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E. M. Ait Ben Hassi, M. Jakhoukh, L. Maniar and W. Zouhair

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# Internal null controllability for the one-dimensional heat equation with
dynamic boundary conditions

El Mustapha Ait Ben Hassi  Mariem Jakhoukh  and Lahcen Maniar  Department of
mathematics, Cadi Ayyad University, Faculty of Sciences Semlalia, LMDP-UMISCO,
B.P. 2390, 40000, Marrakesh, Morocco  Walid Zouhair* 0000-0002-5939-2737
Department of Mathematics, Ibn Zohr University, Faculty of Applied Sciences
Ait Melloul, B.P. 6146, Agadir, Morocco  Dedicated to the memory of Professor
Hammadi Bouslous

(2021; 2021; Date; Date; Date)

###### Abstract

The primary focus of this paper is to establish the internal null
controllability for the one-dimensional heat equation featuring dynamic
boundary conditions. This achievement is realized by introducing a new
Carleman estimate and an observability inequality for the corresponding
backward system. In conclusion, the paper includes a set of numerical
experiments that serve to confirm the validity of the theoretical findings and
underscore the effectiveness of the designed control with a minimal
$L^{2}$-norm.

###### keywords:

1-D heat equation, dynamic boundary conditions, Carleman estimate, null
controllability, observability inequality, Hilbert Uniqueness Method.

##  1 Introduction

The heat equation is a fundamental partial differential equation used to model
the diffusion of heat or temperature in various physical systems. It describes
how temperature changes over time and space due to heat conduction, making it
a crucial tool in fields such as physics, engineering, and materials science.
The equation considers factors like the initial temperature distribution and
the behavior of temperature at the boundaries of the region under study.
Through analytical or numerical solutions, it enables us to predict how
temperature evolves over time and find steady-state solutions, which are
valuable for designing systems that reach thermal equilibrium.

The controllability problem of the heat equation is a fascinating and
challenging topic within control theory and partial differential equations. It
addresses the question of whether, and to what extent, it is possible to
manipulate the temperature distribution in a given domain over time by
applying suitable control inputs, see for instance [1, 3, 15]. This problem
has numerous practical applications, such as designing heating or cooling
systems, optimizing thermal processes, and controlling the temperature in
various industrial and scientific setups. One of the key comments on the
controllability of the heat equation is that it depends on the boundary
conditions, which is one of the main interests of this work focusing on
dynamic boundary conditions that introduces an intriguing dimension into the
modeling of heat transfer phenomena. In contrast to traditional boundary
conditions that are often static, dynamic boundary conditions consider time-
varying factors at the domain’s boundary. This concept is particularly
valuable in scenarios where the temperature distribution within a system is
influenced by evolving external conditions or time-dependent inputs at the
boundaries. Let $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$ be an open bounded
domain, with a smooth boundary $\Gamma=\partial\Omega$, the dynamic boundary
condition is modelized by the equation

$\partial_{t}y_{\Gamma}-\delta\Delta_{\Gamma}y_{\Gamma}+\partial_{\nu}y=0,$

where $y_{\Gamma}=y_{|\Gamma}$ is the trace of the state $y$ on the boundary
$\Gamma$, $\Delta_{\Gamma}$ is the Laplace-Beltrami and $\partial_{\nu}y$ is
the normal derivative with respect to the outward unit normal vector field
$\nu$. We invite the interested reader to see [17, 13, 16].

In all the previously cited works, it has been mandatory for the diffusion
term on the boundary to be present, i.e. $\delta>0$, see [14, Remark 3.3].
This situation is notably relevant in one-dimensional scenarios. For instance
the one-dimensional heat equation with dynamic boundary conditions, which can
be exemplified by the cooling process of a uniformisotropic cylindrical solid
bar. In this scenario, both ends of the bar are in contact with a liquid
medium. The dynamic boundary conditions for this one-dimensional heat transfer
model are established based on the principle of energy conservation. According
to this principle, the heat dissipated by the bar through its ends equals the
heat absorbed by the surrounding liquid. For a more comprehensive exposition
of this concept, we refer to [12, 5].

Recently, in [10], the authors have treated this case in particular situations
where the coefficients do not depend neither in space nor time variables, and
the control functions are of separate variables, using the moments method. For
the best of our knowledge, there is no existing research that has studied
controllability (theoretically or numerically) of such system using Carleman
estimates.

In this paper, we consider the following controlled heat equation with
potential and dynamic boundary conditions

$\displaystyle\begin{aligned}
&\partial_{t}y-\partial_{x}(d(x)\partial_{x}y)+a(t,x)y=1_{\omega}v(t,x)&&\text{
in }\,Q_{T}=(0,T)\times(0,1)\\\
&\partial_{t}y(t,j)+d(j)(-1)^{j+1}\partial_{x}y(t,j)+b_{j}(t)y(t,j)=0&&\text{
on }\,(0,T),\;j=0,1\\\
&(y,y(t,0),y(t,1))_{|t=0}=(y_{0},\beta_{0},\beta_{1})&&\text{ in
}\,(0,1),\end{aligned}$ (1.1)

where the initial data $(y_{0},\beta_{0},\beta_{1})\in
L^{2}(0,1)\times\mathbb{R}^{2}$, and the coefficients $a\in
L^{\infty}(Q_{T})$, $b_{0},b_{1}\in L^{\infty}(0,T)$. Additionally, we assume
that the diffusion coefficient is a bounded function $d\in W^{1,\infty}(0,1)$
such that $d(x)\geq M\quad\text{for all}\quad x\in[0,1],$ where $M$ is a
strictly positive constant. The main purpose of this article is to establish
the null controllability of the above system. In other words, for a given
$T>0$, $\omega\Subset(0,1)$ and initial data $(y_{0},\beta_{0},\beta_{1})\in
L^{2}(0,1)\times\mathbb{R}^{2}$, we look for a control $v\in
L^{2}((0,T)\times\omega)$ such that the corresponding solution satisfies

$\left(y(T,\cdot),y(T,0),y(T,1)\right)=0\quad\text{in }(0,1).$

##  2 Results

We state our main result ensuring the null controllability of the one-
dimensional heat equation with dynamic boundary conditions.

######  Theorem 1.

For each $T>0$, each nonempty open set $\omega\Subset(0,1)$ and all initial
data $(y_{0},\beta_{0},\beta_{1})\in L^{2}(0,1)\times\mathbb{R}^{2}$ there is
a control $v\in L^{2}((0,T)\times\omega)$ such that the unique mild solution
$\left(y,y(\cdot,0),y(\cdot,1)\right)$ of the system (1.1) satisfies

$\left(y(T,\cdot),y(T,0),y(T,1)\right)=0\quad\text{in }(0,1).$

It is currently established that the null controllability of a linear system
aligns with an observability inequality for its adjoint system. To elaborate,
consider $\varphi$, the solution to the following adjoint system corresponding
to (1.1).

$\displaystyle\begin{aligned}
&-\partial_{t}\varphi-\partial_{x}(d(x)\partial_{x}\varphi)+a(t,x)\varphi=0&&\text{
in }Q_{T}\\\
&-\partial_{t}\varphi(t,j)+(-1)^{j+1}d(j)\partial_{x}\varphi(t,j)+b_{j}(t)\varphi(t,j)=0&&\text{
on }(0,T),\;j=0,1\\\
&(\varphi,\varphi(t,0),\varphi(t,1))_{|t=T}=(\varphi_{T},\gamma_{0},\gamma_{1})&&\text{
in }(0,1).\end{aligned}$ (2.1)

The strategy is to establish a new Carleman estimate for the above adjoint
system, leading to the following observability estimate.

$\|\varphi(0,\cdot)\|_{L^{2}(0,1)}^{2}+\sum_{j=0,1}|\varphi(0,j)|^{2}\leq
C\int_{0}^{T}\int_{\omega}|\varphi|^{2}\,dx\,dt,$

and thus the null controllability of the system (1.1).

The rest of the paper is organized as follows, In Section 3, the functional
setting and some useful results associated to the heat equation with dynamic
boundary conditions are presented along with the well-posedness of the
corresponding systems. In Section 4, we elaborate a new Carleman estimate for
the system (2.1). Section 5 is devoted to establish the internal null
controllability. In the final section 6, we present some numerical tests to
illustrate our controllability results.

##  3 Well-posedness of the system

In this section, we present the well-posedness of the following inhomogeneous
system

$\displaystyle\begin{aligned}
&\partial_{t}y-\partial_{x}(d(x)\partial_{x}y)+a(t,x)y=f(t,x),&&\text{ in
}Q_{T},\\\
&\partial_{t}y(t,j)+(-1)^{j+1}d(j)\partial_{x}y(t,j)+b_{j}(t)y(t,j)=g_{j}(t),&&\text{on
}(0,T),\;j=0,1,\\\
&(y,y(t,0),y(t,1))_{|t=0}=(y_{0},\beta_{0},\beta_{1}),&&\text{ in
}(0,1),\end{aligned}$ (3.1)

where $d\in W^{1,\infty}(0,1)$, $a\in L^{\infty}(Q_{T})$, and $b_{j}\in
L^{\infty}(0,T),j=0,1.$ For this end, let us introduce the following spaces:

$\displaystyle\mathbb{L}^{2}:=L^{2}(0,1)\times\mathbb{R}^{2},\qquad\mathbb{H}^{k}:=\\{(y,y(0),y(1)),\;y\in
H^{k}(0,1)\\}\qquad\text{for \ }k\in{\mathbb{N}},$
$\displaystyle\mathbb{E}_{1}(t_{0},t_{1}):=H^{1}(t_{0},t_{1};\mathbb{L}^{2})\cap\mathbb{L}^{2}(t_{0},t_{1};\mathbb{H}^{2})\,\text{for
\ }t_{1}>t_{0}\in\mathbb{R};\,\mathbb{E}_{1}:=\mathbb{E}_{1}(0,T).$

The scalar product on $\mathbb{L}^{2}$ is given by

$\langle(u,\alpha_{0},\alpha_{1}),(v,\beta_{0},\beta_{1})\rangle_{\mathbb{L}^{2}}=\langle
u,v\rangle_{L^{2}(0,1)}+\sum_{j=0,1}\alpha_{j}\beta_{j},$

where $u,v\in L^{2}(0,1)$ and $\alpha_{j},\beta_{j}\in\mathbb{R}$ for j=0,1.
Moreover,

$\|(u,\alpha_{0},\alpha_{1})\|_{\mathbb{L}^{2}}=\big{(}\|u\|_{L^{2}(0,1)}^{2}+\sum_{j=0,1}|\alpha_{j}|^{2}\big{)}^{\frac{1}{2}}$

is the associated norm to the defined scalar product.

The equation (3.1) can be rewritten as an abstract Cauchy problem on
$\mathbb{L}^{2}$ as follows

$\displaystyle\begin{aligned}
&\mathbf{Y}^{\prime}(t)=(A+B(t))\mathbf{Y}(t)+F(t),\qquad t\in(0,T]\\\
&\mathbf{Y}(0)=\mathbf{Y}_{0}=\begin{pmatrix}y_{0},\beta_{0},\beta_{1}\end{pmatrix},\end{aligned}$
(3.2)

where $\mathbf{Y}(t):=\begin{pmatrix}y(t,\cdot),y(t,0),y(t,1)\end{pmatrix}$
and the linear operator
$A:D(A)\subset\mathbb{L}^{2}\longrightarrow\mathbb{L}^{2}$ is given by

$A:=\begin{pmatrix}\partial_{x}(d(x)\partial_{x})&0&&0&\\\ \\\
d(0)\partial_{x}|_{x=0}&0&&0&\\\ \\\
-d(1)\partial_{x}|_{x=1}&0&&0&\end{pmatrix},\qquad D(A)=\mathbb{H}^{2},$

and

$B(t):=\begin{pmatrix}-a(t,\cdot)&0&0\\\ \\\ 0&-b_{0}(t)&0\\\ \\\
0&0&-b_{1}(t)\end{pmatrix},\quad F(t):=\begin{pmatrix}f(t,\cdot)\\\ \\\
g_{1}(t)\\\ \\\ g_{2}(t)\end{pmatrix}.$

The following first generation result is provided in [10, Proposition 5.2.1].

######  Proposition 2.

The operator $A$ is densely defined, self-adjoint and generates a self-adjoint
compact analytic $C_{0}$-semigroup $(e^{tA})_{t\geq 0}$ of angle
$\frac{\pi}{2}$ on $\mathbb{L}^{2}$.

Next, we recall the existence and regularity results from [14, Proposition
2.5].

######  Proposition 3.

Let $f\in L^{2}(Q_{T})$, $g_{0},g_{1}\in L^{2}(0,T),$
$(y_{0},\beta_{0},\beta_{1})\in\mathbb{L}^{2}$ and
$(\varphi_{T},\gamma_{0},\gamma_{1})\in\mathbb{L}^{2}$. Then, the following
assertions are true.

  1. (a)

There is a unique mild solution $\mathbf{Y}:=(y,y(\cdot,0),y(\cdot,1))\in
C([0,T];\mathbb{L}^{2})$ of (3.1). Moreover, $\mathbf{Y}$ belongs to
$\mathbb{E}_{1}(\tau,T)$ and solves (3.1) strongly on $(\tau,T)$ with initial
data $\mathbf{Y}(\tau):=(y(\tau,\cdot)y(\tau,0),y(\tau,1))$, for each
$\tau\in(0,T)$. There are bounded linear operators $S(t,s)$ on
$\mathbb{L}^{2}$ depending strongly continuous on $0\leq s\leq t\leq T$ such
that

$\mathbf{Y}(t)=S(t,0)\mathbf{Y}_{0}+\int_{0}^{t}S(t,s)(f(s,\cdot),g_{0}(s),g_{1}(s))ds;\quad
t\in[0,T].$

  2. (b)

Given $R>0$, there is a constant $C=C(R)>0$ such that for all $a$, $b_{0}$ and
$b_{1}$ with $\|a\|_{\infty}$, $\|b_{0}\|_{\infty}$, $\|b_{1}\|_{\infty}\leq
R$ and all data the mild solution $\mathbf{Y}$ satisfies

$\|\mathbf{Y}\|_{C([0,T],\mathbb{L}^{2})}\leq
C(\|\mathbf{Y}_{0}\|_{\mathbb{L}^{2}}+\|f\|_{L^{2}(Q_{T})}+\sum_{j=0,1}\|g_{j}\|_{L^{2}(0,T)}).$

  3. (c)

If $\mathbf{Y}_{0}\in\mathbb{H}^{1}$, then the mild solution from (a) is the
strong one.

  4. (d)

The backward system of (3.1) with $f=g_{0}=g_{1}=0$ admits a unique mild
solution given by
$(\varphi(t),\varphi(t,0),\varphi(t,1))=S^{*}(T,t)(\varphi_{T},\gamma_{0},\gamma_{1})$
where $S^{*}(T,t)$ denotes the adjoint of $S(T,t)$. It is the unique strong
solution if $(\varphi_{T},\gamma_{0},\gamma_{1})\in\mathbb{H}^{1}$.

##  4 The Carleman estimate

In this section we prove a new Carleman estimate for the following backward
adjoint problem

$\displaystyle\begin{aligned}
&-\partial_{t}\varphi-\partial_{x}(d(x)\partial_{x}\varphi)+a(t,x)\varphi=f(t,x)&&\text{
in }Q_{T},\\\
&-\partial_{t}\varphi(t,j)+(-1)^{j+1}d(j)\partial_{x}\varphi(t,j)+b_{j}(t)\varphi(t,j)=g_{j}(t)&&\text{
on }(0,T),\;j=0,1,\\\
&(\varphi,\varphi(t,0),\varphi(t,1))_{|t=T}=(\varphi_{T},\gamma_{0},\gamma_{1})&&\text{
in }(0,1).\end{aligned}$ (4.1)

We first need to introduce the weight functions, which are based on the
following auxiliary function $\eta_{0}$.

######  Lemma 1.

Given a nonempty open set $\omega^{\prime}\Subset(0,1)$, there is a function
$\eta_{0}$ that satisfies:

$\displaystyle\eta_{0}\in C^{2}(0,1),\quad\eta_{0}>0\;\;\text{ in
}(0,1),\quad\eta_{0}(0)=\eta_{0}(1)=0,$
$\displaystyle|\eta^{\prime}_{0}|>0\;\;\text{ in
}\overline{(0,1)\backslash\omega^{\prime}},\quad\partial_{\nu}\eta_{0}(j)=(-1)^{j+1}\eta_{0}^{\prime}(j)<0.$

Let $\omega^{\prime}\Subset(0,1)$, $\lambda\geq 1$, $m>1$ and $\eta_{0}$ as in
Lemma 1. Following [6], we define the weight functions $\alpha$ and $\xi$ as
follows:

$\displaystyle\alpha(t,x)$ $\displaystyle=(t(T-t))^{-1}\big{(}e^{2\lambda
m\|\eta_{0}\|_{\infty}}-e^{\lambda(m\|\eta_{0}\|_{\infty}+\eta_{0}(x))}\big{)},$
$\displaystyle\xi(t,x)$
$\displaystyle=(t(T-t))^{-1}e^{\lambda(m\|\eta_{0}\|_{\infty}+\eta_{0}(x))}$

for $(t,x)\in(0,T)\times[0,1]$. Note that $\alpha$ and $\xi$ are $C^{2}(0,1)$
positive on $(0,T)\times[0,1]$, and blow up at $t=0,$ $t=T$.

######  Remark 1.

: The properties of $\eta_{0}$ gives the following results

$\displaystyle\alpha(t,0)=\alpha(t,1)$
$\displaystyle:=\alpha(t),\quad\xi(t,0)=\xi(t,1):=\xi(t),$
$\displaystyle\partial_{x}\alpha$
$\displaystyle=-\partial_{x}\xi=-\lambda\xi\eta_{0}^{\prime},$
$\displaystyle\partial_{xx}\alpha$
$\displaystyle=-\lambda^{2}\xi\eta_{0}^{\prime
2}-\lambda\xi\eta_{0}^{\prime\prime}.$ (4.2)

Here, we state the Carleman estimate for our adjoint system (4.1).

######  Lemma 2.

Let $T>0$, $\omega$ a nonempty open subset of $(0,1)$, $a\in
L^{\infty}(Q_{T})$, and $b_{0},\,b_{1}\in L^{\infty}(0,T)$ such that, for a
given $R>0$ we have
$\|a\|_{\infty},\,\|b_{0}\|_{\infty},\,\|b_{1}\|_{\infty}\leq R$. Choose a
nonempty open set $\omega^{\prime}\Subset\omega$. Define $\eta_{0}$, $\alpha$
and $\xi$ as above with respect to $\omega^{\prime}$. Then, there exist
constants $C>0$, and $\lambda_{1},\,s_{1}\geq 1$ such that

$\displaystyle
s^{-1}\int_{Q_{T}}\xi^{-1}e^{-2s\alpha}|\partial_{t}\varphi|^{2}\,dxdt+s^{-1}\int_{Q_{T}}\xi^{-1}e^{-2s\alpha}|\partial_{x}(d(x)\partial_{x}\varphi)|^{2}\,dxdt$
$\displaystyle+s^{-1}\sum_{j=0,1}\int_{0}^{T}\xi^{-1}e^{-2s\alpha}|\partial_{t}\varphi(t,j)|^{2}\,dt+s^{3}\lambda^{4}\int_{Q_{T}}\xi^{3}e^{-2s\alpha}|\varphi|^{2}\,dx\,dt$
$\displaystyle+s\lambda^{2}\int_{Q_{T}}\xi
e^{-2s\alpha}|\partial_{x}\varphi|^{2}\,dx\,dt+s^{3}\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi^{3}e^{-2s\alpha}|\varphi(t,j)|^{2}\,dt$
$\displaystyle+s\lambda\sum_{j=0,1}\int_{0}^{T}\xi
e^{-2s\alpha}|\partial_{x}\varphi(t,j)|^{2}\,dt\leq
C\left(\int_{Q_{T}}e^{-2s\alpha}|L\varphi|^{2}\,dxdt\right.$ (4.3)
$\displaystyle+\left.\sum_{j=0,1}\int_{0}^{T}e^{-2s\alpha}|K\varphi(t,j)|^{2}\,dt+s^{3}\lambda^{4}\int_{(0,T)\times\omega}\xi^{3}e^{-2s\alpha}|\varphi|^{2}\,dx\,dt\right)$

for all $\lambda\geq\lambda_{1}$, $s\geq s_{1}$ and
$(\varphi,\varphi(\cdot,0),\varphi(\cdot,1))\in\mathbb{E}_{1}$, where

$L\varphi=\partial_{t}\varphi+\partial_{x}(d(x)\partial_{x}\varphi)-a\varphi$

and

$K\varphi(t,j)=\partial_{t}\varphi(t,j)-(-1)^{j+1}d(j)\partial_{x}\varphi(t,j)-b_{j}(t)\varphi(t,j),\qquad
j=0,1.$

###### Proof.

We shall present the strategy of the proof step by step following some ideas
in [14] in a modified form. By a density argument, it suffices to consider
smooth solutions $\varphi\in C^{\infty}([0,T]\times[0,1])$. We recall that the
constant $C$ is generic and may change from a line to another.

Step 1. Change of variables. Let $\varphi\in C^{\infty}([0,T]\times[0,1])$,
$\lambda\geq 1$ and $s\geq 1.$ We will define the operators to conjugate in
$(0,T)\times(0,1)$ and $(0,T)\times\\{0,1\\}$. For this matter, we consider
the variable change $\varphi=e^{s\alpha}\psi$ and we compute

$M\psi=e^{-s\alpha}L(e^{s\alpha}\psi)\,\text{ and
}\,N\psi(t,j)=e^{-s\alpha}K(e^{s\alpha}\psi(t,j)),\qquad j=0,1.$

Adopting the same decomposition in [14], we rewrite the operators as

$M_{1}\psi+M_{2}\psi=\tilde{f}\quad\text{in }Q_{T},\qquad
N_{1}\psi(t,j)+N_{2}\psi(t,j)=\tilde{g}_{j}\quad\text{for }j=0,1,$ (4.4)

with the abbreviations

$\displaystyle M_{1}\psi$ $\displaystyle=-s\lambda^{2}\xi\eta_{0}^{\prime
2}d(x)\psi-2s\lambda\xi\eta_{0}^{\prime}d(x)\partial_{x}\psi+\partial_{t}\psi=\sum_{i=1,2,3}(M_{1}\psi)_{i},$
(4.5) $\displaystyle M_{2}\psi$
$\displaystyle=s^{2}\lambda^{2}\xi^{2}\eta_{0}^{\prime
2}d(x)\psi+\partial_{x}(d(x)\partial_{x}\psi)+s\psi\partial_{t}\alpha=\sum_{k=1,2,3}(M_{2}\psi)_{k},$
$\displaystyle\tilde{f}$
$\displaystyle=e^{-s\alpha}L\varphi+s\lambda\xi\eta_{0}^{\prime\prime}d(x)\psi+s\lambda\xi\eta^{\prime}_{0}\partial_{x}d(x)\psi+a\psi,$
$\displaystyle N_{1}\psi(t,j)$
$\displaystyle=\partial_{t}\psi(t,j)+s\lambda\xi(t)\partial_{\nu}\eta_{0}(j)d(j)\psi(t,j)=\sum_{i=1,2}(N_{1}\psi(t,j))_{i},$
$\displaystyle N_{2}\psi(t,j)$
$\displaystyle=s\psi(t,j)\partial_{t}\alpha(t)-(-1)^{j+1}d(j)\partial_{x}\psi(t,j)=\sum_{k=1,2}(N_{2}\psi(t,j))_{k},$
$\displaystyle\tilde{g}_{j}(t)$
$\displaystyle=e^{-s\alpha}K\varphi(t,j)+b_{j}\psi(t,j)\qquad j=0,1.$

Applying the norms $\|\cdot\|_{L^{2}(Q_{T})}^{2}$ (resp.
$\|\cdot\|_{L^{2}(0,T)}^{2}$) to the equations in (4.4) and adding the
resulting identities, we obtain

$\displaystyle\|\tilde{f}\|_{L^{2}(Q_{T})}^{2}$
$\displaystyle+\sum_{j=0,1}\|\tilde{g}_{j}\|_{L^{2}(0,T)}^{2}=\|M_{1}\psi\|_{L^{2}(Q_{T})}^{2}+\|M_{2}\psi\|_{L^{2}(Q_{T})}^{2}$
$\displaystyle+\sum_{j=0,1}\|N_{1}\psi(\cdot,j)\|_{L^{2}(0,T)}^{2}+\sum_{j=0,1}\|N_{2}\psi(\cdot,j)\|_{L^{2}(0,T)}^{2}$
(4.6)
$\displaystyle+2\sum_{i,k=1,2,3}\langle(M_{1}\psi)_{i},(M_{2}\psi)_{k}\rangle_{L^{2}(Q_{T})}$
$\displaystyle+2\sum_{i,k=1,2}\sum_{j=0,1}\langle(N_{1}\psi(\cdot,j))_{i},(N_{2}\psi(\cdot,j))_{k}\rangle_{L^{2}(0,T)}.$

Step 2. Calculating the scalar product in (4.6). Step 2a. We start with the
negative term

$\langle(M_{1}\psi)_{1},(M_{2}\psi)_{1}\rangle=-s^{3}\lambda^{4}\int_{Q_{T}}\eta_{0}^{\prime
4}\xi^{3}d^{2}|\psi|^{2}\,dx\,dt.$

Using integration by parts and Remark 1, we further derive

$\displaystyle\langle(M_{1}\psi)_{2}\,,(M_{2}\psi)_{1}\rangle=3s^{3}\lambda^{4}\int_{Q_{T}}\xi^{3}\eta_{0}^{\prime
4}d^{2}|\psi|^{2}\,dx\,dt+3s^{3}\lambda^{3}\int_{Q_{T}}\xi^{3}\eta_{0}^{\prime
2}\eta_{0}^{\prime\prime}d^{2}|\psi|^{2}\,dx\,dt$
$\displaystyle+2s^{3}\lambda^{3}\int_{Q_{T}}\xi^{3}\eta^{\prime
2}_{0}d\partial_{x}d|\psi|^{2}\,dx\,dt-s^{3}\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi^{3}(t)(\partial_{\nu}\eta_{0}(j))^{3}d(j)|\psi(t,j)|^{2}\,dt.$

Integrating by parts in time, we obtain

$\displaystyle\langle(M_{1}\psi)_{3},(M_{2}\psi)_{1}\rangle$
$\displaystyle=\frac{1}{2}s^{2}\lambda^{2}\int_{Q_{T}}\eta_{0}^{\prime
2}\xi^{2}d\partial_{t}(|\psi|^{2})\,dx\,dt$
$\displaystyle=-s^{2}\lambda^{2}\int_{Q_{T}}\eta_{0}^{\prime
2}\partial_{t}\xi\xi d|\psi|^{2}\,dx\,dt.$

Hence,

$\displaystyle\sum_{i=1,2,3}\langle(M_{1}\psi)_{i},(M_{2}\psi)_{1}\rangle=2s^{3}\lambda^{4}\int_{Q_{T}}\eta_{0}^{\prime
4}\xi^{3}d^{2}|\psi|^{2}\,dx\,dt$
$\displaystyle+3s^{3}\lambda^{3}\int_{Q_{T}}\xi^{3}\eta_{0}^{\prime
2}\eta_{0}^{\prime\prime}d^{2}|\psi|^{2}\,dx\,dt+2s^{3}\lambda^{3}\int_{Q_{T}}\xi^{3}\eta^{\prime
2}_{0}d\partial_{x}d|\psi|^{2}\,dx\,dt$
$\displaystyle-s^{2}\lambda^{2}\int_{Q_{T}}\eta_{0}^{\prime
2}\partial_{t}\xi\xi
d|\psi|^{2}\,dx\,dt-s^{3}\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi^{3}(t)(\partial_{\nu}\eta_{0}(j))^{3}d(j)|\psi(t,j)|^{2}\,dt.$

Step 2b. Integration by parts yields

$\displaystyle\langle(M_{1}\psi)_{1},(M_{2}\psi)_{2}\rangle$
$\displaystyle=-s\lambda^{2}\int_{Q_{T}}\eta_{0}^{\prime 2}\xi
d\psi\partial_{x}(d(x)\partial_{x}\psi)\,dx\,dt$
$\displaystyle=s\lambda^{2}\int_{Q_{T}}\eta_{0}^{\prime 2}\xi
d^{2}|\partial_{x}\psi|^{2}\,dx\,dt+2s\lambda^{2}\int_{Q_{T}}\xi\eta_{0}^{\prime}\eta_{0}^{\prime\prime}d^{2}\partial_{x}\psi\psi\,dx\,dt$
$\displaystyle+s\lambda^{3}\int_{Q_{T}}\xi\eta_{0}^{\prime
3}d^{2}\psi\partial_{x}\psi\,dx\,dt+s\lambda\int_{Q_{T}}\xi\eta^{\prime
2}_{0}d\partial_{x}d\psi\partial_{x}\psi\,dx\,dt$
$\displaystyle-s\lambda^{2}\sum_{j=0,1}\int_{0}^{T}\eta_{0}^{\prime
2}(j)\xi(t)d^{2}(j)\psi(t,j)\partial_{\nu}\psi(t,j)\,dt.$

On the other hand, we have

$\displaystyle\langle(M_{1}\psi)_{2},$
$\displaystyle(M_{2}\psi)_{2}\rangle=-2s\lambda\int_{Q_{T}}\eta_{0}^{\prime}\xi\partial_{x}(d(x)\partial_{x}\psi)d(x)\partial_{x}\psi\,dx\,dt$
$\displaystyle=s\lambda\int_{Q_{T}}\xi\eta_{0}^{\prime\prime}d^{2}|\partial_{x}\psi|^{2}\,dx\,dt+s\lambda^{2}\int_{Q_{T}}\xi\eta_{0}^{\prime
2}d^{2}|\partial_{x}\psi|^{2}\,dx\,dt$
$\displaystyle\qquad-s\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)\partial_{\nu}\eta_{0}(j)d^{2}(j)|\partial_{x}\psi(t,j)|^{2}\,dt.$

$\partial_{x}\psi$ vanishes at $t=0$ and $t=T$. So, we obtain

$\displaystyle\langle(M_{1}\psi)_{3},(M_{2}\psi)_{2}\rangle$
$\displaystyle=\int_{Q_{T}}\partial_{t}\psi\partial_{x}(d(x)\partial_{x}\psi)\,dx\,dt$
$\displaystyle=\sum_{j=0,1}\int_{0}^{T}\partial_{t}\psi(t,j)d(j)\partial_{\nu}\psi(t,j)\,dt.$

All this yield

$\displaystyle\sum_{i=1,2,3}\langle(M_{1}\psi)_{i},(M_{2}\psi)_{2}\rangle=2s\lambda^{2}\int_{Q_{T}}\eta_{0}^{\prime
2}\xi
d^{2}|\partial_{x}\psi|^{2}\,dx\,dt+s\lambda\int_{Q_{T}}\xi\eta_{0}^{\prime\prime}d^{2}|\partial_{x}\psi|^{2}\,dx\,dt$
$\displaystyle+s\lambda^{3}\int_{Q_{T}}\xi\eta_{0}^{\prime
3}d^{2}\psi\partial_{x}\psi\,dx\,dt+s\lambda\int_{Q_{T}}\xi\eta^{\prime
2}_{0}d\partial_{x}d\psi\partial_{x}\psi\,dx\,dt$
$\displaystyle+2s\lambda^{2}\int_{Q_{T}}\xi\eta_{0}^{\prime}\eta_{0}^{\prime\prime}d^{2}\psi\partial_{x}\psi\,dx\,dt-s\lambda^{2}\sum_{j=0,1}\int_{0}^{T}\eta_{0}^{\prime
2}(j)\xi(t)d^{2}(j)\psi(t,j)\partial_{\nu}\psi(t,j)\,dt$
$\displaystyle+\sum_{j=0,1}\int_{0}^{T}\partial_{t}\psi(t,j)d(j)\partial_{\nu}\psi(t,j)\,dt-s\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)\partial_{\nu}\eta_{0}(j)d^{2}(j)|\partial_{x}\psi(t,j)|^{2}\,dt.$

Step 2c. We have,

$\displaystyle\langle(M_{1}\psi)_{1}$
$\displaystyle,(M_{2}\psi)_{3}\rangle=-s^{2}\lambda^{2}\int_{Q_{T}}\eta_{0}^{\prime
2}\partial_{t}\alpha\xi d|\psi|^{2}\,dx\,dt.$

By integrating by parts and making use of (1), we obtain

$\displaystyle\langle(M_{1}\psi)_{2},(M_{2}\psi)_{3}\rangle$
$\displaystyle=-s^{2}\lambda\int_{Q_{T}}\partial_{t}\alpha\xi\eta_{0}^{\prime}d\partial_{x}(|\psi|^{2})\,dx\,dt$
$\displaystyle=s^{2}\lambda\int_{Q_{T}}\partial_{x}(\partial_{t}\alpha)\eta_{0}^{\prime}\xi
d|\psi|^{2}\,dx\,dt+s^{2}\lambda^{2}\int_{Q_{T}}\partial_{t}\alpha\eta_{0}^{\prime
2}\xi d|\psi|^{2}\,dx\,dt$
$\displaystyle+s^{2}\lambda\int_{Q_{T}}\partial_{t}\alpha\eta_{0}^{\prime\prime}\xi
d|\psi|^{2}\,dx\,dt+s^{2}\lambda\int_{Q_{T}}\xi\eta^{\prime}_{0}\partial_{t}\alpha\partial_{x}d|\psi|^{2}\,dx\,dt$
$\displaystyle-s^{2}\lambda\sum_{j=0,1}\int_{0}^{T}\partial_{t}\alpha(t)\xi(t)\partial_{\nu}\eta_{0}(j)d(j)|\psi(t,j)|^{2}\,dt.$

Integrating by parts with respect to time, we derive

$\displaystyle\langle(M_{1}\psi)_{3},(M_{2}\psi)_{3}\rangle$
$\displaystyle=\frac{s}{2}\int_{Q_{T}}\partial_{t}\alpha\partial_{t}(|\psi|^{2})\,dx\,dt=-\frac{s}{2}\int_{Q_{T}}\partial_{tt}\alpha|\psi|^{2}\,dx\,dt.$

This provides

$\displaystyle\sum_{i=1,2,3}\langle(M_{1}\psi)_{i},(M_{2}\psi)_{3}\rangle=s^{2}\lambda\int_{Q_{T}}\partial_{x}(\partial_{t}\alpha)\eta_{0}^{\prime}\xi
d|\psi|^{2}\,dx\,dt$
$\displaystyle+s^{2}\lambda\int_{Q_{T}}\partial_{t}\alpha\xi\eta_{0}^{\prime\prime}d|\psi|^{2}\,dx\,dt+s^{2}\lambda\int_{Q_{T}}\xi\eta^{\prime}_{0}\partial_{t}\alpha\partial_{x}d|\psi|^{2}\,dx\,dt$
$\displaystyle-\frac{s}{2}\int_{Q_{T}}\partial_{tt}\alpha|\psi|^{2}\,dx\,dt-s^{2}\lambda\sum_{j=0,1}\int_{0}^{T}\partial_{t}\alpha(t)\xi(t)\partial_{\nu}\eta_{0}(j)d(j)|\psi(t,j)|^{2}\,dt.$

Step 2d. Next, we consider the boundary terms $N_{1}$ and $N_{2}$.

$\displaystyle\sum_{j=0,1}\langle(N_{1}\psi(\cdot,j))_{1},(N_{2}\psi(\cdot,j))_{1}$
$\displaystyle=\frac{s}{2}\sum_{j=0,1}\int_{0}^{T}\partial_{t}\alpha(t)\partial_{t}(|\psi(t,j)|^{2})\,dt,$
$\displaystyle=-\frac{s}{2}\sum_{j=0,1}\int_{0}^{T}\partial_{tt}\alpha(t)|\psi(t,j)|^{2}\,dt.$
$\displaystyle\sum_{j=0,1}\langle(N_{1}\psi(\cdot,j))_{2},(N_{2}\psi(\cdot,j))_{1}\rangle=s^{2}\lambda\sum_{j=0,1}\int_{0}^{T}\partial_{t}\alpha(t)\xi(t)\partial_{\nu}\eta_{0}(j)d(j)|\psi(t,j)|^{2}\,dt.$

Finally, we have the summand

$\displaystyle\sum_{j=0,1}\langle(N_{1}\psi(\cdot,j)$
$\displaystyle)_{1},(N_{2}\psi(\cdot,j))_{2}\rangle=-\sum_{j=0,1}\int_{0}^{T}d(j)\partial_{t}\psi(t,j)\partial_{\nu}\psi(t,j)\,dt,$

and

$\displaystyle\sum_{j=0,1}\langle(N_{1}\psi(\cdot,j))_{2},(N_{2}\psi(\cdot,j))_{2}\rangle=-s\lambda\sum_{j=0,1}\int_{0}^{T}\partial_{\nu}\eta_{0}(j)\xi(t)d^{2}(j)\psi(t,j)\partial_{\nu}\psi(t,j)\,dt.$

Thus,

$\displaystyle\sum_{i,k=1,2}\sum_{j=0,1}\langle(N_{1}\psi(\cdot,j))_{i},(N_{2}\psi(\cdot,j))_{k}\rangle=$
$\displaystyle-\frac{s}{2}\sum_{j=0,1}\int_{0}^{T}\partial_{tt}\alpha(t)|\psi(t,j)|^{2}\,dt-\sum_{j=0,1}\int_{0}^{T}d(t,j)\partial_{t}\psi(t,j)\partial_{\nu}\psi(t,j)\,dt$
$\displaystyle+s^{2}\lambda\sum_{j=0,1}\int_{0}^{T}\partial_{t}\alpha(t)\xi(t)\partial_{\nu}\eta_{0}(j)d(j)|\psi(t,j)|^{2}\,dt$
$\displaystyle-s\lambda\sum_{j=0,1}\int_{0}^{T}\partial_{\nu}\eta_{0}(j)\xi(t)d^{2}(j)\psi(t,j)\partial_{\nu}\psi(t,j)\,dt.$

Step 3. Estimating the terms from below. We gather the ultimate equalities in
Steps 2a–2d.

$\displaystyle\sum_{i,k=1,2,3}\langle(M_{1}\psi)_{i},(M_{2}\psi)_{k}\rangle+\sum_{i,k=1,2}\sum_{j=0,1}\langle(N_{1}\psi(\cdot,j))_{i},(N_{2}\psi(\cdot,j))_{k}\rangle$
$\displaystyle=2s^{3}\lambda^{4}\int_{Q_{T}}\eta_{0}^{\prime
4}\xi^{3}d^{2}|\psi|^{2}\,dx\,dt+3s^{3}\lambda^{3}\int_{Q_{T}}\xi^{3}\eta_{0}^{\prime
2}\eta_{0}^{\prime\prime}d^{2}|\psi|^{2}\,dx\,dt$
$\displaystyle+2s^{3}\lambda^{3}\int_{Q_{T}}\xi^{3}\eta^{\prime
2}_{0}d\partial_{x}d|\psi|^{2}\,dx\,dt-s^{2}\lambda^{2}\int_{Q_{T}}\eta_{0}^{\prime
2}\partial_{t}\xi\xi d|\psi|^{2}\,dx\,dt$
$\displaystyle+s^{2}\lambda\int_{Q_{T}}\partial_{t}\alpha\xi\eta_{0}^{\prime\prime}d|\psi|^{2}\,dx\,dt+s^{2}\lambda\int_{Q_{T}}\xi\eta^{\prime}_{0}\partial_{t}\alpha\partial_{x}d|\psi|^{2}\,dx\,dt$
$\displaystyle+s^{2}\lambda\int_{Q_{T}}\partial_{x}(\partial_{t}\alpha)\eta_{0}^{\prime}\xi
d|\psi|^{2}\,dx\,dt-\frac{s}{2}\int_{Q_{T}}\partial_{tt}\alpha|\psi|^{2}\,dx\,dt$
$\displaystyle+2s\lambda^{2}\int_{Q_{T}}\eta_{0}^{\prime 2}\xi
d^{2}|\partial_{x}\psi|^{2}\,dx\,dt+s\lambda\int_{Q_{T}}\xi\eta_{0}^{\prime\prime}d^{2}|\partial_{x}\psi|^{2}\,dx\,dt$
$\displaystyle+2s\lambda^{2}\int_{Q_{T}}\xi\eta_{0}^{\prime}\eta_{0}^{\prime\prime}d^{2}\psi\partial_{x}\psi\,dx\,dt+s\lambda^{3}\int_{Q_{T}}\xi\eta_{0}^{\prime
3}d^{2}\psi\partial_{x}\psi\,dx\,dt$ (4.7)
$\displaystyle+s\lambda\int_{Q_{T}}\xi\eta^{\prime
2}_{0}d\partial_{x}d\psi\partial_{x}\psi\,dx\,dt-s^{3}\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi^{3}(t)(\partial_{\nu}\eta_{0}(j))^{3}d(j)|\psi(t,j)|^{2}\,dt$
$\displaystyle-\frac{s}{2}\sum_{j=0,1}\int_{0}^{T}\partial_{tt}\alpha(t)|\psi(t,j)|^{2}\,dt-s\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)\partial_{\nu}\eta_{0}(j)d^{2}(j)|\partial_{x}\psi(t,j)|^{2}\,dt$
$\displaystyle-s\lambda^{2}\sum_{j=0,1}\int_{0}^{T}\eta_{0}^{\prime
2}(j)\xi(t)d^{2}(j)\psi(t,j)\partial_{\nu}\psi(t,j)\,dt$
$\displaystyle-s\lambda\sum_{j=0,1}\int_{0}^{T}\partial_{\nu}\eta_{0}(j)\xi(t)d^{2}(j)\psi(t,j)\partial_{\nu}\psi(t,j)\,dt=\sum_{l=1}^{13}I_{l}+\sum_{l=1}^{5}B_{l}.$

To estimate those terms, we will use the following basic pointwise estimates
on $[0,1]$

$\displaystyle e^{2\lambda m\|\eta_{0}\|_{\infty}}\leq
e^{2\lambda(m\|\eta_{0}\|_{\infty}+\eta_{0}(x))},\quad|\partial_{x}\alpha|,\,|\partial_{x}\xi|\leq
C\lambda\xi,\quad|\partial_{t}\alpha|,\,|\partial_{t}\xi|\leq CT\xi^{2}$
$\displaystyle|\partial_{tt}\alpha|\leq
CT^{2}\xi^{3},\quad|\partial_{x}(\partial_{t}\alpha)|\leq C\lambda
T\xi^{2}\quad\text{ and }\quad|\partial_{xx}\alpha|\leq C\lambda^{2}\xi,\text{
for }\lambda\geq 1,$ (4.8)

as well as the fact that $d\in W^{1,\infty}(0,1)$. From Lemma 1, there exists
a constant $C>0$ such that

$\displaystyle\sum_{l=1}^{3}I_{l}$ $\displaystyle\geq
2s^{3}\lambda^{4}\int_{(0,T)\times(0,1)\setminus\omega^{\prime}}\eta_{0}^{\prime
4}\xi^{3}d^{2}|\psi|^{2}\,dx\,dt+3s^{3}\lambda^{3}\int_{Q_{T}}\xi^{3}\eta_{0}^{\prime
2}\eta_{0}^{\prime\prime}d^{2}|\psi|^{2}\,dx\,dt$
$\displaystyle+2s^{3}\lambda^{3}\int_{Q_{T}}\xi^{3}\eta^{\prime
2}_{0}d\partial_{x}d|\psi|^{2}\,dx\,dt$ $\displaystyle\geq
Cs^{3}\lambda^{4}\int_{(0,T)\times(0,1)\setminus\omega^{\prime}}\xi^{3}|\psi|^{2}\,dx\,dt-
Cs^{3}\lambda^{3}\int_{Q_{T}}\xi^{3}|\psi|^{2}\,dx\,dt$ $\displaystyle\geq
Cs^{3}\lambda^{4}\int_{Q_{T}}\xi^{3}|\psi|^{2}\,dx\,dt-
Cs^{3}\lambda^{4}\int_{(0,T)\times\omega^{\prime}}\xi^{3}|\psi|^{2}\,dx\,dt$
$\displaystyle-Cs^{3}\lambda^{3}\int_{Q_{T}}\xi^{3}|\psi|^{2}\,dx\,dt$
$\displaystyle\geq Cs^{3}\lambda^{4}\int_{Q_{T}}\xi^{3}|\psi|^{2}\,dx\,dt-
Cs^{3}\lambda^{4}\int_{(0,T)\times\omega^{\prime}}\xi^{3}|\psi|^{2}\,dx\,dt$

for $\lambda\geq\lambda_{0}$ and $s\geq s_{0}$ large enough. We finally
obtain, from Lemma 1

$\displaystyle\sum_{l=1}^{3}I_{l}+B_{1}$ $\displaystyle\geq
Cs^{3}\lambda^{4}\int_{Q_{T}}\xi^{3}|\psi|^{2}\,dx\,dt-2Cs^{3}\lambda^{4}\int_{(0,T)\times\omega^{\prime}}\xi^{3}|\psi|^{2}\,dx\,dt$
$\displaystyle+Cs^{3}\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi^{3}(t)|\psi(t,j)|^{2}\,dt.$

From the estimations in (4), we obtain

$\displaystyle\sum_{l=4}^{8}I_{l}+B_{2}$ $\displaystyle\geq-
Cs^{2}\lambda^{2}(T+T^{2})\int_{Q_{T}}\xi^{3}|\psi|^{2}\,dx\,dt-
CsT^{2}\sum_{j=0,1}\int_{0}^{T}\xi^{3}(t)|\psi(t,j)|^{2}\,dt$

for $s\geq 1$ and $\lambda\geq 1$. Using again Lemma 1, we obtain

$\displaystyle I_{9}+I_{10}$ $\displaystyle\geq
Cs\lambda^{2}\int_{Q_{T}}\xi|\partial_{x}\psi|^{2}\,dx\,dt-
Cs\lambda^{2}\int_{(0,T)\times\omega^{\prime}}\xi
d^{2}|\partial_{x}\psi|^{2}\,dx\,dt.$

Next, we apply Young’s inequality to $I_{11}$, $I_{12}$ and $I_{13}$.

$\displaystyle|I_{11}|$ $\displaystyle\leq
Cs\lambda^{4}\int_{Q_{T}}\xi|\psi|^{2}\,dx\,dt+Cs\int_{Q_{T}}\xi|\partial_{x}\psi|^{2}\,dx\,dt,$
$\displaystyle|I_{12}|$ $\displaystyle\leq
Cs^{2}\lambda^{4}\int_{Q_{T}}\xi^{2}|\psi|^{2}\,dx\,dt+C\lambda^{2}\int_{Q_{T}}|\partial_{x}\psi|^{2}\,dx\,dt$

and

$\displaystyle|I_{13}|$ $\displaystyle\leq
Cs^{2}\int_{Q_{T}}\xi^{2}|\psi|^{2}\,dx\,dt+C\lambda^{2}\int_{Q_{T}}|\partial_{x}\psi|^{2}\,dx\,dt.$

It follows that

$\displaystyle\sum_{l=11}^{13}I_{l}$ $\displaystyle\geq-
Cs\lambda^{4}\int_{Q_{T}}\xi|\psi|^{2}\,dx\,dt-
Cs\int_{Q_{T}}\xi|\partial_{x}\psi|^{2}\,dx\,dt$ $\displaystyle-
Cs^{2}\lambda^{4}\int_{Q_{T}}\xi^{2}|\psi|^{2}\,dx\,dt-C\lambda^{2}\int_{Q_{T}}|\partial_{x}\psi|^{2}\,dx\,dt.$

Using Young’s inequality again on $B_{4}$ and $B_{5}$, we obtain for
$\varepsilon>0$ small enough

$\displaystyle|B_{4}|$ $\displaystyle\leq
C_{\varepsilon}s\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi(t)|\psi(t,j)|^{2}\,dt+C\varepsilon
s\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)|\partial_{x}\psi(t,j)|^{2}\,dt,$

and

$\displaystyle|B_{5}|$ $\displaystyle\leq
C_{\varepsilon}s\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)|\psi(t,j)|^{2}\,dt+C\varepsilon
s\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)|\partial_{x}\psi(t,j)|^{2}\,dt.$

It follows that

$\displaystyle B_{4}+B_{5}\geq-
C_{\varepsilon}s\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi(t)|\psi(t,j)|^{2}\,dt-C\varepsilon
s\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)|\partial_{x}\psi(t,j)|^{2}\,dt$

for $\lambda\geq 1$. Subsequently, we estimate equality (4) from below

$\displaystyle\sum_{i,k=1,2,3}\langle(M_{1}\psi)_{i},(M_{2}\psi)_{k}\rangle+\sum_{i,k=1,2}\sum_{j=0,1}\langle(N_{1}\psi(\cdot,j))_{i},(N_{2}\psi(\cdot,j))_{k}\rangle$
$\displaystyle\geq Cs^{3}\lambda^{4}\int_{Q_{T}}\xi^{3}|\psi|^{2}\,dx\,dt-
Cs^{2}\lambda^{2}(T+T^{2})\int_{Q_{T}}\xi^{3}|\psi|^{2}\,dx\,dt$
$\displaystyle-Cs\lambda^{4}\int_{Q_{T}}\xi|\psi|^{2}\,dx\,dt-
Cs^{2}\lambda^{4}\int_{Q_{T}}\xi^{2}|\psi|^{2}\,dx\,dt$ $\displaystyle-
Cs^{3}\lambda^{4}\int_{(0,T)\times\omega^{\prime}}\xi^{3}|\psi|^{2}\,dx\,dt+Cs\lambda^{2}\int_{Q_{T}}\xi|\partial_{x}\psi|^{2}\,dx\,dt$
$\displaystyle-
Cs\lambda\int_{Q_{T}}\xi|\partial_{x}\psi|^{2}\,dx\,dt-C\int_{Q_{T}}(s\xi+\lambda^{2})|\partial_{x}\psi|^{2}\,dx\,dt$
$\displaystyle-Cs\lambda^{2}\int_{(0,T)\times\omega^{\prime}}\xi
d^{2}|\partial_{x}\psi|^{2}\,dx\,dt+Cs^{3}\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi^{3}(t)|\psi(t,j)|^{2}\,dt$
$\displaystyle-CsT^{2}\sum_{j=0,1}\int_{0}^{T}\xi^{3}(t)|\psi(t,j)|^{2}\,dt-
C_{\varepsilon}s\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi(t)|\psi(t,j)|^{2}\,dt$
$\displaystyle+Cs\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)|\partial_{x}\psi(t,j)|^{2}\,dt-C\varepsilon
s\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)|\partial_{x}\psi(t,j)|^{2}\,dt.$

Hence, we obtain the inequality

$\displaystyle\sum_{i,k=1,2,3}\langle(M_{1}\psi)_{i},(M_{2}\psi)_{k}\rangle+\sum_{i,k=1,2}\sum_{j=0,1}\langle(N_{1}\psi(\cdot,j))_{i},(N_{2}\psi(\cdot,j))_{k}\rangle$
$\displaystyle\geq
Cs^{3}\lambda^{4}\int_{Q_{T}}\xi^{3}|\psi|^{2}\,dx\,dt+Cs\lambda^{2}\int_{Q_{T}}\xi|\partial_{x}\psi|^{2}\,dx\,dt$
$\displaystyle+Cs^{3}\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi^{3}(t)|\psi(t,j)|^{2}\,dt+Cs\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)|\partial_{x}\psi(t,j)|^{2}\,dt$
$\displaystyle-
Cs^{3}\lambda^{4}\int_{(0,T)\times\omega^{\prime}}\xi^{3}|\psi|^{2}\,dx\,dt-
Cs\lambda^{2}\int_{(0,T)\times\omega^{\prime}}\xi
d^{2}|\partial_{x}\psi|^{2}\,dx\,dt$

for $s\geq C(T+T^{2})$ and $\lambda\geq\lambda_{0}$ large enough.

Step 4. The transformed estimate. Combining the above estimation with (4.6),
we obtain

$\displaystyle\|\tilde{f}\|_{L^{2}(Q_{T})}^{2}$
$\displaystyle+\sum_{j=0,1}\|\tilde{g}_{j}\|_{L^{2}(0,T)}^{2}\geq\|M_{1}\psi\|_{L^{2}(Q_{T})}^{2}+\|M_{2}\psi\|_{L^{2}(Q_{T})}^{2}$
$\displaystyle+\sum_{j=0,1}\|N_{1}\psi(\cdot,j)\|_{L^{2}(0,T)}^{2}+\sum_{j=0,1}\|N_{2}\psi(\cdot,j)\|_{L^{2}(0,T)}^{2}$
$\displaystyle+Cs^{3}\lambda^{4}\int_{Q_{T}}\xi^{3}|\psi|^{2}\,dx\,dt+Cs\lambda^{2}\int_{Q_{T}}\xi|\partial_{x}\psi|^{2}\,dx\,dt$
$\displaystyle+Cs^{3}\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi^{3}(t)|\psi(t,j)|^{2}\,dt+Cs\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)|\partial_{x}\psi(t,j)|^{2}\,dt$
$\displaystyle-
Cs^{3}\lambda^{4}\int_{(0,T)\times\omega^{\prime}}\xi^{3}|\psi|^{2}\,dx\,dt-
Cs\lambda^{2}\int_{(0,T)\times\omega^{\prime}}\xi
d^{2}|\partial_{x}\psi|^{2}\,dx\,dt.$

The expressions for $\tilde{f}$ and $\tilde{g}_{j}$ lead to additional lower
order terms which can be absorbed to the left-hand side for large $\lambda$
and $s$. Indeed, we obtain

$\displaystyle\|\tilde{f}\|_{L^{2}(Q_{T})}^{2}$ $\displaystyle\leq
C\int_{Q_{T}}e^{-2s\alpha}|L\varphi|^{2}\,dxdt+Cs^{2}\lambda^{2}\int_{Q_{T}}\xi^{2}|\psi|^{2}\,dxdt+C\int_{Q_{T}}|\psi|^{2}\,dxdt$

and

$\displaystyle\sum_{j=0,1}\|\tilde{g}_{j}\|_{L^{2}(0,T)}^{2}$
$\displaystyle\leq
C\sum_{j=0,1}\int_{0}^{T}e^{-2s\alpha}|K\varphi(t,j)|^{2}\,dt+C\sum_{j=0,1}\int_{0}^{T}|\psi(t,j)|^{2}\,dt.$

Thus, for $\lambda\geq\lambda_{0}$ and $s\geq CT^{2}$, we ultimately derive
the following estimation.

$\displaystyle\|M_{1}\psi\|_{L^{2}(Q_{T})}^{2}+\|M_{2}\psi\|_{L^{2}(Q_{T})}^{2}+\sum_{j=0,1}\|N_{1}\psi(\cdot,j)\|_{L^{2}(0,T)}^{2}+\sum_{j=0,1}\|N_{2}\psi(\cdot,j)\|_{L^{2}(0,T)}^{2}$
$\displaystyle+Cs^{3}\lambda^{4}\int_{Q_{T}}\xi^{3}|\psi|^{2}\,dx\,dt+Cs\lambda^{2}\int_{Q_{T}}\xi|\partial_{x}\psi|^{2}\,dx\,dt$
(4.9)
$\displaystyle+Cs^{3}\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi^{3}(t)|\psi(t,j)|^{2}\,dt+Cs\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)|\partial_{x}\psi(t,j)|^{2}\,dt$
$\displaystyle\leq
C\int_{Q_{T}}e^{-2s\alpha}|L\varphi|^{2}\,dxdt+C\sum_{j=0,1}\int_{0}^{T}e^{-2s\alpha}|K\varphi(t,j)|^{2}\,dt$
$\displaystyle+Cs^{3}\lambda^{4}\int_{(0,T)\times\omega^{\prime}}\xi^{3}|\psi|^{2}\,dx\,dt+Cs\lambda^{2}\int_{(0,T)\times\omega^{\prime}}\xi
d|\partial_{x}\psi|^{2}\,dx\,dt.$

On the other hand, from the decomposition (4.5), We achieve

$\displaystyle s^{-1}\int_{Q_{T}}\xi^{-1}|\partial_{t}\psi|^{2}\,dxdt$
$\displaystyle\leq\|M_{1}\psi\|_{L^{2}(Q_{T})}^{2}+Cs\lambda^{2}\int_{Q_{T}}\xi|\partial_{x}\psi|^{2}\,dx\,dt$
$\displaystyle+Cs\lambda^{4}\int_{Q_{T}}\xi|\psi|^{2}\,dx\,dt$ (4.10)

for $\lambda\geq 1$ and $s\geq CT^{2}$. Combining the inequalities (4) and
(4), we obtain

$\displaystyle
s^{-1}\int_{Q_{T}}\xi^{-1}|\partial_{t}\psi|^{2}\,dxdt+\|M_{2}\psi\|_{L^{2}(Q_{T})}^{2}+\sum_{j=0,1}\|N_{1}\psi(\cdot,j)\|_{L^{2}(0,T)}^{2}$
$\displaystyle+\sum_{j=0,1}\|N_{2}\psi(\cdot,j)\|_{L^{2}(0,T)}^{2}+Cs^{3}\lambda^{4}\int_{Q_{T}}\xi^{3}|\psi|^{2}\,dx\,dt+Cs\lambda^{2}\int_{Q_{T}}\xi|\partial_{x}\psi|^{2}\,dx\,dt$
$\displaystyle+Cs^{3}\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi^{3}(t)|\psi(t,j)|^{2}\,dt+Cs\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)|\partial_{x}\psi(t,j)|^{2}\,dt$
$\displaystyle\leq
C\int_{Q_{T}}e^{-2s\alpha}|L\varphi|^{2}\,dxdt+C\sum_{j=0,1}\int_{0}^{T}e^{-2s\alpha}|K\varphi(t,j)|^{2}\,dt$
$\displaystyle\qquad\qquad+Cs^{3}\lambda^{4}\int_{(0,T)\times\omega^{\prime}}\xi^{3}|\psi|^{2}\,dx\,dt+Cs\lambda^{2}\int_{(0,T)\times\omega^{\prime}}\xi
d|\partial_{x}\psi|^{2}\,dx\,dt.$

Similarly, one can handle the identical terms on $Q_{T}$ and in
$(0,T)\times\\{0,1\\}$, which yields to

$\displaystyle
s^{-1}\int_{Q_{T}}\xi^{-1}|\partial_{t}\psi|^{2}\,dxdt+s^{-1}\int_{Q_{T}}\xi^{-1}|\partial_{x}(d(x)\partial_{x}\psi)|^{2}\,dxdt$
$\displaystyle+s^{-1}\sum_{j=0,1}\int_{0}^{T}\xi^{-1}(t)|\partial_{t}\psi(t,j)|^{2}\,dt+s^{3}\lambda^{4}\int_{Q_{T}}\xi^{3}|\psi|^{2}\,dx\,dt+s\lambda^{2}\int_{Q_{T}}\xi|\partial_{x}\psi|^{2}\,dx\,dt$
$\displaystyle+s^{3}\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi^{3}(t)|\psi(t,j)|^{2}\,dt+s\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)|\partial_{x}\psi(t,j)|^{2}\,dt$
$\displaystyle\leq
C\int_{Q_{T}}e^{-2s\alpha}|L\varphi|^{2}\,dxdt+C\sum_{j=0,1}\int_{0}^{T}e^{-2s\alpha}|K\varphi(t,j)|^{2}\,dt$
$\displaystyle+Cs^{3}\lambda^{4}\int_{(0,T)\times\omega^{\prime}}\xi^{3}|\psi|^{2}\,dx\,dt+Cs\lambda^{2}\int_{(0,T)\times\omega^{\prime}}\xi
d|\partial_{x}\psi|^{2}\,dx\,dt,$

for $\lambda\geq\lambda_{0}$ and $s\geq C(T+T^{2})$. Also using
$\omega^{\prime}\Subset\omega$, we absorb the local derivative term on the
right-hand side by the integral on $Q_{T}$ on the left-hand side. For this
matter, we introduce the function $\rho$ defined by

$\rho\in\mathcal{C}_{c}^{2}(\omega),\qquad\rho\equiv 1\text{ in
}\omega^{\prime},\qquad 0\leq\rho\leq 1.$

Two integration by parts yield

$\displaystyle s\lambda^{2}\int_{(0,T)\times\omega^{\prime}}\xi
d|\partial_{x}\psi|^{2}\,dx\,dt\leqslant
s\lambda^{2}\int_{(0,T)\times\omega}\rho\xi d|\partial_{x}\psi|^{2}\,dx\,dt$
$\displaystyle=-s\lambda^{2}\int_{(0,T)\times\omega}\xi\partial_{x}\rho
d\partial_{x}\psi\psi\,dx\,dt-s\lambda^{3}\int_{(0,T)\times\omega}\xi\rho\eta^{\prime}_{0}d\partial_{x}\psi\psi\,dx\,dt$
$\displaystyle-s\lambda^{2}\int_{(0,T)\times\omega}\rho\xi\partial_{x}(d(x)\partial_{x}\psi)\psi\,dx\,dt.$

By the same manner, for $\varepsilon>0$ we obtain

$\displaystyle-s\lambda^{2}\int_{(0,T)\times\omega}\xi\partial_{x}\rho
d\partial_{x}\psi\psi\,dx\,dt$ $\displaystyle\leq\varepsilon
s\lambda^{2}\int_{(0,T)\times\omega}\xi|\partial_{x}\psi|^{2}\,dx\,dt$
$\displaystyle+C_{\varepsilon}s\lambda^{2}\int_{(0,T)\times\omega}\xi|\psi|^{2}\,dx\,dt$

and

$\displaystyle-s\lambda^{2}\int_{(0,T)\times\omega}\rho\partial_{x}\xi
d\partial_{x}\psi\psi\,dx\,dt$ $\displaystyle\leq\varepsilon
s\lambda^{2}\int_{(0,T)\times\omega}\xi|\partial_{x}\psi|^{2}\,dx\,dt$
$\displaystyle+C_{\varepsilon}s\lambda^{4}\int_{(0,T)\times\omega}\xi|\psi|^{2}\,dx\,dt,$

also,

$\displaystyle-s\lambda^{2}\int_{(0,T)\times\omega}\rho\xi\partial_{x}(d\partial_{x}\psi)\psi\,dx\,dt$
$\displaystyle\leq\varepsilon
s^{-1}\int_{(0,T)\times\omega}\xi^{-1}|\partial_{x}(d(x)\partial_{x}\psi)|^{2}\,dx\,dt$
$\displaystyle+C_{\varepsilon}s^{3}\lambda^{4}\int_{(0,T)\times\omega}\xi^{3}|\psi|^{2}\,dx\,dt.$

We finally obtain, for a sufficiently small $\varepsilon$

$\displaystyle
s^{-1}\int_{Q_{T}}\xi^{-1}|\partial_{t}\psi|^{2}\,dxdt+s^{-1}\int_{Q_{T}}\xi^{-1}|\partial_{x}(d(x)\partial_{x}\psi)|^{2}\,dxdt$
$\displaystyle+s^{-1}\sum_{j=0,1}\int_{0}^{T}\xi^{-1}(t)|\partial_{t}\psi(t,j)|^{2}\,dt+s^{3}\lambda^{4}\int_{Q_{T}}\xi^{3}|\psi|^{2}\,dx\,dt$
$\displaystyle+s\lambda^{2}\int_{Q_{T}}\xi|\partial_{x}\psi|^{2}\,dx\,dt+s^{3}\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi^{3}(t)|\psi(t,j)|^{2}\,dt$
(4.11)
$\displaystyle+s\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)|\partial_{x}\psi(t,j)|^{2}\,dt\leq
C\left(\int_{Q_{T}}e^{-2s\alpha}|L\varphi|^{2}\,dxdt\right.$
$\displaystyle\left.+\sum_{j=0,1}\int_{0}^{T}e^{-2s\alpha}|K\varphi(t,j)|^{2}\,dt+s^{3}\lambda^{4}\int_{(0,T)\times\omega}\xi^{3}|\psi|^{2}\,dx\,dt\right)$

for $\lambda\geq\lambda_{0}$ and $s\geq C(T+T^{2})$. Finally, inserting
$\psi=e^{-s\alpha}\varphi$ into (4), we obtain

$\displaystyle
s^{-1}\int_{Q_{T}}\xi^{-1}e^{-2s\alpha}|\partial_{t}\varphi|^{2}\,dxdt+s^{-1}\int_{Q_{T}}\xi^{-1}e^{-2s\alpha}|\partial_{x}(d(x)\partial_{x}\varphi)|^{2}\,dxdt$
$\displaystyle+s^{-1}\sum_{j=0,1}\int_{0}^{T}\xi^{-1}(t)e^{-2s\alpha(t)}|\partial_{t}\varphi(t,j)|^{2}\,dt+s^{3}\lambda^{4}\int_{Q_{T}}\xi^{3}e^{-2s\alpha}|\varphi|^{2}\,dx\,dt$
$\displaystyle+s\lambda^{2}\int_{Q_{T}}\xi
e^{-2s\alpha}|\partial_{x}\varphi|^{2}\,dx\,dt+s^{3}\lambda^{3}\sum_{j=0,1}\int_{0}^{T}\xi^{3}(t)e^{-2s\alpha(t)}|\varphi(t,j)|^{2}\,dt$
$\displaystyle+s\lambda\sum_{j=0,1}\int_{0}^{T}\xi(t)e^{-2s\alpha(t)}|\partial_{x}\varphi(t,j)|^{2}\,dt\leq
C\left(\int_{Q_{T}}e^{-2s\alpha}|L\varphi|^{2}\,dxdt\right.$
$\displaystyle\left.+\sum_{j=0,1}\int_{0}^{T}e^{-2s\alpha}|K\varphi(t,j)|^{2}\,dt+s^{3}\lambda^{4}\int_{(0,T)\times\omega}\xi^{3}e^{-2s\alpha}|\varphi|^{2}\,dx\,dt\right)$

for $\lambda\geq\lambda_{1}=\max\\{\lambda_{0},1\\}$ and $s\geq
s_{1}=C(T+T^{2})$. ∎

##  5 Internal Null controllability

In this section, we utilize the Carleman estimate to establish null
controllability for (1.1). To achieve this, our initial step involves deriving
an observability estimate for the backward system (2.1).

######  Proposition 4.

There is a constant $C>0$ such that for all
$(\varphi_{T},\gamma_{0},\gamma_{1})\in\mathbb{L}^{2}$ the mild solution
$(\varphi,\varphi(\cdot,0),\varphi(\cdot,1))$ of the backward problem (2.1)
satisfies

$\displaystyle\|\varphi(0,\cdot)\|^{2}_{L^{2}(0,1)}+\sum_{j=0,1}|\varphi(0,j)|^{2}\leq
Ce^{(T+\frac{1}{T})}\int_{(0,T)\times\omega}|\varphi|^{2}\,dx\,dt.$

Moreover, for $(y_{0},\beta_{0},\beta_{1})\in\mathbb{L}^{2}$ the mild solution
$(y,y(\cdot,0),y(\cdot,1))$ of the forward problem (1.1) satisfies

$\displaystyle\|y(T,\cdot)\|^{2}_{L^{2}(0,1)}+\sum_{j=0,1}|y(T,j)|^{2}\leq
Ce^{(T+\frac{1}{T})}\int_{(0,T)\times\omega}|y|^{2}\,dx\,dt.$

###### Proof.

The proof mainly follows from the Carleman estimate (2) with an adaptation of
the same ideas in [6] and [14, Proposition 4.1]. Due to the similarity of
concepts, we omit the details. ∎

For the case of inhomogeneous one-dimensional heat equation, we introduce the
weighted $L^{2}$-spaces

$Z_{(0,1)}=\\{f\in L^{2}(Q_{T}):e^{s\alpha}\xi^{-3/2}f\in L^{2}(Q_{T})\\},$

provided with scalar product given by

$\langle
f_{1},f_{2}\rangle_{Z_{(0,1)}}=\int_{Q_{T}}f_{1}f_{2}e^{-2s\alpha}\xi^{3}\,dxdt,$

and

$Z_{j}=\\{g_{j}\in:e^{s\alpha}\xi^{-3/2}g_{j}\in L^{2}(0,T)\\},$

with the following scalar product

$\langle
g^{1}_{j},g^{2}_{j}\rangle_{Z_{j}}=\int_{0}^{T}g^{1}_{j}g^{2}_{j}e^{-2s\alpha}\xi^{3}\,dt.$

We note that those inhomogeneities are exponentially decaying at $t=0$ and
$t=T$. Then, we obtain the following result for which the proof can be derived
by applying similar ideas of [14, Theorem 4.2].

######  Theorem 5.

For each $T>0$, each nonempty open set $\omega\Subset(0,1)$ and all initial
data $(y_{0},\beta_{0},\beta_{1})\in\mathbb{L}^{2}$, $f\in Z_{(0,1)}$ and
$g_{j}\in Z_{j}$, there is a control $v\in L^{2}((0,T)\times\omega)$ such that
the unique mild solution $\left(y,y(\cdot,0),y(\cdot,1)\right)$ of the system
(3.1) satisfies

$\left(y(T,\cdot),y(T,0),y(T,1)\right)=0\quad\text{in}\quad(0,1).$

##  6 Numerical experiments

In the following section we shall present some numerical tests to illustrate
our controllability result for the following one-dimensional heat equation
with potential and dynamic boundary conditions

$\displaystyle\begin{aligned}
&\partial_{t}y(t,x)-\partial_{xx}y(t,x)+y(t,x)=1_{\omega}v(t,x)&&\,\text{ in
}Q_{T},\\\ &\partial_{t}y(t,0)-\partial_{x}y(t,0)=0&&\,\text{on }(0,T),\\\
&\partial_{t}y(t,1)+\partial_{x}y(t,1)=0&&\,\text{on }(0,T),\\\
&\left(y\left(0,\cdot\right),y\left(0,0\right),y\left(0,1\right)\right)=\left(y^{0},c,d\right):=\vartheta^{0}&&\,\text{
in }(0,1),\\\ \end{aligned}$ (6.1)

where $T>0$ is the final time, $\omega\Subset(0,1)$ is a nonempty open subset,
$\vartheta^{0}\in\mathbb{L}^{2}$ denotes the initial data, $a\in
L^{\infty}(Q_{T})$ and $v\in L^{2}((0,T)\times\omega)$ is the control
function.

Let $\varepsilon>0$ be fixed, we define the cost functional
$J_{T,\varepsilon}:\mathbb{L}^{2}\rightarrow\mathbb{R}$ as follows:

$J_{T,\varepsilon}(\varphi^{0})=\frac{1}{2}\int_{0}^{T}\int_{\omega}\varphi^{2}(t,x)\,\,dx\,dt+\frac{\varepsilon}{2}\|\Phi^{0}\|^{2}_{\mathbb{L}^{2}}+\big{\langle}\vartheta^{0},\Phi(\cdot,T)\big{\rangle}_{\mathbb{L}^{2}},$
(6.2)

where $\Phi:=(\varphi,\varphi(\cdot,0),\varphi(\cdot,1))$ is the solution of
the following homogeneous system

$\displaystyle\begin{aligned}
&\partial_{t}\varphi(t,x)-\partial_{xx}\varphi(t,x)+\varphi(t,x)=0&&\,\text{
in }Q_{T},\\\ &\partial_{t}\varphi(t,0)-\partial_{x}\varphi(t,0)=0&&\,\text{
on }(0,T),\\\ &\partial_{t}\varphi(t,1)+\partial_{x}\varphi(t,1)=0&&\,\text{
on }(0,T),\\\
&\left(\varphi\left(0,\cdot\right),\varphi\left(0,0\right),\varphi\left(0,1\right)\right)=\left(\varphi^{0},c,d\right):=\Phi^{0}&&\,\text{
in }(0,1).\\\ \end{aligned}$ (6.3)

The numerical method used here is designed to compute the HUM control ( see,
[2] for a detailed description). This will be done based on a penalized HUM
approach combined with a Conjugate Gradient Algorithm, see [7].

For the numerical resolution of systems with dynamic boundary conditions, we
discretize only the space domain $[0,1]$ into a uniform grid
$(x_{j})_{j=0}^{N_{x}}$ of step $\rho=\frac{1}{N_{x}}$, and leave the time
variable continuous.

In the sequel, we will present some numerical experiments. For convenience,
the system parameters are taken as follows

$T=0.03,\quad N_{x}=25,\quad\omega=(0.2,0.8),$

we consider the initial datum and the potential terms as

$\vartheta^{0}(x)=7(1-x)\log(1+x^{2}),\,\,\text{in}\,\,[0,1].$

Let us set the regularization parameter $\varepsilon=\frac{1}{25^{2}}$, and
stopping parameter $e_{\mathcal{J}}=10^{-6}$. The algorithm stops at iteration
$n=70$.

Next, we plot the controlled and the uncontrolled solutions

Figure 1: The controlled and the uncontrolled solutions. Figure 2: The final
state for controlled and uncontrolled solutions. Figure 3: The computed
control $v$. Table 1: Numerical results for $T=0.03$ and
$e_{\mathcal{J}}=\frac{1}{25^{3}}$

.  $\epsilon$ $10^{-2}$ $10^{-4}$ $10^{-6}$ $N_{\text{iter}}$ $15$ $45$ $70$
$\|\vartheta(T)\|$ $1.226\times 10^{-2}$ $5.402\times 10^{-3}$ $5.397\times
10^{-3}$ $\|v\|$ $5.007$ $5.047$ $5.048$

Based on the outcomes of the numerical experiment, we notice that the
minimally distributed control we’ve developed exhibits high efficiency. It
swiftly drives the solution of the heat equation with dynamic boundary
conditions toward zero. Furthermore, we observe that the solution’s norm
decreases as the number of iterations increases, a behavior influenced by the
reduction in the stopping criterion for the algorithm. This behavior is
entirely expected and normal.

######  Remark 2.

To simplify the numerical aspect, we have assumed that $d(\cdot)=1$,
$b_{j}(\cdot)=0$, and $a(\cdot,\cdot)=1$. However, it’s noteworthy that the
devised algorithm continues to converge even without these assumptions, albeit
requiring more iterations.

##  7 Conclusion and Perspectives

This study established an observability estimate for the one-dimensional heat
equation with dynamic boundary conditions, subsequently applying it to obtain
the internal null controllability of the system (1.1). The proof is based on
the Carleman approach, and a numerical simulation is carried out to visually
validate the theoretical findings on the internal null controllability.

In the existing literature, the controllability of semi-linear systems with
static boundary conditions has been extensively studied, see for instance [8,
9] and the references therein. But in the context of dynamic boundary
conditions, there is little work available in this area, which opens the door
to possible extension. Furthermore, this work opens up possibilities for
generalization, including the examination of perturbations such as
nonlinearities, delays, non-local conditions. Exploring these aspects would
contribute significantly to our understanding of these complex problems.

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# Altering the magnetic ordering of Fe3Ga4 via thermal annealing and
hydrostatic pressure

Brandon Wilfong  Physics Department, United States Naval Academy, Annapolis,
MD 20899, USA  Vaibhav Sharma  Mechanical and Nuclear Engineering, Virginia
Commonwealth University, Richmond, VA 23220, USA  Jared Naphy  Physics
Department, United States Naval Academy, Annapolis, MD 20899, USA  Omar Bishop
Mechanical and Nuclear Engineering, Virginia Commonwealth University,
Richmond, VA 23220, USA  Steven P. Bennett  Material Science and Technology
Division, U.S. Naval Research Laboratory, Washington, DC, 20375, USA  Joseph
Prestigiacomo  Material Science and Technology Division, U.S. Naval Research
Laboratory, Washington, DC, 20375, USA  Radhika Barua  Mechanical and Nuclear
Engineering, Virginia Commonwealth University, Richmond, VA 23220, USA
Michelle E. Jamer  Physics Department, United States Naval Academy, Annapolis,
MD 20899, USA<EMAIL_ADDRESS>
###### Abstract

The effects of post-synthesis annealing temperature on arc-melted samples of
Fe3Ga4 has been studied to investigate changes in crystallographic and
magnetic properties induced by annealing. Results show a significant trend in
the evolution of the (incommensurate spin density wave) ISDW-FM
(ferromagnetic) transition temperature as a function of the refined unit cell
volume in annealed samples. Strikingly, this trend allowed for the tuning of
the transition temperature down to room-temperature (300 K) whilst maintaining
a sharp transition in temperature, opening the door to the use of Fe3Ga4 in
functional devices. Crystallographic analysis through Rietveld refinement of
high-resolution x-ray diffraction data has showed that arc-melted
stoichiometric Fe3Ga4 is multi-phase regardless of annealing temperature with
a minor phase of FeGa3 decreasing in phase fraction at higher annealing
temperature. In order to validate the trend in ISDW-FM transition temperature
with regard to unit cell volume, high pressure magnetometry was performed.
This showed that the FM-ISDW ($\sim$ 68 K) and ISDW-FM ($\sim$ 360 K)
transition temperatures could be tuned, increased and decreased respectively,
linearly with external pressure. Thus, external pressure and the ensuing
crystallographic changes minimize the temperature range of the stability of
the ISDW pointing toward the importance of structural properties on the
mechanism for the formation of the intermediate ISDW phase. These results show
how this model system can be tuned as well as highlighting the need for future
high-pressure crystallography and related single crystal measurements to
understand the mechanism and nature of the intermediate ISDW phase to be
exploited in future devices.

magnetically ordered materials, transition metal alloys and compounds, high-
pressure, X-ray diffraction, metamagnetism

##  I Introduction

There has been a push in recent years to find materials for use in the next-
generation magnetic devices for spintronic and multicaloric applications.
Hoffmann ; Gschneidnerj ; Jungwirth ; Stern ; Barua2  One class of interesting
materials are antiferromagnets (AFM) with room temperature or near-room
temperature magnetic transitions such as FeRh or metallic antiferromagnets.
Manekar ; Barua ; Cress ; Bennett ; Roy ; Yu  Metallic antiferromagnets allow
for exotic interplay between spin, charge, and magnetization dynamics, which
is a very exciting prospect for future devices. Hu ; Siddiqui  Both avenues
offer promise for useful innovations and a concerted effort toward material
discovery is required to identify materials with ideal properties. To that
end, one recently re-discovered material Fe3Ga4 exhibits both types of
behavior which makes it a exemplary candidate for investigation from a
fundamental point-of-view in order to understand the complex magnetic and
physical properties as well as for possible use in multi-functional devices.
Mendez

Fe3Ga4 crystallizes in the $C2/m$ monoclinic space group with four unique Fe
atoms which house a large amount of possibly significant nearest and next-
nearest neighbor interactions which drive the magnetic properties in this
system. Philippe  Previous studies on Fe3Ga4 have indicated that it displays
multiple magnetic transitions from a ferromagnetic (FM) ground state to an
intermediate AFM phase at 68 K to a re-entrant FM phase at 360 K, and a
transition at 420 K to a paramagnetic (PM) state. Mendez ; Al1995 ; Samatham ;
Kawamiya  The high temperature transition from AFM to FM at 360 K is of
particular interest due to its proximity to room temperature as well as the
unknown nature of the intermediate AFM phase. WuThesis ; DuijnThesis  Recent
work has shown evidence that the FM ground state and re-entrant FM state
between 360 K and 420 K are the same state which means the intermediate AFM
state is formed through a instability in the FM ground state. This has been
supported by neutron diffraction and first-principles calculation which have
shown evidence that the true nature of the intermediate AFM phase is an
incommensurate spin density (ISDW) wave which is formed through a Fermi
surface instability of the FM state. Wu  Unlike other examples where an ISDW
forms from a FM state,Bhattacharyya ; Abdul ; Niklowitz  the ISDW state in
Fe3Ga4 has a large temperature range of stability which makes it an ideal
platform to study this exotic magnetic state. There are example of similar
behavior in $f$-electron systems with a re-entrant FM state following an
intermediate AFM state, Kumar ; Wangf ; Tomka  but Fe3Ga4 is the only example
of an only $d$-electron system.

Figure 1: a) High-resolution synchrotron powder XRD pattern of Fe3Ga4 annealed
at 1000 ∘C for 48 hours at room temperature (about 295 K). Tick marks
representing the corresponding binary phases are shown below the calculated,
observed, and difference curves from Rietveld analysis. Fit statistics are
summarized in the figure. b) Crystal structure of monoclinic Fe3Ga4 with the
four unique Fe atoms distinguished with different colors and Fe-Fe bonds under
3 Å shown to emphasize nearest-neighbor Fe interactions.

This intermediate phase has been explored through pressure, doping, and high
magnetic fields in order to better understand its origin and related
properties. It has been shown that doping can effectively tune the IDSW to FM
transition temperature as well as the low temperature FM to ISDW transition.
Al1995 ; Al2000 ; Duijn ; Kobeissi ; Al1998  Similarly, previous work has
shown that external pressure is effective at tuning the low temperature FM-
ISDW transition temperature, but the effect on the high temperature ISDW-FM
transition temperature has not been reported. DuijnThesis ; DujinPressure
From these works, it appears that intermediate IDSW is sensitive to effects
caused by doping through electron count changes and/or structural changes
which in turn effect the electronic structure in this regime; although this
has not be rigorously studied. The transition from FM to ISDW is accompanied
by a very small volume change but no structural transition is observed which
further supports the SDW instability hypothesis. DujinPressure ; DuijnThesis ;
Benavides  Thus, the observed effects of pressure on the magnetic properties
are likely due to electronic structure changes concomitant with applied
pressure.

The object of this study is to look at both the effects of pressure and
annealing temperature to determine if these can be used to tune the transition
temperatures of Fe3Ga4 effectively. Previous work has stated that annealing
temperature does effect the magnetic properties without substantial evidence
to understand what drives these changes. Mendez  In this study,
polycrystalline ingots of Fe3Ga4 were annealed at various temperatures, which
lead to a controllable trend in the approximate composition of the Fe3Ga4 with
respect to the known secondary phase, FeGa3 which is non-magnetic and
semiconducting. Gippius ; Hadano ; Haeussermann  The observation and
quantification of the impurity FeGa3 phase in polycrystalline arc-melted
Fe3Ga4 has not been reported previously. We have used high-resolution
synchrotron x-ray diffraction to determine the changes caused by annealing
temperature and formation of secondary phases which has previously not been
quantified. Magnetic measurements on these annealed samples at ambient
pressure and high pressure are combined to further elucidate the cause of
magnetic property changes accompanying structural changes due to synthesis
method and external pressure. Combined, this work has shown that the magnetic
properties of Fe3Ga4 can be tuned with unit cell volume either through
synthetic procedure or external pressure. This observable change allows for
the exotic re-entrant AFM-FM magnetic transition to be tuned to room
temperature for possible use in future devices.

##  II Experimental details

###  II.1 Synthesis procedure

Bulk samples of Fe3Ga4 were synthesized via arc-melting of elements in
corresponding stoichiometric ratios using an Edmund Buehler MAM-1 under ultra-
high purity Ar atmosphere. The arc-melted ingots were loaded into quartz tubes
and sealed under moderate vacuum ($<10^{-2}$ Torr). The samples were then
annealed at various temperatures of 550, 700, 800, 900, 925, 950, 975, and
1000 ∘C for 48 hours allowed to cool to room temperature naturally.

###  II.2 X-ray diffraction and elemental analysis

High resolution synchrotron X-ray diffraction was performed on powders of
ground as-recovered ingots at Beamline 11-BM at the Advanced Photon Source at
Argonne National Lab. Ground powders were packed in 0.4 mm Kapton capillary
tubes and sealed with clay under atmospheric conditions. Diffraction data was
collected between 0.5 and 46 degrees with a step size of 0.0001∘ using a
constant wavelength of 0.457921 Å at 300 K. Rietveld refinements and data
analysis was performed using the GSAS software suite. GSAS  EDS spectra were
collected on a LEO SEM/EDAX system at a beam energy of 20kV. The tabulated
compositions were obtained as an averages of 5 different sites collected from
a cleaved edge surface of each specimen.

###  II.3 Magnetic measurements

A Quantum Design Dynacool Physical Property Measurement System (PPMS) was used
to measure the magnetization of the Fe3Ga4 samples in magnetic fields up to 5
T and in the temperature interval of 300-800 K. Magnetic measurements under
applied hydrostatic pressure (P) was performed in a commercial CuBe
cylindrical pressure cell (Quantum Design). Daphne 7373 oil was used as the
pressure transmitting medium. The applied pressure value was calibrated by
measuring the shift of the superconducting transition temperature of Pb, which
was used as a reference manometer. Errors due to differential thermal
contraction between the pressure cell components and the pressure transmitting
the medium was minimized by setting a 1 K/min temperature sweep-rate. Ambient
pressure thermomagnetic characterization was carried out on all annealed
samples using a Quantum Design SQUID MPMS system. Samples were first saturated
in a field of 1500 Oe, then the field was removed and the sample was cooled to
5 K. A field of 100 Oe was then applied and the magnetic moment was measured
while heating to 400 K to obtain the zero-field-cooled (ZFC) curve. The sample
was then cooled back to 5 K and the magnetic moment was then measured again in
a 100 Oe field while heating to 400 K to obtained the field-cooled curve (FC).

Temp. (∘C) |  $a$ (Å) |  $b$ (Å) |  $c$ (Å) |  $\beta$ (∘) | Volume (Å3) | % Fe3Ga4 | % FeGa3 | Calc. Stoich. | T1 (K) | T2 (K)
---|---|---|---|---|---|---|---|---|---|---
550 | 10.1023(1) | 7.6692(2) | 7.8750(5) | 106.29(1) | 585.66(1) | 86.8(1) | 13.20(3) | Fe3Ga3.99(2) | 67 | 387
700 | 10.1013(1) | 7.6694(2) | 7.8755(5) | 106.28(2) | 585.66(2) | 83.1(1) | 16.84(3) | Fe3Ga3.94(2) | 53 | 385
800 | 10.0971(2) | 7.6719(2) | 7.8771(5) | 106.26(1) | 585.77(1) | 82.3(1) | 17.61(3) | Fe3Ga3.92(2) | 65 | 377
900 | 10.0982(3) | 7.6881(2) | 7.8726(5) | 106.29(1) | 585.18(1) | 92.1(1) | 7.87(3) | Fe3Ga3.95(2) | 52 | 299
925 | 10.0997(1) | 7.6692(1) | 7.8756(3) | 106.28(2) | 585.56(1) | 90.1(1) | 9.89(3) | Fe3Ga3.96(2) | 61 | 379
950 | 10.1000(2) | 7.6687(1) | 7.8751(4) | 106.28(1) | 585.50(1) | 93.5(2) | 6.47(3) | Fe3Ga3.97(2) | 56 | 368
975 | 10.0986(1) | 7.6685(1) | 7.8751(2) | 106.28(1) | 585.38(1) | 94.3(2) | 5.69(3) | Fe3Ga3.95(2) | 57 | 320
1000 | 10.0988(1) | 7.6689(2) | 7.8736(1) | 106.29(1) | 585.30(1) | 96.8(2) | 3.11(2) | Fe3Ga3.98(2) | 55 | 318
Table 1: Compositional analysis of arc-melted and annealed Fe3Ga4 from
Rietveld refinement of synchrotron powder XRD data. The first five columns
display the structural parameters for Fe3Ga4 in each annealing temperature,
the next two columns show the phase fraction of Fe3Ga4 and FeGa3 in each
sample, and the final two columns show the calculated stoichiometry of Fe3Ga4
from Rietveld refinement of Fe/Ga occupancy and EDS elemental analysis
results. All errors shown in parentheses for corresponding quantities.

##  III Results and Discussion

###  III.1 Crystallographic and compositional analysis

Fe3Ga4 crystallizes in the monoclinic C2/m space group with a complex network
of four unique Fe atoms within the unit cell. These unique Fe atoms are shown
in Figure 1 with a representative XRD powder pattern and refinement for the
1000 ∘C annealed Fe3Ga4 sample. The XRD and refinement show that no unknown
phases are present. Previous reports have stated that crystallographic changes
caused by annealing, vacancy defects, or anti-site defects can cause changes
in the magnetic order of this system as these changes alter the Fe atom
network within the unit cell, but a systematic study has not been reported.
Mendez  We have synthesized polycrystalline ingots of Fe3Ga4 through arc-
melting of stoichiometric elements and subsequently annealed under vaccuum
conditions at various temperature for 48 hours. High-resolution synchrotron
x-ray diffraction (XRD) was employed in order to understand the
crystallographic changes induced by annealing temperature and relate those to
changes in the magnetic ordering of the system. Table 1 shows the results of
Rietveld refinement on all annealed Fe3Ga4 samples. From this data, we can
observe that the secondary impurity phase FeGa3 appears at all annealing
temperature but the phase fraction is significantly decreased with annealing
temperatures above 900 ∘C, but is always present. This is expected as Fe3Ga4
does not have a congruent melting point within the Fe-Ga phase diagram. This
is important to note as the synthesis method performed for this work is in
line with previous polycrystalline work and this quantitative phase analysis
of impurity FeGa3 has not been performed in previous work.Al1995 ; Samatham ;
Kawamiya  The phase fraction of Fe3Ga4 begins to significantly increase above
900 ∘C due to the melting point of FeGa3 and Fe3Ga4 occurring at 824 and 906
∘C respectively. Additionally for stoichiometric mixture of Fe:Ga in 3:4
ratio, the liquidus line is accessible at temperatures above $\sim
950^{\circ}$C and this decreases with higher Ga ratio in the system. FeGa3 is
well-reported as a non-magnetic semiconductorGippius ; Hadano ; Haeussermann
and thus its phase fraction should have no direct impact on the magnetic
signal for these ingots.

It would also appear that the phase fraction of FeGa3 does not have a
systematic impact on the stoichiometry of the Fe3Ga4 phase. This is determined
from XRD refinement which is an average structure analysis - microstructure
analysis could be important to understand how intergrown FeGa3 grains affect
the magnetic properties of Fe3Ga4 polycrystalline material. The refinement
process allowed for Fe atom vacancies, Ga atom vacancies, and anti-site mixing
after all structural and instrumental parameters have been refined to their
optimum values. This process showed that forced Fe vacancies and anti-site
mixing always negatively impacted the fit statistics and fit profile; however,
Ga vacancies on the order of < 3 % on some sites improved the refinement
statistics. This makes sense as the starting material was stoichiometric and
any non-zero amount of FeGa3 impurity should affect the Ga stoichiometry of
the Fe3Ga4 phase. The only other structural parameter which shows change
across these annealing temperature is the unit cell volume which changes on
the order of 0.1% between these samples. However, the driving force for unit
cell changes cannot be determined as its volume does not systematically trend
with Ga vacancies or annealing temperature. Attempts to refine other
crystallographic factors such as crystallite strain for example were not
useful with the current data due to the multiphase nature of the samples.
Future work on single phase Fe3Ga4 which has been post-annealed at various
temperatures could offer insight into the affects of crystallite strain on the
magnetic properties of the system. Lastly, the four unique Fe atoms in the
monoclinic unit cell host a large number of nearest and next-nearest-neighbor
interactions which all may play a critical role in the manifestation of
magnetic order in this system. Rietveld analysis allows Fe-Fe bond distances
to be determined and they are reported in Supplementary Information; however,
they show no direct trends with annealing temperature.

Figure 2: Temperature-dependent magnetization of Fe3Ga4 samples annealed at
900 ∘C (a) and 1000 ∘C (b) from 400 K - 1.8 K with an 100 Oe applied external
field. Field-dependent magnetization of Fe3Ga4 samples annealed at 900 ∘C (c)
and 1000 ∘C (d) from 0 - 3 T at various temperatures.

###  III.2 Ambient pressure magnetometry

We have measured the temperature and field-dependent magnetization for all
annealed samples and the data can be found in Supplementary Information with
summarized transition temperatures in Table 1. For the remainder of this
manuscript, we focus on the two most important samples which catalogue the
magnetic properties of Fe3Ga4 at two different annealing temperature, 900 and
1000 ∘C. These samples show significant differences in magnetic behavior.
Figure 2 shows the temperature-dependent magnetization at 100 Oe applied
external field from 400 K to base temperature in the 900 and 1000 ∘C annealed
samples. The differences in the magnetic response of the two samples are clear
- both samples have the same general behavior with a ferromagnetic (FM) ground
state transitioning to an intermediate ISDW phase before returning to a re-
entrant FM phase at high temperature. However, the ordering temperatures of
the two are much different. The 1000 ∘C sample shows expected behavior with a
FM to ISDW transition around 68 K and an ISDW to FM transition in the range of
previous work. Mendez ; Al1995  The 900 ∘C shows similar behavior for the FM-
ISDW transition with a transition temperature of about 68 K, but the ISDW-FM
transition is suppressed down to about 299 K and this is accompanied by a much
sharper increase in magnetization upon heating through this range compared to
the 1000 ∘C sample.

The change in ISDW-FM transition temperature between the 900 and 1000 ∘C
samples points toward a significant change in the energy of the ISDW phase
relative to the FM phase. Figure 2 shows the field-dependent magnetization at
various temperatures for the 900 and 1000 ∘C samples which further illustrates
the differences between the two samples especially in the ISDW-FM transition
temperature range. Previous work has shown that within the ISDW phase, a
metamagnetic transition is observed with applied field and the required field
to induce this metamagentic transition increases with temperature. Samatham ;
Mendez  Our data supports this observation for both samples. The key
difference between the two samples is the response to field prior to the
metamagnetic transition. At all temperatures within the ISDW phase prior to
the metamagnetic transition, the change in magnetization with respect to
applied field is much sharper for the 900 ∘C sample compared to the 1000 ∘C
sample and at similar temperatures the required metamagnetic field is smaller
for the 900 ∘C sample. This can be seen at the 150 K isotherm in which the 900
∘C sample displays a metamagnetic transition at 1 T whereas the 1000 ∘C
displays the same transition at 1.5 T applied field. Similar phenomena was
observed in single crystals oriented with the external field applied in
different crystallographic directions - where field applied perpendicular to
the $c$-axis induced the metamagnetic transition with smaller applied field
than with field parallel to the $c$-axis. Mendez  Crystallographic anisotropy
is unlikely to be the cause of this deviation from normal behavior seen in the
900 ∘C sample since these samples are not single crystals. Therefore, when
considering the decrease in ISDW-FM transition temperature as well as changes
in the magnetic field response within the ISDW phase for the 900 ∘C sample,
there must be some external factor driving these observed changes in magnetism
within this system.

From Table 1, both samples contain a small amount of FeGa3 and are Ga
deficient on the order of less than 2%, which is similar for the other
annealed samples. The magnetic behavior of the other annealed samples is shown
in Supplementary Information and from these, it is clear there is no direct
trend between annealing temperature, phase fraction of FeGa3 impurity and the
ISDW-FM and FM-ISDW magnetic transition temperatures. However, from the
Rietveld refinements of the annealed samples we can observe that there is a
direct trend between the volume of the Fe3Ga4 unit cell and the ISDW-FM
transition; we have shown this trend in Supplementary Information (all
transition temperatures were determined by the inflection point of the
derivative of magnetization with respect to temperature in the transition
temperature range). This is important as it shows that this transition
temperature can be altered to room temperature to allow for the possibility of
this ISDW-FM phase transition to be incorporated for use in devices.
Importantly, previous work showed that volumetric changes in Fe3Ga4 from above
400 K to 2 K is less than 0.5% and the change from across the ISDW-FM
transition is less than 0.1% such that the observed change in unit cell volume
cannot be solely attributed to structural changes between the ISDW and FM
states and no structural transition accompanies the ISDW-FM magnetic
transition.DuijnThesis ; Benavides  Future work is necessary to determine the
factors which drive these measured changes in the unit cell volume caused by
annealing or changes in the synthesis method in this system. So, although we
cannot currently tune unit cell volume from a growth perspective through
annealing or defects/occupancy, we can simulate unit cell volume changes
through external pressure and determine the changes in magnetic order of the
system, especially within the ISDW phase.

Figure 3: a) Temperature-dependent magnetization of 1000 ∘C annealed sample
from 400 - 1.8 K with an applied field of 100 Oe under various external
pressures. b) Extracted transition temperature of the FM-ISDW (T1), orange)
transition and the ISDW-FM (T2), teal) at the different applied external
pressure. Transition temperatures were determined by the inflection point in
derivative of magnetization with respect to temperature.

###  III.3 High-pressure magnetometry

In order to reliably probe the affect of unit cell volume changes on the
magnetic order of the system, we have measured the magnetic properties of the
1000 ∘C sample under high pressure. Figure 3 shows the magnetization as a
function of temperature at various applied external pressure. We observe that
external pressure tunes both the low temperature FM-ISDW transition as well as
the high temperature ISDW-FM transition and these trends are summarized in Fig
3b for both transitions. In particular, the ISDW-FM transition is suppressed
from around 320 K at ambient pressure down to below 295 K with 10 kbar of
applied pressure which brings the transition temperature in line with the
observed transition for the 900 ∘C sample. Concurrently, the low temperature
FM-ISDW transition temperature is increased meaning that the temperature
regime for the stability of the ISDW phase is growing smaller with applied
pressure. Combined, these two factors show that unit cell volume, which is
directly being changed through external pressure and was inherently smaller in
the 900 ∘C sample, can be used to tune the intermediate ISDW phase in this
system. This points toward a significant impact of crystallographic parameters
on the stability of the ISDW phase with regard to its mechanism of formation
and stabilization. Previous neutron work has suggested that the high
temperature FM phase and low temperature FM phase are identical and the
intermediate ISDW phase arises from an Fermi surface instability. Wu  This
work showed that itinerant magnetism is co-existing with rather large
localized magnetic moments of the Fe sites in the Fe3Ga4 lattice and thus a
coupling mechanism between itinerant spins and the localized moments must play
a role in stabilizing this magnetic order in this system. In previous systems,
the spin density wave formation is driven by Fermi surface nesting by which
different portions of the Fermi surface are coupled by the interaction of
itinerant spins and local moments via the RKKY interaction. Feng ; Overhauser
; Overhauser2  If this type of mechanism holds for Fe3Ga4 then the effects of
pressure and unit cell volume changes should affect the stability of the
intermediate ISDW phase through electronic structure and/or magnetic
interaction changes. Further work would be necessary to uncover the
crystallographic changes caused by external high pressure to relate those
changes to the changes in ISDW ordering temperature and stability.

Figure 4: a) Temperature-dependent magnetization of 1000 ∘C annealed sample
from 400 - 1.8 K at various applied external fields showing the suppression of
the intermediate ISDW at high applied fields. b) Temperature-dependent
magnetization of 1000 ∘C annealed sample from 400 - 1.8 K at various applied
external fields with inset showing the change in FM-PM transition temperature
as a function of applied field.

We have shown that the intermediate ISDW phase is sensitive to applied
external pressure or internal pressure due to crystallographic factors and
previous work has shown that this phase can be affected by applied field as
well, but only below 400 K has been reported. Mendez ; Samatham ; Al1995
Figure 4a shows the temperature-dependent magnetization at various applied
fields below 400 K. At 0.1 T applied field, the aforementioned low temperature
FM-ISDW transition and high temperature ISDW-FM transition are clear and match
well with previous reports. As the applied field is increased, the FM-ISDW
transition temperature is increased in temperature with applied field and the
ISDW-FM transition is broadened greatly with applied field. At high enough
field, > 2 T, which is the maximum metamagnetic field required in the ISDW
phase, the ISDW phase is seemingly fully suppressed so that the entire
magnetization signal below 400 K is ferromagnetic owning from the FM-PM
transition at 420 K above the measured range. From this, it appears that the
intermediate ISDW can be fully suppressed to stabilize the ferromagnetic
ground state across the entire temperature regime below the initial FM-PM
transition with high applied field. Furthermore, we have measured the same
properties in the range from 400 - 700 K in order to understand the effects of
applied field for the FM-PM transition. Figure 4b shows the temperature-
dependent magnetization at various applied fields from 300 - 700 K. At low
fields (0.1 T), the ISDW-FM transition is clear as well as the FM-PM
transition which occurs at 420 K, matching previous reports. Mendez  As
expected, higher applied fields suppresses the ISDW-FM transition until it is
no longer observable. Curiously, the FM-PM transition temperature increases
with applied field as shown by Figure 4b inset. It is unclear what is driving
this change but similar behavior has been noted in other systems and discussed
as the observation of itinerant metamagnetism in the region of the
ferromagnetic to paramagnetic transition. CoS1 ; CoS2 ; CoS3 ; levitin ; Goto
; Yoshimura  For this reason, this anomalous behavior in the region of the FM-
PM transition requires further investigation to understand the field-
dependence especially above the proposed ferromagnetic transition.

##  IV Conclusions

In summary, we have studied the affects of variation in annealing temperature
on crystallographic and magnetic properties of Fe3Ga4 through post-synthesis
modification of arc-melted samples. High-resolution synchrotron XRD and
subsequent Rietveld refinement was performed on all annealed samples to reveal
variations in composition and structural parameters induced by annealing
temperature. All annealed samples are multiphase containing Fe3Ga4 and a small
phase fraction of non-magnetic semiconducting FeGa3Gippius ; Hadano ;
Haeussermann  where the phase fraction of the impurity phase could be
suppressed to a minimum at 1000 ∘C annealing temperature. Due to non-zero
phase fraction of FeGa3, all Fe3Ga4 phases were found to be Ga deficient on
the order of less an 2% without any observable anti-site mixing or Fe
vacancies. Ambient pressure magnetic property measurements showed general
behavior for Fe3Ga4 in agreement with previous work. Mendez ; Al1995  In
particular, previous work showed metamagnetic transition to a field-polarized
paramagnetic state in the ISDW phase and this metamagnetic field increased
with temperature. Our work showed that at high enough fields, the ISDW-FM
state could be completely suppressed yielding one observable FM-PM transition
above 420 K that can also be altered by applied magnetic field. We have shown
the ISDW-FM transition temperature is dependent on refined unit cell volume
and can be altered from its reported value of 360 K to room temperature (299
K) upon a decrease in unit cell volume. In order to further probe this change,
we have measured the magnetic properties of the 1000 ∘C sample under high
pressure to simulate the effect of systematic unit cell volume decrease. High
pressure magnetometry revealed both the FM-ISDW and the ISDW-FM transition
temperatures are pressure dependent where the FM-ISDW transition temperature
increases with pressure and ISDW-FM transition temperature decreases. The net
result of these effects is to decrease the temperature range of stability of
the intermediate ISDW. This points toward the importance of crystallographic
properties to the proposed mechanism for the existence of the ISDW phase in
this compound. Future work on high pressure XRD to study the crystallographic
changes under high pressure would be very useful to discern the structural
changes concomitant with the changes in magnetic properties. Furthermore,
single crystal growth and corresponding strain/pressure experiments could also
help elucidate the mechanism of the ISDW phase and how it can be tuned for use
in devices.

## Supplemental Information

See the supplementary material for the additional data. Additional data
contains high-resolution XRD data for all Fe3Ga4 samples with accompanying
Rietveld refinements with data summarized in Table I of the main text.
Magnetometry data for all Fe3Ga4 samples is included as well, with
temperature-dependent magnetization with magnetic transition temperatures
extracted and field-dependent magnetization at various temperatures for all
samples. Evolution of the FM-ISDW (T1) and ISDW-FM (T2) transitions as a
function of refined unit cell volume is included to show the linear change of
T2 with respect to unit cell volume.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

###### Acknowledgements.

Research at the United States Naval Academy was supported by the NSF DMR-EPM
1904446 and ONR 1400844839. Use of the Advanced Photon Source at Argonne
National Laboratory was supported by the U.S. Department of Energy, Office of
Science, Office of Basic Energy Sciences, under Contract No. DE-
AC02-06CH11357. This work was supported by the U.S. Office of Naval Research
through the Naval Research Laboratory’s basic research program. Work at VCU
was partially funded by National Science Foundation, Award Number: 1726617.

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Supplementary Material for
"Altering the magnetic ordering of Fe3Ga4 via thermal annealing and
hydrostatic pressure"

Results of Rietveld powder X-ray diffraction analysis

X-ray diffraction for the Fe3Ga4 samples annealed at various temperatures were
taken at Beamline 11-BM at the Advanced Photon Source at Argonne National Lab
between 0.5 and 46 degrees with a step size of 0.0001 using a constant
wavelength of 0.457921 Å at 300 K. The sample was rotated continuously. The
experimental data is (points) and fit (pink curve) are shown. The lower blue
curve plots the experimental data minus the Rietveld fit. Tick marks in each
figure represent the peak positions for each binary in the samples are
determined from Rietveld analysis. The corresponding table (Table S1)
catalogues Fe-Fe bond distances less than 3 Å in the Fe3Ga4 unit cell for all
annealed samples as extracted from Rietveld refinement.

Temp. (∘C) | Fe1-Fe1 (Å) | Fe1-Fe2 (Å) | Fe1-Fe3 (Å) | Fe2-Fe2 (Å) | Fe2-Fe3 (Å) | Fe2-Fe4 (Å)
---|---|---|---|---|---|---
550 | 2.9735 | 2.7716 | 2.7546 | 2.9300 | 2.5848 | 2.5598
700 | 2.9738 | 2.7718 | 2.7543 | 2.9302 | 2.5848 | 2.5597
800 | 2.9478 | 2.7710 | 2.7543 | 2.9080 | 2.6152 | 2.5625
900 | 2.9848 | 2.7607 | 2.7558 | 2.9080 | 2.5899 | 2.5655
925 | 2.9573 | 2.7716 | 2.7539 | 2.9224 | 2.5875 | 2.5667
950 | 2.9653 | 2.7636 | 2.7557 | 2.9293 | 2.5858 | 2.5760
975 | 2.9571 | 2.7677 | 2.7418 | 2.9031 | 2.6138 | 2.5639
1000 | 2.9534 | 2.7579 | 2.7552 | 2.9147 | 2.6003 | 2.5755
Table S1: Extracted bond distances from Rietveld refinement for annealed
Fe3Ga4 samples between Fe atoms with less than 3 Å distance. Labels for Fe
atoms correspond to the positions noted in the unit cell depiction in the main
text. FM-ISDW (T1) and ISDW-FM (T2) transition tempertures for each sample.
Figure S1: Evolution of FM-ISDW (T1, orange) and ISDW-FM (T2, blue) magnetic
transition temperatures with respect to refined unit cell volume from Rietveld
analysis of high-resolution powder XRD from 11-BM at Argonne National
Laboratory. A direct trend is observed for the evolution of the ISDW-FM
magnetic transition which is supported by accompanying high pressure
magnetometry of the 1000 ∘C sample presented in the main text. Dotted line as
a guide to the eyes for the volume evolution of the T2 ISDW-FM transition.
Figure S2: High-resolution synchrotron powder XRD pattern of Fe3Ga4 annealed
at 550 ∘C for 48 hours at room temperature (about 295 K). Tick marks
representing the corresponding binary phases are shown below the calculated,
observed, and difference curves from Rietveld analysis. Fit statistics are
summarized in the figure Figure S3: High-resolution synchrotron powder XRD
pattern of Fe3Ga4 annealed at 700 ∘C for 48 hours at room temperature (about
295 K). Tick marks representing the corresponding binary phases are shown
below the calculated, observed, and difference curves from Rietveld analysis.
Fit statistics are summarized in the figure Figure S4: High-resolution
synchrotron powder XRD pattern of Fe3Ga4 annealed at 800 ∘C for 48 hours at
room temperature (about 295 K). Tick marks representing the corresponding
binary phases are shown below the calculated, observed, and difference curves
from Rietveld analysis. Fit statistics are summarized in the figure Figure S5:
High-resolution synchrotron powder XRD pattern of Fe3Ga4 annealed at 900 ∘C
for 48 hours at room temperature (about 295 K). Tick marks representing the
corresponding binary phases are shown below the calculated, observed, and
difference curves from Rietveld analysis. Fit statistics are summarized in the
figure Figure S6: High-resolution synchrotron powder XRD pattern of Fe3Ga4
annealed at 925 ∘C for 48 hours at room temperature (about 295 K). Tick marks
representing the corresponding binary phases are shown below the calculated,
observed, and difference curves from Rietveld analysis. Fit statistics are
summarized in the figure Figure S7: High-resolution synchrotron powder XRD
pattern of Fe3Ga4 annealed at 950 ∘C for 48 hours at room temperature (about
295 K). Tick marks representing the corresponding binary phases are shown
below the calculated, observed, and difference curves from Rietveld analysis.
Fit statistics are summarized in the figure Figure S8: High-resolution
synchrotron powder XRD pattern of Fe3Ga4 annealed at 975 ∘C for 48 hours at
room temperature (about 295 K). Tick marks representing the corresponding
binary phases are shown below the calculated, observed, and difference curves
from Rietveld analysis. Fit statistics are summarized in the figure Figure S9:
High-resolution synchrotron powder XRD pattern of Fe3Ga4 annealed at 1000 ∘C
for 48 hours at room temperature (about 295 K). Tick marks representing the
corresponding binary phases are shown below the calculated, observed, and
difference curves from Rietveld analysis. Fit statistics are summarized in the
figure

Figure S10: (right) Temperature-dependent magnetization of Fe3Ga4 samples
annealed at 550 ∘C with inset showing the derivative of magnetization with
respect to temperature in the vicinity of the FM-ISDW (T1) and ISDW-FM (T2)
magnetic transitions. (left) Field-dependent magnetization of Fe3Ga4 samples
annealed at 550 ∘C from 0 - 3 T at various temperatures.

Figure S11: (right) Temperature-dependent magnetization of Fe3Ga4 samples
annealed at 700 ∘C with inset showing the derivative of magnetization with
respect to temperature in the vicinity of the FM-ISDW (T1) and ISDW-FM (T2)
magnetic transitions. (left) Field-dependent magnetization of Fe3Ga4 samples
annealed at 700 ∘C from 0 - 3 T at various temperatures.

Figure S12: (right) Temperature-dependent magnetization of Fe3Ga4 samples
annealed at 800 ∘C with inset showing the derivative of magnetization with
respect to temperature in the vicinity of the FM-ISDW (T1) and ISDW-FM (T2)
magnetic transitions. (left) Field-dependent magnetization of Fe3Ga4 samples
annealed at 800 ∘C from 0 - 3 T at various temperatures.

Figure S13: (right) Temperature-dependent magnetization of Fe3Ga4 samples
annealed at 900 ∘C with inset showing the derivative of magnetization with
respect to temperature in the vicinity of the FM-ISDW (T1) and ISDW-FM (T2)
magnetic transitions. (left) Field-dependent magnetization of Fe3Ga4 samples
annealed at 900 ∘C from 0 - 3 T at various temperatures.

Figure S14: (right) Temperature-dependent magnetization of Fe3Ga4 samples
annealed at 925 ∘C with inset showing the derivative of magnetization with
respect to temperature in the vicinity of the FM-ISDW (T1) and ISDW-FM (T2)
magnetic transitions. (left) Field-dependent magnetization of Fe3Ga4 samples
annealed at 925 ∘C from 0 - 3 T at various temperatures.

Figure S15: (right) Temperature-dependent magnetization of Fe3Ga4 samples
annealed at 950 ∘C with inset showing the derivative of magnetization with
respect to temperature in the vicinity of the FM-ISDW (T1) and ISDW-FM (T2)
magnetic transitions. (left) Field-dependent magnetization of Fe3Ga4 samples
annealed at 950 ∘C from 0 - 3 T at various temperatures.

Figure S16: (right) Temperature-dependent magnetization of Fe3Ga4 samples
annealed at 975 ∘C with inset showing the derivative of magnetization with
respect to temperature in the vicinity of the FM-ISDW (T1) and ISDW-FM (T2)
magnetic transitions. (left) Field-dependent magnetization of Fe3Ga4 samples
annealed at 975 ∘C from 0 - 3 T at various temperatures.

Figure S17: (right) Temperature-dependent magnetization of Fe3Ga4 samples
annealed at 1000 ∘C with inset showing the derivative of magnetization with
respect to temperature in the vicinity of the FM-ISDW (T1) and ISDW-FM (T2)
magnetic transitions. (left) Field-dependent magnetization of Fe3Ga4 samples
annealed at 1000 ∘C from 0 - 3 T at various temperatures.