# Stackelberg exact controllability of a class of nonlocal parabolic equations

Landry Djomegne Cyrille Kenne University of Dschang, BP 67 Dschang, Cameroon,
West region, email<EMAIL_ADDRESS>Laboratoire LAMIA, Université
des Antilles, Campus Fouillole, 97159 Pointe-à-Pitre Guadeloupe
(FWI)-Laboratoire L3MA, UFR STE et IUT, Université des Antilles, Schoelcher,
Martinique, email<EMAIL_ADDRESS>
###### Abstract

This paper deals with a multi-objective control problem for a class of
nonlocal parabolic equations, where the non-locality is expressed through an
integral kernel. We present the Stackelberg strategy that combines the
concepts of controllability to trajectories with optimal control. The strategy
involves two controls: a main control (the leader) and a secondary control
(the follower). The leader solves a controllability to trajectories problem
which consists to drive the state of the system to a prescribed target at a
final time while the follower solves an optimal control problem which consists
to minimize a given cost functional. The paper considers two cases: in the
first case, both the leader and the follower act in the interior of the
domain, and in the second case, the leader acts in the interior of the domain
and the follower acts on a small part of the boundary. These results are
applied to both linear and nonlinear nonlocal parabolic systems.

2010 Mathematics Subject Classification. 35K05; 49J20; 93B05; 93B07; 93C20.

Key-words : Parabolic equation; Nonlocal terms; Carleman estimates; Null
controllability; Stackelberg strategy.

##  1 Introduction

####

Let $\Omega\subset\mathbb{R}^{N},\ N\geq 1$, be a bounded open set with a
regular boundary $\partial\Omega$. Let $\omega$ and $\mathcal{O}$ be two
arbitrary nonempty open sets of $\Omega$ such that
$\omega\cap\mathcal{O}=\emptyset$ and let $\Gamma$ be a nonempty open subset
of $\partial\Omega$. For the real number $T>0$, we denote
$Q=(0,T)\times\Omega$ and $\Sigma=(0,T)\times\partial\Omega$. Let
$K(t,x,\theta)\in L^{\infty}(Q\times\Omega)$ represents an integral kernel.

In the sequel, we will denote the norm in $L^{\infty}(Q\times\Omega)$ by
$\|\cdot\|_{\infty}$ and the symbol $C$ is used to design a generic positive
constant whose value can change even from line to line. We also denote by
$\nu=\nu(x)$ the outward unit normal vector at the point $x\in\partial\Omega$.
We consider the following linear nonlocal parabolic systems:

$\left\\{\begin{array}[]{rllll}\displaystyle y_{t}-\Delta
y+\int_{\Omega}K(t,x,\theta)y(t,\theta)\
d\theta&=&f\chi_{\omega}+v\chi_{\mathcal{O}}&\mbox{in}&Q,\\\ \displaystyle
y&=&0&\mbox{on}&\Sigma,\\\ \displaystyle
y(0,\cdot)&=&y^{0}&\mbox{in}&\Omega\end{array}\right.$ (1.1)

and

$\left\\{\begin{array}[]{rllll}\displaystyle q_{t}-\Delta
q+\int_{\Omega}K(t,x,\theta)q(t,\theta)\
d\theta&=&g\chi_{\omega}&\mbox{in}&Q,\\\ \displaystyle
q&=&u\textbf{1}_{\Gamma}&\mbox{on}&\Sigma,\\\ \displaystyle
q(0,\cdot)&=&q^{0}&\mbox{in}&\Omega,\end{array}\right.$ (1.2)

where $y=y(t,x)$ and $q=q(t,x)$ denote the states, the functions $f,\ g,\ u$
and $v$ are the controls given in appropriate spaces, $y^{0}$ and $q^{0}$ are
given initial datum. Here, $\chi_{\mathcal{A}}$ denotes the characteristic
function of the set $\mathcal{A}$ and
$\textbf{1}_{\Gamma}\in\mathcal{C}^{2}(\partial\Omega)$ is a smooth
nonnegative function such that supp $\textbf{1}_{\Gamma}=\overline{\Gamma}$.
We denote by $y_{t}$ the partial derivative of $y$ with respect to $t$.

In this work, we develop the Stackelberg strategy [23] for the nonlocal heat
equations. We assume that we can act on the systems through a hierarchic of
controls. More precisely, we combine the optimal control strategy with the
exact controllability to the trajectories of systems (1.1) and (1.2).

The concept of Stackelberg competition introduced in 1934 [31] is a game-
theoretic approach where one player (the leader) makes a strategic decision
first, and the second player (the follower) then respond based on the leader’s
decision.

In the framework of PDEs, the hierarchic control was introduced by J-L. Lions
in 1994 [23, 22] to study a bi-objective control problem for the wave and heat
equation respectively. In theses works, the author acted on the different
systems with two controls. The leader solving an approximate controllability
problem while the follower solving an optimal control problem. In the past
years, many other researchers have used hierarchic control in the sense of
Lions (see for instance [11, 10, 20, 25, 26, 27, 30, 13]). Recently, the
authors in [9, 15] used the hierarchic control combining the concept of
controllability with robustness.

All the previous work deal with hierarchic strategy with distributed controls.
The first paper which solve with the boundary controls is the one given by
[3]. In that paper, the authors dealt with the Stackelberg-Nash exact
controllability of parabolic equations with the possibility of the leader and
the followers being placed on the boundary. In [16], the authors extend and
discuss the results concerning the robust hierarchic strategy for the heat
equation using boundary controls. Recently, the authors in [5] considered a
multi-objective control problem for the Kuramoto-Sivashinsky equation with a
distributed control called leader and two boundary controls called followers.

In all the above cited works, the hierarchic strategy were applied to local
systems. In this paper, we extend the results concerning the hierarchic
strategy to nonlocal parabolic equations. This extension introduces additional
difficulties, particularly when establishing new observability inequalities of
Carleman for adjoint systems associated with equations (1.1) and (1.2). To the
best of our knowledge, this has not been done before.

In this paper, we extend the Stackelberg control of Lions to nonlocal systems
with an integral kernel (1.1) and (1.2). To fix ideas, we begin by explaining
the strategy control to system (1.1). To be more specific, we introduce a
nonempty open set $\mathcal{O}_{d}\subset\Omega$, representing an observation
domain of the follower and define the cost functional

$J(f;v)=\frac{1}{2}\int_{0}^{T}\int_{\mathcal{O}_{d}}|y-y_{d}|^{2}\
dxdt+\frac{\mu}{2}\int_{0}^{T}\int_{\mathcal{O}}|v|^{2}\ dxdt,$ (1.3)

where $\mu$ is a positive constant and $y_{d}\in
L^{2}((0,T)\times\mathcal{O}_{d})$ is given. The structure of the control
process we will follow is described as follows:

  1. 1.

Once the leader $f$ has been fixed, we look for a control $\hat{v}$ depending
of $f$ which solves the following problem

$\begin{array}[]{rllll}\displaystyle J(f;\hat{v})=\inf_{v}J(f;v).\end{array}$
(1.4)

The function $\hat{v}$ satisfied (1.4) is called an optimal control for $J$.
Since the functional $J$ is convex (because the system (1.1) is linear), then
$\hat{v}$ is an optimal control for $J$ if and only if

$\frac{\partial J}{\partial w}(f;\hat{v})\cdot w=0,\ \ \forall w\in
L^{2}((0,T)\times\mathcal{O}).$ (1.5)

  2. 2.

Let us consider an uncontrolled trajectory of (1.1), that is a function
$\bar{y}=\bar{y}(t,x)$ solution of

$\left\\{\begin{array}[]{rllll}\displaystyle\bar{y}_{t}-\Delta\bar{y}+\int_{\Omega}K(t,x,\theta)\bar{y}(t,\theta)\
d\theta&=&0&\mbox{in}&Q,\\\ \displaystyle\bar{y}&=&0&\mbox{on}&\Sigma,\\\
\displaystyle\bar{y}(0,\cdot)&=&\bar{y}^{0}&\mbox{in}&\Omega.\end{array}\right.$
(1.6)

Once the optimal control has been identified and fixed for each $f$, we look
for a control $\hat{f}$ that satisfies

$y(T,x)=\bar{y}(T,x)=0\ \mbox{in}\ \Omega.$ (1.7)

Now, we are also interested to study the system (1.2). The same methodology
presented above can be used here. The cost functional (1.3) should be replaced
by

$\widetilde{J}(g;u)=\frac{1}{2}\int_{0}^{T}\int_{\mathcal{O}_{d}}|q-q_{d}|^{2}\
dxdt+\frac{\mu}{2}\int_{0}^{T}\int_{\Gamma}|u|^{2}\ d\sigma dt,$ (1.8)

where again $\mu$ is a positive constant and $q_{d}\in
L^{2}((0,T)\times\mathcal{O}_{d})$ is a given functional. The control process
can be described as follows.

  1. 1.

For each leader $g$, we find a control $\hat{u}$ which solves the following
problem

$\begin{array}[]{rllll}\displaystyle\widetilde{J}(g;\hat{u})=\inf_{u}\widetilde{J}(g;u).\end{array}$
(1.9)

The function $\hat{u}$ satisfied (1.9) is called an optimal control for
$\widetilde{J}$ given by (1.8). Since the functional $\widetilde{J}$ is
convex, then $\hat{u}$ is an optimal control for $\widetilde{J}$ if and only
if

$\frac{\partial\widetilde{J}}{\partial w}(g;\hat{u})\cdot w=0,\ \ \forall w\in
L^{2}((0,T)\times\Gamma).$ (1.10)

  2. 2.

Let us fix an uncontrolled trajectory of system (1.2), that is a function
$\bar{q}=\bar{q}(t,x)$ solution to

$\left\\{\begin{array}[]{rllll}\displaystyle\bar{q}_{t}-\Delta\bar{q}+\int_{\Omega}K(t,x,\theta)\bar{q}(t,\theta)\
d\theta&=&0&\mbox{in}&Q,\\\ \displaystyle\bar{y}&=&0&\mbox{on}&\Sigma,\\\
\displaystyle\bar{q}(0,\cdot)&=&\bar{q}^{0}&\mbox{in}&\Omega.\end{array}\right.$
(1.11)

We look for a control $\hat{g}$ verifying

$q(T,x)=\bar{q}(T,x)=0\ \mbox{in}\ \Omega.$ (1.12)

As mentioned in [12], the study of the controllability of systems (1.1) and
(1.2) is motivated by many relevant applications from physics, chemotaxis,
biology and ecology; we can see [19] for instance where many models are
introduced. Such systems like (1.1) and (1.2) can appear for instance in
population dynamics where the state $y(t,x)$ represents the density of the
species at time $t$ and position $x$, while the nonlocal term
$\int_{\Omega}K(t,x,\theta)y(t,\theta)\ d\theta$ is considered as the rate of
reproduction. This integral term is a way to express that the evolution of the
species in a point of space depends on the total amount of the species [1].

Some results are available in the literature concerning the controllability of
parabolic nonlocal systems. The first result concerning this subject is the
one studied in [6], where the authors established the exact controllability of
the Schrödinger equation with a nonlocal term. In [12], the authors proved an
interior null controllability result for the system like (1.1) when the kernel
is time-independent and analytic. They used some compactness-uniqueness
arguments in order to obtain the unique continuation properties. A similar
result for the wave equation has also been derived. The results of [12] have
been extended in [24] to a general coupled parabolic system. Moreover, similar
results have been obtained in [28] for a one-dimensional equation with a time-
independent kernel in separated variables, by means of spectral analysis
techniques. Later on, the authors in [4] have extended these mentioned
results, by relaxing the assumptions on the kernel. The authors studied both
the linear and the semilinear case, by using a Carleman approach. Recently, in
[1], the authors have proved the null controllability property for the two
degenerate nonlocal systems using the Kakutani’s fixed point Theorem. The
authors in [17] have proved the internal null-controllability of a heat
equation with Dirichlet boundary conditions and perturbed by a semilinear
nonlocal term.

###  1.1 Main results

In this paper, we use the Carleman approach to solve the null controllability
of the nonlocal systems (1.1) and (1.2). Using the same approach as the
authors of [4], we assume that the integral kernel $K\in
L^{\infty}(Q\times\Omega)$ satisfies the following assumption:

$\mathcal{K}=:\sup_{(t,x)\in\overline{Q}}\exp\left(\frac{\sigma^{-}}{l^{4}(t)}\right)\int_{\Omega}|K(t,x,\theta)|\
d\theta<+\infty,$ (1.13)

where the functions $\sigma^{-}$ and $l(t)$ are defined by (2.20b) and (2.17),
respectively. The assumption (1.13) means that, the kernel $K=K(t,x,\theta)$
is bounded and decrease exponentially when $t$ goes to $0^{+}$ and to $T^{-}$
[4].

We firstly address the Stackelberg control of system (1.1) where all the
controls are localized on the interior of the domain. We have the following
result.

###### Theorem 1.1

Assume that $\displaystyle\mathcal{O}_{d}\cap\omega\neq\emptyset$ and the
kernel $K$ satisfies (1.13). Let $\bar{y}$ be the unique solution to (1.6)
associated to the initial state $\bar{y}^{0}$. Suppose that $y^{0}\in
L^{2}(\Omega)$ and $\mu$ is large enough. Then, there exists a positive real
weight function $\varpi_{1}=\varpi_{1}(t)$ blowing up at $t=T$ such that for
any $y_{d}\in L^{2}((0,T)\times\mathcal{O}_{d})$ satisfying

$\int_{0}^{T}\int_{\mathcal{O}_{d}}\varpi_{1}^{2}|\bar{y}-y_{d}|^{2}\
dxdt<+\infty,$ (1.14)

there exist a control $\hat{f}\in L^{2}((0,T)\times\omega)$ and a unique
optimal control $\hat{v}\in L^{2}((0,T)\times\mathcal{O})$ such that the
corresponding solution to (1.1) satisfies (1.7).

Now, we are interested to the system (1.2) where the leader control is applied
on the interior of the domain and the follower is localized on the boundary.
In the following, we need the Hilbert spaces

$H^{r,s}(Q)=L^{2}((0,T);H^{r}(\Omega))\cap H^{s}((0,T);L^{2}(\Omega)),$ (1.15)

for $r,s\in\mathbb{R}_{+}$ endowed with norms

$\|z\|_{H^{r,s}(Q)}=\left(\|z\|^{2}_{L^{2}((0,T);H^{r}(\Omega))}+\|z\|^{2}_{H^{s}((0,T);L^{2}(\Omega))}\right)^{1/2}.$
(1.16)

###### Remark 1

We can replace in (1.16) the domain $Q$ and $\Omega$ by $\Sigma$ and
$\partial\Omega$, respectively. Then, we obtain the analogous space for
functions defined on the boundary.

We have the following result.

###### Theorem 1.2

Assume that $\displaystyle\mathcal{O}_{d}\cap\omega\neq\emptyset$ and the
kernel $K$ satisfies (1.13). Assume that $q^{0}\in L^{2}(\Omega)$ and $\mu$ is
sufficiently large. Let $\bar{q}$ be the unique solution to (1.11) associated
to the initial state $\bar{q}^{0}$. Then, there exists a positive real weight
function $\varpi_{2}=\varpi_{2}(t)$ blowing up at $t=T$ such that for any
$q_{d}\in L^{2}((0,T)\times\mathcal{O}_{d})$ verifying

$\int_{0}^{T}\int_{\mathcal{O}_{d}}\varpi_{2}^{2}|\bar{q}-q_{d}|^{2}\
dxdt<+\infty,$ (1.17)

we can find a leader control $\hat{g}\in L^{2}((0,T)\times\omega)$ and a
unique optimal control $\hat{v}\in H^{1/2,1/4}((0,T)\times\Gamma)$ such that
the corresponding solution to (1.2) satisfies (1.12).

The rest of the paper is organized as follows. In Section 2, we analyse the
Stackelberg strategy concerning the case of distributed leader and follower,
that is, the case of system (1.1). In this part, we firstly prove the
existence and characterization of optimal control, then we prove some suitable
Carleman inequalities and deduce the exact controllability result. Finally, we
extend the results to the semilinear case. The Section 3 concerns the control
of system (1.2) with distributed leader and boundary follower. In this
section, we follow similar arguments as in Section 2.

##  2 Exact controllability with distributed leader and follower

The first step of this part is to reduce the problem of exact controllability
to trajectory to a null controllability problem. So, let us consider
$z:=y-\bar{y}$, where $y$ and $\bar{y}$ are respectively solutions to (1.1)
and (1.6). The property (1.7) is equivalent to a null controllability property
for $z$, that is

$z(T,x)=0\ \mbox{in}\ \Omega,$ (2.1)

where $z$ is solution of the system

$\left\\{\begin{array}[]{rllll}\displaystyle z_{t}-\Delta
z+\int_{\Omega}K(t,x,\theta)z(t,\theta)\
d\theta&=&f\chi_{\omega}+v\chi_{\mathcal{O}}&\mbox{in}&Q,\\\ \displaystyle
z&=&0&\mbox{on}&\Sigma,\\\ \displaystyle
z(0,\cdot)&=&z^{0}&\mbox{in}&\Omega,\end{array}\right.$ (2.2)

with $z^{0}:=y^{0}-\bar{y}^{0}$. From now on, we are going to work with system
(2.2) instead of (1.1).

The functional $J$ given by (1.3) can be rewritten as follows

$J(f;v)=\frac{1}{2}\int_{0}^{T}\int_{\mathcal{O}_{d}}|z-z_{d}|^{2}\
dxdt+\frac{\mu}{2}\int_{0}^{T}\int_{\mathcal{O}}|v|^{2}\ dxdt,$ (2.3)

where $z_{d}:=y_{d}-\bar{y}$.

Let us introduce the Hilbert space

$W(Q)=\left\\{z\in L^{2}((0,T);H^{1}_{0}(\Omega));z_{t}\in
L^{2}\left((0,T);H^{-1}(\Omega)\right)\right\\},$ (2.4)

equipped with the norm given by

$\|z\|^{2}_{W(Q)}=\|z\|^{2}_{L^{2}((0,T);H^{1}_{0}(\Omega))}+\|z_{t}\|^{2}_{L^{2}\left((0,T);H^{-1}(\Omega)\right)}.$
(2.5)

It is well known (see, for instance [21, Page 37]) that for any $y^{0}\in
L^{2}(\Omega)$, $f\in L^{2}((0,T)\times\omega)$ and $v\in
L^{2}((0,T)\times\mathcal{O})$, the system (2.2) admits a unique weak solution
$z\in W(Q)$. Moreover there exists a constant $C=C(\|K\|_{\infty},T)>0$ such
that the following estimation holds:

$\begin{array}[]{llllll}\displaystyle\|z(T,\cdot)\|^{2}_{L^{2}(\Omega)}+\|z\|^{2}_{W(Q)}\leq
C\left(\|f\|^{2}_{L^{2}((0,T)\times\omega)}+\|v\|^{2}_{L^{2}((0,T)\times\mathcal{O})}+\|z^{0}\|^{2}_{L^{2}(\Omega)}\right).\end{array}$
(2.6)

###  2.1 Existence, uniqueness and characterization of the optimal control

In this section, we are interested in the classical optimal control problem
(1.4). To this end, we first prove the existence and the uniqueness of a
minimizer and secondly, we characterize it through a suitable optimality
system.

Due to the fact that the functional $J$ given by (2.3) is quadratic and the
system (2.2) is linear, it is not difficult to prove that $J$ is continuous
and strictly convex. Moreover, we have

$J(f;v)\geq\frac{\mu}{2}\|v\|^{2}_{L^{2}((0,T)\times\mathcal{O})}.$

The above inequality allow us to say that $J$ is coercive. These three
properties guarantee the existence of a unique solution $\hat{v}\in
L^{2}((0,T)\times\mathcal{O})$ to the optimal control problem (1.4)-(2.3).

In order to characterize the optimal control $\hat{v}$, we use the Euler-
Lagrange optimality condition (1.5), i.e.

$\lim\limits_{\lambda\rightarrow 0}\frac{J(f;\hat{v}+\lambda
w)-J(f;\hat{v})}{\lambda}=0,\ \ \mbox{for all}\ \ w\in
L^{2}((0,T)\times\mathcal{O}).$ (2.7)

After some calculations, (2.7) gives

$\begin{array}[]{rll}&&\displaystyle\int_{0}^{T}\int_{\mathcal{O}_{d}}(z-z_{d})h\
dx\,dt+\mu\int_{0}^{T}\int_{\mathcal{O}}\hat{v}w\ dx\,dt=0,\ \mbox{for all}\
w\in L^{2}((0,T)\times\mathcal{O}),\end{array}$ (2.8)

where $h$ is the solution to

$\left\\{\begin{array}[]{rllll}\displaystyle h_{t}-\Delta
h+\int_{\Omega}K(t,x,\theta)h(t,\theta)\
d\theta&=&w\chi_{\mathcal{O}}&\mbox{in}&Q,\\\ \displaystyle
h&=&0&\mbox{on}&\Sigma,\\\ \displaystyle
h(0,\cdot)&=&0&\mbox{in}&\Omega.\end{array}\right.$ (2.9)

Now, we introduce the adjoint state to system (2.9):

$\left\\{\begin{array}[]{rllll}\displaystyle-p_{t}-\Delta
p+\int_{\Omega}K(t,x,\theta)p(t,\theta)\
d\theta&=&(z-z_{d})\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\ \displaystyle
p&=&0&\mbox{on}&\Sigma,\\\ \displaystyle
p(T,\cdot)&=&0&\mbox{in}&\Omega.\end{array}\right.$

If we multiply the first equation of this latter system by $h$ solution of
(2.9) and we integrate by parts over $Q$, one obtain using the Fubini’s
theorem

$\begin{array}[]{rll}\displaystyle\int_{0}^{T}\int_{\mathcal{O}_{d}}(z-z_{d})h\
dx\,dt=\int_{0}^{T}\int_{\mathcal{O}}wp\ dx\,dt.\end{array}$

Combining this above equality with (2.8), we obtain

$\begin{array}[]{rll}\displaystyle\int_{0}^{T}\int_{\mathcal{O}}w\left(p+\mu\hat{v}\right)\
dx\,dt=0\,\,\mbox{for all}\ w\in L^{2}((0,T)\times\mathcal{O}),\end{array}$

from which we deduce $p+\mu\hat{v}=0\ \ \mbox{in}\ \ (0,T)\times\mathcal{O}$.

We have proved the following result giving the optimality system that
characterizes the optimal control for the cost functional $J$.

###### Proposition 2.1

Let $z^{0}\in L^{2}(\Omega)$ and $f\in L^{2}((0,T)\times\omega)$ be given.
Then, the optimal control problem (1.4)-(2.3) has a unique solution
$\hat{v}\in L^{2}((0,T)\times\mathcal{O})$. Furthermore,

$\hat{v}=-\frac{1}{\mu}p\ \ \mbox{in}\ \ \ (0,T)\times\mathcal{O},$ (2.10)

where $(z,p)$ is the solution to the following optimality systems

$\left\\{\begin{array}[]{rllll}\displaystyle z_{t}-\Delta
z+\int_{\Omega}K(t,x,\theta)z(t,\theta)\ d\theta&=&\displaystyle
f\chi_{\omega}-\frac{1}{\mu}p\chi_{\mathcal{O}}&\mbox{in}&Q,\\\ \displaystyle
z&=&0&\mbox{on}&\Sigma,\\\ \displaystyle
z(0,\cdot)&=&z^{0}&\mbox{in}&\Omega\end{array}\right.$ (2.11)

and

$\left\\{\begin{array}[]{rllll}\displaystyle-p_{t}-\Delta
p+\int_{\Omega}K(t,x,\theta)p(t,\theta)\
d\theta&=&(z-z_{d})\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\ \displaystyle
p&=&0&\mbox{on}&\Sigma,\\\ \displaystyle
p(T,\cdot)&=&0&\mbox{in}&\Omega.\end{array}\right.$ (2.12)

###### Remark 2

Using Proposition $2.2$ [11], we can prove that there exists a positive
constant $C=C(\|K\|_{\infty},T,\mu)$ such that the following estimate holds

$\|\hat{v}\|_{L^{2}((0,T)\times\mathcal{O})}\leq
C\left(\|f\|_{L^{2}((0,T)\times\omega)}+\|z^{0}\|_{L^{2}(\Omega)}\right).$
(2.13)

###  2.2 Carleman inequalities

In this section, we establish a suitable observability inequality useful for
solving the null controllability of the cascade linear system (2.11)-(2.12).
It is well known by now that, solve the null controllability of the system
(2.11)-(2.12) can be reduced to find an observability inequality for the
solutions of the following adjoint systems:

$\left\\{\begin{array}[]{rllll}\displaystyle-\rho_{t}-\Delta\rho+\int_{\Omega}K(t,x,\theta)\rho(t,\theta)\
d\theta&=&\displaystyle\psi\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\
\displaystyle\rho&=&0&\mbox{on}&\Sigma,\\\
\displaystyle\rho(T,\cdot)&=&\rho^{T}&\mbox{in}&\Omega\end{array}\right.$
(2.14)

and

$\left\\{\begin{array}[]{rllll}\displaystyle\psi_{t}-\Delta\psi+\int_{\Omega}K(t,x,\theta)\psi(t,\theta)\
d\theta&=&-\displaystyle\frac{1}{\mu}\rho\chi_{\mathcal{O}}&\mbox{in}&Q,\\\
\displaystyle\psi&=&0&\mbox{on}&\Sigma,\\\
\displaystyle\psi(0,\cdot)&=&0&\mbox{in}&\Omega,\end{array}\right.$ (2.15)

for $\rho^{T}\in L^{2}(\Omega)$.

Now, we introduce some notations and show some preliminary results that will
be useful in the sequel.

Since $\mathcal{O}_{d}\cap\omega\neq\emptyset$, then there exists a nonempty
set $\omega^{\prime}$ such that
$\omega^{\prime}\subset\subset\mathcal{O}_{d}\cap\omega$. Let
$\eta^{0}\in\mathcal{C}^{2}(\overline{\Omega})$ be a function satisfying

$\eta^{0}(x)>0\ \mbox{in}\ \Omega,\ \eta^{0}=0\ \mbox{on}\ \partial\Omega,\
|\nabla\eta^{0}|>0\ \mbox{in}\ \overline{\Omega}\setminus\omega^{\prime}.$
(2.16)

The existence of such a function is proved in [14, Lemma 1.1]. Let
$l\in\mathcal{C}^{\infty}([0,T])$ be a function satisfying

$l(t)>0,\ \forall t\in(0,T),\ \ l(t)=t,\ \forall t\in[0,T/4],\ \ l(t)=T-t,\
\forall t\in[3T/4,T].$ (2.17)

For any $\lambda\geq 1$ and for any $(t,x)\in Q$, we define the following
(positive) weight functions

$\eta(t,x)=\frac{\sigma(x)}{l^{4}(t)},\ \
\varphi(t,x)=\frac{e^{\lambda(2\|\eta^{0}\|_{L^{\infty}(\Omega)}+\eta^{0}(x))}}{l^{4}(t)},$
(2.18)

where

$\sigma(x)=e^{4\lambda\|\eta^{0}\|_{L^{\infty}(\Omega)}}-e^{\lambda(2\|\eta^{0}\|_{L^{\infty}(\Omega)}+\eta^{0}(x))}.$
(2.19)

In the sequel, we use the following notation defined as in [4]:

$\displaystyle\sigma^{+}$ $\displaystyle:=$
$\displaystyle\displaystyle\max_{x\in\overline{\Omega}}\sigma(x)=e^{4\lambda\|\eta^{0}\|_{L^{\infty}(\Omega)}}-e^{2\lambda\|\eta^{0}\|_{L^{\infty}(\Omega)}},$
(2.20a) $\displaystyle\sigma^{-}$ $\displaystyle:=$
$\displaystyle\displaystyle\min_{x\in\overline{\Omega}}\sigma(x)=e^{4\lambda\|\eta^{0}\|_{L^{\infty}(\Omega)}}-e^{3\lambda\|\eta^{0}\|_{L^{\infty}(\Omega)}}.$
(2.20b)

Using the definition of $\varphi$, $\eta$ and with a simple computation, we
show the following estimates

$\nabla\eta=-\nabla\varphi\leq C\lambda\varphi,\ \
|\eta_{t}|=|\varphi_{t}|\leq C(T)\varphi^{2},\ \ \varphi^{-1}\leq C(T).$
(2.21)

For $s\geq 1$, we introduce for a suitable function $z$, the following
notation to abridge the estimate

$\mathcal{M}(z)=s\lambda^{2}\int_{Q}e^{-2s\eta}\varphi|\nabla
z|^{2}dx\,dt+s^{3}\lambda^{4}\int_{Q}e^{-2s\eta}\varphi^{3}|z|^{2}dx\,dt.$
(2.22)

For any $F\in L^{2}(Q)$ and $z^{T}\in L^{2}(\Omega)$, we consider the
following system:

$\left\\{\begin{array}[]{rllll}\displaystyle-z_{t}-\Delta
z&=&F&\mbox{in}&Q,\\\ \displaystyle z&=&0&\mbox{on}&\Sigma,\\\ \displaystyle
z(T,\cdot)&=&z^{T}&\mbox{in}&\Omega.\end{array}\right.$ (2.23)

We state the classical Carleman estimate due to [14] for the solutions to the
heat equation (2.23).

###### Lemma 2.1

Let $\varphi$ and $\eta$ be defined by (2.18). Then, there exist positive
constants $s_{1}\geq 1$, $\lambda_{1}\geq 1$ and $C(\Omega,\omega)>0$ such
that, for any $\lambda\geq\lambda_{1}$, $s\geq s_{1}$, and for any $z$
solution of (2.23), we have

$\begin{array}[]{llll}\displaystyle\mathcal{M}(z)\leq\displaystyle
C(\Omega,\omega)\left(\int_{Q}e^{-2s\eta}|F|^{2}dx\,dt+s^{3}\lambda^{4}\int_{0}^{T}\int_{\omega^{\prime}}e^{-2s\eta}\varphi^{3}|z|^{2}dx\,dt\right).\end{array}$
(2.24)

###### Remark 3

If we make a change of variable $t$ for $T-t$ in (2.23), we have

$\left\\{\begin{array}[]{rllll}\displaystyle\frac{\partial{\tilde{z}}}{\partial{t}}-\Delta\tilde{z}&=&\widetilde{F}&\mbox{in}&Q,\\\
\displaystyle\tilde{z}&=&0&\mbox{on}&\Sigma,\\\
\displaystyle\tilde{z}(0,\cdot)&=&z^{T}&\mbox{in}&\Omega,\end{array}\right.$
(2.25)

where $\tilde{z}(t,x)=z(T-t,x)$ and $\widetilde{F}(t,x)=F(T-t,x)$. Then, the
global Carleman inequality (2.24) is also valid for any $\tilde{z}$ solution
of (2.25).

Now, we state the following result having a similar proof to that of the
Proposition $2.3$ [4].

###### Proposition 2.2

For any fixed $\lambda>0$ and $s>1$, the following inequality holds

$\exp\left(-\frac{(1+s)\sigma^{-}}{l^{4}(t)}\right)<\exp\left(-\frac{s\sigma^{+}}{l^{4}(t)}\right).$
(2.26)

###### Remark 4

In [4, 14], the authors used the function $l(t)=t(T-t)$ to prove Lemma 2.1 and
Proposition 2.2, respectively. Since in this paper we propose a positive
function $l(t)$ defined by (2.17) with the property of going to $0$ as $t$
tends to $0^{+}$ and $T^{-}$, the proof of Lemma 2.1 and Proposition 2.2
doesn’t change if we use the weight functions defined by (2.18) and the
function $l(t)$ defined by (2.17).

Now, using the estimate (2.24), we prove a new Carleman estimate for the
solutions to system (2.14)-(2.15).

###### Proposition 2.3

Assume that $\omega\cap\mathcal{O}_{d}\neq\emptyset$ and $\mu$ is large
enough. Then, there exists a positive constant $C=C(\Omega,\omega,T)>0$ such
that the solution $(\rho,\psi)$ to (2.14)-(2.15) satisfies

$\begin{array}[]{lll}\displaystyle\mathcal{M}(\rho)+\mathcal{M}(\psi)\leq\displaystyle
Cs^{7}\lambda^{8}\int_{0}^{T}\int_{\omega}e^{-2s\eta}\varphi^{7}|\rho|^{2}dx\,dt,\end{array}$
(2.27)

for every $s>0$ large enough.

Proof. We proceed in two steps.
Step 1. We prove that for $s>0$ sufficiently large, we have

$\begin{array}[]{lll}\displaystyle\mathcal{M}(\rho)+\mathcal{M}(\psi)\leq\displaystyle
C(\Omega,\omega)\left(s^{3}\lambda^{4}\int_{0}^{T}\int_{\omega^{\prime}}e^{-2s\eta}\varphi^{3}\left(|\rho|^{2}+|\psi|^{2}\right)dx\,dt\right).\end{array}$
(2.28)

We apply Carleman inequality (2.24) to each equation of (2.14) and (2.15) and
add them up, we have

$\begin{array}[]{lll}\mathcal{M}(\rho)+\mathcal{M}(\psi)&\leq&\displaystyle
C(\Omega,\omega)s^{3}\lambda^{4}\int_{0}^{T}\int_{\omega^{\prime}}e^{-2s\eta}\varphi^{3}\left(|\rho|^{2}+|\psi|^{2}\right)dx\,dt\\\
&+&\displaystyle
C(\Omega,\omega)\left(\int_{Q}e^{-2s\eta}|\psi|^{2}dx\,dt+\int_{Q}e^{-2s\eta}\left|-\frac{1}{\mu}\rho\chi_{\mathcal{O}}\right|^{2}dx\,dt\right)\\\
&+&\displaystyle
C(\Omega,\omega)\int_{Q}e^{-2s\eta}\left|\int_{\Omega}K(t,x,\theta)\rho(t,\theta)\
d\theta\right|^{2}dx\,dt\\\ &+&\displaystyle
C(\Omega,\omega)\int_{Q}e^{-2s\eta}\left|\int_{\Omega}K(t,x,\theta)\psi(t,\theta)\
d\theta\right|^{2}dx\,dt.\end{array}$

Taking the parameter $s$ large enough, we can absorb the second expression in
the right-hand side of above expression and we obtain

$\begin{array}[]{lll}\mathcal{M}(\rho)+\mathcal{M}(\psi)&\leq&\displaystyle
C(\Omega,\omega)s^{3}\lambda^{4}\int_{0}^{T}\int_{\omega^{\prime}}e^{-2s\eta}\varphi^{3}\left(|\rho|^{2}+|\psi|^{2}\right)dx\,dt\\\
&+&\displaystyle
C(\Omega,\omega)\int_{Q}e^{-2s\eta}\left|\int_{\Omega}K(t,x,\theta)\rho(t,\theta)\
d\theta\right|^{2}dx\,dt\\\ &+&\displaystyle
C(\Omega,\omega)\int_{Q}e^{-2s\eta}\left|\int_{\Omega}K(t,x,\theta)\psi(t,\theta)\
d\theta\right|^{2}dx\,dt.\end{array}$ (2.29)

Now, arguing as in [4], we estimate the two last terms in (2.29). We fix the
parameter $\lambda=\lambda_{1}$ sufficiently large. We have

$\begin{array}[]{lll}\displaystyle\left|\int_{\Omega}K(t,x,\theta)\rho(t,\theta)\
d\theta\right|&=&\displaystyle\left|\int_{\Omega}e^{\frac{\sigma^{-}}{l^{4}(t)}}K(t,x,\theta)e^{-\frac{\sigma^{-}}{l^{4}(t)}}\rho(t,\theta)\
d\theta\right|\\\
&\leq&\displaystyle\left[\left(\int_{\Omega}e^{\frac{2\sigma^{-}}{l^{4}(t)}}|K(t,x,\theta)|^{2}\
d\theta\right)\left(\int_{\Omega}e^{-\frac{2\sigma^{-}}{l^{4}(t)}}|\rho(t,\theta)|^{2}\
d\theta\right)\right]^{1/2}.\end{array}$

Using the above expression, one get

$\int_{Q}e^{-2s\eta}\left|\int_{\Omega}K(t,x,\theta)\rho(t,\theta)\
d\theta\right|^{2}dx\,dt\leq\mathcal{K}^{2}\int_{Q}e^{-2s\eta}\left(\int_{\Omega}e^{-\frac{2\sigma^{-}}{l^{4}(t)}}|\rho(t,\theta)|^{2}\
d\theta\right)\ dxdt.$ (2.30)

We proceed to estimate the term in the right-hand side of the above
inequality. We use Fubini’s theorem and we obtain

$\begin{array}[]{lll}\displaystyle\int_{Q}e^{-2s\eta}\left(\int_{\Omega}e^{-\frac{2\sigma^{-}}{l^{4}(t)}}|\rho(t,\theta)|^{2}\
d\theta\right)\
dxdt=\displaystyle\int_{Q}e^{-\frac{2\sigma^{-}}{l^{4}(t)}}|\rho(t,\theta)|^{2}\left(\int_{\Omega}e^{-2s\eta}\
dx\right)\ d\theta dt.\end{array}$

Using the definition of $\sigma^{-}$ given by (2.20b), we get

$\int_{\Omega}e^{-2s\eta}\ dx\leq|\Omega|e^{-\frac{2s\sigma^{-}}{l^{4}(t)}},$
(2.31)

where $|\Omega|$ denotes the measure of $\Omega$. Due to Proposition 2.2, the
definition of $\sigma^{+}$ given by (2.20a) and the estimate (2.31), we have

$\begin{array}[]{lll}\displaystyle\int_{Q}e^{-2s\eta}\left(\int_{\Omega}e^{-\frac{2\sigma^{-}}{l^{4}(t)}}|\rho(t,\theta)|^{2}\
d\theta\right)\
dxdt&\leq&C\displaystyle\int_{Q}e^{-\frac{2(1+s)\sigma^{-}}{l^{4}(t)}}|\rho(t,\theta)|^{2}\
d\theta\ dt\\\ &\leq&\displaystyle
C\displaystyle\int_{Q}e^{-\frac{2s\sigma^{+}}{l^{4}(t)}}|\rho(t,\theta)|^{2}\
d\theta\ dt\\\ &\leq&\displaystyle
C\displaystyle\int_{Q}e^{-2s\eta}|\rho(t,\theta)|^{2}\ d\theta\
dt.\end{array}$

Putting it all together, the expression (2.30) becomes

$\int_{Q}e^{-2s\eta}\left|\int_{\Omega}K(t,x,\theta)\rho(t,\theta)\
d\theta\right|^{2}dx\,dt\leq C\mathcal{K}^{2}\int_{Q}e^{-2s\eta}|\rho|^{2}\
dxdt.$ (2.32)

Proceeding as above, we prove the following estimate

$\int_{Q}e^{-2s\eta}\left|\int_{\Omega}K(t,x,\theta)\psi(t,\theta)\
d\theta\right|^{2}dx\,dt\leq C\mathcal{K}^{2}\int_{Q}e^{-2s\eta}|\psi|^{2}\
dxdt.$ (2.33)

Combining (2.32) and (2.33) and substituting in (2.29), then taking $s$
sufficiently large, we deduce estimate (2.28).
Step 2.  Now, we want to eliminate the local term corresponding to $\psi$ on
the right hand side of the estimate (2.28). We argue as the same as in [7].
Consider an open set $\omega_{0}$ such that
$\omega^{\prime}\subset\subset\omega_{0}\subset\subset\mathcal{O}_{d}\cap\omega$
and a function $\xi\in C^{\infty}_{0}(\Omega)$ satisfying

$\displaystyle\displaystyle 0\leq\xi\leq 1,\,\,\xi=1\hbox{ in
}\omega^{\prime},\,\,\xi=0\hbox{ in }\Omega\setminus\omega_{0},$ (2.34a)
$\displaystyle\displaystyle\frac{\Delta\xi}{\xi^{1/2}}\in
L^{\infty}(\omega_{0}),\,\,\frac{\nabla\xi}{\xi^{1/2}}\in[L^{\infty}(\omega_{0})]^{N}.$
(2.34b)

We define $u=s^{3}\lambda^{4}\varphi^{3}e^{-2s\eta}$. Using the definition of
$\varphi$ and $\eta$, we get $u(T)=u(0)=0$. Due to estimates (2.21), the
following estimations hold:

$\begin{array}[]{rll}\displaystyle|u\xi|\leq
s^{3}\lambda^{4}\varphi^{3}e^{2s\eta}\xi,\ \ \ \ \ \
\displaystyle\left|(u\xi)_{t}\right|\leq
C(T)s^{4}\lambda^{4}\varphi^{5}e^{2s\eta}\xi,\\\ \\\
\displaystyle|\nabla(u\xi)|\leq Cs^{4}\lambda^{5}\varphi^{4}e^{2s\eta}\xi,\ \
\ \ \ \ \displaystyle|\Delta(u\xi)|\leq
Cs^{5}\lambda^{6}\varphi^{5}e^{2s\eta}\xi,\end{array}$ (2.35)

where $C$ is a positive constant.
Multiplying the first equation of (2.14) by $u\xi\psi$, integrating by parts
over $Q$ and using Fubini’s theorem, we obtain

$J_{1}+J_{2}+J_{3}+J_{4}=\int_{Q}u\xi|\psi|^{2}\chi_{\mathcal{O}_{d}}\
dx\,dt,$ (2.36)

where

$\begin{array}[]{lllll}J_{1}=\displaystyle-\frac{1}{\mu}\int_{Q}u\xi|\rho|^{2}\chi_{\mathcal{O}}\
dx\,dt\ dx\,dt,\,J_{2}=\int_{Q}\rho\psi(u\xi)_{t}\ dx\,dt,\\\
J_{3}=\displaystyle-2\int_{Q}\rho\nabla\psi.\nabla(u\xi)\ dx\,dt,\
\displaystyle J_{4}=\displaystyle-\int_{Q}\rho\psi\Delta(u\xi)\
dx\,dt.\end{array}$

Let us estimate each $J_{i},\ i=1,\cdots,4$. From the Hölder’s and Young’s
inequalities and estimates (2.35), we have

$J_{1}\leq\displaystyle\frac{1}{\mu}\int_{Q}s^{3}\lambda^{4}\varphi^{3}e^{-2s\eta}|\rho|^{2}\
dx\,dt.$
$J_{2}\leq\frac{\delta_{1}}{2}\int_{Q}s^{3}\lambda^{4}\varphi^{3}e^{-2s\eta}|\psi|^{2}dx\,dt+\frac{C(T)}{\delta_{1}}\int_{0}^{T}\int_{\omega_{0}}s^{5}\lambda^{4}\varphi^{7}e^{-2s\eta}|\rho|^{2}dx\,dt,$

for any $\delta_{1}>0$.

$J_{3}\leq\frac{\delta_{2}}{2}\int_{Q}s\lambda^{2}\varphi
e^{-2s\eta}|\nabla\psi|^{2}dx\,dt+\frac{C}{\delta_{2}}\int_{0}^{T}\int_{\omega_{0}}s^{7}\lambda^{8}\varphi^{7}e^{-2s\eta}|\rho|^{2}dx\,dt,$

for any $\delta_{2}>0$.

$J_{4}\leq\frac{\delta_{3}}{2}\int_{Q}s^{3}\lambda^{4}\varphi^{3}e^{-2s\eta}|\psi|^{2}dx\,dt+\frac{C}{\delta_{3}}\int_{0}^{T}\int_{\omega_{0}}s^{7}\lambda^{8}\varphi^{7}e^{-2s\eta}|\rho|^{2}dx\,dt,$

for any $\delta_{3}>0$.

Finally, choosing the constants $\delta_{i}$ such that
$\displaystyle\delta_{2}=\frac{1}{C(\Omega,\omega)}$ and
$\displaystyle\delta_{1}=\delta_{3}=\frac{1}{2C(\Omega,\omega)}$, where
$C(\Omega,\omega)$ is the constant obtained in (2.28), it follows from (2.36),
the previous inequalities and (2.35) that

$\begin{array}[]{rll}\displaystyle\int_{0}^{T}\int_{\omega^{\prime}}s^{3}\lambda^{4}\varphi^{3}e^{-2s\eta}|\psi|^{2}dx\,dt\displaystyle&\leq&\displaystyle\frac{1}{2}\mathcal{M}(\psi)+\frac{1}{\mu}\int_{Q}s^{3}\lambda^{4}\varphi^{3}e^{-2s\eta}|\rho|^{2}\
dx\,dt\\\ &+&\displaystyle
C(\Omega,\omega,T)\int_{0}^{T}\int_{\omega_{0}}s^{7}\lambda^{8}\varphi^{7}e^{-2s\eta}|\rho|^{2}\
dx\,dt.\end{array}$ (2.37)

Combining (2.28) with (2.37), we deduce that

$\begin{array}[]{lll}\displaystyle\mathcal{M}(\rho)+\mathcal{M}(\psi)\leq\displaystyle
C(\Omega,\omega,T)\int_{0}^{T}\int_{\omega_{0}}s^{7}\lambda^{8}\varphi^{7}e^{-2s\eta}|\rho|^{2}\
dx\,dt+\frac{1}{\mu}\int_{Q}s^{3}\lambda^{4}\varphi^{3}e^{-2s\eta}|\rho|^{2}\
dx\,dt.\end{array}$

Since $\mu$ is sufficiently large, we can absorb the last term of the above
inequality in the left hand side. Using the fact that
$\omega_{0}\subset\omega$, we deduce (2.27).

###  2.3 Observability and null controllability

To prove the needed observability inequality, we are going to improve the
Carleman inequality (2.27). To this end, we introduce some new weight
functions, similar to the ones in (2.18), that do not vanish at $t=0$

Let us consider firstly the function

$\bar{l}(t)=\left\\{\begin{array}[]{lllll}\displaystyle\|l\|_{L^{\infty}((0,T))}\
\ \mbox{for}\ \ 0\leq t\leq T/2,\\\ \\\ \displaystyle l(t)\ \ \mbox{for}\ \
T/2\leq t\leq T,\end{array}\right.$ (2.38)

and the functions

$\begin{array}[]{rll}&&\displaystyle\alpha(t,x)=\frac{e^{4\lambda\|\eta^{0}\|_{L^{\infty}(\Omega)}}-e^{\lambda(2\|\eta^{0}\|_{L^{\infty}(\Omega)}+\eta^{0}(x))}}{\bar{l}^{4}(t)},\
\
\zeta(t,x)=\frac{e^{\lambda(2\|\eta^{0}\|_{L^{\infty}(\Omega)}+\eta^{0}(x))}}{\bar{l}^{4}(t)},\\\
\\\
&&\displaystyle\alpha^{*}(t)=\displaystyle\max_{x\in\overline{\Omega}}\alpha(t,x),\
\ \
\zeta^{*}(t)=\displaystyle\min_{x\in\overline{\Omega}}\zeta(t,x).\end{array}$
(2.39)

With these definitions, we have the following result.

###### Proposition 2.4

Under the assumptions of Proposition 2.3, there exist a positive constant
$C=C(\Omega,\omega,\|K\|_{\infty},T)$ and a positive weight function
$\varpi_{1}$ such that

$\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{Q}\frac{1}{\varpi_{1}^{2}}|\psi|^{2}\,dx\,dt\leq
C\int_{0}^{T}\int_{\omega}|\rho|^{2}\,dx\,dt,$ (2.40)

where $(\rho,\psi)$ is the solution of (2.14)-(2.15).

Proof. We fix the parameter $s$ large enough and we proceed in two steps.
Step 1. By construction, $\eta=\alpha$ and $\varphi=\zeta$ in
$[T/2,T]\times\Omega$, hence, we obtain

$\begin{array}[]{llll}\displaystyle\int_{T/2}^{T}\int_{\Omega}e^{-2s\alpha}\zeta^{3}|\rho|^{2}\
dxdt+\int_{T/2}^{T}\int_{\Omega}e^{-2s\alpha}\zeta^{3}|\psi|^{2}\ dxdt\\\
=\displaystyle\int_{T/2}^{T}\int_{\Omega}e^{-2s\eta}\varphi^{3}|\rho|^{2}\
dxdt+\int_{T/2}^{T}\int_{\Omega}e^{-2s\eta}\varphi^{3}|\psi|^{2}\ dxdt\\\
\leq\displaystyle
C(\Omega,\omega,T)\int_{0}^{T}\int_{\omega}e^{-2s\eta}\varphi^{7}|\rho|^{2}\,dx\,dt,\end{array}$
(2.41)

where we have used the Carleman estimate (2.27).
Step 2. On the other hand, we use the classical energy estimate for the system
(2.14) on the domain $(0,T/2)\times\Omega$. Let us introduce a function
$\beta\in\mathcal{C}^{1}([0,T])$ such that

$0\leq\beta\leq 1,\ \beta(t)=1\hbox{ for }t\in[0,T/2],\ \beta(t)=0\hbox{ for
}t\in[3T/4,T],\ |\beta^{\prime}(t)|\leq C/T.$ (2.42)

For any $(t,x)\in Q$, we set

$\begin{array}[]{lll}z(t,x)=\beta(t)e^{-r(T-t)}\rho(t,x),\end{array}$

where $r>0$. Then in view of (2.14), the function $z$ is solution of

$\left\\{\begin{array}[]{rllll}\displaystyle-z_{t}-\Delta
z+\int_{\Omega}K(t,x,\theta)z(t,\theta)\ d\theta+rz&=&\displaystyle\beta
e^{-r(T-t)}\psi\chi_{\mathcal{O}_{d}}-\beta^{\prime}e^{-r(T-t)}\rho&\mbox{in}&Q,\\\
\displaystyle z&=&0&\mbox{on}&\Sigma,\\\ \displaystyle
z(T,\cdot)&=&0&\mbox{in}&\Omega.\end{array}\right.$ (2.43)

From the classical energy estimate for the solution $z$ to system (2.43) and
using the definition of $\beta$ and $z$, we obtain

$\begin{array}[]{llll}\displaystyle\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T/2}\int_{\Omega}|\rho|^{2}\
dxdt\displaystyle\leq
C(\|K\|_{\infty},T)\left(\int_{0}^{3T/4}\int_{\Omega}|\psi|^{2}\
dx\,dt+\int_{T/2}^{3T/4}\int_{\Omega}|\rho|^{2}\ dx\,dt\right).\end{array}$

Adding the term $\displaystyle\int_{0}^{T/2}\int_{\Omega}|\psi|^{2}\ dx\,dt$
on both sides of above estimate, we have

$\begin{array}[]{llll}\displaystyle\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T/2}\int_{\Omega}|\rho|^{2}\
dxdt+\int_{0}^{T/2}\int_{\Omega}|\psi|^{2}\ dx\,dt\\\ \displaystyle\leq
C(\|K\|_{\infty},T)\left(\int_{T/2}^{3T/4}\int_{\Omega}|\psi|^{2}\
dx\,dt+\int_{T/2}^{3T/4}\int_{\Omega}|\rho|^{2}\
dx\,dt+\int_{0}^{T/2}\int_{\Omega}|\psi|^{2}\ dx\,dt\right).\end{array}$
(2.44)

In order to eliminate the last term in the right hand side of (2.44), we use
the classical energy estimates for the system (2.15) and we obtain:

$\begin{array}[]{llll}\displaystyle\int_{0}^{T/2}\int_{\Omega}|\psi|^{2}\
dxdt\displaystyle\leq\frac{1}{\mu^{2}}C(\|K\|_{\infty},T)\int_{0}^{T/2}\int_{\Omega}|\rho|^{2}\
dxdt.\end{array}$ (2.45)

Replacing (2.45) in (2.44) and taking $\mu$ large enough, we obtain

$\begin{array}[]{llll}\displaystyle\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T/2}\int_{\Omega}|\rho|^{2}\
dxdt+\int_{0}^{T/2}\int_{\Omega}|\psi|^{2}\ dx\,dt\\\ \displaystyle\leq
C(\|K\|_{\infty},T)\left(\int_{T/2}^{3T/4}\int_{\Omega}|\psi|^{2}\
dx\,dt+\int_{T/2}^{3T/4}\int_{\Omega}|\rho|^{2}\ dx\,dt\right).\end{array}$

Using the definition (2.39), the weight functions $\alpha$ and $\zeta$ have
lower and upper bounds for $(t,x)\in[0,3T/4]$, then we obtain

$\begin{array}[]{llll}\displaystyle\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T/2}\int_{\Omega}e^{-2s\alpha}\zeta^{3}|\rho|^{2}\
dxdt+\int_{0}^{T/2}\int_{\Omega}e^{-2s\alpha}\zeta^{3}|\psi|^{2}\ dx\,dt\\\
\displaystyle\leq
C(\|K\|_{\infty},T)\left(\int_{T/2}^{3T/4}\int_{\Omega}e^{-2s\alpha}\zeta^{3}|\psi|^{2}\
dx\,dt+\int_{T/2}^{3T/4}\int_{\Omega}e^{-2s\alpha}\zeta^{3}|\rho|^{2}\
dx\,dt\right).\end{array}$ (2.46)

Using estimate (2.41), the previous estimate becomes

$\begin{array}[]{llll}\displaystyle\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T/2}\int_{\Omega}e^{-2s\alpha}\zeta^{3}|\rho|^{2}\
dxdt+\int_{0}^{T/2}\int_{\Omega}e^{-2s\alpha}\zeta^{3}|\psi|^{2}\ dx\,dt\\\
\displaystyle\leq
C(\Omega,\omega,\|K\|_{\infty},T)\int_{0}^{T}\int_{\omega}e^{-2s\eta}\varphi^{7}|\rho|^{2}\,dx\,dt.\end{array}$
(2.47)

Adding estimates (2.41) and (2.47), and using the definitions of $\alpha^{*}$
and $\zeta^{*}$ given by (2.39), we have

$\begin{array}[]{llll}\displaystyle\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T}\int_{\Omega}e^{-2s\alpha^{*}}(\zeta^{*})^{3}|\rho|^{2}\
dxdt+\int_{0}^{T}\int_{\Omega}e^{-2s\alpha^{*}}(\zeta^{*})^{3}|\psi|^{2}\
dx\,dt\\\ \displaystyle\leq
C(\Omega,\omega,\|K\|_{\infty},T)\int_{0}^{T}\int_{\omega}e^{-2s\eta}\varphi^{7}|\rho|^{2}\,dx\,dt.\end{array}$
(2.48)

We obtain the observability inequality (2.40) due to the fact that
$\displaystyle e^{-2s\eta}\varphi^{7}\in L^{\infty}(Q)$ and taking

$\varpi_{1}(t)=e^{s\alpha^{*}}(\zeta^{*})^{-3/2}.$ (2.49)

Now, we end the proof of Theorem 1.1 by proving the null controllability of
system (2.11)-(2.12). Thanks to the Proposition 2.4, the following result
holds.

###### Proposition 2.5

Assume that $\mathcal{O}_{d}\cap\omega\neq\emptyset$. Let $\varpi_{1}$ be
defined as in Proposition 2.4. Let also $z^{0}\in L^{2}(\Omega)$ and $z_{d}\in
L^{2}((0,T)\times\mathcal{O}_{d})$ such that (1.14) holds. Then, there exists
a leader control $\hat{f}\in L^{2}((0,T)\times\omega)$ such that the
associated solution $(z,p)$ to (2.11)-(2.12) satisfies (1.7). Furthermore;
there exists a constant $C=C(\Omega,\omega,\|K\|_{\infty},T)$ such that

$\begin{array}[]{ccc}\displaystyle\|\hat{f}\|_{L^{2}((0,T)\times\omega)}\leq
C\displaystyle\left(\|\varpi_{1}z_{d}\|_{L^{2}((0,T)\times\mathcal{O}_{d})}+\|y^{0}\|_{L^{2}(\Omega)}\right).\end{array}$
(2.50)

Proof. Let $\varepsilon>0$ and consider the following cost function

$J_{\varepsilon}(f)=\displaystyle\frac{1}{2\varepsilon}\int_{\Omega}|z(T,x)|^{2}\
dx+\frac{1}{2}\int_{0}^{T}\int_{\omega}|f|^{2}\ dx\,dt,$ (2.51)

where $z$ is the solution to system (2.11)-(2.12). Then, we consider the
optimal control problem:

$\inf_{\atop f\in L^{2}((0,T)\times\omega)}J_{\varepsilon}(f).$ (2.52)

We can prove that $J_{\varepsilon}$ is continuous, coercive and strictly
convex. Then, the optimization problem (2.52) admits a unique solution
$f_{\varepsilon}$ and arguing as in [8], we prove that

$f_{\varepsilon}=\rho_{\varepsilon}\ \ \mbox{in}\ \ (0,T)\times\omega,$ (2.53)

with $(\rho_{\varepsilon},\psi_{\varepsilon})$ solution of the following
systems

$\left\\{\begin{array}[]{rllll}\displaystyle-\rho_{\varepsilon,t}-\Delta\rho_{\varepsilon}+\int_{\Omega}K(t,x,\theta)\rho_{\varepsilon}(t,\theta)\
d\theta&=&\displaystyle\psi_{\varepsilon}\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\
\displaystyle\rho_{\varepsilon}&=&0&\mbox{on}&\Sigma,\\\
\displaystyle\rho_{\varepsilon}(T,\cdot)&=&-\displaystyle\frac{1}{\varepsilon}z_{\varepsilon}(T,\cdot)&\mbox{in}&\Omega\end{array}\right.$
(2.54)

and

$\left\\{\begin{array}[]{rllll}\displaystyle\psi_{\varepsilon,t}-\Delta\psi_{\varepsilon}+\int_{\Omega}K(t,x,\theta)\psi_{\varepsilon}(t,\theta)\
d\theta&=&\displaystyle-\frac{1}{\mu}\rho_{\varepsilon}\chi_{\mathcal{O}}&\mbox{in}&Q,\\\
\displaystyle\psi_{\varepsilon}&=&0&\mbox{on}&\Sigma,\\\
\displaystyle\psi_{\varepsilon}(0,\cdot)&=&0&\mbox{in}&\Omega,\end{array}\right.$
(2.55)

where $(z_{\varepsilon},p_{\varepsilon})$ is the solution of system
(2.11)-(2.12) associated to the control $f_{\varepsilon}$.

Multiplying the first equation of (2.54) and (2.55) by $z_{\varepsilon}$ and
$p_{\varepsilon}$ respectively and integrating by parts over $Q$, we obtain
from (2.53) and the Young’s inequality

$\begin{array}[]{llll}\displaystyle\|f_{\varepsilon}\|^{2}_{L^{2}((0,T)\times\omega)}+\frac{1}{\varepsilon}\|z_{\varepsilon}(T,\cdot)\|^{2}_{L^{2}(\Omega)}\\\
=\displaystyle-\int_{\Omega}y^{0}(x)\rho_{\varepsilon}(0,x)\
dx+\int_{0}^{T}\int_{\mathcal{O}_{d}}z_{d}\psi_{\varepsilon}\ dx\,dt\\\
\leq\displaystyle\left(\left\|\varpi_{1}z_{d}\right\|^{2}_{L^{2}((0,T)\times\mathcal{O}_{d})}+\|z^{0}\|^{2}_{L^{2}(\Omega)}\right)^{1/2}\times\left(\left\|\varpi_{1}^{-1}\psi_{\varepsilon}\right\|^{2}_{L^{2}(Q)}+\|\rho_{\varepsilon}(0,\cdot)\|^{2}_{L^{2}(\Omega)}\right)^{1/2}.\end{array}$
(2.56)

Using the observability inequality (2.40), we deduce from (2.56) the existence
of a constant $C=C(\Omega,\omega,\|K\|_{\infty},T)$ such that

$\begin{array}[]{rll}\displaystyle\|f_{\varepsilon}\|_{L^{2}((0,T)\times\omega)}\leq
C\left(\left\|\varpi_{1}z_{d}\right\|_{L^{2}((0,T)\times\mathcal{O}_{d})}+\|z^{0}\|_{L^{2}(\Omega)}\right)\end{array}$
(2.57)

and

$\begin{array}[]{rll}\displaystyle\|z_{\varepsilon}(T,\cdot)\|_{L^{2}(\Omega)}\leq\displaystyle
C\sqrt{\varepsilon}\left(\left\|\varpi_{1}z_{d}\right\|_{L^{2}((0,T)\times\mathcal{O}_{d})}+\|z^{0}\|_{L^{2}(\Omega)}\right).\end{array}$
(2.58)

Using (2.57)-(2.58) and classical energy estimates to systems (2.11)-(2.12)
associated to the control $f_{\varepsilon}$, we can extract subsequences still
denoted by $f_{\varepsilon},\ z_{\varepsilon}$ and $p_{\varepsilon}$ such that
when $\varepsilon\rightarrow 0$, one has

$\displaystyle f_{\varepsilon}$ $\displaystyle\rightharpoonup$
$\displaystyle\hat{f}$ weakly in $\displaystyle L^{2}((0,T)\times\omega),$
(2.59a) $\displaystyle z_{\varepsilon}$ $\displaystyle\rightharpoonup$
$\displaystyle y$ weakly in $\displaystyle L^{2}((0,T);H^{1}_{0}(\Omega)),$
(2.59b) $\displaystyle p_{\varepsilon}$ $\displaystyle\rightharpoonup$
$\displaystyle p$ weakly in $\displaystyle L^{2}((0,T);H^{1}_{0}(\Omega)),$
(2.59c) $\displaystyle z_{\varepsilon}(T,\cdot)$
$\displaystyle\longrightarrow$ $\displaystyle 0$ strongly in $\displaystyle\
L^{2}(\Omega).$ (2.59d)

Arguing as in [8, 29] while using convergences (2.59), we prove that $(z,p)$
is a solution of (2.11)-(2.12) corresponding to the control $\hat{f}$ and also
$z$ satisfies (1.7). Furthermore, using the convergence (2.59a), we have that
$\hat{f}$ satisfies (2.50).

###  2.4 The semilinear case

In this section, we apply the Stackelberg strategy for the following nonlocal
semilinear system

$\left\\{\begin{array}[]{rllll}\displaystyle y_{t}-\Delta
y+\int_{\Omega}K(t,x,\theta)y(t,\theta)\
d\theta&=&G(y)+f\chi_{\omega}+v\chi_{\mathcal{O}}&\mbox{in}&Q,\\\
\displaystyle y&=&0&\mbox{on}&\Sigma,\\\ \displaystyle
y(0,\cdot)&=&y^{0}&\mbox{in}&\Omega,\end{array}\right.$ (2.60)

where $G$ is a globally Lipschitz continuous function.

In the semilinear case, the convexity of the functional $J$ defined by (1.3)
is not guaranteed and we loose the uniqueness of the optimal control
$\hat{v}$. In the following proposition, we prove that if $\mu$ is
sufficiently large, then the functional (1.3) is indeed strictly convex for
small data.

###### Proposition 2.6

Let us assume that $y_{d}\in L^{\infty}((0,T);\mathcal{O}_{d})$, $f\in
L^{2}((0,T)\times\omega)$ are fixed and $\mu$ is sufficiently large. Suppose
that $y^{0}\in L^{2}(\Omega)$, $N\leq 12$, $G\in\mathcal{C}^{2}(\mathbb{R})$
and there exists a constant $L>0$ such that
$\displaystyle|G^{\prime}(s)|+|G^{\prime\prime}(s)|\leq L,\ \forall
s\in\mathbb{R}$. Then, if $\hat{v}$ satisfies (1.5), there exists a constant
$C>0$ independent of $\mu$ such that

$\displaystyle D^{2}J(f;\hat{v})\cdot(w^{1},w^{1})\geq
C\|w^{1}\|^{2}_{L^{2}((0,T)\times\mathcal{O})},\ \forall w^{1}\in
L^{2}((0,T)\times\mathcal{O}),\ w^{1}\neq 0.$ (2.61)

In particular, the functional $J$ is strictly convex in $\hat{v}$.

Proof. The technique of the proof is inspired by [2, Proposition 1.4].

Let $f\in L^{2}((0,T)\times\omega)$ be given and let $\hat{v}$ be the
associated optimal control solution of (1.4). For any $w^{1},w^{2}\in
L^{2}((0,T)\times\mathcal{O})$ and $s\in\mathbb{R}$, we denote by $y^{s}$ the
solution of the following system

$\left\\{\begin{array}[]{rllll}\displaystyle y^{s}_{t}-\Delta
y^{s}+\int_{\Omega}K(t,x,\theta)y^{s}(t,\theta)\
d\theta&=&G(y^{s})+f\chi_{\omega}+(\hat{v}+sw^{1})\chi_{\mathcal{O}}&\mbox{in}&Q,\\\
\displaystyle y^{s}&=&0&\mbox{on}&\Sigma,\\\ \displaystyle
y^{s}(0,\cdot)&=&y^{0}&\mbox{in}&\Omega,\end{array}\right.$ (2.62)

and we set $y:=y^{s}|_{s=0}$.

Then, we have

$\begin{array}[]{llll}&&\displaystyle DJ(f;\hat{v}+sw^{1})\cdot
w^{2}-DJ(f;\hat{v})\cdot w^{2}=s\mu\int_{0}^{T}\int_{\mathcal{O}}w^{1}w^{2}\
dxdt\\\ &&\displaystyle+\int_{0}^{T}\int_{\mathcal{O}_{d}}(y^{s}-y_{d})z^{s}\
dxdt-\int_{0}^{T}\int_{\mathcal{O}_{d}}(y-y_{d})z\ dxdt,\\\ \end{array}$
(2.63)

where $z^{s}$ is the derivative of $y^{s}$ with respect to $\hat{v}$ in the
direction $w^{2}$, that is $z^{s}$ is the solution to

$\left\\{\begin{array}[]{rllll}\displaystyle z_{t}^{s}-\Delta
z^{s}+\int_{\Omega}K(t,x,\theta)z^{s}(t,\theta)\
d\theta&=&G^{\prime}(y^{s})+w^{2}\chi_{\mathcal{O}}&\mbox{in}&Q,\\\
\displaystyle z^{s}&=&0&\mbox{on}&\Sigma,\\\ \displaystyle
z^{s}(0,\cdot)&=&0&\mbox{in}&\Omega.\end{array}\right.$ (2.64)

We will also use the notation $z:=z^{s}|_{s=0}$.

Let us introduce the adjoint system of (2.64)

$\left\\{\begin{array}[]{rllll}\displaystyle-p_{t}^{s}-\Delta
p^{s}+\int_{\Omega}K(t,x,\theta)p^{s}(t,\theta)\
d\theta&=&G^{\prime}(y^{s})p^{s}+\left(y^{s}-y_{d}\right)\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\
\displaystyle p^{s}&=&0&\mbox{on}&\Sigma,\\\ \displaystyle
p^{s}(T,\cdot)&=&0&\mbox{in}&\Omega\end{array}\right.$ (2.65)

and let us use the notation $p:=p^{s}|_{s=0}$.
Multiplying the first equation of (2.64) by $p^{s}$ solution of (2.65) and
integrating by parts over $Q$, we obtain

$\int_{0}^{T}\int_{\mathcal{O}_{d}}(y^{s}-y_{d})z^{s}\
dxdt=\int_{Q}w^{2}p^{s}\chi_{\mathcal{O}}\ dxdt.$ (2.66)

From (2.63) and (2.66), we have

$\begin{array}[]{llll}\displaystyle DJ(f;\hat{v}+sw^{1})\cdot
w^{2}-DJ(f;\hat{v})\cdot w^{2}&=&\displaystyle
s\mu\int_{0}^{T}\int_{\mathcal{O}}w^{1}w^{2}\ dxdt\\\
\displaystyle&+&\displaystyle\int_{0}^{T}\int_{\mathcal{O}}(p^{s}-p)w^{2}\
dxdt.\end{array}$ (2.67)

Notice that

$\displaystyle\displaystyle-\frac{\partial}{\partial
t}(p^{s}-p)-\Delta(p^{s}-p)+\int_{\Omega}K(t,x,\theta)(p^{s}(t,\theta)-p^{s}(t,\theta))\
d\theta$
$\displaystyle\displaystyle=\left[G^{\prime}(y^{s})-G^{\prime}(y)\right]p^{s}+G^{\prime}(y)(p^{s}-p)+(y^{s}-y)\chi_{\mathcal{O}_{d}}$

and

$\displaystyle\displaystyle\frac{\partial}{\partial
t}(y^{s}-y)-\Delta(y^{s}-y)+\int_{\Omega}K(t,x,\theta)(y^{s}(t,\theta)-y^{s}(t,\theta))\
d\theta=\left[G(y^{s})-G(y)\right]+sw^{1}\chi_{\mathcal{O}}.$

Consequently, the following limits exist

$\eta=\lim\limits_{s\to 0}\frac{1}{s}(p^{s}-p)\ \ \mbox{and}\ \
\phi=\lim\limits_{s\to 0}\frac{1}{s}(y^{s}-y)$

and verify the following systems:

$\left\\{\begin{array}[]{rllll}\displaystyle\phi_{t}-\Delta\phi+\int_{\Omega}K(t,x,\theta)\phi(t,\theta)\
d\theta&=&G^{\prime}(y)\phi+w^{1}\chi_{\mathcal{O}}&\mbox{in}&Q,\\\
\displaystyle\phi&=&0&\mbox{on}&\Sigma,\\\
\displaystyle\phi(0,\cdot)&=&0&\mbox{in}&\Omega\end{array}\right.$ (2.68)

and

$\left\\{\begin{array}[]{rllll}\displaystyle-\eta_{t}-\Delta\eta+\int_{\Omega}K(t,x,\theta)\eta(t,\theta)\
d\theta&=&G^{\prime}(y)\eta+G^{\prime\prime}(y)\phi
p+\phi\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\
\displaystyle\eta&=&0&\mbox{on}&\Sigma,\\\
\displaystyle\eta(T,\cdot)&=&0&\mbox{in}&\Omega.\end{array}\right.$ (2.69)

Thus, from (2.67)-(2.69) with $w^{2}=w^{1}$, we have

$\displaystyle
D^{2}J(f;\hat{v})\cdot(w^{1},w^{1})=\int_{0}^{T}\int_{\mathcal{O}}\eta w^{1}\
dxdt+\mu\int_{0}^{T}\int_{\mathcal{O}}|w^{1}|^{2}\ dxdt.$ (2.70)

Now, we want to show that, for some constant $C>0$ independent of $f,\ \eta,\
\phi,\ w^{1}$ and $\mu$, one has

$\left|\int_{0}^{T}\int_{\mathcal{O}}\eta w^{1}\ dxdt\right|\leq
C\|w^{1}\|_{L^{2}((0,T)\times\mathcal{O})}.$ (2.71)

Indeed, given $w_{1}\in L^{2}((0,T)\times\mathcal{O})$ and since
$G^{\prime}\in L^{\infty}(Q)$, then, we have the following energy estimate
associated to (2.68)

$\|\phi\|^{2}_{L^{2}(Q)}+\|\nabla\phi\|^{2}_{L^{2}(Q)}\leq
C\|w^{1}\|_{L^{2}((0,T)\times\mathcal{O})}.$ (2.72)

Using systems (2.68)-(2.69), we have

$\begin{array}[]{llll}\displaystyle\int_{0}^{T}\int_{\mathcal{O}}\eta w^{1}\
dxdt&=&\displaystyle\int_{Q}\eta\left(\phi_{t}-\Delta\phi+\int_{\Omega}K(t,x,\theta)\phi(t,\theta)\
d\theta-G^{\prime}(y)\phi\right)\ dxdt\\\
&=&\displaystyle\int_{Q}\phi\left(-\eta_{t}-\Delta\eta+\int_{\Omega}K(t,x,\theta)\eta(t,\theta)\
d\theta-G^{\prime}(y)\eta\right)\ dxdt\\\
&=&\displaystyle\int_{Q}\phi\left(G^{\prime\prime}(y)\phi
p+\phi\chi_{\mathcal{O}_{d}}\right)\ dxdt\\\
&=&\displaystyle\int_{Q}G^{\prime\prime}(y)|\phi|^{2}p\
dxdt+\int_{Q}|\phi|^{2}\chi_{\mathcal{O}_{d}}\ dxdt.\end{array}$ (2.73)

Using Hölder inequality in the last expression of (2.73) and the fact that
$G^{\prime\prime}\in L^{\infty}(Q)$, we get

$\begin{array}[]{llll}\displaystyle\left|\int_{0}^{T}\int_{\mathcal{O}}\eta
w^{1}\
dxdt\right|\leq\displaystyle\|G^{\prime\prime}\|_{L^{\infty}(Q)}\|\phi\|^{2}_{L^{2r^{\prime}}((0,T);L^{2s^{\prime}}(\Omega))}\|p\|_{L^{r}((0,T);L^{s}(\Omega))}\displaystyle+\|\phi\|^{2}_{L^{2}((0,T)\times\mathcal{O}_{d})},\end{array}$
(2.74)

where $r^{\prime}$ and $s^{\prime}$ are the conjugate of $r$ and $s$,
respectively. To bound the right-hand side of this latter inequality, the idea
is to find $r$ and $s$ such that

$p\in L^{r}((0,T);L^{s}(\Omega)),\ \ \phi\in
L^{2r^{\prime}}((0,T);L^{2s^{\prime}}(\Omega)).$

First, we have that $\phi\in L^{2}((0,T);H^{2}(\Omega))\cap
L^{\infty}((0,T);H^{1}(\Omega))$. It is reasonable to look for which values of
$d$ and $b$ the following embedding holds:

$L^{2}((0,T);H^{2}(\Omega))\cap L^{\infty}((0,T);H^{1}(\Omega))\hookrightarrow
L^{d}((0,T);L^{b}(\Omega)).$ (2.75)

Using interpolation results, we deduce that

$\frac{1}{d}=\frac{\theta}{2},\ \ 0<\theta<1.$ (2.76)

From Sobolev embedding results, we have

$\displaystyle\displaystyle H^{2}(\Omega)$ $\displaystyle\hookrightarrow$
$\displaystyle\displaystyle L^{\frac{2N}{N-4}}(\Omega),$ (2.77a)
$\displaystyle\displaystyle H^{1}(\Omega)$ $\displaystyle\hookrightarrow$
$\displaystyle\displaystyle L^{\frac{2N}{N-2}}(\Omega).$ (2.77b)

Then, the space $L^{b}(\Omega)$ is an intermediate space with respect to
(2.77a) and (2.77b) if

$\frac{1}{b}=\frac{(N-4)\theta}{2N}+\frac{(N-2)(1-\theta)}{2N},\ \
0<\theta<1.$ (2.78)

Taking $d=2r^{\prime}$ and $b=2s^{\prime}$, it follows that appropriate values
of $r$ and $s$ are

$r=\frac{d}{d-2}\ \ \mbox{and}\ \ s=\frac{dN}{2(d+2)}.$

On the other hand, since $f\in L^{2}((0,T)\times\omega)$, $v\in
L^{2}((0,T)\times\mathcal{O})$ and $y^{0}\in L^{2}(\Omega)$, we have that

$y\in L^{2}((0,T);H^{1}(\Omega))\cap
L^{\infty}((0,T);L^{2}(\Omega))\hookrightarrow
L^{\bar{d}}((0,T);L^{\bar{b}}(\Omega)).$ (2.79)

Using the interpolation argument, we obtain that

$\bar{b}=\frac{2\bar{d}N}{\bar{d}N-4}.$

From parabolic regularity, we have

$\phi\in L^{\bar{d}}((0,T);W^{2,\bar{b}}(\Omega))\hookrightarrow
L^{\bar{d}}((0,T);L^{\frac{N\bar{b}}{N-2\bar{b}}}(\Omega))=L^{\bar{d}}((0,T);L^{\frac{2\bar{d}N}{\bar{d}(N-4)-4}}(\Omega)).$
(2.80)

Taking, $\bar{d}=r$, it follows that $\displaystyle\phi\in
L^{\bar{d}}((0,T);L^{\frac{2dN}{d(N-8)+8}}(\Omega))$ and, in order to have
$\phi\in L^{r}((0,T);L^{s}(\Omega))$, we need

$\frac{dN}{2(d+2)}\leq\frac{2dN}{d(N-8)+8},$

which is true if and only if $N\leq 12$.

Now, using (2.72) and the standard energy estimate of systems (2.62) for $s=0$
and (2.65) for $s=0$, the term (2.74) becomes

$\left|\int_{0}^{T}\int_{\mathcal{O}}\eta w^{1}\ dxdt\right|\leq
C\|w^{1}\|_{L^{2}((0,T)\times\mathcal{O})},$ (2.81)

where $C$ is a positive constant independent of $\mu$.

Combining (2.70) together with (2.81), it follows that

$\displaystyle
D^{2}J(f;\hat{v})\cdot(w^{1},w^{1})\geq\left(\mu-C\right)\int_{0}^{T}\int_{\mathcal{O}}|w^{1}|^{2}\
dxdt,\ \forall w^{1}\in L^{2}((0,T)\times\mathcal{O}),\ w^{1}\neq 0.$

For $\mu$ sufficiently large, the functional $J$ given by (1.3) is strictly
convex and thus, we have the uniqueness of the optimal control $\hat{v}$
solution of (1.4).

Arguing as in the Proposition 2.1, the optimal control $\hat{v}$ is given by

$\hat{v}=-\frac{1}{\mu}p\ \ \mbox{in}\ \ \ (0,T)\times\mathcal{O},$ (2.82)

where $(y,p)$ is the solution to the following optimality system

$\left\\{\begin{array}[]{ccl}\displaystyle y_{t}-\Delta
y+\int_{\Omega}K(t,x,\theta)y(t,\theta)\ d\theta=\displaystyle
G(y)+f\chi_{\omega}-\frac{1}{\mu}p\chi_{\mathcal{O}}&\mbox{in}&Q,\\\
\displaystyle-p_{t}-\Delta p+\int_{\Omega}K(t,x,\theta)p(t,\theta)\
d\theta=G^{\prime}(y)p+(y-y_{d})\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\
\displaystyle y=0,\ \ \ p=0&\mbox{on}&\Sigma,\\\ \displaystyle
y(0,\cdot)=y^{0},\ \ \ p(T,\cdot)=0&\mbox{in}&\Omega.\end{array}\right.$
(2.83)

In the following result, using the same ideas as in the linear case, we prove
the null controllability of the optimality system (2.82)-(2.83).

###### Theorem 2.1

Under the assumptions of Proposition 2.6, there exists a positive weight
function $\varpi_{1}=\varpi_{1}(t)$ blowing up at $t=T$ such that for any
$y_{d}\in L^{2}((0,T)\times\mathcal{O}_{d})$ satisfying (1.14), there exist a
control $\hat{f}\in L^{2}((0,T)\times\omega)$ and a unique optimal control
$\hat{v}\in L^{2}((0,T)\times\mathcal{O})$ such that the corresponding
solution to (2.83) satisfies (1.7).

Proof. We give a sketch of the proof of Theorem 2.1 which combines the results
obtained in the linear case with a fixed point argument.

We introduce the change of variables that allows us to simplify the
controllability of trajectories (1.7) to null controllability property. By
introducing the change of variable $z=y-\bar{y}$, we can write (2.83) as
follows

$\left\\{\begin{array}[]{ccl}\displaystyle z_{t}-\Delta
z+\int_{\Omega}K(t,x,\theta)z(t,\theta)\ d\theta=\displaystyle
a(z)z+f\chi_{\omega}-\frac{1}{\mu}p\chi_{\mathcal{O}}&\mbox{in}&Q,\\\
\displaystyle-p_{t}-\Delta p+\int_{\Omega}K(t,x,\theta)p(t,\theta)\
d\theta=b(z)p+(z-z_{d})\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\ \displaystyle
z=0,\ \ \ p=0&\mbox{on}&\Sigma,\\\ \displaystyle z(0,\cdot)=z^{0},\ \ \
p(T,\cdot)=0&\mbox{in}&\Omega,\end{array}\right.$ (2.84)

where $z^{0}:=y^{0}-\bar{y}^{0}$, $z_{d}:=y_{d}-\bar{y}$ and

$a(z)=\int_{0}^{1}G^{\prime}(\bar{y}+sz)\ ds,\ \ b(z)=G^{\prime}(z+\bar{y}).$

The condition (1.7) is equivalent to a null controllability property for $z$,
that is

$z(T,x)=0\ \mbox{in}\ \Omega.$ (2.85)

For each $w\in L^{2}(Q)$, we introduce the linearized system associated to
(2.84)

$\left\\{\begin{array}[]{ccl}\displaystyle z_{t}-\Delta
z+\int_{\Omega}K(t,x,\theta)z(t,\theta)\ d\theta=\displaystyle
a(w)z+f\chi_{\omega}-\frac{1}{\mu}p\chi_{\mathcal{O}}&\mbox{in}&Q,\\\
\displaystyle-p_{t}-\Delta p+\int_{\Omega}K(t,x,\theta)p(t,\theta)\
d\theta=b(w)p+(z-z_{d})\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\ \displaystyle
z=0,\ \ \ p=0&\mbox{on}&\Sigma,\\\ \displaystyle z(0,\cdot)=z^{0},\ \ \
p(T,\cdot)=0&\mbox{in}&\Omega.\end{array}\right.$ (2.86)

Thanks to the hypothesis on $G$, there exists $M>0$ such that

$\|a(w)\|_{L^{\infty}(Q)},\ \|b(w)\|_{L^{\infty}(Q)}\leq M,\ \forall w\in
L^{2}(Q).$

Arguing as in the proof of Proposition 2.4, we can obtain the following
observability inequality

$\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{Q}\frac{1}{\varpi_{1}^{2}}|\psi|^{2}\,dx\,dt\leq
C\int_{0}^{T}\int_{\omega}|\rho|^{2}\,dx\,dt,$ (2.87)

where
$C=C(\Omega,\omega,\|K\|_{\infty},\|a(w)\|_{L^{\infty}(Q)},\|b(w)\|_{L^{\infty}(Q)},T)>0$,
the function $\varpi_{1}$ is given by (2.49) and $(\rho,\psi)$ is the solution
to the following adjoint system associated to (2.86):

$\left\\{\begin{array}[]{ccl}\displaystyle-\rho_{t}-\Delta\rho+\int_{\Omega}K(t,x,\theta)\rho(t,\theta)\
d\theta=\displaystyle a(w)\rho+\psi\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\
\displaystyle\psi_{t}-\Delta\psi+\int_{\Omega}K(t,x,\theta)\psi(t,\theta)\
d\theta=b(w)\psi-\displaystyle\frac{1}{\mu}\rho\chi_{\mathcal{O}}&\mbox{in}&Q,\\\
\displaystyle\rho=0,\ \ \ \psi=0&\mbox{on}&\Sigma,\\\
\displaystyle\rho(T,\cdot)=\rho^{T},\ \ \
\psi(0,\cdot)=0&\mbox{in}&\Omega.\end{array}\right.$ (2.88)

Thanks to estimate (2.87) and arguing as in the Proposition 2.5, we prove the
existence of a leader control $\hat{f}\in L^{2}((0,T)\times\omega)$ solution
to the null controllability problem (2.85)-(2.86). The leader control verifies
the following estimate

$\begin{array}[]{ccc}\displaystyle\|\hat{f}\|_{L^{2}((0,T)\times\omega)}\leq
C\displaystyle\left(\|\varpi_{1}z_{d}\|_{L^{2}((0,T)\times\mathcal{O}_{d})}+\|y^{0}\|_{L^{2}(\Omega)}\right),\end{array}$
(2.89)

where
$C=C(\Omega,\omega,\|K\|_{\infty},\|a(w)\|_{L^{\infty}(Q)},\|b(w)\|_{L^{\infty}(Q)},T)>0$.

Now, we consider a nonlinear map

$S:L^{2}(Q)\rightarrow L^{2}(Q)$

such that, for every $w\in L^{2}(Q)$, $S(w)=y$ where $(y,p)$ are solutions of
(2.86). Proving that $S$ has a fixed point $y\in L^{2}(Q)$ will allows us to
prove that $y$ is solution of (2.60). To this end, we use the Schauder fixed-
point theorem. Proceeding as in [8, Proposition 3.2], we can prove the
following properties for every $w\in L^{2}(Q)$:

  1. 1.

$S$ is continuous,

  2. 2.

$S$ is compact,

  3. 3.

The range of $S$ is bounded; i.e., $\exists M>0:\ \|S(w)\|_{L^{2}(Q)}\leq M,\
\forall w\in L^{2}(Q).$

This end the proof of Theorem 2.1 and furthermore the proof of Stackelberg
control of system (2.60).

##  3 Exact controllability with distributed leader and boundary follower

In this section, we establish a controllability result for the system (1.2).
Let $z:=q-\bar{q}$, where $q$ and $\bar{q}$ are solutions to (1.2) and (1.11)
respectively. Then, $z$ is the solution of the following system

$\left\\{\begin{array}[]{rllll}\displaystyle z_{t}-\Delta
z+\int_{\Omega}K(t,x,\theta)z(t,\theta)\
d\theta&=&g\chi_{\omega}&\mbox{in}&Q,\\\ \displaystyle
z&=&u\textbf{1}_{\Gamma}&\mbox{on}&\Sigma,\\\ \displaystyle
z(0,\cdot)&=&z^{0}&\mbox{in}&\Omega,\end{array}\right.$ (3.1)

where $z^{0}:=q^{0}-\bar{q}^{0}$. Thus, the property (1.12) is equivalent to

$z(T,x)=0\ \mbox{in}\ \Omega.$ (3.2)

The cost functional $\widetilde{J}$ defined by (1.8) can be rewritten as
follows

$\widetilde{J}(g;u)=\frac{1}{2}\int_{0}^{T}\int_{\mathcal{O}_{d}}|z-z_{d}|^{2}\
dxdt+\frac{\mu}{2}\int_{0}^{T}\int_{\Gamma}|u|^{2}\ d\sigma dt,$ (3.3)

where $z_{d}:=q_{d}-\bar{q}$.

Adapting the arguments of Section 2.1, we can prove the existence and
uniqueness of an optimal control $\hat{u}$ solution of (1.9). Indeed, for any
fixed leader $g$ and $z^{0}\in L^{2}(\Omega)$, one can prove that

$\hat{u}=\frac{1}{\mu}\frac{\partial p}{\partial\nu}\ \ \mbox{on}\ \ \
(0,T)\times\Gamma,$ (3.4)

where $(z,p)$ is the solution of optimality system

$\left\\{\begin{array}[]{ccl}\displaystyle z_{t}-\Delta
z+\int_{\Omega}K(t,x,\theta)z(t,\theta)\ d\theta=\displaystyle
g\chi_{\omega}&\mbox{in}&Q,\\\ \displaystyle-p_{t}-\Delta
p+\int_{\Omega}K(t,x,\theta)p(t,\theta)\
d\theta=(z-z_{d})\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\ \displaystyle
z=\frac{1}{\mu}\frac{\partial p}{\partial\nu}\textbf{1}_{\Gamma},\ \ \
p=0&\mbox{on}&\Sigma,\\\ \displaystyle z(0,\cdot)=z^{0},\ \ \
p(T,\cdot)=0&\mbox{in}&\Omega.\end{array}\right.$ (3.5)

Now, we state a regularity result for the solutions to (3.5).

###### Proposition 3.1

Assume that $g\in L^{2}((0,T)\times\omega)$, $z^{0}\in L^{2}(\Omega)$,
$z_{d}\in L^{2}((0,T)\times\mathcal{O}_{d})$ and $\mu$ is large enough. Then,
the system (3.5) has a unique solution $(z,p)\in W(Q)\times H^{2,1}(Q)$.
Moreover, $\displaystyle\frac{\partial p}{\partial\nu}\in
H^{1/2,1/4}(\Sigma)$, where the Hilbert spaces $H^{1/2,1/4}(\Sigma)$ and
$W(Q)$ are given by (1.15) and (2.4) respectively.

The proof of the above proposition can be obtained by proceeding as in [3,
Proposition 4] or [16, Proposition 5].

###### Remark 5

  1. 1.

Using the Proposition 3.1 and from the characterization of the optimal control
given by (3.4), we have that

$\hat{u}\textbf{1}_{\Gamma}\in H^{1/2,1/4}(\Sigma).$

  2. 2.

The regularity of the follower in $H^{1/2,1/4}(\Sigma)$ instead of
$L^{2}(\Sigma)$ is very important because, with this regularity, we can use
the Carleman inequality with non homogeneous boundary condition belonging to
$H^{1/2,1/4}(\Sigma)$.

  3. 3.

If in the system (3.5), we consider $p(T,\cdot)=p^{T}$ with $p^{T}\in
H^{1}_{0}(\Omega)$, then we can obtain the same well-posedness result as in
the Proposition 3.1.

Now, we want to study the null controllability of the system (3.5). Arguing as
in Section 2.3, solving the null controllability problem of the system (3.5)
require the proof of a suitable observability inequality for the following
adjoint system

$\left\\{\begin{array}[]{ccl}\displaystyle-\rho_{t}-\Delta\rho+\int_{\Omega}K(t,x,\theta)\rho(t,\theta)\
d\theta=\displaystyle\psi\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\
\displaystyle\psi_{t}-\Delta\psi+\int_{\Omega}K(t,x,\theta)\psi(t,\theta)\
d\theta=0&\mbox{in}&Q,\\\ \displaystyle\rho=0,\ \ \
\psi=\frac{1}{\mu}\frac{\partial\rho}{\partial\nu}\textbf{1}_{\Gamma}&\mbox{on}&\Sigma,\\\
\displaystyle\rho(T,\cdot)=\rho^{T},\ \ \
\psi(0,\cdot)=0&\mbox{in}&\Omega,\end{array}\right.$ (3.6)

where $\rho^{T}\in L^{2}(\Omega)$.

The observability estimate associated to (3.6) is given in the following
proposition.

###### Proposition 3.2

Assume that $\mathcal{O}_{d}\cap\omega\neq\emptyset$ and $\mu$ is large
enough. Then, there exist a positive constant $C$ and a weight function
$\varpi_{2}=\varpi_{2}(t)$ blowing up at $t=T$ such that for any $\rho^{T}\in
L^{2}(\Omega)$, the following inequality holds

$\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{Q}\frac{1}{\varpi_{2}^{2}}|\psi|^{2}\,dx\,dt\leq
C\int_{0}^{T}\int_{\omega}|\rho|^{2}\,dx\,dt,$ (3.7)

where the adjoint variables $(\rho,\psi)$ are the solution of (3.6).

Thanks to Proposition 3.2 and proceeding as in the Proposition 2.5, we can
prove that for each $z^{0}\in L^{2}(\Omega)$, there exists a control
$\hat{g}\in L^{2}((0,T)\times\omega)$ such that the associated solution
$(z,p)$ to (3.5) satisfies (3.2).

###### Remark 6

Due to the density of $H^{1}_{0}(\Omega)$ in $L^{2}(\Omega)$, it can be
assumed that $\rho^{T}\in H^{1}_{0}(\Omega)$. Since systems (3.5) and (3.6)
have similar structures, according to Remark 5, the system (3.6) admits a
unique solution $(\rho,\psi)\in H^{2,1}(Q)\times W(Q)$.

Now, we want to prove the inequality (3.7). Thanks to hypothesis
$\mathcal{O}_{d}\cap\omega\neq\emptyset$, then, there exists a nonempty set
$\omega^{\prime}$ such that
$\omega^{\prime}\subset\subset\mathcal{O}_{d}\cap\omega$ and
$\eta^{0}\in\mathcal{C}^{2}(\overline{\Omega})$ a function satisfying (2.16).
For $(t,x)\in Q$, we consider the following functions

$\begin{array}[]{rll}\displaystyle\eta^{*}(t)=\displaystyle\max_{x\in\overline{\Omega}}\eta(t,x),\
\ \
\varphi^{*}(t)=\displaystyle\min_{x\in\overline{\Omega}}\varphi(t,x),\end{array}$
(3.8)

where $\eta$ and $\varphi$ are the weight functions defined by (2.18).

We need the following notation in the sequel

$\mathcal{N}(z)=s^{-1}\int_{Q}e^{-2s\eta}\varphi^{-1}|\nabla
z|^{2}dxdt+s\lambda^{2}\int_{Q}e^{-2s\eta}\varphi|z|^{2}dxdt.$ (3.9)

For $F\in L^{2}(Q)$, $z_{0}\in L^{2}(\Omega)$ and $h\in H^{1/2,1/4}(\Sigma)$,
we consider the following system

$\left\\{\begin{array}[]{rllll}\displaystyle\frac{\partial{z}}{\partial{t}}-\Delta
z&=&F&\mbox{in}&Q,\\\ \displaystyle z&=&h&\mbox{on}&\Sigma,\\\ \displaystyle
z(0,\cdot)&=&z_{0}&\mbox{in}&\Omega.\end{array}\right.$ (3.10)

The observability inequality (3.7) is established via a global Carleman
estimate given in [18] for the solution of system (3.10). More precisely, we
have the following result.

###### Lemma 3.1

Let $F\in L^{2}(Q)$, $z_{0}\in L^{2}(\Omega)$ and $h\in H^{1/2,1/4}(\Sigma)$.
Then, there exists $s_{0}\geq 1$ and $\lambda_{0}\geq 1$ and there exists a
constant $C_{0}>0$ independent of $s$ and $\lambda$ such that for all $s\geq
s_{0}$ and $\lambda\geq\lambda_{0}$, and for any $z\in W(Q)$ solution of
(3.10), we have

$\begin{array}[]{llll}\displaystyle\mathcal{N}(z)&\leq&\displaystyle
C_{0}\left(s^{-\frac{1}{2}}\|\varphi^{-\frac{1}{4}}he^{-s\eta}\|^{2}_{H^{1/2,1/4}(\Sigma)}+s^{-2}\lambda^{-2}\int_{Q}e^{-2s\eta}\varphi^{-2}|F|^{2}\
dxdt\right.\\\
&&\displaystyle\left.+s\lambda^{2}\int_{0}^{T}\int_{\omega^{\prime}}e^{-2s\eta}\varphi|z|^{2}\
dxdt\right).\end{array}$ (3.11)

Thanks to Lemma 2.1 and Lemma 3.1, we can prove the Carleman inequality
associated to adjoint system (3.6). We have the following result.

###### Proposition 3.3

Assume that $\omega\cap\mathcal{O}_{d}\neq\emptyset$ and $\mu$ is large
enough. Then, there exist a positive constant $C=C(\Omega,\omega,T)>0$ such
that the solution $(\rho,\psi)$ to (3.6) satisfies

$\begin{array}[]{lll}\displaystyle\mathcal{M}(\rho)+\mathcal{N}(\psi)\leq\displaystyle
Cs^{5}\lambda^{6}\int_{0}^{T}\int_{\omega}e^{-2s\eta}\varphi^{5}|\rho|^{2}dx\,dt,\end{array}$
(3.12)

for every $s>0$ large enough.

Proof. We apply the estimate (2.24) to the first equation of (3.6), then we
apply the estimate (3.11) to the second equation for (3.6) and add them up, we
obtain

$\begin{array}[]{lll}\mathcal{M}(\rho)+\mathcal{N}(\psi)\\\ \leq\displaystyle
C\left(\int_{0}^{T}\int_{\omega^{\prime}}e^{-2s\eta}\left(s^{3}\lambda^{4}\varphi^{3}|\rho|^{2}+s\lambda^{2}\varphi|\psi|^{2}\right)dx\,dt+\int_{Q}e^{-2s\eta}|\psi|^{2}dx\,dt\right)\\\
+\displaystyle
C\left(\int_{Q}e^{-2s\eta}\left|\int_{\Omega}K(t,x,\theta)\rho(t,\theta)\
d\theta\right|^{2}dx\,dt+\int_{Q}e^{-2s\eta}\left|\int_{\Omega}K(t,x,\theta)\psi(t,\theta)\
d\theta\right|^{2}dx\,dt\right)\\\ +\displaystyle
Cs^{-\frac{1}{2}}\left\|\varphi^{-\frac{1}{4}}e^{-s\eta}\frac{1}{\mu}\frac{\partial\rho}{\partial\nu}\textbf{1}_{\Gamma}\right\|^{2}_{H^{1/2,1/4}(\Sigma)},\end{array}$

for any $s\geq\hat{s}$, $\lambda\geq\hat{\lambda}$, where
$\hat{s}=\max(s_{0},s_{1})$, $\hat{\lambda}=\max(\lambda_{0},\lambda_{1})$ and
the constant $C=C(C_{0},\Omega,\omega)>0$.

Now, we fix the parameter $\lambda=\hat{\lambda}$ sufficiently large. From
(2.32) and (2.33), we can absorb some terms in the right hand side of the
above estimate and we obtain

$\begin{array}[]{lll}\mathcal{M}(\rho)+\mathcal{N}(\psi)\\\ \leq\displaystyle
C\left(\int_{0}^{T}\int_{\omega^{\prime}}e^{-2s\eta}\left(s^{3}\lambda^{4}\varphi^{3}|\rho|^{2}+s\lambda^{2}\varphi|\psi|^{2}\right)dx\,dt+\displaystyle
s^{-\frac{1}{2}}\left\|\varphi^{-\frac{1}{4}}e^{-s\eta}\frac{1}{\mu}\frac{\partial\rho}{\partial\nu}\textbf{1}_{\Gamma}\right\|^{2}_{H^{1/2,1/4}(\Sigma)}\right),\end{array}$
(3.13)

where $C=C(C_{0},\Omega,\omega)>0$ and $C_{0}$ is the constant defined to
Lemma 3.1.
Now, we want to estimate the last term in (3.13). Firstly, note that the map
$\displaystyle\rho\longmapsto\frac{\partial\rho}{\partial\nu}\textbf{1}_{\Gamma}$
is a continuous and linear from $H^{2,1}(Q)$ to $H^{1/2,1/4}(\Sigma)$.
Secondly, due to the definition of weight functions given in (2.18) and (3.8),
we have that $\eta$ and $\varphi$ are equals to $\eta^{*}$ and $\varphi^{*}$
on the boundary. Then, we can write

$\begin{array}[]{lll}\displaystyle
s^{-\frac{1}{2}}\left\|\varphi^{-\frac{1}{4}}e^{-s\eta}\frac{1}{\mu}\frac{\partial\rho}{\partial\nu}\textbf{1}_{\Gamma}\right\|^{2}_{H^{1/2,1/4}(\Sigma)}\leq\displaystyle\frac{1}{\mu}s^{-\frac{1}{2}}\left\|(\varphi^{*})^{-\frac{1}{4}}e^{-s\eta^{*}}\rho\right\|^{2}_{H^{2,1}(Q)}.\end{array}$
(3.14)

The strategy now is to estimate the term in the right hand side of (3.14). We
set $\widehat{\rho}=\sigma\rho$ with $\sigma\in\mathcal{C}^{\infty}([0,T])$
defined by

$\sigma=e^{-s\eta^{*}}(s\varphi^{*})^{-1/4}.$ (3.15)

Observing that $\sigma(T)=0$ and using the definition of weight functions
$\eta^{*}$ and $\varphi^{*}$, we obtain for $s$ large enough

$|\sigma_{t}|\leq Ce^{-s\eta^{*}}(s\varphi^{*}).$ (3.16)

Using the first equation of (3.6), $\widehat{\rho}$ satisfies the following
system

$\left\\{\begin{array}[]{rllll}\displaystyle-\widehat{\rho}_{t}-\Delta\widehat{\rho}+\int_{\Omega}K(t,x,\theta)\widehat{\rho}(t,\theta)\
d\theta&=&\displaystyle\sigma\psi\chi_{\mathcal{O}_{d}}-\sigma_{t}\rho&\mbox{in}&Q,\\\
\displaystyle\widehat{\rho}&=&0&\mbox{on}&\Sigma,\\\
\displaystyle\widehat{\rho}(T,\cdot)&=&0&\mbox{in}&\Omega.\end{array}\right.$
(3.17)

Using the Proposition 3.1, we deduce that the system (3.17) admits a unique
solution $\widehat{\rho}\in H^{2,1}(Q)$. Furthermore, the classical energy
estimate associated to $\widehat{\rho}$ is given by

$\begin{array}[]{llllll}\displaystyle\|\widehat{\rho}\|^{2}_{H^{2,1}(Q)}\leq
C(\|K\|_{\infty},T)\left(\|\sigma\psi\|^{2}_{L^{2}(Q)}+\|\sigma_{t}\rho\|^{2}_{L^{2}((Q)}\right).\end{array}$
(3.18)

Combining (3.15) and (3.16) together with estimate (3.18), we have

$\begin{array}[]{llllll}\displaystyle
s^{-\frac{1}{2}}\left\|(\varphi^{*})^{-\frac{1}{4}}e^{-s\eta^{*}}\rho\right\|^{2}_{H^{2,1}(Q)}\\\
\displaystyle\leq
C(\|K\|_{\infty},T)\left(\int_{Q}e^{-2s\eta^{*}}(s\varphi^{*})^{-1/2}|\psi|^{2}\
dxdt+\int_{Q}e^{-2s\eta^{*}}(s\varphi^{*})^{2}|\rho|^{2}\
dxdt\right).\end{array}$ (3.19)

Using the definition of the weight functions $\eta^{*}$ and $\varphi^{*}$,
then combining (3.19) and (3.14), we get

$\begin{array}[]{llllll}\displaystyle
s^{-\frac{1}{2}}\left\|\varphi^{-\frac{1}{4}}e^{-s\eta}\frac{1}{\mu}\frac{\partial\rho}{\partial\nu}\textbf{1}_{\Gamma}\right\|^{2}_{H^{1/2,1/4}(\Sigma)}\\\
\displaystyle\leq
C(\|K\|_{\infty},\mu,T)\left(\int_{Q}e^{-2s\eta}s\varphi|\psi|^{2}\
dxdt+\int_{Q}e^{-2s\eta}(s\varphi)^{2}|\rho|^{2}\ dxdt\right).\end{array}$
(3.20)

Substituting (3.20) in (3.13) and taking $s$ sufficiently large, we have

$\begin{array}[]{lll}\mathcal{M}(\rho)+\mathcal{N}(\psi)\leq\displaystyle
C(C_{0},\Omega,\omega)\int_{0}^{T}\int_{\omega^{\prime}}e^{-2s\eta}\left(s^{3}\lambda^{4}\varphi^{3}|\rho|^{2}+s\lambda^{2}\varphi|\psi|^{2}\right)dx\,dt.\end{array}$
(3.21)

The last step is to eliminate the local term corresponding to $\psi$ on the
right hand side of the estimate (3.21). We proceed as in the second step of
the proof of Proposition 2.3. We consider the open set $\omega_{0}$ and the
function $\xi$ defined by (2.34).

We define $u=s\lambda^{2}\varphi e^{-2s\eta}$. Using the definition of
$\varphi$ and $\eta$ given in (2.18), we get $u(T)=u(0)=0$. Due to the
estimates (2.21), we have the following:

$\begin{array}[]{rll}\displaystyle|u\xi|\leq s\lambda^{2}\varphi
e^{2s\eta}\xi,\ \ \ \ \ \ \displaystyle\left|(u\xi)_{t}\right|\leq
C(T)s^{2}\lambda^{2}\varphi^{3}e^{2s\eta}\xi,\\\ \\\
\displaystyle|\nabla(u\xi)|\leq Cs^{3}\lambda^{3}\varphi^{2}e^{2s\eta}\xi,\ \
\ \ \ \ \displaystyle|\Delta(u\xi)|\leq
Cs^{3}\lambda^{4}\varphi^{3}e^{2s\eta}\xi,\end{array}$ (3.22)

where $C$ is a positive constant.
Multiplying the first equation of (3.6) by $u\xi\psi$, integrating by parts
over $Q$ and using the Fubini’s theorem, we obtain

$I_{1}+I_{2}+I_{3}=\int_{Q}u\xi|\psi|^{2}\chi_{\mathcal{O}_{d}}\ dx\,dt,$
(3.23)

where

$I_{1}=\displaystyle\int_{Q}\rho\psi(u\xi)_{t}\
dx\,dt\leq\displaystyle\frac{\gamma_{1}}{2}\int_{Q}s\lambda^{2}\varphi
e^{-2s\eta}|\psi|^{2}dx\,dt+\frac{C(T)}{\gamma_{1}}\int_{0}^{T}\int_{\omega_{0}}s^{3}\lambda^{2}\varphi^{5}e^{-2s\eta}|\rho|^{2}dx\,dt,$
$I_{2}=\displaystyle-\int_{Q}\rho\psi\Delta(u\xi)\
dx\,dt\leq\frac{\gamma_{2}}{2}\int_{Q}s\lambda^{2}\varphi
e^{-2s\eta}|\psi|^{2}dx\,dt+\frac{C}{\gamma_{2}}\int_{0}^{T}\int_{\omega_{0}}s^{5}\lambda^{6}\varphi^{5}e^{-2s\eta}|\rho|^{2}dx\,dt,$
$I_{3}=\displaystyle-2\int_{Q}\rho\nabla\psi.\nabla(u\xi)\
dx\,dt\leq\frac{\gamma_{3}}{2}\int_{Q}s^{-1}\varphi^{-1}e^{-2s\eta}|\nabla\psi|^{2}dx\,dt+\frac{C}{\gamma_{3}}\int_{0}^{T}\int_{\omega_{0}}s^{5/2}\lambda^{6}\varphi^{e}{5/2}^{-2s\eta}|\rho|^{2}dx\,dt,$

for any $\gamma_{i}>0,\ i=1,2,3$.

By making the right choice of $\gamma_{i},\ i=1,2,3$, we obtain that

$\begin{array}[]{rll}\displaystyle\int_{0}^{T}\int_{\omega^{\prime}}s\lambda^{2}\varphi
e^{-2s\eta}|\psi|^{2}dx\,dt\displaystyle\leq\displaystyle\frac{1}{2}\mathcal{N}(\psi)+\displaystyle
C(T)\int_{0}^{T}\int_{\omega_{0}}s^{5}\lambda^{6}\varphi^{5}e^{-2s\eta}|\rho|^{2}\
dx\,dt.\end{array}$ (3.24)

We replace (3.24) in (3.21) and since $\omega_{0}\subset\omega$, we deduce
(3.12).

Now, we are ready to prove the observability inequality (3.7).

####

Proof of Proposition 3.2. We proceed as in the proof of Proposition 2.4. The
idea is to improve (3.12) in the sense that the weight functions do not
degenerate at $t=0$. To this end, we consider functions $\alpha,\ \zeta,\
\alpha^{*}$ and $\zeta^{*}$ defined in (2.39).

We fix the parameter $s$ to a sufficiently large fixed value. Firstly, by
construction, $\eta=\alpha$ and $\varphi=\zeta$ in $[T/2,T]\times\Omega$,
hence,

$\begin{array}[]{llll}\displaystyle\int_{T/2}^{T}\int_{\Omega}e^{-2s\alpha}\zeta^{3}|\rho|^{2}\
dxdt+\int_{T/2}^{T}\int_{\Omega}e^{-2s\alpha}\zeta|\psi|^{2}\ dxdt\\\
=\displaystyle\int_{T/2}^{T}\int_{\Omega}e^{-2s\eta}\varphi^{3}|\rho|^{2}\
dxdt+\int_{T/2}^{T}\int_{\Omega}e^{-2s\eta}\varphi|\psi|^{2}\ dxdt\\\
\leq\displaystyle
C(\Omega,\omega,T)\int_{0}^{T}\int_{\omega}e^{-2s\eta}\varphi^{5}|\rho|^{2}\,dx\,dt,\end{array}$
(3.25)

where we have used the Carleman estimate (3.12).

On the other hand, proceeding as in the second step of the proof of
Proposition 2.4, we obtain

$\begin{array}[]{llll}\displaystyle\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T/2}\int_{\Omega}|\rho|^{2}\
dxdt\\\ \displaystyle\leq
C(\|K\|_{\infty},T)\left(\int_{0}^{3T/4}\int_{\Omega}|\psi|^{2}\
dx\,dt+\int_{T/2}^{3T/4}\int_{\Omega}|\rho|^{2}\ dx\,dt\right).\end{array}$

Adding the term $\displaystyle\int_{0}^{T/2}\int_{\Omega}|\psi|^{2}\ dx\,dt$
on both sides of above estimate, we arrive to

$\begin{array}[]{llll}\displaystyle\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T/2}\int_{\Omega}|\rho|^{2}\
dxdt+\int_{0}^{T/2}\int_{\Omega}|\psi|^{2}\ dx\,dt\\\ \displaystyle\leq
C(\|K\|_{\infty},T)\left(\int_{T/2}^{3T/4}\int_{\Omega}(|\psi|^{2}+|\rho|^{2})\
dx\,dt+\int_{0}^{T/2}\int_{\Omega}|\psi|^{2}\ dx\,dt\right).\end{array}$
(3.26)

In order to eliminate the last term in the right hand side of (3.26), we use
the classical energy estimates for the second equation of system (3.6) in the
domain $(0,T/2)\times\Omega$ and we obtain:

$\begin{array}[]{llll}\displaystyle\int_{0}^{T/2}\int_{\Omega}|\psi|^{2}\
dxdt\displaystyle&\leq&\displaystyle\frac{1}{\mu^{2}}C(\|K\|_{\infty},T)\left\|\frac{\partial\rho}{\partial\nu}\textbf{1}_{\Gamma}\right\|^{2}_{H^{1/2,1/4}((0,T/2)\times\partial\Omega)},\\\
&\leq&\displaystyle\frac{1}{\mu^{2}}C(\|K\|_{\infty},T)\left\|\rho\right\|^{2}_{H^{2,1}((0,T/2)\times\Omega)},\\\
&\leq&\displaystyle\frac{1}{\mu^{2}}C(\|K\|_{\infty},T)\left\|(\zeta^{*})^{-\frac{1}{4}}e^{-s\alpha^{*}}\rho\right\|^{2}_{H^{2,1}((0,T/2)\times\Omega)}\\\
&\leq&\displaystyle\frac{1}{\mu^{2}}C(\|K\|_{\infty},T)\left\|(\zeta^{*})^{-\frac{1}{4}}e^{-s\alpha^{*}}\rho\right\|^{2}_{H^{2,1}(Q)}\end{array}$
(3.27)

because the map
$\displaystyle\rho\longmapsto\frac{\partial\rho}{\partial\nu}\textbf{1}_{\Gamma}$
is continuous and linear from $H^{2,1}(Q)$ to $H^{1/2,1/4}(\Sigma)$ and the
functions $\alpha^{*}$ and $\zeta^{*}$ are constants in both variables in
$(0,T/2)\times\Omega$.

Replacing (3.27) in (3.26), and using the fact that the weight functions
$\alpha$ and $\zeta$ defined in (2.39) are bounded in $[0,3T/4]\times\Omega$,
we obtain

$\begin{array}[]{llll}\displaystyle\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T/2}\int_{\Omega}e^{-2s\alpha}\zeta^{3}|\rho|^{2}\
dxdt+\int_{0}^{T/2}\int_{\Omega}e^{-2s\alpha}\zeta|\psi|^{2}\ dx\,dt\\\
\displaystyle\leq
C(\|K\|_{\infty},T)\int_{T/2}^{3T/4}\int_{\Omega}e^{-2s\alpha}(\zeta|\psi|^{2}+\zeta^{3}|\rho|^{2})\
dx\,dt+\frac{1}{\mu^{2}}C(\|K\|_{\infty},T)\left\|(\zeta^{*})^{-\frac{1}{4}}e^{-s\alpha^{*}}\rho\right\|^{2}_{H^{2,1}(Q)}.\end{array}$
(3.28)

Adding estimates (3.25) and (3.28) and using definitions of $\alpha^{*}$ and
$\zeta^{*}$, we have

$\begin{array}[]{llll}\displaystyle\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T}\int_{\Omega}e^{-2s\alpha^{*}}(\zeta^{*})^{3}|\rho|^{2}\
dxdt+\int_{0}^{T}\int_{\Omega}e^{-2s\alpha^{*}}\zeta^{*}|\psi|^{2}\ dx\,dt\\\
\displaystyle\leq\displaystyle
C(\Omega,\omega,T)\int_{0}^{T}\int_{\omega}e^{-2s\eta}\varphi^{5}|\rho|^{2}\,dx\,dt+\frac{1}{\mu^{2}}C(\|K\|_{\infty},T)\left\|(\zeta^{*})^{-\frac{1}{4}}e^{-s\alpha^{*}}\rho\right\|^{2}_{H^{2,1}(Q)}.\end{array}$
(3.29)

To conclude, we want to estimate the last norm in the previous inequality. To
this end, we use system (3.17) and estimate (3.18) with
$\displaystyle\sigma=e^{-s\alpha^{*}}(\zeta^{*})^{-1/4}$ and we obtain

$\begin{array}[]{llllll}\displaystyle\left\|e^{-s\alpha^{*}}\zeta^{*})^{-\frac{1}{4}}\rho\right\|^{2}_{H^{2,1}(Q)}\\\
\displaystyle\leq
C(\|K\|_{\infty},T)\left(\int_{Q}e^{-2s\alpha^{*}}(\zeta^{*})^{-1/2}|\psi|^{2}\
dxdt+\int_{Q}e^{-2s\alpha^{*}}(\zeta^{*})^{2}|\rho|^{2}\
dxdt\right).\end{array}$ (3.30)

Replacing (3.30) in (3.29), using the fact that $(\zeta^{*})^{-1},\
(\zeta^{*})^{-3/2}\in L^{\infty}(Q)$ and taking $\mu$ large enough, we obtain

$\begin{array}[]{llll}\displaystyle\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T}\int_{\Omega}e^{-2s\alpha^{*}}(\zeta^{*})^{3}|\rho|^{2}\
dxdt+\int_{0}^{T}\int_{\Omega}e^{-2s\alpha^{*}}\zeta^{*}|\psi|^{2}\ dx\,dt\\\
\displaystyle\leq\displaystyle
C(\Omega,\omega,T)\int_{0}^{T}\int_{\omega}e^{-2s\eta}\varphi^{5}|\rho|^{2}\,dx\,dt.\end{array}$
(3.31)

We obtain the observability inequality (3.7) due to the fact that
$\displaystyle e^{-2s\eta}\varphi^{5}\in L^{\infty}(Q)$ and taking

$\varpi_{2}(t)=e^{s\alpha^{*}}(\zeta^{*})^{-1/2}.$ (3.32)

###  3.1 The semilinear case

In the same spirit of Section 2.4, we consider a more general version of
system (2.60) given by

$\left\\{\begin{array}[]{rllll}\displaystyle y_{t}-\Delta
y+\int_{\Omega}K(t,x,\theta)G(y(t,\theta))\
d\theta&=&g\chi_{\omega}+v\chi_{\mathcal{O}}&\mbox{in}&Q,\\\ \displaystyle
y&=&u\textbf{1}_{\Gamma}&\mbox{on}&\Sigma,\\\ \displaystyle
y(0,\cdot)&=&y^{0}&\mbox{in}&\Omega,\end{array}\right.$ (3.33)

where $G$ is a globally Lipschitz continuous function.

As in the Section 2.4, in the semilinear case, we lose the convexity of the
functional $\widetilde{J}$ defined by (1.8). For this, we state the following
result whose the proof can be obtained by proceeding in the same way as in the
Proposition 2.6.

###### Proposition 3.4

Let us assume that $q_{d}\in L^{\infty}((0,T);\mathcal{O}_{d})$, $g\in
L^{2}((0,T)\times\omega)$ are fixed and $\mu$ is sufficiently large. Suppose
that $q^{0}\in L^{2}(\Omega)$, $N\leq 6$, $G\in\mathcal{C}^{2}(\mathbb{R})$
and there exists a constant $L>0$ such that
$\displaystyle|G^{\prime}(s)|+|G^{\prime\prime}(s)|\leq L,\ \forall
s\in\mathbb{R}$. Then, if $\hat{u}$ satisfies (1.10), there exists a constant
$C>0$ independent of $\mu$ such that

$\displaystyle D^{2}\widetilde{J}(g;\hat{u})\cdot(w^{1},w^{1})\geq
C\|w^{1}\|^{2}_{L^{2}((0,T)\times\Gamma)},\ \forall w^{1}\in
L^{2}((0,T)\times\Gamma),\ w^{1}\neq 0.$ (3.34)

In particular, the functional $\widetilde{J}$ is strictly convex in $\hat{u}$.

Proceeding as in the Proposition 2.1, the optimal control $\hat{u}$ is given
by

$\hat{u}=\frac{1}{\mu}\frac{\partial p}{\partial\nu}\ \ \mbox{on}\ \ \
(0,T)\times\Gamma,$ (3.35)

where $(q,p)$ is the solution of optimality system

$\left\\{\begin{array}[]{ccl}\displaystyle q_{t}-\Delta
q+\int_{\Omega}K(t,x,\theta)G(q(t,\theta))\ d\theta=\displaystyle
g\chi_{\omega}&\mbox{in}&Q,\\\ \displaystyle-p_{t}-\Delta
p+\int_{\Omega}K(t,x,\theta)G^{\prime}(q(t,\theta))p(t,\theta)\
d\theta=(q-q_{d})\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\ \displaystyle
q=\frac{1}{\mu}\frac{\partial p}{\partial\nu}\textbf{1}_{\Gamma},\ \ \
p=0&\mbox{on}&\Sigma,\\\ \displaystyle q(0,\cdot)=q^{0},\ \ \
p(T,\cdot)=0&\mbox{in}&\Omega.\end{array}\right.$ (3.36)

The following result holds.

###### Theorem 3.1

Under the assumptions of Proposition 3.4, there exists a positive weight
function $\varpi_{2}=\varpi_{2}(t)$ blowing up at $t=T$ such that for any
$q_{d}\in L^{2}((0,T)\times\mathcal{O}_{d})$ satisfying (1.17), there exist a
control $\hat{g}\in L^{2}((0,T)\times\omega)$ and a unique optimal control
$\hat{u}\in H^{1/2,1/4}((0,T)\times\Gamma)$ such that the corresponding
solution to (3.36) satisfies (1.12).

Proof. In order to prove Theorem 3.1, we can use the ideas of Theorem 2.1
given in the Section 2.4. Let us take $z=q-\bar{q}$, where $q$ is the solution
of (1.2) and $\bar{q}$ is the solution of (1.11). Let us rewrite (3.36) of the
form

$\left\\{\begin{array}[]{ccl}\displaystyle z_{t}-\Delta
z+\int_{\Omega}K(t,x,\theta)a(z(t,\theta))z(t,\theta)\ d\theta=\displaystyle
g\chi_{\omega}&\mbox{in}&Q,\\\ \displaystyle-p_{t}-\Delta
p+\int_{\Omega}K(t,x,\theta)b(z(t,\theta))p(t,\theta)\
d\theta=(z-z_{d})\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\ \displaystyle
z=\frac{1}{\mu}\frac{\partial p}{\partial\nu}\textbf{1}_{\Gamma},\ \ \
p=0&\mbox{on}&\Sigma,\\\ \displaystyle z(0,\cdot)=z^{0},\ \ \
p(T,\cdot)=0&\mbox{in}&\Omega,\end{array}\right.$ (3.37)

where $z^{0}:=q^{0}-\bar{q}^{0}$, $z_{d}:=q_{d}-\bar{q}$ and

$a(z)=\int_{0}^{1}G^{\prime}(\bar{y}+sz)\ ds,\ \ b(z)=G^{\prime}(z+\bar{y}).$

The condition (1.12) is equivalent to a null controllability property for $z$,
that is

$z(T,x)=0\ \mbox{in}\ \Omega.$ (3.38)

Let us consider for each $w\in L^{2}(Q)$ the following linear system
associated to (3.37)

$\left\\{\begin{array}[]{ccl}\displaystyle z_{t}-\Delta
z+a(w)\int_{\Omega}K(t,x,\theta)z(t,\theta)\ d\theta=\displaystyle
g\chi_{\omega}&\mbox{in}&Q,\\\ \displaystyle-p_{t}-\Delta
p+b(w)\int_{\Omega}K(t,x,\theta)p(t,\theta)\
d\theta=(z-z_{d})\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\ \displaystyle
z=\frac{1}{\mu}\frac{\partial p}{\partial\nu}\textbf{1}_{\Gamma},\ \ \
p=0&\mbox{on}&\Sigma,\\\ \displaystyle z(0,\cdot)=z^{0},\ \ \
p(T,\cdot)=0&\mbox{in}&\Omega,\end{array}\right.$ (3.39)

Thanks to the hypothesis on $G$ given by the Proposition 3.4, there exists a
constant $M>0$ such that

$\|a(w)\|_{L^{\infty}(Q)},\ \|b(w)\|_{L^{\infty}(Q)}\leq M,\ \forall w\in
L^{2}(Q).$

Thanks to Proposition 3.2, we can prove the following observability inequality

$\|\rho(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{Q}\frac{1}{\varpi_{2}^{2}}|\psi|^{2}\,dx\,dt\leq
C\int_{0}^{T}\int_{\omega}|\rho|^{2}\,dx\,dt,$ (3.40)

where
$C=C(\Omega,\omega,\|K\|_{\infty},\|a(w)\|_{L^{\infty}(Q)},\|b(w)\|_{L^{\infty}(Q)},T)>0$,
the function $\varpi_{2}$ is given by (3.32) and $(\rho,\psi)$ is the solution
to the following adjoint system associated to (3.39):

$\left\\{\begin{array}[]{ccl}\displaystyle-\rho_{t}-\Delta\rho+a(w)\int_{\Omega}K(t,x,\theta)\rho(t,\theta)\
d\theta=\displaystyle\psi\chi_{\mathcal{O}_{d}}&\mbox{in}&Q,\\\
\displaystyle\psi_{t}-\Delta\psi+b(w)\int_{\Omega}K(t,x,\theta)\psi(t,\theta)\
d\theta=0&\mbox{in}&Q,\\\ \displaystyle\rho=0,\ \ \
\psi=\frac{1}{\mu}\frac{\partial\rho}{\partial\nu}\textbf{1}_{\Gamma}&\mbox{on}&\Sigma,\\\
\displaystyle\rho(T,\cdot)=\rho^{T},\ \ \
\psi(0,\cdot)=0&\mbox{in}&\Omega.\end{array}\right.$ (3.41)

Thanks to estimate (3.40) and arguing as in the end of the proof of Theorem
2.1, we obtain the controllability result of the semilinear system (3.33).

## References

  * [1] B. Allal, G. Fragnelli and J Saihi. Null controllability for degenerate parabolic equations with a nonlocal space term. Discrete and Continuous Dynamical Systems- S. Doi: 10.3934/dcdss.2022164 (2022).
  * [2] F. D. Araruna, E. Fernàndez- Cara and M. C. Santos. Stackelberg-Nash exact controllability for linear and semilinear parabolic equations. ESAIM: Control, Optimization and Calculus of Variations. 21(3), 835-856 (2015).
  * [3] F. D. Araruna, E. Fernàndez- Cara and L. C. da. Silva. Hierarchical exact controllability of semilinear parabolic equations with distributed and boundary controls. Communications in Contemporary Mathematics. 22(7), 1950034 (2019), https://doi.org/10.1142/S0219199719500342.
  * [4] U. Biccari and V. Hernández-Santamaría. Null controllability of linear and semilinear nonlocal heat equations with an additive integral kernel. SIAM J. Control Optim. 57(4), 2924-2938 (2019).
  * [5] N. Carreño and M. C. Santos. Stackelberg-Nash exact controllability for the Kuramoto-Sivashinsky equation with boundary and distributed controls. Journal of Differential Equations Control. 343, 1-63 (2023).
  * [6] M. De Leo, S. F. de la Vega and D. R. Diego. Controllability of Schrödinger equation with a nonlocal term. ESAIM Control Optim. Calc. Var., 20, 23-41 (2014).
  * [7] L. de Teresa.  Insensitizing controls for a semilinear heat equation: Semilinear heat equation. Communications in Partial Differential Equations. 25(1-2), 39-72 (2000).
  * [8] Djomegne Njoukoué, L. L. Hierarchic control for a nonlinear parabolic equation in an unbounded domain. Applicable Analysis, 1-34 (2021). DOI: 10.1080/00036811.2021.1991327.
  * [9] L. L. Djomegne Njoukoué and G. Deugoué. Stackelberg control in an unbounded domain for a parabolic equation. Journal of Nonlinear Evolution Equations and Applications.Vol. 2021, No. 5, pp. 95-118, (2021).
  * [10] L. L. Djomegne Njoukoué, C. Kenne, R. Dorville and P. Zongo. Stackelberg–Nash Null Controllability for a Non Linear Coupled Degenerate Parabolic Equations. Applied Mathematics $\&$ Optimizations. 87, 1-41 (2023).
  * [11] L. L. Djomegne Njoukoué, G. Mophou and G. Deugoué. Stackelberg control of a backward linear heat equation. Advance in Evolution Equations: Evolutionary Processes and Applications. 10, 127-149 (2019).
  * [12] E. Fernàndez- Cara, Q. Lü and E. Zuazua. Null controllability of linear heat and wave equations with nonlocal spatial terms. SIAM J. Control Optim. 54(4), 2009–2019 (2016).
  * [13] R. G. Foko Tiomela, G. Mophou, and G. N’guérékata. Hierarchic control of a linear heat equation with missing data. Math. Appl. Sci. Vol 43, No 10 pp. 1-22. (2020).
  * [14] A. V. Fursikov. and Y. O. Imanuvilov. Controllability of evolution equations. Lecture Notes Series, Research Institute of Mathematics. Seoul National University, Seoul, Korea, 34 (1996).
  * [15] V. Hernández-Santamaría and L. de Teresa. Robust Stackelberg controllability for linear and semilinear heat equations. Evolution Equation and Control Theory. Vol 7, No 2, pp. 247-273, (2018).
  * [16] V. Hernández-Santamaría and L. Peralta. Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary. Discrete and Continuous Dynamical Systems Series B. 25(1), 161-190 (2020).
  * [17] V. Hernández-Santamaría and K. Le Balc’h. Null controllability of a nonlocal semilinear heat equation. Appl. Math. Optim. 84, 1435–1483 (2021).
  * [18] Y. O. Imanuvilov, J. P. Puel and M. Yamamoto. Carleman estimates for second order non homogeneous parabolic equations. to appear, (2010).
  * [19] N. I. Kavallaris and T. Suzuki. Non-local Partial Differential Equations for Engineering and Biology, Volume 31 of Mathematics for Industry (Tokyo). Mathematical Modeling and Analysis. Springer, Cham (2018).
  * [20] M. Kéré, M. Mercan and G. Mophou.  Control of Stackelberg for a coupled parabolic equations, J. Dyn. Control Syst. Vol. 23, No 4, pp. 709–733, (2017).
  * [21] J. L. Lions.  Equations différentielles opérationnelles: et problèmes aux limites. 111. Springer-Verlag. (2013).
  * [22] J. L. Lions. Some remarks on Stackelberg’s optimization, Mathematical Model and Methods in Applied Sciences Vol 4, No 4, pp 477-487, (1994).
  * [23] J. L. Lions. Hierarchic control. Proc. Math. Sci., 104(1), 295-304 (1994).
  * [24] P. Lissy and E. Zuazua. Internal controllability for parabolic systems involving analytic non-local terms. Chin. Ann. Math. Ser. B 39(2), 281–296 (2018).
  * [25] M. Mercan. Optimal control for distributed linear systems subjected to null-controllability. Applicable Analysis, Vol 92, No 9, pp 1928-1943, (2013).
  * [26] M. Mercan. Optimal Control for Distributed Linear Systems Subjected to Null Controllability with Constraints on the State. Vol. 37 of the series Springer Proceedings in Mathematics & Statistics pp 213-232, (2013).
  * [27] M. Mercan and O. Nakoulima. Control of Stackelberg for a two stroke problem. Dynamics of continuous, Discrete and Impulsive systems: Ser. B Applications & Algorithms, 22, 441-463 (2015).
  * [28] S. Micu and T. Takahashi. Local controllability to stationary trajectories of a Burgers equation with nonlocal viscosity. J. Differential Equations, 264, 3664–3703 (2018).
  * [29] G. Mophou, M. Kéré and L. L. Djomegne Njoukoué. Robust hierarchic control for a population dynamics model with missing birth rate. Math Controls Signals Syst. 32(2), 209-239 (2020).
  * [30] O. Nakoulima. Optimal control for distributed systems subject to null controllability. Application to discriminating sentinels. ESAIM: COCV. 13(4), 623-638 (2007).
  * [31] H. von Stackelberg. Markform undGleichgewicht. Springer, Berlin, Germany. Vol 4. (1934).