# Boundary Controllability of Riemann–Liouville
Fractional Semilinear Equations††This is a preprint version of the paper
published open access in ’Commun. Nonlinear Sci. Numer. Simul.’
[https://doi.org/10.1016/j.cnsns.2023.107814]. Submitted 26/Jul/2022; Revised
08/Dec/2022 and 16/Oct/2023; Accepted for publication 30/Dec/2023; Available
online 03/Jan/2024.

Asmae Tajani1,2
https://orcid.org/0000-0002-3631-5376
<EMAIL_ADDRESS>Fatima-Zahrae El Alaoui2
https://orcid.org/0000-0001-8912-4031
<EMAIL_ADDRESS>Delfim F. M. Torres1,
https://orcid.org/0000-0001-8641-2505
<EMAIL_ADDRESS>Corresponding author: telephone: +351 234 370 668; fax: +351 234
370 066; e-mail<EMAIL_ADDRESS>
(1Center for Research and Development in Mathematics and Applications (CIDMA),
Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
2TSI Team, Department of Mathematics, Faculty of Sciences,
Moulay Ismail University, 11201 Meknes, Morocco)

###### Abstract

We study the boundary regional controllability of a class of Riemann–Liouville
fractional semilinear sub-diffusion systems with boundary Neumann conditions.
The result is obtained by using semi-group theory, the fractional Hilbert
uniqueness method, and Schauder’s fixed point theorem. Conditions on the order
of the derivative, internal region, and on the nonlinear part are obtained.
Furthermore, we present appropriate sufficient conditions for the considered
fractional system to be regionally controllable and, therefore, boundary
regionally controllable. An example of a population density system with
diffusion is given to illustrate the obtained theoretical results. Numerical
simulations show that the proposed method provides satisfying results
regarding two cases of the control operator.

Keywords: Time-fractional systems; Semilinear systems; Boundary regional
controllability; Fractional diffusion logistic growth law model.

##  1 Introduction

Mathematical control theory may be seen as an interdisciplinary branch of
engineering and mathematics that deals with the behavior of dynamic systems
through many perceptions, one of them being controllability. In short,
controllability means that one can govern the system to any desired state of
the dynamical system, in finite time, using a suitable control function into
the system. The need to control a system has been visualized in many
phenomena, such as the design of electric circuits, manufacturing processes,
communication systems, and biological networks [1, 2, 3, 4]. Therefore, the
controllability question has been widely studied by many authors, for linear
and nonlinear systems, in finite and Hilbert state spaces, etc. (see, e.g.,
[5, 6, 7, 8] and references therein).

In many phenomena, we are interested to govern the system into a desired state
in a subregion of the whole domain and, in some cases, we are also interested
to impose some computational requirements. For this reason, an increasing of
interest related to regional controllability can be observed in the literature
[9, 10, 11, 12].

Regional and approximate controllability of partial differential equations
(PDEs) has been the object of intensive researches during the last decade [13,
14, 15, 16]. In particular, fractional PDEs become more and more interesting
for modeling anomalous phenomena in complex systems theory. Indeed, memory
effects, that appear in some complex processes, can be better described by
using time-fractional systems than using integer-order equations [17, 18, 19].
This is in particular true for population models, where the fractional
operators make the increasing behavior of a population slower. This has been
well documented, for instance, by Mirzazadeh et al., who presented a nonlinear
time fractional biological population model and established an efficient
method to find its exact solution [20].

Due to the large number of applications of fractional-order systems, several
papers on controllability of fractional control systems have appeared in the
literature [21, 22, 23, 24]. For the regional controllability problem, Ge et
al. studied the regional controllability of a linear diffusion system with a
Caputo derivative of order $\alpha\in(0,1]$ in two cases: when the control
operator is bounded, by using the Hilbert Uniqueness Method (HUM) approach;
and when one has an unbounded operator, by using a compactness condition [25].
Tajani et al. established sufficient conditions for regional controllability
of Riemann–Liouville fractional semilinear sub-diffusion systems using two
approaches: an analytical approach, based on the contraction mapping theorem;
and using the HUM approach, where the regional approximate controllability of
the associated linear system is assumed to hold [26, 27].

Our goal is to establish the regional boundary controllability, that is, to
provide a suitable control that is able to bring the state of the system to a
desired one in the boundary of the whole domain. That is the case, for
example, of a biological reactor in which the concentration regulation of a
substratum at the bottom of the reactor is expected. A fractional system of
Riemann–Liouville type can effectively capture the complex behavior of the
biological reactor, where the fractional derivative represents the asymmetric
heat transfer and the hysteresis effect of temperature variation. Tusset et
al. studied the controllability of a cooling fluid flow that passes through
the fermented jacket, in order to maintain the ideal temperature in the
biological reactor in two cases: integer-order system and Riemann–Liouville
type fractional system, in addition to some successful numerical results in
both cases [28]. To do the regional boundary controllability, our approach
consists to study the regional controllability in an internal subregion that
contains a part of the boundary, based on a generalization of the
Gronwall–Bellman inequality, and then to bring the state to the boundary by
projection.

Motivated by the above arguments, and to see the deviation from the classical
behavior, which is studied numerically for some real phenomena, for example
the cholera outbreak studied in [29], when Baleanu et al. demonstrate that the
fractional model of order approximated 1 is more consistent with the real data
when compared to the classical model. Our new techniques allow us to establish
the regional boundary controllability of a wider class of fractional
Riemann–Liouville systems with fractional order in $]\frac{2}{3},1]$, under a
more general assumption in the nonlinear part, that allows one to cover new
problems with relevance in some real problems, for example processes with
impulsive type initial condition to describe certain characteristics of
viscoelastic materials [30].

The text is organized in the following way. In Section 2, we present the
considered problem, we recall some definitions of fractional operators
involving Riemann–Liouville and Caputo derivatives, and we also recall the
definition of mild-solution, respectively for Riemann–Liouville fractional
systems (with a left sided derivative) and Caputo systems with right-sided
fractional derivatives. Some useful properties are also introduced, as the
regional controllability concept for the kind of systems under investigation.
In Section 3, after extending the notion of regional controllability to the
boundary case, we establish a relation between regional and boundary regional
controllability. We prove our main result under an assumption in the nonlinear
term that often is satisfied in applications of Biology. The result is proved
by extending the steps of the HUM method to the fractional-order case under
study and by using Schauder’s fixed point theorem. To illustrate the obtained
result numerically, in Section 4 we apply the proposed method to study the
controllability on a part of the boundary of a Riemann–Liouville fractional
version of the diffusion logistic growth law model. Our example shows the
importance of choosing an appropriate region $\omega$ and the location of the
actuator to obtain more efficient results on $\varGamma$ with
$\varGamma\subset\partial\omega$. We end with Section 5 of conclusions,
pointing also some possible directions of future research.

##  2 Problem Setting and Preliminaries

We start by recalling some necessary definitions from fractional calculus and
semigroup theory.

######  Definition 2.1 (See [31]).

The left sided Riemann–Liouville integral of order $\alpha>0$ for a given
integrable function $y$ is defined as

$\begin{array}[]{lll}\mathrm{I}_{0^{+}}^{\alpha}y(t)&=&\dfrac{1}{\Gamma(\alpha)}\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}y(s)ds,\qquad
t>0,\end{array}$

where $\Gamma$ is the gamma function.

######  Definition 2.2 (See [31]).

The left sided Riemann–Liouville derivative of order $0<\alpha<1$ is defined
as

$\begin{array}[]{lll}{}^{RL}\mathrm{D}_{0^{+}}^{\alpha}&=&\dfrac{d}{dt}{\mathrm{I}_{0^{+}}^{1-\alpha}}.\end{array}$

######  Definition 2.3 (See [31]).

The right sided Caputo derivative of order $0<\alpha<1$ for a given
differentiable function $y$ is defined as

$\begin{array}[]{lll}{}^{C}\mathrm{D}_{T^{-}}^{\alpha}y(t)&=&\dfrac{-1}{\Gamma(1-\alpha)}\displaystyle\int_{t}^{T}(s-t)^{\alpha-1}\dfrac{dy(s)}{ds}ds,\qquad\
0\leq t<T.\end{array}$

######  Definition 2.4 (See [32]).

Let $\mathrm{X}$ be a Banach space. We define a $C_{0}$-semigroup (a strongly
continuous semigroup) to be a family $(S(t))_{t\in\mathbb{R}^{+}}$ of bounded
linear operators from $\mathrm{X}$ into $\mathrm{X}$ that satisfies:

  * i.

$S(0)=I$, where $I$ is the identity operator of $\mathrm{X}$;

  * ii.

$S(t+s)=S(t)+S(s)$ for all $t,s\in\mathbb{R}^{+}$;

  * iii.

$\lim\limits_{t\longrightarrow 0^{+}}||S(t)x-x||_{X}=0$ for all
$x\in\mathrm{X}$.

######  Definition 2.5 (See [32]).

The infinitesimal generator of a $C_{0}$-semigroup
$(S(t))_{t\in\mathbb{R}^{+}}$ is the following linear unbounded operator:

$\mathrm{A}x=\lim\limits_{t\longrightarrow 0^{+}}\dfrac{S(t)x-x}{t},$

defined for every $x$ in its domain given by

$D(\mathrm{A})=\left\\{x\in\mathrm{X}:\lim\limits_{t\longrightarrow
0^{+}}\dfrac{S(t)x-x}{t}\ \ \text{ exists }\right\\}.$

Now we introduce the considered problem in this work. Let $\Omega$ be an open
bounded and regular subset of $\mathbb{R}^{n}$, $n\leq 3$, with a smooth
boundary $\partial\Omega$. For $T>0$, we denote $Q=\Omega\times[0,T]$ and
$\Sigma=\partial\Omega\times[0,T]$. We consider the following semilinear time-
fractional sub-diffusion system with Riemann–Liouville fractional derivative
of order $\alpha\in\,]1/2,1]$:

$\left\\{\begin{array}[]{lllll}{}^{RL}\mathrm{D}^{\alpha}_{0^{+}}y(x,t)&=&\mathrm{A}y(x,t)+Ny(x,t)+\mathcal{B}u(t)&\mbox{in}&Q,\\\
\quad\lim\limits_{t\rightarrow
0^{+}}\mathrm{I}^{1-\alpha}_{0^{+}}y(x,t)&=&y_{0}&\mbox{in}&\Omega,\\\
\dfrac{\partial
y(\xi,t)}{\partial\nu_{A}}&=&0&\mbox{on}&\Sigma,\end{array}\right.$ (1)

where ${}^{RL}\mathrm{D}^{\alpha}_{0^{+}}$ and $\mathrm{I}^{1-\alpha}_{0^{+}}$
denote, respectively, the Riemann–Liouville fractional derivative of order
$\alpha$ and the Riemann–Liouville fractional integral of order $(1-\alpha)$.
The operator $\mathrm{A}$ is an infinitesimal generator of a $C_{0}$-semigroup
$(\mathcal{S}(t)_{t>0})$ on the Hilbert space $\mathrm{X}=H^{1}(\Omega)$. In
addition, $y\in L^{2}(0,T;X)$, the initial datum $y_{0}$ is in $\mathrm{X}$,
$\nu_{A}$ is the outward unit normal to $\partial\Omega$ of the operator
$\mathrm{A}$, and $\mathcal{B}$ is the control operator, which is a bounded
(possibly unbounded) linear operator from $\mathbb{R}^{m}$ into $\mathrm{X}$,
where $m$ is the number of actuators,
$u\in\mathcal{U}:=\mathrm{L}^{2}(0,T;\mathbb{R}^{m})$, and $N$ is a Lipschitz
continuous nonlinear operator. The Riemann–Liouville fractional semilinear
system (1) is defined for $\alpha\in\,]1/2,1]$ because this is a condition for
existence and uniqueness of a mild solution.

We proceed by recalling the necessary results and notions to be applied
throughout the paper.

######  Lemma 2.6 (See [31]).

For all $\beta\geq 0$, the probability density function
$\mathrm{\phi}_{\alpha}$ satisfies

$\int_{0}^{\infty}\theta^{\beta}\mathrm{\phi}_{\alpha}(\theta)d\theta=\dfrac{\Gamma(1+\beta)}{\Gamma(1+\alpha\beta)}.$

######  Lemma 2.7 (See [33]).

Suppose that $f\in L^{2}(0,T;\mathrm{X})$ where $\mathrm{X}$ is a Banach
space. Then, for any initial datum $y_{0}$, the problem

$\left\\{\begin{array}[]{lllll}{}^{RL}\mathrm{D}^{\alpha}_{0^{+}}y(t)&=&\mathrm{A}y(t)+f(t),\\\
\quad\lim\limits_{t\rightarrow
0^{+}}\mathrm{I}^{1-\alpha}_{0^{+}}y(t)&=&y_{0}\in D(A),\end{array}\right.$
(2)

has a unique mild-solution $y\in L^{2}(0,T;\mathrm{X})$ given by

$y(t)=\mathbb{P}_{\alpha}(t)y_{0}+\displaystyle\int_{0}^{t}\mathbb{P}_{\alpha}(t-s)f(s)ds,$

where

$\mathbb{P}_{\alpha}(t)=\alpha\displaystyle\int_{0}^{\infty}\theta
t^{\alpha-1}\mathrm{\phi}_{\alpha}(\theta)\mathcal{S}(t^{\alpha}\theta)d\theta$

with $\mathrm{\phi}_{\alpha}$ the probability density function.

Now, consider the following Caputo fractional differential equation:

$\left\\{\begin{array}[]{lllll}{{}^{C}}D_{T^{-}}^{\alpha}p(t)&=&Ap(t),\\\
p(T)&=&p_{T}.\end{array}\right.$ (3)

We consider the following definition of a mild solution for (3).

######  Definition 2.8 (See [26]).

The function $p(t)\in L^{2}(0,T;X)$ defined by

$p(t)=\mathbb{S}_{\alpha}(T-t)p_{T}$

is a mild solution of system (3), where

$\mathbb{S}_{\alpha}(t)=\displaystyle\int_{0}^{\infty}\mathrm{\phi}_{\alpha}(\theta)\mathcal{S}(t^{\alpha}\theta)d\theta.$

At this point, we can define the mild solution $y(x,t,u):=y_{u}(t)$ of system
(1) by the following integral equation:

$\begin{array}[]{ll}y_{u}(t)=\mathbb{P}_{\alpha}(t)y_{0}+\displaystyle\int_{0}^{t}\mathbb{P}_{\alpha}(t-\tau)\left(Ny_{u}(\tau)+\mathcal{B}u(\tau)\right)d\tau.\end{array}$
(4)

######  Lemma 2.9 (See [34]).

For $\alpha\in(0,1)$ and $t\in[0,T]$ one has

$\mathrm{I}_{0^{+}}^{1-\alpha}(\mathbb{P}^{*}_{\alpha}(t))=\mathrm{S}^{*}_{\alpha}(t),$
(5)

where $\mathbb{P}^{*}_{\alpha}$ and $\mathrm{S}^{*}_{\alpha}$ are the adjoint
operators of $\mathbb{P}_{\alpha}$ and $\mathrm{S}_{\alpha}$, respectively.

######  Lemma 2.10 (See [35]).

For any $t>0$, $\mathrm{K}_{\alpha}(t):=t^{1-\alpha}\mathbb{P}_{\alpha}(t)$ is
a continuous linear bounded operator, i.e., there exists $M>0$ such that

$\|\mathrm{K}_{\alpha}(t)\|_{\mathcal{L}(\mathrm{X},\mathrm{X})}\leq\dfrac{M}{\Gamma(\alpha)}.$

######  Lemma 2.11 (See [36]).

For $\gamma\in(0,1]$ and $0<t_{1}\leq t_{2}$, the following inequality holds:

$|t_{1}^{\gamma}-t_{2}^{\gamma}|\leq(t_{2}-t_{1})^{\gamma}.$

Let us consider $\omega\subseteq\Omega$ and denote the projection operator on
$\omega$ by the following map:

$\begin{array}[]{llll}\chi_{{}_{\omega}}:&H^{1}(\Omega)&\longrightarrow&H^{1}(\omega)\\\
&y&\longmapsto&y_{|_{\omega}}.\end{array}$

We denote by $\chi^{*}_{{}_{\omega}}$ its adjoint operator.

We finish this section with the notion of internal regional controllability.

######  Definition 2.12 (See [35]).

(i) System (1) is said to be $\omega$-exactly controllable if for all
$y_{d}\in\mathrm{X}$ there exists a control $u\in\mathcal{U}$ such that
$\chi_{{}_{\omega}}y_{u}(T)=y_{d}$. (ii) System (1) is said to be
$\omega$-approximately controllable if for all $y_{d}\in\leavevmode\nobreak\
\mathrm{X}$, given $\varepsilon\geq 0$, there exist a control
$u\in\mathcal{U}$ such that

$\|\chi_{{}_{\omega}}y_{u}(T)-y_{d}\|_{H^{1}(\omega)}\leq\varepsilon.$

The main result of our work is to study the regional boundary controllability
in the semilinear case, which is the subject of Section 3. To obtain our
results for the semilinear case, we assume that the linear part of our
semilinear system is regionally controllable and we use available results in
the literature about the linear controllability. According to the choice of
the supports of actuators, for the controllability of the associated linear
part in a subregion $\omega$ we can use Proposition 2.13.

######  Proposition 2.13 (See [37]).

If we denote

$\mathrm{H}^{\alpha}_{\omega}u=\displaystyle\int_{0}^{T}\chi_{\omega}\mathbb{P}_{\alpha}(T-s)\mathcal{B}u(s)ds,$

then the following two properties are equivalent:

  1. 1.

the linear system associated to (1) is approximately $\omega$-controllable at
time $T$;

  2. 2.

$\overline{Im\mathrm{H}^{\alpha}_{\omega}}=H^{1}(\omega)$.

##  3 The Boundary Regional Controllability Problem

In this section, we explore the Hilbert Uniqueness Method (HUM), first
introduced by Lions in [38, 39], and further developed in 2016 and 2017 to
cover the case of fractional linear systems [40, 25]. Our goal is to find a
control that ensures the boundary regional controllability. For that, we make
use of a relation between internal and boundary regional controllability.

Let us consider the trace operator of order zero $\gamma_{0}:H^{1}(\Omega)\to
H^{\frac{1}{2}}(\partial\Omega)$ such that $\gamma_{0}y=y|_{\partial\Omega}$.
The onto operator $\gamma_{0}$ is linear and continuous. Let $\gamma_{0}^{*}$
denote its adjoint operator, which is the solution operator of the Laplace
equation

$\left\\{\begin{array}[]{lllll}\Delta u&=&0&\mbox{in}&\Omega,\\\
u+\gamma_{0}u&=&0&\mbox{on}&\partial\Omega.\end{array}\right.$ (6)

The solution of (6) exists in general. However, an explicit expression for
$\gamma_{0}^{*}$ is not available. For more details and properties about the
trace and adjoint operators, we refer the reader to [41].

If $\mathrm{\varGamma}\subseteq\partial\Omega$, then we consider
$\chi_{{}_{\mathrm{\varGamma}}}$ given by

$\begin{array}[]{rlll}\chi_{{}_{\mathrm{\varGamma}}}:&H^{\frac{1}{2}}(\partial\Omega)&\longrightarrow&H^{\frac{1}{2}}(\mathrm{\varGamma})\\\
&y&\longmapsto&y_{|_{\mathrm{\varGamma}}},\end{array}$

and we denote by $\chi^{*}_{{}_{\mathrm{\varGamma}}}$ its associated adjoint
operator.

######  Definition 3.1.

We say that system (1) is exactly (respectively approximately) boundary
controllable on $\mathrm{\Gamma}$ ($\mathcal{B}$-controllable on
$\mathrm{\Gamma}$) if for all $y_{d}\in H^{\frac{1}{2}}(\mathrm{\varGamma})$
and for all $\varepsilon>0$, there exists a control $u\in\mathcal{U}$ such
that

$\chi_{{}_{\mathrm{\varGamma}}}(\gamma_{0}y_{u}(T))=y_{d}\quad\left(\mbox{respectively
}\|\chi_{{}_{\mathrm{\varGamma}}}(\gamma_{0}y_{u}(T))-y_{d}\|_{H^{\frac{1}{2}}(\mathrm{\varGamma})}\leq\varepsilon\right).$

The following lemma gives a useful relation between internal and boundary
regional controllability on $\varGamma$.

######  Lemma 3.2 (See [37, 26]).

If $\mathrm{\varGamma}\subseteq\partial\omega$ and system (1) is
$\omega$-exactly (respectively $\omega$-approximately) controllable, then it
is exactly (respectively approximately) $\mathcal{B}$-controllable on
$\mathrm{\varGamma}$.

Throughout the rest of the paper we assume that
$\mathrm{\varGamma}\subseteq\partial\omega$, so that it is sufficient, by
Lemma 3.2, to prove the controllability results on a constructive internal
part $\omega$ of the evolution domain $\Omega$ such that
$\mathrm{\varGamma}\subseteq\partial\omega$.

To obtain regional controllability in $\omega$ of system (1), we consider the
following hypotheses:

  * $({H_{0}})$

The nonlinear operator $N$ satisfies:

$\exists(L,K)\neq(0,0)\ \mbox{ a positive pair,}\ \mbox{ such that }\
\|Ny\|_{X}\leq L\|y\|_{X}+K\|y\|^{2}_{X}.$ (7)

  * $({H_{1}})$

The fractional order derivative satisfies $\dfrac{2}{3}<\alpha\leq 1$.

The condition $\alpha\in(2/3,1]$ of hypothesis $({H_{1}})$ is necessary in
order to deal with the class of semi-linear systems that we are studying (see
the proof of Theorem 3.3). Moreover, the interval $(2/3,1]$ represents a
region that includes the classical model ($\alpha=1$) and is large enough to
describe well the deviation of the classical behavior in several applications
[29].

While system (1) is well defined for $1/2<\alpha\leq 1$, we are only able to
prove our regional controllability result under hypotheses $({H_{0}})$ and
$({H_{1}})$. We can see that $(H_{0})$ is a kind of generalization of the
conditions used in the HUM approach, and that there are many nonlinear
operators satisfying the $(H_{0})$ condition, which means that several real
phenomena are included in our model. For example, $({H_{0}})$ holds for some
important nonlinear operators that appear in biological systems, including the
logistic growth law model [42, 20].

######  Theorem 3.3.

Assume that $(H_{0})$ and $(H_{1})$ hold. In addition, let the associate
linear system to (1) be approximately $\omega$-controllable at time $T$. Then
the semilinear system (1) is exactly controllable in $\omega$ by the control
function

$u^{*}(t)=\mathcal{B}^{*^{RL}}\mathrm{D}_{T^{-}}^{1-\alpha}\varphi(t),$

where $\varphi$ is the mild solution of the following retrograde system:

$\left\\{\begin{array}[]{lllll}{{}^{C}}D_{T^{-}}^{\alpha}\varphi(t)&=&A^{*}\varphi(t),\\\
\varphi(T)&=&\chi_{{}_{\omega}}^{*}\varphi_{T}:=\varphi_{0}\in
D(A^{*}).\end{array}\right.$ (8)

In order to prove Theorem 3.3, we recall the steps of the fractional HUM
approach, that transfers our controllability problem to a solvability problem
(a fixed point problem to obtain $\varphi_{0}$).

Let us consider

$G=\left\\{y\in\mathrm{X}\quad\mbox{such that}\ y=0\ \mbox{in}\
\Omega\backslash\omega\right\\},$

in which we define the norm [26]:

$\|\varphi_{0}\|_{G}=\|\mathcal{B}^{*^{RL}}\mathrm{D}_{T^{-}}^{1-\alpha}\varphi(\cdot)\|_{U}.$

Consider system (1) controlled by $u^{*}(t)=\mathcal{B}^{*\
^{RL}}\mathrm{D}_{T^{-}}^{1-\alpha}\varphi(t)$:

$\left\\{\begin{array}[]{lllll}{}^{RL}\mathrm{D}^{\alpha}_{0^{+}}\phi(x,t)&=&\mathrm{A}\phi(x,t)+N\phi(x,t)+\mathcal{B}\mathcal{B}^{*\
^{RL}}\mathrm{D}_{T^{-}}^{1-\alpha}\varphi(t)&\mbox{in}&Q,\\\
\quad\lim\limits_{t\rightarrow
0^{+}}\mathrm{I}^{1-\alpha}_{0^{+}}\phi(x,t)&=&y_{0}&\mbox{in}&\Omega,\\\
\dfrac{\partial\phi(\xi,t)}{\partial\nu_{A}}&=&0&\mbox{on}&\Sigma,\end{array}\right.$
(9)

where $\varphi(t)=\mathbb{S}_{\alpha}^{*}(T-t)\varphi_{0}$ is the mild
solution of system (8). We decompose system (9) into three parts: the first
being linear,

$\left\\{\begin{array}[]{lll}{}^{RL}\mathrm{D}_{0^{+}}^{\alpha}\phi_{0}(t)&=&\mathrm{A}\phi_{0}(t),\\\
\lim\limits_{t\rightarrow
0^{+}}\mathrm{I}^{1-\alpha}_{0^{+}}\phi_{0}(t)&=&y_{0},\end{array}\right.$
(10)

the second linear controlled by $u^{*}$,

$\qquad\qquad\ \ \
\left\\{\begin{array}[]{lll}{}^{RL}\mathrm{D}_{0^{+}}^{\alpha}\phi_{1}(t)&=&\mathrm{A}\phi_{1}(t)+\mathcal{B}\mathcal{B}^{*^{RL}}\mathrm{D}_{T^{-}}^{1-\alpha}\varphi(t),\\\
\lim\limits_{t\rightarrow
0^{+}}\mathrm{I}^{1-\alpha}_{0^{+}}\phi_{1}(t)&=&0,\end{array}\right.$ (11)

and the third being semilinear,

$\qquad\ \
\qquad\left\\{\begin{array}[]{lll}{}^{RL}\mathrm{D}_{0^{+}}^{\alpha}\phi_{2}(t)&=&\mathrm{A}\phi_{2}(t)+N(\phi_{0}(t)+\phi_{1}(t)+\phi_{2}(t)),\\\
\lim\limits_{t\rightarrow
0^{+}}\mathrm{I}^{1-\alpha}_{0^{+}}\phi_{2}(t)&=&0,\end{array}\right.$ (12)

such that $\phi=\phi_{0}+\phi_{1}+\phi_{2}$. Let us now define the operators

$L_{o}=\chi_{\omega}^{*}\chi_{\omega}(\phi_{0}(T)),\qquad
C_{l}\varphi_{0}=\chi_{\omega}^{*}\chi_{\omega}(\phi_{1}(T)),\qquad\mbox{and}\qquad
O_{N}\varphi_{0}=\chi_{\omega}^{*}\chi_{\omega}(\phi_{2}(T)).$

Thus, the controllability problem reduces to solve

$C_{l}\varphi_{0}=\chi_{\omega}^{*}y_{d}-L_{o}-O_{N}\varphi_{0}.$ (13)

Since the associated linear system of (1) is $\omega$-approximately
controllable, then, by Lemma 3.5 in [37], $C_{l}$ is an invertible operator.
If we denote

$\mathcal{K}\varphi_{0}:=C_{l}^{-1}\chi_{\omega}^{*}y_{d}-C_{l}^{-1}L_{o}-C_{l}^{-1}O_{N}\varphi_{0},$
(14)

then the equivalent solvability problem is to find a fixed point of the
operator $\mathcal{K}$. We are now ready to prove Theorem 3.3.

###### Proof.

In order to prove Theorem 3.3, we use Schauder’s fixed point theorem. For that
we need to prove that $\mathcal{K}$ defined by (14) maps
$B(0,r):=B_{r}\subseteq G$ into itself for $r>0$ and, moreover, $\mathcal{K}$
is a compact operator. According to (14), $\mathcal{K}$ is compact if, and
only if, $O_{N}$ is compact. Let us now show that
$O_{N}(B_{r})=\\{O_{N}\varphi_{0}\quad|\quad\varphi_{0}\in B_{r}\\}$ is
relatively compact. The fact that

$O_{N}(B_{r})\subset\mathrm{E}_{N}:=\\{\chi_{\omega}^{*}\chi_{\omega}(\phi_{2}(t))\quad|\varphi_{0}\in
B_{r},\ \ t\in[0,T]\\},$

tell us that it is sufficient to show that $\mathrm{E}_{N}$ is relatively
compact. Using the Arzelà–Ascoli theorem, we can prove the relative
compactness of $\mathrm{E}_{N}$ in two steps:

  * (i)

$\mathrm{E}_{N}$ is uniformly bounded. For simplify, we denote
$P_{\omega}=\chi_{\omega}^{*}\chi_{\omega}$ and, by the linearity and
continuity of $P_{\omega}$, we have that

$\exists c_{\omega}>0\ \ \mbox{such that}\ \|P_{\omega}(\phi_{2}(t))\|_{G}\leq
c_{\omega}\|\phi_{2}(t)\|_{\mathrm{X}}.$

Moreover, $\phi_{2}(t)$ is the mild solution of (12), which is written as

$\phi_{2}(t)=\displaystyle\int_{0}^{t}\mathbb{P}_{\alpha}(t-\tau)N(\phi_{0}(\tau)+\phi_{1}(\tau)+\phi_{2}(\tau))d\tau.$

From Lemma 2.10 and hypothesis ($H_{0}$), we have

$\begin{array}[]{lll}\|\phi_{2}(t)\|_{\mathrm{X}}&\leq&\displaystyle\int_{0}^{t}\|(t-s)^{\alpha-1}K_{\alpha}(t-s)N[\phi_{0}(s)+\phi_{1}(s)+\phi_{2}(s)]\|_{\mathrm{X}}ds\\\
&\leq&\dfrac{M}{\Gamma(\alpha)}\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}\|N[\phi_{0}(s)+\phi_{1}(s)+\phi_{2}(s)]\|_{\mathrm{X}}ds\\\
&\leq&\dfrac{ML}{\Gamma(\alpha)}\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}\|\phi_{0}(s)+\phi_{1}(s)+\phi_{2}(s)\|_{\mathrm{X}}ds\\\
&&+\dfrac{MK}{\Gamma(\alpha)}\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}\|\phi_{0}(s)+\phi_{1}(s)+\phi_{2}(s)\|^{2}_{\mathrm{X}}ds,\end{array}$

and, by the same argument used in [26], we get that

$\begin{array}[]{llll}\|\phi_{2}(t)\|_{\mathrm{X}}&\leq&\dfrac{ML}{\Gamma(\alpha)}\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}\left[\|\phi_{0}(s)\|_{\mathrm{X}}+\|\phi_{1}(s)\|_{\mathrm{X}}+\|\phi_{2}(s)\|_{\mathrm{X}}\right]ds\\\
&&+\dfrac{3MK}{\Gamma(\alpha)}\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}\left[\|\phi_{0}(s)\|^{2}_{\mathrm{X}}+\|\phi_{1}(s)\|^{2}_{\mathrm{X}}+\|\phi_{2}(s)\|^{2}_{\mathrm{X}}\right]ds\\\
&\leq&\mathcal{C}(T,\varphi_{0})+\dfrac{ML}{\Gamma(\alpha)}\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}\|\phi_{2}(s)\|_{\mathrm{X}}ds\\\
&&+\dfrac{3MK}{\Gamma(\alpha)}\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}\|\phi_{2}(s)\|^{2}_{\mathrm{X}}ds,\end{array}$

where

$\begin{array}[]{lll}\mathcal{C}(T,\varphi_{0})&=&\dfrac{M^{2}LT^{2\alpha-1}}{(\Gamma(\alpha))^{2}}\left[\|y_{0}\|_{\mathrm{X}}\mathbf{B}(\alpha,\alpha)+\dfrac{T}{\alpha}\|\mathcal{B}\|_{U}\|\varphi_{0}\|_{G}\mathbf{B}(\alpha,\alpha+1)\right]\\\
&&+\dfrac{3M^{3}KT^{3\alpha-2}}{(\Gamma(\alpha))^{3}}\left[\|y_{0}\|^{2}_{\mathrm{X}}\mathbf{B}(\alpha,2\alpha-1)+\dfrac{T^{2}}{\alpha^{2}}\|\mathcal{B}\|^{2}_{U}\|\varphi_{0}\|^{2}_{G}\mathbf{B}(\alpha,2\alpha+1)\right].\end{array}$

Recalling that

$\phi_{0}(t)=t^{\alpha-1}\mathrm{K}_{\alpha}(t)y_{0},$

and

$\phi_{1}(t)=\displaystyle\int_{0}^{t}(t-\tau)^{\alpha-1}\mathrm{K}_{\alpha}(t-\tau)\mathcal{B}\mathcal{B}^{*^{RL}}\mathrm{D}_{T^{-}}^{1-\alpha}\varphi(\tau)d\tau,$

we obtain the following inequalities:

$\begin{array}[]{llll}\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}\|\phi_{0}(s)\|_{\mathrm{X}}ds&\leq&\dfrac{M}{\Gamma(\alpha)}\|y_{0}\|_{\mathrm{X}}T^{2\alpha-1}\mathbf{B}(\alpha,\alpha),\\\
\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}\|\phi_{0}(s)\|^{2}_{\mathrm{X}}ds&\leq&\left(\dfrac{M}{\Gamma(\alpha)}\right)^{2}\|y_{0}\|^{2}_{\mathrm{X}}T^{3\alpha-2}\mathbf{B}(\alpha,2\alpha-1),\\\
\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}\|\phi_{1}(s)\|_{\mathrm{X}}ds&\leq&\dfrac{M}{\Gamma(1+\alpha)}\|\mathcal{B}\|_{U}\|\varphi_{0}\|_{G}T^{2\alpha}\mathbf{B}(\alpha,\alpha+1),\\\
\displaystyle\int_{0}^{t}(t-s)^{\alpha-1}\|\phi_{1}(s)\|^{2}_{\mathrm{X}}ds&\leq&\left(\dfrac{M}{\Gamma(1+\alpha)}\right)^{2}\|\mathcal{B}\|^{2}_{U}\|\varphi_{0}\|^{2}_{G}T^{3\alpha}\mathbf{B}(\alpha,2\alpha)+1,\end{array}$

which are only valid under assumption $({H_{1}})$.

By applying the generalized Gronwall’s lemma in [43] with

$f(s)=\dfrac{ML}{\Gamma(\alpha)}(t-s)^{\alpha-1},\quad
g(s)=\dfrac{MK}{\Gamma(\alpha)}(t-s)^{\alpha-1},$

and $n=2$, we obtain that

$\begin{array}[]{lll}\|\phi_{2}(t)\|_{\mathrm{X}}&\leq&\dfrac{\mathcal{C}(T,\varphi_{0})\exp\left({\dfrac{MKT^{\alpha}}{\Gamma(1+\alpha)}}\right)}{1-\dfrac{3MT^{\alpha}K}{\Gamma(1+\alpha)}\exp\left({\dfrac{MLT^{\alpha}}{\Gamma(1+\alpha)}}\right)\mathcal{C}(T,\varphi_{0})},\end{array}$

such that $T$ satisfies the following condition:

$1-\dfrac{3MT^{\alpha}K}{\Gamma(1+\alpha)}\exp\left({\dfrac{MLT^{\alpha}}{\Gamma(1+\alpha)}}\right)\mathcal{C}(T,\varphi_{0})>0.$

It follows that

$\begin{array}[]{lll}\|\mathrm{P}(\psi_{2}(t))\|_{G}&\leq&\dfrac{c_{\omega}\mathcal{C}(T,\varphi_{0})\exp\left({\dfrac{MKT^{\alpha}}{\Gamma(1+\alpha)}}\right)}{1-\dfrac{3MT^{\alpha}K}{\Gamma(1+\alpha)}\exp\left({\dfrac{MLT^{\alpha}}{\Gamma(1+\alpha)}}\right)\mathcal{C}(T,\varphi_{0})}.\end{array}$

Thus, $\mathrm{E}_{N}$ is uniformly bounded.

  * (ii)

$\mathrm{E}_{N}$ is equicontinuous. For $\varphi_{0}\in\mathrm{E}_{N}$, let us
consider $t_{i},t_{s}$ such that $0<t_{i}<t_{s}\leq T$. Then,

$\|\phi_{2}(t_{s})-\phi_{2}(t_{i})\|_{\mathrm{X}}\leq\mathrm{J}_{1}+\mathrm{J}_{2}+\mathrm{J}_{3},$

where

$\begin{array}[]{lll}\mathrm{J}_{1}&=&\left\|\displaystyle\int_{0}^{t_{i}}[(t_{s}-\tau)^{\alpha-1}-(t_{i}-\tau)^{\alpha-1}]\mathrm{K}_{\alpha}(t_{s}-\tau)N(\phi_{0}(\tau)+\phi_{1}(\tau)+\phi_{2}(\tau))d\tau\right\|_{\mathrm{X}},\\\
\mathrm{J}_{2}&=&\left\|\displaystyle\int_{0}^{t_{i}}(t_{i}-\tau)^{\alpha-1}[\mathrm{K}_{\alpha}(t_{s}-\tau)-\mathrm{K}_{\alpha}(t_{i}-\tau)]N(\phi_{0}(\tau)+\phi_{1}(\tau)+\phi_{2}(\tau))d\tau\right\|_{\mathrm{X}},\\\
\mathrm{J}_{3}&=&\left\|\displaystyle\int_{t_{i}}^{t_{s}}(t_{s}-\tau)^{\alpha-1}\mathrm{K}_{\alpha}(t_{s}-\tau)N(\phi_{0}(\tau)+\phi_{1}(\tau)+\phi_{2}(\tau))d\tau\right\|_{\mathrm{X}}.\end{array}$

The goal in this step is to show that $J_{1}$, $J_{2}$ and $J_{3}$ tend to
zero as $t_{s}-t_{i}\longrightarrow 0$, independently of the choice of
$\varphi_{0}$ in $\mathrm{E}_{N}$, so that we get that $\mathrm{E}_{N}$ is
equicontinuous. According to Lemmas 2.10 and 2.11 and assumption (7), we
obtain that

$\begin{array}[]{lll}\mathrm{J}_{1}&\leq&\dfrac{M(t_{2}-t_{1})^{1-\alpha}}{\Gamma(\alpha)}\displaystyle\int_{0}^{t_{i}}(t_{s}-\tau)^{\alpha-1}(t_{i}-\tau)^{\alpha-1}L(\|\phi_{0}(\tau)\|_{\mathrm{X}}+\|\phi_{1}(\tau)\|_{\mathrm{X}}+\|\phi_{2}(\tau)\|_{\mathrm{X}})d\tau\\\
&+&\dfrac{M(t_{2}-t_{1})^{1-\alpha}}{\Gamma(\alpha)}\displaystyle\int_{0}^{t_{i}}(t_{s}-\tau)^{\alpha-1}(t_{i}-\tau)^{\alpha-1}K(\|\phi_{0}(\tau)\|^{2}_{\mathrm{X}}+\|\phi_{1}(\tau)\|^{2}_{\mathrm{X}}+\|\phi_{2}(\tau)\|^{2}_{\mathrm{X}})d\tau.\end{array}$

Thus, $\underset{t_{s}-t_{i}\longrightarrow 0}{\mathrm{J}_{1}\longrightarrow
0}$. On the other hand, for $t_{i}>0$ and $\nu>0$ small enough, we have

$\begin{array}[]{lll}\mathrm{J}_{2}&\leq&\underset{\tau\in[0,t_{i}-\nu]}{\sup}\|\mathrm{K}_{\alpha}(t_{s}-\tau)-\mathrm{K}_{\alpha}(t_{i}-\tau)\|_{\mathcal{L}(\mathrm{X},\mathrm{X})}\displaystyle\int_{0}^{t_{i}-\nu}(t_{i}-\tau)^{\alpha-1}\|N(\phi_{0}+\phi_{1}+\phi_{2})(\tau)\|_{\mathrm{X}}d\tau\\\
&+&\dfrac{2M}{\Gamma(\alpha)}\displaystyle\int_{t_{i}-\nu}^{t_{i}}(t_{i}-\tau)^{\alpha-1}\|N((\phi_{0}+\phi_{1}+\phi_{2})(\tau))\|_{\mathrm{X}}d\tau\\\
&\leq&\underset{\tau\in[0,t_{i}-\nu]}{\sup}\|\mathrm{K}_{\alpha}(t_{s}-\tau)-\mathrm{K}_{\alpha}(t_{i}-\tau)\|_{\mathcal{L}(\mathrm{X},\mathrm{X})}\dfrac{(t_{i}-\nu)^{\alpha-\frac{1}{2}}}{(2\alpha-1)^{\frac{1}{2}}}\\\
&\times&(L\|(\phi_{0}+\phi_{1}+\phi_{2})(\tau)\|_{L^{2}(0,T;\mathrm{X})}+K\|(\phi_{0}+\phi_{1}+\phi_{2})(\tau)\|^{2}_{L^{4}(0,T;\mathrm{X})})\\\
&+&\dfrac{2M\nu^{\alpha-\frac{1}{2}}}{(2\alpha-1)^{\frac{1}{2}}\Gamma(\alpha)}(L\|(\phi_{0}+\phi_{1}+\phi_{2})(\tau)\|_{L^{2}(0,T;\mathrm{X})}+K\|(\phi_{0}+\phi_{1}+\phi_{2})(\tau))\|^{2}_{L^{4}(0,T;\mathrm{X})}.\end{array}$

Therefore, by the continuity of the operator $\mathrm{K}_{\alpha}$, it can be
seen that $\mathrm{J}_{2}$ tends to zero as $t_{s}-t_{i}\longrightarrow 0$,
$\nu\longrightarrow 0$ for all $\varphi_{0}$ in $\mathrm{E}_{N}$. For
$\mathrm{J}_{3}$, we can also obtain the following inequality, which ensures
that the limit of $\mathrm{J}_{3}$ when $t_{s}-t_{i}$ tends to zero is zero:

$\mathrm{J}_{3}\leq\dfrac{M(t_{s}-t_{i})^{\alpha-\frac{1}{2}}}{(2\alpha-1)^{\frac{1}{2}}\Gamma(\alpha)}(L\|(\phi_{0}+\phi_{1}+\phi_{2})(\tau)\|_{L^{2}(0,T;\mathrm{X})}+K\|(\phi_{0}+\phi_{1}+\phi_{2})(\tau)\|^{2}_{L^{4}(0,T;\mathrm{X})}).$

Finally, we prove that $\mathcal{K}(B_{r})\subseteq B_{r}$. If this is not the
case, then, for each $r>0$, it is possible to find a $\varphi_{0}\in B_{r}$
such that $\|\mathcal{K}(\varphi_{0})\|_{G}>r$. From the definition of
$\mathcal{K}$, we have that

$\begin{array}[]{lll}r<\|\mathcal{K}(\varphi_{0})\|_{G}&\leq&\|C_{l}^{-1}\chi_{\omega}^{*}y_{d}-C_{l}^{-1}L_{o}\|_{G}+\|C_{l}^{-1}O_{N}\varphi_{0}\|_{G}\\\
&\leq&\|C_{l}^{-1}\chi_{\omega}^{*}y_{d}-C_{l}^{-1}L_{o}\|_{G}+\dfrac{c_{\omega}\mathcal{C}(T,r)\|C_{l}^{-1}\|_{{\mathcal{L}(G,G)}}\exp\left(\dfrac{MKT^{\alpha}}{\Gamma(1+\alpha)}\right)}{1-\dfrac{3MT^{\alpha}K}{\Gamma(1+\alpha)}\exp\left({\dfrac{MLT^{\alpha}}{\Gamma(1+\alpha)}}\right)\mathcal{C}(T,r)}.\end{array}$

Dividing both sides by $r$, and taking the lower limit as
$r\rightarrow+\infty$, we obtain a contradiction. We conclude that
$\mathcal{K}(B_{r})\subseteq B_{r}$. ∎

It should be noted that Theorem 3.3 is not a consequence of known results and
that its proof employs a novel method. Here we study the boundary regional
controllability for a class of Riemann–Liouville fractional semilinear systems
under a condition on the nonlinear part of the system that appears in some
real models, for example in the nonlinear growth population model, that is not
covered by the results of [26, 27]. To do so, we use in the proof of Theorem
3.3 the HUM approach and a generalization of the Gronwall–Bellman inequality,
which contrasts with the method used in papers [26, 27] that is based on the
standard Gronwall’s lemma.

##  4 Algorithm and Simulations

Now we consider the two-dimensional diffusive logistic population growth
nonlinear model with a Riemann–Liouville time derivative of fractional order
$0<\alpha<1$:

$\left\\{\begin{array}[]{llll}{}^{{}^{RL}}\mathrm{D}_{0^{+}}^{{}^{\alpha}}N(x,y,t)&=&\Delta
N(x,y,t)+CN(x,y,t)\left(1-\dfrac{N(x,y,t)}{K}\right)+Bu(t)&\text{ in
}\Omega\times\left]0,2\right],\\\ \dfrac{\partial
N(\xi,\mu,t)}{\partial\nu}&=&0&\mbox{on}\quad\partial\Omega\times\left]0,2\right],\\\
\lim\limits_{t\rightarrow
0^{+}}\mathcal{I}^{1-\alpha}_{0^{+}}N(x,y,t)&=&N_{0}(x,y)&\text{ in
}\Omega,\end{array}\right.$ (15)

where $\Omega=]0,\pi[\times]0,\pi[$, $N$ is the population density, $\partial
N/\partial\nu=\nu\cdot\nabla N|_{\partial\Omega}$, $t$ is in $[0,2]$, $C$ and
$K$ are positive constants, and where we consider the initial data $N_{0}$ to
be non negative and nonzero.

Note that the Neumann boundary conditions signify that the number of
individuals is fixed in the domain. Moreover, the logistic growth operator
satisfies our hypothesis. In this case, the operators $K_{\alpha}$ and
$S_{\alpha}$ have the following form:

$\mathrm{S}_{\alpha}(t)y=\sum_{j=1}^{+\infty}\sum_{k=1}^{r_{j}}\mathrm{E}_{\alpha}(\lambda_{j}t^{\alpha})<y,\xi_{jk}>_{X}\xi_{jk}(x)\
\mbox{and}\ \
\mathrm{K}_{\alpha}(t)y=\sum_{j=1}^{+\infty}\sum_{k=1}^{r_{j}}\mathrm{E}_{\alpha,\alpha}(\lambda_{j}t^{\alpha})<y,\xi_{jk}>_{X}\xi_{jk}(x),$

where $(\xi_{jk})_{j\geq 1,1\leq k\leq r_{j}}$ is a complete system of
eigenfunctions in $X$ of the Laplace operator and $\lambda_{j}$ are the
associated eigenvalues with the multiplicity $r_{j}$.

Algorithm 1

Initialization:

– Fractional order of derivative $\alpha$, time $T$, initial data $y_{0}$,
$\varGamma$ as a part of the boundary $\partial\Omega$, the control operator
$B$ (zonal or pointwise), and the desired state $y_{d}$ on $\varGamma$.

– Choose a region $\omega$ such that $\varGamma\subseteq\partial\omega$ and an
extension of $y_{d}$ defined in $\omega$.

– Choose $\varepsilon$ small enough.

– Choose an initial function $\varphi_{0}^{1}$.

– Let $k=1$.

Repeat

– Solve (8) and obtain $\varphi^{k}$.

– Solve (10) and obtain $\phi_{0}^{k}$.

– Solve (11) and obtain $\phi_{1}^{k}$.

– Solve (12) and obtain $\phi_{2}^{k}$.

– Put $\varphi_{0}^{k+1}=\mathcal{K}(\varphi_{0}^{k})$, $k:=k+1$.

Until

the control is
$u^{*}(t)=\mathcal{B}^{*^{RL}}\mathrm{D}_{T^{-}}^{1-\alpha}\varphi(t)$.

By applying Algorithm 1, we show, through a simulation study of the diffusion
logistic population growth law model, the effectiveness of our proposed method
to reach the steady state at time $T$ by means of a suitable control function
(zonal or pointwise actuator).

We can calculate the expression of $\varphi$ using the formula
$\varphi(t)=S_{\alpha}^{*}(T-t)\varphi_{0}$. Consequently, after the
decomposition of the mild solution $N(x,y,t)$, we obtain the expression of
$\psi_{0}(t)$ using the formula of $K_{\alpha}$. For obtaining $\psi_{1}(t)$,
we use the expression of $K_{\alpha}$ and $\mathrm{B}^{*}$ according to the
choice of the actuators, while for $\psi_{2}(t)$ we solve the system using the
predictor-corrector method and the explicit expression of $\psi_{0}$ and
$\psi_{1}$.

###  4.1 Example 1: Zonal Actuator

We consider system (15) with $\alpha=0.75$, $B=\chi_{{}_{D}}$,$C=1$, $K=100$
and $N_{0}(x,y)=\sqrt{xy}$. The domain of the actuator is
$D=[0.5,1]\times[0.7,1]$, the considered subregion being
$\omega=[0,0.3]\times[0,0.5]$, and the boundary subregion in $\partial\omega$
is $\varGamma=\\{0\\}\times[0,0.5]$.

Let us consider the desired state $z_{d}(y)$ on $\varGamma$, which is the
steady state $H(y)$ of the logistic growth model
$z_{d}(y):=H(y)=(1-\mu\cos(2y))$ and let
$N_{d}(x,y)=(1-\mu\cos(2x))(1-\delta\cos(2y))$ be the extension of the desired
state in $\omega$. By choosing $\mu=0.5$, we obtained Figures 1(a) and 1(b).

(a) The desired and reached states in $\Omega$ for the example of Section 4.1.

(b) The desired and reached states on $\varGamma$ for the example of Section
4.1.

Precisely, Figures 1(a) and 1(b) display the desired and the reached states in
$\omega$ and also on $\varGamma$. They show that the reached state is very
closed to the desired state with an error in $\omega$ equal to $8.0\times
10^{-3}$ and an error of order $10^{-5}$ on $\varGamma$. Our simulation
results, including the error on $\varGamma$ as a function of the chosen
subregion and the actuator location, are summarized in Table 1, which shows
that the choice of the region $\omega$ is crucial to obtain the boundary
regional controllability.

Table 1: Relation Error–Actuator–Region for the example of Section 4.1. Actuator | Region $\mathbf{\omega}$ | Error on $\mathbf{\varGamma}$
---|---|---
$[0.5,1]\times[0.7,1]$ |  $[0,0.5]\times[0,0.7]$ | $0.1179$
| $[0,1]\times[0,1]$ | $0.6392$
$[0.3,0.5]\times[0.7,1]$ |  $[0,0.3]\times[0,0.5]$ | $0.0554$
| $[0,0.1]\times[0,0.7]$ | $0.0069$
$[0,0.3]\times[0,0.1]$ |  $[0,0.5]\times[0,0.5]$ | $0.0953$
| $[0,0.5]\times[0,0.7]$ | $0.1081$

###  4.2 Example 2: Pointwise Actuator

In this example, we fix $\alpha=0.8$, $B=\delta(b_{1},b_{2})$ (pointwise
actuator), $C=1$, $K=1$, and

$N_{0}=\sqrt{xy}\exp(xy).$

Let us consider $(b_{1},b_{2})=(0,0.5)$, the desired state

$z_{d}(y)=100\left(1-\dfrac{1}{2}\cos(2y)\right)$

on $\varGamma=\\{0\\}\times[0,0.4]$ and let
$N_{d}(x,y)=2\left(1-\dfrac{1}{2}\cos(2x)\right)\left(1-\dfrac{1}{2}\cos(2y)\right)$
be the extension of the desired state in $\omega=[0,0.3]\times[0,0.4]$.

(a) The desired and reached states in $\Omega$ for the example of Section 4.2.

(b) The desired and reached states on $\varGamma$ for the example of Section
4.2.

The results obtained by the application of our method are reported in Figures
2(a) and 2(b). They show that the desired state and the steady state are close
in $\omega$, differing by $9.31\times 10^{-2}$, and on $\varGamma$ by only
$6.95\times 10^{-5}$.

Table 2: Relation Error–Actuator–Region for the example of Section 4.2. Region $\mathbf{\omega}$ | Pointwise actuator | Error on $\mathbf{\varGamma}$
---|---|---
$[0,0.3]\times[0,0.4]$ |  $(0,1)$ | $0.3884$
| $(0,0.3)$ | $0.1022$
$[0,0.5]\times[0,1]$ |  $(0,0.5)$ | $0.3969$
| $(0,0.1)$ | $0.0639$
$[0,1]\times[0,0.5]$ |  $(0,0.5)$ | $0.0363$
| $(0,0.3)$ | $0.0552$

We observe in Table 2 that the proposed method is highly sensitive with
respect to the region $\omega$ such that $\varGamma\subseteq\partial\omega$,
as well as the actuator location.

##  5 Conclusion

The Fractional Hilbert Uniqueness Method (FHUM) has been proposed for solving
a boundary regional controllability problem for a new class of semilinear
Riemann–Liouville fractional systems. We studied the regional controllability
on a subregion of the boundary evolution domain under some assumptions on the
nonlinear operator and the order of the derivative. We developed a numerical
simulation to illustrate the obtained result through an example of the
logistic growth law model with different actuators. An interesting open
question concerns the study of the gradient regional controllability of
semilinear fractional systems for fractional operators with respect to another
function. This is under investigation and will be the objective of our future
work.

## Acknowledgments

This research is part of first author’s post-doc, which is carried out at
University of Aveiro, Portugal, within the project “Mathematical Modelling of
Multiscale Control Systems: Applications to Human Diseases” (CoSysM3),
Reference 2022.03091.PTDC, financially supported by national funds (OE)
through FCT/MCTES. Tajani and Torres are also supported by Portuguese funds
through FCT-Portugal within project UIDB/04106/2020 (CIDMA) with DOI
https://doi.org/10.54499/UIDB/04106/2020. The authors are very grateful to
four anonymous referees and to an Associate Editor for their suggestions and
invaluable comments, which helped them to improve the manuscript.

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