Null
controllability “up to a finite-dimensional space” in the space
$H^{s}(0,1)\times L^{2}(0,1),s>\frac{1}{2}$ has been shown by a velocity
control in time $T>1$ with periodic-Dirichlet boundary set up.

Next, we will state our boundary controllability results for the system (1.9).
Observe that, if the system (1.9)-(1.10) is boundary exactly controllable then
necessarily

$\displaystyle\int_{0}^{2\pi}\rho_{0}(x)\,{\rm d}x+$ $\displaystyle
u_{s}\int_{0}^{T}q(t){\rm d}t=0,\quad\int_{0}^{2\pi}u_{0}(x)\,{\rm
d}x+b\int_{0}^{T}q(t)\,{\rm d}t=0,\quad\int_{0}^{2\pi}S_{0}(x)\,{\rm d}x=0,$

which imply

$b\int_{0}^{2\pi}\rho_{0}(x)\,{\rm d}x=u_{s}\int_{0}^{2\pi}u_{0}(x)\,{\rm
d}x,\quad\int_{0}^{2\pi}S_{0}(x)\,{\rm d}x=0.$

For the sake of simplicity, we take $\int_{0}^{2\pi}\rho_{0}(x)\,{\rm
d}x=\int_{0}^{2\pi}u_{0}(x)\,{\rm d}x=0.$ Hence we consider the following
space for our study:

$L_{m}^{2}(0,2\pi)\times L_{m}^{2}(0,2\pi)\times L_{m}^{2}(0,2\pi).$

######  Theorem 1.7.

Let us assume (1.5). Then the system (1.9)-(1.10) is exactly controllable in
$L_{m}^{2}(0,2\pi)\times L_{m}^{2}(0,2\pi)\times L_{m}^{2}(0,2\pi)$ at time
$T>T_{0}$ (same as in 1.4), by a boundary control $q\in L^{2}(0,T)$ for the
density.

######  Remark 1.8.

If the control acts in one of the boundary, then we get the exact
controllability of the system (1.9):

  1. (1)

Let us assume (1.5). Then the control system (1.9)-(1.11) is exactly
controllable in $L_{m}^{2}(0,2\pi)\times L_{m}^{2}(0,2\pi)\times
L_{m}^{2}(0,2\pi)$ at time $T>T_{0}$, by a boundary control $r\in L^{2}(0,T)$
for the velocity.

  2. (2)

Let us assume (1.5). Then the control system (1.9)-(1.12) is exactly
controllable in $L_{m}^{2}(0,2\pi)\times L_{m}^{2}(0,2\pi)\times
L_{m}^{2}(0,2\pi)$ at time $T>T_{0}$, by a boundary control $p\in L^{2}(0,T)$
for the stress.

The proof of the exact controllability result relies on an observability
inequality and the spectral analysis of the linearized operator. The spectrum
of the linear operator consists of three sequences of complex eigenvalues
whose real parts converge to three distinct finite numbers, and the imaginary
parts behave as $n$ for $|n|\rightarrow\infty$ (see, Section 2.1). Therefore
the system behaves like a hyperbolic system. Hence the geometric control
condition is needed for the controllability of the system using localized
control. Moreover, the eigenfunctions of the linearized operator and its
adjoint form Riesz bases on $L^{2}(0,2\pi)\times L^{2}(0,2\pi)\times
L^{2}(0,2\pi)$. Then using the series representation of the solution of the
adjoint problem and a hyperbolic type Ingham inequality, we can prove Theorem
1.2. The proof of the Ingham-type inequality relies on the construction of a
family biorthogonal to the family of exponentials
$\\{e^{-{\lambda_{k}^{l}}t},k\in\mathbb{Z}^{*},l=1,2,3\\}$, see Section 7. The
technique we have adapted is inspired from [28] and [5]. The entire study
relies on the theory of logarithmic integral and complex analysis. One can
refer to the book [42] for more details. Moreover, it is worthwhile to mention
that a modification (for example [10, Proposition 3.1] or [1, Proposition
8.1]) to the general hyperbolic Ingham inequality (see [23], [29, Theorem
4.1]) is not enough to prove the required Ingham-type inequality for our case.

This paper also deals with the lack of controllability of the system (1.4).
For the influence of the transport component of the system, it is relevant to
guess that the system (1.4) cannot be steered to zero (as exact and null
controllability are equivalent for our system) at small time (see Theorem
5.1). More precisely, in Section 5, we have established that, the system is
not exactly controllable at small time by means of localized interior control.
We have utilized the work [3, Section 3] to establish this result. The same
result can be proved for boundary case also (for example, see [8], [27]).

###  1.4. Stabilizability

In this paper, we also study the boundary feedback stabilization of linearized
compressible Navier-Stokes system with Maxwell’s law (1.9) with the control
acting only in the any boundary of density (1.10), velocity (1.11) and stress
(1.12). More precisely, we construct a feedback control law which forces the
solution of (1.9)-(1.10) to decay exponentially towards the origin with any
order. In finite-dimensional cases, the notions of controllability and
complete stabilizability (stabilization with any decay rate) are equivalent.
Whereas for infinite-dimension cases, this does not hold in general. For
bounded control operator, it is known that controllability immediately implies
stabilizability, see Theorem 3.3 in [43], [14]. Thus for the case of interior
control, our linearized Maxwell’s system is exponentially stabilizable with
any decay rate. However, for the boundary control cases, the corresponding
control operator is unbounded. Thus our work adds one more example where
controllability implies stabilizability for the unbounded control operator,
like [7], [8].

In our case, we prove the exponential stabilization result by an explicit
feedback law following a method demonstrated by J.M. Urquiza [36] which relies
on the Gramian approach and the Riccati equations, see [38], [25] for more
details. This method addresses the exponential stabilization issue for an
exactly controllable system with an operator that generates an infinitesimal
group of continuous operator. This approach has been successfully utilized to
study rapid exponential stabilization of KdV equation [7], Boussinesq system
of KdV-KdV type in [6], linearized compressible Navier-Stokes equation in the
case of creeping flow in [8]. Our work regarding exponential stabilization is
inspired from [7], [6] and [8]. Recently, classical moment method has been
extensively utilized to establish the exact controllability and
stabilizability of some dispersive system, see [39], [40]. In these works,
utilizing the group structure of the corresponding system, the authors
conclude the stabilizability from the controllability in some periodic Sobolev
spaces.

Let us state the main stabilization result for the closed-loop system
(1.9)-(1.10).

######  Theorem 1.9.

Let us assume (1.5). Then the control system (1.9)-(1.10) is completely
stabilizable in $L_{m}^{2}(0,2\pi)\times L_{m}^{2}(0,2\pi)\times
L_{m}^{2}(0,2\pi)$ by boundary feedback control $q\in
L^{2}\left(0,\infty\right)$ for the density. That means, for any initial data
$(\rho_{0},u_{0},S_{0})\in L_{m}^{2}(0,2\pi)\times L_{m}^{2}(0,2\pi)\times
L_{m}^{2}(0,2\pi)$ and any $\nu>0$, there exists a continuous linear map $\Pi$
such that the system (1.9)-(1.10) with
$q(t)=\Pi(\left(\rho(t,\cdot),u(t,\cdot),S(t,\cdot)\right))$ satisfies the
following

$\displaystyle\left\|\rho(t,\cdot)\right\|_{L^{2}(0,2\pi)}+\left\|u(t,\cdot)\right\|_{L^{2}(0,2\pi)}+\left\|S(t,\cdot)\right\|_{L^{2}(0,2\pi)}$
$\displaystyle\leqslant Ce^{-\nu
t}\bigg{[}\left\|\rho_{0}\right\|_{L^{2}(0,2\pi)}+\left\|u_{0}\right\|_{L^{2}(0,2\pi)}+\left\|S_{0}\right\|_{L^{2}(0,2\pi)}\bigg{]},$

for all $t>0$, where $C=C(T)$ is a positive constant, independent of
$\rho_{0},u_{0},S_{0},q$ and $t$.

###  1.5. Organization of the paper

The plan of the paper is as follows. In Section 2, we study the well-posedness
of the system (1.4) using semigroup theory, and we determine the adjoint of
the linear operator associated to the coupled system. Then in Section 2.1, we
analyze the behavior of the spectrum of the linearized operator associated to
the system (1.4). After that we show that the eigenfunctions of the linear
operator form a Riesz basis in $(L^{2}(0,2\pi))^{3}$. The exact
controllability result Theorem 1.2 is proved in Section 3. The proof of
boundary exact controllability result Theorem 1.7 is given in Section 4. In
Section 5, we will present a further discussion on the lack of controllability
of the system (1.4) in small time. After that, we study the rapid exponential
stabilization (Theorem 1.9) in Section 6. Section 7 is devoted to the proof of
the Ingham-type inequality 7.22.

##  2\. Linearized operator

Let us define $\mathcal{{Z}}=L^{2}(0,2\pi)\times L^{2}(0,2\pi)\times
L^{2}(0,2\pi)$ and the positive constant

$b=a\gamma{\rho_{s}}^{\gamma-2}.$ (2.1)

Let $\mathcal{{Z}}$ be endowed with the inner product

$\left\langle\begin{pmatrix}\rho\\\ u\\\
S\end{pmatrix},\begin{pmatrix}\sigma\\\ v\\\
\tilde{S}\end{pmatrix}\right\rangle_{\mathcal{{Z}}}=b\int_{0}^{2\pi}\rho\bar{\sigma}\,{\rm
d}x+\rho_{s}\int_{0}^{2\pi}u\bar{v}\,{\rm
d}x+\frac{\kappa}{\mu}\int_{0}^{2\pi}S\bar{\tilde{S}}\,{\rm d}x.$ (2.2)

We now define the unbounded operator
$\left(\mathcal{A},\mathcal{D}(\mathcal{A};\mathcal{{Z}})\right)$ in
$\mathcal{{Z}}$ by

$\mathcal{D}(\mathcal{A};\mathcal{{Z}})=\left\\{\begin{pmatrix}\rho\\\ u\\\
S\end{pmatrix}\in\mathcal{{Z}}\>:\>\left(\rho,u,S\right)^{\top}\in
H^{1}(0,2\pi)\times H^{1}(0,2\pi)\times H^{1}(0,2\pi),\>\begin{aligned}
&\rho(0)=\rho(2\pi),\\\ &u(0)=u(2\pi),\\\
&S(0)=S(2\pi).\end{aligned}\right\\}$

and

$\mathcal{A}=\begin{bmatrix}-u_{s}\frac{d}{dx}\hskip
2.84526pt&-\rho_{s}\frac{d}{dx}\hskip 2.84526pt&0\vspace{2mm}\\\
-b\frac{d}{dx}&-u_{s}\frac{d}{dx}&\frac{1}{\rho_{s}}\frac{d}{dx}\vspace{2mm}\\\
0&\frac{\mu}{\kappa}\frac{d}{dx}&-\frac{1}{\kappa}\end{bmatrix}.$ (2.3)

The control operator $\mathcal{B}\in\mathcal{L}(\mathcal{{Z}};\mathcal{{Z}})$
is defined by

$\mathcal{B}f=\left(\mathbbm{1}_{\mathcal{O}_{1}}f_{1},\mathbbm{1}_{\mathcal{O}_{2}}f_{2},\mathbbm{1}_{\mathcal{O}_{3}}f_{3}\right)^{\top},\qquad
f=(f_{1},f_{2},f_{3})^{\top}\in\mathcal{{Z}}.$ (2.4)

With the above introduced notations, the system (1.4) can be rewritten as

$\dot{z}(t)=\mathcal{A}z(t)+\mathcal{B}f(t),\quad t\in(0,T),\qquad
z(0)=z_{0},$ (2.5)

where, we have set $z(t)=(\rho(t,\cdot),u(t,\cdot),S(t,\cdot))^{\top},$
$z_{0}=(\rho_{0},u_{0},S_{0})^{\top},\text{ and
}f(t)=(f_{1}(t,\cdot),f_{2}(t,\cdot),f_{3}(t,\cdot))^{\top}.$

Next, we will prove the well-posedness of the system (2.5).

######  Proposition 2.1.

The operator $(\mathcal{A},\mathcal{D}(\mathcal{A};\mathcal{{Z}}))$ is the
infinitesimal generator of a strongly continuous semigroup
$\left\\{\mathbb{T}_{t}\right\\}_{t\geq 0}$ on $\mathcal{{Z}}.$ Further, for
any $f\in L^{2}(0,T;\mathcal{{Z}})$ and for any $z_{0}\in\mathcal{{Z}}$, (2.5)
admits a unique solution $(\rho,u,S)\in C([0,T];\mathcal{{Z}})$ with

$\|(\rho,u,S)\|_{C([0,T];\mathcal{{Z}})}\leqslant
C\Big{(}\|z_{0}\|_{\mathcal{{Z}}}+\|f\|_{L^{2}(0,T;\mathcal{{Z}})}\Big{)}.$

###### Proof.

We rewrite $\mathcal{A}:=\mathcal{A}_{1}+\mathcal{A}_{2},$ with

$\mathcal{A}_{1}=\begin{bmatrix}-u_{s}\frac{d}{dx}\hskip
2.84526pt&-\rho_{s}\frac{d}{dx}\hskip 2.84526pt&0\vspace{2mm}\\\
-b\frac{d}{dx}&-u_{s}\frac{d}{dx}&\frac{1}{\rho_{s}}\frac{d}{dx}\vspace{2mm}\\\
0&\frac{\mu}{\kappa}\frac{d}{dx}&0\end{bmatrix},\quad\mathcal{A}_{2}=\begin{bmatrix}0&0&0\\\
0&0&0\\\ 0&0&-\frac{1}{\kappa}\end{bmatrix}.$

Note that

$\left\|\mathcal{A}_{2}\begin{pmatrix}\rho\\\ u\\\
S\end{pmatrix}\right\|_{\mathcal{{Z}}}\leqslant\frac{1}{{\kappa}}\left\|\begin{pmatrix}\rho\\\
u\\\ S\end{pmatrix}\right\|_{\mathcal{{Z}}}\quad\mbox{ for all
}\begin{pmatrix}\rho\\\ u\\\ S\end{pmatrix}\in\mathcal{{Z}}.$

Thus $\mathcal{A}_{2}$ is a bounded perturbation of the operator
$\mathcal{A}_{1}$ on $\mathcal{{Z}}$. Therefore, it is sufficient to show that
$\mathcal{A}_{1}$ generates a $C^{0}$-semigroup on $\mathcal{{Z}},$ (for
details, see [35, Theorem 2.11.2]).

The adjoint $\mathcal{A}^{*}_{1}$ of $\mathcal{A}_{1}$ with the domain
$\mathcal{D}(\mathcal{A}_{1}^{*};\mathcal{{Z}})=\mathcal{D}(\mathcal{A};\mathcal{{Z}})$
is given by

$\mathcal{A}^{*}_{1}=\begin{bmatrix}u_{s}\frac{d}{dx}\hskip
2.84526pt&\rho_{s}\frac{d}{dx}\hskip 2.84526pt&0\vspace{2mm}\\\
b\frac{d}{dx}&u_{s}\frac{d}{dx}&-\frac{1}{\rho_{s}}\frac{d}{dx}\vspace{2mm}\\\
0&-\frac{\mu}{\kappa}\frac{d}{dx}&0\end{bmatrix}=-\mathcal{A}_{1}.$

Thus $\mathcal{A}_{1}$ is a skew-adjoint operator. Then using Stone Theorem
([35, Theorem 3.8.6]), we get that $\mathcal{A}_{1}$ generates a unitary group
on $\mathcal{{Z}}$. ∎

Thus, we have the following well-posedness result for the system (1.4).

######  Theorem 2.2.

For any $(\rho_{0},u_{0},S_{0})^{\top}\in\mathcal{{Z}}$ and $f_{i}\in
L^{2}(0,T;L^{2}(0,2\pi)),\,i=1,2,3,$ the system (1.4) admits a unique solution
$(\rho,u,S)^{\top}\in C\left([0,T];\mathcal{{Z}}\right)$ with

$\|(\rho,u,S)\|_{C([0,T];\mathcal{{Z}})}\leqslant
C\Big{(}\|(\rho_{0},u_{0},S_{0})\|_{\mathcal{{Z}}}+\|(f_{1},f_{2},f_{3})\|_{L^{2}(0,T;\mathcal{{Z}})}\Big{)}.$

Note that, the exact controllability of the pair $(\mathcal{A},\mathcal{B})$
is equivalent to the observability of the pair
$(\mathcal{A}^{*},\mathcal{B}^{*}),$ where $\mathcal{A}^{*}$ and $B^{*}$ are
the adjoint operators of $\mathcal{A}\text{ and }\mathcal{B}$ respectively
(see [13, Chapter 2.3] for details). Thus it is important to derive the
adjoint of the operator $\mathcal{A}:$

######  Proposition 2.3.

The adjoint of $(\mathcal{A},\mathcal{D}(\mathcal{A};\mathcal{{Z}}))$ in
$\mathcal{{Z}}$ is defined by

$\mathcal{D}(\mathcal{A}^{*};\mathcal{{Z}})=\mathcal{D}(\mathcal{A};\mathcal{{Z}}),$
(2.6)

and

$\mathcal{A}^{*}=\begin{bmatrix}u_{s}\frac{d}{dx}\hskip
2.84526pt&\rho_{s}\frac{d}{dx}\hskip 2.84526pt&0\vspace{2mm}\\\
b\frac{d}{dx}&u_{s}\frac{d}{dx}&-\frac{1}{\rho_{s}}\frac{d}{dx}\vspace{2mm}\\\
0&-\frac{\mu}{\kappa}\frac{d}{dx}&-\frac{1}{\kappa}\end{bmatrix}.$ (2.7)

Moreover, $(\mathcal{A}^{*},\mathcal{D}(\mathcal{A}^{*};\mathcal{{Z}}))$ is
the infinitesimal generator of a strongly continuous semigroup
$\left\\{\mathbb{T}^{*}_{t}\right\\}_{t\geq 0}$ on $\mathcal{{Z}}.$

As mentioned in Section 1.2, the controllability results for (1.4) is expected
in the subspaces of $\mathcal{Z}$ satisfying (1.7). Thus, we need to have that
the restriction of the operator $\mathcal{A}$ on those subspaces of
$\mathcal{Z}$ also generates a semigroup. From [35, Proposition 2.4.4, Section
2.4, Chapter 2], the following result holds.

######  Proposition 2.4.

Let us define the spaces

$\displaystyle{\mathcal{Z}_{m}}=$ $\displaystyle L^{2}(0,\pi)\times
L_{m}^{2}(0,\pi)\times L_{m}^{2}(0,\pi),$ (2.8)
$\displaystyle{\mathcal{Z}_{m,m}}=$ $\displaystyle L_{m}^{2}(0,\pi)\times
L_{m}^{2}(0,\pi)\times L_{m}^{2}(0,\pi).$ (2.9)

where the spaces ${\mathcal{Z}_{m}},{\mathcal{Z}_{m,m}}$ are equipped with the
inner product (2.2).

Then $\mathcal{Z}_{m}$ is invariant under the semigroups $\mathbb{T}$ and
$\mathbb{T}^{*}$. The restriction of $\mathbb{T}$ to $\mathcal{Z}_{m}$ is a
strongly continuous semigroup in $\mathcal{Z}_{m}$ generated by
$(\mathcal{A},\mathcal{D}(\mathcal{A};\mathcal{Z}_{m}))$, where
$\mathcal{D}(\mathcal{A};\mathcal{Z}_{m})=\mathcal{D}(\mathcal{A},\mathcal{Z})\cap\mathcal{Z}_{m}$.
Also, the restriction of $\mathbb{T}^{*}$ to $\mathcal{Z}_{m}$ is a strongly
continuous semigroup in $\mathcal{Z}_{m}$ generated by
$(\mathcal{A}^{*},\mathcal{D}(\mathcal{A}^{*};\mathcal{Z}_{m}))$, where
$\mathcal{D}(\mathcal{A}^{*};\mathcal{Z}_{m})=\mathcal{D}(\mathcal{A}^{*},\mathcal{Z})\cap\mathcal{Z}_{m}$.

Similarly, $\mathcal{Z}_{m,m}$ is invariant under the semigroups $\mathbb{T}$
and $\mathbb{T}^{*}$. The restriction of $\mathbb{T}$ to $\mathcal{Z}_{m,m}$
is a strongly continuous semigroup in $\mathcal{Z}_{m,m}$ generated by
$(\mathcal{A},\mathcal{D}(\mathcal{A};\mathcal{Z}_{m,m}))$, where
$\mathcal{D}(\mathcal{A};\mathcal{Z}_{m,m})=\mathcal{D}(\mathcal{A},\mathcal{Z})\cap\mathcal{Z}_{m,m}$.
Also, the restriction of $\mathbb{T}^{*}$ to $\mathcal{Z}_{m,m}$ is a strongly
continuous semigroup in $\mathcal{Z}_{m,m}$ generated by
$(\mathcal{A}^{*},\mathcal{D}(\mathcal{A}^{*};\mathcal{Z}_{m,m}))$, where
$\mathcal{D}(\mathcal{A}^{*};\mathcal{Z}_{m,m})=\mathcal{D}(\mathcal{A}^{*},\mathcal{Z})\cap\mathcal{Z}_{m,m}$.

###  2.1. Spectral analysis of the linearized operator

We define a Fourier basis
$\left\\{\phi_{0,1},\>\phi_{n,l},\>l=1,2,3\right\\}_{n\in\mathbb{Z}^{*}}$
$\left(\>\mathbb{Z}^{*}=\mathbb{Z}\setminus\\{0\\}\right)$ in
$\mathcal{{Z}}_{m}$ as follows :

$\displaystyle\phi_{0,1}(x)=\frac{1}{\sqrt{2\pi b}}\begin{pmatrix}1\\\ 0\\\
0\end{pmatrix},\>\phi_{n,1}(x)=\frac{1}{\sqrt{2\pi
b}}\begin{pmatrix}e^{inx}\\\ 0\\\ 0\end{pmatrix},$
$\displaystyle\phi_{n,2}(x)=\frac{1}{\sqrt{2\pi\rho_{s}}}\begin{pmatrix}0\\\
e^{inx}\\\
0\end{pmatrix},\>\phi_{n,3}(x)=\sqrt{\frac{\mu}{2\pi\kappa}}\begin{pmatrix}0\\\
0\\\ e^{inx}\end{pmatrix},\quad n\in\mathbb{Z}^{*}.$ (2.10)

Let us define the following finite dimensional spaces

$\mathbf{V}_{0}=\text{span}\left\\{\phi_{0,1}\right\\},\>\mathbf{V}_{n}=\text{span}\left\\{\phi_{n,l},\>l=1,2,3\right\\},n\in\mathbb{Z}^{*},$

where ’span’ stands for the vector space generated by those functions. One can
verify that $\mathcal{{Z}}_{m}=\oplus_{n\in\mathbb{Z}}\mathbf{V}_{n}.$

######  Lemma 2.5.

For all $n\in\mathbb{Z}^{*}$, $\mathbf{V}_{n}$ is invariant under
$\mathcal{A}$ and

$\mathcal{A}_{n}=\mathcal{A}|_{\mathbf{V}_{n}}\in\mathcal{L}(\mathbf{V}_{n}),$

has the following matrix representation

$\begin{pmatrix}-inu_{s}&-in\sqrt{b\rho_{s}}&0\\\
-in\sqrt{b\rho_{s}}&-inu_{s}&in\sqrt{\frac{\mu}{\kappa\rho_{s}}}\\\
0&in\sqrt{\frac{\mu}{\kappa\rho_{s}}}&-\frac{1}{\kappa}\end{pmatrix}$

in the basis $\left\\{\phi_{n,l},\>l=1,2,3\right\\}_{n\in\mathbb{Z}^{*}}$ of
$\mathbf{V}_{n}$.

######  Proposition 2.6.

The spectrum of the operator
$(\mathcal{A},\mathcal{D}(\mathcal{A};\mathcal{{Z}}_{m}))$ consists of $0$ and
three sequences $\lambda_{n}^{1}$, $\lambda_{n}^{2}$ and $\lambda_{n}^{3}$ of
eigenvalues. Furthermore, we have the following asymptotic behaviors:

$\left.\begin{aligned}
\lambda_{n}^{1}=&-\omega_{1}+i\beta_{1}n+O\left(\frac{1}{|n|}\right),\\\
\lambda_{n}^{2}=&-\omega_{2}+i\beta_{2}n+O\left(\frac{1}{|n|}\right),\\\
\lambda_{n}^{3}=&-\omega_{3}+i\beta_{3}n+O\left(\frac{1}{|n|}\right),\end{aligned}\right\\}$
(2.11)

for $|n|$ large enough, where $\beta_{j},j=1,2,3$ are the distinct real roots
of the equation

$r^{3}+2u_{s}r^{2}+\left(u_{s}^{2}-b\rho_{s}-\frac{\mu}{\kappa\rho_{s}}\right)r-\frac{\mu
u_{s}}{\kappa\rho_{s}}=0,$ (2.12)

and

$\omega_{j}=\frac{\beta_{j}^{2}+2u_{s}\beta_{j}+u_{s}^{2}-b\rho_{s}}{\kappa\left(3\beta_{j}^{2}+4u_{s}\beta_{j}+u_{s}^{2}-b\rho_{s}-\mu/\kappa\rho_{s}\right)}\neq
0,\quad j=1,2,3.$ (2.13)

###### Proof.

We see that $\mathcal{A}\phi_{0,1}=0\cdot\phi_{0,1}$ and
$\phi_{0,1}\neq\mathbf{0}$. Thus $0$ is an eigenvalue of $\mathcal{A}$.

From 2.5, the characteristic equation is given by:

$\displaystyle\mathcal{F}_{n}(\lambda):=\lambda^{3}+\left(\frac{1}{\kappa}+2inu_{s}\right)\lambda^{2}+$
$\displaystyle\left(-u_{s}^{2}n^{2}+b\rho_{s}n^{2}+\frac{\mu
n^{2}}{\kappa\rho_{s}}+\frac{2iu_{s}n}{\kappa}\right)\lambda$ $\displaystyle+$
$\displaystyle\left(-\frac{u_{s}^{2}n^{2}}{\kappa}+\frac{b\rho_{s}n^{2}}{\kappa}+\frac{i\mu
u_{s}n^{3}}{\kappa\rho_{s}}\right)=0,\quad n\in\mathbb{Z}^{*}.$ (2.14)

Let $\lambda_{n}^{l}=\eta_{n}^{l}+i\tau_{n}^{l}$ with
$\eta_{n}^{l}\in\mathbb{R}$, $\tau_{n}^{l}\in\mathbb{R}$, and $l=1,2,3$ are
the roots of (2.1). Using [44, Theorem 3.2], we can show that all the roots of
(2.1) have negative real parts, i.e.,

$\eta_{n}^{l}<0,\text{ for }l=1,2,3\text{ and }n\in\mathbb{Z}^{*}.$ (2.15)

From the relation between roots and coefficients of the equation (2.1), we get

$\displaystyle\eta_{n}^{1}+\eta_{n}^{2}+\eta_{n}^{3}=-\frac{1}{\kappa},$
(2.16) $\displaystyle\tau_{n}^{1}+\tau_{n}^{2}+\tau_{n}^{3}=-2u_{s}n,$ (2.17)
$\displaystyle\left(\eta_{n}^{1}\eta_{n}^{2}+\eta_{n}^{1}\eta_{n}^{3}+\eta_{n}^{2}\eta_{n}^{3}\right)-\left(\tau_{n}^{1}\tau_{n}^{2}+\tau_{n}^{1}\tau_{n}^{3}+\tau_{n}^{2}\tau_{n}^{3}\right)=\left(b\rho_{s}+\frac{\mu}{\kappa\rho_{s}}-u^{2}_{s}\right)n^{2},$
(2.18)
$\displaystyle\eta_{n}^{1}\left(\tau_{n}^{2}+\tau_{n}^{3}\right)+\eta_{n}^{2}\left(\tau_{n}^{1}+\tau_{n}^{3}\right)+\eta_{n}^{3}\left(\tau_{n}^{1}+\tau_{n}^{2}\right)=\frac{2u_{s}}{\kappa}n,$
(2.19)
$\displaystyle\eta_{n}^{1}\eta_{n}^{2}\eta_{n}^{3}-\eta_{n}^{1}\tau_{n}^{2}\tau_{n}^{3}-\eta_{n}^{2}\tau_{n}^{1}\tau_{n}^{3}-\eta_{n}^{3}\tau_{n}^{1}\tau_{n}^{2}=\frac{u^{2}_{s}-b\rho_{s}}{\kappa}n^{2},$
(2.20)
$\displaystyle\tau_{n}^{1}\tau_{n}^{2}\tau_{n}^{3}-\tau_{n}^{1}\eta_{n}^{2}\eta_{n}^{3}-\eta_{n}^{1}\tau_{n}^{2}\eta_{n}^{3}-\eta_{n}^{1}\eta_{n}^{2}\tau_{n}^{3}=\frac{\mu
u_{s}}{\kappa\rho_{s}}n^{3}.$ (2.21)

We will prove (2.11) in the following steps.

Step 1: Asymptotic behavior of real part of eigenvalues:

Since $\eta_{n}^{l}<0$, for $l=1,2,3$ and $n\in\mathbb{Z}^{*}$, from (2.16),
we observe that the sequence
$\left\\{\eta_{n}^{l}\right\\}_{n\in\mathbb{Z}^{*}}$ is bounded for $l=1,2,3.$
Thus we get $\frac{\eta_{n}^{l}}{n}\rightarrow 0$ as $|n|\rightarrow\infty$,
for $l=1,2,3.$

Step 2: Asymptotic behavior of imaginary part of eigenvalues:

Thanks to (2.16), (2.17) and (2.18), we have

$\displaystyle\left(\left(\eta_{n}^{1}\right)^{2}+\left(\eta_{n}^{2}\right)^{2}+\left(\eta_{n}^{3}\right)^{2}\right)-\left(\left(\tau_{n}^{1}\right)^{2}+\left(\tau_{n}^{2}\right)^{2}+\left(\tau_{n}^{3}\right)^{2}\right)=\frac{1}{\kappa^{2}}-2\left(u_{s}^{2}+b\rho_{s}+\frac{\mu}{\kappa\rho_{s}}\right)n^{2}.$
(2.22)

Set $\tilde{\tau}_{n}^{l}=\frac{\tau_{n}^{l}}{n}$, $l=1,2,3$. Then using Step
1 and (2.22), it follows that

$\lim\limits_{|n|\rightarrow\infty}\left(\left(\tilde{\tau}_{n}^{1}\right)^{2}+\left(\tilde{\tau}_{n}^{2}\right)^{2}+\left(\tilde{\tau}_{n}^{3}\right)^{2}\right)=2\left(u_{s}^{2}+b\rho_{s}+\frac{\mu}{\kappa\rho_{s}}\right).$
(2.23)

Thus the sequence $\left\\{\tilde{\tau}_{n}^{l}\right\\}_{n\in\mathbb{Z}^{*}}$
is bounded for $l=1,2,3.$

Also from (2.17), (2.18) and (2.21), we deduce that

$\left.\begin{aligned}
\tilde{\tau}_{n}^{1}+\tilde{\tau}_{n}^{2}+\tilde{\tau}_{n}^{3}=&-2u_{s},\\\
\lim\limits_{|n|\rightarrow\infty}\left(\tilde{\tau}_{n}^{1}\tilde{\tau}_{n}^{2}+\tilde{\tau}_{n}^{1}\tilde{\tau}_{n}^{3}+\tilde{\tau}_{n}^{2}\tilde{\tau}_{n}^{3}\right)=&u_{s}^{2}-b\rho_{s}-\frac{\mu}{\kappa\rho_{s}},\\\
\lim\limits_{|n|\rightarrow\infty}\tilde{\tau}_{n}^{1}\tilde{\tau}_{n}^{2}\tilde{\tau}_{n}^{3}=&\frac{\mu
u_{s}}{\kappa\rho_{s}}.\end{aligned}\right\\}$ (2.24)

Thus from (2.24), it follows that

$\left.\begin{aligned}
\tilde{\tau}_{n}^{1}+\tilde{\tau}_{n}^{2}+\tilde{\tau}_{n}^{3}=&-2u_{s},\\\
\tilde{\tau}_{n}^{1}\tilde{\tau}_{n}^{2}+\tilde{\tau}_{n}^{1}\tilde{\tau}_{n}^{3}+\tilde{\tau}_{n}^{2}\tilde{\tau}_{n}^{3}=&u_{s}^{2}-b\rho_{s}-\frac{\mu}{\kappa\rho_{s}}+\delta^{1}_{n},\\\
\tilde{\tau}_{n}^{1}\tilde{\tau}_{n}^{2}\tilde{\tau}_{n}^{3}=&\frac{\mu
u_{s}}{\kappa\rho_{s}}+\delta^{2}_{n},\end{aligned}\right\\}$ (2.25)

where $\left|\delta^{l}_{n}\right|\rightarrow 0$, as $|n|\rightarrow\infty$,
for $l=1,2$. Therefore, $\tilde{\tau}_{n}^{l},l=1,2,3$ satisfies the following
equation:

$\mathcal{P}_{n}\left(r\right):=r^{3}+2u_{s}r^{2}+\left(u_{s}^{2}-b\rho_{s}-\frac{\mu}{\kappa\rho_{s}}+\delta^{1}_{n}\right)r-\left(\frac{\mu
u_{s}}{\kappa\rho_{s}}+\delta^{2}_{n}\right)=0,\quad n\in\mathcal{Z}^{*}.$
(2.26)

Set

$\mathcal{P}\left(r\right):=r^{3}+2u_{s}r^{2}+\left(u_{s}^{2}-b\rho_{s}-\frac{\mu}{\kappa\rho_{s}}\right)r-\frac{\mu
u_{s}}{\kappa\rho_{s}}=0.$ (2.27)

Let us denote the roots of (2.27) by $\beta_{l}$, $l=1,2,3$. Observe that the
coefficients of $\mathcal{P}_{n}$ converge to the coefficients of
$\mathcal{P}$, as $|n|\rightarrow\infty$. Then using the fact that the roots
of a polynomial continuously depend on the coefficients of the polynomial (see
[20]), we get that the roots of the polynomial $\mathcal{P}_{n}$ converge to
the roots of the polynomial $\mathcal{P}$, as $|n|\rightarrow\infty$, i.e.,
$\frac{\tau_{n}^{l}}{n}\rightarrow\beta_{l}$ as $|n|\rightarrow\infty$, for
$l=1,2,3.$

Step 3: Behavior of roots of the polynomial $\mathcal{P}$:

The discriminant of the cubic polynomial $\mathcal{P}$ is given by

$\displaystyle D=\frac{9\mu^{2}u_{s}^{2}}{\kappa^{2}\rho_{s}^{2}}+\frac{36\mu
u_{s}^{2}}{\kappa\rho_{s}}\left(b\rho_{s}-u_{s}^{2}\right)+\frac{32\mu
u_{s}^{4}}{\kappa\rho_{s}}+4u_{s}^{2}\left(u_{s}^{2}-b\rho_{s}-\frac{\mu}{\kappa\rho_{s}}\right)^{2}+4\left(b\rho_{s}-u_{s}^{2}+\frac{\mu}{\kappa\rho_{s}}\right)^{3}.$
(2.28)

Using (1.5) and (2.1), we get $D>0$, i.e., the roots of the polynomial
$\mathcal{P}$ are real and distinct, which imply

$\mathcal{P}^{\prime}\left(\beta_{l}\right)=3\beta_{l}^{2}+4u_{s}\beta_{l}+u_{s}^{2}-b\rho_{s}-\frac{\mu}{\kappa\rho_{s}}\neq
0,\text{ for }l=1,2,3.$ (2.29)

Step 4: Asymptotic behavior of the eigenvalues:

Using Step 1 and Step 2, we can rewrite the roots of (2.1) as

$\lambda_{n}=i\beta n+\epsilon_{n},$ (2.30)

where $\beta$ is a root of the equation (2.27) and

$\epsilon_{n}=n\theta_{n}^{1}+in\theta_{n}^{2},\quad\theta_{n}^{l}\rightarrow
0\text{ as }\left|n\right|\rightarrow\infty,\text{ for }l=1,2.$ (2.31)

From (2.31), it follows that

$\frac{\epsilon_{n}}{n}\rightarrow 0\text{ as
}\left|n\right|\rightarrow\infty.$ (2.32)

Now, we will see the asymptotic behavior of $\epsilon_{n}$. From (2.1) and
(2.30), we get that $\epsilon_{n}$ satisfy the following equation:

$\displaystyle\epsilon_{n}^{3}+\left(\frac{1}{\kappa}+\left(3\beta+2u_{s}\right)in\right)\epsilon_{n}^{2}+\left(\left(-3\beta^{2}-4u_{s}\beta-
u_{s}^{2}+b\rho_{s}+\frac{\mu}{\kappa\rho_{s}}\right)n^{2}+2\frac{\beta+u_{s}}{\kappa}in\right)\epsilon_{n}$
$\displaystyle+\frac{b\rho_{s}-u_{s}^{2}-2u_{s}\beta-\beta^{2}}{\kappa}n^{2}=0.$

Using (2.29) and the fact that $\beta$ is root of (2.27), we can rewrite the
above equation in the following form

$\displaystyle\epsilon_{n}=-\frac{\frac{\epsilon_{n}^{2}}{\kappa
n^{2}}-\frac{b\rho_{s}-u_{s}^{2}-2u_{s}\beta-\beta^{2}}{\kappa}}{\left(\frac{2u_{s}+3\beta}{2}+\frac{\epsilon_{n}}{in}\right)^{2}+\mathcal{P}^{\prime}(\beta)-\left(\frac{\left(2u_{s}+3\beta\right)^{2}}{4}+2i\frac{u_{s}+\beta}{\kappa
n}\right)}.$ (2.33)

Denoting numerator of (2.33) by $c_{n}$ and denominator of (2.33) by $d_{n}$,
then using (2.29) and (2.32), we get

$\displaystyle
c_{n}\rightarrow\frac{b\rho_{s}-u_{s}^{2}-2u_{s}\beta-\beta^{2}}{\kappa}\text{
as }\left|n\right|\rightarrow\infty,$ (2.34) $\displaystyle
d_{n}\rightarrow\mathcal{P}^{\prime}(\beta)\neq 0\text{ as
}\left|n\right|\rightarrow\infty.$ (2.35)

Therefore (2.33) is well-defined for $\left|n\right|$ large enough. Thus from
(2.33), (2.34) and (2.35), it follows that

$\epsilon_{n}\rightarrow-\frac{\beta^{2}+2u_{s}\beta+u_{s}^{2}-b\rho_{s}}{\kappa\mathcal{P}^{\prime}(\beta)}:=-\omega(\text{say}),\text{
as }\left|n\right|\rightarrow\infty.$ (2.36)

Next, we will find the explicit asymptotic of the eigenvalues. For that, let
us define

$F\left(\zeta,\eta\right)=\zeta+\frac{-\frac{1}{\kappa}\zeta^{2}\eta^{2}-\frac{b\rho_{s}-u_{s}^{2}-2u_{s}\beta-\beta^{2}}{\kappa}}{\left(\frac{2u_{s}+3\beta}{2}-{\zeta\eta}\right)^{2}+\mathcal{P}^{\prime}(\beta)-\left(\frac{\left(2u_{s}+3\beta\right)^{2}}{4}+2\frac{u_{s}+\beta}{\kappa}\eta\right)},\quad\zeta,\eta\in\mathbb{C}.$
(2.37)

Using (2.29), we can see that $F$ is well defined and analytic in a
neighborhood of $\left(-\omega,0\right)$. Moreover

$F\left(-\omega,0\right)=\lim\limits_{\eta\rightarrow
0}F\left(-\omega,\eta\right)=0,$

and

$\displaystyle\frac{\partial
F}{\partial\zeta}\left(-\omega,0\right)=\lim\limits_{\eta\rightarrow
0}\frac{\partial F}{\partial\zeta}\left(-\omega,\eta\right)=1\neq 0.$

Then using the Implicit Function Theorem, there exist neighborhoods
$N\left(-\omega\right)$ and $N\left(0\right)$ such that the equation
$F\left(\zeta,\eta\right)=0$ has a unique root $\zeta=G\left(\eta\right)$ in
$N\left(-\omega\right)$ for any given $\eta\in N\left(0\right)$. Moreover, the
function $G\left(\eta\right)$ is single-valued and analytic in
$N\left(0\right)$, and satisfies the condition $G\left(0\right)=-\omega$.
Therefore we obtain

$\displaystyle\zeta=G\left(\eta\right)=$ $\displaystyle
G\left(0\right)+G^{\prime}\left(0\right)\eta+\dots$ $\displaystyle=$
$\displaystyle-\omega+O\left(\eta\right)\quad\text{ in }N\left(0\right).$
(2.38)

From (2.33), (2.36), (2.37) and (2.1), it follows that, for $|n|$ large
enough, $\epsilon_{n}\in N\left(-\omega\right)$, $\frac{i}{n}\in N(0)$,
$F\left(\epsilon_{n},\frac{i}{n}\right)=0$ and

$\epsilon_{n}=-\omega+O\left(\frac{1}{|n|}\right).$

Thus from (2.30), we get

$\lambda_{n}=-\omega+i\beta n+O\left(\frac{1}{|n|}\right),$ (2.39)

for $|n|$ large enough.

Since the equation (2.27) has three distinct real roots, we get three distinct
real values of $\beta$ and for each $\beta$, from (2.36), we get a real
$\omega.$ Furthermore, we can show that
$\omega_{1}\neq\omega_{2}\neq\omega_{3}\neq 0$. Indeed, suppose that
$\omega_{j}=0$, which implies

$\beta_{j}^{2}+2u_{s}\beta_{j}+u_{s}^{2}-b\rho_{s}=0.$ (2.40)

Since $\beta_{j}$ is a root of the equation (2.27), we get

$\displaystyle\beta_{j}^{3}+2u_{s}\beta_{j}^{2}+\left(u_{s}^{2}-b\rho_{s}-\frac{\mu}{\kappa\rho_{s}}\right)\beta_{j}-\frac{\mu
u_{s}}{\kappa\rho_{s}}=0,$

which along with (2.40) implies $\beta_{j}=-u_{s}$. This is a contradiction,
since $-u_{s}$ is not a root of the equation (2.27) $\left(\text{ as
}\mathcal{P}\left(-u_{s}\right)=b\rho_{s}u_{s}>0\right)$.

Moreover, from (2.29) and (2.36), we have

$\omega_{j}-\omega_{l}=\frac{\beta_{j}-\beta_{l}}{\kappa\mathcal{P}^{\prime}(\beta_{j})\mathcal{P}^{\prime}(\beta_{l})}\left[-\frac{2\mu
u_{s}^{2}}{\kappa\rho_{s}\beta_{p}}-\beta_{p}\left(2b\rho_{s}-2u_{s}^{2}-\frac{\mu}{\kappa\rho_{s}}\right)-2u_{s}(b\rho_{s}-u_{s}^{2})\right],$
(2.41)

where $j,l,p\in\\{1,2,3\\},j\neq l\neq p.$ Using the fact that $\beta_{j}$ are
distinct and using (1.5), from (2.41), one can show that
$\omega_{j}\neq\omega_{l}$. Thus 2.6 follows. ∎

######  Remark 2.7.

From the asymptotic behaviors (2.11) of the eigenvalues, it is clear that all
eigenvalues are simple at least for large $n$. Thus if there exist multiple
eigenvalues depending on the system parameters $\rho_{s},u_{s},b,\kappa,\mu$,
they are finite in numbers.

Figure 1. Eigenvalues of $\mathcal{A}$ in the complex plane for $|n|$ varies
from $1$ to $30$ when $\mu=\rho_{s}=u_{s}=b=1$ and k=1.

For the case of multiple eigenvalues the following results follow by
introducing generalized eigenfunctions suitably (for more details see [1,
Section 7]). From the previous remark, we see that the multiple eigenvalues of
$\mathcal{A}$, if at all they occur, are only finitely many. Since the
asymptotic behaviors of the coefficients of the eigen vectors are needed for
our later analysis, we avoid the explicit computation of the generalized
eigenfunctions.

Thus, now onwards, we assume the following :

The spectrum of $\mathcal{A}$ has simple eigenvalues on $\mathcal{{Z}}_{m}$.
(${{\mathcal{H}}}$)

We define the family
$\left\\{\xi_{0}\right\\}\cup\left\\{\xi_{n,l}\>|\>1\leqslant l\leqslant
3,n\in\mathbb{Z}^{*}\right\\}$ as follows :

We choose a normalized eigenfunctions of $\mathcal{A}$ for the eigenvalue
$\lambda_{0}=0$ defined by

$\xi_{0}=\frac{1}{\sqrt{2b\pi}}\begin{pmatrix}1\\\ 0\\\ 0\end{pmatrix},$

and for the eigenvalue $\lambda_{n}^{l}$, the normalized eigenfunction defined
by

$\xi_{n,l}=\frac{1}{\theta_{n,l}}\begin{pmatrix}-1\vspace{1mm}\\\
\frac{\lambda_{n}^{l}+inu_{s}}{{in\rho_{s}}}\vspace{1mm}\\\
\frac{\mu\left(\lambda_{n}^{l}+inu_{s}\right)}{\rho_{s}\left(1+\kappa\lambda_{n}^{l}\right)}\end{pmatrix}e^{inx},\>l\in\left\\{1,2,3\right\\},\>n\in\mathbb{Z}^{*},$
(2.42)

where

$\theta_{n,l}=\sqrt{2{\pi}\left(b+\frac{\left|\lambda_{n}^{l}+inu_{s}\right|^{2}}{\rho_{s}n^{2}}+\frac{\kappa\mu\left|\lambda_{n}^{l}+inu_{s}\right|^{2}}{\rho_{s}^{2}\left|1+\kappa\lambda_{n}^{l}\right|^{2}}\right)}.$
(2.43)

$\left|1+\kappa\lambda_{n}^{l}\right|\neq 0\left(l=1,2,3\right)$, as
$-1/\kappa$ is not a root of the characteristic polynomial (2.1).
Similarly, we choose
$\left\\{\xi^{*}_{0}\right\\}\cup\left\\{\xi^{*}_{n,l}\>|\>1\leqslant
l\leqslant 3,n\in\mathbb{Z}^{*}\right\\}$ as follows :

$\xi^{*}_{0}$ is an eigenfunction of $\mathcal{A}^{*}$ with the eigenvalue
$\lambda_{0}=0$ defined by

$\xi^{*}_{0}=\frac{1}{\sqrt{2b\pi}}\begin{pmatrix}1\\\ 0\\\ 0\end{pmatrix},$
(2.44)

and for the eigenvalue $\overline{\lambda_{n}^{l}}$, the eigenfunction defined
by

$\xi^{*}_{n,l}=\frac{1}{\psi_{n,l}}\begin{pmatrix}1\vspace{1mm}\\\
\frac{\overline{\lambda_{n}^{l}}-inu_{s}}{{in\rho_{s}}}\vspace{1mm}\\\
-\frac{\mu\left(\overline{\lambda_{n}^{l}}-inu_{s}\right)}{\rho_{s}\left(1+\kappa\overline{\lambda_{n}^{l}}\right)}\end{pmatrix}e^{inx},\>l\in\left\\{1,2,3\right\\},\>n\in\mathbb{Z}^{*},$
(2.45)

where

$\psi_{n,l}=\frac{\sqrt{{2\pi}}\left(-b+\frac{\left(\overline{\lambda_{n}^{l}+inu_{s}}\right)^{2}}{\rho_{s}n^{2}}-\frac{\mu\kappa\left(\overline{\lambda_{n}^{l}+inu_{s}}\right)^{2}}{\rho_{s}^{2}\left(1+\kappa\overline{\lambda_{n}^{l}}\right)^{2}}\right)}{\sqrt{\left(b+\frac{\left|\lambda_{n}^{l}+inu_{s}\right|^{2}}{\rho_{s}n^{2}}+\frac{\kappa\mu\left|\lambda_{n}^{l}+inu_{s}\right|^{2}}{\rho_{s}^{2}\left|1+\kappa\lambda_{n}^{l}\right|^{2}}\right)}}.$
(2.46)

Here for all $n\in\mathbb{Z}^{*}$, $\psi_{n,l}\neq 0$ for $l=1,2,3$, because
of the assumption (${{\mathcal{H}}}$).

The choice of this family of eigenfunctions of $\mathcal{A}$ and
$\mathcal{A}^{*}$ ensures the following lemma.

######  Lemma 2.8.

Under the assumption (${{\mathcal{H}}}$), the families
$\left\\{\xi_{0}\right\\}\cup\left\\{\xi_{n,l}\>|\>1\leqslant l\leqslant
3,n\in\mathbb{Z}^{*}\right\\}$ and
$\left\\{\xi^{*}_{0}\right\\}\cup\left\\{\xi^{*}_{n,l}\>|\>1\leqslant
l\leqslant 3,n\in\mathbb{Z}^{*}\right\\}$ satisfy the following bi-
orthonormality relations :

$\displaystyle\left\langle\xi_{0},\xi^{*}_{k,p}\right\rangle_{\mathcal{Z}}=0,\quad\left\langle\xi_{n,l},\xi^{*}_{k,p}\right\rangle_{\mathcal{Z}}=\delta_{k}^{n}\delta_{p}^{l},\>l,p\in\left\\{1,2,3\right\\},\>k,n\in\mathbb{Z}^{*},$
(2.47)
$\displaystyle\left\langle\xi_{0},\xi^{*}_{0}\right\rangle_{\mathcal{Z}}=1,\quad\left\langle\xi_{n,l},\xi^{*}_{0}\right\rangle_{\mathcal{Z}}=0,\>l\in\left\\{1,2,3\right\\},\>n\in\mathbb{Z}^{*},$
(2.48)

where

$\delta_{n}^{l}=\begin{cases}0,\>n\neq l\\\ 1,\>n=l.\end{cases}$

Moreover, the asymptotic behaviors of $\theta_{n,l}$ and $\psi_{n,l}$ are as
follows :

$\displaystyle\left|\theta_{n,l}\right|,\left|\psi_{n,l}\right|\rightarrow\sqrt{2{\pi}\left(b+\frac{\left(\beta_{l}+u_{s}\right)^{2}}{\rho_{s}}+\frac{\mu\left(\beta_{l}+u_{s}\right)^{2}}{\kappa\rho_{s}^{2}\beta_{l}^{2}}\right)},\text{
for }l=1,2,3\text{ as }|n|\rightarrow\infty.$ (2.49)

Also if we write the eigenfunctions of $\mathcal{A}^{*}$ in the following way

$\xi_{n,l}^{*}=\frac{1}{\psi_{n,l}}\begin{pmatrix}\alpha_{n,l}^{1}\\\
\alpha_{n,l}^{2}\\\ \alpha_{n,l}^{3}\end{pmatrix}e^{inx},\quad
n\in\mathbb{Z}^{*},l=1,2,3,$ (2.50)

then we have the following asymptotic behaviors

$\displaystyle\alpha_{n,l}^{1}=1,\text{ for }l=1,2,3,\text{ for all
}n\in\mathbb{Z}^{*},$ (2.51)
$\displaystyle\left|\alpha_{n,l}^{2}\right|\rightarrow\left|\frac{\beta_{l}+u_{s}}{\rho_{s}}\right|,\text{
for }l=1,2,3,\text{ as }|n|\rightarrow\infty,$ (2.52)
$\displaystyle\left|\alpha_{n,l}^{3}\right|\rightarrow\left|\frac{\mu\left(\beta_{l}+u_{s}\right)}{\kappa\rho_{s}\beta_{l}}\right|,\text{
for }l=1,2,3,\text{ as }|n|\rightarrow\infty.$ (2.53)

Now, we show that the eigenfunctions of $\mathcal{A}$ form a Riesz basis in
$\mathcal{Z}_{m}$. Recall from (2.1) that
$\left\\{\phi_{0,1},\phi_{n,1},\phi_{n,2},\phi_{n,3}|n\in\mathbb{Z}^{*}\right\\}$
forms an orthonormal basis in $\mathcal{Z}_{m}$.

For the definition of Riesz basis in $\mathcal{Z}_{m}$, we refer [35,
Definition 2.5.1, Chapter 2, Section 2.5]. In other words, the sequence
$\left\\{\xi_{0}\right\\}\cup\left\\{\xi_{n,l}\>|\>1\leqslant l\leqslant
3\right\\}_{n\in\mathbb{Z}^{*}}$ forms a Riesz basis in $\mathcal{Z}_{m}$ if
there exists an invertible operator
$Q\in\mathcal{L}\left(\mathcal{Z}_{m}\right)$ such that

$Q\phi_{0,1}=\xi_{0},\>Q\phi_{n,1}=\xi_{n,1},\>Q\phi_{n,2}=\xi_{n,2},\>Q\phi_{n,3}=\xi_{n,3},\>\forall
n\in\mathbb{Z}^{*}.$

######  Lemma 2.9.

Assume (${{\mathcal{H}}}$). Let us set
$\mathcal{B}_{n}^{1}=\left\\{\phi_{n,l}\>|l=1,2,3\right\\}$ and
$\mathcal{B}_{n}^{2}=\left\\{\xi_{n,l}\>|l=1,2,3\right\\}$ for all
$n\in\mathbb{Z}^{*}$, where $\left\\{\phi_{n,l}\right\\}_{1\leqslant
l\leqslant 3}$ and $\left\\{\xi_{n,l}\right\\}_{1\leqslant l\leqslant 3}$ are
defined as in (2.1) and (2.42). Let $z_{n}\in\mathbf{V}_{n}$ be expressed in
the basis $\mathcal{B}_{n}^{1}$ and $\mathcal{B}_{n}^{2}$ as follows :

$z_{n}=\sum\limits_{p=1}^{3}c_{n,p}\phi_{n,p}=\sum\limits_{l=1}^{3}d_{n,l}\xi_{n,l},$
(2.54)

and let the operator $\Gamma_{n}\in\mathbb{C}^{3\times 3}$ be defined by

$\Gamma_{n}\begin{pmatrix}c_{n,1}\\\ c_{n,2}\\\
c_{n,3}\end{pmatrix}=\begin{pmatrix}d_{n,1}\\\ d_{n,2}\\\
d_{n,3}\end{pmatrix}.$ (2.55)

Then there exist positive constants $l_{1}$ and $l_{2}$ independent of $n$
such that

$\left\|\Gamma_{n}\right\|<l_{1},\quad\left\|\Gamma_{n}^{-1}\right\|<l_{2},\text{
for all }n\in\mathbb{Z}^{*}.$

###### Proof.

Let $n\in\mathbb{Z}^{*}$ and $z_{n}$ as in (2.54). Taking inner product of
$z_{n}$ with each $\xi_{n,l}^{*}$ for $1\leqslant l\leqslant 3$, and using the
bi-orthonormality relation (2.47), we get

$\displaystyle d_{n,l}=$ $\displaystyle\left\langle
z_{n},\xi_{n,l}^{*}\right\rangle_{\mathcal{{Z}}}=\sum\limits_{p=1}^{3}c_{n,p}\left\langle\phi_{n,p},\xi_{n,l}^{*}\right\rangle_{\mathcal{{Z}}}$
$\displaystyle=$ $\displaystyle\sqrt{2\pi
b}\frac{\overline{\alpha_{n,l}^{1}}}{\overline{\psi_{n,l}}}c_{n,1}+\sqrt{2\pi\rho_{s}}\frac{\overline{\alpha_{n,l}^{2}}}{\overline{\psi_{n,l}}}c_{n,2}+\sqrt{\frac{2\pi\kappa}{\mu}}\frac{\overline{\alpha_{n,l}^{3}}}{\overline{\psi_{n,l}}}c_{n,3}.$

Then the matrix $\Gamma_{n}$ in (2.55) is given by

$\Gamma_{n}=\begin{pmatrix}\sqrt{2\pi
b}\frac{\overline{\alpha_{n,1}^{1}}}{\overline{\psi_{n,1}}}&\sqrt{2\pi\rho_{s}}\frac{\overline{\alpha_{n,1}^{2}}}{\overline{\psi_{n,1}}}&\sqrt{\frac{2\pi\kappa}{\mu}}\frac{\overline{\alpha_{n,1}^{3}}}{\overline{\psi_{n,1}}}\\\
\sqrt{2\pi
b}\frac{\overline{\alpha_{n,2}^{1}}}{\overline{\psi_{n,2}}}&\sqrt{2\pi\rho_{s}}\frac{\overline{\alpha_{n,2}^{2}}}{\overline{\psi_{n,2}}}&\sqrt{\frac{2\pi\kappa}{\mu}}\frac{\overline{\alpha_{n,2}^{3}}}{\overline{\psi_{n,2}}}\\\
\sqrt{2\pi
b}\frac{\overline{\alpha_{n,3}^{1}}}{\overline{\psi_{n,3}}}&\sqrt{2\pi\rho_{s}}\frac{\overline{\alpha_{n,3}^{2}}}{\overline{\psi_{n,3}}}&\sqrt{\frac{2\pi\kappa}{\mu}}\frac{\overline{\alpha_{n,3}^{3}}}{\overline{\psi_{n,3}}}\end{pmatrix}.$
(2.56)

Then using (${{\mathcal{H}}}$), we obtain

$\displaystyle\det\left(\Gamma_{n}\right)=$ $\displaystyle\frac{2\pi\sqrt{2\pi
b\rho_{s}\kappa\mu}}{in\rho_{s}^{2}\overline{\psi_{n,1}}\overline{\psi_{n,2}}\overline{\psi_{n,3}}}\begin{vmatrix}1&\lambda_{n}^{1}+inu_{s}&\frac{\lambda_{n}^{1}+inu_{s}}{1+\kappa\lambda_{n}^{1}}\\\
1&\lambda_{n}^{2}+inu_{s}&\frac{\lambda_{n}^{2}+inu_{s}}{1+\kappa\lambda_{n}^{2}}\\\
1&\lambda_{n}^{3}+inu_{s}&\frac{\lambda_{n}^{3}+inu_{s}}{1+\kappa\lambda_{n}^{3}}\end{vmatrix}$
$\displaystyle=$ $\displaystyle\frac{2\pi\kappa\sqrt{2\pi
b\rho_{s}\kappa\mu}}{in\rho_{s}^{2}\overline{\psi_{n,1}}\overline{\psi_{n,2}}\overline{\psi_{n,3}}}\frac{\left(\lambda_{n}^{1}-\lambda_{n}^{2}\right)\left(\lambda_{n}^{1}-\lambda_{n}^{3}\right)\left(\lambda_{n}^{2}-\lambda_{n}^{3}\right)\left(1-\kappa
inu_{s}\right)}{\left(1+\kappa\lambda_{n}^{1}\right)\left(1+\kappa\lambda_{n}^{2}\right)\left(1+\kappa\lambda_{n}^{3}\right)}\neq
0.$ (2.57)

Using (2.11) and (2.49), we get

$\left|\det\left(\Gamma_{n}\right)\right|\rightarrow C\text{ as
}|n|\rightarrow\infty,$

where $C$ is a positive constant. Thus, for each
$n\in\mathbb{Z}^{*},\Gamma_{n}$ is invertible with the inverse
$\Gamma_{n}^{-1}$.

From the matrix $\Gamma_{n}$, using (2.49), (2.51), (2.52) and (2.53), we get
a positive constant $l_{1}$ independent of $n$ such that

$\left\|\Gamma_{n}\right\|<l_{1},\>\forall\>n\in\mathbb{Z}^{*}.$

Similarly, we can get a positive constant $l_{2}$ independent of $n$ such that

$\left\|\Gamma_{n}^{-1}\right\|<l_{2},\>\forall\>n\in\mathbb{Z}^{*}.$

∎

Using 2.9 and from the second part of [35, Proposition 2.5.3, Chapter 2,
Section 2.5], we can see that the eigenfunctions of $\mathcal{A}$,
$\big{\\{}\xi_{0},\xi_{n,l}\>|1\leqslant l\leqslant
3,\>n\in\mathbb{Z}^{*}\big{\\}}$ forms a Riesz basis in $\mathcal{Z}_{m}$.
Also, using [35, Proposition 2.8.6, Chapter 2, Section 2.8], [35, Definition
2.6.1, Chapter 2, Section 2.6] and 2.8, we obtain
$\left\\{\xi_{0}^{*},\xi^{*}_{n,l}\>|1\leqslant l\leqslant
3,\>n\in\mathbb{Z}^{*}\right\\}$ forms a Riesz basis in $\mathcal{Z}_{m}$.

######  Proposition 2.10.

Assume (${{\mathcal{H}}}$) holds. Then any $z\in{\mathcal{{Z}}_{m}}$ can be
uniquely represented as

$z=\sum\limits_{n\in\mathbb{Z}^{*}}\sum\limits_{l=1}^{3}\left\langle
z,\xi^{*}_{n,l}\right\rangle_{\mathcal{{Z}}}\xi_{n,l}+\left\langle
z,\xi^{*}_{0}\right\rangle_{\mathcal{{Z}}}\xi_{0}.$ (2.58)

There exist positive numbers $C_{1}$ and $C_{2}$ such that

$C_{1}\left(\left|\left\langle
z,\xi^{*}_{0}\right\rangle\right|^{2}+\sum_{n\in\mathbb{Z}^{*}}\sum_{l=1}^{3}\left|\left\langle
z,\xi^{*}_{n,l}\right\rangle\right|^{2}\right)\leqslant\left\|z\right\|^{2}_{{\mathcal{{Z}}}}\leqslant
C_{2}\left(\left|\left\langle
z,\xi^{*}_{0}\right\rangle\right|^{2}+\sum_{n\in\mathbb{Z}^{*}}\sum_{l=1}^{3}\left|\left\langle
z,\xi^{*}_{n,l}\right\rangle\right|^{2}\right).$ (2.59)

Similarly, any $z\in\mathcal{{Z}}_{m}$ can be uniquely represented in the
basis $\left\\{\xi_{0}^{*}\right\\}\cup\\{\xi^{*}_{n,l}\>|1\leqslant
l\leqslant 3,\>n\in\mathbb{Z}^{*}\\}$ by

$z=\sum\limits_{n\in\mathbb{Z}^{*}}\sum\limits_{l=1}^{3}\left\langle
z,\xi_{n,l}\right\rangle_{\mathcal{{Z}}}\xi^{*}_{n,l}+\left\langle
z,\xi_{0}\right\rangle_{\mathcal{{Z}}}\xi^{*}_{0},$ (2.60)

and

$\frac{1}{C_{2}}\left(\left|\left\langle
z,\xi_{0}\right\rangle\right|^{2}+\sum_{n\in\mathbb{Z}^{*}}\sum_{l=1}^{3}\left|\left\langle
z,\xi_{n,l}\right\rangle\right|^{2}\right)\leqslant\left\|z\right\|^{2}_{{\mathcal{{Z}}}}\leqslant\frac{1}{C_{1}}\left(\left|\left\langle
z,\xi_{0}\right\rangle\right|^{2}+\sum_{n\in\mathbb{Z}^{*}}^{\infty}\sum_{l=1}^{3}\left|\left\langle
z,\xi_{n,l}\right\rangle\right|^{2}\right).$ (2.61)

As a consequence of the above proposition, from [35, Remark 2.6.4, Chapter 2,
Section 2.6], we get the following result of spectrum of $\mathcal{A}$ and
$\mathcal{A}^{*}$.

######  Theorem 2.11.

Assume (${{\mathcal{H}}}$) holds. Then the spectrum of the operator
$(\mathcal{A},\mathcal{D}(\mathcal{A};\mathcal{{Z}}_{m}))$ and
$(\mathcal{A}^{*},\mathcal{D}(\mathcal{A}^{*};\mathcal{{Z}}_{m}))$ are given
by

$\displaystyle\sigma(\mathcal{A})$
$\displaystyle=\left\\{\lambda_{0}=0,\lambda_{n}^{j},\>n\in\mathbb{Z}^{*},\>j=1,2,3\right\\},$
$\displaystyle\sigma(\mathcal{A}^{*})$
$\displaystyle=\left\\{\lambda_{0},\overline{\lambda_{n}^{j}},\>n\in\mathbb{Z}^{*},\>j=1,2,3\right\\}.$

##  3\. Interior controllability

This section is devoted to the exact controllability of the system (1.4) by
means of localized interior control acting in any of the equations. Our proof
relies on the duality between the controllability of the system and the
observability of the corresponding adjoint system. Here we only prove the
exact controllability result when the control acts only in the density
equation. For the other two cases, the proofs follow in a similar fashion.

We recall the final state observability of
$(\mathcal{A}^{*},\mathcal{B}^{*}):$ For
$(\mathcal{A}^{*},\mathcal{D}(\mathcal{A}^{*};\mathcal{{Z}}_{m}))$ defined in
(2.6)-(2.7) and $(\sigma_{T},v_{T},\tilde{S}_{T})^{\top}\in\mathcal{{Z}}_{m},$
we set

$(\sigma(t),v(t),\tilde{S}(t))^{\top}=\mathbb{T}_{T-t}^{*}(\sigma_{T},v_{T},\tilde{S}_{T})^{\top}\qquad(t\in[0,T]),$

where $\mathbb{T}^{*}$ is the $C^{0}$-semigroup generated by
$(\mathcal{A}^{*},\mathcal{D}(\mathcal{A}^{*};\mathcal{{Z}}_{m}))$ on
$\mathcal{{Z}}_{m}$. In view of 2.3, $(\sigma,v,\tilde{S})^{\top}$ belongs to
$C([0,T];\mathcal{{Z}}_{m})$ and satisfies:

$\begin{cases}\partial_{t}\sigma+u_{s}\partial_{x}\sigma+{\rho_{s}}\partial_{x}v=0&\mbox{
in }(0,T)\times(0,2\pi),\\\
\partial_{t}v+b\partial_{x}\sigma+u_{s}\partial_{x}v-\frac{1}{\rho_{s}}\partial_{x}\tilde{S}=0&\mbox{
in }(0,T)\times(0,2\pi),\\\
\partial_{t}\tilde{S}-\frac{1}{\kappa}\tilde{S}-\frac{\mu}{\kappa}\partial_{x}v=0&\mbox{
in }(0,T)\times(0,2\pi),\\\
\sigma(t,0)=\sigma(t,2\pi),v(t,0)=v(t,2\pi),\>\tilde{S}(t,0)=\tilde{S}(t,2\pi),&\mbox{
in }(0,T),\\\ \sigma(T,x)=\sigma_{T}(x),\quad
v(T,x)=v_{T}(x),\quad\tilde{S}(T,x)=\tilde{S}_{T}(x)&\mbox{ in
}(0,2\pi).\end{cases}$ (3.1)

Here, at first we discuss a standard approach to deduce the observability
inequality for the adjoint system (3.1) which essentially gives the exact
controllability of the main system (1.4). Let us consider the linear map
$\mathcal{F}_{T}:L^{2}(0,T,L^{2}(0,2\pi)\to(L^{2}(0,2\pi))^{3}$ by
$\mathcal{F}_{T}(f_{1})=\left(\rho(T,\cdot),u(T,\cdot),S(T,\cdot)\right)^{\top}$
where $(\rho,u,S)^{\top}$ is the solution of the system (1.4) with
$(\rho_{0},u_{0},S_{0})^{\top}=(0,0,0)^{\top}$ and $f_{2}=f_{3}=0$. It is
clear that exact controllability of (1.4) is equivalent to the surjectivity of
the linear map $\mathcal{F}_{T}$. Note that the map $\mathcal{F}_{T}$ is
surjective if and only if there exists a constant $C>0$ such that the
following inequality holds:

$\displaystyle\left\|\mathcal{F}_{T}^{*}{z}\right\|_{L^{2}(0,T,L^{2}(0,2\pi)}\geq
C\left\|z\right\|_{(L^{2}(0,2\pi))^{3}},\,\text{ for all
}z\in(L^{2}(0,2\pi))^{3}.$ (3.2)

A direct computation shows that
$\mathcal{F}_{T}^{*}((\sigma_{T},v_{T},\tilde{S}_{T})^{\top})=\mathbbm{1}_{\mathcal{O}_{1}}\sigma$,
where $(\sigma,v,\tilde{S})$ is the solution of the adjoint system (3.1) with
the terminal data $(\sigma_{T},v_{T},\tilde{S}_{T})$. Thus the exact
controllability of the system (1.4) is equivalent to the following
observability inequality:

######  Proposition 3.1.

Let $T>0$. Assume $f_{2}=0=f_{3}$. Then the system (1.4) is exactly
controllable in $\mathcal{{Z}}_{m}$ at time $T>0$ using a control $f_{1}$ in
$L^{2}(0,T;L^{2}(0,2\pi))$ with support in $\mathcal{O}_{1}$ acting only in
the density equation, if and only if, there exists a positive constant
$C_{T}>0$ such that for any
$(\sigma_{T},v_{T},\tilde{S}_{T})^{\top}\in\mathcal{{Z}}_{m}$,
$(\sigma,v,\tilde{S})^{\top}$, the solution of (3.1), satisfies the following
observability inequality:

$\int_{0}^{2\pi}|\sigma_{T}(x)|^{2}\ {\rm
d}x\;+\;\int_{0}^{2\pi}|v_{T}(x)|^{2}\ {\rm
d}x+\;\int_{0}^{2\pi}|\tilde{S}_{T}(x)|^{2}\ {\rm d}x\leqslant
C_{T}\int_{0}^{T}\int_{\mathcal{O}_{1}}|\sigma(t,x)|^{2}\,{\rm d}x\,{\rm d}t.$
(3.3)

Next, we prove Theorem 1.2 by showing the observability inequality (3.3) using
the following Ingham-type inequality.

######  Proposition 3.2.

Let
$T>{2\pi}\left(\frac{1}{|\beta_{1}|}+\frac{1}{|\beta_{2}|}+\frac{1}{|\beta_{3}|}\right)$
and $M\in\mathbb{N}$. Then there exist positive constants $C$ and $C_{1}$
depending on $T$ such that for $g(t)=\displaystyle\sum_{|n|\geq
M}\sum_{l=1}^{3}{a_{n}^{l}e^{\overline{\lambda_{n}^{l}}(T-t)}}$ with
$\displaystyle{\sum_{|n|\geq M}}\sum_{l=1}^{3}|a_{n}^{l}|^{2}<\infty$, the
following inequality holds:

$C\sum_{|n|\geq
M}\sum_{l=1}^{3}|a_{n}^{l}|^{2}\leqslant\int_{0}^{T}|g(t)|^{2}\,{\rm
d}t\leqslant C_{1}\sum_{|n|\geq M}\sum_{l=1}^{3}|a_{n}^{l}|^{2}.$ (3.4)

###### Proof.

The proof follows trivially from (7.22) of Section 7 by suitable choices of
$\\{a_{n}^{l}\\}$ and a change of variable. ∎

###  3.1. Proof of Theorem 1.2:

Let $(\sigma_{T},v_{T},\tilde{S}_{T})^{\top}\in\mathcal{{Z}}_{m}.$ From 2.10,
we have

$\begin{pmatrix}\sigma_{T}\\\ v_{T}\\\
\tilde{S}_{T}\end{pmatrix}=\sum\limits_{n\in\mathbb{Z}^{*}}\sum\limits_{l=1}^{3}c_{n,l}\xi^{*}_{n,l}+c_{0}\xi_{0}^{*},\quad\mathrm{with}\quad\sum\limits_{n\in\mathbb{Z}^{*}}\sum\limits_{l=1}^{3}\left|c_{n,l}\right|^{2}+\left|c_{0}\right|^{2}<\infty.$
(3.5)

Then the corresponding solution $(\sigma(t),v(t),\tilde{S}(t))^{\top}$ of the
adjoint problem (3.1) can be written as

$\begin{pmatrix}\sigma\\\ v\\\
\tilde{S}\end{pmatrix}(t,x)=\sum\limits_{n\in\mathbb{Z}^{*}}\sum\limits_{l=1}^{3}c_{n,l}e^{\overline{\lambda_{n}^{l}}(T-t)}\xi^{*}_{n,l}(x)+c_{0}e^{\overline{\lambda_{0}}(T-t)}\xi_{0}^{*}(x).$
(3.6)

In particular, using the expression of $\xi^{*}_{0}$ and $\xi^{*}_{n,l}$ ( eq.
(2.44), (2.45) and (2.50)), from (3.6) we get

$\displaystyle\sigma(t,x)=\sum\limits_{n\in\mathbb{Z}^{*}}^{\infty}\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}e^{\overline{\lambda_{n}^{l}}(T-t)}e^{inx}+\frac{c_{0}}{\sqrt{2b\pi}},\quad\forall\,t\in(0,T),\quad
x\in(0,2\pi),$ (3.7) $\displaystyle
v(t,x)=\sum\limits_{n\in\mathbb{Z}^{*}}^{\infty}\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\alpha_{n,l}^{2}e^{\overline{\lambda_{n}^{l}}(T-t)}e^{inx},\quad\forall\,t\in(0,T),\quad
x\in(0,2\pi),$ (3.8)
$\displaystyle\tilde{S}(t,x)=\sum\limits_{n\in\mathbb{Z}^{*}}^{\infty}\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\alpha_{n,l}^{3}e^{\overline{\lambda_{n}^{l}}(T-t)}e^{inx},\quad\forall\,t\in(0,T),\quad
x\in(0,2\pi).$ (3.9)

Since $\left\\{e^{inx},x\in(0,2\pi)\right\\}_{n\in\mathbb{Z}}$ is an
orthonormal basis in $L^{2}\left(0,2\pi\right)$ and $\\{e^{inx},$
$x\in(0,2\pi)\\}_{n\in\mathbb{Z}^{*}}$ is an orthonormal basis in
$L_{m}^{2}\left(0,2\pi\right)$, then using Parseval’s identity, we get

$\displaystyle\int_{0}^{2\pi}|\sigma^{T}(x)|^{2}\ {\rm
d}x\;+\;\int_{0}^{2\pi}|v^{T}(x)|^{2}\ {\rm
d}x+\;\int_{0}^{2\pi}|\tilde{S}^{T}(x)|^{2}\ {\rm d}x$ $\displaystyle=$
$\displaystyle\sum\limits_{n\in\mathbb{Z}^{*}}^{\infty}\left(\left|\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}+\frac{c_{0}}{\sqrt{2b\pi}}\right|^{2}+\left|\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\alpha_{n,l}^{2}\right|^{2}+\left|\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\alpha_{n,l}^{3}\right|^{2}\right).$
(3.10)

Now using (2.52) and (2.53), we get a positive constant $C$, independent of
$n$ and a large $N\in\mathbb{N}$, such that

$\displaystyle\sum\limits_{|n|>N}\left(\left|\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}+\frac{c_{0}}{\sqrt{2b\pi}}\right|^{2}+\left|\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\alpha_{n,l}^{2}\right|^{2}+\left|\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\alpha_{n,l}^{3}\right|^{2}\right)$
$\displaystyle\leqslant$ $\displaystyle
C\left(\sum\limits_{|n|>N}\sum\limits_{l=1}^{3}\left|\frac{c_{n,l}}{\psi_{n,l}}\right|^{2}+\left|\frac{c_{0}}{\sqrt{2b\pi}}\right|^{2}\right).$
(3.11)

Let us define

$a_{n}^{l}(x)=\frac{c_{n,l}}{\psi_{n,l}}e^{inx},\text{ for
}|n|>N,l=1,2,3,\text{ and }x\in(0,2\pi).$

Using (2.49) and (3.5), we have $\displaystyle{\sum_{|n|\geq
N}}\sum_{l=1}^{3}|a_{n}^{l}(x)|^{2}<\infty$. Thanks to 3.2, for
$T>{2\pi}\left(\frac{1}{|\beta_{1}|}+\frac{1}{|\beta_{2}|}+\frac{1}{|\beta_{3}|}\right)$,
we have

$\displaystyle\sum\limits_{|n|>N}\sum\limits_{l=1}^{3}\left|\frac{c_{n,l}}{\psi_{n,l}}e^{inx}\right|^{2}\leqslant
C\int_{0}^{T}\left|\sum\limits_{\left|n\right|>N}\sum_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}e^{inx}e^{\overline{\lambda_{n}^{l}}(T-t)}\right|^{2}\,{\rm
d}t.$

Integrating both sides over $\mathcal{O}_{1}$, we get

$\displaystyle\sum\limits_{|n|>N}\sum\limits_{l=1}^{3}\left|\frac{c_{n,l}}{\psi_{n,l}}\right|^{2}\leqslant
C\int_{0}^{T}\int_{\mathcal{O}_{1}}\left|\sum\limits_{\left|n\right|>N}\sum_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}e^{inx}e^{\overline{\lambda_{n}^{l}}(T-t)}\right|^{2}\,{\rm
d}x\,{\rm d}t.$ (3.12)

Therefore only finitely many terms are leaving. Since by ${{\mathcal{H}}}$,
all the eigenvalues are distinct, then the missing finitely many exponential (
for $|n|\leqslant N$) can be added one by one in the inequality (3.12) as the
required gap condition for the eigenvalues holds. For similar details one can
see [29, Chapter 4, Theorem 4.3 ].

Thus for
$T>{2\pi}\left(\frac{1}{|\beta_{1}|}+\frac{1}{|\beta_{2}|}+\frac{1}{|\beta_{3}|}\right)$,
we get

$\displaystyle\sum\limits_{n\in\mathbb{Z}^{*}}\sum\limits_{l=1}^{3}\left|\frac{c_{n,l}}{\psi_{n,l}}\right|^{2}+\left|\frac{c_{0}}{\sqrt{2b\pi}}\right|^{2}$
$\displaystyle\leqslant
C\int_{0}^{T}\int_{\mathcal{O}_{1}}\left|\sum\limits_{n\in\mathbb{Z}^{*}}\sum_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}e^{inx}e^{\overline{\lambda_{n}^{l}}(T-t)}+\frac{c_{0}}{\sqrt{2b\pi}}\right|^{2}{\rm
d}x\,{\rm d}t$
$\displaystyle=C\int_{0}^{T}\int_{\mathcal{O}_{1}}\left|\sigma(t,x)\right|^{2}{\rm
d}x\,{\rm d}t.\>(\text{ using }\eqref{nmaxrepsigma})$ (3.13)

Therefore using (3.1), (3.1) and (3.1), we obtain (3.3). Then by 3.1, we
conclude Theorem 1.2.∎

######  Remark 3.3.

In the proof of Theorem 1.2, i.e, in the proof of the observability inequality
(3.1), we assume that all the eigenvalues of $\mathcal{A}$ are simple to get
the Ingham-type inequality for full class of exponentials
$\\{e^{-\overline{\lambda_{n}^{l}}t}:n\in\mathbb{Z}^{*},l\in\\{1,2,3\\}\\}.$
In 2.7, we have mentioned that the spectrum of $\mathcal{A}$ may have finite
number of multiple eigenvalues. In this case also, one can prove the
observability inequality with a slight modification of the technique for
simple eigenvalue case. Details analysis can be found in [10, Section 4.1].
One can also refer to the book [26, Remarks, Page 178] for a version of Ingham
inequality with repeated eigenvalues.

##  4\. Boundary controllability

In this section, we will discuss the boundary controllability of the
compressible Navier–Stokes system with Maxwell’s law (1.9) with any of the
boundary conditions (1.10), (1.11) and (1.12). At first, we study the boundary
controllability of the system (1.9), when the control acts only in density.

###  4.1. Control in density

Let us consider the system (1.9) with (1.10). For the reader’s convenience, we
recall the system:

$\left.\begin{aligned}
&\partial_{t}\rho+u_{s}\partial_{x}\rho+\rho_{s}\partial_{x}u=0,\qquad&&\text{in
}(0,T)\times(0,2\pi),\\\
&\partial_{t}u+u_{s}\partial_{x}u+a\gamma{\rho_{s}}^{\gamma-2}\partial_{x}\rho-\frac{1}{\rho_{s}}\partial_{x}S=0,\qquad&&\text{in
}(0,T)\times(0,2\pi),\\\
&\partial_{t}S+\frac{1}{\kappa}S-\frac{\mu}{\kappa}\partial_{x}u=0,\qquad&&\text{in
}(0,T)\times(0,2\pi),\\\
&\rho(t,0)=\rho(t,2\pi)+q(t),\>u(t,0)=u(t,2\pi),\>S(t,0)=S(t,2\pi),\qquad&&t\in(0,T),\\\
&\rho(0,x)=\rho_{0}(x),\quad u(0,x)=u_{0}(x),\quad
S(0,x)=S_{0}(x),\qquad&&x\in(0,2\pi).\end{aligned}\right\\}$ (4.1)

####  4.1.1. Well-posedness

Like interior controllability cases, at first let us write the system (4.1) in
the following abstract form:

$\dot{z}(t)=\mathcal{A}z(t)+\mathcal{B}_{\rho}q(t),\quad t\in(0,T),\qquad
z(0)=z_{0},$ (4.2)

where we have set $z(t)=(\rho(t,\cdot),u(t,\cdot),S(t,\cdot))^{\top},$
$z_{0}=(\rho_{0},u_{0},S_{0})^{\top}$. One can identify the operator
$\mathcal{A},\mathcal{B}$ through its adjoint by taking the scalar product of
(4.2) with a vector of smooth functions $(\phi,\psi,\xi)$ and comparing it
with (4.1). $\mathcal{A}$ is the unbounded operator given by (2.3) with its
domain

$\mathcal{D}(\mathcal{A};\mathcal{Z}_{m,m})=\left\\{\begin{pmatrix}\rho\\\
u\\\ S\end{pmatrix}\in\mathcal{Z}_{m,m}\>:\>\left(\rho,u,S\right)^{\top}\in
H^{1}(0,2\pi)\times H^{1}(0,2\pi)\times H^{1}(0,2\pi),\>\begin{aligned}
&\rho(0)=\rho(2\pi),\\\ &u(0)=u(2\pi),\\\
&S(0)=S(2\pi).\end{aligned}\right\\}.$

The control operator
$\mathcal{B}_{\rho}\in\mathcal{L}(\mathbb{C};\mathcal{D}(\mathcal{A}^{*})^{\prime})$
defined by

$\left<{\mathcal{B}_{\rho}q(t)},{(\phi,\psi,\xi)}\right>_{\mathcal{D}(\mathcal{A}^{*})^{\prime},\mathcal{D}(\mathcal{A}^{*})}=q(t)\left(bu_{s}\overline{\phi(2\pi)}+b\rho_{s}\overline{\psi(2\pi)}\right).$

Clearly $\mathcal{B}_{\rho}$ is well-defined as $\mathcal{B}_{\rho}q$ is
continuous on $H^{1}(0,2\pi)\times H^{1}(0,2\pi)\times H^{1}(0,2\pi)$ (by the
embedding theorem $H^{1}(0,2\pi)\hookrightarrow C[0,2\pi]$). Its adjoint
$\mathcal{B}_{\rho}^{*}\in\mathcal{L}(\mathcal{D}(\mathcal{A}^{*});\mathbb{C})$
is

$\mathcal{B}_{\rho}^{*}(\phi,\psi,\xi)=\mathbbm{1}_{\\{x=2\pi\\}}\begin{pmatrix}bu_{s}\\\
b\rho_{s}\\\
0\end{pmatrix}(\phi,\psi,\xi)=bu_{s}{\phi(2\pi)}+b\rho_{s}{\psi(2\pi)}.$ (4.3)

One can prove that the operator $\mathcal{B}_{{\rho}}$ satisfies the following
so-called admissibility condition

$\int_{0}^{T}|\mathcal{B}_{\rho}^{*}\mathbb{T}_{T-t}^{*}\Phi|^{2}\,{\rm
d}t\leqslant
C\left\|\Phi\right\|_{{\mathcal{Z}}},\forall\,\Phi\in\mathcal{D}(\mathcal{A}^{*}),$
(4.4)

where $C$ is some positive constant. The proof relies on the explicit
expression of the solution of the adjoint (3.1) and the right hand side
inequality of (3.4). Since $\mathcal{A}$ generates a contraction semigroup and
$\mathcal{B}_{\rho}$ is an admissible operator, we can prove the following
well-posedness result (see Theorem $2.37$ in page 53 of [13], for more
details):

######  Proposition 4.1.

Let $T>0$, $(\rho_{0},u_{0},S_{0})^{\top}\in L_{m}^{2}(0,2\pi)\times
L_{m}^{2}(0,2\pi)\times L_{m}^{2}(0,2\pi)$ and $q\in L^{2}(0,T)$. Then the
system (4.1) has a unique solution $(\rho,u,S)^{\top}\in
C\left([0,T];(L^{2}(0,2\pi)^{3})\right)$ and the solution satisfies

$\displaystyle\|(\rho,u,S)\|_{C([0,T];\mathcal{{Z}})}\leqslant
C\Big{(}\|(\rho_{0},u_{0},S_{0})\|_{\mathcal{{Z}}}+\|q\|_{L^{2}(0,T)}\Big{)},$

where $C=C(T)$ is a positive constant, independent of $\rho_{0},u_{0},S_{0},q$
and $t$.

####  4.1.2. Exact Controllability

In this section, we prove the exact controllability of the system (4.1). As
interior control case, here at first we discuss the classical approach to
deduce the observability inequality for the adjoint system (3.1) which
essentially gives the exact controllability of the main system (4.1). Let us
consider the linear map $\mathcal{F}_{T}:L^{2}(0,T)\to(L^{2}(0,2\pi))^{3}$ by
$\mathcal{F}_{T}(q)=\left(\rho(T,\cdot),u(T,\cdot),S(T,\cdot)\right)^{\top}$,
where $(\rho,u,S)$ is the solution of the system (4.1) with
$(\rho_{0},u_{0},S_{0})=(0,0,0)$. It is clear that exact controllability for
(4.1) is equivalent to the surjectivity of the map $\mathcal{F}_{T}$. Note
that the map $\mathcal{F}_{T}$ is surjective if and only if there exists a
constant $C>0$ such that the following inequality holds:

$\displaystyle\left\|\mathcal{F}_{T}^{*}\mathbf{z}\right\|_{L^{2}(0,T)}\geq
C\left\|\mathbf{z}\right\|_{(L^{2}(0,2\pi))^{3}},\,\text{ for all
}\mathbf{z}\in(L^{2}(0,2\pi))^{3}.$ (4.5)

A direct computation shows that
$\mathcal{F}_{T}^{*}((\sigma_{T},v_{T},\tilde{S}_{T}))=bu_{s}\sigma(t,2\pi)+b\rho_{s}v(t,2\pi)$,
where $(\sigma,v,\tilde{S})$ is the solution of the adjoint system (3.1) with
the terminal data $(\sigma_{T},v_{T},\tilde{S}_{T})$. Thus we get the
following proposition.

######  Proposition 4.2.

The system (4.1) is exactly controllable in $\mathcal{Z}_{m,m}$ at time $T>0$
using a control $q$ in $L^{2}(0,T)$ acting only in the density, if and only
if, there exists a positive constant $C_{T}>0$ such that for any
$(\sigma_{T},v_{T},\tilde{S}_{T})\in\mathcal{Z}_{m,m}$,
$(\sigma,v,\tilde{S})$, the solution of (3.1), satisfies the following
observability inequality:

$\displaystyle\int_{0}^{2\pi}|\sigma_{T}(x)|^{2}\,{\rm
d}x+\int_{0}^{2\pi}|v_{T}(x)|^{2}\,{\rm
d}x+\int_{0}^{2\pi}|\tilde{S}_{T}(x)|^{2}\,{\rm d}x\leqslant
C_{T}\int_{0}^{T}|bu_{s}\sigma(t,2\pi)+b\rho_{s}v(t,2\pi)|^{2}\,{\rm d}t.$
(4.6)

Before giving the proof of the above observability inequality, we mention the
following lemma which ensures that the observation term is nonzero.

######  Lemma 4.3.

Let us recall the eigen functions
$\mathcal{E}(\mathcal{A}^{*})=\left\\{\xi^{*}_{n,l}\>|\>1\leqslant l\leqslant
3,n\in\mathbb{Z}^{*}\right\\}$ of the unbounded operator $\mathcal{A}^{*}.$
Then we have the following result:

$\mathcal{B}_{\rho}^{*}\xi\neq 0,\forall\,\xi\in\mathcal{E}(\mathcal{A}^{*}),$

where $\mathcal{B}_{\rho}^{*}$ is the observation operator associated to the
system (4.1), defined by (4.3).

###### Proof.

Observe that,
$\mathcal{B}_{\rho}^{*}\xi^{*}_{n,l}=\frac{b}{{\psi}_{n}^{l}}\left({u_{s}\alpha^{1}_{n,l}+\rho_{s}\alpha^{2}_{n,l}}\right).$
From the first equation of the eigen equations
$\mathcal{A}^{*}\xi^{*}_{n,l}=\overline{\lambda_{n}^{l}}\xi^{*}_{n,l}$, we
obtain

$\frac{b}{{\psi}_{n}^{l}}\left({u_{s}\alpha^{1}_{n,l}+\rho_{s}\alpha^{2}_{n,l}}\right)=\frac{b\alpha_{n,l}^{1}\overline{\lambda_{n}^{l}}}{{\psi}_{n}^{l}in}=\frac{\overline{\lambda_{n}^{l}}}{{\psi}_{n}^{l}in}\neq
0\,(\text{ using }\eqref{negrealpart},\,\eqref{nmaxequ-psi_nl}).$ (4.7)

Thus $\mathcal{B}_{\rho}^{*}\xi^{*}_{n,l}\neq 0.$ ∎

####  4.1.3. Proof of Theorem 1.7

###### Proof.

Let $(\sigma_{T},v_{T},\tilde{S}_{T})^{\top}\in\mathcal{Z}_{m,m}.$ From 2.10,
we have

$\begin{pmatrix}\sigma_{T}\\\ v_{T}\\\
\tilde{S}_{T}\end{pmatrix}=\sum\limits_{n\in\mathbb{Z}^{*}}\sum\limits_{l=1}^{3}c_{n,l}\,\xi^{*}_{n,l}\quad\mathrm{with}\quad\sum\limits_{n\in\mathbb{Z}^{*}}\sum\limits_{l=1}^{3}\left|c_{n,l}\right|^{2}<\infty.$
(4.8)

Then the corresponding solution $(\sigma(t),v(t),\tilde{S}(t))^{\top}$ of the
adjoint problem (3.1) can be written as

$\begin{pmatrix}\sigma\\\ v\\\
\tilde{S}\end{pmatrix}(t,x)=\sum\limits_{n\in\mathbb{Z}^{*}}\sum\limits_{l=1}^{3}c_{n,l}e^{\overline{\lambda_{n}^{l}}(T-t)}\xi^{*}_{n,l}(x)$
(4.9)

In particular, using the expression of $\xi^{*}_{n,l}$ (4.9), we get

$\displaystyle\sigma(t,x)=\sum\limits_{n\in\mathbb{Z}^{*}}^{\infty}\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}e^{\overline{\lambda_{n}^{l}}(T-t)}e^{inx},\quad\forall\,t\in(0,T),\quad
x\in(0,2\pi),$ (4.10) $\displaystyle
v(t,x)=\sum\limits_{n\in\mathbb{Z}^{*}}^{\infty}\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\alpha_{n,l}^{2}e^{\overline{\lambda_{n}^{l}}(T-t)}e^{inx},\quad\forall\,t\in(0,T),\quad
x\in(0,2\pi),$ (4.11)
$\displaystyle\tilde{S}(t,x)=\sum\limits_{n\in\mathbb{Z}^{*}}^{\infty}\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\alpha_{n,l}^{3}e^{\overline{\lambda_{n}^{l}}(T-t)}e^{inx},\quad\forall\,t\in(0,T),\quad
x\in(0,2\pi).$ (4.12)

then, in a similar technique used in the proof of Theorem 1.2, we get a
positive constant $C$, independent of $n$, and a large $N\in\mathbb{N}$, such
that

$\displaystyle\sum\limits_{|n|>N}\left(\left|\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\right|^{2}+\left|\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\alpha_{n,l}^{2}\right|^{2}+\left|\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\alpha_{n,l}^{3}\right|^{2}\right)$
$\displaystyle\leqslant$ $\displaystyle
C\sum\limits_{|n|>N}\sum\limits_{l=1}^{3}\left|\frac{c_{n,l}}{\psi_{n,l}}\right|^{2}.$
(4.13)

Putting $x=2\pi$ in (4.10) and (4.11), we have

$\displaystyle\sigma(t,2\pi)=\sum\limits_{n\in\mathbb{Z}^{*}}^{\infty}\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}e^{\overline{\lambda_{n}^{l}}(T-t)},\quad\forall\,t\in(0,T),$
(4.14) $\displaystyle
v(t,2\pi)=\sum\limits_{n\in\mathbb{Z}^{*}}^{\infty}\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\alpha_{n,l}^{2}\,e^{\overline{\lambda_{n}^{l}}(T-t)},\quad\forall\,t\in(0,T),$
(4.15) $\displaystyle
S(t,2\pi)=\sum\limits_{n\in\mathbb{Z}^{*}}^{\infty}\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\alpha_{n,l}^{3}\,e^{\overline{\lambda_{n}^{l}}(T-t)},\quad\forall\,t\in(0,T).$
(4.16)

Now the observation term becomes

$\displaystyle|bu_{s}\sigma(t,2\pi)+b\rho_{s}v(t,2\pi)|=\left|\sum\limits_{n\in\mathbb{Z}^{*}}^{\infty}\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}(bu_{s}+\alpha_{n,l}^{2}b\rho_{s})e^{\overline{\lambda_{n}^{l}}(T-t)}\right|.$
(4.17)

Using 3.2, along with (4.1.3) and (4.17), for
$T>{2\pi}\left(\frac{1}{|\beta_{1}|}+\frac{1}{|\beta_{2}|}+\frac{1}{|\beta_{3}|}\right)$,
we have the following inequality:

$\displaystyle\sum\limits_{|n|>N}\left(\left|\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\right|^{2}+\left|\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\alpha_{n,l}^{2}\right|^{2}+\left|\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}\alpha_{n,l}^{3}\right|^{2}\right)$
$\displaystyle\quad\quad\leqslant
C\int_{0}^{T}\left|\sum\limits_{|n|>N}\sum\limits_{l=1}^{3}\frac{c_{n,l}}{\psi_{n,l}}(u_{s}+\alpha_{n,l}^{2}b)e^{\overline{\lambda_{n}^{l}}(T-t)}\right|^{2}\,{\rm
d}t.$ (4.18)

Then by ${{\mathcal{H}}}$ and 4.3, the missing finitely many exponential ( for
$|n|\leqslant N$) can be added one by one in the inequality (4.1.3) as the
required gap condition for the eigenvalues holds. Thus finally we deduce the
following observability inequality

$\displaystyle\int_{0}^{2\pi}|\sigma_{T}(x)|^{2}\ {\rm
d}x\;+\;\int_{0}^{2\pi}|v_{T}(x)|^{2}\ {\rm
d}x+\;\int_{0}^{2\pi}|\tilde{S}_{T}(x)|^{2}\ {\rm d}x$
$\displaystyle\quad\quad\quad\quad\leqslant
C\int_{0}^{T}\left|u_{s}\sigma(t,2\pi)+bv(t,2\pi)\right|^{2}\,{\rm d}t.$
(4.19)

provided
$T>{2\pi}\left(\frac{1}{|\beta_{1}|}+\frac{1}{|\beta_{2}|}+\frac{1}{|\beta_{3}|}\right)$.
Hence Theorem 1.7 is proved. ∎

In a similar approach as density control case, we can also prove the exact
controllability result (see 1.8) of the system (1.9), when control acts only
on the velocity or stress. In the following sections, we just indicate the
changes according to the system (1.9)-(1.11) and (1.9)-(1.12).

###  4.2. Control in velocity

Here the control operator
$\mathcal{B}_{u}\in\mathcal{L}(\mathbb{C};\mathcal{D}(\mathcal{A}^{*})^{\prime})$
is defined by

$\left<{\mathcal{B}_{u}r(t)},{(\phi,\psi,\xi)}\right>_{\mathcal{D}(\mathcal{A}^{*})^{\prime},\mathcal{D}(\mathcal{A}^{*})}=r\left(b\rho_{s}\overline{\phi(2\pi)}+\rho_{s}u_{s}\overline{\psi(2\pi)}-\overline{\xi(2\pi)}\right).$

Clearly $\mathcal{B}_{u}$ is well-defined as $\mathcal{B}_{u}r$ is continuous
on $H^{1}(0,2\pi)\times H^{1}(0,2\pi)\times H^{1}(0,2\pi)$ (by the embedding
theorem $H^{1}(0,2\pi)\hookrightarrow C^{0}[0,2\pi]$). Its adjoint
$\mathcal{B}_{u}^{*}\in\mathcal{L}(\mathcal{D}(\mathcal{A}^{*});\mathbb{C})$
is given by

$\mathcal{B}_{u}^{*}(\phi,\psi,\xi)=\mathbbm{1}_{\\{x=2\pi\\}}\begin{pmatrix}b\rho_{s}\\\
\rho_{s}u_{s}\\\
-1\end{pmatrix}(\phi,\psi,\xi)=b\rho_{s}{\phi(2\pi)}+\rho_{s}u_{s}{\psi(2\pi)}-{\xi(2\pi)}.$
(4.20)

As in the density control case, here also one can prove that the control
operator satisfies the admissibility condition (4.4). Thus for any
$(\rho_{0},u_{0},S_{0})^{\top}\in\mathcal{Z}_{m,m}$ and $r\in L^{2}(0,T)$, the
system (1.9)-(1.11) has a unique solution and the solution satisfies:

$\displaystyle\|(\rho,u,S)\|_{C([0,T];(L^{2}(0,2\pi)^{3})}\leqslant
C\Big{(}\|(\rho_{0},u_{0},S_{0})\|_{\mathcal{Z}}+\|r\|_{L^{2}(0,T)}\Big{)},$

where $C=C(T)$ is a positive constant, independent of $\rho_{0},u_{0},S_{0},r$
and $t$.

######  Proposition 4.4.

The system (1.9)-(1.11) is exactly controllable in $\mathcal{{Z}}_{m,m}$ at
time $T>0$ using a boundary control $r$ in $L^{2}(0,T)$ acting only in the
velocity, if and only if, there exists a positive constant $C_{T}>0$ such that
for any $(\sigma_{T},v_{T},\tilde{S}_{T})^{\top}\in\mathcal{Z}_{m,m}$,
$(\sigma,v,\tilde{S})$, the solution of (3.1), satisfies the following
observability inequality:

$\int_{0}^{2\pi}|\sigma_{T}(x)|^{2}\,{\rm
d}x+\int_{0}^{2\pi}|v_{T}(x)|^{2}\,{\rm
d}x+\int_{0}^{2\pi}|\tilde{S}_{T}(x)|^{2}\,{\rm d}x\leqslant
C_{T}\int_{0}^{T}|b\rho_{s}\sigma(t,2\pi)\\\
+\rho_{s}u_{s}v(t,2\pi)-\tilde{S}(t,2\pi)|^{2}\,{\rm d}t.$ (4.21)

######  Lemma 4.5.

Let us recall the eigen functions
$\mathcal{E}(\mathcal{A}^{*})=\left\\{\xi^{*}_{n,l}\>|\>1\leqslant l\leqslant
3,n\in\mathbb{Z}^{*}\right\\}$ of the unbounded operator $\mathcal{A}^{*}.$
Then we have the following result:

$\mathcal{B}_{u}^{*}\xi\neq 0,\forall\,\xi\in\mathcal{E}(\mathcal{A}^{*}).$

###### Proof.

From (4.20) and (2.50), we have

$\mathcal{B}_{u}^{*}\xi^{*}_{n,l}=\frac{1}{\psi_{n,l}}(b\rho_{s}+\alpha_{n,l}^{2}\rho_{s}u_{s}-\alpha_{n,l}^{3}).$
(4.22)

The second equation of the eigen-equations
$\mathcal{A}^{*}\xi^{*}_{n,l}=\overline{\lambda_{n}^{l}}\xi^{*}_{n,l}$ gives

$b\rho_{s}+\alpha_{n,l}^{2}\rho_{s}u_{s}-\alpha_{n,l}^{3}=\frac{\rho_{s}\alpha_{n,l}^{2}\overline{\lambda_{n}^{l}}}{in}.$
(4.23)

Thus from (4.22), (4.23) and (2.45), we have

$\mathcal{B}_{u}^{*}\xi^{*}_{n,l}=-\frac{\rho_{s}\overline{\lambda_{n}^{l}}(\overline{\lambda_{n}^{l}}-inu_{s})}{n^{2}\psi_{n,l}\rho_{s}}\neq
0,$

since all the eigenvalues have negative real part. ∎

###  4.3. Control in stress

The control operator
$\mathcal{B}_{S}\in\mathcal{L}(\mathbb{C};\mathcal{D}(\mathcal{A}^{*})^{\prime})$
defined by

$\left<{\mathcal{B}_{S}p(t)},{(\phi,\psi,\xi)}\right>_{\mathcal{D}(\mathcal{A}^{*})^{\prime},\mathcal{D}(\mathcal{A}^{*})}=-p\psi(2\pi).$

Clearly $\mathcal{B}_{S}$ is well-defined as $\mathcal{B}_{S}p$ is continuous
on $H^{1}(0,2\pi)\times H^{1}(0,2\pi)\times H^{1}(0,2\pi)$ (by the embedding
theorem $H^{1}(0,2\pi)\hookrightarrow C^{0}[0,2\pi]$). Its adjoint
$\mathcal{B}_{S}^{*}\in\mathcal{L}(\mathcal{D}(\mathcal{A}^{*});\mathbb{C})$
is

$\mathcal{B}_{S}^{*}(\phi,\psi,\xi)=\mathbbm{1}_{\\{x=2\pi\\}}\begin{pmatrix}0\\\
-1\\\ 0\end{pmatrix}(\phi,\psi,\xi)=-{\psi(2\pi)},$ (4.24)

and also $\mathcal{B}_{S}^{*}\xi\neq
0,\forall\,\xi\in\mathcal{E}(\mathcal{A}^{*}).$

##  5\. Small time lack of controllability

In this section, we study the lack of exact controllability of the system
(1.4) when the localized interior control is acting on the density equation.
Similar result holds true for the remaining two cases (control in velocity or
control in stress). We establish the result by violating the observability
inequality on an appropriate solution of the corresponding adjoint system. Our
proof is in the same spirit of [3, Section 3]. We first find a candidate
function as a terminal data of the transport equation and exploit the lack of
null controllability of the same in small time.

Without loss of generality, we consider the control domain
$\mathcal{O}_{1}=(l_{1},l_{2}),0\leqslant l_{1}<l_{2}<2\pi.$

######  Theorem 5.1.

Let us take $|\beta_{1}|T<\max\\{l_{1},2\pi-l_{2}\\}$. Then the system (1.4)
is not exactly controllable in time $T$ by means of a control $f_{1}\in
L^{2}\left(0,T;L^{2}(\mathcal{O}_{1})\right)$ acting on density equation.

###### Proof.

Since $|\beta_{1}|T<\max\\{l_{1},2\pi-l_{2}\\}$, there exists a nontrivial
function $\hat{\sigma}_{T}\in C_{c}^{\infty}(0,2\pi)$ such that the solution
of the following equation

$\begin{cases}\hat{\sigma}_{t}-\beta_{1}\hat{\sigma}_{x}-\omega_{1}\hat{\sigma}=0,&\mbox{
in }(0,T)\times(0,2\pi),\\\ \hat{\sigma}(t,0)=\hat{\sigma}(t,2\pi),&\mbox{ in
}(0,T),\\\ \hat{\sigma}(T,x)=\hat{\sigma}_{T}(x),&\mbox{ in
}(0,2\pi),\end{cases}$ (5.1)

satisfies that
$\text{supp}(\hat{\sigma})\cap[0,T]\times\mathcal{O}_{1}=\emptyset$ (see also
[3, Section 3]). We will use the asymptotic behavior of the spectrum of the
linearized operator to show the lack of controllability. At first, we
construct a high-frequency version of the candidate function
$\hat{\sigma}_{T}$. Let $N>0$ be a fixed integer. Next, we define the
polynomial

$P^{N}(X):=\prod_{j=-N}^{N}(X-j),$

and the function

$\hat{\sigma}_{T}^{N}:=P^{N}\left(-i\frac{d}{dx}\right)\hat{\sigma}_{T}.$

Since $\hat{\sigma}_{T}^{N}$ is the image of $\hat{\sigma}_{T}$ by a
differential operator, we have
$\text{supp}({\hat{\sigma}_{T}^{N}})\subset\text{supp}(\hat{\sigma}_{T})$. Let
us write

$\hat{\sigma}_{T}=\sum_{n\in\mathbb{Z}}a_{n}e^{inx}\text{, where
}a_{n}=\int_{0}^{2\pi}e^{inx}\hat{\sigma}_{T}(x)dx,\forall n\in\mathbb{Z}.$

Then using the definition of $P^{N}$ and $\hat{\sigma}_{T}^{N}$, we can write
$\hat{\sigma}_{T}^{N}$ in the following:

$\hat{\sigma}_{T}^{N}(x)=\sum_{n\in\mathbb{Z}}a_{n}\prod_{j=-N}^{N}\left(-i\frac{d}{dx}-j\right)e^{inx}=\sum_{n\in\mathbb{Z}}a_{n}\prod_{j=-N}^{N}\left(n-j\right)e^{inx}=\sum_{n\in\mathbb{Z}}a_{n}P^{N}(n)e^{inx},$

for $(t,x)\in(0,T)\times(0,2\pi)$. Note that $P^{N}(n)=0$ for all
$|{n}|\leqslant N$ and therefore

$\hat{\sigma}_{T}^{N}(x)=\sum_{{|n|}\geq N+1}a_{n}P^{N}(n)e^{inx}.$ (5.2)

Next, we denote the function $\hat{\sigma}^{N}$ is the solution of the
following equation

$\begin{cases}\hat{\sigma}^{N}_{t}-\beta_{1}\hat{\sigma}^{N}_{x}-\omega_{1}\hat{\sigma}^{N}=0,&\mbox{
in }(0,T)\times(0,2\pi),\\\
\hat{\sigma}^{N}(t,0)=\hat{\sigma}^{N}(t,2\pi),&\mbox{ in }(0,T),\\\
\hat{\sigma}^{N}(T,x)=\hat{\sigma}_{T}^{N}(x),&\mbox{ in
}(0,2\pi).\end{cases}$ (5.3)

Let $(\sigma^{N},v^{N},\tilde{S}^{N})$ be the solution of the following
adjoint system:

$\begin{cases}\partial_{t}\sigma^{N}+u_{s}\partial_{x}\sigma^{N}+{\rho_{s}}\partial_{x}v^{N}=0,&\mbox{
in }(0,T)\times(0,2\pi),\\\
\partial_{t}v^{N}+b\partial_{x}\sigma^{N}+u_{s}\partial_{x}v^{N}-\frac{1}{\rho_{s}}\partial_{x}\tilde{S}^{N}=0,&\mbox{
in }(0,T)\times(0,2\pi),\\\
\partial_{t}\tilde{S}^{N}-\frac{1}{\kappa}\tilde{S}^{N}-\frac{\mu}{\kappa}\partial_{x}v^{N}=0,&\mbox{
in }(0,T)\times(0,2\pi),\\\
\sigma^{N}(t,0)=\sigma^{N}(t,2\pi),v^{N}(t,0)=v^{N}(t,2\pi),\>\tilde{S}^{N}(t,0)=\tilde{S}^{N}(t,2\pi),&\mbox{
in }(0,T),\\\ \sigma^{N}(T,x)=\hat{\sigma}^{N}_{T}(x),\quad
v^{N}(T,x)=\hat{v}^{N}_{T}(x),\quad\tilde{S}^{N}(T,x)=\hat{S}^{N}_{T}(x),&\mbox{
in }(0,2\pi),\end{cases}$ (5.4)

where
$\displaystyle(\hat{\sigma}^{N}_{T},\hat{v}^{N}_{T},\hat{S}^{N}_{T})=\sum_{|n|\geq
N+1}c^{1}_{n}\,\xi^{*}_{n,1},\text{ where
}\frac{c^{1}_{n}}{\psi_{n,1}}=a_{n}^{N}=a_{n}P^{N}(n)$ and
$\psi_{n,1},\,\xi^{*}_{n,1}$ are defined in (2.50). We write the solutions of
the systems (5.3) and (5.4) respectively as

$\displaystyle\hat{\sigma}^{N}(t,x)$ $\displaystyle=\sum_{|{n}|\geq
N+1}a^{N}_{n}\,e^{(-\beta_{1}in-\omega_{1})(T-t)}e^{inx},$ (5.5)
$\displaystyle\sigma^{N}(t,x)$ $\displaystyle=\sum_{|{n}|\geq
N+1}a^{N}_{n}\,e^{\overline{\lambda_{n}^{1}}(T-t)}e^{inx}.$ (5.6)

Now, we prove that the solution component $\sigma^{N}$ of (5.4) approximates
the solution $\hat{\sigma}^{N}$ of (5.3). Indeed,

$\displaystyle\left\|\sigma^{N}(t,\cdot)-\hat{\sigma}^{N}(t,\cdot)\right\|_{L^{2}(0,2\pi)}^{2}$
$\displaystyle\leqslant C\sum_{|{n}|\geq
N+1}|{a^{N}_{n}}|^{2}\left|e^{\overline{\lambda_{n}^{1}}(T-t)}-e^{\left({-\omega_{1}-i\beta_{1}n}\right)(T-t)}\right|^{2}$
$\displaystyle\leqslant C\sum_{|{n}|\geq
N+1}\frac{1}{|{n}|^{2}}|{a^{N}_{n}}|^{2},$

for all $t\in[0,T]$ and therefore performing integration over $[0,T]$ we have

$\left\|\sigma^{N}-\hat{\sigma}^{N}\right\|_{L^{2}(0,T,L^{2}(0,\pi))}^{2}\leqslant\frac{C}{|{N}|^{2}}\sum_{|{n}|\geq
N+1}|{a^{N}_{n}}|^{2}.$

Using triangle inequality, we deduce

$\displaystyle\left\|\sigma^{N}\right\|_{L^{2}(0,T,L^{2}(\mathcal{O}_{1}))}^{2}$
$\displaystyle\leqslant\left\|\sigma^{N}-\hat{\sigma}^{N}\right\|_{L^{2}(0,T,L^{2}(\mathcal{O}_{1})}^{2}+\left\|\hat{\sigma}^{N}\right\|_{L^{2}(0,T,L^{2}(\mathcal{O}_{1}))}^{2}.$

Since the support of $\hat{\sigma}_{N}$ does not intersect the domain
$[0,T]\times\mathcal{O}_{1}$, the second term of the right hand sides
vanishes. Thus we get

$\left\|\sigma^{N}\right\|_{L^{2}(0,T,L^{2}(\mathcal{O}_{1}))}^{2}\leqslant\frac{C}{|{N}|^{2}}\sum_{|{n}|\geq
N+1}|{a^{N}_{n}}|^{2}.$ (5.7)

Next, if possible we assume that the observability for the system (5.4) holds.
Therefore we have the following:

$\int_{0}^{2\pi}|\hat{\sigma}^{N}_{T}(x)|^{2}\ {\rm
d}x\;+\;\int_{0}^{2\pi}|\hat{v}^{N}_{T}(x)|^{2}\ {\rm
d}x+\;\int_{0}^{2\pi}|\hat{S}^{N}_{T}(x)|^{2}\ {\rm d}x\leqslant
C_{T}\int_{0}^{T}\int_{\mathcal{O}_{1}}|\sigma^{N}(t,x)|^{2}\,{\rm d}x\,{\rm
d}t.$ (5.8)

Using (5.2), (5.7) (5.8), we have

$\int_{0}^{2\pi}|\hat{\sigma}^{N}_{T}(x)|^{2}\ {\rm
d}x\leqslant\left\|\hat{\sigma}^{N}\right\|_{L^{2}(0,T,L^{2}(\mathcal{O}_{1}))}^{2}\leqslant\frac{C}{|{N}|^{2}}\sum_{|{n}|\geq
N+1}|{a^{N}_{n}}|^{2}=\frac{C}{|{N}|^{2}}\int_{0}^{2\pi}|\hat{\sigma}^{N}_{T}(x)|^{2}\
{\rm d}x,$ (5.9)

which gives a contradiction. ∎

##  6\. Rapid exponential stabilization

Let us recall the control system (1.9). The goal of this section is to study
the boundary stabilization issues for (1.9) with single boundary control
force. More precisely, we construct a stationary feedback law $q(t)$, $r(t)$
or $p(t)$) of the form $\Pi\left(\rho(t,\cdot),u(t,\cdot),S(t,\cdot)\right)$
such that the solution of the closed loop system (1.9) decays exponentially to
zero at any prescribed decay rate. At first we describe Urquiza’s approach
[36] which is the key argument of this section.

###  6.1. Urquiza’s method

Let us consider an abstract control system

$\begin{cases}\dot{y}(t)=\mathcal{A}y(t)+\mathcal{B}q(t),\quad t\in(0,T),\\\
y(0)=y_{0},\end{cases}$ (6.1)

where $y(t)\in\mathcal{H},y_{0}\in\mathcal{H},q\in L^{2}(0,T)$, $\mathcal{B}$
is an unbounded operator from $\mathbb{C}$ to $\mathcal{H}$.
$\mathcal{A}:D(\mathcal{A})\subset\mathcal{H}\to\mathcal{H}$ is an unbounded
operator and $\mathcal{D}(\mathcal{A})$ is dense in $\mathcal{H}.$ To employ
the method of Urquiza, one needs to take the following assumptions on the
operator $\mathcal{A}$ and $\mathcal{B}$

  * (H1)

$\mathcal{A}$ is the infinitesimal generator of a strongly continuous group
$\\{e^{t\mathcal{A}}\\}_{t\in\mathbb{R}}$ on $\mathcal{H}$.

  * (H2)

$\mathcal{B}:\mathbb{C}\to\mathcal{D}(\mathcal{A}^{*})^{\prime}$ is linear and
continuous.

  * (H3)

Regularity property. For all $T>0$ there exists $C(T)>0$ such that

$\int_{0}^{T}\left|\mathcal{B}^{*}e^{-t\mathcal{A}^{*}}y_{T}\right|^{2}\leqslant
C\left\|y_{T}\right\|^{2}_{\mathcal{H}},\quad\forall\,y_{T}\in\mathcal{D}(\mathcal{A}^{*}).$

  * (H4)

Controllability property. There are two constants $T>0$ and $c(T)>0$ such that

$\int_{0}^{T}\left|\mathcal{B}^{*}e^{-t\mathcal{A}^{*}}y_{T}\right|^{2}\geq
c\left\|y_{T}\right\|^{2}_{\mathcal{H}},\quad\forall\,y_{T}\in\mathcal{D}(\mathcal{A}^{*}).$

These hypotheses lead to the the following stabilization result. Its proof
relies on algebraic Riccati equation associated with the linear quadratic
regulator problem [18]. Let us introduce the growth bound of the semigroup
$\\{e^{\mathcal{A}t}\\}_{t\geq 0}$ of continuous linear operator as follows:

$g(\mathcal{A})=\inf_{t>0}\frac{1}{t}\log\left\|e^{t\mathcal{A}}\right\|_{\mathcal{L}(\mathcal{H})}.$

######  Theorem 6.1.

[Urquiza [36], Theorem $2.1$] Consider operators $\mathcal{A}$ and
$\mathcal{B}$ under assumptions (H1)-(H4). For any
$\omega>\max\left(g(-\mathcal{A}),0\right)$, we have

  * (i)

The symmetric positive operator $\Lambda_{\omega}$ defined by

$\left<{\Lambda_{\omega}x},{z}\right>_{\mathcal{H}}=\int_{0}^{\infty}\left<{\mathcal{B}^{*}e^{-\tau(\mathcal{A}+\omega
I)^{*}}x},{\mathcal{B}^{*}e^{-\tau(\mathcal{A}+\omega
I)^{*}}z}\right>_{\mathbb{C}}d\tau,\,\forall\,x,z\in\mathcal{H}$

is coercive and is an isomorphism on $\mathcal{H}$.

  * (ii)

Let $\Pi_{\omega}:=-\mathcal{B}^{*}\Lambda_{\omega}^{-1}$. The operator
$\mathcal{A}+\mathcal{B}\Pi_{\omega}$ with
$\mathcal{D}(\mathcal{A}+\mathcal{B}\Pi_{\omega})=\Lambda_{\omega}(\mathcal{D}(\mathcal{A}^{*}))$
is the infinitesimal generator of a strongly continuous semigroup on
$\mathcal{H}$.

  * (iii)

The closed-loop system (6.1) with $q=\Pi_{\omega}(y)$ is exponentially stable
that is,

$\left\|e^{t(\mathcal{A}+\mathcal{B}\Pi_{\omega})}x\right\|_{\mathcal{H}}\leqslant
Ce^{\left(-2\omega+g(-\mathcal{A})\right)t}\left\|x\right\|_{\mathcal{H}},\forall
x\in\mathcal{H},$

where $C$ is a positive constant.

We utilize Theorem 6.1 to prove the complete exponential stabilization of the
linearized compressible Navier-Stokes equation with Maxwell’s law (1.9)-(1.10)
with boundary feedback law.

###  6.2. Control in density

To apply the method mentioned above we need to show that all the assumptions
(H1)-(H4) hold true for our system (4.1). Let us recall the corresponding
spatial operator $(\mathcal{A},\mathcal{D}(\mathcal{A};\mathcal{Z}_{m,m})$,
where $\mathcal{Z}_{m,m}=L_{m}^{2}(0,2\pi)\times L_{m}^{2}(0,2\pi)\times
L_{m}^{2}(0,2\pi)$.

It can be check that the operator
$(\mathcal{A},\mathcal{D}(\mathcal{A};\mathcal{Z}_{m,m})$ generates a strongly
continuous group $\\{\mathbb{T}_{t}\\}_{t\in\mathbb{R}}$ of continuous linear
operator. Hence (H1) holds. Also (H2) is true, see (4.3). We get (H3) and (H4)
by Ingham inequality.

###  6.3. Feedback control law and proof of Theorem 1.9

In this section we employ Urquiza’s method to construct the feedback law for
our system (4.1). Let us take
$U^{1}_{0}=(\sigma^{1}_{0},v^{1}_{0},\tilde{S}^{1}_{0}),U^{2}_{0}=(\sigma^{1}_{0},v^{1}_{0},\tilde{S}^{1}_{0})\in\mathcal{Z}_{m,m}.$
We consider the bilinear form

$a_{w}(U^{1}_{0},U^{2}_{0})=\int_{0}^{\infty}e^{-2\omega
t}\left(bu_{s}{\sigma^{1}(t,2\pi)}+b\rho_{s}{v^{1}(t,2\pi)}\right)\overline{\left(bu_{s}{\sigma^{2}(t,2\pi)}+b\rho_{s}{v^{2}(t,2\pi)}\right)}\,{\rm
d}t,$

where $U^{1}=(\sigma^{1},v^{1},\tilde{S}^{1})$ and
$U^{2}=(\sigma^{2},v^{2},\tilde{S}^{2})$ are the solutions of the following
systems respectively

$\begin{cases}\partial_{t}\sigma^{1}+u_{s}\partial_{x}\sigma^{1}+{\rho_{s}}\partial_{x}v^{1}=0&\mbox{
in }(0,T)\times(0,2\pi),\\\
\partial_{t}v^{1}+b\partial_{x}\sigma^{1}+u_{s}\partial_{x}v^{1}-\frac{1}{\rho_{s}}\partial_{x}\tilde{S}^{1}=0&\mbox{
in }(0,T)\times(0,2\pi),\\\
\partial_{t}\tilde{S}^{1}-\frac{1}{\kappa}\tilde{S}^{1}-\frac{\mu}{\kappa}\partial_{x}v^{1}=0&\mbox{
in }(0,T)\times(0,2\pi),\\\
\sigma^{1}(t,0)=\sigma^{1}(t,2\pi),v^{1}(t,0)=v^{1}(t,2\pi),\>\tilde{S}^{1}(t,0)=\tilde{S}^{1}(t,2\pi),&\mbox{
in }(0,T),\\\
\left(\sigma^{1}(T,x),v^{1}(T,x),\tilde{S}^{1}(T,x)\right)=\mathbb{T}_{-T}\left(\sigma^{1}_{0}(x),v^{1}_{0}(x),\tilde{S}^{1}_{0}(x)\right)&x\in(0,2\pi).\end{cases}$
(6.2)

and

$\begin{cases}\partial_{t}\sigma^{2}+u_{s}\partial_{x}\sigma^{2}+{\rho_{s}}\partial_{x}v^{2}=0&\mbox{
in }(0,T)\times(0,2\pi),\\\
\partial_{t}v^{2}+b\partial_{x}\sigma^{2}+u_{s}\partial_{x}v^{2}-\frac{1}{\rho_{s}}\partial_{x}\tilde{S}^{2}=0&\mbox{
in }(0,T)\times(0,2\pi),\\\
\partial_{t}\tilde{S}^{2}-\frac{1}{\kappa}\tilde{S}^{2}-\frac{\mu}{\kappa}\partial_{x}v^{2}=0&\mbox{
in }(0,T)\times(0,2\pi),\\\
\sigma^{2}(t,0)=\sigma^{2}(t,2\pi),v^{2}(t,0)=v^{2}(t,2\pi),\>\tilde{S}^{2}(t,0)=\tilde{S}^{2}(t,2\pi),&\mbox{
in }(0,T),\\\
\left(\sigma^{2}(T,x),v^{2}(T,x),\tilde{S}^{2}(T,x)\right)=\mathbb{T}_{-T}\left(\sigma^{2}_{0}(x),v^{2}_{0}(x),\tilde{S}^{2}_{0}(x)\right)&x\in(0,2\pi).\end{cases}$
(6.3)

Let us define the operator
$\Lambda_{\omega}:\mathcal{Z}_{m,m}\to\mathcal{Z}_{m,m}$ satisfying the
following

$\left<{\Lambda_{\omega}U^{1}_{0}},{U^{2}_{0}}\right>_{\mathcal{Z}_{m,m}}=a_{\omega}(U^{1}_{0},U^{2}_{0}),\,U^{1}_{0},U^{2}_{0}\in\mathcal{Z}_{m,m}.$
(6.4)

Next we see that

$\displaystyle a_{w}(U^{1}_{0},U^{2}_{0})=$
$\displaystyle\int_{0}^{\infty}e^{-2\omega
t}\left(bu_{s}{\sigma^{1}(t,2\pi)}+b\rho_{s}{v^{1}(t,2\pi)}\right)\,\,\overline{\left(bu_{s}{\sigma^{2}(t,2\pi)}+b\rho_{s}{v^{2}(t,2\pi)}\right)}\,{\rm
d}t$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}e^{-2\omega
t}{\mathcal{B}^{*}_{\rho}}U^{1}(t)\,\,\overline{{\mathcal{B}^{*}_{\rho}}U^{2}(t)}\,{\rm
d}t$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}e^{-2\omega
t}\left({\mathcal{B}^{*}_{\rho}}\mathbb{T}^{*}_{T-t}\mathbb{T}^{*}_{-T}U^{1}_{0}\right)\overline{\left({\mathcal{B}^{*}_{\rho}}\mathbb{T}^{*}_{T-t}\mathbb{T}^{*}_{-T}U^{2}_{0}\right)}\,{\rm
d}t$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}e^{-2\omega
t}\left({\mathcal{B}^{*}_{\rho}}\mathbb{T}^{*}_{-t}U^{1}_{0}\right)\overline{\left({\mathcal{B}^{*}_{\rho}}\mathbb{T}^{*}_{-t}U^{2}_{0}\right)}\,{\rm
d}t.$

Therefore from (6.4), we have

$\left<{\Lambda_{\omega}U^{1}_{0}},{U^{2}_{0}}\right>_{\mathcal{Z}_{m,m}}=\int_{0}^{\infty}e^{-2\omega
t}\left<{{\mathcal{B}^{*}_{\rho}}\mathbb{T}^{*}_{-t}U^{1}_{0}},{\,{\mathcal{B}^{*}_{\rho}}\mathbb{T}^{*}_{-t}U^{2}_{0}}\right>_{\mathbb{C}}\,{\rm
d}t.$ (6.5)

Thanks to Theorem 6.1, the operator $\Lambda_{\omega}$ defined by (6.4) is
coercive and isomorphism. Finally, let us define the operator
${\Pi}_{\omega}:\mathcal{Z}_{m,m}\to\mathbb{C}$ by

$\displaystyle{\Pi}_{\omega}(\mathbf{z})=-\left(bu_{s}{\sigma^{1}_{0}(2\pi)}+b\rho_{s}{v^{1}_{0}(2\pi)}\right),$

where $U^{1}_{0}=(\sigma^{1}_{0},v^{1}_{0},\tilde{S}^{1}_{0})$ is the solution
of the following Lax-Milgram problem

$a_{\omega}(U^{1}_{0},U^{2}_{0})=\left<{\mathbf{z}},{U^{2}_{0}}\right>,\,\forall\,U^{2}_{0}\in\mathcal{Z}_{m,m}.$

Hence we obtain
$\left<{\Lambda_{\omega}U^{1}_{0}},{U^{2}_{0}}\right>_{\mathcal{Z}_{m,m}}=\left<{\mathbf{z}},{U^{2}_{0}}\right>,\forall\,U^{2}_{0}\in\mathcal{Z}_{m,m}.$
This gives $\Lambda_{\omega}U^{1}_{0}=\mathbf{z}.$ It follows that
$U^{1}_{0}=\Lambda_{\omega}^{-1}\mathbf{z}.$ Thus we have
${\Pi}_{\omega}=-{\mathcal{B}}^{*}\Lambda_{\omega}^{-1}.$ Thanks to Theorem
6.1, rapid exponential stabilization for the system (4.1) is established by
means of the feedback law
$q(t)={\Pi}_{\omega}(\rho(t,\cdot),v(t,\cdot),S(t,\cdot))$. More precisely, we
get a positive constant $C$ such that the solution of (4.1) satisfies the
following estimate

$\displaystyle\left\|\rho(t,\cdot)\right\|_{L^{2}(0,2\pi)}+\left\|u(t,\cdot)\right\|_{L^{2}(0,2\pi)}+\left\|S(t,\cdot)\right\|_{L^{2}(0,2\pi)}$
$\displaystyle\leqslant Ce^{-\nu
t}\bigg{[}\left\|\rho_{0}\right\|_{L^{2}(0,2\pi)}+\left\|u_{0}\right\|_{L^{2}(0,2\pi)}+\left\|S_{0}\right\|_{L^{2}(0,2\pi)}\bigg{]}.$
(6.6)

A similar process will give the complete stabilization for the control systems
(1.9)-(1.11) and (1.9)-(1.12).

##  7\. Construction of the biorthogonal family and Ingham-type inequality

This section is devoted to the proof of a suitable Ingham-type inequality
which essentially helps to derive the required observability inequality. The
proof of the Ingham-type inequality relies on the construction of a
biorthogonal family of
$\\{e^{-{\lambda_{k}^{l}}t}\\}_{k\in\mathbb{Z}^{*},l\in\\{1,2,3\\}}$. All the
constants used in this section while finding the relevant estimates are
generic, which may vary from line to line.

Let us assume (${{\mathcal{H}}}$). We recall the expressions of the
eigenvalues of $\mathcal{A}$ in the following manner.

$\left.\begin{aligned}
\lambda_{n}^{1}=&-\omega_{1}+i\beta_{1}n+O\left(\frac{1}{|n|}\right),&n\in\mathbbm{Z}^{*},\\\
\lambda_{n}^{2}=&-\omega_{2}+i\beta_{2}n+O\left(\frac{1}{|n|}\right),&n\in\mathbbm{Z}^{*},\\\
\lambda_{n}^{3}=&-\omega_{3}+i\beta_{3}n+O\left(\frac{1}{|n|}\right)&n\in\mathbbm{Z}^{*}.\end{aligned}\right\\}$
(7.1)

We further denote the following notation:

$\Sigma=\\{(n,j):n\in\mathbb{Z}^{*},1\leqslant j\leqslant 3\\}.$

######  Lemma 7.1 (Gap properties of the spectrum).

For any $(n,j),(k,l)\in\Sigma$, there exists $\hat{C}>0$ depending on
$\omega_{1},\omega_{2},\omega_{3},\beta_{1},\beta_{2},\beta_{3}$, such that

$\left|\lambda_{n}^{j}-\lambda_{k}^{l}\right|\geq\hat{C}.$ (7.2)

###### Proof.

Note that $\omega_{1},\omega_{2},\omega_{3}$ are distinct and
$\beta_{1},\beta_{2},\beta_{3}$ are also distinct. Thanks to the asymptotic
behavior of the eigenvalues (7.1), we have a large $N\in\mathbb{N}$ such that
for $(n,j),(k,j)\in\Sigma,\text{ with }n\neq k$ and $|n|,|k|>N$,

$\left|\lambda_{n}^{j}-\lambda_{k}^{j}\right|\geq\left|\Im({\lambda_{n}^{j}-\lambda_{k}^{j}})\right|\geq
C\left|\beta_{j}(n-k)\right|\geq C|\beta_{j}|.$

Analogously, for $(n,j),(k,l)\in\Sigma,j\neq l$, $|n|,|k|>N$, we have

$\left|\lambda_{n}^{j}-\lambda_{k}^{l}\right|\geq\left|\Re({\lambda_{n}^{j}-\lambda_{k}^{l}})\right|\geq
C\left|\omega_{j}-\omega_{l}\right|.$

As the eigenvalues are simple, we also find the gap condition (7.2) for finite
number of the lower frequencies. ∎

Let us first define the exponential type and sine type functions.

######  Definition 7.2 (Entire functions of exponential type).

An entire function $f$ is said to be of exponential type $A$ if there exist
positive constants $A,B$ such that

$\displaystyle|f(z)|\leqslant Be^{A|z|},\,z\in\mathbb{C},$

and of exponential type at most $A$, if for any $\epsilon>0$, there exits
$B_{\epsilon}>0$ such that

$\displaystyle|f(z)|\leqslant
B_{\epsilon}\,e^{(A+\epsilon)|z|},\,z\in\mathbb{C}.$

######  Definition 7.3.

(Sine type function) An entire function $f$ of exponential type $\pi$ is said
to be of sine type if

  * •

the zeros of $f(z)$, say $\mu_{k}$ satisfy the gap condition, i.e., there
exists $\delta>0$ such that $|\mu_{k}-\mu_{l}|>\delta$ for $k\neq l$, and

  * •

there exist positive constants $C_{1}$,$C_{2}$ and $C_{3}$ such that

$C_{1}e^{\pi|y|}\leqslant|f(x+iy)|\leqslant
C_{2}e^{\pi|y|},\,\forall\,x,y\in\mathbb{R}\text{ with }|y|\geq C_{3}.$

The following proposition states some important properties of sine-type
functions:

######  Proposition 7.4.

Let $f$ be a sine type function, and let $\\{\mu_{k}\\}_{k\in\mathcal{I}}$
with $\mathcal{I}\subset\mathbb{Z}$ be its sequence of zeros. Then, we have:

  * •

for any $\epsilon>0$, there exist constants
$K_{\epsilon},\tilde{K}_{\epsilon}>0$ such that

$K_{\epsilon}e^{\pi|y|}\leqslant|f(x+iy)|\leqslant\tilde{K}_{\epsilon}e^{\pi|y|},\text{
if dist$\left(x+iy,\\{\mu_{k}\\}\right)>\epsilon$},$

  * •

there exist some constants $K_{1},K_{2}>0$ such that

$K_{1}<|f^{\prime}(\mu_{k})|<K_{2},\quad\forall k\in\mathcal{I}.$

The main goal of this section is to find a class of entire functions
$\mathcal{G}=\\{\Psi_{k}^{j}\\}_{(k,j)\in\Sigma}$ with the following
properties:

  * 1.

The family $\mathcal{G}$ contains entire functions of exponential type
$\frac{T}{2}$. That means there exists a positive constant $C$ such that

$\displaystyle\left|\Psi_{k}^{j}(z)\right|\leqslant
Ce^{\frac{T}{2}\left|z\right|},\quad\forall
z\in\mathbb{C},\,\,(k,j)\in\Sigma.$ (7.3)

  * 2.

All the members of $\mathcal{G}$ are square-integrable on the real line, i.e.,

$\displaystyle\int_{\mathbb{R}}\left|\Psi_{k}^{j}(x)\right|^{2}\,{\rm
d}x<\infty.$ (7.4)

Let us now state the celebrated Paley-Wiener theorem, from which we can
conclude about the desired biorthogonal family using the family $\mathcal{G}$.

######  Theorem 7.5 (Paley-Wiener).

Let $f$ be an entire function of exponential type $A$ and suppose

$\displaystyle\int_{-\infty}^{\infty}|f(x)|^{2}\,{\rm d}x<\infty.$

Then there exists a function $\phi\in L^{2}(-A,A)$ with the following
representation

$\displaystyle f(z)=\int_{-A}^{A}e^{izt}\phi(t)\,{\rm d}t,\,z\in\mathbb{C}.$

Thus if we have the existence of such family $\mathcal{G}$ of entire functions
satisfying (7.3)-(7.4), then applying the Paley-Wiener theorem for the same,
one can get a family of functions
$\mathcal{J}=\\{\Theta_{k}^{j},\\}_{(k,j)\in\Sigma}$ supported in
$[-\frac{T}{2},\frac{T}{2}]$, such that the following representation holds

$\displaystyle\Psi_{k}^{j}(z)=\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{izt}\Theta_{k}^{j}(t)dt,\,z\in\mathbb{C}.$
(7.5)

Clearly, $\Theta_{k}^{j}$ are the Fourier transformations of $\Psi_{k}^{j}$
respectively, for $(k,j)\in\Sigma.$ Also, by Plancharel’s Theorem, we have

$\displaystyle\int_{\mathbb{R}}\left|\Psi_{k}^{j}(x)\right|^{2}=2\pi\int_{-\frac{T}{2}}^{\frac{T}{2}}\left|\Theta_{k}^{j}(t)\right|^{2}\,{\rm
d}t.$

Now we are in the position of formulating the construction of the family
$\mathcal{G}$ satisfying (7.3)-(7.4). Let us first introduce the following
entire function, which has simple zeros exactly at
$i\overline{\lambda_{k}^{j}}:$

$\displaystyle
P(z)=z^{3}\prod_{(k,j)\in\Sigma}\left(1-\frac{z}{i\overline{\lambda_{k}^{j}}}\right).$
(7.6)

######  Proposition 7.6.

Let $P$ be the canonical product defined in (7.6). Then $P$ is an entire
function of exponential type
$\pi\left(\frac{1}{|\beta_{1}|}+\frac{1}{|\beta_{2}|}+\frac{1}{|\beta_{3}|}\right)$,
which satisfies the following properties:

  * •

There exists a positive constant $C>0$ such that

$|P(x)|\leqslant C,\quad\forall x\in\mathbb{R}.$ (7.7)

  * •

There exists constant $C_{2}>0$ such that

$\left|P^{\prime}\left(i\overline{\lambda_{k}^{j}}\right)\right|\geq
C_{2},\forall(k,j)\in\Sigma.$ (7.8)

###### Proof.

Let us denote $\nu_{n}^{j}=\frac{1}{\beta_{j}}\lambda_{n}^{j}.$ thus, we have:

$\begin{cases}\nu_{n}^{1}=-\frac{\omega_{1}}{\beta_{1}}+in+O\left(\frac{1}{|n|}\right),\\\
\nu_{n}^{2}=-\frac{\omega_{2}}{\beta_{2}}+in+O\left(\frac{1}{|n|}\right),\\\
\nu_{n}^{3}=-\frac{\omega_{3}}{\beta_{3}}+in+O\left(\frac{1}{|n|}\right).\end{cases}$
(7.9)

Let us denote the products $P_{j},j\in\\{1,2,3\\}$ by

$P_{j}(z)=z\prod_{k\in\mathbb{Z}^{*}}\left(1-\frac{z}{i\overline{\nu_{k}^{j}}}\right).$
(7.10)

Thus the canonical product $P$ can be written in the following form

$P(z)=\prod_{j\in\\{1,2,3\\}}\beta_{j}P_{j}\left(\frac{z}{\beta_{j}}\right).$
(7.11)

######  Lemma 7.7 (Young[42], Rosier [28]).

Let $\Lambda_{k}=k+d_{k}$, where $d_{k}=d+O(k^{-1}),$ as
$\left|k\right|\to\infty$ for some constant $d\in\mathbb{C}$, and that
$\Lambda_{k}\neq\Lambda_{l}$ for $k\neq l.$ Then
$f(z)=z\prod_{k\in\mathbb{Z}^{*}}\left(1-\frac{z}{\Lambda_{k}}\right)$ is an
entire function of sine type.

Note that,
$i\overline{\nu_{k}^{j}}=-i\frac{\omega_{j}}{\beta_{j}}+n+O\left(\frac{1}{|n|}\right),j\in\\{1,2,3\\}$.
Also $i\overline{\nu_{k}^{j}}\neq i\overline{\nu_{l}^{j}},$ for $l\neq k$.
Thanks to 7.7, each $P_{j}$ is a sine type function. Moreover, each of these
functions is an entire function of exponential type $\pi$. Hence, $P$ is an
entire function of the exponential type
$\pi\left(\frac{1}{|\beta_{1}|}+\frac{1}{|\beta_{2}|}+\frac{1}{|\beta_{3}|}\right)$.

Thanks to the definition of sine-type function, for any $\epsilon>0$ there
exist positive constants $C_{1},C_{2},C_{3},C_{4},$ $C_{5}$, where $C_{2}$ and
$C_{3}$ depend on $\epsilon$, such that

$\displaystyle\left|P_{j}(z)\right|\leqslant C_{1}^{j}e^{\pi\left|z\right|},$
$\displaystyle\>\>z\in\mathbb{C},\quad j\in\\{1,2,3\\},$ (7.12) $\displaystyle
C_{2}^{j}e^{\pi\left|y\right|}\leqslant\left|P_{j}(x+iy)\right|\leqslant
C_{3}^{j}e^{\pi\left|y\right|},\>\>$ $\displaystyle\text{if
}dist\left(x+iy,\\{i\overline{\nu_{j}^{k}}\\}\right)>\epsilon,$ (7.13)
$\displaystyle
C_{4}^{j}<\left|P_{j}^{\prime}\left(i\overline{\nu_{k}^{j}}\right)\right|<C_{5}^{j},$
$\displaystyle\>\>\forall(k,j)\in\Sigma.$ (7.14)

Now using the above inequality (7.13) and continuity of $P_{j}$, we get a
positive constant $C^{j}$ such that

$\left|P_{j}(x)\right|\leqslant C^{j},\quad\forall\,x\in\mathbb{R}.$ (7.15)

Therefore the estimate (7.7) holds.

Using (7.11) and (7.14) we have:

$\displaystyle\left|P^{\prime}\left(i\overline{\lambda_{n}^{j}}\right)\right|=\left|P_{j}^{\prime}\left(i\overline{\nu_{n}^{j}}\right)\prod_{\begin{subarray}{c}l\in\\{1,2,3\\}\\\
l\neq
j\end{subarray}}\beta_{l}P_{l}\left(\frac{i\overline{\lambda_{n}^{j}}}{\beta_{l}}\right)\right|\geq\min_{1\leqslant
j\leqslant 3}\\{C_{4}^{j}\\}\prod_{\begin{subarray}{c}l\in\\{1,2,3\\}\\\ l\neq
j\end{subarray}}\left|\beta_{l}P_{l}\left(\frac{i\overline{\lambda_{n}^{j}}}{\beta_{l}}\right)\right|.$
(7.16)

Therefore to prove the estimate (7.8), we need to compute the estimate of the
quantities
$P_{l}\left(\frac{i\overline{\lambda_{n}^{j}}}{\beta_{l}}\right),l\neq j.$
Thanks to 7.1, we have

$\displaystyle\left|\frac{i\overline{\lambda_{n}^{j}}}{\beta_{l}}-i\overline{\nu_{k}^{l}}\right|=\frac{1}{|\beta_{l}|}\left|{\overline{\lambda_{n}^{j}}}-\beta_{l}\overline{\nu_{k}^{l}}\right|=\frac{1}{|\beta_{l}|}\left|{\overline{\lambda_{n}^{j}}}-\overline{\lambda_{k}^{l}}\right|>\hat{C}.$

Therefore using the estimate (7.13), we finally have the estimate

$\left|P_{l}\left(\frac{i\overline{\lambda_{n}^{j}}}{\beta_{l}}\right)\right|\geq
C_{2}^{l}e^{\pi\left|\frac{\omega_{j}}{\beta_{l}}\right|}.$ (7.17)

Hence (7.16) along with (7.17) provide the estimate (7.8). ∎

######  Theorem 7.8.

Let
$T>2\pi\left(\frac{1}{|\beta_{1}|}+\frac{1}{|\beta_{2}|}+\frac{1}{|\beta_{3}|}\right).$
Then there exists a family $\\{\Theta_{n}^{j}\\}_{(n,j)\in\Sigma}$ which is
biorthogonal to the family of exponentials
$\\{e^{-{\lambda_{k}^{l}}t}\\}_{(k,l)\in\Sigma}$ in
$L^{2}\left(-\frac{T}{2},\frac{T}{2}\right)$, i.e.,

$\int_{-\frac{T}{2}}^{\frac{T}{2}}\Theta_{n}^{j}(t)e^{-\overline{\lambda_{k}^{l}}t}\,{\rm
d}t=\delta_{nk}\delta_{jl}.$ (7.18)

Moreover, there exists a positive constant $C>0$ such that the following
estimate holds:

$\left\|\sum_{(n,j)\in\Sigma}a_{n}^{j}\Theta_{n}^{j}\right\|_{L^{2}\left(-\frac{T}{2},\frac{T}{2}\right)}^{2}\leqslant
C\sum_{(n,j)\in\Sigma}|a_{n}^{j}|^{2},$ (7.19)

for any finite sequence of complex numbers $\\{a_{n}^{j}\\}_{(n,j)\in\Sigma}$.

###### Proof.

Let us first define the entire function

$\Psi_{n}^{j}(z):=\frac{P(z)}{(z-i\overline{\lambda_{n}^{j}})P^{\prime}\left(i\overline{\lambda_{n}^{j}}\right)}.$

Clearly, $\Psi_{n}^{j}$ is a collection of entire functions of exponential
type at most
$\pi\left(\frac{1}{|\beta_{1}|}+\frac{1}{|\beta_{2}|}+\frac{1}{|\beta_{3}|}\right).$
Moreover, it is easy to check that

$\displaystyle\int_{\mathbb{R}}\left|\Psi_{n}^{j}(x)\right|^{2}\,{\rm d}x<C.$
(7.20)

Therefore by Paley-Wiener Theorem (Theorem 7.5), there exists a collection of
functions $\\{\Theta_{n}^{j}\\}_{(n,j)\in\Sigma}$ supported in
$[-\frac{T^{\prime}}{2},\frac{T^{\prime}}{2}]$, with
$T^{\prime}=2\pi\left(\frac{1}{|\beta_{1}|}+\frac{1}{|\beta_{2}|}+\frac{1}{|\beta_{3}|}\right)$
such that the following relation holds

$\displaystyle\Psi_{n}^{j}(z)=\int_{-\frac{T^{\prime}}{2}}^{\frac{T^{\prime}}{2}}e^{izt}\Theta_{n}^{j}(t)\,{\rm
d}t,\,z\in\mathbb{C}.$ (7.21)

Also, we can deduce

$\Psi_{n}^{j}(i\overline{\lambda_{k}^{l}})=\delta_{nk}\delta_{jl},\,\,(n,j),(k,l)\in\Sigma.$

Thus we have the family $\\{\Theta_{n}^{j}\\}_{(n,j)\in\Sigma}$ which is
biorthogonal to the family of exponentials
$\\{e^{-{\lambda_{k}^{l}}t}\\}_{(k,l)\in\Sigma}$ in
$L^{2}\left(-\frac{T^{\prime}}{2},\frac{T^{\prime}}{2}\right)$ and by
Plancharel’s Theorem and (7.20), we have

$2\pi\int_{-\frac{T^{\prime}}{2}}^{\frac{T^{\prime}}{2}}\left|\Theta_{k}^{j}(t)\right|^{2}\,{\rm
d}t=\int_{\mathbb{R}}\left|\Psi_{k}^{j}(x)\right|^{2}\,{\rm d}x\leqslant C.$

Then using [35, Proposition 8.3.9], we have for any $T>T^{\prime}$, there
exists a biorthogonal family $\\{\Theta_{n}^{j}\\}_{(n,j)\in\Sigma}$ of
$\\{e^{-{\lambda_{k}^{l}}t}\\}_{(k,l)\in\Sigma}$ in
$L^{2}\left(-\frac{T}{2},\frac{T}{2}\right)$ such that the estimate (7.19)
holds. ∎

The following corollary is an immediate consequence of the above theorem which
will essentially give us the required Ingham-type inequality.

######  Corollary 7.9.

Let us assume
$T>2\pi\left(\frac{1}{|\beta_{1}|}+\frac{1}{|\beta_{2}|}+\frac{1}{|\beta_{3}|}\right).$
Then for any finite sequence of scalars $\\{a_{n}^{j}\\}_{(n,j)\in\Sigma}$,
there exist two positive constants $C_{1},C_{2}$ such that the following
inequality holds:

$C_{1}\sum_{(n,j)\in\Sigma}|a_{n}^{j}|^{2}\leqslant\left\|\sum_{(n,j)\in\Sigma}a_{n}^{j}e^{-\overline{\lambda_{n}^{j}}t}\right\|_{L^{2}\left(-\frac{T}{2},\frac{T}{2}\right)}^{2}\leqslant
C_{2}\sum_{(n,j)\in\Sigma}|a_{n}^{j}|^{2}.$ (7.22)

###### Proof.

Using (7.18), we can deduce

$\displaystyle\sum_{(n,j)\in\Sigma}|a_{n}^{j}|^{2}=$
$\displaystyle\int_{-\frac{T}{2}}^{\frac{T}{2}}{\left(\sum_{(n,j)\in\Sigma}a_{n}^{j}e^{-\overline{\lambda_{n}^{j}}t}\right)}\left(\sum_{(k,l)\in\Sigma}\overline{a_{k}^{l}}\Theta_{k}^{l}(t)\right)\,{\rm
d}t$ $\displaystyle\leqslant$
$\displaystyle\left\|\sum_{(n,j)\in\Sigma}a_{n}^{j}e^{-\overline{\lambda_{n}^{j}}t}\right\|_{L^{2}\left(-\frac{T}{2},\frac{T}{2}\right)}\left\|\sum_{(n,j)\in\Sigma}\overline{a_{k}^{l}}\Theta_{k}^{l}\right\|_{L^{2}\left(-\frac{T}{2},\frac{T}{2}\right)}.$
(7.23)

The inequality (7) along with (7.19) establish the left inequality of (7.22).

Straightforward estimate provides the right hand inequality. Indeed,

$\displaystyle\left\|\sum_{(n,j)\in\Sigma}a_{n}^{j}e^{-\overline{\lambda_{n}^{j}}t}\right\|_{L^{2}\left(-\frac{T}{2},\frac{T}{2}\right)}^{2}\leqslant
C_{2}\sum_{j=1}^{3}\left\|\sum_{n\in\mathbb{Z}^{*}}a_{n}^{j}e^{-\overline{\lambda_{n}^{j}}t}\right\|_{L^{2}\left(-\frac{T}{2},\frac{T}{2}\right)}^{2}.$

Next, utilizing the Ingham-type inequality [1, Proposition 8.1], one can get
the right hand inequality.∎

## Acknowledgments

The authors would like to thank Dr. Shirshendu Chowdhury and Dr. Debanjana
Mitra for fruitful discussions. Sakil Ahamed expresses his gratitude to the
IISER Kolkata’s Department of Mathematics & Statistics for their hospitality
and assistance during his visit.

## Data availability statement

This article describes entirely theoretical research. Thus, data sharing is
not applicable to this article as no datasets were generated or analysed
during the current study.

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