# Insensitizing controls for a fourth order semi-linear parabolic equations

Bo You***Email address<EMAIL_ADDRESS>
School of Mathematics and Statistics, Xi’an Jiaotong University
Xi’an, 710049, P. R. China
Fang Li†††Email address<EMAIL_ADDRESS>
School of Mathematics and Statistics, Xidian University
Xi’an, 710071, P. R. China

###### Abstract

This paper is concerned with the existence of insensitizing controls for a
fourth order semilinear parabolic equation. Here, the initial data is
partially unknown, we would like to find controls such that a specific
functional is insensitive for small perturbations of the initial data. In
general, this kind of problems can be recast as a null controllability problem
for a nonlinear cascade system. We will first prove a null controllability
result for a linear problem by global Carleman estimates and dual arguments.
Then, by virtue of Leray-Schauder’s fixed points theorem, we conclude the null
controllability for the cascade system in the semi-linear case.

Keywords: Carleman estimates; Insensitizing controls; Null controllability,
Leray-Schauder’s fixed points theorem.

Mathematics Subject Classification (2010) : 35Q93; 49J20; 90C31; 93B05; 93C20;
93C41.

##  1 Introduction

Let $D\subset\mathbb{R}^{n}(n\geq 2)$ be a nonempty bounded connected open set
with smooth boundary $\partial D,$ $T>0$ and $\omega\subset D$ is a small
nonempty open subset which is usually referred to as a control domain. Denote
by $Q=D\times(0,T),$ $\Sigma=\partial D\times(0,T),$
$Q_{\omega}=\omega\times(0,T).$ Let $\mathcal{O}\subset D$ be another open set
which is the so-called observation set.

In this paper, we mainly consider the following semilinear fourth order
parabolic equation with incomplete data:

$\begin{cases}\frac{\partial y}{\partial
t}+\Delta^{2}y+a_{0}y+B_{0}\cdot\nabla y+B:\nabla^{2}y+a_{1}\Delta
y=F(y,\nabla y,\nabla^{2}y)+v\chi_{\omega}+f,\,\,\,\,\forall\,\,\,(x,t)\in
Q,\\\ y=\Delta y=0,\,\,\,\,\,\forall\,\,\,\,(x,t)\in\Sigma,\\\
y(x,0)=y_{0}(x)+\tau\hat{y}_{0}(x),\,\,\,\,\forall\,\,\,\,x\in D.\end{cases}$
(1.1)

Here, the functions $a_{0},$ $a_{1}\in L^{\infty}(Q;\mathbb{R}),$
$B_{0}=(B_{01},B_{02},\cdots,B_{0n})\in L^{\infty}(Q;\mathbb{R}^{n}),$
$B=(B_{ij})_{n\times n}\in L^{\infty}(Q;\mathbb{R}^{n^{2}}),$ $f\in L^{2}(Q)$
is a given externally applied force, the function
$F:\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n^{2}}\rightarrow\mathbb{R}$
is locally Lipschitz continuous, $\chi_{\omega}$ is the characteristic
function of the set $\omega,$ $v\in L^{2}(Q_{\omega})$ is a control function
to be determined and the initial data $y(x,0)$ is partially unknown in the
following sense:

  1. (a)

$y_{0}\in L^{2}(D)$ is known.

  2. (b)

$\hat{y}_{0}\in L^{2}(D)$ is unknown with $\|\hat{y}_{0}\|_{L^{2}(D)}=1.$

  3. (c)

$\tau$ is a small unknown real number.

Let $y$ be the solution of problem (1.1) associated to $\tau$ and $v,$ we
observe the solution of problem (1.1) via some functional $\Phi(y),$ which is
called the sentinel. Here, the sentinel is defined by the square of the local
$L^{2}$-norm of the state variable:

$\displaystyle\Phi(y)=\frac{1}{2}\int_{0}^{T}\int_{\mathcal{O}}|y(x,t)|^{2}\,dxdt.$
(1.2)

A control function $v$ is said to insensitize the functional $\Phi,$ if

$\displaystyle\frac{\partial\Phi(y)}{\partial\tau}|_{\tau=0}=0,\,\,\,\,\forall\,\,\,\hat{y}_{0}\in
L^{2}(D)\,\,\,\textit{with}\,\,\,\|\hat{y}_{0}\|_{L^{2}(D)}=1.$ (1.3)

Thus, the insensitizing control problem is to seek for a control $v,$ such
that the uncertainty in the initial data does not effect the measurement
$\Phi$ at least at the first order.

To the best of our knowledge, this kind of insensitizing control problem was
first considered by J. L. Lions in [31]. Later, in [2, 30], the authors
reformulated the insensitization problem with this kind of the sentinel $\Phi$
as a null controllability problem for a cascade system. Inspired by these
works, there have been many results concerning the existence of insensitizing
controls in all kinds of different contexts. Initially, the existence of an
approximate insensitizing controls (i.e., such that
$\left|\partial_{\tau}\Phi(y)|_{\tau=0}\right|\leq\epsilon$) was proved in [2]
for a semilinear heat system with $\mathcal{C}^{1}$ and globally Lipschitz
nonlinearities. In [16], the author proved for the linear heat equation that
we cannot expect insensitivity to hold for all initial data, except when the
control acts everywhere in $\Omega.$ Regarding the class of initial data that
can be insensitized, the results in [18] also give different results of
positive and negative nature. Later, the results in [2] was generalized in [3,
4, 38] to superlinear heat equation with nonlinear terms depending on the
state and/or its gradient. In particular, there are some results about the
existence of insensitizing controls for the parabolic equation with different
boundary conditions. For example, the authors in [41] proved the existence of
insensitizing controls for the parabolic equations with dynamic boundary
conditions. The existence of a local insensitizing control for the semilinear
parabolic equations with nonlinear Fourier boundary conditions was established
in [5]. Moreover, the author in [32] proved the existence of insensitizing
controls for the quasilinear parabolic equations. The existence of
insensitizing controls for a phase field system was proved in
[6].Additionally, the authors studied the existence of insensitizing controls
for the Navier-Stokes equation and the Boussinesq system (see [7, 10, 11,
26]), the semilinear wave equations (see [1, 37]). It is worthy to mention
that the authors treated the case of a different type of sentinel consisting
of the gradient of the solution of a parabolic equation in [24, 36] and the
case of the curl for the Stokes system in [23].

Adapting the computations in [2] to problem (1.1)-(1.3), we conclude that the
existence of a control $v$ such that (1.3) holds is equivalent to the
existence of a control $v$ such that the solution $(y,q)$ of problem

$\begin{cases}\frac{\partial y}{\partial
t}+\Delta^{2}y+a_{0}y+B_{0}\cdot\nabla y+B:\nabla^{2}y+a_{1}\Delta
y=F(y,\nabla y,\nabla^{2}y)+\chi_{\omega}v+f,\,\,\,\,\forall\,\,\,(x,t)\in
Q,\\\ -\frac{\partial q}{\partial
t}+\Delta^{2}q+a_{0}q-\nabla\cdot(B_{0}q)+\sum_{i,j=1}^{n}\frac{\partial^{2}(B_{ij}q)}{\partial
x_{i}\partial x_{j}}+\Delta(a_{1}q)=F_{y}(y,\nabla y,\nabla^{2}y)q\\\
-\nabla\cdot(\nabla_{p}F(y,\nabla
y,\nabla^{2}y)q)+\sum_{i,j=1}^{n}\frac{\partial^{2}(F_{r_{ij}}(y,\nabla
y,\nabla^{2}y)q)}{\partial x_{i}\partial
x_{j}}+\chi_{\mathcal{O}}y,\,\,\,\,\forall\,\,\,(x,t)\in Q,\\\ y=\Delta
y=0,\,\,\,q=\Delta q=0,\,\,\forall\,\,\,\,(x,t)\in\Sigma,\\\
y(x,0)=y_{0}(x),\,\,q(x,T)=0,\,\,\forall\,\,\,\,x\in D\end{cases}$ (1.4)

verifying

$\displaystyle q(x,0)=0,\,\,\,\,\forall\,\,\,x\in D,$ (1.5)

where $p=\nabla y$ and $r_{ij}=\frac{\partial^{2}y}{\partial x_{i}\partial
x_{j}}.$

In recent several years, there are some results about the controllability for
fourth order parabolic equations in both one dimension (see [8, 9, 12, 13, 14,
22, 27, 33]) and the higher dimensions (see [19, 25, 28, 34, 40]). In
particular, the approximate controllability and non-approximate
controllability of higher order parabolic equations were studied in [19]. The
author in [40] proved the null controllability of fourth order parabolic
equations by using the ideas of [29]. It is worthy to mention that the
Carleman inequality for a fourth order parabolic equation with $n\geq 2$ was
first established in [25]. Later, the author in [28] proved the null
controllability and the exact controllability to the trajectories at any time
$T>0$ for the fourth order semi-linear parabolic equations with a control
function acting at the interior. The null controllability for fourth order
stochastic parabolic equations was proved by duality arguments and a new
global Carleman estimates in [34]. A unified weighted inequality for fourth-
order partial differential operators was given in [15]. Moreover, they applied
it to obtain the log-type stabilization result for the plate equation.
Recently, we in [39] established the global Carleman estimates for the fourth
order parabolic equations with low regularity terms subject to the homogeneous
Dirichlet boundary conditions of $y$ as well as $\Delta y,$ and applied it to
the null controllability. However, there is no results concerning the
existence of insensitizing controls for fourth order semilinear parabolic
equations. Since the insensitizing control problems describe some kind of
stability of system (1.1) with respect to initial data, it is very meaningful
to investigate the existence of insensitizing controls for problem (1.1).

The main objective of this paper is to study the insensitizing controls
problem (1.1)-(1.3). Inspired by the work in [2], we conclude that the
insensitizing controls problem (1.1)-(1.3) is equivalent to the partial null
controllability of problem (1.4).Thus, we need to establish an observation
inequality for the adjoint problem (3.1) of the linearized system for problem
(1.4) based on the duality arguments. But problem (1.4) is coupled, we will
choose some suitable cut-off function and combine the global Carleman
estimates to conclude the following inequality

$\displaystyle\int_{0}^{T}\int_{\mathcal{O}}|\psi|^{2}e^{2s\alpha}\leq
C\int_{Q_{\omega_{1}}}|\varphi|^{2}\,dxdt$

for some suitable subset $\omega_{1}$ of $D$ and weight function $\alpha,$
which will entails the desired observability inequality of problem (3.1).
Throughout this paper, we will always suppose that
$\omega\cap\mathcal{O}\neq\emptyset$, which is a condition that has always
been imposed as long as insensitizing controls are concerned. However, in
[17], it has been proved that this is not a necessary condition for
$\epsilon$-insensitizing controls for some linear parabolic equations (see
also [35]). Thus, we shall assume that $y_{0}\equiv 0$ which is a classical
hypothesis in insensitization problems.

The rest of this paper is organized as follows: in Section 2, we will recall
some global Carleman estimates and prove a technique lemma. In Section 3, we
will prove an observability inequality, which implies the existence of
insensitizing controls for fourth order linear parabolic equations. Section 4
is devoted to the existence of insensitizing controls for the semilinear case.

##  2 Preliminaries

In this section, we will recall the Carleman inequalities of fourth order
parabolic equations and some lemmas used in the sequel. To this purpose, we
first introduce the following weight functions.

######  Lemma 2.1.

([21]) Let $\omega_{0}\subset\subset D$ be an arbitrary fixed subdomain of $D$
such that $\overline{\omega_{0}}\subset\omega.$ Then there exists a function
$\eta\in\mathcal{C}^{4}(\overline{D})$ such that

$\displaystyle\eta(x)>0,\,\,\,\,\forall\,\,\,x\in
D;\,\,\eta(x)=0,\,\,\,\,\forall\,\,\,x\in\partial
D,;\,\,|\nabla\eta(x)|>0,\,\,\,\,\textit{for\,\,\,all}\,\,\,x\in\overline{D\backslash\omega_{0}}.$

In order to state the global Carleman inequality, we define some weight
functions:

$\displaystyle\alpha_{0}(x)=e^{\lambda(2\|\eta\|_{L^{\infty}(D)}+\eta(x))}-e^{4\lambda\|\eta\|_{L^{\infty}(D)}},\,\,\,\,\xi_{0}(x)=e^{\lambda(2\|\eta\|_{L^{\infty}(D)}+\eta(x))},$
(2.1)
$\displaystyle\alpha(x,t)=\frac{\alpha_{0}(x)}{\sqrt{t(T-t)}},\,\,\,\xi(x,t)=\frac{e^{\lambda(2\|\eta\|_{L^{\infty}(D)}+\eta(x))}}{\sqrt{t(T-t)}}.$
(2.2)

Moreover, they possess the following properties:

$\displaystyle\nabla\alpha=\nabla\xi=\lambda\xi\nabla\eta,\,\,\xi^{-1}\leq\frac{T}{2},\,\,|\alpha_{t}|+|\xi_{t}|\leq\frac{T}{2}\xi^{3},\,\,\forall\,\,(x,t)\in
Q.$ (2.3)

######  Lemma 2.2.

(see [25, 39]) Assume that $z_{0}\in L^{2}(D),$ $g\in L^{2}(Q)$ and the
functions $\alpha,$ $\xi$ are defined by (2.2). Then there exists
$\hat{\lambda}>0$ such that for an arbitrary $\lambda\geq\hat{\lambda},$ we
can choose $s_{0}=s_{0}(\lambda)>0$ satisfying: there exists a constant
$C=C(\lambda)>0$ independent of $s,$ such that the solution $z\in L^{2}(Q)$ to
problem

$\begin{cases}L^{*}z=-\frac{\partial z}{\partial
t}+\Delta^{2}z=g,\,\,\,\,\textit{in}\,\,\,Q,\\\ z=\Delta
z=0,\,\,\,\,\,\textit{on}\,\,\,\,\Sigma,\\\
z(x,T)=z_{0}(x),\,\,\,\,\textit{in}\,\,\,\,D,\end{cases}$ (2.4)

satisfies the following inequality:

  1. (i)

If $g\in L^{2}(Q),$ for any $\lambda\geq\hat{\lambda}$ and any $s\geq
s_{0}(\lambda)(\sqrt{T}+T),$ one has

$\displaystyle\int_{Q}e^{2s\alpha}\left(s^{6}\lambda^{8}\xi^{6}|z|^{2}+s^{4}\lambda^{6}\xi^{4}|\nabla
z|^{2}+s^{3}\lambda^{4}\xi^{3}|\Delta
z|^{2}+s^{2}\lambda^{4}\xi^{2}|\nabla^{2}z|^{2}+s\lambda^{2}\xi|\nabla\Delta
z|^{2}\right)\,dxdt$
$\displaystyle+\int_{Q}e^{2s\alpha}\left(\frac{1}{s\xi}(|z_{t}|^{2}+|\Delta^{2}z|^{2})\right)\,dxdt$
$\displaystyle\leq
C\left(\int_{Q_{\omega}}s^{7}\lambda^{8}\xi^{7}|z|^{2}e^{2s\alpha}\,dxdt+\int_{Q}|g|^{2}e^{2s\alpha}\,dxdt\right)$

  2. (ii)

If $g=g_{0}+\sum_{i=1}^{n}\frac{\partial g_{i}}{\partial
x_{i}}-\sum_{i,j=1}^{n}\frac{\partial^{2}(B_{ij}y)}{\partial x_{i}\partial
x_{j}}-\Delta(a_{1}y)$ with $g_{i}\in L^{2}(Q)$ for any $0\leq i\leq n,$ and
$a_{1}\in L^{\infty}(Q;\mathbb{R}),$ $B=(B_{ij})_{n\times n}\in
L^{\infty}(Q;\mathbb{R}^{n^{2}}),$ then

$\displaystyle\int_{Q}e^{2s\alpha}\left(s^{6}\lambda^{8}\xi^{6}|z|^{2}+s^{4}\lambda^{6}\xi^{4}|\nabla
z|^{2}+s^{2}\lambda^{4}\xi^{2}|\Delta
z|^{2}+s^{2}\lambda^{4}\xi^{2}|\nabla^{2}z|^{2}\right)\,dxdt$
$\displaystyle\leq$ $\displaystyle
C\int_{Q}\left(|g_{0}|^{2}+\sum_{i=1}^{n}(s\lambda\xi)^{2}|g_{i}|^{2}\right)e^{2s\alpha}\,dxdt+C\int_{Q_{\omega}}s^{7}\lambda^{8}\xi^{7}|z|^{2}e^{2s\alpha}\,dxdt$

for any
$\lambda\geq\hat{\lambda}(\lambda)(1+\|B\|_{L^{\infty}(Q)}^{\frac{1}{2}}+\|a_{1}\|_{L^{\infty}(Q)}^{\frac{1}{2}})$
and any $s\geq s_{0}(\sqrt{T}+T).$

In what follows, we also prove the following technical lemma, which will be
used to establish an observability inequality.

######  Lemma 2.3.

Let the functions $\alpha_{0},$ $\xi_{0},$ $\alpha,$ $\xi$ be defined by (2.1)
and (2.2), denote by $m_{0}=\min\limits_{x\in D}\alpha_{0}(x),$
$M_{0}=\max\limits_{x\in D}\alpha_{0}(x)<0,$ $n_{0}=\min\limits_{x\in
D}\xi_{0}(x)>0$ and $N_{0}=\max\limits_{x\in D}\xi_{0}(x).$ Then the following
conclusions hold:

  1. (1)

For any $s\geq\frac{4T}{|M_{0}|},$ we have

$\displaystyle s^{16}\xi^{16}e^{2s\alpha}\leq
2^{48}\left(\frac{N_{0}}{M_{0}e}\right)^{16}$

for any $(x,t)\in Q.$

  2. (2)

For any $s\geq 0,$ we have

$\displaystyle s^{6}\xi^{6}e^{2s\alpha}\geq A_{s}e^{-\frac{M_{s}}{\sqrt{t}}}$

for any $(x,t)\in\Omega\times(0,\frac{T}{2}),$ where

$\displaystyle
A_{s}=\frac{(2sn_{0})^{6}}{T^{6}}e^{\frac{-2|m_{0}|s}{T}},\,\,\,\,M_{s}=\frac{2|m_{0}|s}{\sqrt{T}}.$

  3. (3)

For any $s\geq 0,$ we have

$\displaystyle\xi^{-6}e^{-2s\alpha}\leq(2n_{0})^{-6}T^{6}e^{\frac{8|m_{0}|s}{\sqrt{3}T}}$

for any $(x,t)\in\Omega\times(\frac{T}{4},\frac{3T}{4}).$

###### Proof.

  1. (i)

Let $\alpha_{0},$ $\alpha,$ $N_{0}$ and $M_{0}$ be as in the statement. Then
we have

$\displaystyle
s^{16}\xi^{16}e^{2s\alpha}\leq(sN_{0})^{16}e^{-\frac{2|M_{0}|s}{\sqrt{t(T-t)}}}t^{-8}(T-t)^{-8}=f_{s}(t)=\frac{1}{g_{s}(t)}$

for any $s>0$ and any $t\in(0,T).$ In what follows, we will give a lower bound
of $g_{s}(t)$ on $(0,T).$ Thanks to

$\displaystyle
g^{\prime}_{s}(t)=\frac{1}{(sN_{0})^{16}}e^{\frac{2|M_{0}|s}{\sqrt{t(T-t)}}}t^{\frac{13}{2}}(T-t)^{\frac{13}{2}}(T-2t)\left\\{8\sqrt{t(T-t)}-|M_{0}|s\right\\},$

which implies that for any $s\geq\frac{4T}{|M_{0}|},$ the function $g_{s}$ is
strictly decreasing in $(0,\frac{T}{2})$ and strictly increasing in
$(\frac{T}{2},T).$ Thus, we have

$\displaystyle f_{s}(t)\leq f_{s}(\frac{T}{2})=2^{16}T^{-16}N_{0}^{16}G(s)$

for any $t\in(0,T)$ with $G(s)=s^{16}e^{-\frac{4|M_{0}|s}{T}}.$ Thanks to

$\displaystyle
G^{\prime}(s)=4s^{15}e^{-\frac{4|M_{0}|s}{T}}(4-\frac{|M_{0}|s}{T}),$

which entails that the function $G(s)$ is strictly decreases in
$(\frac{4T}{|M_{0}|},+\infty).$ Thus, for every $s\geq\frac{4T}{|M_{0}|},$ we
have

$\displaystyle s^{16}\xi^{16}e^{2s\alpha}\leq
2^{16}T^{-16}N_{0}^{16}G(\frac{4T}{|M_{0}|})=2^{48}e^{-16}\left(\frac{N_{0}}{M_{0}}\right)^{16}$

for any $(x,t)\in Q.$

  2. (ii)

First of all, notice that

$\displaystyle
s^{6}\xi^{6}e^{2s\alpha}\geq(sn_{0})^{6}e^{-\frac{2|m_{0}|s}{\sqrt{Tt}}}t^{-3}(T-t)^{-3}e^{-\frac{2|m_{0}|s}{\sqrt{t(T-t)}}+\frac{2|m_{0}|s}{\sqrt{Tt}}}$

and for any $s\geq 0,$

$\displaystyle-\frac{2|m_{0}|s}{\sqrt{t(T-t)}}+\frac{2|m_{0}|s}{\sqrt{Tt}}=$
$\displaystyle-\frac{2|m_{0}|s\sqrt{t}}{\sqrt{T(T-t)}(\sqrt{T}+\sqrt{T-t})}$
$\displaystyle\geq$ $\displaystyle-\frac{2|m_{0}|s}{T}$

for any $t\in(0,\frac{T}{2}).$ Therefore, for any $s\geq 0,$ we obtain

$\displaystyle s^{6}\xi^{6}e^{2s\alpha}\geq$
$\displaystyle(sn_{0})^{6}e^{-\frac{2|m_{0}|s}{\sqrt{Tt}}}t^{-3}(T-t)^{-3}e^{-\frac{2|m_{0}|s}{T}}$
$\displaystyle\geq$
$\displaystyle\frac{(2sn_{0})^{6}}{T^{6}}e^{-\frac{2|m_{0}|s}{T}}e^{-\frac{2|m_{0}|s}{\sqrt{Tt}}}$

for any $t\in(0,\frac{T}{2}).$

  3. (iii)

Thanks to

$\displaystyle\frac{1}{\sqrt{t(T-t)}}\leq\frac{4}{\sqrt{3}T}$

for any $t\in(\frac{T}{4},\frac{3T}{4}),$ we obtain

$\displaystyle\xi^{-6}e^{-2s\alpha}\leq
n_{0}^{-6}e^{\frac{2|m_{0}|s}{\sqrt{t(T-t)}}}2^{-6}T^{6}\leq(2n_{0})^{-6}T^{6}e^{\frac{8|m_{0}|s}{\sqrt{3}T}}$

for any $(x,t)\in\Omega\times(\frac{T}{4},\frac{3T}{4}).$

∎

##  3 The linear case

In this section, we will always assume that $F\equiv 0$ and prove the
existence of an insensitizing control of problem (1.1) such that (1.3) holds.
To start with, we introduce the adjoint problem of the linearized system of
problem (1.4):

$\begin{cases}-\frac{\partial\psi}{\partial
t}+\Delta^{2}\psi+a_{0}\psi-\nabla\cdot(B_{0}\psi)+\sum_{i,j=1}^{n}\frac{\partial^{2}(B_{ij}\psi)}{\partial
x_{i}\partial
x_{j}}+\Delta(a_{1}\psi)=\chi_{\mathcal{O}}\varphi,\,\,\,\,\forall\,\,\,(x,t)\in
Q,\\\ \frac{\partial\varphi}{\partial
t}+\Delta^{2}\varphi+a_{0}\varphi+B_{0}\cdot\nabla\varphi+B:\nabla^{2}\varphi+a_{1}\Delta\varphi=0,\,\,\,\,\forall\,\,\,(x,t)\in
Q,\\\
\psi=\Delta\psi=0,\,\,\,\varphi=\Delta\varphi=0,\,\,\,\,\,\forall\,\,\,\,(x,t)\in\Sigma,\\\
\psi(x,T)=0,\,\,\,\varphi(x,0)=\varphi_{0}(x),\,\,\,\,\forall\,\,\,\,x\in
D.\end{cases}$ (3.1)

From the regularity of fourth order parabolic equations, we conclude that for
any $\varphi_{0}\in L^{2}(D),$ there exists a unique solution of problem (3.1)
satisfying

$\displaystyle\varphi,\psi\in X=L^{2}(0,T;H_{0}^{1}(D)\cap H^{2}(D))\cap
H^{1}(0,T;(H^{2}(D))^{*}).$

In what follows, we will establish an observability inequality of problem
(3.1), which used to obtain the existence of an insensitizing control such
that the solution of problem (1.1) verifying (1.3) in the linear case.

######  Theorem 3.1.

Assume that $\omega\cap\mathcal{O}\neq\emptyset.$ Then there exist two
positive constants $M$ and $H,$ such that for any $\varphi_{0}\in L^{2}(D),$
the corresponding solution $(\psi,\varphi)$ of problem (3.1) with initial data
$(0,\varphi_{0})$ satisfies

$\displaystyle\int_{Q}e^{-\frac{M}{\sqrt{t}}}|\psi|^{2}\,dxdt\leq
H\int_{Q_{\omega}}|\psi|^{2}\,dxdt.$ (3.2)

More precisely, $M=\frac{2|m_{0}|s}{\sqrt{T}}$ and

$\displaystyle
H=C2^{48}\left(\frac{N_{0}}{M_{0}e}\right)^{16}+C2^{42}\left(\frac{N_{0}}{M_{0}e}\right)^{16}e^{2\beta
T+\frac{8|m_{0}|s}{\sqrt{3}T}}n_{0}^{-6}T^{6}$

for any $s\geq\frac{4T}{|M_{0}|},$ where $C=C(D,\omega,\mathcal{O}).$

###### Proof.

Let $\omega_{1}$ and $\omega_{2}$ be two open subsets such that
$\omega_{1}\subset\subset\omega_{2}\subset\subset\omega\cap\mathcal{O}.$
Applying Lemma 2.2 to the second equation of problem (3.1) with
$g=-a_{0}\varphi-B_{0}\cdot\nabla\varphi-B:\nabla^{2}\varphi-
a_{1}\Delta\varphi$ and $\omega=\omega_{1},$ we conclude that there exists a
positive constant $\hat{\lambda},$ such that for any
$\lambda\geq\hat{\lambda},$ we can choose $s_{0}=s_{0}(\lambda)$ satisfying:
there exist a positive constant $C_{1}=C_{1}(D,\omega_{1}),$ such that

$\displaystyle\int_{Q}e^{2s\alpha}\left(s^{6}\lambda^{8}\xi^{6}|\varphi|^{2}+s^{4}\lambda^{6}\xi^{4}|\nabla\varphi|^{2}+s^{3}\lambda^{4}\xi^{3}|\Delta\varphi|^{2}+s^{2}\lambda^{4}\xi^{2}|\nabla^{2}\varphi|^{2}+s\lambda^{2}\xi|\nabla\Delta\varphi|^{2}\right)\,dxdt$
$\displaystyle\leq$ $\displaystyle
C_{1}\int_{Q}\left(|a_{0}|^{2}|\varphi|^{2}+|B_{0}|^{2}|\nabla\varphi|^{2}+|B|^{2}|\nabla^{2}\varphi|^{2}+|a_{1}|^{2}|\Delta\varphi|^{2}\right)e^{2s\alpha}\,dxdt$
$\displaystyle+C_{1}\int_{Q_{\omega_{1}}}s^{7}\lambda^{8}\xi^{7}|\varphi|^{2}e^{2s\alpha}\,dxdt$

for any $s\geq s_{0}(\lambda)(T+\sqrt{T}),$ which implies that

$\displaystyle\int_{Q}e^{2s\alpha}\left(s^{6}\lambda^{8}\xi^{6}|\varphi|^{2}+s^{4}\lambda^{6}\xi^{4}|\nabla\varphi|^{2}+s^{3}\lambda^{4}\xi^{3}|\Delta\varphi|^{2}+s^{2}\lambda^{4}\xi^{2}|\nabla^{2}\varphi|^{2}+s\lambda^{2}\xi|\nabla\Delta\varphi|^{2}\right)\,dxdt$
$\displaystyle\leq$ $\displaystyle
C\int_{Q_{\omega_{1}}}s^{7}\lambda^{8}\xi^{7}|\varphi|^{2}e^{2s\alpha}\,dxdt$
(3.3)

for any
$\lambda\geq\hat{\lambda}(1+\|a_{0}\|_{L^{\infty}(Q)}^{\frac{1}{4}}+\|B_{0}\|_{L^{\infty}(Q)}^{\frac{1}{3}}+\|B\|_{L^{\infty}(Q)}^{\frac{1}{2}}+\|a_{1}\|_{L^{\infty}(Q)}^{\frac{1}{2}})$
and any $s\geq s_{0}(\lambda)(T+\sqrt{T}).$

Employing again Lemma 2.2 to the first equation of problem (3.1) with
$g=-a_{0}\psi+\nabla\cdot(B_{0}\psi)-\sum_{i,j=1}^{n}\frac{\partial^{2}(B_{ij}\psi)}{\partial
x_{i}\partial x_{j}}-\Delta(a_{1}\psi)$ and $\omega=\omega_{2},$ we deduce
that there exists a positive constant $\hat{\lambda},$ such that for any
$\lambda\geq\hat{\lambda},$ we can choose $s_{0}=s_{0}(\lambda)$ satisfying:
there exist a positive constant $C_{2}=C_{2}(D,\omega_{2}),$ such that

$\displaystyle\int_{Q}e^{2s\alpha}\left(s^{6}\lambda^{8}\xi^{6}|\psi|^{2}+s^{4}\lambda^{6}\xi^{4}|\nabla\psi|^{2}+s^{2}\lambda^{4}\xi^{2}|\Delta\psi|^{2}+s^{2}\lambda^{4}\xi^{2}|\nabla^{2}\psi|^{2}\right)\,dxdt$
$\displaystyle\leq
C_{2}\left(\int_{Q_{\omega_{2}}}s^{7}\lambda^{8}\xi^{7}|\psi|^{2}e^{2s\alpha}\,dxdt+\int_{Q}\left(|a_{0}|^{2}|\psi|^{2}+(s\lambda\xi)^{2}|B_{0}|^{2}|\psi|^{2}+\chi_{\mathcal{O}}|\varphi|^{2}\right)e^{2s\alpha}\,dxdt\right),$

for any
$\lambda\geq\hat{\lambda}(1+\|B\|_{L^{\infty}(Q)}^{\frac{1}{2}}+\|a_{1}\|_{L^{\infty}(Q)}^{\frac{1}{2}})$
and any $s\geq s_{0}(\lambda)(\sqrt{T}+T),$ which entails that

$\displaystyle\int_{Q}e^{2s\alpha}\left(s^{6}\lambda^{8}\xi^{6}|\psi|^{2}+s^{4}\lambda^{6}\xi^{4}|\nabla\psi|^{2}+s^{2}\lambda^{4}\xi^{2}|\Delta\psi|^{2}+s^{2}\lambda^{4}\xi^{2}|\nabla^{2}\psi|^{2}\right)\,dxdt$
$\displaystyle\leq
C\left(\int_{Q_{\omega_{2}}}s^{7}\lambda^{8}\xi^{7}|\psi|^{2}e^{2s\alpha}\,dxdt+\int_{0}^{T}\int_{\mathcal{O}}|\varphi|^{2}e^{2s\alpha}\,dxdt\right)$
(3.4)

for any
$\lambda\geq\hat{\lambda}(\lambda)(1+\|a_{0}\|_{L^{\infty}(Q)}^{\frac{1}{4}}+\|B_{0}\|_{L^{\infty}(Q)}^{\frac{1}{3}}+\|B\|_{L^{\infty}(Q)}^{\frac{1}{2}}+\|a_{1}\|_{L^{\infty}(Q)}^{\frac{1}{2}})$
and any $s\geq s_{0}(T+\sqrt{T}).$ In what follows, we will prove an
inequality which bounds $\varphi$ with respect to $\psi.$ Let
$\theta_{1}\in\mathcal{C}_{0}^{\infty}(\omega_{2})$ be a cut-off function such
that

$\displaystyle 0\leq\theta_{1}\leq
1,\,\,\,\textit{in}\,\,\,\omega_{2};\,\,\,\theta_{1}\equiv
1,\,\,\,\forall\,\,\,x\in\omega_{1}.$ (3.5)

Define

$\displaystyle u=s^{7}\lambda^{8}\xi^{7}e^{2s\alpha}$

for any
$\lambda\geq\hat{\lambda}(1+\|a_{0}\|_{L^{\infty}(Q)}^{\frac{1}{4}}+\|B_{0}\|_{L^{\infty}(Q)}^{\frac{1}{3}}+\|B\|_{L^{\infty}(Q)}^{\frac{1}{2}}+\|a_{1}\|_{L^{\infty}(Q)}^{\frac{1}{2}})$
and any $s\geq s_{0}(\lambda)(T+\sqrt{T}).$ Multiplying the first equation of
problem (3.1) by $u\varphi\theta_{1}$ and integrating by parts, we obtain

$\displaystyle\int_{0}^{T}\int_{\mathcal{O}}s^{7}\lambda^{8}\xi^{7}|\varphi|^{2}e^{2s\alpha}\theta_{1}\,dxdt=\int_{Q}\psi
u_{t}\theta_{1}\varphi+4\nabla\Delta\varphi\cdot\nabla(u\theta_{1})\psi+2\Delta\varphi\Delta(u\theta_{1})\psi\,dxdt$
$\displaystyle+\int_{Q}4\nabla^{2}\varphi:\nabla^{2}(u\theta_{1})\psi+4\nabla\varphi\cdot\nabla\Delta(u\theta_{1})\psi+\Delta^{2}(u\theta_{1})\varphi\psi+B_{0}\cdot\nabla(u\theta_{1})\varphi\psi\,dxdt$
$\displaystyle+\int_{Q}\sum_{i,j=1}^{n}\left(B_{ij}\frac{\partial\varphi}{\partial
x_{i}}\frac{\partial(u\theta_{1})}{\partial
x_{j}}\psi+B_{ij}\frac{\partial\varphi}{\partial
x_{j}}\frac{\partial(u\theta_{1})}{\partial
x_{i}}\psi+B_{ij}\frac{\partial^{2}(u\theta_{1})}{\partial x_{i}\partial
x_{j}}\psi\varphi\right)\,dxdt$
$\displaystyle+\int_{Q}\left(2a_{1}\nabla\varphi\cdot\nabla(u\theta_{1})\psi+a_{1}\varphi\psi\Delta(u\theta_{1})\right)\,dxdt=:\sum_{i=1}^{12}I_{i}.$
(3.6)

In what follows, let $C$ be a positive constant depending only on $D,$
$\omega_{1}$ and $\omega_{2},$ which may change from one line to another, we
will estimate each $I_{i}$ in inequality (3) for $1\leq i\leq 12$ by Hölder’s
inequality, Young’s inequality along with inequality (3).

To begin with, we conclude from the properties of weight functions (2.3) that

$\displaystyle|u_{t}|\leq$ $\displaystyle
Cs^{10}\lambda^{8}\xi^{10}e^{2s\alpha},$
$\displaystyle|\nabla^{k}(u\theta_{1})|\leq$ $\displaystyle
C(s^{7+k}\lambda^{8+k}\xi^{7+k})e^{2s\alpha}\chi_{\omega_{2}},\,\,\,\forall\,\,k\in\mathbb{Z}_{+}$

for any $\lambda\geq\hat{\lambda}$ and any $s\geq s_{0}(1+\sqrt{T}+T).$ Thus,
we obtain

$\displaystyle|I_{1}|\leq$ $\displaystyle
C\int_{Q}s^{10}\lambda^{8}\xi^{10}|\varphi||\psi|\theta_{1}e^{2s\alpha}\,dxdt$
$\displaystyle\leq$
$\displaystyle\frac{1}{14}\int_{Q}s^{7}\lambda^{8}\xi^{7}|\varphi|^{2}\theta_{1}e^{2s\alpha}\,dxdt+C\int_{Q_{\omega_{2}}}s^{13}\lambda^{8}\xi^{13}|\psi|^{2}e^{2s\alpha}\,dxdt,$
(3.7)

$\displaystyle|I_{2}|+|I_{3}|+|I_{4}|\leq$ $\displaystyle
C\int_{Q_{\omega_{2}}}\left(s^{8}\lambda^{9}\xi^{8}|\nabla\Delta\varphi||\psi|+s^{9}\lambda^{10}\xi^{9}|\Delta\varphi||\psi|\right)e^{2s\alpha}\,dxdt$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{Q}\left(s\lambda^{2}\xi|\nabla\Delta\varphi|^{2}+s^{3}\lambda^{4}\xi^{3}|\Delta\varphi|^{2}\right)e^{2s\alpha}\,dxdt\right)^{\frac{1}{2}}\left(\int_{Q_{\omega_{2}}}s^{15}\lambda^{16}\xi^{15}|\psi|^{2}e^{2s\alpha}\,dxdt\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{14}\int_{Q}s^{7}\lambda^{8}\xi^{7}|\varphi|^{2}\theta_{1}e^{2s\alpha}\,dxdt+C\int_{Q_{\omega_{2}}}s^{15}\lambda^{16}\xi^{15}|\psi|^{2}e^{2s\alpha}\,dxdt,$
(3.8)

$\displaystyle|I_{5}|+|I_{6}|\leq$ $\displaystyle
C\int_{Q_{\omega_{2}}}\left(s^{10}\lambda^{11}\xi^{10}|\nabla\varphi||\psi|+s^{11}\lambda^{12}\xi^{11}|\varphi||\psi|\right)e^{2s\alpha}\,dxdt$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{Q}\left(s^{4}\lambda^{6}\xi^{4}|\nabla\varphi|^{2}+s^{6}\lambda^{8}\xi^{6}|\varphi|^{2}\right)e^{2s\alpha}\,dxdt\right)^{\frac{1}{2}}\left(\int_{Q_{\omega_{2}}}s^{16}\lambda^{16}\xi^{16}|\psi|^{2}e^{2s\alpha}\,dxdt\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{14}\int_{Q}s^{7}\lambda^{8}\xi^{7}|\varphi|^{2}\theta_{1}e^{2s\alpha}\,dxdt+C\int_{Q_{\omega_{2}}}s^{16}\lambda^{16}\xi^{16}|\psi|^{2}e^{2s\alpha}\,dxdt,$
(3.9)

$\displaystyle|I_{7}|\leq$ $\displaystyle
C\int_{Q_{\omega_{2}}}|B_{0}|s^{8}\lambda^{9}\xi^{8}|\varphi||\psi|e^{2s\alpha}\,dxdt$
$\displaystyle\leq$ $\displaystyle
C\|B_{0}\|_{L^{\infty}(Q)}\left(\int_{Q}s^{6}\lambda^{8}\xi^{6}|\varphi|^{2}e^{2s\alpha}\,dxdt\right)^{\frac{1}{2}}\left(\int_{Q_{\omega_{2}}}s^{10}\lambda^{10}\xi^{10}|\psi|^{2}e^{2s\alpha}\,dxdt\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{14}\int_{Q}s^{7}\lambda^{8}\xi^{7}|\varphi|^{2}\theta_{1}e^{2s\alpha}\,dxdt+C\|B_{0}\|_{L^{\infty}(Q)}^{2}\int_{Q_{\omega_{2}}}s^{10}\lambda^{10}\xi^{10}|\psi|^{2}e^{2s\alpha}\,dxdt,$
(3.10)

$\displaystyle|I_{8}|+|I_{9}|\leq$ $\displaystyle
C\int_{Q_{\omega_{2}}}|B|s^{8}\lambda^{9}\xi^{8}|\nabla\varphi||\psi|e^{2s\alpha}\,dxdt$
$\displaystyle\leq$ $\displaystyle
C\|B\|_{L^{\infty}(Q)}\left(\int_{Q}s^{4}\lambda^{6}\xi^{4}|\nabla\varphi|^{2}e^{2s\alpha}\,dxdt\right)^{\frac{1}{2}}\left(\int_{Q_{\omega_{2}}}s^{12}\lambda^{12}\xi^{12}|\psi|^{2}e^{2s\alpha}\,dxdt\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{14}\int_{Q}s^{7}\lambda^{8}\xi^{7}|\varphi|^{2}\theta_{1}e^{2s\alpha}\,dxdt+C\|B\|_{L^{\infty}(Q)}^{2}\int_{Q_{\omega_{2}}}s^{12}\lambda^{12}\xi^{12}|\psi|^{2}e^{2s\alpha}\,dxdt,$
(3.11)

$\displaystyle|I_{10}|\leq$ $\displaystyle
C\int_{Q_{\omega_{2}}}|B|s^{9}\lambda^{10}\xi^{9}|\psi||\varphi|e^{2s\alpha}\,dxdt$
$\displaystyle\leq$
$\displaystyle\frac{1}{14}\int_{Q}s^{7}\lambda^{8}\xi^{7}|\varphi|^{2}\theta_{1}e^{2s\alpha}\,dxdt+C\|B\|_{L^{\infty}(Q)}^{2}\int_{Q_{\omega_{2}}}s^{12}\lambda^{12}\xi^{12}|\psi|^{2}e^{2s\alpha}\,dxdt,$
(3.12)

$\displaystyle|I_{11}|+|I_{12}|\leq
C\int_{Q_{\omega_{2}}}|a_{1}|(s^{8}\lambda^{9}\xi^{8}|\nabla\varphi||\psi|+s^{9}\lambda^{10}\xi^{9}|\varphi||\psi|)e^{2s\alpha}\,dxdt$
$\displaystyle\leq$ $\displaystyle
C\|a_{1}\|_{L^{\infty}(Q)}\left(\int_{Q}\left(s^{6}\lambda^{8}\xi^{6}|\varphi|^{2}+s^{4}\lambda^{6}\xi^{4}|\nabla\varphi|^{2}\right)e^{2s\alpha}\,dxdt\right)^{\frac{1}{2}}\left(\int_{Q_{\omega_{2}}}s^{12}\lambda^{12}\xi^{12}|\psi|^{2}e^{2s\alpha}\,dxdt\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{14}\int_{Q}s^{7}\lambda^{8}\xi^{7}|\varphi|^{2}\theta_{1}e^{2s\alpha}\,dxdt+C\|a_{1}\|_{L^{\infty}(Q)}^{2}\int_{Q_{\omega_{2}}}s^{12}\lambda^{12}\xi^{12}|\psi|^{2}e^{2s\alpha}\,dxdt.$
(3.13)

Therefore, we deduce from inequalities (3)-(3) that

$\displaystyle\int_{0}^{T}\int_{\mathcal{O}}s^{7}\lambda^{8}\xi^{7}|\varphi|^{2}e^{2s\alpha}\theta_{1}\,dxdt\leq
C\int_{Q_{\omega_{2}}}s^{16}\lambda^{16}\xi^{16}|\psi|^{2}e^{2s\alpha}\,dxdt$
(3.14)

for any
$\lambda\geq\hat{\lambda}(1+\|a_{0}\|_{L^{\infty}(Q)}^{\frac{1}{4}}+\|B_{0}\|_{L^{\infty}(Q)}^{\frac{1}{3}}+\|B\|_{L^{\infty}(Q)}^{\frac{1}{2}}+\|a_{1}\|_{L^{\infty}(Q)}^{\frac{1}{2}})$
and any $s\geq s_{0}(1+\sqrt{T}+T).$

Thus, in view of inequality (3) and inequality (3.14), yields

$\displaystyle\int_{Q}e^{2s\alpha}s^{6}\xi^{6}|\varphi|^{2}\leq$
$\displaystyle
C_{1}\int_{Q_{\omega_{1}}}s^{7}\xi^{7}|\varphi|^{2}e^{2s\alpha}\,dxdt$
$\displaystyle\leq$
$\displaystyle\int_{0}^{T}\int_{\mathcal{O}}s^{7}\xi^{7}|\varphi|^{2}e^{2s\alpha}\theta_{1}\,dxdt$
$\displaystyle\leq$ $\displaystyle
C\int_{Q_{\omega_{2}}}s^{16}\xi^{16}|\psi|^{2}e^{2s\alpha}\,dxdt$ (3.15)

for any fixed
$\lambda\geq\hat{\lambda}(1+\|a_{0}\|_{L^{\infty}(Q)}^{\frac{1}{4}}+\|B_{0}\|_{L^{\infty}(Q)}^{\frac{1}{3}}+\|B\|_{L^{\infty}(Q)}^{\frac{1}{2}}+\|a_{1}\|_{L^{\infty}(Q)}^{\frac{1}{2}})$
and any $s\geq s_{0}(1+\sqrt{T}+T).$

Combining inequalities (3) with inequality (3), we obtain

$\displaystyle\int_{Q}s^{6}\xi^{6}|\psi|^{2}e^{2s\alpha}\,dxdt\leq
C\int_{Q_{\omega_{2}}}s^{16}\xi^{16}|\psi|^{2}e^{2s\alpha}\,dxdt$ (3.16)

for any fixed
$\lambda\geq\hat{\lambda}(1+\|a_{0}\|_{L^{\infty}(Q)}^{\frac{1}{4}}+\|B_{0}\|_{L^{\infty}(Q)}^{\frac{1}{3}}+\|B\|_{L^{\infty}(Q)}^{\frac{1}{2}}+\|a_{1}\|_{L^{\infty}(Q)}^{\frac{1}{2}})$
and any $s\geq s_{0}(1+\sqrt{T}+T).$

Finally, we will combining energy estimates with inequalities (3)-(3.16) to
obtain the desired observability inequality. At this point, applying classical
estimates of the fourth order parabolic equation to systems (3.1), we obtain
for any $t_{1},$ $t_{2}\in[0,T]$ with $t_{1}<t_{2}$ and any $t\in[0,T],$

$\displaystyle\|\varphi(t_{2})\|_{L^{2}(D)}^{2}\leq
e^{2\beta(t_{2}-t_{1})}\|\varphi(t_{1})\|_{L^{2}(D)}^{2}$ (3.17)

and

$\displaystyle\|\psi(t)\|_{L^{2}(D)}^{2}\leq\int_{t}^{T}e^{2\beta(s-t)}\|\varphi(s)\|_{L^{2}(\mathcal{O})}^{2}\,ds,$
(3.18)

where

$\displaystyle\beta=2+\|a_{0}\|_{L^{\infty}(Q)}^{2}+\|B_{0}\|_{L^{\infty}(Q)}^{2}+\|B\|_{L^{\infty}(Q)}^{2}+\|a_{1}\|_{L^{\infty}(Q)}^{2}.$

In particular, we have

$\displaystyle\|\varphi(t+\frac{T}{4})\|_{L^{2}(D)}^{2}\leq e^{\frac{\beta
T}{2}}\|\varphi(t)\|_{L^{2}(D)}^{2}$

for any $t\in[\frac{T}{4},\frac{3T}{4}],$ which implies that

$\displaystyle\int_{\frac{T}{2}}^{T}\|\varphi(t)\|_{L^{2}(D)}^{2}\,dt\leq
e^{\frac{\beta
T}{2}}\int_{\frac{T}{4}}^{\frac{3T}{4}}\|\varphi(t)\|_{L^{2}(D)}^{2}.$ (3.19)

On the other hand, we deduce from inequality (3.18) that

$\displaystyle\int_{t}^{T}\|\psi(s)\|_{L^{2}(D)}^{2}\,ds\leq(T-t)e^{\beta
T}\int_{t}^{T}\|\varphi(s)\|_{L^{2}(\mathcal{O})}^{2}\,ds$

for any $t\in[\frac{T}{2},T],$ which entails that

$\displaystyle\int_{\frac{T}{2}}^{T}\|\psi(s)\|_{L^{2}(D)}^{2}\,ds\leq
e^{(1+\beta)T}\int_{\frac{T}{2}}^{T}\|\varphi(s)\|_{L^{2}(\mathcal{O})}^{2}\,ds.$
(3.20)

Denote by $m_{0}=\min\limits_{x\in\overline{D}}\alpha_{0}(x)$ and
$M_{0}=\max\limits_{x\in\overline{D}}\alpha_{0}(x),$ we deduce from Lemma 2.3
that for any $s\geq 0,$

$\displaystyle\int_{Q}s^{6}\xi^{6}|\psi|^{2}e^{2s\alpha}\,dxdt\geq
A_{s}\int_{0}^{\frac{T}{2}}\int_{D}e^{-\frac{M_{s}}{\sqrt{t}}}|\psi|^{2}\,dxdt$
(3.21)

with $A_{s}$ and $M_{s}$ given in Lemma 2.3.

In what follows, we will Bounding the right hand side of inequality (3.16) by
using Lemma 2.3, we obtain

$\displaystyle\int_{Q}s^{6}\xi^{6}|\psi|^{2}e^{2s\alpha}\,dxdt\leq$
$\displaystyle
C\int_{Q_{\omega_{2}}}s^{16}\xi^{16}|\psi|^{2}e^{2s\alpha}\,dxdt$
$\displaystyle\leq$ $\displaystyle
C2^{48}\left(\frac{N_{0}}{M_{0}e}\right)^{16}\int_{Q_{\omega_{2}}}|\psi|^{2}\,dxdt$
(3.22)

for any $s\geq\frac{4T}{|M_{0}|}.$ Thus, we obtain

$\displaystyle\int_{0}^{\frac{T}{2}}\int_{D}e^{-\frac{M_{s}}{\sqrt{t}}}|\psi|^{2}\,dxdt\leq
C\frac{2^{30}}{e^{16}}e^{\frac{2|m_{0}|s}{T}}\left(\frac{N_{0}^{16}}{M_{0}^{10}n_{0}^{6}}\right)\int_{Q_{\omega_{2}}}|\psi|^{2}\,dxdt.$
(3.23)

Along with inequality (3.20) and inequality (3.23), yields

$\displaystyle\int_{Q}e^{-\frac{M_{s}}{\sqrt{t}}}|\psi|^{2}\,dxdt\leq$
$\displaystyle\int_{0}^{\frac{T}{2}}\int_{D}e^{-\frac{M_{s}}{\sqrt{t}}}|\psi|^{2}\,dxdt+\int_{\frac{T}{2}}^{T}\int_{D}|\psi|^{2}\,dxdt$
$\displaystyle\leq$ $\displaystyle
C2^{48}\left(\frac{N_{0}}{M_{0}e}\right)^{16}\int_{Q_{\omega_{2}}}|\psi|^{2}\,dxdt$
$\displaystyle+e^{(1+\beta)T}\int_{\frac{T}{2}}^{T}\|\varphi(s)\|_{L^{2}(\mathcal{O})}^{2}\,ds.$
(3.24)

In view of inequalities (3.16), (3.19) and Lemma 2.3, yields

$\displaystyle\int_{\frac{T}{2}}^{T}\|\varphi(s)\|_{L^{2}(D)}^{2}\,ds\leq$
$\displaystyle e^{\frac{\beta
T}{2}}\int_{\frac{T}{4}}^{\frac{3T}{4}}\|\varphi(t)\|_{L^{2}(D)}^{2}$
$\displaystyle\leq$ $\displaystyle e^{\frac{\beta
T}{2}+\frac{8|m_{0}|s}{\sqrt{3}T}}(2n_{0})^{-6}T^{6}\int_{\frac{T}{4}}^{\frac{3T}{4}}\int_{D}\xi^{6}e^{2s\alpha}|\varphi(t)|^{2}\,dxdt$
$\displaystyle\leq$ $\displaystyle Ce^{\frac{\beta
T}{2}+\frac{8|m_{0}|s}{\sqrt{3}T}}(2n_{0})^{-6}T^{6}\int_{\frac{T}{4}}^{\frac{3T}{4}}\int_{Q_{\omega_{2}}}s^{16}\xi^{16}|\psi|^{2}e^{2s\alpha}\,dxdt$
$\displaystyle\leq$ $\displaystyle
C2^{42}\left(\frac{N_{0}}{M_{0}}\right)^{16}e^{\frac{\beta
T}{2}+\frac{8|m_{0}|s}{\sqrt{3}T}-16}n_{0}^{-6}T^{6}\int_{\frac{T}{4}}^{\frac{3T}{4}}\int_{Q_{\omega_{2}}}|\psi|^{2}\,dxdt$
(3.25)

for any $s\geq\frac{4T}{|M_{0}|}.$

Combining inequality (3) with inequality (3), we obtain

$\displaystyle\int_{Q}e^{-\frac{M_{s}}{\sqrt{t}}}|\psi|^{2}\,dxdt\leq
H\int_{Q_{\omega}}|\psi|^{2}\,dxdt$ (3.26)

for any $s\geq\frac{4T}{|M_{0}|},$ where

$\displaystyle
H=C2^{48}\left(\frac{N_{0}}{M_{0}e}\right)^{16}+C2^{42}\left(\frac{N_{0}}{M_{0}e}\right)^{16}e^{2\beta
T+\frac{8|m_{0}|s}{\sqrt{3}T}}n_{0}^{-6}T^{6}.$

∎

In the following, we will prove the existence of an insensitizing control such
that the solution of problem (1.1) verifies condition (1.3), i.e., we will
prove the null-controllability of problem (1.4).

######  Theorem 3.2.

Assume that $\omega\cap\mathcal{O}\neq\emptyset,$ $y_{0}=0$ and the positive
constants $M$ and $H$ are defined as in Theorem 3.1. If $f\in L^{2}(Q)$
satisfies

$\displaystyle\int_{Q}e^{\frac{M}{\sqrt{t}}}|f|^{2}\,dxdt<+\infty,$

then there exists a control $v\in L^{2}(Q_{\omega}),$ such that the solution
$(y,q)$ of problem (1.4) satisfies

$\displaystyle q(x,0)\equiv 0,\,\,\forall\,\,x\in D.$ (3.27)

Moreover, we also have

$\displaystyle\|v\|_{L^{2}(Q_{\omega})}\leq
2\sqrt{H}\left(\int_{Q}e^{\frac{M}{\sqrt{t}}}|f|^{2}\,dxdt\right)^{\frac{1}{2}}.$

###### Proof.

In what follows, we will prove the null controllability of problem (1.1) by
the similar method in [20]. To this purpose, for any $\epsilon>0,$ we
introduce a functional defined on $L^{2}(D):$

$\displaystyle\mathcal{J}(\varphi_{0})=\frac{1}{2}\int_{Q_{\omega}}|\psi|^{2}\,dxdt+\epsilon\|\varphi_{0}\|_{L^{2}(D)}+\int_{Q}f\psi\,dxdt,$

where $(\psi,\varphi)$ is the solution of problem (3.1) with initial data
$\psi(0)=0$ and $\varphi(0)=\varphi_{0}\in L^{2}(D).$

In view of Theorem 3.1, we conclude that the functional
$\mathcal{J}(\varphi_{0})$ is continous, strictly convex and coercive on
$L^{2}(D).$ Therefore, for any $\epsilon>0,$ there exists a unique minimum
point $\varphi_{0\epsilon}\in L^{2}(D)$ of $\mathcal{J},$ which implies that

$\displaystyle
0=\mathcal{J}(0)\geq\mathcal{J}(\varphi_{0\epsilon})=\frac{1}{2}\int_{Q_{\omega}}|\psi_{\epsilon}|^{2}\,dxdt+\epsilon\|\varphi_{0\epsilon}\|_{L^{2}(D)}+\int_{Q}f\psi_{\epsilon}\,dxdt,$
(3.28)

where $(\psi_{\epsilon},\varphi_{\epsilon})$ solves problem (3.1) with initial
data $(0,\varphi_{0\epsilon}).$

Therefore, we deduce from inequalities (3.2), (3.28) and Hölder’s inequality
that

$\displaystyle\frac{1}{2}\int_{Q_{\omega}}|\psi_{\epsilon}|^{2}\,dxdt+\epsilon\|\varphi_{0\epsilon}\|_{L^{2}(D)}\leq$
$\displaystyle-\int_{Q}f\psi_{\epsilon}\,dxdt$ $\displaystyle\leq$
$\displaystyle\left(\int_{Q}e^{\frac{M}{\sqrt{t}}}|f|^{2}\,dxdt\right)^{\frac{1}{2}}\left(\int_{Q}e^{-\frac{M}{\sqrt{t}}}|\psi_{\epsilon}|^{2}\,dxdt\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle\sqrt{H}\left(\int_{Q}e^{\frac{M}{\sqrt{t}}}|f|^{2}\,dxdt\right)^{\frac{1}{2}}\left(\int_{Q_{\omega}}|\psi_{\epsilon}|^{2}\,dxdt\right)^{\frac{1}{2}}$

for any $s\geq\frac{4T}{|M_{0}|}.$

Employing Young’s inequality, yields

$\displaystyle\int_{Q_{\omega}}|\psi_{\epsilon}|^{2}\,dxdt+4\epsilon\|\varphi_{0\epsilon}\|_{L^{2}(D)}\leq
4H\int_{Q}e^{\frac{M}{\sqrt{t}}}|f|^{2}\,dxdt$

for any $s\geq\frac{4T}{|M_{0}|}.$

If $\varphi_{0\epsilon}\neq 0,$ then $\mathcal{J}$ satisfies the optimality
condition

$\displaystyle\int_{Q_{\omega}}\psi_{\epsilon}\psi\,dxdt+\int_{Q}f\psi\,dxdt+\frac{\epsilon}{\|\varphi_{0\epsilon}\|_{L^{2}(D)}}\int_{D}\varphi_{0\epsilon}\varphi_{0}\,dx=0$
(3.29)

for any $\varphi_{0}\in L^{2}(D),$ where $(\psi,\varphi)$ is the solution of
problem (3.1) with initial data $(0,\varphi_{0}).$

Now, let $v_{\epsilon}=\psi_{\epsilon}$ and let $(y_{\epsilon},q_{\epsilon})$
be the solution of problem (1.4), then we infer from problem (1.4) and problem
(3.1) that

$\displaystyle\int_{Q}\chi_{\mathcal{O}}\varphi
y_{\epsilon}\,dxdt=\int_{Q}(\chi_{\omega}v_{\epsilon}+f)\psi\,dxdt$ (3.30)

and

$\displaystyle\int_{D}q_{\epsilon}(x,0)\varphi_{0}\,dx=\int_{Q}\chi_{\mathcal{O}}\varphi
y_{\epsilon}\,dxdt.$ (3.31)

Thus, along with inequalities (3.29)-(3.31) and the fact that
$v_{\epsilon}=\psi_{\epsilon},$ we obtain

$\displaystyle\int_{D}q_{\epsilon}(x,0)\varphi_{0}\,dx=-\frac{\epsilon}{\|\varphi_{0\epsilon}\|_{L^{2}(D)}}\int_{D}\varphi_{0\epsilon}\varphi_{0}\,dx$
(3.32)

for any $\varphi_{0}\in L^{2}(D),$ which implies that

$\displaystyle\|q_{\epsilon}(0)\|_{L^{2}(D)}\leq\epsilon.$ (3.33)

If $\varphi_{0\epsilon}=0,$ then

$\displaystyle\lim_{t\rightarrow 0}\frac{\mathcal{J}(t\varphi_{0})}{t}\geq 0$

for any $\varphi_{0}\in L^{2}(D),$ i.e.,

$\displaystyle\epsilon\|\varphi_{0}\|_{L^{2}(D)}+\int_{Q}f\psi\,dxdt\geq 0,$
(3.34)

where $(\psi,\varphi)$ solves problem (3.1) with initial data
$(0,\varphi_{0}).$ Consequently, we can also conclude from inequalities
(3.30)-(3.31), (3.34) and the fact that $v_{\epsilon}=\psi_{\epsilon}=0$ that

$\displaystyle\epsilon\|\varphi_{0}\|_{L^{2}(D)}+\int_{D}q_{\epsilon}(x,0)\varphi_{0}\,dx\geq
0$

for any $\varphi_{0}\in L^{2}(D),$ which also implies that

$\displaystyle\|q_{\epsilon}(0)\|_{L^{2}(D)}\leq\epsilon.$

Therefore, the solution $(y_{\epsilon},q_{\epsilon})$ of problem (1.4)
associated with $v_{\epsilon}$ satisfies inequality

$\displaystyle\|q_{\epsilon}(0)\|_{L^{2}(D)}\leq\epsilon.$ (3.35)

Moreover, we obtain

$\displaystyle\int_{Q_{\omega}}|v_{\epsilon}|^{2}\,dxdt\leq
4H\int_{Q}e^{\frac{M}{\sqrt{t}}}|f|^{2}\,dxdt$

for any $s\geq\frac{4T}{|M_{0}|},$ which entails that the controls
$\\{v_{\epsilon}\\}_{\epsilon>0}$ are uniformly bounded in
$L^{2}(Q_{\omega}).$ Without loss of generality, we can assume that
$v_{\epsilon}\rightharpoonup v$ weakly in $L^{2}(Q_{\omega})$ and

$\displaystyle(y_{\epsilon},q_{\epsilon})\rightharpoonup(y,q),\,\,\,\,\,\textit{weakly\,\,\,in}\,\,\,X\times
X,$

where $(y,q)$ is the solution of problem (1.4) with $v.$ In particular, we
have the weak convergence of $q_{\epsilon}(0)$ in $L^{2}(D).$ Thus, we
conclude from inequality (3.35) that $q(0)\equiv 0,$ i.e., $v$ is the desired
control. Moreover, we have

$\displaystyle\int_{Q_{\omega}}|v_{\epsilon}|^{2}\,dxdt\leq
4H\int_{Q}e^{\frac{M}{\sqrt{t}}}|f|^{2}\,dxdt$

for any $s\geq\frac{4T}{|M_{0}|}.$ ∎

##  4 The semi-linear case

In this section, under the assumptions that $F\in
W^{1,\infty}(\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n^{2}};\mathbb{R})$
and $y_{0}=0,$ we will prove the existence of an insensitizing control of
problem

$\begin{cases}\frac{\partial y}{\partial
t}+\Delta^{2}y+a_{0}y+B_{0}\cdot\nabla y+B:\nabla^{2}y+a_{1}\Delta
y=F(y,\nabla y,\nabla^{2}y)+v\chi_{\omega}+f,\,\,\,\,\forall\,\,\,(x,t)\in
Q,\\\ -\frac{\partial q}{\partial
t}+\Delta^{2}q+a_{0}q-\nabla\cdot(B_{0}q)+\sum_{i,j=1}^{n}\frac{\partial^{2}(B_{ij}q)}{\partial
x_{i}\partial x_{j}}+\Delta(a_{1}q)=F_{y}(y,\nabla y,\nabla^{2}y)q\\\
-\nabla\cdot(\nabla_{p}F(y,\nabla
y,\nabla^{2}y)q)+\sum_{i,j=1}^{n}\frac{\partial^{2}(F_{r_{ij}}(y,\nabla
y,\nabla^{2}y)q)}{\partial x_{i}\partial
x_{j}}+\chi_{\mathcal{O}}y,\,\,\,\,\forall\,\,\,(x,t)\in Q,\\\ y=\Delta
y=0,\,\,\,q=\Delta q=0,\,\,\forall\,\,\,\,(x,t)\in\Sigma,\\\
y(x,0)=0,\,\,q(x,T)=0,\,\,\forall\,\,\,\,x\in D\end{cases}$ (4.1)

such that

$\displaystyle q(x,0)\equiv 0,\,\,\,\forall x\in D.$ (4.2)

From the regularity of fourth order parabolic equations, we conclude that
there exists a unique solution of problem (4.1) satisfying

$\displaystyle y\in Y=L^{2}(0,T;H_{0}^{1}(D)\cap H^{4}(D))\cap
H^{1}(0,T;L^{2}(D)),$ $\displaystyle q\in X=L^{2}(0,T;H_{0}^{1}(D)\cap
H^{2}(D))\cap H^{1}(0,T;(H^{2}(D))^{*}).$

In what follows, we will establish the existence of an insensitizing control
such that the solution of problem (4.1) verifying (4.2) in the semi-linear
case.

######  Theorem 4.1.

Assume that $\omega\cap\mathcal{O}\neq\emptyset,$ $y_{0}=0,$ $F\in
W^{1,\infty}(\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n^{2}};\mathbb{R}),$
the assumption on $f$ is given as in Theorem 3.2. Then there exists a control
$v\in L^{2}(Q_{\omega}),$ such that the solution $(y,q)$ of problem (4.1)
satisfies (4.2).

###### Proof.

Let $z\in L^{2}(0,T;H_{0}^{1}(D)\cap H^{2}(D))$ be given, consider the
following problem

$\begin{cases}\frac{\partial y}{\partial
t}+\Delta^{2}y+a_{0}y+B_{0}\cdot\nabla y+B:\nabla^{2}y+a_{1}\Delta
y=G_{1}(z,\nabla z,\nabla^{2}z)y+G_{2}(z,\nabla z,\nabla^{2}z)\cdot\nabla y\\\
+G_{3}(z,\nabla
z,\nabla^{2}z):\nabla^{2}y+F(0,0,0)+v\chi_{\omega}+f,\,\,\,\,(x,t)\in Q,\\\
-\frac{\partial q}{\partial
t}+\Delta^{2}q+a_{0}q-\nabla\cdot(B_{0}q)+\sum_{i,j=1}^{n}\frac{\partial^{2}(B_{ij}q)}{\partial
x_{i}\partial x_{j}}+\Delta(a_{1}q)=F_{y}(z,\nabla z,\nabla^{2}z)q\\\
-\nabla\cdot(\nabla_{p}F(z,\nabla
z,\nabla^{2}z)q)+\sum_{i,j=1}^{n}\frac{\partial^{2}(F_{r_{ij}}(z,\nabla
z,\nabla^{2}z)q)}{\partial x_{i}\partial
x_{j}}+y\chi_{\mathcal{O}},\,\,\,\,(x,t)\in Q,\\\ y=\Delta y=0,\,\,\,q=\Delta
q=0,\,\,(x,t)\in\Sigma,\\\ y(x,0)=0,\,\,q(x,T)=0,\,\,\,x\in D,\end{cases}$
(4.3)

where

$\displaystyle G_{1}(w,\nabla w,\nabla^{2}w)=\int_{0}^{1}\frac{\partial
F}{\partial y}(\tau w,\tau\nabla w,\tau\nabla^{2}w)\,d\tau,$ $\displaystyle
G_{2}(w,\nabla w,\nabla^{2}w)=\int_{0}^{1}\nabla_{p}F(\tau w,\tau\nabla
w,\tau\nabla^{2}w)\,d\tau,$ $\displaystyle G_{3}^{ij}(w,\nabla
w,\nabla^{2}w)=\int_{0}^{1}\frac{\partial F}{\partial r_{ij}}(\tau
w,\tau\nabla w,\tau\nabla^{2}w)\,d\tau.$

Since $F\in
W^{1,\infty}(\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n^{2}},\mathbb{R}),$
there exists a positive constant $M,$ such that

$\displaystyle|G_{1}(u,p,r)|+|G_{2}(u,p,r)|+|G_{3}(u,p,r)|\leq
M,\,\,\,\,\,\forall\,\,\,\,(u,p,r)\in\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n^{2}}$

and

$\displaystyle|F_{y}(u,p,r)|+|\nabla_{p}F(u,p,r)|+\sum_{i,j=1}^{n}\left|\frac{\partial
F}{\partial r_{ij}}(u,p,r)\right|\leq
M,\,\,\,\,\,\forall\,\,\,\,(u,p,r)\in\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n^{2}}.$

From Theorem 3.1, we conclude that there exists at least one control $v\in
L^{2}(Q_{\omega}),$ such that the solution $(y^{z},q^{z})$ of problem (4.3)
satisfies

$\displaystyle q^{z}(x,0)\equiv 0,\,\,\forall\,\,x\in D.$ (4.4)

Moreover, we also have

$\displaystyle\|v^{z}\|_{L^{2}(Q_{\omega})}\leq
2\sqrt{H}\left(\int_{Q}e^{\frac{M}{\sqrt{t}}}|f|^{2}\,dxdt\right)^{\frac{1}{2}}.$
(4.5)

In what follows, we denote by $v^{z}$ the control with the minimal
$L^{2}(Q_{\omega})$-norm in the set of the controls such that the solution
$(y^{z},q^{z})$ of problem (4.3) corresponding to $z$ satisfies (4.4).

From the regularity theory of parabolic equations, we conclude that there
exists a unique weak solution $(y^{z},q^{z})\in Y\times X.$ Moreover, since
$F\in
W^{1,\infty}(\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n^{2}};\mathbb{R}),$
there exists a positive constant $C$ independent of $z,$ such that

$\displaystyle\|y^{z}\|_{Y}+\|q^{z}\|_{X}\leq$ $\displaystyle
C(\|F(0,0,0)+v^{z}\chi_{\omega}+f\|_{L^{2}(Q)})$ $\displaystyle\leq$
$\displaystyle C(1+\|v^{z}\|_{L^{2}(Q_{\omega})}+\|f\|_{L^{2}(Q)}).$ (4.6)

Thus, along with inequalities (4.5)-(4), we deduce that there exists a
positive constant $\mathcal{L}_{1}$ independent of $z,$ such that

$\displaystyle\|y^{z}\|_{Y}+\|q^{z}\|_{X}\leq\mathcal{L}_{1}\left(1+\|e^{\frac{M}{2\sqrt{t}}}f\|_{L^{2}(Q)}\right).$
(4.7)

Define $\Lambda:L^{2}(0,T;H_{0}^{1}(D)\cap H^{2}(D))\rightarrow
L^{2}(0,T;H_{0}^{1}(D)\cap H^{2}(D))$ by

$\displaystyle\Lambda(z)=y^{z},$

then the mapping $\Lambda$ is well-defined. In what follows, we will prove the
existence of a fixed point for the operator $\Lambda$ by the Leray-Schauder’s
fixed points Theorem. To this purpose, we will first prove that $\Lambda$ is
continuous, i.e., if $z_{k}\rightarrow z$ in $L^{2}(0,T;H_{0}^{1}(D)\cap
H^{2}(D)),$ we have $\Lambda(z_{k})\rightarrow\Lambda(z).$

Denote by $y^{k}=\Lambda(z_{k}),$ where $(y^{k},q^{k})$ is the solution of
problem

$\begin{cases}\frac{\partial y^{k}}{\partial
t}+\Delta^{2}y^{k}+a_{0}y^{k}+B_{0}\cdot\nabla
y^{k}+B:\nabla^{2}y^{k}+a_{1}\Delta y^{k}=G_{1}(z_{k},\nabla
z_{k},\nabla^{2}z_{k})y^{k}\\\ +G_{2}(z_{k},\nabla
z_{k},\nabla^{2}z_{k})\cdot\nabla y^{k}+G_{3}(z_{k},\nabla
z_{k},\nabla^{2}z_{k}):\nabla^{2}y^{k}+F(0,0,0)+v_{z_{k}}\chi_{\omega}+f,\,\,\,\,(x,t)\in
Q,\\\ -\frac{\partial q^{k}}{\partial
t}+\Delta^{2}q^{k}+a_{0}q^{k}-\nabla\cdot(B_{0}q^{k})+\sum_{i,j=1}^{n}\frac{\partial^{2}(B_{ij}q^{k})}{\partial
x_{i}\partial x_{j}}+\Delta(a_{1}q^{k})=F_{y}(z_{k},\nabla
z_{k},\nabla^{2}z_{k})q^{k}\\\ -\nabla\cdot(\nabla_{p}F(z_{k},\nabla
z_{k},\nabla^{2}z_{k})q^{k})+\sum_{i,j=1}^{n}\frac{\partial^{2}(F_{r_{ij}}(z_{k},\nabla
z_{k},\nabla^{2}z_{k})q^{k})}{\partial x_{i}\partial
x_{j}}+y^{k}\chi_{\mathcal{O}},\,\,\,\,(x,t)\in Q,\\\ y^{k}=\Delta
y^{k}=0,\,\,\,q^{k}=\Delta q^{k}=0,\,\,(x,t)\in\Sigma,\\\
y^{k}(x,0)=0,\,\,q^{k}(x,T)=0,\,\,\,x\in D.\end{cases}$ (4.8)

It follows from inequality (4.7) and the fact that $z_{k}\rightarrow z$ in
$L^{2}(0,T;H_{0}^{1}(D)\cap H^{2}(D))$ that

$\displaystyle\\{(y^{k},q^{k})\\}_{k=1}^{\infty}\,\,\,\textit{is\,\,\,uniformly\,\,\,bounded\,\,in}\,\,Y\times
X,$
$\displaystyle\\{v^{z_{k}}\\}_{k=1}^{\infty}\,\,\,\textit{is\,\,\,uniformly\,\,\,bounded\,\,in}\,\,L^{2}(Q_{\omega}),$

which entails that there exists a subsequence of $\\{y^{k}\\}_{k=1}^{\infty},$
$\\{q^{k}\\}_{k=1}^{\infty},$ $\\{v^{z_{k}}\\}_{k=1}^{\infty}$ (still denote
by themselves) and $y\in Y,$ $q\in X,$ $v\in L^{2}(Q_{\omega}),$ such that

$\displaystyle y^{k}\rightharpoonup
y\,\,\,\textit{in}\,\,Y\,\,\,\textit{as}\,\,k\rightarrow+\infty,$
$\displaystyle y^{k}\rightarrow
y\,\,\,\textit{in}\,\,L^{2}(0,T;H_{0}^{1}(D)\cap
H^{2}(D))\,\,\,\textit{as}\,\,k\rightarrow+\infty,$ $\displaystyle
q^{k}\rightharpoonup
q\,\,\,\textit{in}\,\,X\,\,\,\textit{as}\,\,k\rightarrow+\infty,$
$\displaystyle v^{z_{k}}\rightharpoonup
v\,\,\,\textit{in}\,\,L^{2}(Q_{\omega})\,\,\,\textit{as}\,\,k\rightarrow+\infty.$

Since $F\in
W^{1,\infty}(\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n^{2}},\mathbb{R}),$
we conclude that there exists a subsequence of $\\{G_{1}(z_{k},\nabla
z_{k},\nabla^{2}z_{k})\\}_{k=1}^{\infty},$ $\\{G_{2}(z_{k},\nabla
z_{k},\nabla^{2}z_{k})\\}_{k=1}^{\infty},$ $\\{G_{3}(z_{k},\nabla
z_{k},\nabla^{2}z_{k})\\}_{k=1}^{\infty},$ $\\{F_{y}(z_{k},\nabla
z_{k},\nabla^{2}z_{k})\\}_{k=1}^{\infty},$ $\\{\nabla_{p}F(z_{k},\nabla
z_{k},\nabla^{2}z_{k})\\}_{k=1}^{\infty},$ $\\{(F_{r_{ij}}(z_{k},\nabla
z_{k},\nabla^{2}z_{k}))_{1\leq i,j\leq n}\\}_{k=1}^{\infty},$ (still denote by
themselves), such that

$\displaystyle G_{1}(z_{k},\nabla z_{k},\nabla^{2}z_{k})\rightarrow
G_{1}(z,\nabla
z,\nabla^{2}z)\,\,\,\textit{weakly\,\,star\,\,in}\,\,L^{\infty}(Q),\,\,\textit{as}\,\,k\rightarrow+\infty,$
$\displaystyle G_{2}(z_{k},\nabla z_{k},\nabla^{2}z_{k})\rightarrow
G_{2}(z,\nabla
z,\nabla^{2}z)\,\,\,\textit{weakly\,\,star\,\,in}\,\,L^{\infty}(Q),\,\,\textit{as}\,\,k\rightarrow+\infty,$
$\displaystyle G_{3}(z_{k},\nabla z_{k},\nabla^{2}z_{k})\rightarrow
G_{3}(z,\nabla
z,\nabla^{2}z)\,\,\,\textit{weakly\,\,star\,\,in}\,\,L^{\infty}(Q),\,\,\textit{as}\,\,k\rightarrow+\infty,$
$\displaystyle F_{y}(z_{k},\nabla z_{k},\nabla^{2}z_{k})\rightarrow
F_{y}(z,\nabla
z,\nabla^{2}z)\,\,\,\textit{weakly\,\,star\,\,in}\,\,L^{\infty}(Q),\,\,\textit{as}\,\,k\rightarrow+\infty,$
$\displaystyle\nabla_{p}F(z_{k},\nabla
z_{k},\nabla^{2}z_{k})\rightarrow\nabla_{p}F(z,\nabla
z,\nabla^{2}z)\,\,\,\textit{weakly\,\,star\,\,in}\,\,L^{\infty}(Q),\,\,\textit{as}\,\,k\rightarrow+\infty,$
$\displaystyle F_{r_{ij}}(z_{k},\nabla z_{k},\nabla^{2}z_{k})\rightarrow
F_{r_{ij}}(z,\nabla
z,\nabla^{2}z)\,\,\,\textit{weakly\,\,star\,\,in}\,\,L^{\infty}(Q),\,\,\textit{as}\,\,k\rightarrow+\infty.$

Let $k\rightarrow+\infty$ in problem (4.8), we obtain

$\begin{cases}\frac{\partial y}{\partial
t}+\Delta^{2}y+a_{0}y+B_{0}\cdot\nabla y+B:\nabla^{2}y+a_{1}\Delta
y=G_{1}(z,\nabla z,\nabla^{2}z)y+G_{2}(z,\nabla z,\nabla^{2}z)\cdot\nabla y\\\
+G_{3}(z,\nabla
z,\nabla^{2}z):\nabla^{2}y+F(0,0,0)+v\chi_{\omega}+f,\,\,\,\,(x,t)\in Q,\\\
-\frac{\partial q}{\partial
t}+\Delta^{2}q+a_{0}q-\nabla\cdot(B_{0}q)+\sum_{i,j=1}^{n}\frac{\partial^{2}(B_{ij}q)}{\partial
x_{i}\partial x_{j}}+\Delta(a_{1}q)=F_{y}(z,\nabla z,\nabla^{2}z)q\\\
-\nabla\cdot(\nabla_{p}F(z,\nabla
z,\nabla^{2}z)q)+\sum_{i,j=1}^{n}\frac{\partial^{2}(F_{r_{ij}}(z,\nabla
z,\nabla^{2}z)q)}{\partial x_{i}\partial
x_{j}}+y\chi_{\mathcal{O}},\,\,\,\,(x,t)\in Q,\\\ y=\Delta y=0,\,\,\,q=\Delta
q=0,\,\,(x,t)\in\Sigma,\\\ y(x,0)=y_{0}(x),\,\,q(x,T)=0,\,\,\,x\in
D\end{cases}$ (4.9)

and

$\displaystyle q(x,0)\equiv 0,\,\,\forall\,\,x\in D,$ (4.10)

which entails that $y=\Lambda(z).$ Thus, we have proved that
$\Lambda(z_{k})\rightarrow\Lambda(z)$ in $L^{2}(0,T;H_{0}^{1}(D)\cap
H^{2}(D)),$ i.e., the mapping $\Lambda:L^{2}(0,T;H_{0}^{1}(D)\cap
H^{2}(D))\rightarrow L^{2}(0,T;H_{0}^{1}(D)\cap H^{2}(D))$ is continuous.
Thanks to the compactness of $X\subset L^{2}(0,T;H_{0}^{1}(D)\cap H^{2}(D))$
and inequality (4.7), we conclude that the mapping
$\Lambda:L^{2}(0,T;H_{0}^{1}(D)\cap H^{2}(D))\rightarrow
L^{2}(0,T;H_{0}^{1}(D)\cap H^{2}(D))$ is compact. Denote by

$\displaystyle\mathcal{R}_{1}=\mathcal{L}_{1}\left(1+\|e^{\frac{M}{2\sqrt{t}}}f\|_{L^{2}(Q)}\right)$

and

$\displaystyle B=\\{u\in L^{2}(0,T;H_{0}^{1}(D)\cap
H^{2}(D)):\|u\|_{L^{2}(0,T;H_{0}^{1}(D)\cap H^{2}(D))}\leq\mathcal{R}_{1}\\},$

then $\Lambda:B\rightarrow B.$ Thus, we can employ the Leray-Schauder’s fixed
points Theorem to conclude that the operator $\Lambda$ possesses at least one
fixed point $y\in L^{2}(0,T;H_{0}^{1}(D)\cap H^{2}(D)).$ That is, for any
$y_{0}\in L^{2}(\Omega),$ there exist at least one control $v\in
L^{2}(Q_{\omega}),$ such that the corresponding solutions to problem (4.1)
satisfy $q(x,0)\equiv 0$ for any $x\in D$ ∎

######  Corollary 4.2.

Assume that $\omega\cap\mathcal{O}\neq\emptyset,$ $y_{0}=0,$ $F\in
W^{1,\infty}(\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}^{n^{2}};\mathbb{R}),$
the assumption on $f$ is given as in Theorem 3.2. Then there exists a control
$v\in L^{2}(Q_{\omega})$ insensitizing the functional $\Phi$ defined by (1.2)
for problem (1.1).

######  Remark 4.3.

Under the same assumptions as in Corollary 4.2. If problem (1.1) is subject to
the homogeneous Dirichlet boundary conditions $y|_{\Sigma}=\frac{\partial
y}{\partial\vec{n}}|_{\Sigma}=0,$ the same conclusion as in Corollary 4.2
remains true.

## Acknowledgement

This work was supported by the National Science Foundation of China Grant
(11801427, 11871389) and the Fundamental Research Funds for the Central
Universities (xzy012022008, JB210714).

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# Channel-Adaptive Wireless Image Transmission with OFDM

Haotian Wu, Yulin Shao, Krystian Mikolajczyk, and Deniz Gündüz The authors are
with the Department of Electrical and Electronic Engineering, Imperial College
London, London SW7 2AZ, U.K. (e-mail: haotian.wu17@imperial.ac.uk). D. Gündüz
is also with the Department of Engineering ‘Enzo Ferrari’, University of
Modena and Reggio Emilia (UNI- MORE), Italy.This work was supported by the
European Research Council (ERC) under Grant 677854, and by the UK EPSRC
(EP/W035960/1 and EP/S032398/1) under the CHIST-ERA program (CHIST-
ERA-20-SICT-004).

###### Abstract

We present a learning-based channel-adaptive joint source and channel coding
(CA-JSCC) scheme for wireless image transmission over multipath fading
channels. The proposed method is an end-to-end autoencoder architecture with a
dual-attention mechanism employing orthogonal frequency division multiplexing
(OFDM) transmission. Unlike the previous works, our approach is adaptive to
channel-gain and noise-power variations by exploiting the estimated channel
state information (CSI). Specifically, with the proposed dual-attention
mechanism, our model can learn to map the features and allocate transmission-
power resources judiciously to the available subchannels based on the
estimated CSI. Extensive numerical experiments verify that CA-JSCC achieves
state-of-the-art performance among existing JSCC schemes. In addition, CA-JSCC
is robust to varying channel conditions and can better exploit the limited
channel resources by transmitting critical features over better subchannels.

###### Index Terms:

Joint source channel coding, deep neural networks, OFDM, image communications.

##  I Introduction

Shannon’s separation theorem states that it is optimal to design source and
channel codes separately for an infinite block-length [1]. However, an
increasing number of wireless applications, such as Internet-of-things and
edge intelligence [2, 3, 4], require the efficient transmission of large
volumes of data under strict delay constraints, resulting in an increasing
interest in joint source channel coding (JSCC) in recent years. Recently,
inspired by the success of deep learning techniques, researchers have started
to exploit deep neural networks to design novel and competitive JSCC schemes
to transmit high information content signals, such as images or videos, over
wireless channels [5, 6, 7, 8, 9, 10, 11, 12, 13]. This approach has been
pioneered in [5], where an autoencoder-based JSCC architecture is proposed for
wireless image transmission, which outperformed conventional compression and
channel coding schemes over additive white Gaussian noise (AWGN) and Rayleigh
fading channels. This was later extended to feedback channels in [6] and to
bandwidth-adaptive transmission in [7]. In [8], authors consider JSCC over
orthogonal frequency division multiplexing (OFDM) channels. An alternative
generative architecture is considered in [9, 10, 11].

Figure 1: Our proposed channel-adaptive CA-JSCC scheme. Figure 2: Basic blocks
of our dual-attention encoder and decoder architectures.

However, adaptability to various channel conditions is still a challenge for
deep-learning-based JSCC. Methods in [5, 6, 7] are either trained for a
specific signal-to-noise ratio (SNR), or for a range of channel SNRs. The
former requires significant storage memory to store different network
parameters for different channel conditions, while the latter sacrifices
performance and does not exploit the channel state information (CSI). In
conventional digital communication systems, CSI at the transmitter can allow
power allocation to boost the communication rate. In [12, 8, 13], CSI is used
in a similar manner in the context of learning-aided design, mainly to adjust
the feature weights according to the CSI; however, in the case of OFDM, CSI
will be instrumental not only for power control, but also to decide the
mapping of the features to different subcarriers according to their relative
qualities. For example, more critical features of the input image can be
mapped to more reliable subcarriers.

We introduce a channel-adaptive JSCC (CA-JSCC) scheme, which employs a dual-
attention mechanism to adjust its features in the multi-scale intermediate
layers according to the estimated CSI at the encoder and decoder. Our dual-
attention mechanism employs both channel-wise attention and spatial attention,
and jointly learns to map the features to the subcarriers and to allocate
power judiciously. Our method achieves state-of-the-art performance and can
adapt to different channel conditions.

Our main contributions can be summarized as:

  * •

To the best of our knowledge, channel adaptability for JSCC with OFDM has not
been studied before. All previous methods require the training and testing
SNRs to match without fully exploiting the CSI.

  * •

We present a CA-JSCC scheme with state-of-the-art performance in various SNR
and bandwidth scenarios. We propose a dual-attention mechanism to
simultaneously exploit the estimated CSI to aid the allocation of features and
power resources to adapt to time-varying channel conditions.

##  II System Model

We consider OFDM-based JSCC of images over a multipath fading channel with
$L_{t}$ paths. We transmit each input image using $N_{s}$ OFDM symbols
accompanied with $N_{p}$ pilots for channel estimation of $L_{f}$ OFDM
subcarriers. As shown in Fig 1, an encoding function ${E}_{\bm{\theta}}$ first
maps the input image $\bm{x}\in\mathbb{R}^{c\times h\times w}$ into a complex
matrix $\bm{Y}\in\mathbb{C}^{N_{s}\times L_{f}}$, where $c,h$ and $w$ denote
the color, height and width. The generated channel input can be denoted by
$\bm{Y}={E}_{\bm{\theta}}(\bm{x,\hat{h}})$, where $\bm{\hat{h}}$ is the
estimated CSI vector available at the transmitter.

Channel input $\bm{Y}$ is subject to an average power constraint $P_{s}$:
$\frac{1}{N_{s}L_{f}}\mathbb{E}\big{[}\|\bm{Y}\|^{2}_{\text{F}}\big{]}\leq
P_{s}$, where the expectation is taken over the input images, and
$\|\cdot\|_{\text{F}}$ denotes the Frobenius norm. Without loss of generality,
we set $P_{s}=1$.

Each OFDM symbol is passed through the inverse discrete Fourier transform
(IFFT) module, appended with the cyclic prefix (CP), and transmitted to the
receiver over the multipath channel. The transfer function of the multipath
fading channel with $L_{t}$ paths is defined as:
$\bm{\hat{y}}=h_{c}(\bm{y})=\bm{h_{t}}\ast\bm{y}+\bm{w}$, where $\bm{y}$ and
$\bm{\hat{y}}$ denote the input and output vectors, respectively; $\ast$ is
the linear convolution operation, $\bm{h_{t}}\in\mathbb{C}^{L}$ is the channel
impulse response, and $\bm{w}$ is the AWGN term.

The receiver first demodulates $\bm{\hat{y}}$ by removing the CP and applying
fast Fourier transform (FFT). The equivalent frequency-domain transfer
function of $h_{c}$ can be written as:

$\hat{Y}[i,k]=H[k]\bar{Y}[i,k]+W[i,k],$ (1)

where the frequency-domain channel matrix
$\bm{H}\in\mathbb{C}^{(L_{f},L_{f})}$ is a diagonal matrix with the $k$-th
diagonal element being $H[k]$ (frequency-domain channel response of the $k$-th
subcarrier). Let $\bar{\bm{Y}}\in\mathbb{C}^{(N_{s},L_{f})}$ denote the output
of the power normalization module at the transmitter (i.e., the inputs to the
IFFT module), where $\bar{Y}[i,k]$ denotes the symbol at the $k$-th subcarrier
of the $i$-th OFDM symbol. $\bm{W}\in\mathbb{C}^{(N_{s},L_{f})}$ is the
frequency domain noise matrix, where $W[i,k]\sim\mathcal{CN}(0,\sigma^{2})$
and independent of each other.

Given the FFT output of the pilots at the receiver, we use minimum mean square
error (MMSE) or least square (LS) channel estimator to estimate the CSI
($H[k]$) in frequency-domain (Eqn. (1)). The estimated CSI vector
$\bm{\hat{h}}$ is then used to equalize the data (MMSE equalizer). We have

$\bm{\hat{h}}\triangleq\left[|\hat{h}_{1}|,|\hat{h}_{2}|,\cdots,|\hat{h}_{L_{f}}|,\mu\right]^{\top},$
(2)

where $\hat{h}_{i}$, $i=1,...,L_{f}$, is the estimated channel gain of the
$i$-th subcarrier; while $\mu$ is the average SNR defined as
$\mu=10\log_{10}\frac{P_{s}}{\sigma^{2}}$ dB for transmit power $P_{s}$ and
noise power $\sigma^{2}$.

Based on the equalized data $\bm{\hat{Y}_{e}}$ and the estimated CSI vector
$\bm{\hat{h}}$, the decoder $D_{\bm{\phi}}$ reconstructs the transmitted image
as $\bm{\hat{x}}$, i.e.,
$\bm{\hat{x}}=D_{\bm{\phi}}(\bm{\hat{Y}_{e}},\bm{\hat{h}})$. The performance
indicator is the peak signal-to-noise ratio (PSNR), defined as
$\text{PSNR}=10\log_{10}\frac{(\max{\bm{x}})^{2}}{\text{MSE}(\bm{x},\bm{\hat{x}})}~{}(\text{dB})$,
where $\max{\bm{x}}$ denotes the maximum possible value of the input signal
$\bm{x}$, $\text{MSE}(\bm{x},\bm{\hat{x}})\triangleq
E[\|\bm{x}-\bm{\hat{x}}\|^{2}_{2}]$ and the expectation is taken over all
pixels.

We then jointly train the encoder and decoder to minimize the loss function
$\mathcal{L}(\bm{\theta},\bm{\phi})=\mathbb{E}\big{[}\text{PSNR}(\bm{x},\bm{\hat{x}})\big{]}$,
where the expectation is taken over the randomness both in the source and
channel distributions.

(a)

(b)

(c)

Figure 3: (a) Comparison of our CA-JSCC scheme with Exp-JSCC when the values
of the train and test SNRs are the same. (b) Ablation experiments for the
attention strategy. (c) Visualization of the power allocation for the CA-JSCC
and the Exp-JSCC.

##  III Dual-attention mechanism

In OFDM systems, different subcarriers face different channel gains, and a
judicious transmission scheme should be able to allocate power and features
across appropriate subcarriers to adapt to channel variations. To fully
exploit the estimated CSI, we propose a dual-attention based CA-JSCC scheme.

The architecture of our method is shown in Fig 2, both the encoder and the
decoder have five feature learning (FL) and four channel learning (CL) modules
in a comb structure, allowing modulating each feature with a different scale.
The FL module consisting of 2D convolution/deconvolution, batch normalization,
and PReLU layers, is designed to learn an intermediate representation of the
input. The dual-attention based CL module is designed to learn an attention
mask to map the intermediate features to appropriate subcarriers based on the
estimated CSI and input features. The CL module consists of a channel-wise
attention module and a spatial attention module. Its operation is presented in
Algorithm 1 in detail.

Algorithm 1 Dual-attention based CL

Input: $\bm{F_{in}}\in\mathbb{R}^{c\times h\times w}$,
$\bm{\hat{h}}\in\mathbb{R}^{L_{f}+1}$
Output: $\bm{F_{out}}\in\mathbb{R}^{c\times h\times w}$
Stage 1: Channel-wise attention

1:$\bm{F_{ca}}=Ave_{c}(\bm{F_{in}})\in\mathbb{R}^{(c)}$

2:$\bm{i_{c}}=concatenate(\bm{F_{ca}},\bm{\hat{h}})\in\mathbb{R}^{(c+L_{f}+1)}$

3:$\bm{S_{c}}=f_{c}(\bm{i_{c}})\in\mathbb{R}^{(c,1,1)}$

4:for i= 0:1:c do

5:
$\bm{F_{cout}}[i,h,w]=\bm{S_{c}}[i]\odot\bm{F_{in}}[i,h,w]\in\mathbb{R}^{(c,h,w)}$

6:end for

Stage 2: Spatial attention

1:$\bm{F_{sa}}=Ave_{s}(\bm{F_{cout}})\in\mathbb{R}^{(1,h,w)}\Rightarrow\mathbb{R}^{(hw)}$

2:$\bm{i_{s}}=concatenate(\bm{F_{sa}},\bm{\hat{h}})\in\mathbb{R}^{(hw+L_{f}+1)}$

3:$\bm{S_{s}}=f_{s}(\bm{i_{s}})\in\mathbb{R}^{(hw)}\Rightarrow\mathbb{R}^{(h,w)}$

4:for j= 0:1:h do

5: for k= 0:1:w do

6:
$\bm{F_{out}}[c,j,k]=\bm{S_{s}}[j,k]\odot\bm{F_{cout}}[c,j,k]\in\mathbb{R}^{(c,h,w)}$

7: end for

8:end for

####  III-1 Channel-wise attention module

Our channel-wise attention module is inspired by[12], which adapts to a single
SNR value in an AWGN channel model. Instead, CA-JSCC learns an attention mask
to allocate and map features based on the estimated CSI of all $N_{s}$
subcarriers.

We first apply average pooling $Ave_{c}(\bm{F_{in}})$ on input features
$\bm{F_{in}}$ along the spatial direction to get vector $\bm{F_{ca}}$, where
$Ave_{c}(\bm{F_{in}})\triangleq\frac{1}{hw}\sum_{j=1}^{h}\sum_{k=1}^{w}F_{in}[c,h,w]$.
$\bm{F_{ca}}$ is then concatenated with $\bm{\hat{h}}$ to get the intermediate
vector $\bm{i_{c}}$ to compute the channel-wise attention mask $\bm{S_{c}}$ by
several fully connected (FC) layers: $S_{c}=f_{c}(\bm{i_{c}})$, where $f_{c}$
represents the FC layers followed by PReLU functions. Finally, we get the
output of our channel-wise attention module as
$\bm{F_{cout}}=\bm{S_{c}}\odot\bm{F_{in}}$.

Our channel-wise attention module learns to map features from the input to the
subcarriers based on the estimated CSI, allowing JSCC to dynamically adjust to
different channel SNRs. But the spatial information is ignored when computing
the channel attention mask by the average pooling operation. So we design a
spatial attention module to compensate for spatial information.

####  III-2 Spatial attention module

Our spatial attention module learns to match the more critical spatial
features along the $h$ and $w$ dimensions with better channel conditions
depending on the estimated CSI.

This time we firstly apply average pooling $Ave_{s}(\bm{F_{cout}})$ on the
$\bm{F_{cout}}$ along the channel direction to get $\bm{F_{sa}}$, where
$Ave_{s}(\bm{F_{cout}})\triangleq\frac{1}{c}\sum_{j=1}^{c}F_{cout}[c,h,w]$.
Then, $\bm{\hat{h}}$ and $\bm{F_{sa}}$ are concatenated to get the
intermediate feature $\bm{i_{s}}$, which is used to compute the spatial mask
$\bm{S_{s}}$ by several FC layers. We compute the final output feature vector
as $\bm{F_{out}}=\bm{S_{s}}\odot\bm{F_{cout}}$.

The spatial attention module can further improve the PSNR performance by
exploiting the spatial information and helping JSCC encoder to do more
adaptive power allocation, which matches critical features with better
channels.

(a) $R=1/12$

(b) $R=1/6$

(c) $R=1/3$

Figure 4: Performance of our CA-JSCC model comparied with the Exp-JSCC model
of different bandwidth ratios.

##  IV Training and evaluation

This section presents numerical experiment results to evaluate the performance
of our CA-JSCC scheme. The Exp-JSCC scheme introduced in [8] is the most
related work to the current study in the literature. We use Exp-JSCC trained
on different channel conditions as a benchmark to compare with the proposed
CA-JSCC scheme.

###  IV-A Experimental setup

If not specified otherwise, all experiments were performed on the CIFAR-10
dataset [14] with PyTorch. Models were trained until the performance on a
validation set (selected separately from the training dataset) stops
improving. The Adam optimizer is used to perform backpropagation.

We set the number of subcarriers to $L_{f}=64$. The Zadoff-Chu (ZC) sequence
[15], denoted by $\bm{Y_{p}}\in\mathbb{C}^{(2,64)}$, is used as the pilot. The
values of channel gains $\\{H[k]:k=1,2,...,L_{f}\\}$ are sampled from a
complex Gaussian distribution $\mathcal{CN}(0,1)$. Unless specified otherwise,
the frequency-domain channel responses in the experiments are estimated by an
MMSE estimator. We also sort the channels based on their estimated CSI to make
training process easier; that is, we have
$|H[1]|^{2}\geq\cdots\geq|H[L]|^{2}$. Following[6], we define the bandwidth
ratio (i.e., bandwidth usage to source symbol ratio) as
$R\triangleq\frac{N_{s}L_{f}}{c\times h\times w}$, where $N_{s}L_{f}$ is the
number of symbols transmitted per image.

###  IV-B Channel-gain adaptability

We first verify the adaptability of the CA-JSCC scheme to channel-gain
variations. Specifically, under a fixed bandwidth ratio and a given SNR, we
want to see if our dual-attention mechanism can instruct the transmitter to
exploit better channels and allocate power to different subcarriers
judiciously.

The experimental results are shown in Fig 3a, where we set the number of OFDM
symbols to $N_{s}=8$ and the bandwidth ratio to $R=1/6$. The Exp-JSCC and CA-
JSCC models are both trained at a fixed SNR, and tested at the same SNR. As
can be seen, by feeding the estimated CSI to the transmitter, CA-JSCC can
exploit the channel-gain information and adaptively allocate power to
different subcarriers. We can see a significant gain compared to Exp-JSCC at
all SNRs. This can be attributed to the advantage of our method in better
exploiting the channel gains and allocating power to different subcarriers.

###  IV-C SNR adaptability

Next, we evaluate the SNR adaptability of our scheme. If not specified
otherwise, we train the CA-JSCC model at random SNR values of each training
episode chosen uniformly from $[0,20]$ dB and test the well-trained model at
different SNRs.

Compared with the CA-JSCC scheme trained at specific SNRs in Fig 3a, we
observe that there is a slight performance degradation when it is trained at
random SNR values. We conclude that, while CA-JSCC can learn to adapt to
different channel SNRs, this flexibility comes at the expense of some loss in
the PSNR (up to $1$dB). However, this CA-JSCC model trained at random SNR
values still outperforms the Exp-JSCC models trained at specific SNR values.

We also compare the performance of dual-attention based architecture CA-JSCC
with an alternative using only channel-wise attention, called (CA-JSCC-CH), as
an ablation study. As shown in Fig 3b, for three different bandwidth ratios,
CA-JSCC architecture outperforms CA-JSCC-CH at all SNR values, which shows
that the spatial attention mechanism is essential to achieve the improved
performance provided by CA-JSCC. We also observe larger gains by our dual-
attention method at higher bandwidth ratios and $\text{SNR}_{\text{test}}$
values, where more spatial information and better CSI adaptability benefit
both feature mapping and power allocation. To visualize the power allocation
executed by CA-JSCC and Exp-JSCC, we plot the average channel gain and the
average power allocated to each subcarrier in Fig. 3c, where we set SNR$=1$dB
and R=$1/6$. The channel gains are ordered in an increasing manner in the
plot. Compared with Exp-JSCC, Fig. 3c shows that CA-JSCC generally allocates
more power for the subcarrier with better channel conditions, as one would
desire.

It is worth noting that the Exp-JSCC scheme is not SNR-adaptive, which means
the training and test SNRs of Exp-JSCC must match to achieve a sound
performance, as shown in Fig. 3a. Fig 4 presents the PSNR versus test SNR
results for bandwidth ratios of $R=1/12,1/6,1/3$ (we set $L_{f}=64$ and vary
$N_{s}$ to attain different $R$ values). As stated above, our CA-JSCC scheme
can be trained with random SNRs, and yields a single model for each bandwidth
ratio to be verified on a range of test SNRs. The Exp-JSCC scheme, on the
other hand, is trained at five different SNRs, yielding five different models
under each bandwidth ratio. The Exp-JSCC scheme performs the best when the
training and test SNRs match. However, our CA-JSCC scheme is SNR-adaptive and
consistently outperforms Exp-JSCC at all SNRs and bandwidth ratios with a
considerable margin.

Additional experiments by training over the ImageNet dataset is shown in Fig.
5a. We train the models with randomly cropped $64\times 64$ patches from
ImageNet, and evaluate the models on the Kodak dataset. Results show that
training on a sufficiently large dataset (ImageNet) can allow our CA-JSCC
model to perform well on a never-seen dataset (Kodak). CA-JSCC can still
achieve state-of-the-art performance with the additional capability of channel
adaptability.

(a)

(b)

Figure 5: (a) Additional experiments with training on the ImageNet dataset
tested on the Kodak dataset. (b) Comparison of CA-JSCC with different channel
estimation methods.

###  IV-D Impact of CSI estimation errors

In the above experiments, we have assumed MMSE estimated CSI. In this
subsection, we look into the effect of channel estimation errors on the
performance of CA-JSCC. We repeat the experiment in Fig 4b with three types of
CSI: i) perfect CSI, $H_{per}$; ii) MMSE estimated CSI, $H_{mmse}$; and iii)
LS estimated CSI, $H_{ls}$. We remark that $H_{mmse}$ provides a more accurate
estimate than $H_{ls}$.

The experimental results are shown in Fig 5b, where we train our CA-JSCC
models with $H_{mmse}$ and $H_{per}$, respectively, and evaluate these two
models with $H_{per}$, $H_{mmse}$ and $H_{ls}$, respectively. As expected, the
model trained and tested with $H_{per}$ achieves the best performance. On the
other hand, the model trained with perfect CSI is not robust to CSI errors
during test time. Its performance gets worse as the quality of channel
estimation degrades. Instead, we see that models trained with $H_{mmse}$
perform better, since during training they learn to compensate for CSI
estimation errors. We conclude from these results that a more accurate CSI
during testing is generally beneficial, and the performance improves if
training is done with the same type of CSI.

##  V Conclusion

We presented the CA-JSCC scheme for wireless image transmission over OFDM
channels. CA-JSCC employs a dual-attention mechanism to exploit the available
CSI to simultaneously aid the allocation of features and power resources
adaptively. The dual-attention mechanism comprises a channel-wise attention
module, which learns to map features to subcarriers according to the CSI, and
a spatial attention module, which learns high-level spatial features, allowing
the JSCC encoder to match the most important features with the most reliable
subcarriers. Numerical experiments show that our method achieves state-of-the-
art performance in various SNR and bandwidth scenarios. Besides, our method
simultaneously uses the estimated CSI to adapt to time-varying channel
conditions.

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