# Analysis and approximations of an optimal control problem for the Allen-Cahn
equation

KONSTANTINOS CHRYSAFINOS<EMAIL_ADDRESS>and  DIMITRA PLAKA
<EMAIL_ADDRESS>
###### Abstract.

The scope of this paper is the analysis and approximation of an optimal
control problem related to the Allen-Cahn equation. A tracking functional is
minimized subject to the Allen-Cahn equation using distributed controls that
satisfy point-wise control constraints. First and second order necessary and
sufficient conditions are proved. The lowest order discontinuous Galerkin - in
time - scheme is considered for the approximation of the control to state and
adjoint state mappings. Under a suitable restriction on maximum size of the
temporal and spatial discretization parameters $k$, $h$ respectively in terms
of the parameter $\epsilon$ that describes the thickness of the interface
layer, a-priori estimates are proved with constants depending polynomially
upon $1/\epsilon$. Unlike to previous works for the uncontrolled Allen-Cahn
problem our approach does not rely on a construction of an approximation of
the spectral estimate, and as a consequence our estimates are valid under low
regularity assumptions imposed by the optimal control setting. These estimates
are also valid in cases where the solution and its discrete approximation do
not satisfy uniform space-time bounds independent of $\epsilon$. These
estimates and a suitable localization technique, via the second order
condition (see [4, 17, 18, 15]), allows to prove error estimates for the
difference between local optimal controls and their discrete approximation as
well as between the associated state and adjoint state variables and their
discrete approximations

###### 2020 Mathematics Subject Classification:

Primary 65M60, 49J20, 49K20, 65N30, 35B25, 35K58.

Department of Mathematics, School of Mathematical and Physical Sciences,
National Technical University of Athens, Zografou 15780, Greece and IACM,
FORTH, 20013 Heraklion, Crete, Greece.

Department of Mathematics, School of Mathematical and Physical Sciences,
National Technical University of Athens, Zografou 15780, Greece.

##  1\. Introduction

We consider the following distributed optimal control problem that is governed
by the Allen-Cahn equation: Minimize

(1.1) $\displaystyle J(u)$ $\displaystyle=\
\frac{1}{2}\int_{0}^{T}{\int_{\Omega}{\lvert
y_{u}(t,x)-y_{d}(t,x)\rvert^{2}}\mathop{}\\!\mathrm{d}{x}}\mathop{}\\!\mathrm{d}{t}+\frac{\gamma}{2}\int_{\Omega}{\lvert
y_{u}(T,x)-y_{\Omega}(x)\rvert^{2}}\mathop{}\\!\mathrm{d}{x}$
$\displaystyle\quad\ +\frac{{\mu}}{2}\int_{0}^{T}{\int_{\Omega}{\lvert
u(t,x)\rvert^{2}}\mathop{}\\!\mathrm{d}{x}}\mathop{}\\!\mathrm{d}{t},$

subject to

(1.2) $\displaystyle y_{u,t}-\Delta
y_{u}+\frac{1}{{\epsilon}^{2}}(y^{3}_{u}-y_{u})$ $\displaystyle=u$ in
$\displaystyle\ \Omega_{T}=\Omega\times(0,T),$ $\displaystyle y_{u}$
$\displaystyle=0$ on $\displaystyle\ \Sigma_{T}=\partial\Omega\times(0,T),$
$\displaystyle y_{u}(\cdot,0)$ $\displaystyle=y_{0}$ in $\displaystyle\
\Omega.$

The Allen-Cahn equation can be viewed as a prototype equation that describes a
phase separation process. Indeed, denoting the nonlinear term by
$F:\mathbb{R}\to\mathbb{R},\ F(s):=\frac{1}{\epsilon^{2}}(s^{3}-s)$, we
observe that $F(s)$ represents the derivative of the classical double-well
potential. Hence, at the uncontrolled case, (i.e. when $u=0$), it is well
known that the solution (denoted by) $y$ of (1.2) develops time-dependent
interfaces $\Gamma_{t}:=\\{x\in\Omega:y(t,x)=0\\}$ that separate regions where
$y\approx 1$ from regions where $y\approx-1$. Furthermore, it is well
understood that the solution moves from one region to the other very fast,
with typical length of “diffuse interface” ${\mathcal{O}}(\epsilon)$. As a
consequence, the “smallness” of the parameter $\epsilon<<1$ should be taken
into consideration in both the analysis as well as the numerical analysis of
problems involving the Allen-Cahn equation. We refer the reader to [28] for an
overview of various phase field models, their analysis and approximations.

Our work is devoted to the analysis and numerical approximation of an optimal
control problem related to (1.2). In particular, our goal is to intervene to
the dynamics of (1.2) by using a control function $u$, in order to guarantee
that the solution $y_{u}$ will be as close as possible to a given target
$y_{d}$. Here $\mu>0$ is the typical Tikhonov regularization term, while the
inclusion of the terminal tracking term in the functional (with $\gamma\geq
0$) is necessary in order to obtain effective approximations near the end
point of the time interval. Throughout this work the set of admissible
controls is defined as

$U_{ad}=\Big{\\{}u\in L_{2}\left(0,T;L_{2}(\Omega)\right);\ u_{a}\leq
u(t,x)\leq u_{b}\ \ \text{for a.e.}\ (t,x)\in\Omega_{T}\Big{\\}},$

and the optimal control problem is formulated, in the standard reduced
functional form, as

(1.3) $\begin{cases}&\min J(u)\\\ &u\in U_{ad}.\end{cases}$

Note that the above optimal control problem is non-convex. Thus, it is
necessary to distinguish between local and global solutions. A control
$\bar{u}\in U_{ad}$ is said to be a local optimal control of (1.3) in the
sense of $L_{2}(0,T;L_{2}(\Omega))$, if there exists $\alpha>0$ such that
$J(\bar{u})\leq J(u)$ for all $u\in U_{ad}\cap B_{\alpha}(\bar{u})$, where
$B_{\alpha}(\bar{u})$ is the open ball of $L_{2}(0,T;L_{2}(\Omega))$ centered
at $\bar{u}$ with radius $\alpha$. We shall use the notation
$\bar{y}:=y_{\bar{u}}$ and $\bar{\varphi}:=\varphi_{\bar{u}}$ for the
associated state and adjoint state solutions, respectively.

Optimal control problems having states constrained to semi-linear parabolic
PDEs have been extensively studied. We refer the reader to [44] (see also
references within) for an overview of optimal control problems related to
classical elliptic and parabolic semi-linear PDEs. In fact, various issues
such as existence, first and second order necessary and sufficient conditions
have been considered even for nonstandard optimal control problems related to
semi-linear parabolic PDEs: see for instance [14] (BV controls), [16] (control
problems in absence of Tikhonov regularization term), [13, 19, 20, 21] (and
references within) (control problems with sparse controls), [39] (control
problems for non smooth semilinear parabolic PDEs).

The numerical analysis of optimal control problems with semi-linear parabolic
PDEs as constraints have been considered in [40] (control constraints,
piecewise constants / linear controls / variational discretization approach),
[24] (discontinuous in time schemes, no constraints), and [16] (error
estimates in absence of Tikhonov term). In these works, error estimates of
fully-discrete approximations have been presented under a monotonicity
assumption on the semilinear term. In [8, 9, 10, 11] the numerical analysis of
optimal control problems related to the evolutionary Navier-Stokes equations,
including error estimates, where presented.

Finally, for related works for other nonlinear parabolic PDEs, we refer the
reader to cite [34] (time-discrete two-phase flows), [35] (a-priori estimates
for a coupled semilinear PDE-ODE system).

A common ingredient in the error analysis of fully-discrete schemes for
optimal control problems related to nonlinear parabolic PDEs, under control
constraints (such as [40, 16, 8, 9, 10, 11, 35]) is the use of results
regarding the Lipschitz continuity of the control to state and of state to
adjoint mappings, the derivation of first and second order necessary and
sufficient optimality conditions, and detailed error estimates of the
corresponding control to state, and state to adjoint mappings that allow the
classical localization argument of [4, 17, 18, 15] (developed for error
analysis of discretization schemes for semilinear elliptic PDE constrained
optimization problems) to work under the prescribed regularity assumptions.

However, to our best knowledge none of the above key results are present in
case of the Allen-Cahn equation. For instance, observe that the Allen-Cahn
equation involves an nonmonotone nonlinearity that satisfies
$\frac{1}{\epsilon^{2}}F^{\prime}(s):=\frac{1}{\epsilon^{2}}(3s^{2}-1)\geq-\frac{1}{\epsilon^{2}}.$
As a consequence, for realistic values of $\epsilon$ the classical approaches
for proving the Lipschitz continuity of the control to state mapping as well
as its numerical analysis, fail since they introduce constants depending
exponentially upon $\frac{1}{\epsilon^{2}}$. For the numerical analysis of the
uncontrolled Allen-Cahn equation, i.e. for $u=0$, this difficulty was first
circumvented in the seminal works of [33], [36], [31], [7], [6], where error
estimates (a-priori and a-posteriori) were established for the homogeneous
Allen-Cahn equation with constants that dependent polynomially upon
$\frac{1}{\epsilon}$ based on suitable discrete approximation of the spectral
estimate and a nonstandard continuation argument of the form of a nonlinear
Gronwall Lemma. Unfortunately, such analysis typically requires regularity
assumptions on the state (as well as to its fully-discrete counterpart) that
are not available within the optimal control context. For example, the
assumption $y\in L_{\infty}(0,T;H^{2}(\Omega))\cap
H^{1}(0,T;H^{2}(\Omega))\cap H^{2}(0,T;L_{2}(\Omega))$ (used in the above
mentioned works for a-priori estimates of the uncontrolled Allen-Cahn
equation) is not present in our optimal control setting. In addition, similar
difficulties arise when dealing with analysis and numerical analysis of the
state to adjoint mapping. In fact, it turns out that the analysis can be even
harder due to the absence of the cubic nonlinear term that generates the
$L_{4}(0,T;L_{4}(\Omega))$ norm when testing (1.2) with $y.$

Our work aims to present an error estimate for a fully-discrete scheme based
on the discontinuous -in time- Galerkin dG(0) framework, (piece-wise constant
approximations in time, finite elements based on piecewise linear polynomials
approximations in space). First, we present a detailed analysis of the control
to state, and of the state to adjoint mappings and we derive first and second
order optimality conditions. In particular, under a closeness assumption on
the controls, we establish the Lipschitz continuity of the control to state
mapping, with Lipschitz constant that is independent of $\epsilon$ in the
$L_{\infty}(0,T;L_{2}(\Omega))$ norm by exploiting the presence of
$L_{4}(0,T;L_{4}(\Omega))$ critical term, the spectral estimate and the
nonlinear Gronwall Lemma. For the state to adjoint mapping, using similar
technical tools we are able to obtain Lipschitz results with constants that
depend polynomially upon $\frac{1}{\epsilon}$ (see Section 3).

The numerical analysis of the control to state under low regularity
assumptions is based on a slight but critical modification of the approach by
[7, 6] that is the construction of a globally space-time projection as the
dG(0) solution of a heat equation with right-hand side $y_{t}-\Delta y$
(similar to earlier works of [26] (uncontrolled Navier-Stokes equations), [8]
(for controlled Navier-Stokes) and [23] (for the uncontrolled Allen-Cahn). In
our approach we do not assume any point-wise space-time bound of the fully-
discrete solution of the control to state mapping and most crucially we do not
construct a discrete approximation of the spectral estimate, resulting an
estimate that is valid under the limited regularity assumptions imposed by our
optimal control setting.

For the numerical analysis of the discrete adjoint state mapping, the key
difficulty involves the development of discrete stability and error bounds
that depend polynomially upon $1/\epsilon$. Indeed, we note that the spectral
estimate is not longer valid if we replace $y$ by its discretization and the
direct application of the nonlinear Gronwall Lemma will lead to severe
restrictions on size of $\|y_{u}-y_{d}\|_{L_{2}(0,T;L_{2}(\Omega))}$ and
$\|y(T)-y_{\Omega}\|_{L_{2}(\Omega)}$ in terms of $\epsilon$ that are not
appropriate in the optimal control setting. Our approach is based on a pseudo
duality argument that avoids the construction of a discrete approximation of
the spectral estimate and the use of a nonlinear Gronwall Lemma resulting to
discrete stability estimates. Then, for the derivation of error estimates for
the discrete state to adjoint mapping, we employ a boot-strap argument. Note
that the error analysis of the state to adjoint mapping, is of independent
interest since it concerns a linear singularly perturbed problem under low
regularity assumptions on the given data. Combining these estimates, we are
able to proceed similar to [8] to establish the desired estimates for the
difference between local optimal controls and their discrete approximation as
well as estimates for the differences between the corresponding state and
adjoint state and theirs discrete approximations.

##  2\. Preliminary setting

Let $\Omega\subset\mathbb{R}^{d}$ be a convex, polygonal ($d=2$) or polyhedral
($d=3$) domain of the Euclidean space $\mathbb{R}^{d}$ and
$T\in\mathbb{R}^{+}$ the final time. We denote by $\Omega_{T}$ the space-time
cylinder. We use standard notation for $L_{p}(\Omega)$ spaces with $1\leq
p\leq\infty$ and their norns. We set $W^{k,p}(\Omega)$ for the $k$th order of
Sobolev spaces based on $L_{p}(\Omega)$, and we denote by
$H^{k}(\Omega):=W^{k,2}(\Omega)$, $k\geq 0$, for the Hilbertian space along
with the corresponding norms $\lVert\cdot\rVert_{W^{k,p}(\Omega)}$ and
$\lVert\cdot\rVert_{H^{k}(\Omega)}$, respectively. We denote
$H_{0}^{1}(\Omega):=\\{\varv\in H^{1}(\Omega):\varv|_{\partial\Omega}=0\\}$.
For a given Banach space $X$ we consider the Bochner spaces, $L_{p}(0,T;X)$,
endowed with the norms:

$\displaystyle\lVert\varv\rVert_{L_{p}(0,T;X)}=\Big{(}\int_{0}^{T}\lVert\varv\rVert^{p}_{X}\mathop{}\\!\mathrm{d}t\Big{)}^{1/p},\
p\in[1,+\infty),\quad\lVert\varv\rVert_{L_{\infty}(0,T;X)}=\operatorname*{ess.sup}_{t\in[0,T]}\lVert\varv\rVert_{X}.$

We also consider the functional space

$W^{2,1}_{p}\left(\Omega_{T}\right)=\Big{\\{}\varv\in
L_{p}\left(\Omega_{T}\right):\frac{\partial\varv}{\partial
x_{i}},\frac{\partial^{2}\varv}{\partial x_{i}\partial
x_{j}},\frac{\partial\varv}{\partial t}\in L_{p}\left(\Omega_{T}\right),\
1\leq i,j\leq d\Big{\\}},$

with the respective norm

$\displaystyle\lVert\varv\rVert^{p}_{W^{2,1}_{p}\left(\Omega_{T}\right)}$
$\displaystyle{=}\int_{\Omega_{T}}{\left(\lvert\varv\rvert^{p}{+}\Big{\lvert}\frac{\partial\varv}{\partial
t}\Big{\rvert}^{p}\right)}\mathop{}\\!\mathrm{d}{x}\mathop{}\\!\mathrm{d}{t}{+}\sum_{i=1}^{d}\int_{\Omega_{T}}{\bigg{\lvert}\frac{\partial\varv}{\partial
x_{i}}\bigg{\rvert}^{p}}\mathop{}\\!\mathrm{d}{x}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle{+}\sum_{i,j=1}^{d}\int_{\Omega_{T}}{\bigg{\lvert}\frac{\partial^{2}\varv}{\partial
x_{i}\partial
x_{j}}\bigg{\rvert}^{p}}\mathop{}\\!\mathrm{d}{x}\mathop{}\\!\mathrm{d}{t}.$

We adopt the notation:
$H^{2,1}\left(\Omega_{T}\right):=W^{2,1}_{2}\left(\Omega_{T}\right)$.
Furthermore, we consider the set $W(0,T):=\Big{\\{}\varv\in
L_{2}(0,T;H^{1}_{0}(\Omega)),\ \varv_{t}\in
L_{2}(0,T;H^{-1}(\Omega))\Big{\\}}.$ We will use extensively the classical
interpolation inequality  for all $\varv\in L_{4}(\Omega),$

(2.1) $\displaystyle\lVert\varv\rVert^{3}_{L_{3}(\Omega)}$
$\displaystyle\leq\lVert\varv\rVert_{L_{2}(\Omega)}\lVert\varv\rVert^{2}_{L_{4}(\Omega)},\
\ \text{for }d=2,3,$

and Gagliardo-Nirenberg-Ladyzhenskaya inequalities (GNL) for all $\varv\in
H^{1}_{0}(\Omega)$

(2.2) $\displaystyle\lVert\varv\rVert_{L_{4}(\Omega)}$
$\displaystyle\leq\tilde{c}\lVert\varv\rVert^{1/2}_{L_{2}(\Omega)}\lVert\nabla\varv\rVert^{1/2}_{L_{2}(\Omega)},\
\text{for }d=2,$ (2.3) $\displaystyle\lVert\varv\rVert_{L_{3}(\Omega)}$
$\displaystyle\leq\tilde{c}\lVert\varv\rVert^{1/2}_{L_{2}(\Omega)}\lVert\nabla\varv\rVert^{1/2}_{L_{2}(\Omega)},\
\text{for }d=3,$ (2.4) $\displaystyle\lVert\varv\rVert_{L_{4}(\Omega)}$
$\displaystyle\leq\tilde{c}\lVert\varv\rVert^{1/4}_{L_{2}(\Omega)}\lVert\nabla\varv\rVert^{3/4}_{L_{2}(\Omega)},\
\text{for }d=3,$

with $\tilde{c}>0$, independent of $\varv$. Moreover, we consider the Young’s
inequality reading: For any $\delta>0$, $a,b\geq 0$, $p,q>1$ and for some
$C(p,q)>0$, it holds

$\displaystyle ab\leq\delta
a^{p}+C(p,q)\delta^{-\frac{q}{p}}b^{q},\quad\text{where}\quad 1/p+1/q=1.$

Throughout this paper $C$ denotes a positive constant depending only on the
domain.

Assume that $u\in L_{2}(0,T;L_{2}(\Omega))$, $y_{0}\in L_{2}(\Omega)$,
$y_{\Omega}\in H^{1}_{0}(\Omega)$. The weak formulation of the state equation
(1.2) becomes: We seek $y\in W(0,T)$ such that for a.e. $t\in(0,T)$,

(2.5) $\displaystyle\langle y_{t},\varv\rangle+\left(\nabla
y,\nabla\varv\right)+\epsilon^{-2}\langle y^{3}-y,\varv\rangle$
$\displaystyle=\left(u,\varv\right),$ $\displaystyle(y(0),\varv)$
$\displaystyle=(y_{0},\varv),$

for all $\varv\in H_{0}^{1}(\Omega).$ Here, $\left(\cdot,\cdot\right)$ denotes
the standard $L_{2}(\Omega)$ inner product, and $\langle\cdot,\cdot\rangle$
the associated duality pairing between $H^{-1}(\Omega)$ and
$H^{1}_{0}(\Omega).$ For any $\epsilon>0$, and $y_{0}\in H^{1}_{0}(\Omega)$,
it easy to show that $y\in H^{2,1}\left(\Omega_{T}\right)\cap
C\left(0,T;H^{1}_{0}(\Omega)\right)$ (see [43]). We refer the readers to [43,
47] for straightforward techniques that prove the enhanced regularity results.
The following lemma quantifies the dependence upon $\epsilon$ of various
norms.

######  Lemma 2.1.

$1.$ Let $u\in L_{2}(0,T;L_{2}(\Omega))$ and $y_{0}\in L_{2}(\Omega).$ Then,
there exists a constant $C$ independent of $\epsilon$ such that:

$\displaystyle\lVert y\rVert_{L_{2}(0,T;L_{2}(\Omega))}{+}\lVert
y\rVert^{2}_{L_{4}(0,T;L_{4}(\Omega))}{\leq}C\left(|\Omega_{T}|^{\frac{1}{2}}{+}\epsilon\lVert
y_{0}\rVert_{L_{2}(\Omega)}{+}\epsilon^{2}\lVert
u\rVert_{L_{2}(0,T;L_{2}(\Omega))}\right){=:}C_{\text{st},1},$
$\displaystyle\lVert y\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}{+}\lVert
y\rVert_{L_{2}(0,T;H^{1}(\Omega))}\leq\frac{C_{\text{st},1}}{\epsilon}.$

$2.$ Let $u\in L_{2}(0,T;L_{2}(\Omega)$ and $y_{0}\in H^{1}_{0}(\Omega)$.
Then, there exists a constant $C$ independent of $\epsilon$, such that the
following estimates hold:

$\displaystyle\lVert y\rVert_{L_{\infty}(0,T;H^{1}(\Omega))}{+}\lVert
y_{t}\rVert_{L_{2}(0,T;L_{2}(\Omega))}{+}\frac{1}{2\epsilon}\|(y^{2}-1)^{2}\|^{1/2}_{L_{\infty}(0,T;L_{1}(\Omega))}$
$\displaystyle\leq C\left(\|\nabla
y_{0}\|_{L_{2}(\Omega)}{+}\frac{1}{2\epsilon}\|(y^{2}_{0}-1)^{2}\|^{1/2}_{L_{1}(\Omega)}{+}\|u\|_{L_{2}(0,T;L_{2}(\Omega))}\right){:=}C_{\text{st},2},$
$\displaystyle\lVert y\rVert_{L_{2}(0,T;H^{2}(\Omega))}\leq
C\left(\frac{T^{1/2}C_{\text{st},2}}{\epsilon}{+}\|\nabla
y_{0}\|_{L_{2}(\Omega)}{+}\|u\|_{L_{2}(0,T;L_{2}(\Omega))}\right){:=}C_{\text{st},3}.$

###### Proof.

These estimates are standard (see e.g. [33, 5, 23]). For completeness we state
the main steps that lead to the estimates. For the first result testing (2.5)
with $y$, and using Young’s inequalities, we obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}\|y\|^{2}_{L_{2}(\Omega)}+\|\nabla
y\|^{2}_{L_{2}(\Omega)}+\frac{1}{\epsilon^{2}}\|y\|^{4}_{L_{4}(\Omega)}\leq\frac{2}{\epsilon^{2}}\|y\|^{2}_{L_{2}(\Omega)}+\frac{\epsilon^{2}}{4}\|u\|^{2}_{L_{2}(\Omega)}$
$\displaystyle\leq\frac{1}{2\epsilon^{2}}\|y\|^{4}_{L_{4}(\Omega)}+\frac{2}{\epsilon^{2}}|\Omega|+\frac{\epsilon^{2}}{4}\|u\|^{2}_{L_{2}(\Omega)}.$

Integrating from $0$ to $T$ and standard algebra we get the estimate on
$\|{\cdot}\|_{L_{4}(0,T;L_{4}(\Omega))}.$ Integrating from $0$ to $t$, and
using the bound on $\|{\cdot}\|_{L_{4}(0,T;L_{4}(\Omega))}$, we deduce the
estimate on $\|{\cdot}\|_{L_{2}(0,T;L_{2}(\Omega))}$. Note that Hölder and
Young’s inequalities yield,

$\|y\|_{L_{2}(0,T;L_{2}(\Omega))}\leq|\Omega|^{1/4}T^{1/4}\|y\|_{L_{4}(0,T;L_{4}(\Omega))}\leq
C\left(|\Omega|^{1/2}T^{1/2}+\|y\|^{2}_{L_{4}(0,T;L_{4}(\Omega))}\right).$

Now we test (2.5) with $y_{t}$, and we use Young’s inequality to get

$\displaystyle\|y_{t}\|^{2}_{L_{2}(\Omega)}+\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}{t}}\Big{(}\frac{1}{2}\|\nabla
y\|^{2}_{L_{2}(\Omega)}+\frac{1}{4\epsilon^{2}}\|(y^{2}{-}1)^{2}\|_{L_{1}(\Omega)}\Big{)}\leq\frac{1}{2}\|y_{t}\|^{2}_{L_{2}(\Omega)}+\frac{1}{2}\|u\|^{2}_{L_{2}(\Omega)}.$

Integrating with respect to $t\in(0,\uptau)$,

(2.6) $\displaystyle\|y_{t}\|^{2}_{L_{2}(0,\uptau;L_{2}(\Omega))}+\|\nabla
y(\uptau)\|^{2}_{L_{2}(\Omega)}+\frac{1}{2\epsilon^{2}}\|(y^{2}(\uptau){-}1)^{2}\|_{L_{1}(\Omega)}$
$\displaystyle\ \leq\|u\|^{2}_{L_{2}(0,\uptau;L_{2}(\Omega))}+\|\nabla
y(0)\|^{2}_{L_{2}(\Omega)}+\frac{1}{2\epsilon^{2}}\|(y^{2}(0){-}1)^{2}\|_{L_{1}(\Omega)},$

which implies the desired estimate. Multiplying (1.2) with $-\Delta y,$
integrating by parts with respect to space, and Young’s inequality yield,

$\displaystyle\frac{1}{2}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}{t}}\|\nabla
y\|^{2}_{L_{2}(\Omega)}{+}\|\Delta
y\|^{2}_{L_{2}(\Omega)}{+}\frac{3}{\epsilon^{2}}\|y\nabla
y\|^{2}_{L_{2}(\Omega)}$ $\displaystyle\leq\frac{1}{2}\left(\|\Delta
y\|^{2}_{L_{2}(\Omega)}{+}\|u\|^{2}_{L_{2}(\Omega))}\right){+}\frac{1}{\epsilon^{2}}\|\nabla
y\|^{2}_{L_{2}(\Omega)},$

The estimate is completed after integrating with respect to time and
substituting the bounds of $\|\nabla y\|^{2}_{L_{\infty}(0,T;L_{2}(\Omega))}$.
∎

######  Remark 2.2.

Note that if $\|\nabla
y(0)\|_{L_{2}(\Omega)}{+}\frac{1}{2\epsilon}\|(y^{2}(0){-1})^{2}\|^{1/2}_{L_{1}(\Omega)}\leq
D$, with $D$ independent of $\epsilon$, holds then $C_{st,2}$ is bounded
independent of $\epsilon$ and hence we deduce $\|y\|_{H^{2,1}(\Omega_{T})}\leq
C\epsilon^{-1}(D{+}\|u\|_{L_{2}(0,T;L_{2}(\Omega))})$ where $C$ is an
algebraic constant depending only on the domain. The condition $\|\nabla
y(0)\|^{2}_{L_{2}(\Omega)}{+}\frac{1}{4\epsilon^{2}}\|(y^{2}(0){-1})^{2}\|_{L_{1}(\Omega)}\leq
D^{2}$ relates to the assumption that the associated Ginzburg-Landau energy is
bounded at $0$ (see for instance [33]) and it is commonly used in the
literature. Indeed, for the uncontrolled problem with zero Neumann boundary
data and initial data $y_{0}\in H^{1}_{0}(\Omega)$, the solution of (1.2)
satisfies, for any $t\geq 0$,

(2.7) $\frac{d}{dt}E(t)+\|y_{t}(t)\|^{2}_{L_{2}(\Omega)}=0$

where $E(t)$ denotes the associated Ginzburg-Landau energy i.e., for a.e.
$t\in(0,T]$,

$E(t)=\int_{\Omega}\left(\frac{1}{2}|\nabla
y|^{2}+\frac{1}{4\epsilon^{2}}(y^{2}-1)^{2}\right)dx.$

As a consequence, (2.7), implies that $E(s)\leq E(0),\ \forall s\in(0,T].$

We conclude our preliminary section by stating three key results that are
necessary for the analysis as well as the numerical analysis of the Allen-Cahn
equation. A crucial idea is the _spectral estimate_ of the principal
eigenvalue of the linearized Allen-Cahn operator about the state solution,
$y$,

(2.8) $-\lambda(t):={\displaystyle\inf_{\varv\in
H^{1}_{0}(\Omega)\setminus\\{{0\\}}}}\frac{\lVert\nabla\varv\rVert^{2}_{L_{2}(\Omega)}+\epsilon^{-2}\left((3y^{2}-1)\varv,\varv\right)}{\lVert\varv\rVert^{2}_{L_{2}(\Omega)}}.$

The celebrated works [22, 27, 3] showed that $\lambda\in L_{\infty}(0,T)$ can
be bounded independently of $\epsilon$ for the case of smooth, evolved
interfaces. We state generalizations of the continuous and discrete Gronwall
lemmas that are required for the upcoming analysis on Section 3. The proofs
are presented in details at [7, 5].

######  Lemma 2.3.

Let the nonnegative functions $w_{1}\in C(0,T)$, $w{}_{2},w_{3}\in
L_{1}(0,T)$, $\alpha\in L_{\infty}(0,T)$ and the real number $A\geq 0$
satisfy, for all $t\in[0,T]$,

$w_{1}(t)+\int^{t}_{0}w_{2}(s)\mathop{}\\!\mathrm{d}{s}\leq
A+\int^{t}_{0}\alpha(s)w_{1}(s)\mathop{}\\!\mathrm{d}{s}+\int^{t}_{0}w_{3}(s)\mathop{}\\!\mathrm{d}{s}.$

Assume also that for $B\geq 0$, $\beta>0$ and for every $t\in[0,T]$, it holds,

$\int^{t}_{0}w_{3}(s)\mathop{}\\!\mathrm{d}{s}\leq
B\sup_{s\in[0,t]}w^{\beta}_{1}(s)\int^{t}_{0}\left(w_{1}(s)+w_{2}(s)\right)\mathop{}\\!\mathrm{d}{s}.$

If in addition, $4AE\leq\left(4B(T+1)E\right)^{-1/\beta},$ with
$E:=\exp{\left(\int_{0}^{T}{\alpha(t)}\mathop{}\\!\mathrm{d}{t}\right)},$ it
holds,

$\sup_{t\in[0,T]}w_{1}(t)+\int_{0}^{T}{w_{2}(t)}\mathop{}\\!\mathrm{d}{t}\leq
4AE.$

######  Lemma 2.4.

Let $k>0$ and suppose that the nonnegative real sequences
$\\{w^{n}_{j}\\}^{N}_{n=0}$, $j=1,2,3$, $\\{\alpha^{n}\\}^{N}_{n=0}$ and the
real number $A\geq 0$ satisfy

$\displaystyle w^{m}_{1}+k\sum^{m}_{n=1}w^{n}_{2}\leq
A+k\sum^{m}_{n=1}\alpha^{n}w^{n}_{1}+k\sum^{m-1}_{n=1}w^{n}_{3},$

for all $m=1,\ldots,N$. Let $\sup_{n=1,\ldots,N}k\alpha^{n}\leq 1/2$ and
$Nk\leq T$. Assume also that for $B\geq 0$, $\beta>0$ and for every
$m=1,\ldots,N$ we have,

$\displaystyle k\sum^{m-1}_{n=1}w^{n}_{3}\leq
B\sup_{n=1,\ldots,m-1}\left(w^{n}_{1}\right)^{\beta}k\sum^{m-1}_{n=1}\left(w^{n}_{1}+w^{n}_{2}\right).$

If in addition, $4AE\leq\left(4B(T+1)E\right)^{-1/{\beta}}$ where
$E:=\exp\big{(}2k\sum^{N}_{n=1}\alpha^{n}\big{)},$ then it holds,

$\displaystyle\sup_{n=1,\ldots,N}w^{n}_{1}+k\sum^{N}_{n=1}w^{n}_{2}\leq 4AE.$

##  3\. Optimality conditions

This section is devoted to the analysis of optimal control problem
$\eqref{P}$. Since we are dealing with a non-convex problem, the first order
necessary optimality conditions are no longer sufficient.

###  3.1. Continuity

We begin by studying the continuity of the relation between the control and
the state. The emphasis here is on quantifying the dependence of various
Lipschitz constants upon $\epsilon$.

######  Definition 3.1.

The mapping $G:L_{2}(0,T;L_{2}(\Omega))\rightarrow
H^{2,1}\left(\Omega_{T}\right)\cap C\left(0,T;H^{1}_{0}(\Omega)\right)$ that
assigns each control function $u$ to the corresponding state
$y_{u}=y(u)=G(u)$, is called control to state operator.

Throughout the analysis we use the abbreviation: $F(y):=y^{3}{-}y$ and for
$u_{i}\in L_{2}(0,T;L_{2}(\Omega))$, $i=1,2$, we denote by
$y_{i}=G(u_{i}):=y_{u_{i}}$.

######  Theorem 3.2.

Assume that it holds,

(3.1) $\displaystyle\lVert u_{1}{-}u_{2}\rVert_{L_{2}(0,T:L_{2}(\Omega))}\leq\
\frac{\epsilon^{3}\ \lVert
y\rVert^{-1}_{L_{\infty}(0,T;L_{\infty}(\Omega))}}{12\tilde{c}(T{+}1)}\
E^{-1}_{d},\ \ \qquad\mbox{ for $d=2$,}$ $\displaystyle\lVert
u_{1}{-}u_{2}\rVert_{L_{2}(0,T:L_{2}(\Omega))}\leq\ \frac{\epsilon^{4}\ \lVert
y\rVert^{-1}_{L_{\infty}(0,T;L_{\infty}(\Omega))}}{48\tilde{c}(T{+}1)}\
E^{-3/2}_{d},\qquad\mbox{ for $d=3$,}$

where
$E_{d}{:=}\exp(\int_{0}^{T}{2\lambda(t)(1{-}\epsilon^{2}){+}(6{-}d)}\mathop{}\\!\mathrm{d}{t}).$
Then, with $L_{1}{:=}2E^{1/2}_{d}$ it holds,

$\displaystyle\sup_{t\in[0,T]}\lVert
y_{1}{-}y_{2}\rVert_{L_{2}(\Omega)}{+}\epsilon\lVert
y_{1}{-}y_{2}\rVert_{L_{2}(0,T:H^{1}_{0}(\Omega))}{+}\epsilon^{-1}\lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{4}(0,T:L_{4}(\Omega))}$ $\displaystyle\ \leq
L_{1}\lVert u_{1}{-}u_{2}\rVert_{L_{2}(0,T:L_{2}(\Omega))}.$

###### Proof.

Subtracting the equations satisfied by $y_{1}$ and $y_{2}$, it yields $\forall
v\in H^{1}_{0}(\Omega)$,

(3.2)
$\displaystyle\langle\left(y_{1}{-}y_{2}\right)_{t},v\rangle{+}\left(\nabla\left(y_{1}{-}y_{2}\right),\nabla
v\right){+}\epsilon^{-2}\langle F(y_{1}){-}F(y_{2}),v\rangle$
$\displaystyle=(u_{1}{-}u_{2},v)$ $\displaystyle y_{1}(0){-}y_{2}(0)$
$\displaystyle=0.$

Rewriting the nonlinear part,

$\displaystyle F(y_{1}){-}F(y_{2})$
$\displaystyle=y^{3}_{1}{-}y^{3}_{2}{-}(y_{1}{-}y_{2}){=}(y_{1}{-}y_{2})^{3}{+}3y_{1}y_{2}(y_{1}{-}y_{2}){-}(y_{1}{-}y_{2})$
$\displaystyle=(y_{1}{-}y_{2})^{3}{+}(3y^{2}_{1}{-}1)(y_{1}{-}y_{2}){-}3y_{1}(y_{1}{-}y_{2})^{2},$

relation $\eqref{subtract1}$ becomes

(3.3)
$\displaystyle\langle\left(y_{1}{-}y_{2}\right)_{t},v\rangle{+}\left(\nabla\left(y_{1}{-}y_{2}\right),\nabla
v\right)+\epsilon^{-2}\langle(y_{1}{-}y_{2})^{3}{+}(3y^{2}_{1}{-}1)(y_{1}{-}y_{2}),v\rangle$
$\displaystyle\ =(u_{1}{-}u_{2},v){+}3\epsilon^{-2}\langle
y_{1}(y_{1}{-}y_{2})^{2},v\rangle.$

Testing with $y_{1}{-}y_{2}$, and noting that
$F^{\prime}(y_{1})=3y^{2}_{1}{-}1$,

$\displaystyle\frac{1}{2}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{2}(\Omega)}{+}\lVert\nabla(y_{1}{-}y_{2})\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{-2}\lVert
y_{1}{-}y_{2}\rVert^{4}_{L_{4}(\Omega)}$
$\displaystyle+\epsilon^{-2}\big{(}F^{\prime}(y_{1})(y_{1}{-}y_{2}),y_{1}{-}y_{2}\big{)}=\left(u_{1}{-}u_{2},y_{1}{-}y_{2}\right){+}3\epsilon^{-2}\left(y_{1}(y_{1}{-}y_{2})^{2},y_{1}{-}y_{2}\right).$

Recall the spectral estimate (2.8) for $\varv=y_{1}{-}y_{2}\in
H_{0}^{1}(\Omega)$ to deduce that

$\displaystyle\lVert\nabla(y_{1}{-}y_{2})\rVert^{2}_{L_{2}(\Omega)}+\epsilon^{-2}\big{(}F^{\prime}(y_{1})(y_{1}{-}y_{2}),y_{1}{-}y_{2}\big{)}$
$\displaystyle\geq{-}{\lambda}(t)(1{-}\epsilon^{2})\lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{2}\lVert\nabla(y_{1}{-}y_{2})\rVert^{2}_{L_{2}(\Omega)}+\big{(}F^{\prime}(y_{1})(y_{1}{-}y_{2}),y_{1}{-}y_{2}\big{)}.$

Combining the last two relations, and using Hölder and Young’s inequalities it
yields,

$\displaystyle\frac{1}{2}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{2}\lVert\nabla(y_{1}{-}y_{2})\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{-2}\lVert
y_{1}{-}y_{2}\rVert^{4}_{L_{4}(\Omega)}$ $\displaystyle\leq\frac{1}{2}\lVert
u_{1}{-}u_{2}\rVert^{2}_{L_{2}(\Omega)}{+}\big{(}{\lambda}(t)(1{-}\epsilon^{2}){+}\frac{3}{2}\big{)}\lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{2}(\Omega)}{+}3\epsilon^{-2}\lVert
y_{1}\rVert_{L_{\infty}(\Omega)}\lVert
y_{1}{-}y_{2}\rVert^{3}_{L_{3}(\Omega)}.$

Integrating over $t\in(0,\uptau)$, we have

(3.4)
$\displaystyle\lVert(y_{1}{-}y_{2})(\uptau)\rVert^{2}_{L_{2}(\Omega)}{+}2\epsilon^{2}\int_{0}^{\uptau}{\lVert\nabla(y_{1}{-}y_{2})\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}2\epsilon^{-2}\int_{0}^{\uptau}{\lVert
y_{1}{-}y_{2}\rVert^{4}_{L_{4}(\Omega)}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\leq\lVert(y_{1}{-}y_{2})(0)\rVert^{2}_{L_{2}(\Omega)}{+}\int_{0}^{\uptau}{\lVert
u_{1}{-}u_{2}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\int_{0}^{\uptau}{\left(2{\lambda}(t)(1{-}\epsilon^{2}){+}3\right)\lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\quad{+}6\epsilon^{-2}\lVert
y_{1}\rVert_{L_{\infty}(0,T;L_{\infty}(\Omega))}\int_{0}^{\uptau}{\lVert
y_{1}{-}y_{2}\rVert^{3}_{L_{3}(\Omega)}}\mathop{}\\!\mathrm{d}{t}.$

The interpolation inequality (2.1), the embedding $H^{1}_{0}(\Omega)\subset
L_{4}(\Omega)$ and Poincaré inequality yield $\lVert
y_{1}{-}y_{2}\rVert^{3}_{L_{3}(\Omega)}\leq\frac{C}{\epsilon^{2}}\lVert
y_{1}{-}y_{2}\rVert_{L_{2}(\Omega)}\Big{(}\lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{2}\lVert\nabla(y_{1}{-}y_{2})\rVert^{2}_{L_{2}(\Omega)}\Big{)}.$
Then, for

$\displaystyle
w_{1}(\uptau){=}\lVert(y_{1}{-}y_{2})(\uptau)\rVert^{2}_{L_{2}(\Omega)},\
w_{2}(t)=2\epsilon^{2}\lVert\nabla(y_{1}{-}y_{2})\rVert^{2}_{L_{2}(\Omega)}{+}2\epsilon^{-2}\lVert
y_{1}{-}y_{2}\rVert^{4}_{L_{4}(\Omega)},$ $\displaystyle w_{3}(t){=}\lVert
y_{1}{-}y_{2}\rVert^{3}_{L_{3}(\Omega)},\ A{=}\lVert
u_{1}{-}u_{2}\rVert^{2}_{L_{2}(0,T;L_{2}(\Omega))},\ B{=}6C\epsilon^{-4}\lVert
y_{1}\rVert_{L_{\infty}(0,T;L_{\infty}(\Omega))},$ $\displaystyle\beta{=}1/2,\
\alpha(t){=}2\lambda(t)(1-\epsilon^{2}){+}3,$

Lemma 2.3 implies the result for $d=3$. For $d=2$, using (2.1) and the GNL
inequality (2.2), we deduce, $\lVert
y_{1}{-}y_{2}\rVert^{3}_{L_{3}(\Omega)}\leq{\tilde{c}}\ \lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{2}(\Omega)}\lVert\nabla(y_{1}{-}y_{2})\rVert_{L_{2}(\Omega)}$.
Therefore, substituting the above inequality into (3.4), and using Young’s
inequality,

$\displaystyle\lVert(y_{1}{-}y_{2})(\uptau)\rVert^{2}_{L_{2}(\Omega)}+2\epsilon^{2}\int_{0}^{\uptau}{\lVert\nabla(y_{1}{-}y_{2})\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}+2\epsilon^{-2}\int_{0}^{\uptau}{\lVert
y_{1}{-}y_{2}\rVert^{4}_{L_{4}(\Omega)}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\leq\lVert(y_{1}{-}y_{2})(0)\rVert^{2}_{L_{2}(\Omega)}+\int_{0}^{\uptau}{\lVert
u_{1}{-}u_{2}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}+\int_{0}^{\uptau}{\left(2{\lambda}(t)(1{-}\epsilon^{2}){+}3\right)\lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle{+}\frac{9{\tilde{c}}^{2}}{\epsilon^{4}}\lVert
y_{1}\rVert^{2}_{L_{\infty}(0,T;L_{\infty}(\Omega))}\int_{0}^{\uptau}{\lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{2}(\Omega)}\|\nabla(y_{1}{-}y_{2})\|^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\int_{0}^{\uptau}{\|y_{1}{-}y_{2}\|^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}.$

The result now follows upon choosing $\beta=1,\
B=9{\tilde{c}}^{2}\epsilon^{-6}\lVert
y_{1}\rVert^{2}_{L_{\infty}(0,T;L_{\infty}(\Omega))}$ and
$\alpha(t)=2\lambda(t)(1-\epsilon^{2})+4$ in Lemma 2.3. ∎

###  3.2. Differentiability

Next, we determine the first and second order derivatives of the control to
state mapping G of Definition 3.1.

######  Theorem 3.3.

Let $u,\varv\in L_{2}(0,T;L_{2}(\Omega))$. The mapping
$G:L_{2}(0,T;L_{2}(\Omega))\rightarrow H^{2,1}(\Omega_{T})\cap
C(0,T;H^{1}_{0}(\Omega))$, such that $y_{u}:=G(u)$, is of class $C^{\infty}$.
We denote by $z_{\varv}=G^{\prime}(u)\varv$ and
$z_{\varv\varv}=G{{}^{\prime\prime}}(u){\varv}^{2}$, the unique solutions to
the following equations

(3.5) $\displaystyle z_{\varv,t}{-}\Delta
z_{\varv}{+}\epsilon^{-2}\left(3y^{2}_{u}{-}1\right)z_{\varv}{=}\varv\quad\text{in}\
\Omega_{T},$ $\displaystyle z_{\varv}{=}0\quad\text{on}\ \Sigma_{T},\
z_{\varv}(0){=}0\quad\text{in}\ \Omega,$ (3.6) $\displaystyle
z_{\varv\varv,t}{-}\Delta
z_{\varv\varv}{+}\epsilon^{-2}\left(3y^{2}_{u}{-}1\right)z_{\varv\varv}{=}{-}6\epsilon^{-2}y_{u}z^{2}_{\varv}\quad\text{in}\
\Omega_{T},$ $\displaystyle z_{\varv,\varv}{=}0\quad\text{on}\ \Sigma_{T},\
z_{\varv,\varv}(0){=}0\quad\text{in}\ \Omega.$

###### Proof.

The proof follows using similar arguments as in [8, 10]. ∎

The above differentiability properties of G imply that the reduced cost
functional $J:L_{2}(0,T;L_{2}(\Omega))\rightarrow\mathbb{R}$ is of class
$C^{\infty}$, as well.

######  Lemma 3.4.

For any $u,v\in L_{2}(0,T;L_{2}(\Omega))$, it holds that

(3.7) $J^{\prime}(u)\varv{=}\int_{0}^{T}{\int_{\Omega}{(\varphi_{u}{+}\mu
u)\varv}\mathop{}\\!\mathrm{d}{x}}\mathop{}\\!\mathrm{d}{t},$ (3.8)
$\displaystyle J^{\prime\prime}(u)\varv^{2}{=}$
$\displaystyle\int_{0}^{T}{\int_{\Omega}{\lvert
z_{\varv}\rvert^{2}}\mathop{}\\!\mathrm{d}{x}}\mathop{}\\!\mathrm{d}{t}{+}\gamma\int_{\Omega}{\lvert
z_{\varv}(T)\rvert^{2}}\mathop{}\\!\mathrm{d}{x}{+}\mu\int_{0}^{T}{\int_{\Omega}{\lvert\varv\rvert^{2}}\mathop{}\\!\mathrm{d}{x}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\ {-}6\epsilon^{-2}\int_{0}^{T}{\int_{\Omega}{y_{u}\
z^{2}_{\varv}\
\varphi_{u}}\mathop{}\\!\mathrm{d}{x}}\mathop{}\\!\mathrm{d}{t},$

where $z_{\varv}$ is the solution of (3.5) while $\varphi_{u}\in
H^{2,1}(\Omega_{T})\cap C(0,T;H^{1}_{0}(\Omega))$ is the unique solution that
satisfies for all $w\in H^{1}_{0}(\Omega)$ the adjoint sate equation,

(3.9)
$\begin{cases}{-}\left(\varphi_{u,t},w\right){+}\left(\nabla\varphi_{u},\nabla
w\right){+}\epsilon^{-2}\big{(}\left(3y^{2}_{u}{-}1\right)\varphi_{u},w\big{)}{=}\left(y_{u}{-}y_{d},w\right)\\\
(\varphi_{u}(T),w){=}(\gamma(y_{u}(T)-y_{\Omega}),w).\end{cases}$

######  Remark 3.5.

The assumption that $y_{\Omega}\in H^{1}_{0}(\Omega)$ together with the
regularity properties of the state solution $y_{u}$ implies that
$\varphi_{u}(T)\in H^{1}_{0}(\Omega)$. Thus, we deduce that

$\displaystyle\lVert\varphi_{u}(T)\rVert_{H^{1}_{0}(\Omega)}=\lVert\gamma(y_{u}(T){-}y_{\Omega})\rVert_{H^{1}_{0}(\Omega)}\leq\gamma\Big{(}\sup_{t\in[0,T]}\lVert
y_{u}(t)\rVert_{H^{1}_{0}(\Omega)}{+}\lVert
y_{\Omega}\rVert_{H^{1}_{0}(\Omega)}\Big{)}.$

Recalling Lemma 2.1 and the embedding $H^{2,1}(\Omega_{T})\subset
C(0,T;H^{1}(\Omega))$ we obtain
$\lVert\varphi_{u}(T)\rVert_{H^{1}_{0}(\Omega)}\leq C\epsilon^{-2}$ when
$\epsilon\|\nabla
y_{0}\|_{L_{2}(\Omega)}{+}\frac{1}{2}\|(y_{0}^{2}-1)^{2}\|^{1/2}_{L_{1}(\Omega)}\leq
C$. Similarly, we have that
$\lVert\varphi_{u}(T)\rVert_{H^{1}_{0}(\Omega)}\leq C\epsilon^{-1}$ when
$\|\nabla
y_{0}\|_{L_{2}(\Omega)}{+}\frac{1}{2\epsilon}\|(y_{0}^{2}{-}1)^{2}\|^{1/2}_{L_{1}(\Omega)}\leq
D.$

######  Lemma 3.6.

Let $\varphi_{u}$ be the solution to (3.9), $y_{d}\in
L_{2}(0,T;L_{2}(\Omega))$ and $\varphi_{u}(T)\in H^{1}_{0}(\Omega)$. Then,
there exists a constant $C>0$, depending on $\Omega$ such that

(3.10)
$\displaystyle\sup_{t\in[0,T]}\lVert\varphi_{u}(t)\rVert_{L_{2}(\Omega)}{+}\
\epsilon\lVert\nabla\varphi_{u}\rVert_{L_{2}(0,T;L_{2}(\Omega))}{+}\|\varphi_{u}y_{u}\|_{L_{2}(0,T;L_{2}(\Omega))}$
$\displaystyle\ \leq
CC^{1/2}_{\varphi}\left(\lVert\varphi_{u}(T)\rVert_{L_{2}(\Omega)}{+}\lVert
y_{u}{-}y_{d}\rVert_{L_{2}(0,T;L_{2}(\Omega))}\right):=D_{\text{st},1},$
(3.11)
$\displaystyle\lVert\varphi_{u,t}\rVert_{L_{2}(0,T;L_{2}(\Omega))}{+}\sup_{t\in[0,T]}\lVert\nabla\varphi_{u}(t)\rVert_{L_{2}(\Omega)}\leq
C\Big{(}\lVert y_{u}{-}y_{d}\rVert_{L_{2}(0,T;L_{2}(\Omega))}$
$\displaystyle+\lVert\nabla\varphi_{u}(T)\rVert_{L_{2}(\Omega)}{+}\frac{1}{\epsilon^{2}}\left(\lVert
y_{u}\rVert_{L_{\infty}(0,T;L_{\infty}(\Omega))}{+}T^{1/2}\right)D_{\text{st},1}\Big{)}{:=}D_{\text{st},2},$
(3.12) $\displaystyle\lVert\varphi_{u}\rVert_{L_{2}(0,T;H^{2}(\Omega))}\leq
C\Big{(}\lVert y_{u}-y_{d}\rVert_{L_{2}(0,T;L_{2}(\Omega))}+D_{\text{st},2}$
$\displaystyle\ \ +\frac{1}{\epsilon^{2}}\left(1+3\lVert
y_{u}\rVert_{L_{\infty}(0,T;L_{\infty}(\Omega))}\right)D_{\text{st},1}\Big{)},$

where we denote by
$C_{\varphi}:=\exp\left(\int_{0}^{T}{(2\lambda(t)(1{-}\epsilon^{2})+3)}\mathop{}\\!\mathrm{d}{t}\right)$.

###### Proof.

To derive the first stability estimate, we test (3.9) with $w=\varphi_{u}$.

$\displaystyle-\frac{1}{2}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}{t}}\lVert\varphi_{u}\rVert^{2}_{L_{2}(\Omega)}{+}\lVert\nabla\varphi_{u}\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{-2}(F^{\prime}(y_{u})\varphi_{u},\varphi_{u}){=}(y_{u}{-}y_{d},\varphi_{u}).$

Recall (2.8) for $\varv=\varphi_{u}$ about state solution $y_{u}$, apply
Cauchy-Schwarz and Young’s inequalities on the right-hand side and integrate
with respect to $t\in(\uptau,T)$ to obtain

$\displaystyle\frac{1}{2}\lVert\varphi_{u}(\uptau)\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{2}\int^{T}_{\uptau}\lVert\nabla\varphi_{u}\rVert^{2}_{L_{2}(\Omega)}\mathop{}\\!\mathrm{d}{t}{+}3\int^{T}_{\uptau}\lVert\varphi_{u}y_{u}\rVert^{2}_{L_{2}(\Omega)}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\leq\frac{1}{2}\lVert\varphi_{u}(T)\rVert^{2}_{L_{2}(\Omega)}{+}\int^{T}_{\uptau}\Big{(}\lambda(t)(1{-}\epsilon^{2}){+}\frac{3}{2}\Big{)}\lVert\varphi_{u}\rVert^{2}_{L_{2}(\Omega)}\mathop{}\\!\mathrm{d}{t}{+}\frac{1}{2}\int^{T}_{\uptau}\lVert
y_{u}{-}y_{d}\rVert^{2}_{L_{2}(\Omega)}\mathop{}\\!\mathrm{d}{t}.$

The (linear) Gronwall inequality yields the result. Setting
$w{=}{-}\varphi_{u,t}$ into (3.9) we have

(3.13)
$\displaystyle\lVert\varphi_{u,t}\rVert^{2}_{L_{2}(\Omega)}{-}\frac{1}{2}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}{t}}\lVert\nabla\varphi_{u}\rVert^{2}_{L_{2}(\Omega)}{=}(y_{u}{-}y_{d},\varphi_{u,t}){+}\epsilon^{-2}(F^{\prime}(y_{u})\varphi_{u},\varphi_{u,t}).$

Note that using Young’s inequalities, we may bound the last two terms as
follows:

$\displaystyle|(y_{u}{-}y_{d},\varphi_{u,t})|\leq(1/4)\|\varphi_{u,t}\|^{2}_{L_{2}(\Omega)}{+}\|y_{u}{-}y_{d}\|^{2}_{L_{2}(\Omega)},$
$\displaystyle\epsilon^{-2}|(F^{\prime}(y_{u})\varphi_{u},\varphi_{u,t})|=\epsilon^{-2}|\left((3y^{2}_{u}{-}1)\varphi_{u},\varphi_{u.t}\right)|$
$\displaystyle\
\leq(1/4)\|\varphi_{u,t}\|^{2}_{L_{2}(\Omega)}{+}18\epsilon^{-4}\|y_{u}\|^{2}_{L_{\infty}(\Omega)}\|\varphi_{u}y_{u}\|^{2}_{L_{2}(\Omega)}{+}2\epsilon^{-4}\|\phi_{u}\|^{2}_{L_{2}(\Omega)}.$

Substituting the last two inequalities into (3.13) and integrating from
$\uptau$ to $T$ it yields,

$\displaystyle\int^{T}_{\uptau}\lVert\varphi_{u,t}\rVert^{2}_{L_{2}(\Omega)}\mathop{}\\!\mathrm{d}{t}{+}\lVert\nabla\varphi_{u}(\uptau)\rVert^{2}_{L_{2}(\Omega)}\leq
2\int^{T}_{\uptau}\lVert
y_{u}{-}y_{d}\rVert^{2}_{L_{2}(\Omega)}\mathop{}\\!\mathrm{d}{t}{+}\lVert\nabla\varphi_{u}(T)\rVert^{2}_{L_{2}(\Omega)}$
$\displaystyle{+}36\epsilon^{-4}\lVert
y_{u}\rVert^{2}_{L_{\infty}(0,T;L_{\infty}(\Omega))}\int^{T}_{\uptau}\lVert\varphi_{u}y_{u}\rVert^{2}_{L_{2}(\Omega)}\mathop{}\\!\mathrm{d}{t}{+}4\epsilon^{-4}\int^{T}_{\uptau}\|\varphi_{u}\|^{2}_{L^{2}(\Omega)}\mathop{}\\!\mathrm{d}{t}.$

Using the stability bound (3.10) we obtain (3.11). The third estimate follows
using similar techniques by setting $w=-\Delta\varphi_{u}$ into (3.9) and
using the previous bounds to estimate
$\|\varphi_{u,t}\|_{L_{2}(0,T;L_{2}(\Omega))},\lVert\nabla\varphi_{u}\rVert_{L_{2}(0,T;L_{2}(\Omega))}\
\text{and}\ \lVert y_{u}\varphi_{u}\rVert_{L_{2}(0,T;L_{2}(\Omega))}$. ∎

Let $u_{1},u_{2}\in L_{2}(0,T;L_{2}(\Omega))$ be the control functions. Then,
we denote by $y_{i}=y_{u_{i}}$ and $\varphi_{i}=\varphi_{u_{i}}$ the
associated state and adjoint state solutions for $i=1,2$, respectively.

######  Lemma 3.7.

Assume that (3.1) holds. Then, for $d=2$, there exists a constant $C_{T}$
depending only on the domain $\Omega_{T}$ such that,

(3.14)
$\displaystyle\sup_{t\in[0,T]}\lVert\varphi_{1}{-}\varphi_{2}\rVert_{L_{2}(\Omega)}+\epsilon\lVert\varphi_{1}{-}\varphi_{2}\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}$
$\displaystyle\ \leq
C_{T}E^{1/2}_{\varphi}L_{1}\left(1{+}\frac{C_{\infty}{\tilde{c}}D_{\text{st},1}}{\epsilon^{7/2}}\right)\lVert
u_{1}{-}u_{2}\rVert_{L_{2}(0,T;L_{2}(\Omega))}.$

For $d=3$, there exists a constant $C_{T}$ depending only on the domain
$\Omega_{T}$ such that,

(3.15)
$\displaystyle\sup_{t\in[0,T]}\lVert\varphi_{1}{-}\varphi_{2}\rVert_{L_{2}(\Omega)}+\epsilon\lVert\varphi_{1}{-}\varphi_{2}\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}$
$\displaystyle\ \leq
C_{T}E^{1/2}_{\varphi}L_{1}\left(1{+}\frac{C_{\infty}{\tilde{c}}D_{\text{st},1}}{\epsilon^{15/4}}\right)\lVert
u_{1}{-}u_{2}\rVert_{L_{2}(0,T;L_{2}(\Omega))}.$

Here, we denote by $C_{\infty}{:=}\left(C\big{(}\lVert
y_{1}\rVert^{2}_{L_{\infty}(0,T;L_{\infty}(\Omega))}{+}\lVert
y_{2}\rVert^{2}_{L_{\infty}(0,T;L_{\infty}(\Omega))}\big{)}\right)^{1/2}$ with
$C$ depending on $|\Omega|$, and by
$E_{\varphi}{:=}\int_{0}^{T}\left(2\lambda(t)(1{-}\epsilon^{2}){+}4\right)\mathop{}\\!\mathrm{d}{t}.$

###### Proof.

Subtracting the equations satisfied by $\varphi_{1}$ and $\varphi_{2}$, it
yields for $w\in H^{1}_{0}(\Omega)$

$\begin{cases}-\langle\varphi_{1,t}{-}\varphi_{2,t},w\rangle{+}\left(\nabla(\varphi_{1}{-}\varphi_{2}),\nabla
w\right){+}\epsilon^{-2}\langle
F^{\prime}(y_{1})\varphi_{1}{-}F^{\prime}(y_{2})\varphi_{2},w\rangle{=}(y_{1}{-}y_{2},w)\\\
\left(\varphi_{1}{-}\varphi_{1}\right)(T){=}\gamma\left(y_{1}{-}y_{2}\right)(T).\end{cases}$

Inserting the identity,
$F^{\prime}(y_{1})\varphi_{1}{-}F^{\prime}(y_{2})\varphi_{2}{=}\left(3y^{2}_{1}{-}1\right)\left(\varphi_{1}{-}\varphi_{2}\right){+}3\left(y^{2}_{1}{-}y^{2}_{2}\right)\varphi_{2}$
and testing with $w=\varphi_{1}{-}\varphi_{2}$, we deduce that

$\displaystyle-$
$\displaystyle\frac{1}{2}\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}\lVert\varphi_{1}{-}\varphi_{2}\rVert^{2}_{L_{2}(\Omega)}{+}\lVert\nabla(\varphi_{1}{-}\varphi_{2})\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{-2}\left(F^{\prime}(y_{1})(\varphi_{1}{-}\varphi_{2}),\varphi_{1}{-}\varphi_{2}\right)$
$\displaystyle{=}\left(y_{1}{-}y_{2},\varphi_{1}{-}\varphi_{2}\right){-}3\epsilon^{-2}\left((y^{2}_{1}{-}y^{2}_{2})\varphi_{2},\varphi_{1}{-}\varphi_{2}\right){:=}\mathcal{K}_{1}{+}\mathcal{K}_{2}.$

Cauchy-Schwarz and Young’s inequalities yield,
$\mathcal{K}_{1}{\leq}\frac{1}{2}\lVert\varphi_{1}{-}\varphi_{2}\rVert^{2}_{L_{2}(\Omega)}{+}\frac{1}{2}\lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{2}(\Omega)}$. Applying Hölder, Poincaré and
Young’s inequalities on the second term, we obtain

$\displaystyle\mathcal{K}_{2}$ $\displaystyle\leq 3\epsilon^{-2}\lVert
y_{1}{-}y_{2}\rVert_{L_{2}(\Omega)}\left(\lVert
y_{1}\rVert_{L_{\infty}(\Omega)}+\lVert
y_{2}\rVert_{L_{\infty}(\Omega)}\right)\lVert\varphi_{2}\rVert_{L_{4}(\Omega)}\lVert\varphi_{1}{-}\varphi_{2}\rVert_{L_{4}(\Omega)}$
$\displaystyle\leq\frac{C^{2}_{\infty}}{\epsilon^{6}}\lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{2}(\Omega)}\lVert\varphi_{2}\rVert^{2}_{L_{4}(\Omega)}{+}\frac{\epsilon^{2}}{2}\lVert\nabla(\varphi_{1}{-}\varphi_{2})\rVert^{2}_{L_{2}(\Omega)}.$

Substituting the bounds on $\mathcal{K}_{i}$, the spectral estimate (2.8) for
$\varv=\varphi_{1}-\varphi_{2}$, $y=y_{1}$, and integrating from $\uptau$ to
$T$, (3.2) yields,

$\displaystyle\lVert(\varphi_{1}{-}\varphi_{2})(\uptau)\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{2}\int^{T}_{\uptau}\lVert\nabla\left(\varphi_{1}{-}\varphi_{2}\right)\rVert^{2}_{L_{2}(\Omega)}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\leq\lVert\gamma(y_{1}{-}y_{2})(T)\rVert^{2}_{L_{2}(\Omega)}{+}\int_{\uptau}^{T}{\lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\int_{\uptau}^{T}\left(2\lambda(t)(1{-}\epsilon^{2}){+}4\right)\lVert\varphi_{1}{-}\varphi_{2}\rVert^{2}_{L_{2}(\Omega)}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\quad{+}\frac{C^{2}_{\infty}}{\epsilon^{6}}\lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{\infty}(0,T;L_{2}(\Omega))}\int_{\uptau}^{T}\lVert\varphi_{2}\rVert^{2}_{L_{4}(\Omega)}\mathop{}\\!\mathrm{d}{t}.$

For $d=3$, (2.4) and Hölder’s inequalities with $s_{1}=4$ and $s_{2}=4/3$, and
(3.10) imply,

$\displaystyle\int_{\uptau}^{T}\lVert\varphi_{2}\rVert^{2}_{L_{4}(\Omega)}\mathop{}\\!\mathrm{d}{t}\leq{\tilde{c}}^{2}\int_{\uptau}^{T}\|\varphi_{2}\|^{1/2}_{L_{2}(\Omega)}\|\nabla\varphi_{2}\|^{3/2}_{L_{2}(\Omega)}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\leq{\tilde{c}}^{2}\|\varphi_{2}\|^{1/2}_{L_{2}(0,T;L_{2}(\Omega))}\|\nabla\varphi_{2}\|^{3/2}_{L_{2}(0,T;L_{2}(\Omega))}\leq
T^{1/4}{\tilde{c}}^{2}D^{2}_{st,1}\epsilon^{-3/2}.$

Similarly, for $d=2$, we have

$\displaystyle\int_{\uptau}^{T}\lVert\varphi_{2}\rVert^{2}_{L_{4}(\Omega)}\mathop{}\\!\mathrm{d}{t}\leq{\tilde{c}}^{2}\|\varphi_{2}\|_{L_{2}(0,T;L_{2}(\Omega))}\|\nabla\varphi_{2}\|_{L_{2}(0,T;L_{2}(\Omega))}\leq
T^{1/2}{\tilde{c}}^{2}D^{2}_{st,1}\epsilon^{-1}.$

Then, for $d=3$, the (linear) Gronwall inequality implies that

$\displaystyle\sup_{t\in[0,T]}\lVert(\varphi_{1}{-}\varphi_{2})(t)\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{2}\lVert\nabla\left(\varphi_{1}{-}\varphi_{2}\right)\rVert^{2}_{L_{2}(0,T;L_{2}(\Omega))}$
$\displaystyle\leq
E_{\varphi}\Big{(}\lVert\gamma(y_{1}{-}y_{2})(T)\rVert^{2}_{L_{2}(\Omega)}{+}\Big{(}T{+}\frac{T^{1/4}C^{2}_{\infty}D^{2}_{\text{st},1}}{\epsilon^{15/2}}\Big{)}\sup_{t\in[0,T]}\lVert
y_{1}{-}y_{2}\rVert^{2}_{L_{2}(\Omega)}\Big{)}.$

The estimate now follows by using the estimate Lipschitz continuity (3.2).
Working in an identical way, we deduce the estimate for $d=2$ ∎

###  3.3. Necessary and sufficient conditions

Below, we state the optimality conditions. We refer the reader to [8, Theorems
3.4 and 3.3] for the related proofs.

######  Theorem 3.8.

Every locally optimal control $\bar{u}$ for problem (1.3), satisfies, together
with its associated state $\bar{y}\in H^{2,1}(\Omega_{T})$ and adjoint state
$\bar{\varphi}\in H^{2,1}(\Omega_{T})$

(3.16)
$\begin{cases}\langle{\bar{y}}_{t},\varv\rangle{+}\left(\nabla{\bar{y}},\nabla\varv\right){+}\epsilon^{-2}\langle{\bar{y}}^{3}{-}{\bar{y}},\varv\rangle{=}\left({\bar{u}},\varv\right)\quad\forall\varv\in
H^{1}_{0}(\Omega)\\\ {\bar{y}}(0){=}y_{0},\end{cases}$ (3.17)
$\begin{cases}-\langle{\bar{\varphi}}_{t},w\rangle{+}\left(\nabla{\bar{\varphi}},\nabla
w\right){+}\epsilon^{-2}\big{\langle}\left(3{\bar{y}}^{2}{-}1\right){\bar{\varphi}},w\big{\rangle}{=}\left({\bar{y}}{-}y_{d},w\right)\
\forall w\in H^{1}_{0}(\Omega)\\\
{\bar{\varphi}}(T){=}\gamma({\bar{y}}(T){-}y_{\Omega}),\end{cases}$

and the variational inequality (optimality condition)

(3.18)
$\int_{0}^{T}{\int_{\Omega}{\left(\bar{\varphi}{+}\mu\bar{u}\right)\left(u{-}\bar{u}\right)}\mathop{}\\!\mathrm{d}{x}}\mathop{}\\!\mathrm{d}{t}\geq
0\quad\forall u\in U_{ad},$

where $\bar{u}\in L_{2}(0,T;W^{1,p}(\Omega))\cap C(0,T;H^{1}(\Omega))\cap
H^{1}(\Omega_{T})$, for any $1\leq p<\infty.$
Furthermore, let $\epsilon\|\nabla
y_{0}\|_{L_{2}(\Omega)}{+}\frac{1}{2}\|(y^{2}_{0}{-}1)^{2}\|^{1/2}_{L_{1}(\Omega)}\leq
C$. Then ,

$\displaystyle\|\bar{y}\|_{H^{2,1}(\Omega_{T})}\leq\tilde{C}\epsilon^{-2},\
\text{and}\ \
\|\bar{\varphi}\|_{H^{2,1}(\Omega_{T})}\leq\tilde{D}\epsilon^{-3}.$

If in addition, $\|\nabla
y_{0}\|_{L_{2}(\Omega)}{+}\frac{1}{2\epsilon}\|(y^{2}_{0}{-}1)^{2}\|^{1/2}_{L_{1}(\Omega)}\leq
D$ then,

$\displaystyle\|\bar{y}\|_{H^{2,1}(\Omega_{T})}\leq{\tilde{C}}\epsilon^{-1},\
\mbox{and}\
\|\bar{\varphi}\|_{H^{2,1}(\Omega_{T})}\leq\tilde{D}\epsilon^{-2}.$

Here, the constants $\tilde{C}$, $\tilde{D}$ are independent of $\epsilon$ and
depend only on data.

###### Proof.

Note that every local optimal solution satisfies
$J^{\prime}(\bar{u})(u-\bar{u})\geq 0,$ $\forall u\in U_{ad}$. The optimality
system (3.16), (3.17) and (3.18) follows from (3.7). Inequality (3.18) implies
the standard projection formula

(3.19)
$\bar{u}(t,x)=\mathrm{Proj}_{[u_{a},u_{b}]}\Big{(}-\frac{1}{\mu}\bar{\varphi}(t,x)\Big{)}\
\text{for a.e.}\ (t,x)\in\Omega_{T},$

from which we deduce $\bar{u}\in H^{1}({\Omega_{T}})\cap
C(0,T;H^{1}(\Omega))\cap L_{2}(0,T;W^{1,p}(\Omega))$, for all $1\leq
p<\infty$. Therefore, if $\epsilon\|\nabla
y_{0}\|_{L_{2}(\Omega)}+\frac{1}{2}\|(y^{2}_{0}-1)^{2}\|^{1/2}_{L^{1}(\Omega)}\leq
C$, (independent of $\epsilon$) we observe that the constants $C_{st,2}$,
$C_{st,3}$ of Lemma 2.1 satifsy
$C_{st,2}\leq\left(\frac{C}{\epsilon}{+}\|\bar{u}\|_{L_{2}(0,T;L_{2}(\Omega))}\right)$
and
$C_{st,3}{\leq}\left(\frac{C}{\epsilon^{2}}{+}\|\bar{u}\|_{L_{2}(0,T;L_{2}(\Omega))}\right)$
from which we deduce that
$\|\bar{y}\|_{H^{2,1}(\Omega_{T})}{\leq}\tilde{C}\epsilon^{-2}$. Similar, the
estimates of Lemma 3.6, imply that
$\|\bar{\varphi}\|_{H^{2,1}(\Omega_{T})}\leq\tilde{D}\epsilon^{-3}$ when
$\gamma>0$ and
$\|\bar{\varphi}\|_{H^{2,1}(\Omega_{T})}\leq\tilde{D}\epsilon^{-2}$ when
$\gamma=0$. Under the assumption $\|\nabla
y_{0}\|_{L_{2}(\Omega)}+\frac{1}{2\epsilon}\|(y^{2}_{0}-1)^{2}\|^{1/2}_{L_{1}(\Omega)}\leq
D$, a similar boot-strap argument, imply that
$\|\bar{y}\|_{H^{2,1}(\Omega_{T})}\leq\tilde{C}\epsilon^{-1}$ and
$\|\bar{\phi}\|_{H^{2,1}(\Omega_{T})}\leq\tilde{D}\epsilon^{-2}$ (we note here
that Lemma 2.1 implies that the constant $D_{st,1}$ of (3.10) is bounded
independent of $\epsilon$). ∎

In the usual manner, we deduce from (3.18) that for a.e. $(t,x)\in\Omega_{T}$,

$\displaystyle\begin{aligned} &\begin{cases}\bar{u}(t,x)=u_{a}\
\Rightarrow\bar{\varphi}(t,x)+\mu\bar{u}(t,x)\geq 0,\\\ \bar{u}(t,x)=u_{b}\
\Rightarrow\bar{\varphi}(t,x)+\mu\bar{u}(t,x)\leq 0,\\\
\bar{u}(t,x)\in(u_{a},u_{b})\
\Rightarrow\bar{\varphi}(t,x)+\mu\bar{u}(t,x)=0,\\\ \end{cases}\\\
\end{aligned}$ $\displaystyle\begin{aligned}
&\begin{cases}\bar{\varphi}(t,x)+\mu\bar{u}(t,x)>0\Rightarrow\bar{u}(t,x)=u_{a},\\\
\bar{\varphi}(t,x)+\mu\bar{u}(t,x)<0\Rightarrow\bar{u}(t,x)=u_{b}.\\\
\end{cases}\\\ \end{aligned}$

We introduce the cone of critical directions that is necessary to state the
second order conditions.

(3.20) $\mathcal{C}_{\bar{u}}=\big{\\{}\varv\in
L_{2}(0,T;L_{2}(\Omega)):\varv\ \text{satisfies}\ \eqref{cone-
conditions}\big{\\}},$ (3.21) $\begin{cases}\varv(t,x)=0\quad\ \text{if}\ \
\bar{\varphi}(t,x)+\mu\bar{u}(t,x)\neq 0\\\ \varv(t,x)\geq 0\quad\ \text{if}\
\ \bar{u}(t,x)=u_{a}\\\ \varv(t,x)\leq 0\quad\ \text{if}\ \
\bar{u}(t,x)=u_{b}.\end{cases}$

Let us notice that

(3.22) $\displaystyle
J^{\prime}(\bar{u})\varv=\int_{0}^{T}{\int_{\Omega}{\left(\bar{\varphi}(t,x)+\mu\bar{u}(t,x)\right)\varv}\mathop{}\\!\mathrm{d}{x}}\mathop{}\\!\mathrm{d}{t},$
$\displaystyle\left(\bar{\varphi}(t,x)+\mu\bar{u}(t,x)\right)\varv(t,x)=0\
\text{for a.e.}\ (t,x)\in\Omega_{T}\ \text{and}\
\forall\varv\in\mathcal{C}_{\bar{u}}.$

######  Theorem 3.9.

Let $\bar{u}$ be a local solution of problem (1.3). Then, it holds that
$J^{\prime\prime}(\bar{u})\varv^{2}\geq 0$,
$\forall\varv\in\mathcal{C}_{\bar{u}}$ . Conversely, if $\bar{u}\in U_{ad}$
satisfies

(3.23) $\displaystyle J^{\prime}(\bar{u})(u-\bar{u})$ $\displaystyle\geq 0\ \
\forall u\in U_{ad},$ (3.24) $\displaystyle
J^{\prime\prime}(\bar{u})\varv^{2}$ $\displaystyle>0\ \ \forall\varv\in
C_{\bar{u}}\setminus\\{0\\},$

then there exist $\alpha>0$ and $\delta>0$ such that

(3.25) $J(\bar{u})+\frac{\delta}{2}\lVert
u-\bar{u}\rVert^{2}_{L_{2}(0,T;L_{2}(\Omega))}\leq J(u)\ \ \forall u\in
U_{ad}\cap B_{\alpha}(\bar{u}),$

where $B_{\alpha}(\bar{u})$ is the open ball of $L_{2}(0,T;L_{2}(\Omega))$
centered at $\bar{u}$ with radius $\alpha$.

###### Proof.

The proof follows identically to [8, 10] based on arguments of [15] (see also
references within). We note that $\|y_{u}\|_{L_{2}(0,T;L_{2}(\Omega))}$ is
bounded independent of $\epsilon$ for any $u\in{\mathcal{U}}_{ad}$ (see Lemma
2.1). In the remaining, we point out that the constant $\delta>0$ will not
appear at any exponential. ∎

######  Remark 3.10.

The second order sufficient condition (3.24) is equivalent to

(3.26)
$J^{\prime\prime}(\bar{u})\varv^{2}\geq\delta\lVert\varv\rVert^{2}_{L_{2}(0,T;L_{2}(\Omega))}\
\ \forall\varv\in\mathcal{C}_{\bar{u}}.$

##  4\. Approximation of the control problem

Let $\\{\mathcal{T}_{h}\\}_{h>0}$ be a family of triangulations of
$\bar{\Omega}$. We consider each $\mathcal{T}_{h}$ to be a conforming and
shape-regular subdivision such that
$\cup_{\tau\in\mathcal{T}_{h}}\tau=\bar{\Omega}$. With each element
$\tau\in\mathcal{T}_{h}$ we associate two parameters $h_{\tau}$ and
$\rho_{\tau}$, where $h_{\tau}$ is the diameter of $\tau$ while $\rho_{\tau}$
is the diameter of the largest ball contained in $\tau$. Then, we define the
meshsize parameter as $h:=\max_{\tau\in\mathcal{T}_{h}}h_{\tau}$. To each
$\mathcal{T}_{h}$ we associate the finite element space:

$Y_{h}:=\big{\\{}y_{h}\in C(\bar{\Omega});\
y_{h}|_{\tau}\in\mathbb{P}_{1}(\tau),\
\forall\tau\in\mathcal{T}_{h}\big{\\}}\subset H^{1}_{0}(\Omega),$

with $\mathbb{P}_{1}$ denoting the $d$-variate space of linear polynomials. We
assume the following classical inverse estimates:

(4.1) $\lVert\varv_{h}\rVert_{L_{3}(\Omega)}\leq
C_{\text{inv}}h^{-(d/6)}\lVert\varv_{h}\rVert_{L_{2}(\Omega)},\ \text{and}\
\lVert\varv_{h}\rVert_{H^{1}(\Omega)}\leq
C_{\text{inv}}h^{-1}\lVert\varv_{h}\rVert_{L_{2}(\Omega)}.$

Furthermore, we set

$U_{h}=\big{\\{}u_{h}\in L_{2}(0,T;L_{2}(\Omega));\ u_{h}|_{\tau}\equiv
u_{\tau}\in\mathbb{R}\big{\\}}.$

Let $0=t_{0}<t_{1}<\ldots<t_{N}=T$. We partition the time interval $[0,T]$
into subintervals $J_{n}:=(t_{n-1},t_{n}]$ with $k_{n}:=t_{n}-t_{n-1}$,
$n=1,\ldots,N$ each time step. We assume that

(4.2) $\displaystyle\exists C_{0}>0\ \text{s.t.}\ k=\max_{1\leq n\leq
N}k_{n}<C_{0}k_{n}\ \ \forall 1\leq n\leq N\ \text{and}\ \forall k>0.$

Setting $\sigma=(k,h)$, we consider the following fully discrete spaces:

$\displaystyle Y_{\sigma}$ $\displaystyle:=\big{\\{}y_{\sigma}\in
L_{2}(0,T;H^{1}_{0}(\Omega));\ y_{\sigma}|_{J_{n}}\in Y_{h},\ 1\leq n\leq
N\\},$ $\displaystyle U_{\sigma}$ $\displaystyle:=\big{\\{}u_{\sigma}\in
L_{2}(0,T;L_{2}(\Omega));\ u_{\sigma}|_{J_{n}}\in U_{h},\ 1\leq n\leq N\\}.$

The functions of $Y_{\sigma}$ and $U_{\sigma}$ are piecewise constant in time.
We are looking for discrete controls in $U_{\sigma}$ that can be written in
the form:

$\displaystyle
u_{\sigma}=\sum^{N}_{n=1}\sum_{\tau\in\mathcal{T}_{h}}u_{n,\tau}\mathcal{\chi}_{n}\mathcal{\chi}_{\tau},\quad\text{with}\
u_{n,\tau}\in\mathbb{R},$

$\mathcal{\chi}_{n}$, $\mathcal{\chi}_{\tau}$ are the characteristic functions
over $(t_{n-1},t_{n})$ and $\tau$, respectively. We consider the convex subset
of $U_{\sigma}$,

$U_{\sigma,ad}=U_{\sigma}\cap U_{ad}=\big{\\{}u_{\sigma}\in
U_{\sigma}:u_{n,\tau}\in[u_{a},u_{b}]\big{\\}}.$

Every element of $Y_{\sigma}$ can be written in the form,

$\displaystyle
y_{\sigma}=\sum^{N}_{n=1}y_{n,h}\mathcal{\chi}_{n},\quad\text{with}\
y_{n,h}\in Y_{h}.$

We fix $y_{\sigma}(t_{n})=y_{n,h}$ in order $y_{\sigma}$ to be continuous from
the left. Thus, we have $y_{\sigma}(T)=y_{\sigma}(t_{N})=y_{N,h}$.

To introduce the discrete control problem, we need to define the fully
discrete scheme of the state equation (1.2). For any $u\in
L_{2}(0,T;L_{2}(\Omega))$, the backward Euler-finite element method
(discontinuous in time Galerkin dG(0)) reads: For each $n=1,\ldots,N$ and for
all $w_{h}\in Y_{h}$,

(4.3) $\Big{(}\frac{y_{n,h}{-}y_{n-1,h}}{k_{n}},w_{h}\Big{)}{+}\left(\nabla
y_{n,h},\nabla
w_{h}\right){+}\epsilon^{-2}\left(F(y_{n,h}),w_{h}\right){=}(u_{n},w_{h}),$

with $y_{0,h}{:=}y_{0h}$ and

(4.4)
$\displaystyle(u_{n},w_{h})=\frac{1}{k_{n}}\int^{t_{n}}_{t_{n-1}}(u(t),w_{h})\mathop{}\\!\mathrm{d}{t},$
$\displaystyle y_{0h}\in Y_{h}\ \ \text{s.t.}\ \ \lVert
y_{0}-y_{0h}\rVert_{L_{2}(\Omega)}\leq Ch\ \ \text{and}\ \ \lVert
y_{0h}\rVert_{H^{1}(\Omega)}\leq C\ \ \forall h>0.$

Then, we define the discrete control problem as follows,

(4.5) $\begin{cases}&\min J_{\sigma}(u_{\sigma})\\\ &u_{\sigma}\in
U_{\sigma,ad},\end{cases}$

where

(4.6) $\displaystyle
J_{\sigma}(u_{\sigma}){=}\frac{1}{2}\int_{0}^{T}{\int_{\Omega}{\lvert
y_{\sigma}(u_{\sigma}){-}y_{d}\rvert^{2}}\mathop{}\\!\mathrm{d}{x}}\mathop{}\\!\mathrm{d}{t}{+}\frac{\gamma}{2}\int_{\Omega}{\lvert
y_{\sigma}(T){-}y_{\Omega,h}\rvert^{2}}\mathop{}\\!\mathrm{d}{x}$
$\displaystyle\qquad\qquad{+}\frac{\mu}{2}\int_{0}^{T}{\int_{\Omega}{\lvert
u_{\sigma}\rvert^{2}}\mathop{}\\!\mathrm{d}{x}}\mathop{}\\!\mathrm{d}{t},$
$\displaystyle y_{\Omega,h}\in Y_{h}\ \ \text{s.t.}\ \ \lVert
y_{\Omega}{-}y_{\Omega,h}\rVert_{L_{2}(\Omega)}\leq Ch\ \ \text{and}\ \ \lVert
y_{\Omega,h}\rVert_{H^{1}(\Omega)}\leq C\ \ \forall h>0.$

The study of the control problem consists of four steps. We begin with the
analysis and error estimation of the discrete state equation. The choice of
the dG(0) method is due to the low regularity imposed by the optimal control
setting. Other approaches for discretization of the Allen-Cahn can be found in
[1, 2] (BDF and extrapolated Runge-Kutta methods via an auxialiary variable
formulation), [32] (symmetric interior penalty discontinuous Galerkin
methods), [38, 41] (scalar auxiliary variable / Crank-Nickolson scheme) [42,
46] (second order semi-implicit scheme). Recent results regarding discrete
maximum principles can be found in [45, 29, 30] (see also references within).

###  4.1. Analysis of the discrete state equation

Let $y=y_{u}=G(u)$ and $y_{\sigma}=y_{\sigma}(u)\in Y_{\sigma}$ be a solution
to (4.3). We begin by presenting stability estimates.

######  Lemma 4.1.

Let $y_{\sigma}$ be a solution to (4.3) corresponding to the control function
$u\in L_{2}(0,T;L_{2}(\Omega))$ and $y_{0h}{:=}P_{h}y_{0}$. Then, there exists
a constant $C>0$ independent of $\sigma=(k,h)$, $\epsilon$ and $\lVert
y\rVert_{L_{\infty}(0,T;L_{\infty}(\Omega))},$ such that

(4.7)
$\displaystyle\|y_{\sigma}\|_{L_{2}(0,T;L_{2}(\Omega))}{+}\|y_{\sigma}\|^{2}_{L_{4}(0,T;L_{4}(\Omega))}$
$\displaystyle\ \leq
C(|\Omega_{T}|^{\frac{1}{2}}{+}\|y_{0}\|_{L_{2}(\Omega)}{+}\|u\|_{L_{2}(0,T;L_{2}(\Omega))}){:=}C^{\text{dG}}_{\text{st},1},$
(4.8)
$\|y_{\sigma}\|_{L_{\infty}(0,T;L_{2}(\Omega))}{+}\|y_{\sigma}\|_{L_{2}(0,T;H^{1}(\Omega))}{+}\Big{(}\sum^{N}_{n=1}\lVert
y_{n,h}{-}y_{n{-}1,h}\rVert^{2}_{L_{2}(\Omega)}\Big{)}^{1/2}{\leq}\frac{C^{\text{dG}}_{\text{st},1}}{\epsilon}.$

If in addition, $k{\leq}\frac{3C_{0}\epsilon^{2}}{2}$, with $C_{0}$ defined by
(4.2), then the following estimate holds,

(4.9)
$\displaystyle\|y_{\sigma}\|_{L_{\infty}(0,T;H^{1}_{0}(\Omega))}{+}\epsilon^{-1}\|y_{\sigma}\|^{2}_{L_{\infty}(0,T;L_{4}(\Omega))}$
$\displaystyle\leq C\left(\|\nabla
y_{0h}\|_{L_{2}(\Omega)}{+}\epsilon^{-1}\|y_{0h}\|^{2}_{L_{4}(\Omega)}{+}\lvert\Omega\rvert^{1/2}\epsilon^{-1}{+}\|u\|_{L_{2}(0;T;L_{2}(\Omega))}\right){:=}C^{dG}_{st,2}.$

###### Proof.

The first two stability estimates can be derived by setting
$w_{h}{=}y_{\sigma}$ into (4.3) (see also [23, Section 3]). For the proof of
the third, we proceed as follows: We choose
$w_{h}=(y_{n,h}{-}y_{n{-}1,h})/k_{n}$ in (4.3), to get

$\displaystyle\Big{\|}\frac{y_{n,h}{-}y_{n{-}1,h}}{k_{n}}\Big{\|}^{2}_{L_{2}(\Omega)}{+}\frac{1}{k_{n}}(\nabla
y_{n,h},\nabla(y_{n,h}-y_{n-1,h})){+}\frac{1}{\epsilon^{2}k_{n}}(y^{3}_{n,h},y_{n,h}-y_{n-1,h})$
$\displaystyle{=}\frac{1}{k_{n}}(u_{n},y_{n,h}-y_{n-1,h}){+}\frac{1}{\epsilon^{2}k_{n}}(y_{n,h},y_{n,h}-y_{n-1,h}).$

Young’s inequality yields
$\int_{\Omega}|y^{3}_{n,h}||y_{n-1,h}|\mathop{}\\!\mathrm{d}{x}\leq\frac{3}{4}\|y_{n,h}\|^{4}_{L_{4}(\Omega)}{+}\frac{1}{4}\|y_{n-1,h}\|^{4}_{L_{4}(\Omega)},$
for $p=\frac{3}{4}$ and $q=4$. Hence, using Hölder and Young’s inequalities
and after some standard algebra, we deduce that

$\displaystyle\frac{3}{4}\Big{\|}\frac{y_{n,h}{-}y_{n{-}1,h}}{k_{n}}\Big{\|}^{2}_{L_{2}(\Omega)}{+}\frac{1}{2k_{n}}\lVert\nabla
y_{n,h}\rVert^{2}_{L_{2}(\Omega)}{+}\frac{1}{2k_{n}}\lVert\nabla(y_{n,h}{-}y_{n{-}1,h})\rVert^{2}_{L_{2}(\Omega)}$
$\displaystyle+\frac{1}{4\epsilon^{2}k_{n}}\lVert
y_{n,h}\rVert^{4}_{L_{4}(\Omega)}{+}\frac{1}{2\epsilon^{2}k_{n}}\lVert
y_{n{-}1,h}\rVert^{2}_{L_{2}(\Omega)}\leq\lVert
u_{n}\rVert^{2}_{L_{2}(\Omega)}{+}\frac{1}{2k_{n}}\lVert\nabla
y_{n{-}1,h}\rVert^{2}_{L_{2}(\Omega)}$
$\displaystyle+\frac{1}{4\epsilon^{2}k_{n}}\lVert
y_{n{-}1,h}\rVert^{4}_{L_{4}(\Omega)}{+}\frac{1}{2\epsilon^{2}k_{n}}\lVert
y_{n,h}\rVert^{2}_{L_{2}(\Omega)}{+}\frac{1}{2\epsilon^{2}k_{n}}\lVert
y_{n,h}{-}y_{n{-}1,h}\rVert^{2}_{L_{2}(\Omega)}.$

Multiplying by $k_{n}$, summing from $n=1$ up to $m$, and using (4.2), where
$1\leq m\leq N$, it yields that

(4.10)
$\displaystyle\frac{3C_{0}}{2k}\sum_{n=1}^{m}\|y_{n,h}{-}y_{n{-}1,h}\|^{2}_{L_{2}(\Omega)}{+}\lVert\nabla
y_{m,h}\rVert^{2}_{L_{2}(\Omega)}{+}\frac{1}{2\epsilon^{2}}\lVert
y_{m,h}\rVert^{4}_{L_{4}(\Omega)}$ $\displaystyle\leq\lVert\nabla
y_{0h}\rVert^{2}_{L_{2}(\Omega)}{+}\frac{1}{2\epsilon^{2}}\lVert
y_{0h}\rVert^{4}_{L_{4}(\Omega)}{+}\frac{1}{\epsilon^{2}}\lVert
y_{m,h}\rVert^{2}_{L_{2}(\Omega)}{+}2\lVert
u\rVert^{2}_{L_{2}(0,t_{m};L_{2}(\Omega))}$ $\displaystyle\
{+}\frac{1}{\epsilon^{2}}\sum^{m}_{n=1}\lVert
y_{n,h}{-}y_{n{-}1,h}\rVert^{2}_{L_{2}(\Omega)}.$

We use Young’s inequality to bound
$\frac{1}{\epsilon^{2}}\|y_{m,h}\|^{2}_{L_{2}(\Omega)}{\leq}\frac{1}{4\epsilon^{2}}\|y_{m,h}\|^{4}_{L_{4}(\Omega)}{+}\frac{\lvert\Omega\rvert}{\epsilon^{2}}.$
Finally, choosing $k$ in a way to hide the last term of the right-hand side of
(4.10) on the left, i.e. for $\frac{1}{\epsilon^{2}}\leq\frac{3C_{0}}{2k},$ we
deduce the desired estimate. ∎

######  Definition 4.2.

We define the projection operator $P_{h}:L_{2}(\Omega)\rightarrow Y_{h}$
through $\left(P_{h}y,w_{h}\right){=}\left(y,w_{h}\right)\quad\forall w_{h}\in
Y_{h}$. Also, we define $P_{\sigma}:C(0,T;L_{2}(\Omega))\rightarrow
Y_{\sigma}$ by $\left(P_{\sigma}y\right)_{n,h}{=}P_{h}y(t_{n})$, for each
$1\leq n\leq N$.

######  Lemma 4.3.

There exists a constant $C>0$ independent of $\sigma$ such that for every
$y\in H^{2,1}(\Omega_{T})\cap C(0,T;H^{1}_{0}(\Omega))$ it holds that

(4.11) $\displaystyle\lVert
y{-}P_{\sigma}y\rVert_{L_{2}(0,T;L_{2}(\Omega))}{\leq}C\Big{(}k\lVert
y_{t}\rVert_{L_{2}(0,T;L_{2}(\Omega))}{+}h^{2}\lVert
y\rVert_{L_{2}(0,T;H^{2}(\Omega))}\Big{)},$ (4.12) $\displaystyle\lVert
y{-}P_{\sigma}y\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}{\leq}C\Big{(}\sqrt{k}\lVert
y_{t}\rVert_{L_{2}(0,T;L_{2}(\Omega))}{+}(\sqrt{k}{+}h)\lVert
y\rVert_{L_{2}(0,T;H^{2}(\Omega))}\Big{)}.$

Another technical tool is the definition of a global space-time projection
onto $Y_{\sigma}$ as a discrete solution to the following auxiliary linear
parabolic problem. Let $y\in H^{2,1}(\Omega_{T})\cap C(0,T;H^{1}_{0}(\Omega))$
be the solution to (1.2) and $y_{0h}{:=}P_{h}y_{0}.$ We define
$\hat{y}_{\sigma}\in Y_{\sigma}$ that satisfies: For each $n=1,\ldots,N$ and
$\forall w_{h}\in Y_{h}$

(4.13)
$\displaystyle\Big{(}\frac{\hat{y}_{n,h}{-}\hat{y}_{n-1,h}}{k_{n}},w_{h}\Big{)}{+}\left(\nabla\hat{y}_{n,h},\nabla
w_{h}\right)$ $\displaystyle{=}(\hat{f}_{n},w_{h})\mbox{ \quad and
\quad}\hat{y}_{0,h}=y_{0h},$

where,

(4.14) $\displaystyle(\hat{f}_{n},w_{h})$
$\displaystyle{=}\frac{1}{k_{n}}\int^{t_{n}}_{t_{n-1}}\\{(\nabla y(t),\nabla
w_{h}){+}(y_{t}(t),w_{h})\\}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle{=}\frac{1}{k_{n}}\int^{t_{n}}_{t_{n-1}}(\nabla y(t),\nabla
w_{h})\mathop{}\\!\mathrm{d}{t}{+}\left(\frac{y(t_{n})-y(t_{n-1})}{k_{n}},w_{h}\right).$

According to [9, Lemma 4.6], (4.13) has a unique solution $\hat{y}_{\sigma}\in
Y_{\sigma}$. We split the error as follows:
$\hat{e}=y{-}\hat{y}_{\sigma}{=}(y-P_{\sigma}y){+}(P_{\sigma}y{-}\hat{y}_{\sigma}).$
Substituting (4.14) into (4.13), we note that $\hat{e}$ satisfies the
following orthogonality condition, for each $w_{h}\in Y_{h}$, $\forall
n=1,...,N$,

(4.15)
$\displaystyle\Big{(}\hat{e}(t_{n}){-}\hat{e}(t_{n-1}),w_{h}\Big{)}{+}\int_{t_{n-1}}^{t_{n}}\left(\nabla\hat{e},\nabla
w_{h}\right)\mathop{}\\!\mathrm{d}{t}$ $\displaystyle{=}0.$

The following Lemma collects various stability and error estimates.

######  Lemma 4.4.

Suppose that $\hat{y}_{\sigma}\in Y_{\sigma}$ is the solution to (4.13). Then,
there exists $C>0$ independent of $\sigma$ and $\epsilon$, such that,

(4.16) $\begin{cases}&\lVert
y{-}\hat{y}_{\sigma}\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}{+}\lVert
y{-}\hat{y}_{\sigma}\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}\\\ &\qquad\leq
C\left(\sqrt{k}\lVert y_{t}\rVert_{L_{2}(0,T;L_{2}(\Omega))}{+}h\lVert
y\rVert_{L_{2}(0,T;H^{2}(\Omega))}{+}h\lVert
y_{0}\rVert_{H^{1}_{0}(\Omega)}\right),\\\ &\lVert
y{-}\hat{y}_{\sigma}\rVert_{L_{2}(0,T;L_{2}(\Omega))}\leq C\left(k\lVert
y_{t}\rVert_{L_{2}(0,T;L_{2}(\Omega))}{+}h^{2}\lVert
y\rVert_{L_{2}(0,T;H^{2}(\Omega))}\right),\\\
&\lVert\hat{y}_{\sigma}\rVert_{L_{\infty}(0,T;H^{1}_{0}(\Omega))}\leq
C\|y\|_{H^{2,1}(\Omega_{T})}.\end{cases}$

###### Proof.

Recall that we split the error as
$\hat{e}=y-\hat{y}_{\sigma}=(y-P_{\sigma}y)+(P_{\sigma}y-\hat{y}_{\sigma}).$
In view of Lemma 4.3, the first term is bounded immediately. As for the
estimation of the second one, that belongs on $Y_{\sigma}$, we refer the
readers to [25, 26], [12] and [9]. ∎

The following result states the main error estimate for the control to state
mapping. We emphasise that, unlike previous works for the uncontrolled Allen-
Cahn equation, we do not exceed the $H^{2,1}(\Omega_{T})$ regularity. Our
technique employs the spectral estimate at the “continuous level”, and hence
it avoids the construction of discrete approximations, which typically lead to
higher regularity requirements.

######  Theorem 4.5.

Let $u\in L_{2}(0,T;L_{2}(\Omega)),y\in H^{2,1}(\Omega_{T})\cap
C(0,T;H^{1}_{0}(\Omega))$ and $y_{\sigma}\in Y_{\sigma}$ satisfy (1.2) and
(4.3) respectively. Suppose that (2.8) holds with
$\|\lambda\|_{L_{\infty}(0,T)}\leq C.$
1\. For $d=2$, if there exists $C>0$ such that

(4.17) $\displaystyle(\sqrt{k}{+}h)\lVert
y\rVert^{1/2}_{H^{2,1}(\Omega_{T})}{\max}\Big{\\{}\frac{C_{\infty}}{\epsilon},\lVert
y\rVert_{H^{2,1}(\Omega_{T})}\Big{\\}}^{{1/2}}\lVert
y\rVert^{1/2}_{L_{\infty}(0,T;L_{\infty}(\Omega))}{\leq}\epsilon^{2}CE^{-1/2},$

then, the following estimates hold:

(4.18) $\displaystyle\lVert
y{-}y_{\sigma}\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}+\epsilon^{-1}\|y-y_{\sigma}\|^{2}_{L_{4}(0,T;L_{4}(\Omega)}$
$\displaystyle\leq\max\Big{\\{}\frac{C_{I}}{\epsilon^{2}}(\sqrt{k}{+}h),\frac{C_{II}}{\epsilon}(\sqrt{k}{+}h)\lVert
y\rVert_{H^{2,1}(\Omega_{T})},C\Big{\\}}(\sqrt{k}{+}h)\lVert
y\rVert_{H^{2,1}(\Omega_{T})},$ (4.19) $\displaystyle\lVert
y{-}y_{\sigma}\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}$
$\displaystyle{\leq}\max\Big{\\{}\frac{C_{I}}{\epsilon^{3}}(\sqrt{k}{+}h),\frac{C_{II}}{\epsilon^{2}}(\sqrt{k}{+}h)\lVert
y\rVert_{H^{2,1}(\Omega_{T})},C\Big{\\}}(\sqrt{k}{+}h)\lVert
y\rVert_{H^{2,1}(\Omega_{T})}.$

2\. For $d=3$, if there exists $C>0$ such that

(4.20) $\displaystyle(\sqrt{k}{+}h)\lVert
y\rVert^{4/3}_{H^{2,1}(\Omega_{T})}\lVert
y\rVert^{2/3}_{L_{\infty}(0,T;L_{\infty}(\Omega))}{\leq}\epsilon^{3}CE^{-1},$

then, the following estimates hold:

(4.21) $\displaystyle\lVert
y{-}y_{\sigma}\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}+\epsilon^{-1}\|y-y_{\sigma}\|^{2}_{L_{4}(0,T;L_{4}(\Omega))}$
$\displaystyle\leq\max\Big{\\{}\frac{C_{II}}{\epsilon}(\sqrt{k}{+}h)^{1/2}\lVert
y\rVert_{H^{2,1}(\Omega_{T})},C\Big{\\}}(\sqrt{k}{+}h)\|y\|_{H^{2,1}(\Omega_{T})},$
(4.22) $\displaystyle\lVert
y{-}y_{\sigma}\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}$
$\displaystyle\leq\max\Big{\\{}\frac{C_{II}}{\epsilon^{2}}(\sqrt{k}{+}h)^{1/2}\lVert
y\rVert_{H^{2,1}(\Omega_{T})},C\Big{\\}}(\sqrt{k}{+}h)\|y\|_{H^{2,1}(\Omega_{T})}.$

Here, we denote by $C_{I}{:=}2E^{1/2}C_{\infty}$, $C_{II}{:=}CE^{1/2}$and
$E{:=}\exp(2T\alpha)$ where

$\displaystyle C_{\infty}$
$\displaystyle{:=}C\Big{(}1{+}(1{+}\epsilon^{2})\lVert
y\rVert^{2}_{L_{\infty}(0,T;L_{\infty}(\Omega))}\Big{)}^{1/2},\
\alpha{:=}\sup_{t\in[0,T]}(2\lambda(t)(1{-}\epsilon^{2}){+}4),$

and $C{>}0$ an algebraic constant (that might be different in each occurrence)
but independent of $\sigma,\epsilon$, and
$\|y\|_{L_{\infty}(0,T;L_{\infty}(\Omega))}$.

None of the above constants depend on $\lVert
y\rVert_{L_{\infty}(0,T;L_{\infty}(\Omega))}$ exponentially. Indeed, $C_{I}$
depends on the $\lVert y\rVert_{L_{\infty}(0,T;L_{\infty}(\Omega))}$
polynomially while $C>0$ is an algebraic constant independent of $\epsilon.$
We mainly focus on the case where $\lVert
y\rVert_{L_{\infty}(0,T;L_{\infty}(\Omega))}$ is bounded independent
$1/\epsilon$ (see Remark 4.6 for a detailed discussion). However, our results
hold, without any exponential dependence upon $1/\epsilon$ even in more
general cases.

###### Proof.

We begin by splitting the total error:

(4.23)
$e{=}y{-}y_{\sigma}{=}(y{-}\hat{y}_{\sigma}){+}(\hat{y}_{\sigma}{-}y_{\sigma}){=}\hat{e}{+}e_{\sigma}.$

The aim of our analysis is to bound the term $e_{\sigma}$ in terms of
$\hat{e}$, whose bounds are known from Lemma 4.4. From relations (2.5) and
(4.3), for all $w_{h}\in Y_{h}$, $\forall n=1,...,N$ it holds that

$\left(e(t_{n}){-}e(t_{n-1}),w_{h}\right){+}\int^{t_{n}}_{t_{n-1}}\left(\nabla
e(t),\nabla
w_{h}\right)\mathop{}\\!\mathrm{d}{t}{+}\epsilon^{-2}\int^{t_{n}}_{t_{n-1}}\left(F(y){-}F(y_{n,h}),w_{h}\right)\mathop{}\\!\mathrm{d}{t}{=}0.$

Using (4.23), the orthogonality condition (4.14) and choosing
$w_{h}{=}e_{n,h}$ we obtain,

$\left(e_{n,h}{-}e_{n-1,h},e_{n,h}\right){+}\int^{t_{n}}_{t_{n-1}}\lVert\nabla
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}\mathop{}\\!\mathrm{d}{t}{+}\epsilon^{-2}\int^{t_{n}}_{t_{n-1}}\left(F(y){-}F(y_{n,h}),e_{n,h}\right)\mathop{}\\!\mathrm{d}{t}{=}0.$

A standard algebraic manipulation implies that

$\displaystyle
F(y){-}F(y_{n,h}){=}(3y^{2}{-}1)(y{-}y_{n,h}){-}3y(y{-}y_{n,h})^{2}{+}(y{-}y_{n,h})^{3}.$

Using the decomposition (4.23) and elementary identities, we deduce that

(4.24) $\displaystyle\frac{1}{2}\lVert
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}{-}\frac{1}{2}\lVert
e_{n{-}1,h}\rVert^{2}_{L_{2}(\Omega)}{+}\frac{1}{2}\lVert
e_{n,h}{-}e_{n{-}1,h}\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{-2}\int^{t_{n}}_{t_{n-1}}\lVert
e_{n,h}\rVert^{4}_{L_{4}(\Omega)}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle{+}\int^{t_{n}}_{t_{n-1}}\left(\lVert\nabla
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{-2}\left(F^{\prime}(y)e_{n,h},e_{n,h}\right)\right)\mathop{}\\!\mathrm{d}{t}{+}3\epsilon^{-2}\int^{t_{n}}_{t_{n-1}}(\hat{e}^{2},e^{2}_{n,h})\mathop{}\\!\mathrm{d}{t}$
$\displaystyle{=}$
$\displaystyle{-}\epsilon^{-2}\int^{t_{n}}_{t_{n-1}}\left(F^{\prime}(y)\hat{e},e_{n,h}\right)\mathop{}\\!\mathrm{d}{t}{+}3\epsilon^{-2}\int^{t_{n}}_{t_{n-1}}\left(y\hat{e}^{2},e_{n,h}\right)\mathop{}\\!\mathrm{d}{t}{+}6\epsilon^{-2}\int^{t_{n}}_{t_{n-1}}\left(y\hat{e},e^{2}_{n,h}\right)\mathop{}\\!\mathrm{d}{t}$
$\displaystyle{-}\epsilon^{-2}\int^{t_{n}}_{t_{n-1}}\left(\hat{e}^{3},e_{n,h}\right)\mathop{}\\!\mathrm{d}{t}{-}3\epsilon^{-2}\int^{t_{n}}_{t_{n-1}}\left(\hat{e},e^{3}_{n,h}\right)\mathop{}\\!\mathrm{d}{t}{+}3\epsilon^{-2}\int^{t_{n}}_{t_{n-1}}\left(y,e^{3}_{n,h}\right)\mathop{}\\!\mathrm{d}{t}{=:}\sum_{j=1}^{6}\mathcal{I}_{j}.$

First, we recover additional coercivity at the right-hand side based on (2.8).
Indeed,

$\displaystyle\int^{t_{n}}_{t_{n-1}}\left(\lVert\nabla
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{-2}\left(F^{\prime}(y)e_{n,h},e_{n,h}\right)\right)\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\geq\int_{t_{n-1}}^{t_{n}}{\left((\epsilon^{2}{-}1)\lambda(t){-}1\right)\|e_{n,h}\|^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\epsilon^{2}\int_{t_{n-1}}^{t_{n}}{\|\nabla
e_{n,h}\|^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}3\int_{t_{n-1}}^{t_{n}}{\|ye_{n,h}\|^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}.$

We estimate the right-hand side of (4.24). Hölder and Young’s inequalities
yield,

$\displaystyle\mathcal{I}_{1}$
$\displaystyle{\leq}\int_{t_{n-1}}^{t_{n}}{\|ye_{n,h}\|^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\frac{1}{4\epsilon^{4}}\int_{t_{n-1}}^{t_{n}}{\left(9\lVert
y\rVert^{2}_{L_{\infty}(\Omega)}{+}1\right)\lVert\hat{e}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\int_{t_{n-1}}^{t_{n}}{\lVert
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t},$
$\displaystyle\mathcal{I}_{2}$
$\displaystyle{\leq}\frac{1}{2\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\|{\hat{e}}e_{n,h}\|^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\frac{9}{2\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\|y\|^{2}_{L_{\infty}(\Omega)}\lVert{\hat{e}}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t},$
$\displaystyle\mathcal{I}_{3}$
$\displaystyle{\leq}\frac{36}{\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert
y\rVert^{2}_{L_{\infty}(\Omega)}\lVert\hat{e}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\frac{1}{4\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert
e_{n,h}\rVert^{4}_{L_{4}(\Omega)}}\mathop{}\\!\mathrm{d}{t},$
$\displaystyle\mathcal{I}_{4}$
$\displaystyle\leq\frac{3}{4\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert\hat{e}\rVert^{4}_{L_{4}(\Omega)}}\mathop{}\\!\mathrm{d}{t}+\frac{1}{4\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert
e_{n,h}\rVert^{4}_{L_{4}(\Omega)}}\mathop{}\\!\mathrm{d}{t},$
$\displaystyle\mathcal{I}_{5}$
$\displaystyle\leq\frac{3^{7}}{4\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert\hat{e}\rVert^{4}_{L_{4}(\Omega)}}\mathop{}\\!\mathrm{d}{t}+\frac{1}{4\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert
e_{n,h}\rVert^{4}_{L_{4}(\Omega)}}\mathop{}\\!\mathrm{d}{t}.$

The bound of $\mathcal{I}_{6}$ for $d=3$ differs from one for $d=2$. We begin
with $d=3$:

$\displaystyle\mathcal{I}_{6}$
$\displaystyle\leq\frac{3}{\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert
y\rVert_{L_{\infty}(\Omega)}\lVert
e_{n,h}\rVert^{3}_{L_{3}(\Omega)}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\leq\frac{12}{\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert
y\rVert_{L_{\infty}(\Omega)}\lVert
e_{n,h}-e_{n{-}1,h}\rVert^{3}_{L_{3}(\Omega)}}\mathop{}\\!\mathrm{d}{t}+\frac{12}{\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert
y\rVert_{L_{\infty}(\Omega)}\lVert
e_{n{-}1,h}\rVert^{3}_{L_{3}(\Omega)}}\mathop{}\\!\mathrm{d}{t}.$

We want to ensure that the first term of ${\mathcal{I}}_{6}$ can be absorbed
by the term $\frac{1}{2}\lVert e_{n,h}-e_{n{-}1,h}\rVert^{2}_{L_{2}(\Omega)}$
of the left-hand side of (4.24). Using (2.3) and (4.1) it yields that $\lVert
e_{n,h}{-}e_{n{-}1,h}\rVert^{3}_{L_{3}(\Omega)}\leq
h^{-1/2}C_{\text{inv}}\tilde{c}^{3}\lVert
e_{n,h}{-}e_{n{-}1,h}\rVert^{2}_{L_{2}(\Omega)}\lVert\nabla(e_{n,h}{-}e_{n{-}1,h})\rVert_{L_{2}(\Omega)}.$
We can bound the quantity
$\lVert\nabla(e_{n,h}-e_{n{-}1,h})\rVert_{L_{2}(\Omega)}$ through (4.9) and
(4.16). Therefore, it is enough to assume that for all $n=1,\ldots,N$ it holds

(4.25) $\displaystyle 12\tilde{c}^{3}C_{\text{inv}}\lVert
y\rVert_{L_{\infty}(t_{n-1},t_{n};L_{\infty}(\Omega))}\frac{k_{n}}{\epsilon^{2}\sqrt{h}}\left(C^{\text{dG}}_{\text{st,2}}+\|y\|_{H^{2,1}(\Omega_{T})}\right)\leq
1/4.$

Recalling (2.8) for $\varv=e_{n,h}$ and inserting the bounds of
${\mathcal{I}}_{i}$ into (4.24) to deduce,

$\displaystyle\lVert e_{n,h}\rVert^{2}_{L_{2}(\Omega)}{-}\lVert
e_{n{-}1,h}\rVert^{2}_{L_{2}(\Omega)}{+}\frac{1}{2\epsilon^{2}}\int^{t_{n}}_{t_{n-1}}\lVert
e_{n,h}\rVert^{4}_{L_{4}(\Omega)}\mathop{}\\!\mathrm{d}{t}{+}2\epsilon^{2}\int^{t_{n}}_{t_{n-1}}\lVert\nabla
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}$ $\displaystyle\
{+}\frac{5}{\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\|{\hat{e}}e_{n,h}\|^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}4\int_{t_{n-1}}^{t_{n}}{\|ye_{n,h}\|^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}+\frac{1}{2}\|e_{n,h}{-}e_{n-1,h}\|^{2}_{L_{2}(\Omega)}$
$\displaystyle{\leq}\int_{t_{n-1}}^{t_{n}}{\left(\Big{(}\frac{9}{2\epsilon^{4}}{+}\frac{81}{\epsilon^{2}}\Big{)}\lVert
y\rVert^{2}_{L_{\infty}(\Omega)}{+}\frac{1}{2\epsilon^{4}}\right)\lVert\hat{e}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\int_{t_{n-1}}^{t_{n}}{\Big{(}\frac{3}{2\epsilon^{2}}{+}\frac{3^{7}}{2\epsilon^{2}}\Big{)}\lVert\hat{e}\rVert^{4}_{L_{4}(\Omega)}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\quad{+}\int_{t_{n-1}}^{t_{n}}{\left(2\lambda(t)(1{-}\epsilon^{2}){+}4\right)\lVert
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\frac{24}{\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert
y\rVert_{L_{\infty}(\Omega)}\lVert
e_{n{-}1,h}\rVert^{3}_{L_{3}(\Omega)}}\mathop{}\\!\mathrm{d}{t}.$

Summing from $n{=}1$ up to $m$ where $1\leq m\leq N$, and dropping positive
terms, we obtain

(4.26) $\displaystyle\lVert
e_{m,h}\rVert^{2}_{L_{2}(\Omega)}{+}\frac{1}{2\epsilon^{2}}\sum^{m}_{n=1}k_{n}\lVert
e_{n,h}\rVert^{4}_{L_{4}(\Omega)}{+}2\epsilon^{2}\sum^{m}_{n=1}k_{n}\lVert\nabla
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}$
$\displaystyle{\leq}A{+}\sum^{m}_{n=1}k_{n}\alpha\lVert
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}{+}\frac{24}{\epsilon^{2}}\sum^{m-1}_{n=1}\int_{t_{n-1}}^{t_{n}}{\lVert
y\rVert_{L_{\infty}(\Omega)}\lVert
e_{n,h}\rVert^{3}_{L_{3}(\Omega)}}\mathop{}\\!\mathrm{d}{t},$

where we denote by
$\alpha{=}\sup_{t\in[0,T]}(2\lambda(t)(1{-}\epsilon^{2}){+}4)$ and by

$\displaystyle
A{:=}\left(\Big{(}\frac{9}{2\epsilon^{4}}{+}\frac{81}{\epsilon^{2}}\Big{)}\lVert
y\rVert^{2}_{L_{\infty}(0,T;L_{\infty}(\Omega))}{+}\frac{1}{2\epsilon^{4}}\right)\lVert\hat{e}\rVert^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}\left(\frac{3}{2\epsilon^{2}}{+}\frac{3^{7}}{2\epsilon^{2}}\right)\lVert\hat{e}\rVert^{4}_{L_{4}(0,T;L_{4}(\Omega))}.$

Assume that $\sup_{n=1,\ldots,N}k_{n}\alpha\leq 1/2$. Note that for every
$m=1,\ldots,N,$ (2.3) and Young’s inequalities (with $p=4$ and $q=4/3$) and
standard algebra imply that

$\displaystyle\frac{24}{\epsilon^{2}}\sum^{m-1}_{n=1}\int_{t_{n-1}}^{t_{n}}{\lVert
y\rVert_{L_{\infty}(\Omega)}\lVert
e_{n,h}\rVert^{3}_{L_{3}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{\leq}\frac{24\tilde{c}^{3}}{\epsilon^{2}}\sum^{m-1}_{n=1}\int_{t_{n-1}}^{t_{n}}{\lVert
y\rVert_{L_{\infty}(\Omega)}\lVert e_{n,h}\rVert^{3/2}_{L_{2}(\Omega)}\|\nabla
e_{n,h}\|^{3/2}_{L_{2}(\Omega}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle{=}\frac{24\tilde{c}^{3}}{\epsilon^{7/2}}\sum^{m-1}_{n=1}\int_{t_{n-1}}^{t_{n}}{\lVert
y\rVert_{L_{\infty}(\Omega)}\lVert e_{n,h}\rVert_{L_{2}(\Omega)}\left(\lVert
e_{n,h}\rVert^{1/2}_{L_{2}(\Omega)}\epsilon^{3/2}\|\nabla
e_{n,h}\|^{3/2}_{L_{2}(\Omega}\right)}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle{\leq}\frac{24\tilde{c}^{3}}{\epsilon^{7/2}}\lVert
y\rVert_{L_{\infty}(0,T;L_{\infty}(\Omega))}\sup_{n=1,\ldots,m-1}\lVert
e_{n,h}\rVert_{L_{2}(\Omega)}\sum^{m-1}_{n=1}k_{n}\left(\lVert
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}+\epsilon^{2}\lVert\nabla
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}\right).$

Then, applying Lemma 2.4 we deduce that

(4.27) $\sup_{n=1,\ldots,N}\lVert
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}{+}2\epsilon^{2}\sum^{N}_{n=1}k_{n}\lVert\nabla
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}{+}\frac{2}{\epsilon^{2}}\sum^{N}_{n=1}k_{n}\lVert
e_{n,h}\rVert^{4}_{L_{4}(\Omega)}{\leq}4AE.$

The above estimate holds, upon setting $\beta=1/2$ and
$B=24\epsilon^{-7/2}\tilde{c}\lVert
y\rVert_{L_{\infty}(0,T;L_{\infty}(\Omega))}$ in Lemma 2.4 as long as

(4.28) $\displaystyle A{\leq}\epsilon^{7}\left(192\tilde{c}\lVert
y\rVert_{L_{\infty}(0,T;L_{\infty}(\Omega))}(T{+}1)E^{3/2}\right)^{-2}:=\epsilon^{7}C_{T}E^{-3}\|y\|^{-2}_{L_{\infty}(0,T;L_{\infty}(\Omega))}.$

To quantify the relation between $k,h$ and $\epsilon$ from (4.28), observe
that Lemma 4.4 and the embedding $H^{2,1}(\Omega_{T})\subset
C(0,T;H^{1}(\Omega))$ imply
$\|y-\hat{y}_{\sigma}\|_{L_{\infty}(0,T;H^{1}(\Omega))}\leq
C\|y\|_{H^{2,1}(\Omega_{T})}$. Hence, denoting by
$C^{2}_{\infty}:=C(1+(1+\epsilon^{2})\|y\|^{2}_{L_{\infty}(0,T;L_{\infty}(\Omega))})$
where $C$ is an algebraic constant, (2.4) and Lemma 4.4 imply that

$\displaystyle A$
$\displaystyle{\sim}\frac{C^{2}_{\infty}}{\epsilon^{4}}\lVert
y{-}\hat{y}_{\sigma}\rVert^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}\frac{C\tilde{c}^{4}\|y\|_{H^{2,1}(\Omega)}}{\epsilon^{2}}\lVert
y{-}\hat{y}_{\sigma}\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}\lVert
y{-}\hat{y}_{\sigma}\rVert^{2}_{L_{2}(0,T;H^{1}_{0}(\Omega))}$
$\displaystyle{\sim}\frac{C^{2}_{\infty}}{\epsilon^{4}}(k+h^{2})^{2}\lVert
y\rVert^{2}_{H^{2,1}(\Omega_{T})}{+}\frac{C\tilde{c}^{4}}{\epsilon^{2}}(\sqrt{k}+h)^{3}\lVert
y\rVert^{4}_{H^{2,1}(\Omega_{T})}.$

It is clear that if

(4.29)
$\sqrt{k}+h\leq\epsilon^{2}\|y\|^{2}_{H^{2,1}(\Omega_{T})}C^{-2}_{\infty}$

then the second term of (4.1) dominates the above quantity, and hence (4.1)
and (4.28) result to the following restriction:

$(\sqrt{k}+h)^{3}\lVert
y\rVert^{4}_{H^{2,1}(\Omega_{T})}\tilde{c}^{4}C\leq\epsilon^{9}C_{T}E^{-3}\lVert
y\rVert^{-2}_{L_{\infty}(0,T;L_{\infty}(\Omega))},$

where $C_{T}{>}0$ is independent of $k,h,\epsilon$, and
$\|y\|_{L_{\infty}(0,T;L_{\infty}(\Omega))}$, $\|y\|_{H^{2,1}(\Omega_{T})}$.
The estimates (4.21) and (4.22) follow by triangle inequality and the estimate
of Lemma 4.4. Note that if (4.20) is satisfied then both conditions (4.25) and
(4.29) are also satisfied.
For $d=2$, we proceed in a similar way. Using Hölder, (2.3), and Young’s
inequalities we deduce,

$\displaystyle\mathcal{I}_{6}$
$\displaystyle\leq\frac{3\tilde{c}}{\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert
y\rVert_{L_{\infty}(\Omega)}\lVert
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}\lVert\nabla
e_{n,h}\rVert_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\leq\frac{6\tilde{c}}{\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert
y\rVert_{L_{\infty}(\Omega)}\left(\lVert
e_{n,h}{-}e_{n-1,h}\rVert^{2}_{L_{2}(\Omega)}{+}\lVert
e_{n-1,h}\rVert^{2}_{L_{2}(\Omega)}\right)\|\nabla
e_{n,h}\|_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\leq\frac{6\tilde{c}}{\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert
y\rVert_{L_{\infty}(\Omega)}\lVert
e_{n,h}{-}e_{n-1,h}\rVert^{2}_{L_{2}(\Omega)}\|\nabla
e_{n,h}\|_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\quad{+}\frac{36\tilde{c}^{2}}{\epsilon^{6}}\int_{t_{n-1}}^{t_{n}}{\|e_{n-1,h}\|^{4}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\frac{\epsilon^{2}}{4}\int_{t_{n-1}}^{t_{n}}{\|\nabla
e_{n,h}\|^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}.$

Then, through (4.9) and (4.16), we conclude to an analogous assumption to
(4.25),

(4.30) $\displaystyle 6\tilde{c}\lVert
y\rVert_{L_{\infty}(t_{n-1},t_{n};L_{\infty}(\Omega))}\frac{k_{n}}{\epsilon^{2}}\left(C^{\text{dG}}_{\text{st,2}}{+}\|y\|_{H^{2,1}(\Omega_{T})}\right)\leq
1/4.$

Working exactly as in $d=3$-case, we have that

(4.31) $\displaystyle\lVert
e_{m,h}\rVert^{2}_{L_{2}(\Omega)}{+}\frac{1}{2\epsilon^{2}}\sum^{m}_{n=1}k_{n}\lVert
e_{n,h}\rVert^{4}_{L_{4}(\Omega)}{+}\frac{3}{2}\epsilon^{2}\sum^{m}_{n=1}k_{n}\lVert\nabla
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}$ $\displaystyle\leq
A{+}\sum^{m}_{n=1}k_{n}\alpha\lVert
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}{+}\frac{36\tilde{c}^{2}}{\epsilon^{6}}\sum^{m-1}_{n=1}\int_{t_{n-1}}^{t_{n}}{\lVert
y\rVert^{2}_{L_{\infty}(\Omega)}\lVert
e_{n,h}\rVert^{4}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t},$

where $\sup_{n=1,\ldots,N}k_{n}\alpha\leq 1/2.$ Note that for every
$m=1,\ldots,N$

$\displaystyle\frac{36\tilde{c}^{2}}{\epsilon^{6}}\sum^{m-1}_{n=1}\int_{t_{n-1}}^{t_{n}}{\lVert
y\rVert^{2}_{L_{\infty}(\Omega)}\lVert
e_{n,h}\rVert^{4}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\leq\frac{36\tilde{c}^{2}}{\epsilon^{6}}\lVert
y\rVert^{2}_{L_{\infty}(0,T;L_{\infty}(\Omega))}\sup_{n=1,\ldots,m-1}\lVert
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}\sum^{m-1}_{n=1}k_{n}\lVert
e_{n,h}\rVert^{2}_{L_{2}(\Omega)}.$

From Lemma 2.4 for $\beta=1$ and $B=36\epsilon^{-6}\tilde{c}^{2}\lVert
y\rVert^{2}_{L_{\infty}(0,T;L_{\infty}(\Omega))}$, there holds an identical
result to (4.27) as long the assumption

(4.32) $\displaystyle A{\leq}\epsilon^{6}\left(16{\cdot}36\tilde{c}^{4}\lVert
y\rVert^{2}_{L_{\infty}(0,T;L_{\infty}(\Omega))}(T{+}1)E^{2}\right)^{-1}{:=}\epsilon^{6}C_{T}E^{-2}\|y\|^{-2}_{L_{\infty}(0,T;L_{\infty}(\Omega))},$

is satisfied. To quantify the dependence of $k$,$h$ upon $\epsilon$ through
(4.32) observe that (2.2) and Lemma 4.4, implies

$\displaystyle A$ $\displaystyle\sim\frac{C^{2}_{\infty}}{\epsilon^{4}}\lVert
y-\hat{y}_{\sigma}\rVert^{2}_{L_{2}(0,T;L_{2}(\Omega))}+\frac{C\tilde{c}^{4}}{\epsilon^{2}}\lVert
y{-}\hat{y}_{\sigma}\rVert^{2}_{L_{\infty}(0,T;L_{2}(\Omega))}\lVert
y-\hat{y}_{\sigma}\rVert^{2}_{L_{2}(0,T;H^{1}_{0}(\Omega))}$
$\displaystyle\sim\frac{C^{2}_{\infty}}{\epsilon^{4}}(k{+}h^{2})^{2}\lVert
y\rVert^{2}_{H^{2,1}(\Omega_{T})}{+}\frac{C\tilde{c}^{4}}{\epsilon^{2}}(k{+}h^{2})^{2}\lVert
y\rVert^{4}_{H^{2,1}(\Omega_{T})},$

that along with (4.28) imply

$(k{+}h^{2})^{2}\lVert
y\rVert^{2}_{H^{2,1}(\Omega_{T})}\max\Big{\\{}\frac{C^{2}_{\infty}}{\epsilon^{2}},C\tilde{c}^{4}\lVert
y\rVert^{2}_{H^{2,1}(\Omega_{T})}\Big{\\}}\leq\epsilon^{8}C_{T}E^{-2}\lVert
y\rVert^{-2}_{L_{\infty}(0,T;L_{\infty}(\Omega))}.$

Observe that if the above estimate is satisfied then (4.30) also holds. The
estimate follows by triangle inequality and the estimate of Lemma 4.4. ∎

######  Remark 4.6.

Recall that $C_{I}{\sim}\lVert y\rVert_{L_{\infty}(0,T;L_{\infty}(\Omega))}$
and $\lVert y\rVert_{H^{2,1}(\Omega_{T})}{\sim}\epsilon^{-r},$
$r\in\\{1,2\\}.$ Assume that there exists $C{>}0$ independent of
$\sigma,\epsilon$ such that: $\lVert
y\rVert_{L_{\infty}(0,T;L_{\infty}(\Omega))}{\leq}C$. Then, even for $r=2$,
the conditions (4.30), (4.25) are less restrictive than (4.17), (4.20). Now,
we shall indicate the dominant term in the bounds of Theorem 4.5. Indeed, for
$d=2$, (4.17) becomes

(4.33) $\displaystyle(\sqrt{k}{+}h)\lVert y\rVert_{H^{2,1}(\Omega_{T})}\leq
C\epsilon^{2}\ \Longrightarrow\ \sqrt{k}{+}h\leq C\epsilon^{2+r},$

where $C$ is an algebraic constant depending only on the domain, and
$\|\lambda\|_{L_{\infty}(0,T)}$ and hence replacing (4.33) into the estimates
(4.18) and (4.19) we obtain,

$\displaystyle\lVert
y{-}y_{\sigma}\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}{+}\epsilon^{-1}\|y{-}y_{\sigma}\|^{2}_{L_{4}(0,T;L_{4}(\Omega))}$
$\displaystyle\leq C(\sqrt{k}{+}h)\epsilon^{-r},$ $\displaystyle\lVert
y{-}y_{\sigma}\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}$ $\displaystyle\leq
C(\sqrt{k}{+}h)\epsilon^{-r}.$

For $d=3$, we deduce that (4.20) becomes

(4.34) $\displaystyle(\sqrt{k}{+}h)\lVert
y\rVert^{4/3}_{H^{2,1}(\Omega_{T})}\leq C\epsilon^{3}\ \Longrightarrow\
\sqrt{k}{+}h\leq C\epsilon^{3+(4r/3)},$

and the estimates (4.21) and (4.22) can be written as

$\displaystyle\lVert
y{-}y_{\sigma}\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}{+}\epsilon^{-1}\|y{-}y_{\sigma}\|^{2}_{L_{4}(0,T;L_{4}(\Omega))}\leq
C\max\\{\epsilon^{(1/2){-}(r/3)},1\\}(\sqrt{k}{+}h)\epsilon^{-r}$
$\displaystyle\leq\begin{cases}C(\sqrt{k}{+}h)\epsilon^{-1}\quad\quad\
\text{when}\ r=1,\\\ C(\sqrt{k}{+}h)\epsilon^{-13/6}\quad\text{when}\
r=2,\end{cases}\leq C(\sqrt{k}{+}h)\epsilon^{-(7r-1)/6},$ $\displaystyle\lVert
y{-}y_{\sigma}\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}\leq
C(\sqrt{k}{+}h)\epsilon^{-(8r+3)/6}.$

######  Remark 4.7.

If additional regularity is present, i.e, if
$\|y\|_{W^{2,1}_{4}(\Omega_{T})}\approx\epsilon^{-r}$, then observe that using
the maximal parabolic regularity results of [37] we may bound
$\|\hat{e}\|^{4}_{L_{4}(0,T;L_{4}(\Omega))}\sim(\kappa{+}h^{2})^{4}\|y\|^{4}_{W^{2,1}_{4}(\Omega_{T})}{\sim}(\kappa{+}h^{2})^{4}\epsilon^{-4r}$.
Therefore, we easily deduce that
$A\sim\frac{C^{2}_{\infty}}{\epsilon^{4}}(k{+}h^{2})^{2}\lVert
y\rVert^{2}_{H^{2,1}(\Omega_{T})}$ in both $d{=}2,3.$ As a consequence the
conditionality reads as $\sqrt{k}{+}h\sim\epsilon^{2{+}(r/2)}$ when $d{=}2$
and $\sqrt{k}{+}h\sim\epsilon^{(11/4){+}(r/2)}$ when $d{=}3$.

Theorem 4.5 and Remark 4.6 play a crucial role in the derivation of the
following result.

######  Corollary 4.8.

Let $u,\varv\in L_{2}(0,T;L_{2}(\Omega))$, $y_{u}\in H^{2,1}(\Omega_{T})\cap
C(0,T;H^{1}_{0}(\Omega))$ be the solution of (1.2) while $y_{\sigma}(\varv)\in
Y_{\sigma}$ the solution of (4.3) corresponding to the control $\varv$.
Suppose that the assumptions of Theorems 3.2 and 4.5 hold. In addition, let
$\|y_{v}\|_{L_{\infty}(0,T;L_{\infty}(\Omega))}\leq C$, and
$\|y_{v}\|_{H^{2,1}(\Omega_{T})}\leq C\epsilon^{-r}$ with $r\in\\{1,2\\}$,
where $C$ denotes a constant that depends only on data and it is independent
of $\epsilon$ and that (4.33) or (4.34) hold for $d=2$ or $3$, respectively.
Then, for $d{=}2$ there holds

(4.35) $\displaystyle\lVert
y_{u}{-}y_{\sigma}(\varv)\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}\leq
L_{1}\lVert
u{-}\varv\rVert_{L_{2}(0,T;L_{2}(\Omega))}+\frac{C}{\epsilon^{r}}(\sqrt{k}{+}h),$
(4.36) $\displaystyle\lVert
y_{u}{-}y_{\sigma}(\varv)\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}\leq\frac{L_{1}}{\epsilon}\lVert
u{-}\varv\rVert_{L_{2}(0,T;L_{2}(\Omega))}+\frac{C}{\epsilon^{r}}(\sqrt{k}{+}h),$

while for $d=3$, there holds

(4.37) $\displaystyle\lVert
y_{u}{-}y_{\sigma}(\varv)\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}\leq
L_{1}\lVert
u{-}\varv\rVert_{L_{2}(0,T;L_{2}(\Omega))}+\frac{C}{\epsilon^{(7r{-}1)/6}}(\sqrt{k}{+}h),$
(4.38) $\displaystyle\lVert
y_{u}{-}y_{\sigma}(\varv)\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}\leq\frac{L_{1}}{\epsilon}\lVert
u{-}\varv\rVert_{L_{2}(0,T;L_{2}(\Omega))}+\frac{C}{\epsilon^{(8r{+}3)/6}}(\sqrt{k}{+}h).$

Let $u_{\sigma}\in U_{\sigma}$. If $u_{\sigma}\rightharpoonup u$ weakly in
$L_{2}(0,T;L_{2}(\Omega))$ for every $\sigma$, then

(4.39) $\begin{cases}\lVert
y_{u}{-}y_{\sigma}(u_{\sigma})\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}\rightarrow
0,\\\ \lVert
y_{u}{-}y_{\sigma}(u_{\sigma})\rVert_{L_{p}(0,T;L_{2}(\Omega))}\rightarrow
0\quad\forall 1\leq p<\infty,\\\ \lVert
y_{u}(T){-}y_{\sigma}(u_{\sigma})(T)\rVert_{L_{2}(\Omega)}\rightarrow
0.\end{cases}$

###### Proof.

Inequalities (4.35), (4.36), (4.37), (4.38) follow from (3.2) and Remark 4.6
using triangle inequality. For (4.39) we split,
$y_{u}{-}y_{\sigma}(u_{\sigma})=(y_{u}{-}y_{u_{\sigma}}){+}(y_{u_{\sigma}}{-}y_{\sigma}(u_{\sigma}))$.
According to Lemma 2.1 and the boundedness of $\\{u_{\sigma}\\}_{\sigma}$ in
$L_{2}(0,T;L_{2}(\Omega))$ we have that $\lVert
y_{u_{\sigma}}\rVert_{H^{2,1}(\Omega_{T})}\leq C_{\text{st}}\epsilon^{-r}$,
$r\in\\{1,2\\}.$ Then, any weakly convergent subsequence of
$\\{y_{u_{\sigma}}\\}_{\sigma}$ in $H^{2,1}(\Omega_{T})$, converges to
$y_{u}$. Note that the compact embeddings $H^{2,1}(\Omega_{T})\subset
L_{2}(0,T;H^{1}_{0}(\Omega))$, $H^{2,1}(\Omega_{T})\subset
L_{p}(0,T;L_{2}(\Omega))$ for $1\leq p<\infty$ and
$H^{2,1}(\Omega_{T})\hookrightarrow L_{2}(\partial\Omega_{T})$ imply that

$\displaystyle\lVert
y_{u}{-}y_{u_{\sigma}}\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}{+}\lVert
y_{u}{-}y_{u_{\sigma}}\rVert_{L_{p}(0,T;L_{2}(\Omega))}{+}\lVert
y_{u}(T){-}y_{u_{\sigma}}(T)\rVert_{L_{2}(\Omega)}\rightarrow 0.$

For every fixed $\epsilon$ as $\sigma\rightarrow 0$, with $\kappa,h$
satisfying the assumptions of Theorem (4.5) we deduce,

$\displaystyle\begin{cases}\lVert
y_{u_{\sigma}}(T){-}y_{\sigma}(u_{\sigma})(T)\rVert_{L_{2}(\Omega)}{+}\lVert
y_{u_{\sigma}}{-}y_{\sigma}(u_{\sigma})\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}\rightarrow
0,\\\ \lVert
y_{u_{\sigma}}{-}y_{\sigma}(u_{\sigma})\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}\rightarrow
0,\end{cases}$

which completes the proof. ∎

The next Theorem studies the differentiability of the relation between control
and discrete state. The proof follows well known techniques and it is omitted
(see for instance [8, Theorem 4.10]).

######  Theorem 4.9.

Let $u,\varv\in L_{2}(0,T;L_{2}(\Omega))$. The mapping
$G_{\sigma}:L_{2}(0,T;L_{2}(\Omega))\rightarrow Y_{\sigma}$, such that
$y_{\sigma}=y_{\sigma}(u)=G_{\sigma}(u)$, is of class $C^{\infty}$. We denote
by $z_{\sigma}(\varv)=G^{\prime}_{\sigma}(u)\varv$ the unique solution to the
problem: For $n=1,\ldots,N$ and for all $w_{h}\in Y_{h}$,

(4.40)
$\displaystyle\Big{(}\frac{z_{n,h}{-}z_{n-1,h}}{k_{n}},w_{h}\Big{)}{+}\left(\nabla
z_{n,h},\nabla
w_{h}\right){+}\epsilon^{-2}\left((3y^{2}_{n,h}{-}1)z_{n,h},w_{h}\right)$
$\displaystyle\
=\frac{1}{k_{n}}\int^{t_{n}}_{t_{n-1}}(\varv(t),w_{h})\mathop{}\\!\mathrm{d}{t},\quad\textit{with}\quad
z_{0,h}=0.$

######  Remark 4.10.

We note that existence and uniqueness of (4.40) can be proved by standard
techniques for any fixed $\epsilon.$ The main difficulty is to prove bounds
with constants that do not depend exponentially upon $1/\epsilon$. Let us
mention the absence of the cubic semilinear term and that the (2.8) can not
being used for $y_{n,h}$ hence the recovery of stability bounds that are
independent of the exponential of $1/\epsilon$ is not straightforward. A key
part of the remaining of our work is to circumvent this difficulty. Our
approach is demonstrated in Section 4.2 for the discrete adjoint-state
equation, but can be applied to (4.40) in a straightforward manner.

###  4.2. Analysis of the discrete adjoint state equation

The diffrentiability properties of
$G_{\sigma}:L_{2}(0,T;L_{2}(\Omega))\rightarrow Y_{\sigma}$ imply that the
reduced cost functional
$J_{\sigma}:L_{2}(0,T;L_{2}(\Omega))\rightarrow\mathbb{R}$ is of class
$C^{\infty}$, as well. Applying the chain rule, we get

(4.41) $\displaystyle J^{\prime}_{\sigma}(u)\varv{=}$
$\displaystyle\int_{0}^{T}{\int_{\Omega}{(y_{\sigma}{-}y_{d})z_{\sigma}}\mathop{}\\!\mathrm{d}{x}}\mathop{}\\!\mathrm{d}{t}{+}\gamma\int_{\Omega}{(y_{\sigma}(T){-}y_{\Omega,h})z_{\sigma}(T)}\mathop{}\\!\mathrm{d}{x}{+}\mu\int_{0}^{T}{\int_{\Omega}{u\varv}\mathop{}\\!\mathrm{d}{x}}\mathop{}\\!\mathrm{d}{t},$

To eliminate $z_{\sigma}$ from (4.41) we consider the corresponding discrete
scheme of the adjoint state equation (3.9), reading: For each $n=N,\ldots,1$
and for all $w_{h}\in Y_{h}$,

(4.42)
$\displaystyle\big{(}\varphi_{n,h}{-}\varphi_{n+1,h},w_{h}\big{)}{+}\int_{t_{n-1}}^{t_{n}}\left(\left(\nabla\varphi_{n,h},\nabla
w_{h}\right){+}\epsilon^{-2}\left((3y^{2}_{n,h}{-}1)\varphi_{n,h},w_{h}\right)\right)\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\
=\int_{t_{n-1}}^{t_{n}}{\left(y_{n,h}{-}y_{d}(t),w_{h}\right)}\mathop{}\\!\mathrm{d}{t},\
\
\quad\text{with}\quad\varphi_{N+1,h}{=}\gamma\left(y_{N,h}{-}y_{\Omega,h}\right).$

The above (backwards in time) fully-discrete equation is understood as
follows: We begin by computing $\varphi_{N,h}$ using $\varphi_{N+1,h}$ and
then we descent from $n{=}N$ until $n{=}1$. Note that we set
$\varphi_{n,h}{=}\varphi_{\sigma}(t_{n-1})$, $1{\leq}n{\leq}N$. Following
identically [8, Section 4.2], the expression (4.41) can be written as

(4.43)
$J^{\prime}_{\sigma}(u)\varv{=}\int_{0}^{T}{\int_{\Omega}{\left(\varphi_{\sigma}{+}\mu
u\right)\varv}\mathop{}\\!\mathrm{d}{x}}\mathop{}\\!\mathrm{d}{t},\quad\forall\varv\in
L_{2}(0,T;L_{2}(\Omega)).$

The following projection operator $R_{\sigma}$ is the analogue of projection
operator $P_{\sigma}$ suitably modified to handle the backwards in time
problem. Let $R_{\sigma}:C(0,T;L_{2}(\Omega))\rightarrow Y_{\sigma}$ defined
through
$\left(R_{\sigma}w\right)_{n,h}{=}\left(R_{\sigma}w\right)(t_{n-1}){=}P_{h}w(t_{n-1})$,
$1\leq n\leq N$. If $w\in H^{2,1}(\Omega_{T})\cap C(0,T;H^{1}_{0}(\Omega))$
there exists $C>0$ such that:

(4.44) $\displaystyle\lVert w{-}R_{\sigma}w\rVert_{L_{2}(0,T;L_{2}(\Omega))}$
$\displaystyle{\leq}C\left(k\lVert
w_{t}\rVert_{L_{2}(0,T;L_{2}(\Omega))}{+}h^{2}\lVert
w\rVert_{L_{2}(0,T;H^{2}(\Omega))}\right),$ (4.45) $\displaystyle\lVert
w{-}R_{\sigma}w\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}$ $\displaystyle\leq
C\left(\sqrt{k}\lVert w_{t}\rVert_{L_{2}(0,T;L_{2}(\Omega))}{+}h\lVert
w\rVert_{L_{2}(0,T;H^{2}(\Omega))}\right),$ (4.46) $\displaystyle\lVert
w{-}R_{\sigma}w\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}$ $\displaystyle\leq
C\left(\sqrt{k}\lVert w_{t}\rVert_{L_{2}(0,T;L_{2}(\Omega))}{+}h\lVert
w\rVert_{L_{\infty}(0,T;H^{1}(\Omega))}\right).$

The following Lemma provides the basic stability estimates for (4.42). We note
that our approach avoids assumptions regarding the construction of a discrete
approximation of the spectral estimate.

######  Lemma 4.11.

Let $u\in L_{2}(0,T;L_{2}(\Omega))$, $y_{u}\in H^{2,1}(\Omega_{T})\cap
C(0,T;H^{1}_{0}(\Omega))$ be the solution of (1.2) while
$y_{\sigma}(u){:=}y_{\sigma}\in Y_{\sigma}$ the solution of (4.3)
corresponding to the control $u.$ Suppose that the assumptions of Theorem 3.2
hold. Let $C_{\infty}{\sim}\|y_{u}\|_{L_{\infty}(0,T;L_{\infty}(\Omega))},$
and $\|y_{u}\|_{H^{2,1}(\Omega_{T})}\leq C\epsilon^{-r}$ with $r\in\\{1,2\\}$,
where $C$ denotes a constant that depends only on data and it is independent
of $\epsilon$ and
$C_{\zeta}{:=}\exp\left(\int_{0}^{T}2\lambda(t)(1{-}\epsilon^{2}){+}3\mathop{}\\!\mathrm{d}{t}\right),$
and that (4.33) or (4.34) hold for $d=2$ or $3$, respectively. If in addition,

(4.47) $\displaystyle\sqrt{k}+h$
$\displaystyle\leq\begin{cases}\displaystyle\frac{C\epsilon^{2+r}}{C_{\infty}C_{\zeta}}&\text{for}\
d=2,\\\
\displaystyle\frac{C\epsilon^{3+(4r/3)}}{C_{\infty}C_{\zeta}}&\text{for}\
d=3,\end{cases}$

there exists $D^{\text{dG}}_{\text{st},1}>0$ (independent of $\sigma=(k,h)$
and $\epsilon$) such that:

$\displaystyle\|\nabla\varphi_{\sigma}\|_{L_{2}(0,T;L_{2}(\Omega))}+\frac{1}{\epsilon}\|y_{\sigma}\varphi_{\sigma}\|_{L_{2}(0,T;L_{2}(\Omega))}\leq\frac{D^{\text{dG}}_{\text{st},1}}{\epsilon}.$

Here,
$D^{\text{dG}}_{\text{st},1}:=C\left(\gamma\|y_{N,h}{-}y_{\Omega,h}\|_{L_{2}(\Omega)}+\|y_{\sigma}{-}y_{d}\|_{L_{2}(0,T;L_{2}(\Omega))}\right)$
with $C$ denoting an algebraic constant independent of $\epsilon.$

###### Proof.

Standard arguments imply existence and uniqueness of solution
$\varphi_{\sigma}$ of (4.42). As usual, we need to develop stability bounds
with constants independent of $1/\epsilon$. For this purpose, we employ a
duality argument. Given right-hand side $\varphi_{\sigma}$, we define
$\zeta_{\sigma}\in\mathcal{Y}_{\sigma}$ such that for $n{=}1,\cdots,N$, and
for $w_{h}\in\mathcal{Y}_{\sigma}$,

(4.48)
$\displaystyle\big{(}\zeta_{n,h}{-}\zeta_{n-1,h},w_{h}\big{)}{+}\int_{t_{n-1}}^{t_{n}}\left(\left(\nabla\zeta_{n,h},\nabla
w_{h}\right){+}\epsilon^{-2}\left((3y^{2}{-}1)\zeta_{n,h},w_{h}\right)\right)\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\
{=}\int_{t_{n-1}}^{t_{n}}{\left(\varphi_{n,h},w_{h}\right)}\mathop{}\\!\mathrm{d}{t},\quad\text{with}\quad\zeta_{0,h}{=}0.$

Setting $w_{h}{=}\zeta_{n,h}$ and using (2.8), we easily deduce that

(4.49)
$\displaystyle\|\zeta_{\sigma}\|_{L_{\infty}(0,T;L_{2}(\Omega))}{+}\epsilon\|\nabla\zeta_{\sigma}\|_{L_{2}(0,T;L_{2}(\Omega))}{+}\|y\zeta_{\sigma}\|_{L_{2}(0,T;L_{2}(\Omega))}\leq
C_{\zeta}\|\varphi_{\sigma}\|_{L_{2}(0,T;L_{2}(\Omega))},$

where
$C_{\zeta}{:=}C\exp\left(\int_{0}^{T}2\lambda(t)(1{-}\epsilon^{2})+3\mathop{}\\!\mathrm{d}{t}\right)$
(with $C$ an algebraic constant depending on the domain). Setting now
$w_{h}{=}(\zeta_{n,h}{-}\zeta_{n-1,h})/k_{n}$ into (4.48), and using Hölder
and Young’s inequalities, we obtain:

$\displaystyle\frac{1}{k_{n}}\Big{\|}\zeta_{n,h}{-}\zeta_{n{-}1,h}\Big{\|}^{2}_{L_{2}(\Omega)}{+}\left(\|\nabla
z_{n,h}\|^{2}_{L_{2}(\Omega)}{-}\|\nabla
z_{n{-}1,h}\|^{2}_{L_{2}(\Omega)}{+}\|\nabla(\zeta_{n,h}{-}\zeta_{n{-}1,h})\|^{2}_{L_{2}(\Omega)}\right)$
$\displaystyle\leq\left(\frac{3\lVert
y\rVert^{2}_{L_{\infty}(\Omega)}{+}1}{\epsilon^{2}}\
\|\zeta_{n,h}\|_{L_{2}(\Omega)}{+}\|\varphi_{n,h}\|_{L_{2}(\Omega)}\right)\Big{\|}\zeta_{n,h}{-}\zeta_{n-1,h}\Big{\|}_{L_{2}(\Omega)}$
$\displaystyle\leq\frac{1}{2k_{n}}\Big{\|}\zeta_{n,h}{-}\zeta_{n-1,h}\Big{\|}^{2}_{L_{2}(\Omega)}{+}\frac{(3\lVert
y\rVert^{2}_{L_{\infty}(\Omega)}{+}1)^{2}}{\epsilon^{4}}k_{n}\|\zeta_{n,h}\|^{2}_{L_{2}(\Omega)}{+}k_{n}\|\varphi_{n,h}\|^{2}_{L_{2}(\Omega)}.$

Summing the above inequalities and using (4.49) we derive the estimate,

(4.50) $\|\nabla\zeta_{\sigma}\|_{L_{\infty}(0,T;L_{2}(\Omega))}\leq
C_{\zeta}\Big{(}\frac{C^{2}_{\infty}{+}1}{\epsilon^{2}}{+}1\Big{)}\|\varphi_{\sigma}\|_{L_{2}(0,T;L_{2}(\Omega))},$

upon setting $C^{2}_{\infty}:=3\lVert
y\rVert^{2}_{L_{\infty}(0,T;L_{\infty}(\Omega))}$ for brevity. Now we proceed
with the duality argument. Setting $w_{h}{=}\zeta_{n,h}$ into (4.42) we obtain
that

$\displaystyle\Big{(}\varphi_{n,h}{-}\varphi_{n+1,h},\zeta_{n,h}\Big{)}{+}\int_{t_{n-1}}^{t_{n}}{\left(\left(\nabla\varphi_{n,h},\nabla\zeta_{n,h}\right){+}\epsilon^{-2}\left((3y^{2}_{n,h}{-}1)\varphi_{n,h},\zeta_{n,h}\right)\right)}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\
=\int_{t_{n-1}}^{t_{n}}{\left(y_{n,h}{-}y_{d}(t),\zeta_{n,h}\right)}\mathop{}\\!\mathrm{d}{t}.$

Similarly setting $w_{h}{=}\varphi_{n,h}$ into (4.48) respectively

$\displaystyle\Big{(}\zeta_{n,h}{-}\zeta_{n-1,h},\varphi_{n,h}\Big{)}{+}\int_{t_{n-1}}^{t_{n}}{\left(\left(\nabla\varphi_{n,h},\nabla\zeta_{n,h}\right){+}\epsilon^{-2}\left((3y^{2}{-}1)\zeta_{n,h},\varphi_{n,h}\right)\right)}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\
{=}\int_{t_{n-1}}^{t_{n}}{\|\varphi_{n,h}\|^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}.$

Subtracting the last two equalities and summing them from $1$ up to $N$, we
deduce,

(4.51)
$\displaystyle\|\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}{=}(\zeta_{N,h},\varphi_{N+1,h}){+}\int_{0}^{T}\left(y_{\sigma}{-}y_{d},\zeta_{\sigma}\right)\mathop{}\\!\mathrm{d}{t}{+}\frac{3}{\epsilon^{2}}\int_{0}^{T}\left((y^{2}{-}y^{2}_{\sigma})\varphi_{\sigma},\zeta_{\sigma}\right)\mathop{}\\!\mathrm{d}{t},$

where we have used the definition $\zeta_{0,h}=0$ and the calculation

$\sum_{n=1}^{N}\Big{\\{}\left(\zeta_{n,h}{-}\zeta_{n-1,h},\varphi_{n,h}\right){-}\left(\varphi_{n,h}{-}\varphi_{n+1,h},\zeta_{n,h}\right)\Big{\\}}{=}(\zeta_{N,h},\varphi_{N+1,h}).$

Therefore, we may bound the terms of the right-hand side of (4.51), as
follows: Using the estimates of (4.49), and Young’s inequality,

(4.52)
$\displaystyle\int_{0}^{T}\left(y_{\sigma}{-}y_{d},\zeta_{\sigma}\right)\mathop{}\\!\mathrm{d}{t}{+}(\zeta_{N,h},\varphi_{N+1,h})$
$\displaystyle\
\leq\|y_{\sigma}{-}y_{d}\|_{L_{2}(0,T;L_{2}(\Omega))}\|\zeta_{\sigma}\|_{L_{2}(0,T;L_{2}(\Omega))}{+}\|\zeta_{N,h}\|_{L_{2}(\Omega)}\|\varphi_{N+1,h}\|_{L_{2}(\Omega)}$
$\displaystyle\
\leq\frac{1}{4}\|\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}C^{2}_{\zeta}\left(\|y_{\sigma}{-}y_{d}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}\|\varphi_{N+1,h}\|^{2}_{L_{2}(\Omega)}\right).$

For the third term of (4.51), the identity
$y^{2}{-}y^{2}_{\sigma}=2y(y{-}y_{\sigma}){-}(y{-}y_{\sigma})^{2}$ yields,

(4.53)
$\displaystyle\frac{3}{\epsilon^{2}}\int_{0}^{T}\left((y^{2}{-}y^{2}_{\sigma}))\varphi_{\sigma},\zeta_{\sigma}\right)\mathop{}\\!\mathrm{d}{t}$
$\displaystyle=\frac{6}{\epsilon^{2}}\int_{0}^{T}\left(y(y{-}y_{\sigma})\varphi_{\sigma},\zeta_{\sigma}\right)\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\
{-}\frac{3}{\epsilon^{2}}\int_{0}^{T}\left(-(y{-}y_{\sigma})^{2}y_{\sigma}\varphi_{\sigma},\zeta_{\sigma}\right)\mathop{}\\!\mathrm{d}{t}.$

Once again we need to distinguish the cases $d{=}2$ and $d{=}3$. For $d{=}3$
we work as follows: Using Hölder and Young’s inequalities (with an appropriate
$\delta_{1}>0$ to be chosen later), estimate (4.49) and Theorem 4.5 \- Remark
4.6 we may bound the first term of (4.53),

$\displaystyle\frac{6}{\epsilon^{2}}\int_{0}^{T}\left((y{-}y_{\sigma})y\varphi_{\sigma},\zeta_{\sigma}\right)\mathop{}\\!\mathrm{d}{t}\leq\frac{6}{\epsilon^{2}}\int_{0}^{T}{\lVert
y{-}y_{\sigma}\rVert_{L_{6}(\Omega)}\|y\|_{L_{\infty}(\Omega)}\lVert\varphi_{\sigma}\rVert_{L_{3}(\Omega)}\lVert\zeta_{\sigma}\rVert_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\leq\frac{3\tilde{c}^{2}C^{2}_{\infty}}{2\epsilon^{5}\delta_{1}}\lVert\zeta_{\sigma}\rVert^{2}_{L_{\infty}(0,T;L_{2}(\Omega))}\lVert
y{-}y_{\sigma}\rVert^{2}_{L_{2}(0,T;H^{1}_{0}(\Omega))}{+}\epsilon\delta_{1}\lVert\varphi_{\sigma}\rVert^{2}_{L_{2}(0,T;L_{3}(\Omega))}$
$\displaystyle\leq\frac{3C^{2}_{\infty}C^{2}_{\zeta}}{2\delta_{1}\epsilon^{5}}\frac{(k{+}h^{2})}{\epsilon^{(8r/3){+}1}}\|\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}\frac{1}{8}\lVert\varphi_{\sigma}\rVert^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}2\tilde{c}^{2}\epsilon^{2}\delta^{2}_{1}\|\nabla\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}.$

Choosing $k,h$ such that

(4.54)
$\displaystyle\frac{3C^{2}_{\infty}C^{2}_{\zeta}(k{+}h^{2})}{2\delta_{1}\epsilon^{6+(8r/3)}}\leq\frac{1}{8},$

we obtain,

(4.55)
$\frac{6}{\epsilon^{2}}\int_{0}^{T}\left((y{-}y_{\sigma})y\varphi_{\sigma},\zeta_{\sigma}\right)\mathop{}\\!\mathrm{d}{t}{\leq}\frac{1}{4}\|\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}2\tilde{c}^{2}\epsilon^{2}\delta^{2}_{1}\|\nabla\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}.$

For the second term of (4.53), we proceed as follows: For a suitable chosen
$\delta_{2}>0$ (to be determined later) we use Hölder and Young’s
inequalities, the Gagliardo-Nirenberg inequality (2.3), estimates (4.49),
(4.50) and the estimates of Theorem 4.5 \- Remark 4.6 for $y-y_{\sigma}$ to
get,

$\displaystyle\frac{3}{\epsilon^{2}}\int_{0}^{T}\left((y{-}y_{\sigma})^{2}\varphi_{\sigma},\zeta_{\sigma}\right)\mathop{}\\!\mathrm{d}{t}\leq\frac{3}{\epsilon^{2}}\int_{0}^{T}{\lVert
y{-}y_{\sigma}\rVert^{2}_{L_{4}(\Omega)}\lVert\varphi_{\sigma}\rVert_{L_{6}(\Omega)}\lVert\zeta_{\sigma}\rVert_{L_{3}(\Omega)}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\leq\frac{9\tilde{c}^{2}}{\delta_{2}\epsilon^{6}}\lVert\nabla\zeta_{\sigma}\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}\lVert\zeta_{\sigma}\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}\lVert
y{-}y_{\sigma}\rVert^{4}_{L_{4}(0,T;L_{4}(\Omega))}$
$\displaystyle\quad{+}\delta_{2}\epsilon^{2}\lVert\nabla\varphi_{\sigma}\rVert^{2}_{L_{2}(0,T;L_{2}(\Omega))}$
$\displaystyle\leq\frac{9\tilde{c}^{2}}{\delta_{2}\epsilon^{6}}C^{2}_{\zeta}\Big{(}\frac{C^{2}_{\infty}{+}1}{\epsilon^{2}}{+}1\Big{)}\epsilon^{2}\frac{(k{+}h^{2})}{\epsilon^{(7r{-}1)/3}}\|\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}\delta_{2}\epsilon^{2}\lVert\nabla\varphi_{\sigma}\rVert^{2}_{L_{2}(0,T;L_{2}(\Omega))}.$

Choosing $\kappa$, $h$ such that

(4.56)
$\left(\frac{9}{\delta_{2}}\right)\frac{\tilde{c}^{2}}{\epsilon^{4}}C^{2}_{\zeta}\Big{(}\frac{C^{2}_{\infty}{+}1}{\epsilon^{2}}{+}1\Big{)}\frac{(k+h^{2})}{\epsilon^{(7r-1)/3}}\leq\frac{1}{4},$

there holds that

(4.57)
$\frac{3}{\epsilon^{2}}\int_{0}^{T}|\left((y{-}y_{\sigma})y_{\sigma}\varphi_{\sigma},\zeta_{\sigma}\right)|\mathop{}\\!\mathrm{d}{t}{\leq}\frac{1}{4}\|\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}\delta_{2}\epsilon^{2}\|\nabla\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}.$

Substituting (4.55), (4.57) into (4.53) we get,

(4.58)
$\displaystyle\frac{3}{\epsilon^{2}}\int_{0}^{T}\left((y^{2}{-}y^{2}_{\sigma}))\varphi_{\sigma},\zeta_{\sigma}\right)\mathop{}\\!\mathrm{d}{t}{\leq}\frac{1}{2}\|\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}(2\tilde{c}^{2}\delta^{2}_{1}{+}\delta_{2})\epsilon^{2}\|\nabla\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}.$

Thus, returning back to (4.51), to substitute (4.52) and (4.58), it yields,

(4.59)
$\displaystyle\frac{1}{4}\|\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}\leq(2\tilde{c}^{2}\delta^{2}_{1}+\delta_{2})\epsilon^{2}\|\nabla\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}$
$\displaystyle\
+C^{2}_{\zeta}\left(\|y_{\sigma}-y_{d}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}+\|\varphi_{N+1,h}\|^{2}_{L_{2}(\Omega)}\right).$

Next we employ a boot-strap argument. First, we return to (4.42) and set
$w_{h}=\varphi_{n,h}$ to deduce, after standard algebra,

$\displaystyle(1/2)\|\varphi_{0,h}\|^{2}_{L_{2}(\Omega)}+\|\nabla\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}3\epsilon^{-2}\|y_{\sigma}\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}$
$\displaystyle\leq((1/2){+}\epsilon^{-2})\|\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}(1/2)\|y_{\sigma}{-}y_{d}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}(1/2)\|\varphi_{N+1,h}\|^{2}_{L_{2}(\Omega)}.$

We apply (4.59) to substitute the first term of the right-hand side,

$\displaystyle\|\varphi_{0,h}\|^{2}_{L_{2}(\Omega)}{+}2\|\nabla\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}6\epsilon^{-2}\|y_{\sigma}\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}$
$\displaystyle\leq(1{+}2\epsilon^{-2})4(2\tilde{c}^{2}\delta^{2}_{1}{+}\delta_{2})\epsilon^{2}\|\nabla\varphi_{\sigma}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}$
$\displaystyle\ \
{+}\left(\left(1{+}2\epsilon^{-2}\right)4C^{2}_{\zeta}{+}1\right)\left(\|y_{\sigma}{-}y_{d}\|^{2}_{L_{2}(0,T;L_{2}(\Omega))}{+}\|\varphi_{N+1,h}\|^{2}_{L_{2}(\Omega)}\right),$

Choosing $\delta_{1},\delta_{2}>0$ such that
$\left(1{+}\frac{2}{\epsilon^{2}}\right)4(2\tilde{c}^{2}\delta^{2}_{1}{+}\delta_{2})\epsilon^{2}{\leq}1$,
we finally deduce the desired estimate. For the estimate For $d{=}2$, we work
analogously. ∎

######  Theorem 4.12.

Let $u\in L_{2}(0,T;L_{2}(\Omega))$ and $y,\varphi\in H^{2,1}(\Omega_{T})\cap
C(0,T;H^{1}_{0}(\Omega))$ be the associated state solution of (1.2) and the
adjoint state solution of (3.9), respectively. Let, $y_{\sigma}$ be the
associated discrete state solution of (4.3) while $\varphi_{\sigma}$ the
associated discrete adjoint state solution of (4.42). Then, under the
assumptions of Theorem 4.5 and Lemma 4.11 there exists $\hat{C}{>}0$ such that
for $r\in\\{1,2\\}$ there holds that:

(4.60)
$\displaystyle\lVert\varphi{-}\varphi_{\sigma}\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}{+}\epsilon\lVert\varphi{-}\varphi_{\sigma}\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}\leq\begin{cases}\displaystyle\frac{\hat{C}(\sqrt{k}{+}h)}{\epsilon^{4{+}r}}&\text{
for $d=2,$}\\\
\displaystyle\frac{\hat{C}(\sqrt{k}{+}h)}{\epsilon^{4{+}(7r-1)/6}}&\text{ for
$d=3.$}\end{cases}$

Here, we denote by $E{:=}\exp(T\alpha)$ where
$\alpha{:=}\sup_{t\in[0,T]}(2\lambda(t)(1{-}\epsilon^{2}){+}5)$ and

$\hat{C}_{\text{}}{:=}C(E^{1/2}\left(C^{dG}_{st,2}{+}C_{st,2})D^{\text{dG}}_{\text{st},1}\right).$

###### Proof.

We split the total error as follows:

(4.61)
$e{:=}\varphi{-}\varphi_{\sigma}{=}(\varphi{-}R_{\sigma}\varphi){+}(R_{\sigma}\varphi{-}\varphi_{\sigma}){=}\eta{+}\
\xi_{\sigma}.$

For each $n{=}0,\ldots,N{-}1$ we have,

$\displaystyle\eta(t_{n}){=}\varphi(t_{n}){-}(R_{\sigma}\varphi)(t_{n}){=}\varphi(t_{n}){-}\left(R_{\sigma}\varphi\right)_{n+1,h}{=}\varphi(t_{n}){-}P_{h}\varphi(t_{n})$
$\displaystyle\xi_{\sigma}(t_{n}){=}(R_{\sigma}\varphi)(t_{n}){-}\varphi_{\sigma}(t_{n}){=}\left(R_{\sigma}\varphi\right)_{n+1,h}{-}\varphi_{n{+}1,h}\equiv\xi_{n{+}1,h}.$

For $n=N$, note that $(R_{\sigma}\varphi)_{N+1,h}{=}P_{h}\varphi(T)$ and
$\varphi_{N+1,h}{=}\gamma(y_{N,h}{-}y_{\Omega,h})$. From (3.9) and (4.42), we
deduce for each $n=N,\ldots,1$ and for all $w_{h}\in Y_{h}$ that

(4.62)
$\displaystyle\left(e(t_{n{-}1}){-}e(t_{n}),w_{h}\right){+}\int_{t_{n-1}}^{t_{n}}{\left(\nabla
e(t),\nabla
w_{h}\right)}\mathop{}\\!\mathrm{d}{t}{+}\epsilon^{-2}\int_{t_{n-1}}^{t_{n}}{\left((3y^{2}{-}1)\varphi,w_{h}\right)}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle{-}\epsilon^{-2}\int_{t_{n-1}}^{t_{n}}{\left((3y^{2}_{n,h}{-}1)\varphi_{n,h},w_{h}\right)}\mathop{}\\!\mathrm{d}{t}{=}\int_{t_{n-1}}^{t_{n}}{\left(y(t){-}y_{n,h},w_{h}\right)}\mathop{}\\!\mathrm{d}{t}.$

Setting $w_{h}=\xi_{n,h}$, using (4.61) and $(\eta(t_{n}),\xi_{n,h})=0$, and
adding and subtracting appropriate terms, (4.62) yields,

$\displaystyle\left(\xi_{n,h}{-}\xi_{n+1,h},\xi_{n,h}\right){+}\int_{t_{n-1}}^{t_{n}}{\lVert\nabla\xi_{n,h}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\epsilon^{-2}\int_{t_{n-1}}^{t_{n}}{\left((3y^{2}{-}1)(\varphi{-}\varphi_{n,h}),\xi_{n,h}\right)}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle{+}3\epsilon^{-2}\int_{t_{n-1}}^{t_{n}}{\left((y^{2}{-}y^{2}_{\sigma})\varphi_{n,h},\xi_{n,h}\right)}\mathop{}\\!\mathrm{d}{t}{=}\int_{t_{n-1}}^{t_{n}}{\left(y{-}y_{\sigma},\xi_{n,h}\right)}\mathop{}\\!\mathrm{d}{t}{-}\int_{t_{n-1}}^{t_{n}}{\left(\nabla\eta,\nabla\xi_{n,h}\right)}\mathop{}\\!\mathrm{d}{t}.$

Relation (4.61), and standard algebraic manipulations imply that

$\displaystyle\quad\frac{1}{2}\lVert\xi_{n,h}\rVert^{2}_{L_{2}(\Omega)}{-}\frac{1}{2}\lVert\xi_{n+1,h}\rVert^{2}_{L_{2}(\Omega)}{+}\frac{1}{2}\lVert\xi_{n,h}{-}\xi_{n+1,h}\rVert^{2}_{L_{2}(\Omega)}$
$\displaystyle\quad{+}\int_{t_{n-1}}^{t_{n}}{\left(\lVert\nabla\xi_{n,h}\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{-2}\left((3y^{2}{-}1)\xi_{n,h},\xi_{n,h}\right)\right)}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle{=}\int_{t_{n-1}}^{t_{n}}{\left(y-y_{\sigma},\xi_{n,h}\right)}\mathop{}\\!\mathrm{d}{t}{-}\int_{t_{n-1}}^{t_{n}}{\left(\nabla\eta,\nabla\xi_{n,h}\right)}\mathop{}\\!\mathrm{d}{t}{-}\epsilon^{-2}\int_{t_{n-1}}^{t_{n}}{\left((3y^{2}{-}1)\eta,\xi_{n,h}\right)}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\quad{-}3\epsilon^{-2}\int_{t_{n-1}}^{t_{n}}{\left((y{-}y_{\sigma})(y_{\sigma}{+}y)\varphi_{n,h},\xi_{n,h}\right)}\mathop{}\\!\mathrm{d}{t}{:=}\sum^{4}_{j=1}\mathcal{T}_{j}.$

Applying Hölder and Young’s inequalities, we easily get

$\displaystyle\mathcal{T}_{1}$
$\displaystyle{\leq}\frac{1}{2}\int_{t_{n-1}}^{t_{n}}{\lVert
y-y_{\sigma}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\frac{1}{2}\int_{t_{n-1}}^{t_{n}}{\lVert\xi_{n,h}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t},$
$\displaystyle\mathcal{T}_{2}$
$\displaystyle{\leq}\frac{1}{\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert\nabla\eta\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\frac{\epsilon^{2}}{4}\int_{t_{n-1}}^{t_{n}}{\lVert\nabla\xi_{n,h}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t},$
$\displaystyle\mathcal{T}_{3}$
$\displaystyle{\leq}\int_{t_{n-1}}^{t_{n}}{\|y\xi_{n,h}\|^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\frac{1}{4\epsilon^{4}}\int_{t_{n-1}}^{t_{n}}{\left(9\lVert
y\rVert^{2}_{L_{\infty}(\Omega)}{+}1\right)\lVert\eta\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\int_{t_{n-1}}^{t_{n}}{\lVert\xi_{n,h}\rVert^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t}.$

For the term $\mathcal{T}_{4}$, we distinguish two cases. For $d=3$, we note
that using Hölder and Young’s inequalities, and the stability estimates of
$y,y_{\sigma}$ and $\varphi_{\sigma}$, we deduce,

$\displaystyle\mathcal{T}_{4}\leq\frac{3}{\epsilon^{2}}\int_{t_{n-1}}^{t_{n}}{\lVert
y{-}y_{\sigma}\rVert_{L_{2}(\Omega)}\|\varphi_{n,h}\|_{L_{6}(\Omega)}\left(\lVert
y_{\sigma}\rVert_{L_{6}(\Omega)}{+}\|y\|_{L_{6}(\Omega)}\right)\|\xi_{n,h}\|_{L_{6}(\Omega)}}\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\leq\frac{9C(C_{st,2}{+}C^{dG}_{st,2})^{2}}{\epsilon^{6}}\int_{t_{n-1}}^{t_{n}}{\|y{-}y_{\sigma}\|^{2}_{L_{2}(\Omega)}\|\varphi_{n,h}\|^{2}_{L_{6}(\Omega)}}\mathop{}\\!\mathrm{d}{t}+\frac{\epsilon^{2}}{4}\int_{t_{n-1}}^{t_{n}}{\|\nabla\xi_{n,h}\|^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t},$
$\displaystyle\leq\frac{9C(C_{st,2}{+}C^{dG}_{st,2})^{2}}{\epsilon^{6}}\|y{-}y_{\sigma}\|^{2}_{L_{\infty}(0,T;L_{2}(\Omega))}\int_{t_{n-1}}^{t_{n}}{\|\varphi_{n,h}\|^{2}_{L_{6}(\Omega)}}\mathop{}\\!\mathrm{d}{t}{+}\frac{\epsilon^{2}}{4}\int_{t_{n-1}}^{t_{n}}{\|\nabla\xi_{n,h}\|^{2}_{L_{2}(\Omega)}}\mathop{}\\!\mathrm{d}{t},$

where $C$ is a constant depending only on the domain. Using the spectral
estimate (2.8) for $\varv=\xi_{n,h}$, collecting the above bounds, summing
from $n=m$ up to $N$ where $1\leq m\leq N$, using a standard (linear) Gronwall
Lemma, for $\sup_{n=1,\ldots,N}\alpha k_{n}<1$, where
$\alpha:=\sup_{t\in[0,T]}(2\lambda(t)(1-\epsilon^{2})+5)$, we deduce that

$\displaystyle\lVert\xi_{m,h}\rVert^{2}_{L_{2}(\Omega)}{+}\sum_{n=m}^{N}\lVert\xi_{n,h}{-}\xi_{n+1,h}\rVert^{2}_{L_{2}(\Omega)}{+}\epsilon^{2}\sum_{n=m}^{N}k_{n}\lVert\nabla\xi_{n,h}\rVert^{2}_{L_{2}(\Omega)}$
$\displaystyle\leq
E\Bigg{\\{}\lVert\xi_{N+1,h}\rVert^{2}_{L_{2}(\Omega)}{+}\int_{t_{m}}^{T}\left(\frac{9\lVert
y\rVert^{2}_{L_{\infty}(\Omega)}{+}1}{2\epsilon^{4}}\lVert\eta\rVert^{2}_{L_{2}(\Omega)}{+}\frac{2}{\epsilon^{2}}\lVert\nabla\eta\rVert^{2}_{L_{2}(\Omega)}\right)\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\ {+}\int_{t_{m}}^{T}\lVert
y{-}y_{\sigma}\rVert^{2}_{L_{2}(\Omega)}\mathop{}\\!\mathrm{d}{t}{+}\frac{C(C_{st,2}{+}C^{dG}_{st,2})^{2}}{\epsilon^{6}}\|y{-}y_{\sigma}\|^{2}_{L_{\infty}(0,T;L_{2}(\Omega))}\int_{t_{m}}^{T}\|\varphi_{\sigma}\|^{2}_{L_{6}(\Omega)}\mathop{}\\!\mathrm{d}{t}\Bigg{\\}},$

where $E{:=}\exp(T\alpha).$ The estimate follows using the embedding
$H^{1}(\Omega)\subset L_{6}(\Omega)$ and Lemma 4.11 to bound
$\int_{t_{m}}^{T}\|\varphi_{\sigma}\|^{2}_{L_{6}(\Omega)}\mathop{}\\!\mathrm{d}{t}\leq(D^{dG}_{st,1}/\epsilon)^{2}$
and Remark 4.6 to bound
$\|y-y_{\sigma}\|^{2}_{L_{\infty}(0,T;L_{2}(\Omega))}\leq\frac{C}{\epsilon^{2r}}(\kappa+h^{2})$
for $d=2$ and
$\|y-y_{\sigma}\|^{2}_{L_{\infty}(0,T;L_{2}(\Omega))}\leq\frac{C}{\epsilon^{(7r-1)/3}}(\kappa+h^{2})$
for $d=3.$ Note that the term
$\frac{2}{\epsilon^{2}}\int_{t_{m}}^{T}\|\nabla\eta\|^{2}_{L_{2}(\Omega)}\leq\frac{C}{\epsilon^{2}}(k+h^{2})\|\phi\|^{2}_{H^{2,1}(\Omega_{T})}$
dominates all $\eta$ terms. Although, the last term of the right-hand side
dominates all terms. ∎

The following result is analogous to Corollary 4.8 and an immediate
consequence of the Theorem 4.12, Lemma 3.7.

######  Corollary 4.13.

Let $u,\varv\in L_{2}(0,T;L_{2}(\Omega))$ and $\varphi_{u}\in
H^{2,1}(\Omega_{T})\cap C(0,T;H^{1}_{0}(\Omega))$ be the solution of (3.9)
while $\varphi_{\sigma}(\varv)\in Y_{\sigma}$ the solution of (4.42)
corresponding to the control $\varv.$ Assume that Lemmas 3.7 and 4.11 and
Theorem 4.12 hold and let $r\in\\{1,2\\}$. Then, for $d=2$ there holds

(4.63)
$\displaystyle\lVert\varphi_{u}{-}\varphi_{\sigma}(\varv)\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}+\epsilon\lVert\varphi_{u}{-}\varphi_{\sigma}(\varv)\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}$
$\displaystyle\leq\frac{L_{2}}{\epsilon^{7/2}}\lVert
u{-}\varv\rVert_{L_{2}(0,T;L_{2}(\Omega))}+\frac{\hat{C}(\sqrt{k}{+}h)}{\epsilon^{4{+}r}},$

while for $d=3$ it follows that

(4.64)
$\displaystyle\lVert\varphi_{u}{-}\varphi_{\sigma}(\varv)\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}+\epsilon\lVert\varphi_{u}{-}\varphi_{\sigma}(\varv)\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}$
$\displaystyle\leq\frac{L_{3}}{\epsilon^{15/4}}\lVert
u{-}\varv\rVert_{L_{2}(0,T;L_{2}(\Omega))}+\frac{\hat{C}(\sqrt{k}{+}h)}{\epsilon^{4{+}(7r-1)/6}}.$

Here, according to the notation of Lemma 3.7 we denote by

(4.65) $\displaystyle
L_{2}:=L_{1}\left(\epsilon^{7/2}C_{T}E^{1/2}_{\varphi}{+}\tilde{c}C^{1/2}_{\infty}D_{\text{st},1}\right),\
L_{3}:=L_{1}\left(\epsilon^{15/4}C_{T}E^{1/2}_{\varphi}{+}\tilde{c}C^{1/2}_{\infty}D_{\text{st},1}\right).$

###### Proof.

The estimate follows by triangle inequality. ∎

###  4.3. Convergence of the discrete control problem

In this section we study the convergence of the solutions of the discrete
control problem (4.5) towards the solutions of the continuous problem (1.3).
Every discrete problem (4.5) has at least one solution because the
minimisation function is continuous and coercive on a nonempty closed subset
of a finite dimensional space.

######  Theorem 4.14.

For every $\sigma=(k,h)$, let $\bar{u}_{\sigma}$ be a global solution of
problem (4.5). Then, the sequence $\\{\bar{u}_{\sigma}\\}_{\sigma}$ is bounded
in $L_{2}(0,T;L_{2}(\Omega))$ and there exist subsequences denoted in the same
way, converging to a point $\bar{u}$ weakly in $L_{2}(0,T;L_{2}(\Omega))$. Any
of these limit points is a solution of problem (4.5). Moreover, we have

(4.66) $\lim_{\sigma\rightarrow
0}\lVert\bar{u}-\bar{u}_{\sigma}\rVert_{L_{2}(0,T;L_{2}(\Omega))}=0\quad\textit{and}\quad\lim_{\sigma\rightarrow
0}J_{\sigma}(\bar{u}_{\sigma})=J(\bar{u}).$

######  Theorem 4.15.

Let $\bar{u}$ be a strict local minimum of (1.3). Then, there exists a
sequence of local minima of problems (4.5) such that (4.66) holds.

The proofs of the Theorems are presented in [8, Theorems 4.15 and 4.17].

######  Remark 4.16.

It is important to mention that in our case, the above convergence results are
proved for each fixed interface length, $\epsilon$.

###  4.4. Error estimates

In this section we denote by $\bar{u}$ a local solution of (1.3) and by
$\bar{u}_{\sigma}$ a local solution of (4.5) $\forall\sigma.$ From Theorems
4.14 and 4.15, we deduce that
$\lVert\bar{u}{-}\bar{u}_{\sigma}\rVert_{L_{2}(0,T;L_{2}(\Omega))}\rightarrow
0.$ Moreover, let $\bar{y}$ and $\bar{\varphi}$ be the state and adjoint state
associated to $\bar{u}$ while $\bar{y}_{\sigma}$ and $\bar{\varphi}_{\sigma}$
the discrete state and adjoint state associated to $\bar{u}_{\sigma}$.

######  Definition 4.17.

We denote by $u_{\sigma}$ the projection of $\bar{u}$ into $U_{\sigma}$ define
through,

(4.67) $\displaystyle
u_{\sigma}=\sum^{N}_{n=1}\sum_{\tau\in\mathcal{T}_{h}}u_{n,\tau}\mathcal{\chi}_{n}\mathcal{\chi}_{\tau}\quad\textit{where}\quad
u_{n,\tau}=\frac{1}{k_{n}\lvert\tau\rvert}\int_{t_{n-1}}^{t_{n}}{\int_{\tau}\lvert\bar{u}(t,x)\rvert\mathop{}\\!\mathrm{d}{x}}\mathop{}\\!\mathrm{d}{t}.$

######  Lemma 4.18.

Let $\bar{u}\in H^{1}(\Omega_{T}).$ Then, there exists a constant $C>0$
independent of $\sigma=(k,h)$ and $\epsilon$ such that,

(4.68)
$\displaystyle\lVert\bar{u}{-}u_{\sigma}\rVert_{L_{2}(0,T;L_{2}(\Omega))}$
$\displaystyle\leq C(\sqrt{k}{+}h)\lVert\bar{u}\rVert_{H^{1}(\Omega_{T})},$
(4.69) $\displaystyle\lVert\bar{u}{-}u_{\sigma}\rVert_{H^{-1}(\Omega_{T})}$
$\displaystyle\leq
C\left(k{+}h^{2}\right)\lVert\bar{u}\rVert_{H^{1}(\Omega_{T})}.$

###### Proof.

We refer the reader to [8, Lemma 4.17]. ∎

######  Theorem 4.19.

Suppose that the assumptions of Theorems 3.8 and 3.9 and Corollaries 4.8 and
4.13 hold. Then, there exist $\mathcal{C}_{1}:=\sqrt{T}L_{2}$,
$\widetilde{\mathcal{C}}_{1}:=L_{1}\mathcal{C}_{1}$ such that for $d=2$ and
$r=1$

(4.70)
$\displaystyle\lVert\bar{u}{-}\bar{u}_{\sigma}\rVert_{L_{2}(0,T;L_{2}(\Omega))}\leq\frac{\mathcal{C}_{1}}{\epsilon^{7/2}}(\sqrt{k}{+}h)\lVert\bar{u}\rVert_{H^{1}(\Omega_{T})},$
(4.71)
$\displaystyle\lVert\bar{y}{-}\bar{y}_{\sigma}\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}+\epsilon\lVert\bar{y}{-}\bar{y}_{\sigma}\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}\leq\frac{\widetilde{\mathcal{C}}_{1}}{\epsilon^{7/2}}(\sqrt{k}{+}h)\lVert\bar{u}\rVert_{H^{1}(\Omega_{T})},$

while for $r=2$ there exist $\mathcal{C}_{2}:=\sqrt{T}\hat{C}$,
$\widetilde{\mathcal{C}}_{2}:=L_{1}\mathcal{C}_{2}$ such that

(4.72)
$\displaystyle\lVert\bar{u}{-}\bar{u}_{\sigma}\rVert_{L_{2}(0,T;L_{2}(\Omega))}\leq\frac{\mathcal{C}_{2}}{\epsilon^{6}}(\sqrt{k}{+}h),$
(4.73)
$\displaystyle\lVert\bar{y}{-}\bar{y}_{\sigma}\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}+\epsilon\lVert\bar{y}{-}\bar{y}_{\sigma}\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}\leq\frac{\widetilde{\mathcal{C}}_{2}}{\epsilon^{6}}(\sqrt{k}{+}h).$

In addition, for $d=3$ and $r=1$ there exist $\mathcal{C}_{3}:=\sqrt{T}L_{3},$
$\widetilde{\mathcal{C}}_{3}:=L_{1}\mathcal{C}_{3}$ such that

(4.74)
$\displaystyle\lVert\bar{u}{-}\bar{u}_{\sigma}\rVert_{L_{2}(0,T;L_{2}(\Omega))}\leq\frac{\mathcal{C}_{3}}{\epsilon^{15/4}}(\sqrt{k}{+}h)\lVert\bar{u}\rVert_{H^{1}(\Omega_{T})},$
(4.75)
$\displaystyle\lVert\bar{y}{-}\bar{y}_{\sigma}\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}+\epsilon\lVert\bar{y}{-}\bar{y}_{\sigma}\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}\leq\frac{\widetilde{\mathcal{C}}_{3}}{\epsilon^{15/4}}(\sqrt{k}{+}h)\lVert\bar{u}\rVert_{H^{1}(\Omega_{T})}.$

Finally, for $d=3$ and $r=2$ there exist $\mathcal{C}_{4}=\sqrt{T}\hat{C}$,
$\widetilde{\mathcal{C}}_{4}:=L_{1}\mathcal{C}_{4}$ such that

(4.76)
$\displaystyle\lVert\bar{u}{-}\bar{u}_{\sigma}\rVert_{L_{2}(0,T;L_{2}(\Omega))}\leq\frac{\mathcal{C}_{4}}{\epsilon^{4+13/6}}(\sqrt{k}{+}h),$
(4.77)
$\displaystyle\lVert\bar{y}{-}\bar{y}_{\sigma}\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}+\epsilon\lVert\bar{y}{-}\bar{y}_{\sigma}\rVert_{L_{2}(0,T;H^{1}_{0}(\Omega))}\leq\frac{\widetilde{\mathcal{C}}_{4}}{\epsilon^{{4+13/6}}}(\sqrt{k}{+}h).$

###### Proof.

(Sketch:) The proof follows [8, Section 4]. We begin with $d=3$ and $r=1$.
Note that (4.75) is a consequence of (4.74) combined with (4.35) and (4.38).
Thus, the main subject is the proof of (4.74). Suppose that (4.74) is false.
More specifically, this assumption implies that

$\displaystyle\lim_{\sigma\rightarrow
0}\sup\frac{\epsilon^{15/4}\lVert\bar{u}\rVert^{-1}_{H^{1}(\Omega_{T})}}{\mathcal{C}_{3}(\sqrt{k}+h)}\lVert\bar{u}-\bar{u}_{\sigma}\rVert_{L_{2}(0,T;L_{2}(\Omega))}=+\infty.$

This means that there exists a sequence of $\sigma$ such that

(4.78) $\displaystyle\lim_{\sigma\rightarrow
0}\frac{\epsilon^{15/4}\lVert\bar{u}\rVert^{-1}_{H^{1}(\Omega_{T})}}{\mathcal{C}_{3}(\sqrt{k}+h)}\lVert\bar{u}-\bar{u}_{\sigma}\rVert_{L_{2}(0,T;L_{2}(\Omega))}=+\infty.$

We aim to conclude to a contradiction. First of all, the fact that
$\bar{u}_{\sigma}$ is a local minimum of (4.5) combined with the property that
$J_{\sigma}$ is of class $C^{\infty}$ and $u_{\sigma}\in U_{\sigma,ad}$ imply
that $J^{\prime}_{\sigma}(\bar{u}_{\sigma})(u_{\sigma}-\bar{u}_{\sigma})\geq
0$. After basic manipulations, we have that

$\displaystyle
J^{\prime}(\bar{u}_{\sigma})(\bar{u}-\bar{u}_{\sigma})+[J^{\prime}_{\sigma}(\bar{u}_{\sigma})-J^{\prime}(\bar{u}_{\sigma})](\bar{u}-\bar{u}_{\sigma})$
$\displaystyle+[J^{\prime}_{\sigma}(\bar{u}_{\sigma})-J^{\prime}(\bar{u})](u_{\sigma}-\bar{u})+J^{\prime}(\bar{u})(u_{\sigma}-\bar{u})\geq
0.$

Since $\bar{u}$ is an local minimum of (1.3) and $\bar{u}_{\sigma}\in U_{ad}$,
then $J^{\prime}(\bar{u})(\bar{u}_{\sigma}-\bar{u})\geq 0$. Adding this
nonnegative term to the above inequality, it yields that

(4.79)
$\displaystyle[J^{\prime}(\bar{u}_{\sigma})-J^{\prime}(\bar{u})](\bar{u}_{\sigma}-\bar{u})\leq[J^{\prime}_{\sigma}(\bar{u}_{\sigma})-J^{\prime}(\bar{u}_{\sigma})](\bar{u}-\bar{u}_{\sigma})$
$\displaystyle+[J^{\prime}_{\sigma}(\bar{u}_{\sigma})-J^{\prime}(\bar{u})](u_{\sigma}-\bar{u})+J^{\prime}(\bar{u})(u_{\sigma}-\bar{u}).$

Using the second order necessary and sufficient condition of Theorem 3.9 we
derive an estimate from below for the left-hand side and then an upper bound
for the three terms on the right-hand side through the estimates of Sections
3.1, 3.2, 4.1 and 4.2. Indeed, working identically to [8, Lemma 4.18] we
obtain, for a suitable $\sigma_{0}$ independent of $\epsilon$,

$\frac{1}{2}\min\\{\delta,\mu\\}\lVert\bar{u}_{\sigma}-\bar{u}\rVert^{2}_{L_{2}(0,T;L_{2}(\Omega))}\leq[J^{\prime}(\bar{u}_{\sigma})-J^{\prime}(\bar{u})](\bar{u}_{\sigma}-\bar{u})\quad\text{for}\quad\lvert\sigma\rvert\leq\lvert\sigma_{0}\rvert.$

Inserting the above lower bound in (4.79) we obtain that,

(4.80)
$\displaystyle\frac{1}{2}\min\\{\delta,\lambda\\}\lVert\bar{u}_{\sigma}-\bar{u}\rVert^{2}_{L_{2}(0,T;L_{2}(\Omega))}\leq[J^{\prime}_{\sigma}(\bar{u}_{\sigma})-J^{\prime}(\bar{u}_{\sigma})](\bar{u}-\bar{u}_{\sigma})$
$\displaystyle+[J^{\prime}_{\sigma}(\bar{u}_{\sigma})-J^{\prime}(\bar{u})](u_{\sigma}-\bar{u})+J^{\prime}(\bar{u})(u_{\sigma}-\bar{u}).$

We shall estimate from above the three terms on the right-hand side. More
specifically, from (3.7) and (4.43)

(4.81)
$\displaystyle[J^{\prime}_{\sigma}(\bar{u}_{\sigma})-J^{\prime}(\bar{u}_{\sigma})](\bar{u}-\bar{u}_{\sigma})\leq\lVert\bar{\varphi}_{\sigma}-\varphi_{\bar{u}_{\sigma}}\rVert_{L_{2}(0,T;L_{2}(\Omega))}\lVert\bar{u}-\bar{u}_{\sigma}\rVert_{L_{2}(0,T;L_{2}(\Omega))}$
$\displaystyle\
\leq\sqrt{T}\lVert\bar{\varphi}_{\sigma}-\varphi_{\bar{u}_{\sigma}}\rVert_{L_{\infty}(0,T;L_{2}(\Omega))}\lVert\bar{u}-\bar{u}_{\sigma}\rVert_{L_{2}(0,T;L_{2}(\Omega))}$
$\displaystyle\
\leq\frac{\sqrt{T}\hat{C}}{\epsilon^{4+(7r-1)/6}}(\sqrt{k}+h)\lVert\bar{u}-\bar{u}_{\sigma}\rVert_{L_{2}(0,T;L_{2}(\Omega))},$

we used the estimate (4.64) for $u=\varv=\bar{u}_{\sigma}.$ For the second
term on the right-hand side of (4.80), we recall again (4.64) for $u=\bar{u}$
and $\varv=\bar{u}_{\sigma}$ and (4.68)

(4.82)