# Stability analysis of the Navier–Stokes velocity tracking problem with bang-
bang controls††thanks: A.D.C. is supported by the Alexander von Humboldt
Foundation with an Alexander von Humboldt research fellowship and the grant
FWF I4571-N. N.J was supported by the FWF grants P-31400-N32 and I4571-N. Š.
N. and J.S.H.S. have been supported by Praemium Academiæ of Š. Nečasová. Š. N.
was also supported by the Czech Science Foundation (GAČR) through projects
GC22-08633J. The Institute of Mathematics, CAS is supported by RVO:67985840.

Alberto Domínguez Corella  Nicolai Jork  Šarká Nečasová  John Sebastian H.
Simon

###### Abstract

This paper focuses on the stability of solutions for a velocity-tracking
problem associated with the two-dimensional Navier–Stokes equations. The
considered optimal control problem does not possess any regularizer in the
cost, and hence bang-bang solutions can be expected. We investigate
perturbations that account for uncertainty in the tracking data and the
initial condition of the state, and analyze the convergence rate of solutions
when the original problem is regularized by the Tikhonov term. The stability
analysis relies on the Hölder subregularity of the optimality mapping, which
stems from the necessary conditions of the problem.

##  1 Introduction

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded domain and $T>0$, and let
$u_{a},u_{b}:\Omega\times(0,T)\to\mathbb{R}^{2}$ be bounded functions. We
consider the control set

$\displaystyle\mathcal{U}:=\left\\{u\in L^{\infty}(Q)^{2}:\,\,u_{a}(x,t)\leq
u(x,t)\leq u_{b}(x,t)\,\,\,\,\text{for a.e. }(x,t)\in
Q:=\Omega\times(0,T)\right\\}.$ (1)

For each control $u\in\mathcal{U}$, interpreted as a force per unit mass
acting on the fluid, there is an associated state
${y_{u}}:Q\to\mathbb{R}^{2}$, which is the fluid velocity, satisfying the
Navier–Stokes equation

$\displaystyle\left\\{\begin{aligned}
\partial_{t}{y_{u}}-\nu\Delta{y_{u}}+({y_{u}}\cdot\nabla){y_{u}}+\nabla{p_{u}}=u\text{
in }Q,\\\ \operatorname{div}{y_{u}}=0\,\,\,\,\text{in
Q},\,\,{y_{u}}=0\,\,\,\,\text{on
$\Sigma$},\,\,\,{y_{u}}(\cdot,0)=y_{0}\,\,\,\text{in
$\Omega$}.\end{aligned}\right.$ (2)

Here, $y_{0}$ denotes the initial velocity field, $\nu>0$ is the so-called
kinematic viscosity parameter, and $\Sigma:=\partial\Omega\times(0,T)$ the
lateral boundary of the open cylinder $Q$. We consider the classical velocity
tracking problem

$\displaystyle\min_{u\in\mathcal{U}}\frac{1}{2}\int_{0}^{T}\int_{\Omega}|{y_{u}}(x,t)-y_{d}(x,t)|^{2}\,dx\,dt.$
(3)

Optimal control problems often involve a regularizing term that aids in the
formulation and analysis of the control strategies and the numerical
implementation. However, there exist certain scenarios where this regularizing
term is absent, such is the case for affine problems, which leads to a
different class of solutions that pose additional challenges. In particular,
with the absence of the classical regularizing term in problem (1)-(3) the
optimal control may generically exhibit a bang-bang behavior. The absence of
regularizing terms has attracted attention due to its implications for the
solution of the underlying dynamical system (2). While a regularization on the
objective functional could offer certain mathematical advantages, the problem
at hand steers the state solution towards the desired velocity better as
highlighted previously in [23] and [9]. Nevertheless, the analysis without
regularization terms requires careful consideration of bang-bang controls.

This paper focuses on the stability of solutions for problem (1)-(3). In order
to highlight our contributions, we recall previous literature related to the
problem. We begin with some previous works on optimal control of the
Navier–Stokes equations where the regularization term appears. As second-order
conditions are intimately related to stability, we mention the work [32] on
sufficient conditions. In regard to stability, we mention the works [7, 8],
where error estimates for a Galerkin time-stepping scheme were established.

In [23], the authors presented a systematic approach to the mathematical
analysis and numerical approximation of tracking the velocity for
Navier–Stokes flows with bounded distributed controls without the classic
Tikhonov regularization term. This analysis included first-order necessary
conditions as well as discretization schemes. Regarding the second-order
sufficient conditions and stability of this type of problem, we mention the
work [9], where a fully-discrete scheme based on discontinuous (in time)
Galerkin approach combined with conforming finite element subspaces in space,
is proposed and analyzed. Besides this work, to the best of our knowledge,
there are not any other works in stability concerned with bang-bang controls
of the velocity tracking problem (1)-(3).

The study of bang-bang controls for problems constrained by partial
differential equations has not been thoroughly explored. It was only in 2012
that a second-order analysis for a semilinear elliptic equation was presented
[6]. The investigation into the stability analysis of bang-bang minimizers for
problems constrained by partial differential equations originated with [15],
focusing on the error estimates for discretizing the optimization problem.
Since then, several papers have addressed other types of problems involving
bang-bang minimizers. For parabolic problems, the work of N. von Daniels and
M. Hinze [33] explores the accuracy of a variational discretization, while E.
Casas and F. Tröltzsch [12] examined stability with respect to initial data.
In the context of optimal control problems governed by ordinary differential
equations, the stability analysis of bang-bang minimizers was explored in
[28]. This study investigated the stability of the first-order necessary
conditions using the metric regularity property, considering assumptions that
involved $L^{1}$-growths. These assumptions are similar to the classic
coercivity condition but with certain modifications.

This article aims to investigate the effect of perturbations on problem
(1)-(3). Specifically, we focus on perturbations that arise from uncertainties
in tracking data and initial state conditions. Additionally, we consider the
classic Tikhonov regularization term as a perturbation of the original
problem. We provide Hölder estimates for the rates of convergence. To
elaborate on our main result, suppose that we don’t know the exact initial
condition nor the datum we are tracking, however, we have an approximation of
these data, i.e., $w_{0}$ and $w_{d}$. We can then solve the
$\varepsilon$-regularized problem

$\displaystyle\min_{u\in\mathcal{U}}\frac{1}{2}\int_{0}^{T}\int_{\Omega}|{y_{u}}(x,t)-w_{d}(x,t)|^{2}\,dx\,dt+\frac{\varepsilon}{2}\int_{0}^{T}\int_{\Omega}|u(x,t)|^{2}\,dx\,dt$
(4)

subject to (2) with $w_{0}$ as initial datum and obtain a solution
$\widehat{u}$. Our main result (Theorem 5.2) gives the estimate

$\displaystyle\|\widehat{u}-\bar{u}\|_{L^{1}(Q)^{2}}\leq\kappa\Big{(}\|y_{0}-w_{0}\|_{W^{2-\frac{2}{\bar{s}},s}_{0.\sigma}(\Omega)^{2}}+\|y_{d}-w_{d}\|_{L^{2}(Q)}+\varepsilon\Big{)}^{\frac{1}{\mu}},$

for a reference solution $\bar{u}$ of problem (1)-(3), under a growth
assumption, in particular, Assumption 5.1 and the growth assumption in Theorem
4.12. We refer the reader to Theorem 5.2 for more precise details, and the
technicalities surrounding the statement.

In our analysis, we employ the concept of strong Hölder subregularity of a
set-valued mapping associated with the optimality of the optimal control
problem [21, Section 3I]. The stability of the first-order necessary
conditions is investigated as a property of a set-valued mapping that
encapsulates the generalized equation satisfied by local minimizers. This
property is also referred to as strong (metric) $\theta$-subregularity in the
literature, see [14, Section 4]. Previous works have utilized this property to
establish stability results for optimal control problems, particularly those
constrained by partial differential equations. Subregularity results for
semilinear problems can be found in [16, 17]. However, to the best of our
knowledge, such results have not been explored for systems involving the
Navier–Stokes equations, which constitutes one of the key contributions of
this paper. It is important to note that the low regularity solutions of the
Navier–Stokes equations present additional challenges in the analysis, which
demand the development of distinct estimates compared to papers such as [16,
17, 28].

In addition, we dedicate a self-contained appendix to abstract results in
metric subregularity; we formalize the tools used to establish stability in
the presence of perturbations. Importantly, these tools are designed to be
applicable to a wide range of optimal control problems, extending beyond those
constrained by the Navier–Stokes equations. Previous papers that have
addressed the subregularity of the optimality mapping have mainly focused on
providing sufficient conditions. However, in this paper, we present an
abstract result for necessary conditions, see Theorem A.8. This is a novel
contribution and is significant because it was previously unknown whether
subregularity could imply a growth condition. Furthermore, our result clearly
establishes that this type of stability implies the bang-bang nature of
optimal controls. Due to the general nature of this result, it can be applied
to several previously studied optimal control problems, including those
examined in [16, 17, 28].

This article is structured as follows: Section 2 discusses the functional
theoretic tools that are essential for the analysis of the state equations.
Section 3 focuses on the analysis of the Navier–Stokes equations, encompassing
the existence and regularity of solutions, as well as the examination of the
linear system known as the Oseen equations and its dual system. In Section 4,
we delve into the analysis of the optimal control problem, providing first-
order necessary conditions and second-order sufficient conditions. Finally, in
Section 5, we establish the stability of optimal controls in the presence of
perturbations, which include variations in the desired velocity, the initial
state of the governing equations, and a Tikhonov perturbation in the objective
functional. The abstract results on subregularity are presented in the
Appendix.

##  2 Preliminaries

For a given real normed space $X$, its continuous dual is denoted by $X^{*}$
and $\langle\cdot,\cdot\rangle_{X}:X^{*}\times X\to\mathbb{R}$ denotes their
duality pairing. The domain $\Omega$ under consideration is a connected
bounded subset of $\mathbb{R}^{2}$ with boundary $\partial\Omega$ of class
$C^{3}$. The unit normal vector field is denoted by
$n:\partial\Omega\to\mathbb{R}^{2}$.

Sobolev spaces. For a measurable set $E$, we consider the usual Lebesgue
spaces $L^{s}(E)$ of $s$-integrable functions for $s\in[1,\infty)$, and
$L^{\infty}(E)$ the space of essentially bounded functions. These spaces are
endowed with their standard norms, denoted by $\|\cdot\|_{L^{s}(E)^{d}}$ for
$s\in[1,+\infty]$ and $d=1,2,2\times 2$. We shall use the notation
$(\cdot,\cdot)_{E}$ for the $L^{2}(E)^{d}$ inner product, i.e.,
$(\cdot,\cdot)_{E}=\langle\cdot,\cdot\rangle_{L^{2}(E)}$.

For $m\in\mathbb{N}$ and $1\leq s\leq\infty$, $W^{m,s}(\Omega)$ denotes the
space of all functions in $L^{p}(\Omega)$ with all of its weak derivatives of
order $m$ belonging $L^{p}(\Omega)$. We write $W^{0,p}(\Omega):=L^{p}(\Omega)$
and $H^{m}(\Omega):=W^{m,2}(\Omega)$. The norms in these spaces will be
denoted as $\|\cdot\|_{W^{m,s}(\Omega)^{d}}$.

The zero trace Sobolev spaces are denoted by $W_{0}^{m,s}(\Omega)$. Poincaré
inequality implies that the seminorm in $W^{1,s}(\Omega)$, which is defined as
the sum of the $L^{s}$ norms of the first order distributional derivatives of
the elements, is a norm in $W_{0}^{1,s}(\Omega)$ and is equivalent to the
usual norm. Hence, from hereon, when we refer to the norm in
$W_{0}^{1,s}(\Omega)$ we mean it to be the seminorm. As usual, for the
Hilbertian case, we write $H^{1}_{0}(\Omega):=W_{0}^{1,2}(\Omega)$ and
$H^{-1}(\Omega):=H_{0}^{1}(\Omega)^{*}$.

Let us recall some embeddings that are vital for the upcoming analyses.
Rellich–Kondrachov embedding theorem gives us the compact embedding
$W^{m,s_{1}}(\Omega)\hookrightarrow L^{s_{2}}(\Omega)$ if either of the two
cases hold: i. $m>0$ and $1\leq s_{2}<2s_{1}/(2-ms_{2})$; and ii. $2=ms_{1}$
and $1\geq s_{2}<+\infty$. Similarly, we get the continuous embedding
$W^{j+m,s}(\Omega)\hookrightarrow C^{j}(\overline{\Omega})$ if either $ms>2$
or $m=2$ and $s=1$, it becomes compact when $ms>2$ or $ms>m$. For references,
see [1, Theorem 6.3, Part I] for the former, and [1, Theorem 6.3, Part III]
for the latter.

Solenoidal spaces. To take into account the incompressibility condition, we
consider the following solenoidal spaces

$\displaystyle W^{m,s}_{0,\sigma}:=\left\\{\psi\in
W^{m,s}_{0}(\Omega)^{2}:\,\operatorname{div}\psi=0\text{ in }\Omega\right\\},$
$\displaystyle H_{s}:=\left\\{\psi\in
L^{s}(\Omega)^{2}:\operatorname{div}\psi=0\text{ in }L^{s}(\Omega),\,\psi\cdot
n=0\text{ on }\partial\Omega\right\\}.$

We write $V_{s}:=W^{1,s}_{0,\sigma}$ and in the Hilbertian case, we use the
notations $V:=V_{2}$ and $H:=H_{2}$.

The spaces $V$ and $H$ form a Gelfand triple $(V,H,V^{*})$, i.e., the
embeddings

$\displaystyle{V\hookrightarrow H\cong H^{*}\hookrightarrow V^{*}}$ (5)

are dense and continuous. Moreover, the first embedding is compact due to
Rellich–Kondrachov Theorem; and by Schauder’s Theorem, the second one is also
compact.

Auxiliary spatial operators. The orthogonal complement of $H$ can be
characterized as $H^{\bot}=\\{\psi\in L^{2}(\Omega)^{2}:\,\psi=\nabla
q\,\,\,\,\text{for some $q\in H^{1}(\Omega)^{2}$}\\}.$ The representation
$L^{2}(\Omega)^{2}=H\oplus H^{\bot}$ is called the Helmholtz–Leray
decomposition. We define the Leray projection operator $P:L^{2}(\Omega)^{2}\to
H$ by $P\psi=\psi_{1}$, where $\psi_{1}$ is the unique element of $H$ such
that $\psi-\psi_{1}$ belongs to $H^{\bot}$.

The Stokes operator $A:V\cap H^{2}(\Omega)^{2}\subset H\to H$ is thus defined
as $A\psi=-P\Delta\psi$. We can also look at the Stokes operator as linear
operator from $V\cap H^{2}(\Omega)^{2}$ to $V^{*}$, i.e., $\langle
A\psi_{1},\psi_{2}\rangle_{V}=(\nabla\psi_{1},\nabla\psi_{2})_{\Omega}$ for
any $\psi_{1}\in V\cap H^{2}(\Omega)^{2}$ and $\psi_{2}\in V$.

To deal with the nonlinearity caused by the convective term, we introduce the
trilinear form $b:H^{1}(\Omega)^{2}\times H^{1}(\Omega)^{2}\times
H^{1}(\Omega)^{2}\to\mathbb{R}$ defined as
$b(\psi_{1},\psi_{2},\psi_{3})=((\psi_{1}\cdot\nabla)\psi_{2},\psi_{3})_{\Omega}$.
Using Hölder inequality and Rellich–Kondrachov embedding one can see that $b$
is continuous. Furthermore, for any $\psi,\psi_{1},\psi_{2}\in V$ we have
$b(\psi,\psi_{1},\psi_{2})=-b(\psi,\psi_{2},\psi_{1})$. As a consequence of
Hölder and Gagliardo–Nirenberg inequalities we also get

$\displaystyle b(\psi_{1},\psi_{2},\psi_{3})\leq
c\|\psi_{1}\|_{H}^{1/2}\|\psi_{1}\|_{V}^{1/2}\|\psi_{2}\|_{V}\|\psi_{3}\|_{H}^{1/2}\|\psi_{3}\|_{V}^{1/2}{\quad\forall\psi_{1},\psi_{2},\psi_{3}\in
V}.$ (6)

Bochner spaces. Let $T>0$ and $X$ be a Banach space. We use the notation
$C(\overline{I};X)$ for functions from $I$ to $X$ that can be continuously
extended to $[0,T]$. For $s\in[1,\infty]$, we denote by $L^{s}(I;X)$ the usual
space of functions $\psi:I\to X$ such that $t\to\|\psi(t)\|_{X}$ belongs to
$L^{s}(0,T)$. The norm in $L^{s}(I;X)$ will be denoted as
$\|\cdot\|_{L^{s}(X)}$, and the duality pairing of $L^{s}(I;X)$ and its dual
as $\langle\cdot,\cdot\rangle_{L^{s}(X)}$. For $s\in[1,\infty]$ and
$m\in\mathbb{N}$, the space $W^{m,s}(I;X)$ consists of functions $\psi\in
L^{s}(I;X)$ whose distributional time derivative $\partial_{t}^{m}\psi$
belongs to $L^{s}(I;X)$. For $m\in\mathbb{N}$, we write
$H^{m}(I;X):=W^{m,2}(I;X)$. Note that the spaces $L^{s}(I;L^{s}(\Omega)^{2})$
for $1\leq s<\infty$ can be identified as the space $L^{s}(Q)^{2}$.

We also introduce the spaces that take into account the distributional time
derivatives, i.e., we have the space $W^{\alpha}(I):=L^{2}(I;V)\cap
W^{1,\alpha}(I;V^{*})$ with the norm
$\|\cdot\|_{W^{\alpha}(I)}:=\|\cdot\|_{L^{2}(I;V)}+\|\cdot\|_{W^{1,\alpha}(I;V^{*})}$.
For simplicity, we use $W(I)$ when $\alpha=2$.

To aid us in our analyses, we recall some embedding theorems from [2, Theorem
3]. We start with two real Banach spaces $X_{0}$ and $X_{1}$ such that
$X_{1}\hookrightarrow X_{0}$ is dense. We denote by
$X_{\theta,s}:=(X_{0},X_{1})_{\theta,q}$ their real interpolation with
exponential functor $0<\theta<1$, where $1\leq q\leq+\infty$. Amman’s theorem
gives us an embedding for the space
$W^{1}_{s}(I,(X_{0},X_{1})):=L^{s}(I;X_{1})\cap W^{1,s}(I;X_{0})$. In fact, if
$r\in\mathbb{R}$ with $1/s<r<1$ and $0\leq\theta<1-r$ then
$W^{1}_{s}(I,(X_{0},X_{1}))\hookrightarrow C^{0,r-1/s}(I;X_{\theta,1})$.
Furthermore, if $X_{1}\hookrightarrow X_{0}$ is compact then
$W^{1}_{s}(I,(X_{0},X_{1}))\hookrightarrow
C^{0,r-1/s}(\overline{I};X_{\theta,1})$ is also compact. By virtue of trace
theorem one also gets $W^{1}_{s}(I,(X_{0},X_{1}))\hookrightarrow
C(\overline{I};X_{1-1/s,1})$.

Some of the direct consequences of the embedding above are the following:

  * •

when $X_{0}=V^{*}$ and $X_{1}=V$ so that taking $s=2$ on the second embedding
above we have $X_{1/2,1}\hookrightarrow X_{1/2,2}=(V^{*},V)_{1/2,2}=H$ which
implies that the embedding $W(I)=W^{1}_{2}(I,(V^{*},V))\hookrightarrow
C(\overline{I};H)$ is compact;

  * •

when $X_{0}=L^{s}(\Omega)^{2}$ and $X_{1}=W^{2,s}(\Omega)^{2}$, we have, from
[1, Theorem 7.31], that $X_{1-1/s,1}\hookrightarrow W^{2-2/s,s}(\Omega)^{2}$
and hence the continuous embeddings

$\displaystyle W^{2,1}_{s}:=L^{s}(I;W^{2,s}(\Omega)\cap V_{s})\cap
W^{1,s}(I;L^{s}(\Omega)^{2})\hookrightarrow
C(\overline{I};W^{2-2/s,s}_{0,\sigma}(\Omega)^{2}).$ (7)

Because of the divergence-free and zero trace assumptions on the elements of
$W^{2,1}_{s}$, we used $W^{2-2/s,s}_{0,\sigma}(\Omega)^{2}$ instead of
$W^{2-2/s,s}(\Omega)^{2}$. Furthermore, by virtue of Rellich–Kondrachov
embedding theorem, we see that the embedding
$W^{2-2/s,s}(\Omega)^{2}\hookrightarrow C(\overline{\Omega})^{2}$ is compact,
whenever $(2-2/s)s>2$. Hence, the embedding $W^{2,1}_{s}\hookrightarrow
C(\overline{Q})^{2}$ is also compact, whenever $s>2$. Similarly,
$W^{2-2/s,s}(\Omega)^{2}\hookrightarrow C^{1}(\overline{\Omega})^{2}$ is
compact if $(1-2/s)s>2$. Therefore, we have the compact embedding
$W^{2,1}_{s}\hookrightarrow C(\overline{I};C^{1}(\overline{\Omega})^{2})$
whenever $s>4$.

We end this section by introducing some operators which help in the
forthcoming analyses. We begin with the time-dependent extension of the Stokes
operator, i.e., $A:L^{2}(I;V)\to L^{2}(I,V^{*})$ defined as

$\displaystyle\langle Au,v\rangle_{L^{2}(V)}$
$\displaystyle=\int_{0}^{T}\langle(Au)(t),v(t)\rangle_{V}\operatorname*{d\\!}t=\int_{0}^{T}(\nabla
u(t),\nabla v(t))_{\Omega}\operatorname*{d\\!}t.$

In connection with the trilinear form we mentioned previously, we introduce
the bilinear operator $B:\widetilde{W}\times\widetilde{W}\to L^{2}(I;V^{*})$
defined as

$\displaystyle\langle
B(u,v),w\rangle_{L^{2}(V)}=\int_{0}^{T}b(u(t),v(t),w(t))\operatorname*{d\\!}t,$

where $\widetilde{W}:=W^{2}(I)\cap L^{\infty}(I;H)$ endowed with the norm
$\|\cdot\|_{\widetilde{W}}:=\max\\{\|\cdot\|_{W^{2}(I)},\|\cdot\|_{L^{\infty}(H)}\\}$.
Indeed, the membership of $B(u,v)$ to $L^{2}(I;V^{*})$ for
$u,v\in\widetilde{W}$ follows from Hölder’s inequality (6)

$\displaystyle|\langle B(u,v),w\rangle_{L^{2}(V)}|$ $\displaystyle\leq
c\|u\|_{\widetilde{W}}\|v\|_{\widetilde{W}}\|w\|_{L^{2}(V)}.$

We use the notation $B(u)=B(u,u)$, for simplicity. Motivated by the
linearization of the Navier–Stokes equations we introduce the operator
$\widetilde{B}:\widetilde{W}\times\widetilde{W}\to\mathcal{L}(\widetilde{W},L^{2}(I;V^{*}))$
defined as

$\displaystyle\widetilde{B}(y_{1},y_{2})z=B(y_{1},z)+B(z,y_{2})\quad\forall
y_{1},y_{2},z\in\widetilde{W}.$ (8)

Given $\overline{y}\in\widetilde{W}$, we also see that the Fréchet derivative
$B^{\prime}(\overline{y})$ of the operator $B$ at $\overline{y}$ is defined as

$\displaystyle\langle
B^{\prime}(\overline{y})z,v\rangle_{L^{2}(V)}=\langle\widetilde{B}(\overline{y},\overline{y})z,v\rangle_{L^{2}(V)}\quad\forall
z\in\widetilde{W},\,v\in L^{2}(I;V^{*}).$ (9)

The adjoint of the linear operator
$\widetilde{B}(\overline{y}_{1},\overline{y}_{2})$, for
$\overline{y}_{1},\overline{y}_{2}\in\widetilde{W}$, is denoted as
$\widetilde{B}(\overline{y}_{1},\overline{y}_{2})^{*}$ and is defined by

$\langle\widetilde{B}(\overline{y}_{1},\overline{y}_{2})^{*}w,v\rangle_{L^{2}(V)}=\langle
B(\overline{y}_{1},v),w\rangle_{L^{2}(V)}+\langle
B(v,\overline{y}_{2}),w\rangle_{L^{2}(V)}.$

We also see that the adjoint of the linear operator $B^{\prime}(\overline{y})$
given $\overline{y}$ can be written as

$\displaystyle
B^{\prime}(\overline{y})^{*}w=\widetilde{B}(\overline{y},\overline{y})^{*}w\quad\forall
w\in\widetilde{W}.$

##  3 Analysis of the governing equations

In this section, we shall discuss some known well-posedness results for the
Navier–Stokes equations and an Oseen equation (linearization of the
Navier–Stokes equations), as well as its corresponding adjoint system.

Navier–Stokes equations. Given $y_{0}\in H$ and $u\in L^{2}(I;V^{*})$, we say
that $y\in W(I)$ is a weak solution to the Navier–Stokes equations whenever it
satisfies

$\displaystyle\langle\partial_{t}y(t),v\rangle_{V}+\nu(\nabla y(t),\nabla
v)_{\Omega}+b(u(t),u(t),v)=\langle u(t),v\rangle_{V}\quad\forall v\in V,$ (10)

for a.e. $t\in I$, and $y(0)=y_{0}$ in $H$. The evaluation $y(0)$ is well-
defined due to the embedding $W(I)\hookrightarrow C(\overline{I};H)$.

The existence and regularity of solutions has been well studied, for such
results we refer to the book [31].

######  Theorem 3.1.

Suppose that $u\in L^{2}(I;V^{*})$ and $y_{0}\in H$, then there exists a
unique $y\in\widetilde{W}$ such that (10) holds. Furthermore, the energy
estimate

$\displaystyle\|y\|_{\widetilde{W}}\leq
c(\|u\|_{L^{2}(I;V^{*})}+\|y_{0}\|_{H})$ (11)

holds for some constant $c>0$ independent of $u$ and $y_{0}$.

The norm $\|\cdot\|_{\widetilde{W}}$ majorizes the norms
$\|\cdot\|_{L^{\infty}(I;H)}$ and $\|\cdot\|_{L^{2}(I;V)}$. The proof of
Theorem 3.1 can be carried out using the usual Galerkin methods, it is even
possible to prove that the mapping $[(u,y_{0})\mapsto y]:L^{2}(I;V^{*})\times
H\to\widetilde{W}$ is locally Lipschitz continuous.

In the analysis of the optimal control problem, we will employ $L^{s}$-strong
solutions of the Navier–Stokes equations, $s\in[2,\infty)$. We say that $y\in
L^{s}(I;V_{s})$ with $\partial_{t}y\in L^{s}(Q)^{2}$ is an $L^{s}$-strong
solution of the Navier–Stokes equations (2) if $y\in
L^{s}(I;W^{2,s}(\Omega)^{2})$ and

$\displaystyle\int_{0}^{T}(\partial_{t}y(t),v(t))_{\Omega}+\nu(\nabla
y(t),\nabla
v)_{\Omega}+b(y(t),y(t),v(t))\operatorname*{d\\!}t=\int_{0}^{T}(u(t),v(t))\operatorname*{d\\!}t,$
(12)

for all $v\in L^{s^{\prime}}(I;V_{s^{\prime}})$, where $s^{\prime}$ is the
Hölder conjugate of $s$. Due to the embedding $W^{2,1}_{s}\hookrightarrow
C(\overline{I};W^{2-2/s,s}_{0,\sigma}(\Omega)^{2})$, the appropriate space for
the initial data is $W^{2-2/s,s}_{0,\sigma}(\Omega)^{2}$. The following
theorem summarizes the existence of strong solutions for (2) as well as the
recovery of the pressure term.

######  Theorem 3.2.

Let $s\geq 2$, the assumptions $u\in L^{s}(Q)^{2}$ and $y_{0}\in
W_{0,\sigma}^{2-2/s,s}(\Omega)^{2}$ imply the unique existence of the strong
solution $y\in W^{2,1}_{s}$ of (12). Furthermore, there exists $p\in
L^{s}(I;W^{1,s}(\Omega)\cap L^{s}(\Omega)/\mathbb{R})$ such that

$\displaystyle\left\\{\begin{aligned} \partial_{t}y+\nu Ay+By+\nabla
p&=u&&\text{in }L^{s}(I;H_{s}),\\\ y(0)&=y_{0}&&\text{in
}W_{0}^{2-2/s,s}(\Omega)^{2},\end{aligned}\right.$ (13)

and the pair $(y,p)\in W^{2,1}_{s}\times L^{s}(I;W^{1,s}(\Omega)\cap
L^{s}(\Omega)/\mathbb{R})$ satisfies the estimate

$\displaystyle\|y\|_{W^{2,1}_{s}}+\|\nabla p\|_{L^{s}({Q})^{2}}\leq
c(\|y_{0}\|_{L^{s}(\Omega)^{2}}+\|u\|_{L^{s}(Q)^{2}}).$ (14)

We also have a weak-strong convergence for the force-to-velocity operator.

######  Theorem 3.3 ([10, Corollary 2.4]).

Let $\\{u_{k}\\}\subset L^{s}(Q)^{2}$ be a sequence converging weakly to $u\in
L^{s}(Q)^{2}$, where $s>2$. Then $y_{k}\to y$ in $C(\overline{Q})^{2}$, where
$y_{k}$ and $y$ solve (13) with $u_{k}$ and $u$ as the external forces,
respectively.

Oseen equations. As usual, in optimal control problems, one needs to consider
the linearization of the systems governing the states. For this reason, for
known elements $\overline{y}_{1},\overline{y}_{2}$ and external force $v$, we
shall consider the Oseen equations

$\displaystyle\left\\{\begin{aligned} \partial_{t}z-\nu\Delta
z+(\overline{y}_{1}\cdot\nabla)z+(z\cdot\nabla)\overline{y}_{2}+\nabla
q=v\,\,\,\,\text{in }Q,\\\ \operatorname{div}{z}=0\,\,\,\,\text{in
}Q,\,\,{z}=0\,\,\,\,\text{on }\Sigma,\,\,\,{z}(\cdot,0)=z_{0}\,\,\,\text{in
}\Omega.\end{aligned}\right.$ (15)

We can write the weak formulation of (15) as

$\displaystyle\left\\{\begin{aligned} \partial_{t}z+\nu
Az+\widetilde{B}(\overline{y}_{1},\overline{y}_{2})z&=v&&\text{in
}L^{2}(I;V^{*}),\\\ z(0)&=z_{0}&&\text{in }H.\end{aligned}\right.$ (16)

The existence of weak solutions of the linearized system can be proven using
Faedo-Galerkin method just as in the nonlinear system.

######  Theorem 3.4.

Suppose that $v\in L^{2}(I;V^{*})$, $z_{0}\in H$ and
$\overline{y}_{1},\overline{y}_{2}\in L^{2}(I;V)\cap L^{\infty}(I;H)$, then
there exists a unique $z\in\widetilde{W}$ such that (16) holds. Furthermore,
the energy estimate

$\displaystyle\|z\|_{\widetilde{W}}\leq
c\left(\|z_{0}\|_{H}+\|v\|_{L^{2}(I;V^{*})}\right)$ (17)

holds for some constant $c>0$ independent of $z_{0}$ and $v$.

Quite similarly as in the nonlinear case, improved regularity of the external
force and initial datum leads to a more regular solution.

######  Theorem 3.5.

Let $s\geq 2$. The assumptions $v\in L^{s}(Q)^{2}$, $z_{0}\in
W_{0,\sigma}^{2-2/s,s}(\Omega)^{2}$ and $\overline{y}_{1},\overline{y}_{2}\in
W^{2,1}_{s}$ imply the unique existence of the strong solution $z\in
W^{2,1}_{s}$ of (15). The existence of a pressure term $q\in
L^{s}(I;W^{1,s}(\Omega)\cap L^{s}(\Omega)/\mathbb{R})$ is also guaranteed and
is known to satisfy

$\displaystyle\left\\{\begin{aligned} \partial_{t}z+\nu
Az+\widetilde{B}(\overline{y}_{1},\overline{y}_{2})z+\nabla q&=v&&\text{in
}L^{s}(I;H_{s}),\\\ z(0)&=z_{0}&&\text{in
}W_{0,\sigma}^{2-2/s,s}(\Omega)^{2}.\end{aligned}\right.$ (18)

Furthermore, the pair $(z,q)\in W^{2,1}_{s}\times L^{s}(I;W^{1,s}(\Omega)\cap
L^{s}(\Omega)/\mathbb{R})$ satisfies

$\displaystyle\|z\|_{W^{2,1}_{s}}+\|\nabla q\|_{L^{s}(Q)^{2}}\leq
c(\|z_{0}\|_{W_{0,\sigma}^{2-2/s,s}(\Omega)^{2}}+\|v\|_{L^{s}(Q)^{2}}).$ (19)

Dual Oseen equations. Given elements
$\overline{y}_{1},\overline{y}_{2}\in\widetilde{W}$ and $v\in
L^{\alpha}(I;V^{*})$ we are also interested in looking at the following
system, which is also known as the adjoint to the Oseen equations introduced
above

$\displaystyle\left\\{\begin{aligned} -\partial_{t}w-\nu\Delta
w-(\overline{y}_{2}\cdot\nabla)w+(\nabla\overline{y}_{2})^{\top}w+\nabla
r=v\,\,\,\,\text{in Q},\\\ \operatorname{div}{w}=0\,\,\,\,\text{in
Q},\,\,{w}=0\,\,\,\,\text{on $\Sigma$},\,\,\,{w}(\cdot,T)=0\,\,\,\text{in
$\Omega$}.\end{aligned}\right.$ (20)

The variational form of the system above is

$\displaystyle\begin{aligned} -\partial_{t}w+\nu
Aw+\widetilde{B}(\overline{y}_{1},\overline{y}_{2})^{*}w&=v&&\text{in
}L^{\alpha}(I;V^{*})\\\ w(T)&=0&&\text{in }H.\end{aligned}$ (21)

The regularity of weak solutions of the adjoint equations are not the same as
that of (10) and (16). In fact, we get $W^{4/3}(I)$ instead of having the
solutions belong to $W(I)$. We have such a result summarized in the following
theorem.

######  Theorem 3.6.

Let $v\in L^{4/3}(I;V^{*})$, and
$\overline{y}_{1},\overline{y}_{2}\in\widetilde{W}$. Then the unique solution
$w\in W^{4/3}(I)$ of (21) exists and satisfies

$\displaystyle\|w\|_{W^{4/3}(I)}\leq c\|v\|_{L^{4/3}(I;V^{*})}.$ (22)

for some constant $c>0$ independent of $v$.

Despite the lack of time regularity of the weak solution, we could eventually
recover $L^{s}$-regularity of the adjoint solution given additional regularity
of the data.

######  Theorem 3.7.

Suppose that $v\in L^{s}(Q)^{2}$ with $s\geq 2$, and
$\overline{y}_{1},\overline{y}_{2}\in{W}^{2,1}_{s}$. The weak solution of (21)
is a $L^{s}$-strong solution and belongs to $W^{2,1}_{s}$. Furthermore, there
exists $r\in L^{s}(I;W^{1,s}(\Omega)\cap L^{s}(\Omega)/\mathbb{R})$ such that

$\displaystyle\begin{aligned} -\partial_{t}w+\nu
Aw+\widetilde{B}(\overline{y}_{1},\overline{y}_{2})^{*}w+\nabla r&=v&&\text{in
}L^{s}(I;H_{s})\\\ w(T)&=0&&\text{in
}W^{2-2/s,s}_{0}(\Omega)^{2}.\end{aligned}$ (23)

The pair $(w,r)\in{W}^{2,1}_{s}\times L^{s}(I;W^{1,s}(\Omega)\cap
L^{s}(\Omega)/\mathbb{R})$, furthermore, satisfies the energy inequality

$\displaystyle\|w\|_{W^{2,1}_{s}}+\|\nabla r\|_{L^{s}(Q)^{2}}\leq
c\|v\|_{L^{s}(Q)^{2}},$ (24)

from some constant $c>0$ independent of $v$.

We end this section with the following lemma which was inspired by the
$L^{s}$-$L^{1}$ stability of solutions proven in [11, Lemma 2.3] and [17,
Lemma 2].

######  Lemma 3.8.

Let $v\in L^{s}(Q)^{2}$ for $s>2$, and $\overline{y}\in W^{2,1}_{s}$. If
$z_{v}\in W^{2,1}_{s}$ is the strong solution of (18) and $w_{v}\in
W^{2,1}_{s}$ is the solution of (23) both with
$\overline{y}_{1}=\overline{y}_{2}=\overline{y}$. Then for any
$\tilde{s}\in[1,2)$, there exists $c>0$ independent on $v$ such that

$\displaystyle\max\\{\|z_{u}\|_{L^{\tilde{s}}(Q)^{2}},\|w_{u}\|_{L^{\tilde{s}}(Q)^{2}}\\}\leq
c\|v\|_{L^{1}(Q)^{2}}.$ (25)

###### Proof.

Since $v\in L^{s}(Q)^{2}$ for $s>2$, $z_{u}\in C(\overline{\Omega})^{2}$. This
implies that $|z_{v}|^{\tilde{s}-2}z_{v}\in L^{\tilde{s}^{\prime}}(Q)^{2},$
where $\tilde{s}^{\prime}=\tilde{s}/(\tilde{s}-1)>2$. The solution
$\mathfrak{w}$ satisfying

$\displaystyle\begin{aligned} -\partial_{t}\mathfrak{w}+\nu
A\mathfrak{w}+B^{\prime}(\overline{y})^{*}\mathfrak{w}+\nabla\mathfrak{r}&=|z_{v}|^{\tilde{s}-2}z_{v}&&\text{in
}L^{\tilde{s}^{\prime}}(Q)^{2}\\\ \mathfrak{w}(T)&=0&&\text{in
}W_{0}^{2-2/{\tilde{s}^{\prime}},\tilde{s}^{\prime}}(\Omega)^{2},\end{aligned}$
(26)

belongs to $C(\overline{Q})^{2}$, and satisfies — according to Theorem 3.7 —

$\displaystyle\begin{aligned} \|\mathfrak{w}\|_{C(\overline{Q})^{2}}&\leq
c\||z_{v}|^{\tilde{s}-2}z_{v}\|_{L^{\tilde{s}^{\prime}}(Q)^{2}}\leq
c\left(\int_{0}^{T}\int_{\Omega}\left||z_{v}|^{\tilde{s}-2}z_{v}\right|^{\tilde{s}^{\prime}}\operatorname*{d\\!}x\operatorname*{d\\!}t\right)^{1/\tilde{s}^{\prime}}\\\
&\leq
c\left(\int_{0}^{T}\int_{\Omega}|z_{v}|^{\tilde{s}}\operatorname*{d\\!}x\operatorname*{d\\!}t\right)^{(\tilde{s}-1)/\tilde{s}}\leq
c\|z_{v}\|^{\tilde{s}-1}_{L^{\tilde{s}}(Q)^{2}}.\end{aligned}$ (27)

Using integration by parts, and the definition of the adjoint of the
operators, we get

$\displaystyle\begin{aligned}
\|z_{v}\|^{\tilde{s}}_{L^{\tilde{s}}(Q)^{2}}&=\int_{\Omega}|z_{v}|^{\tilde{s}}\operatorname*{d\\!}x\operatorname*{d\\!}t=\big{\langle}|z_{v}|^{\tilde{s}-2}z_{v},z_{v}\big{\rangle}_{L^{2}(Q)^{2}}\\\
&=\big{\langle}-\partial_{t}\mathfrak{w}+\nu
A\mathfrak{w}+B^{\prime}(\overline{y})^{*}\mathfrak{w}+\nabla\mathfrak{r},z_{v}\big{\rangle}_{L^{2}(Q)^{2}}\\\
&=\big{\langle}\mathfrak{w},\partial_{t}z_{v}+\nu
Az_{v}+B^{\prime}(\overline{y})z_{v}+\nabla{q}\big{\rangle}_{L^{2}(Q)^{2}}\\\
&=\big{\langle}\mathfrak{w},v\big{\rangle}_{L^{2}(Q)^{2}}\leq\|v\|_{L^{1}(Q)^{2}}\|\mathfrak{w}\|_{C(\overline{Q})^{2}}\\\
&\leq
c\|v\|_{L^{1}(Q)^{2}}\|z_{u}\|^{\tilde{s}-1}_{L^{\tilde{s}}(Q)^{2}}.\end{aligned}$
(28)

The transition from the second line to the third line in the computation above
used the fact that $\nabla\cdot z_{v}=\nabla\cdot\mathfrak{w}=0$. This cancels
out the term $\mathfrak{r}\nabla\cdot z_{v}$ and allows the addition of
$q\nabla\cdot\mathfrak{w}$. We thus have the estimate for $z_{v}$ in (25).

For the solution $w_{v}\in W^{2,1}_{s}$, we also get $w_{v}\in
C(\overline{Q})^{2}$ from which we get $|w_{v}|^{\tilde{s}-2}w_{v}\in
L^{\tilde{s}^{\prime}}(\Omega)^{2}$ whence the solution $\mathfrak{Z}\in
C(\overline{Q})^{2}$ to the equations

$\displaystyle\begin{aligned} \partial_{t}\mathfrak{Z}+\nu
A\mathfrak{Z}+B^{\prime}(\overline{y})\mathfrak{Z}+\nabla\mathfrak{q}&=|w_{v}|^{\bar{s}-2}w_{v}&&\text{in
}L^{\tilde{s}^{\prime}}(Q)^{2}\\\ \mathfrak{Z}(0)&=0&&\text{in
}W_{0}^{2-2/{\tilde{s}^{\prime}},\tilde{s}^{\prime}}(\Omega)^{2},\end{aligned}$
(29)

satisfies — by virtue of Theorem 3.5 — the estimate
$\|\mathfrak{Z}\|_{C(\overline{Q})^{2}}\leq
c\|w_{v}\|^{\tilde{s}-1}_{L^{\tilde{s}}(Q)^{2}}$. From this, we get the
estimate

$\displaystyle\begin{aligned}
\|w_{v}\|_{L^{\tilde{s}}(Q)^{2}}^{\tilde{s}}&=\int_{\Omega}|w_{v}|^{\tilde{s}}\operatorname*{d\\!}x\operatorname*{d\\!}t=\big{\langle}|w_{v}|^{\tilde{s}-2}w_{v},w_{v}\big{\rangle}_{L^{2}(Q)^{2}}\\\
&=\big{\langle}\partial_{t}\mathfrak{Z}+\nu
A\mathfrak{Z}+B^{\prime}(\overline{y})\mathfrak{Z}+\nabla\mathfrak{q},w_{v}\big{\rangle}_{L^{2}(Q)^{2}}\\\
&=\big{\langle}\mathfrak{Z},-\partial_{t}w_{v}+\nu
Aw_{v}+B^{\prime}(\overline{y})^{*}w_{v}+\nabla
r\big{\rangle}_{L^{2}(Q)^{2}}\\\
&=\big{\langle}\mathfrak{Z},v\big{\rangle}_{L^{2}(Q)^{2}}\leq\|\mathfrak{Z}\|_{C(\overline{Q})^{2}}\|v\|_{L^{1}(Q)^{2}}\\\
&\leq
c\|w_{v}\|^{\tilde{s}-1}_{L^{\tilde{s}}(Q)^{2}}\|v\|_{L^{1}(Q)^{2}}.\end{aligned}$
(30)

This finally establishes estimate (25). ∎

##  4 The velocity tracking problem

In this section, we give a concise statement of the optimal control problem
and specify the data assumptions. We also recall some known results including
the first-order necessary conditions and the second-order sufficient
condition.

We fix positive numbers $\bar{s}$ and $\bar{s}^{\prime}$ such that $\bar{s}>2$
and $1/\bar{s}+1/\bar{s}^{\prime}=1$. For the optimal control problem (P), we
will make the following standing assumption.

######  Assumption 4.1.

The following statements hold.

  * (i)

The set $\Omega$ is an open connected bounded subset of $\mathbb{R}^{2}$with
boundary $\partial\Omega$ of class $C^{3}$;

  * (iii)

the initial datum $y_{0}$ belongs to
$W^{2-\frac{2}{\bar{s}},\bar{s}}_{0.\sigma}(\Omega)^{2}$;

  * (iv)

the tracking datum $y_{d}$ belongs to $L^{\infty}(Q)^{2}$;

  * (v)

the functions $u_{a},u_{b}:\Omega\to\mathbb{R}^{2}$ are bounded and the
kinematic viscosity parameter $\nu$ is positive.

We recall that the control set is given by

$\displaystyle\mathcal{U}=\left\\{u=(u^{1},u^{2})\in
L^{\infty}(Q)^{2}:\,\,u^{j}_{a}\leq u^{j}\leq u_{b}^{j}\,\,\,\,\text{a.e. in
$Q$ for each $j=1,2$}\right\\}.$ (31)

By the box constraints imposed on the controls, $\mathcal{U}$ should be
bounded in $L^{\infty}(Q)$. From which we define

$\displaystyle
M_{\mathcal{U}}:=\sup_{u\in\mathcal{U}}|u|_{L^{\infty}(Q)^{2}}.$ (32)

Denoting the objective functional by $\mathcal{J}:\mathcal{U}\to\mathbb{R}$,
i.e.,

$\displaystyle\mathcal{J}(u):=\frac{1}{2}\int_{0}^{T}\int_{\Omega}\,|{y_{u}}(t,x)-y_{d}(t,x)|^{2}\,dx\,dt,$

we specify the optimal control problem as

$\displaystyle\min_{u\in\mathcal{U}}\mathcal{J}(u)\quad\text{ subject to
}\eqref{dynsys}.$ (P)

######  Definition 4.2.

Let $\bar{u}\in\mathcal{U}$. We define the minimality radius of $\bar{u}$ as

$\displaystyle\bar{r}_{\bar{u}}:=\sup\left\\{\delta\geq
0:\,\,\mathcal{J}(\bar{u})\leq\mathcal{J}(u)\,\,\,\,\text{for all
$u\in\mathcal{U}$ with $\|u-\bar{u}\|_{{L^{1}(Q)^{2}}}\leq\delta$}\right\\}.$

We say that $\bar{u}$ is a local minimizer of problem (P) if
$\bar{r}_{\bar{u}}<+\infty$ and that $\bar{u}$ is a global minimizer of
problem (P) if $\bar{r}_{\bar{u}}=+\infty$.

The existence of at least one global minimizer for problem (P) follows from
the weak sequential lower semi-continuity of the objective functional and the
continuity of the force-velocity operator according to Theorem 3.3.

Force-to-velocity mapping. By Theorem 3.2, to each control and each initial
datum corresponds a unique state. The mapping
$\widetilde{\mathcal{S}}:\mathcal{U}\times W^{2-2/\bar{s},2}_{0,\sigma}\to
W_{\bar{s}}^{2,1}$ given by $\widetilde{\mathcal{S}}(u,y_{0})=y_{u}$ is called
the data-to-velocity mapping. And for a fixed initial datum $y_{0}\in
W^{2-2/\bar{s},2}_{0,\sigma}$ we define $\mathcal{S}:\mathcal{U}\to
W_{\bar{s}}^{2,1}$ as
$\mathcal{S}(\cdot):=\widetilde{\mathcal{S}}(\cdot,y_{0})$ which we call the
force-to-velocity map. Such mapping is infinitely differentiable.

######  Theorem 4.3.

The force-to-velocity mapping $\mathcal{S}:\mathcal{U}\to W_{\bar{s}}^{2,1}$
is of class $C^{\infty}$, i.e., it has Fréchet derivatives of all orders.
Moreover, the following statements hold:

  * (i)

The first derivative is given by

$\displaystyle\mathcal{S}^{\prime}(u)v=z_{u,v}\quad\text{for $u\in\mathcal{U}$
and $v\in L^{\bar{s}}(Q)^{2}$ satisfying $u+v\in\mathcal{U}$,}$

where each $z_{u,v}\in W_{\bar{s}}^{2,1}$ is the unique $L^{\bar{s}}$-strong
solution of the Oseen equations (15) with
$\overline{y}_{1}=\overline{y}_{2}=y_{u}$ and zero initial datum.

  * (ii)

The second derivative is given by

$\displaystyle\mathcal{S}^{\prime\prime}(u)\big{(}v_{1},v_{2}\big{)}=z_{u,(v_{1},v_{2})}\quad\text{for
$u\in\mathcal{U}$ and $v_{1},v_{2}\in L^{\bar{s}}(Q)^{2}$ satisfying
$u+v_{i}\in\mathcal{U}$,}$

where each $z_{u,(v_{1},v_{2})}\in W_{\bar{s}}^{2,1}$ is the unique
$L^{\bar{s}}$-strong solution of the system

$\displaystyle\begin{aligned} \partial_{t}z+\nu Az+B^{\prime}(y)z+\nabla
q&=-B^{\prime\prime}(y_{u})[z_{u,v_{1}},z_{u,v_{2}}]&&\text{in
}L^{\bar{s}}(Q)^{2}\\\ z(0)&=0&&\text{in
}W_{0,\sigma}^{2-2/\bar{s},\bar{s}}(\Omega)^{2}.\end{aligned}$ (33)

where $z_{u,v_{1}}=\mathcal{S}^{\prime}(u)v_{1}$ and
$z_{u,v_{2}}=\mathcal{S}^{\prime}(u)v_{2}$.

Force-to-covelocity mapping. Since
$\mathcal{S}^{\prime}(u)\in\mathcal{L}(L^{\bar{s}}(Q)^{2};W^{2,1}_{\bar{s},0})$,
we have its adjoint denoted as $\mathcal{S}^{\prime}(u)^{*}$. For an arbitrary
force $\bar{v}\in L^{\bar{s}}(Q)^{2}$ the costate
$w_{u,\bar{v}}=\mathcal{S}^{\prime}(u)^{*}\bar{v}$ is the element that solves
(21) with $\overline{y}_{1}=\overline{y}_{2}=y_{u}$ and $\bar{v}$ as the
right-hand side under the appropriate spaces.

To this end, we recall some stability estimates for the maps $\mathcal{S}$,
its linearization $\mathcal{S}^{\prime}$ and the adjoint
$\mathcal{S}^{\prime}(\cdot)^{*}$.

######  Lemma 4.4 ([10, Corollary 2.6]).

Let $u,\bar{u}\in L^{\bar{s}}(Q)^{2}$, $y_{u}=\mathcal{S}(u)\in
W^{2,1}_{\bar{s}}$ and $y_{\bar{u}}=\mathcal{S}(\bar{u})\in W^{2,1}_{\bar{s}}$
be the states corresponding to $u$ and $\bar{u}$, respectively. Then there
exists a constant $c>0$ such that

$\displaystyle\|y_{u}-y_{\bar{u}}\|_{W^{2,1}_{\bar{s}}}\leq
c\|u-\bar{u}\|_{L^{\bar{s}}(Q)^{2}}.$ (34)

######  Lemma 4.5 ([10, Lemma 3.8]).

Let $u,\bar{u}\in L^{\bar{s}}(Q)^{2}$, and $y_{u}=\mathcal{S}(u)\in
W^{2,1}_{\bar{s}}$ and $y_{\bar{u}}=\mathcal{S}(\bar{u})\in W^{2,1}_{\bar{s}}$
be the states respectively corresponding to $u$ and $\bar{u}$, and
$z_{\bar{u},u-\bar{u}}=S^{\prime}(\bar{u})(u-\bar{u})\in W^{2,1}_{\bar{s}}$ be
the linearized state corresponding to the Fréchet derivative of $\mathcal{S}$
at $\bar{u}$ in the direction $u-\bar{u}$. There exists $\delta>0$ such that
whenever $\|u-\bar{u}\|_{L^{1}({Q})^{2}}\leq\delta$ we have

$\displaystyle\|y_{u}-y_{\bar{u}}\|_{L^{2}(Q)^{2}}\leq
2\|z_{\bar{u},u-\bar{u}}\|_{L^{2}(Q)^{2}}\leq
3\|y_{u}-y_{\bar{u}}\|_{L^{2}(Q)^{2}}.$ (35)

######  Lemma 4.6 ([10, Lemma 3.9]).

Let $v,u,\bar{u}\in L^{\bar{s}}(Q)^{2}$, setting
$y_{u}=\mathcal{S}(u),y_{\bar{u}}=\mathcal{S}(\bar{u}),z_{u,v}=\mathcal{S}^{\prime}(u)v,z_{\bar{u},v}=\mathcal{S}^{\prime}(\bar{u})v$
one finds constants $c>0$ such that the following estimates hold

$\displaystyle\max\\{\|z_{u,v}-z_{\bar{u},v}\|_{L^{2}(Q)^{2}},\|\nabla(z_{u,v}-z_{\bar{u},v})\|_{L^{2}(Q)^{2}}\\}\leq
c\|y_{u}-y_{\bar{u}}\|_{C({\overline{Q}})^{2}}\|z_{\bar{u},v}\|_{L^{2}(Q)^{2}}$
(36)

Furthermore, there exists $\delta>0$ such that

$\displaystyle\|z_{\bar{u},v}\|_{L^{2}(Q)^{2}}\leq
2\|z_{u,v}\|_{L^{2}(Q)^{2}}\leq 3\|z_{\bar{u},v}\|_{L^{2}(Q)^{2}}$ (37)

whenever $\|u-{\bar{u}}\|_{L^{1}({Q})^{2}}<\delta$.

######  Remark 4.7.

We note that the lemmata above have been proven analogously in [10], but of
course with a few changes on the proofs. We decided not to write the full
proofs but we mention a few tweaks from the proofs presented in the said
reference to arrive in their current states:

  * •

Lemma 4.4: one can skip using the embedding $W_{\bar{s}}^{2,1}\hookrightarrow
C(\overline{Q})^{2}$ and can stop before using the inequality
$\|u-\bar{u}\|_{L^{\bar{s}}(Q)^{2}}\leq
M_{\mathcal{U}}\|u-\bar{u}\|_{L^{2}(Q)^{2}}^{2/{\bar{s}}}$.

  * •

Lemma 4.5: choose $\varepsilon=\frac{1}{2\bar{M}\sqrt{T}}$ instead of that one
presented in [10, Lemma 3.8].

  * •

Lemma 4.6: one can control the gap of the states by controlling the gap of the
controls to achieve (37) from inequality (3.24) in [10].

######  Lemma 4.8.

Let $u,\bar{u}\in L^{s}(\Omega)^{2}$ for $s>4$,
$y_{u}=\mathcal{S}(u),y_{\bar{u}}=\mathcal{S}(\bar{u}),w_{u}=\mathcal{S}^{\prime}(u)^{*}(y_{u}-y_{d}),$
and $w_{\bar{u}}=\mathcal{S}^{\prime}(\bar{u})^{*}(y_{\bar{u}}-y_{d})$. There
exists $c>0$ such that the following inequality hold

$\displaystyle\begin{aligned}
\|w_{u}-w_{\bar{u}}\|_{C(\overline{I};C^{1}(\overline{\Omega})^{2})}\leq
c\|y_{u}-y_{\bar{u}}\|_{C(\overline{Q})^{2}}.\end{aligned}$ (38)

First-order necessary condition. Let us begin recalling the first and second
variations of the objective functional.

######  Theorem 4.9.

The objective functional $\mathcal{J}:\mathcal{U}\to\mathbb{R}$ is of class
$C^{\infty}$. Moreover, the following statements hold.

  * (i)

The first variation is given by

$\displaystyle\mathcal{J}^{\prime}(u)v=\int_{Q}w_{u}\cdot
v\,dx\,dt\quad\text{for $u\in\mathcal{U}$ and $v\in L^{\bar{s}}(Q)^{2}$ with
$u+v\in\mathcal{U}$},$

where $w_{u}:=\mathcal{S}^{\prime}(u)^{*}(y_{u}-y_{d})\in W_{\bar{s}}^{2,1}$.

  * (i)

The second variation is given by

$\displaystyle\mathcal{J}^{\prime\prime}(u)v^{2}=\int_{Q}\Big{[}|z_{u,v}|^{2}-2\big{(}z_{u,v}\cdot\nabla\big{)}z_{u,v}\cdot
w_{u}\Big{]}\,dx\,dt\quad\text{for $u\in\mathcal{U}$ and $v\in
L^{\bar{s}}(Q)^{2}$ with $u+v\in\mathcal{U}$}.$

where $z_{u,v}=\mathcal{S}^{\prime}(u)v\in W^{2,1}_{\bar{s}}$.

We are now in a position to lay down the first-order necessary condition for
problem (P).

######  Theorem 4.10.

Let $\bar{u}\in\mathcal{U}$ be a local minimizer of problem (P). Then

$\displaystyle\int_{Q}w_{\bar{u}}\cdot\big{(}u-\bar{u}\big{)}\,dx\,dt\geq
0\quad\text{for all }u\in\mathcal{U}.$ (39)

We can immediately see that (39) is a direct consequence of
$\mathcal{J}^{\prime}(\bar{u})(u-\bar{u})\geq 0$ whenever
$\bar{u}\in\mathcal{U}$ is a minimizer. Furthermore, from the first order
necessary condition a local minimizer $\bar{u}=(\bar{u}_{1},\bar{u}_{2})$ must
satisfy

$\displaystyle\bar{u}^{j}(x,t)=\left\\{\begin{array}[]{l}\bar{u}_{a}^{j}(x,t)\,\,\,\,\text{if
$w_{\bar{u}}^{j}(x,t)>0$}\\\ \\\ \bar{u}_{b}^{j}(x,t)\,\,\,\,\text{if
$w_{\bar{u}}^{j}(x,t)<0$},\end{array}\right.$

for $j\in\\{1,2\\}$ and a.e. $(t,x)\in Q$. It is not hard to deduce from the
first order necessary condition that a control $\bar{u}$ is bang-bang if and
only if

$\displaystyle\operatorname{meas}\bigl{\\{}(x,t)\in
Q:\,w_{\bar{u}}^{1}(x,t)=0\,\,\,\text{or}\,\,\,w_{\bar{u}}^{2}(x,t)=0\bigr{\\}}=0.$

We recall that a control $\bar{u}$ is said to be bang-bang if
$\bar{u}^{j}(x,t)\in\\{u_{a}^{j}(x,t),u_{b}^{j}(x,t)\\}$ for a.e. $(x,t)\in
Q$.

A second-order sufficient condition. In order to establish the second-order
sufficient condition, we shall need the following lemma, which deals with the
curvature of the objective functional. Aside from the sufficient condition the
lemma below will be vital for the stability result in the upcoming section.

######  Lemma 4.11.

Let $\mu\in[1,2)$, $\theta\in[0,1]$ and $\bar{u}\in\mathcal{U}$. For every
$\varepsilon>0$ there exists $\delta>0$ such that

$\displaystyle|\mathcal{J}^{\prime\prime}(\bar{u}+\theta(u-\bar{u}))(u-\bar{u})^{2}-\mathcal{J}^{\prime\prime}(\bar{u})(u-\bar{u})^{2}|\leq\varepsilon\|u-\bar{u}\|_{L^{1}(Q)^{2}}^{\mu+1},$

for all $u\in\mathcal{U}$ with $\|u-\bar{u}\|_{L^{1}(Q)^{2}}\leq\delta$.

The proof is postponed to the end of this section but we now give the promised
sufficient condition.

######  Theorem 4.12.

Let $\mu\in[1,2)$ and $\bar{u}\in\mathcal{U}$. Suppose that there exists
$\delta>0$ such that

$\displaystyle\mathcal{J}^{\prime}(\bar{u})(u-\bar{u})+\frac{1}{2}\mathcal{J}^{\prime\prime}(\bar{u})(u-\bar{u})^{2}\geq\|u-\bar{u}\|_{L^{1}(Q)^{2}}^{\mu+1}\quad\text{for
all $u\in\mathcal{U}$ with $\|u-\bar{u}\|_{L^{1}(Q)^{2}}\leq\delta$.}$ (40)

Then there exists $c>0$ such that

$\displaystyle\mathcal{J}(u)\geq\mathcal{J}(\bar{u})+c\|u-\bar{u}\|_{L^{1}(Q)^{2}}^{\mu+1}\quad\text{for
all $u\in\mathcal{U}$ with $\|u-\bar{u}\|_{L^{1}(Q)^{2}}\leq\delta$}.$ (41)

In particular, $\bar{u}$ is a strict local minimizer.

###### Proof.

Define $\bar{u}_{\theta}:=\bar{u}+\theta(u-\bar{u})$ for $\theta\in[0,1]$. By
Taylor’s theorem, there exists a $\theta\in(0,1)$ such that

$\displaystyle\mathcal{J}(u)-\mathcal{J}(\bar{u})$
$\displaystyle=\mathcal{J}^{\prime}(\bar{u})(u-\bar{u})+\frac{1}{2}\mathcal{J}^{\prime\prime}(\bar{u}_{\theta})(u-\bar{u})^{2}$
$\displaystyle\geq\mathcal{J}^{\prime}(\bar{u})(u-\bar{u})+\frac{1}{2}\mathcal{J}^{\prime\prime}(\bar{u})(u-\bar{u})^{2}-\frac{1}{2}\Big{|}\mathcal{J}^{\prime\prime}(\bar{u})(u-\bar{u})^{2}-\mathcal{J}^{\prime\prime}(\bar{u}_{\theta})(u-\bar{u})^{2}\Big{|}.$

Now (41) follows from (40) and Lemma 4.11. ∎

We conclude with the proof of Lemma 4.11.

###### Proof of Lemma 4.11.

Denote by $u_{\theta}=\bar{u}+\theta(u-\bar{u})$,
$y_{\theta}=\mathcal{S}(u_{\theta})$, $y_{\bar{u}}=\mathcal{S}(\bar{u})$,
$z_{u_{\theta},u-\bar{u}}=\mathcal{S}^{\prime}(u_{\theta})(u-\bar{u})$,
$z_{\bar{u},u-\bar{u}}=\mathcal{S}^{\prime}(\bar{u})(u-\bar{u})$, and
$w_{u_{\theta}}=\mathcal{D}(u_{\theta})$. According to (i), we get

$\displaystyle\begin{aligned}
&|[\mathcal{J}^{\prime\prime}(u_{\theta})-\mathcal{J}^{\prime\prime}(\bar{u})](u-\bar{u})^{2}|=\int_{Q}\left\\{|z_{u_{\theta},u-\bar{u}}|^{2}-|z_{\bar{u},u-\bar{u}}|^{2}\right\\}\operatorname*{d\\!}x\operatorname*{d\\!}t\\\
&+\left\\{\langle
B^{\prime\prime}(y_{\bar{u}})[z_{\bar{u},u-\bar{u}},z_{\bar{u},u-\bar{u}}],w_{\bar{u}}\rangle_{L^{2}(V)}-\langle
B^{\prime\prime}(y_{u_{\theta}})[z_{u_{\theta},u-\bar{u}},z_{u_{\theta},u-\bar{u}}],w_{u_{\theta}}\rangle_{L^{2}(V)}\right\\}.\end{aligned}$
(42)

Let us denote by $I_{1}$ and $I_{s}$ the integral corresponding to the square
norms of the linear variables and trilinear form, respectively.

Estimate (36) aids us to get an upperbound for $I_{1}$

$\displaystyle\begin{aligned}
|I_{1}|&=\left|\int_{Q}\left(z_{u_{\theta},u-\bar{u}}+z_{\bar{u},u-\bar{u}}\right)\cdot\left(z_{u_{\theta},u-\bar{u}}-z_{\bar{u},u-\bar{u}}\right)\operatorname*{d\\!}x\operatorname*{d\\!}t\right|\\\
&\leq\left(\|z_{u_{\theta},u-\bar{u}}\|_{L^{2}(Q)^{2}}+\|z_{\bar{u},u-\bar{u}}\|_{L^{2}(Q)^{2}}\right)\|z_{u_{\theta},u-\bar{u}}-z_{\bar{u},u-\bar{u}}\|_{L^{2}(Q)^{2}}\\\
&\leq
c_{1}\left(\|z_{u_{\theta},u-\bar{u}}\|_{L^{2}(Q)^{2}}+\|z_{\bar{u},u-\bar{u}}\|_{L^{2}(Q)^{2}}\right)\|y_{u_{\theta}}-y_{\bar{u}}\|_{C({\overline{Q}})^{2}}\|z_{\bar{u},u-\bar{u}}\|_{L^{2}(Q)^{2}}\end{aligned}$
(43)

Denoting by $\delta_{1}>0$ the constant mentioned in Lemma 4.6 we know that
whenever $\|u_{\theta}-\bar{u}\|_{L^{1}(Q)^{2}}<\delta_{1}$ we get

$\displaystyle\begin{aligned} |I_{1}|&\leq
c_{1}\|y_{u_{\theta}}-y_{\bar{u}}\|_{C({\overline{Q}})^{2}}\|z_{\bar{u},u-\bar{u}}\|_{L^{2}(Q)^{2}}^{2}.\end{aligned}$
(44)

Using for now the notations $z_{\bar{u}}:=z_{\bar{u},u-\bar{u}}$ and
$z_{{u_{\theta}}}:=z_{{u_{\theta}},u-\bar{u}}$, estimates (38) and (50) gives
us an estimate for $I_{2}$,

$\displaystyle\begin{aligned}
|I_{2}|=&\,\left|\int_{Q}[(z_{\bar{u}}\cdot\nabla)z_{\bar{u}}]\cdot
w_{\bar{u}}-[(z_{{u_{\theta}}}\cdot\nabla)z_{{u_{\theta}}}]\cdot
w_{{u_{\theta}}}\operatorname*{d\\!}x\operatorname*{d\\!}t\right|\\\
\leq&\,\left|\int_{Q}[((z_{\bar{u}}-z_{{u_{\theta}}})\cdot\nabla)z_{\bar{u}}]\cdot
w_{\bar{u}}\operatorname*{d\\!}x\operatorname*{d\\!}t\right|+\left|\int_{Q}[(z_{{u_{\theta}}}\cdot\nabla)(z_{\bar{u}}-z_{{u_{\theta}}})]\cdot
w_{\bar{u}}\operatorname*{d\\!}x\operatorname*{d\\!}t\right|\\\
&+\left|\int_{Q}[(z_{{u_{\theta}}}\cdot\nabla)z_{{u_{\theta}}}]\cdot(w_{\bar{u}}-w_{{u_{\theta}}})\operatorname*{d\\!}x\operatorname*{d\\!}t\right|\\\
=&\,\left|\int_{Q}[((z_{\bar{u}}-z_{{u_{\theta}}})\cdot\nabla)w_{\bar{u}}]\cdot
z_{\bar{u}}\operatorname*{d\\!}x\operatorname*{d\\!}t\right|+\left|\int_{Q}[(z_{{u_{\theta}}}\cdot\nabla)w_{\bar{u}}]\cdot(z_{\bar{u}}-z_{{u_{\theta}}})\operatorname*{d\\!}x\operatorname*{d\\!}t\right|\\\
&+\left|\int_{Q}[(z_{{u_{\theta}}}\cdot\nabla)(w_{\bar{u}}-w_{{u_{\theta}}})]\cdot
z_{{u_{\theta}}}\operatorname*{d\\!}x\operatorname*{d\\!}t\right|\\\
\leq&\,\|z_{\bar{u}}-z_{{u_{\theta}}}\|_{L^{2}(Q)^{2}}\|w_{\bar{u}}\|_{C(\overline{I};C^{1}(\overline{\Omega})^{2})}(\|z_{\bar{u}}\|_{L^{2}(Q)^{2}}+\|z_{{u_{\theta}}}\|_{L^{2}(Q)^{2}})\\\
&+\|z_{{u_{\theta}}}\|_{L^{2}(Q)^{2}}^{2}\|w_{\bar{u}}-w_{{u_{\theta}}}\|_{C(\overline{I};C^{1}(\overline{\Omega})^{2})}\\\
\leq&\,\left(2c_{1}M_{p}+\frac{9}{4}c\right)\|y_{\theta}-y_{\bar{u}}\|_{C({\overline{Q}})^{2}}\|z_{\bar{u},u-\bar{u}}\|_{L^{2}(Q)^{2}}^{2}.\end{aligned}$
(45)

We mention that to be able to reach the fifth relation we, once again, used
the assumption that $\|u_{\theta}-\bar{u}\|_{L^{1}(\Omega)^{2}}<\delta_{1}$
where $\delta_{1}>0$ is as in Lemma 4.6.

Let $\tilde{s}\in[1,2)$ and $\tilde{s}^{\prime}=\tilde{s}/(\tilde{s}-1)>2$ be
its Hölder conjugate, from (19), (34) and (25), with the help of (32) we get

$\displaystyle|[\mathcal{J}^{\prime\prime}(u_{\theta})-\mathcal{J}^{\prime\prime}(\bar{u})](u-\bar{u})^{2}|$
$\displaystyle\leq
c\|y_{\theta}-y_{\bar{u}}\|_{C({\overline{Q}})^{2}}\|z_{\bar{u},u-\bar{u}}\|_{L^{2}(Q)^{2}}^{2}$
$\displaystyle\leq
c\|u_{\theta}-\bar{u}\|_{L^{\tilde{s}^{\prime}}(Q)^{2}}\|z_{\bar{u},u-\bar{u}}\|_{C(\overline{Q})^{2}}^{2-\tilde{s}}\|z_{\bar{u},u-\bar{u}}\|_{L^{\tilde{s}}(Q)^{2}}^{\tilde{s}}$
$\displaystyle\leq
c\|u-\bar{u}\|_{L^{\tilde{s}^{\prime}}(Q)^{2}}\|u-\bar{u}\|_{L^{\tilde{s}^{\prime}}(Q)^{2}}^{2-\tilde{s}}\|u-\bar{u}\|_{L^{1}(Q)^{2}}^{\tilde{s}}$
$\displaystyle\leq
cM_{\mathcal{U}}^{\frac{3-\tilde{s}}{\tilde{s}}}\|u-\bar{u}\|_{L^{1}(Q)^{2}}^{\frac{3-\tilde{s}}{\tilde{s}^{\prime}}+\tilde{s}}.$

By choosing $\tilde{s}>3/2$, we can find $\ell_{1},\ell_{2}>0$ such that
$2+\ell_{1}+\ell_{2}=\frac{3-\tilde{s}}{\tilde{s}^{\prime}}+\tilde{s}$. Thus,
if we take $\mu=1+\ell_{1}$ and
$\delta_{2}=\left(\varepsilon/cM_{\mathcal{U}}^{\frac{3-\tilde{s}}{\tilde{s}}}\right)^{1/\ell_{2}}$
we get that whenever
$\|u-\bar{u}\|_{L^{1}(Q)^{2}}<\delta:=\min\\{\delta_{1},\delta_{2}\\}$

$\displaystyle|[\mathcal{J}^{\prime\prime}(u_{\theta})-\mathcal{J}^{\prime\prime}(\bar{u})](u-\bar{u})^{2}|$
$\displaystyle\leq\varepsilon\|u-\bar{u}\|_{L^{1}(Q)^{2}}^{1+\mu}.$

Furthermore, since $\bar{s}\in(3/2,2)$ we have $\mu\in(1,2)$. ∎

##  5 Stability under perturbations

This section deals with the main result of this paper. That is, we present the
effects of perturbations on problem (P).

The perturbed problem. To begin, we consider the datum-perturbed Navier–Stokes
equations

$\displaystyle\left\\{\begin{aligned}
&\partial_{t}{y}-\nu\Delta{y}+({y}\cdot\nabla){y}+\nabla{p}=u\,\,\,\,\text{in
Q},\\\ &\operatorname{div}{y}=0\,\,\,\,\text{in Q},\,\,{y}=0\,\,\,\,\text{on
$\Sigma$},\,\,\,{y}(\cdot,0)=y_{0}+\xi\,\,\,\text{in
$\Omega$},\end{aligned}\right.$ (46)

where the perturbation $\xi\in W^{2-\frac{2}{\bar{s}}}_{0}(\Omega)^{2}$
accounts for possible uncertainty on the initial datum. From Theorem 3.2, we
know that the operator $\mathcal{S}_{\xi}:\mathcal{U}\to W_{\bar{s}}^{2,1}$
defined as $\mathcal{S}_{\xi}(u)=\widetilde{\mathcal{S}}(u,y_{0}+\xi)$ is
well-defined, from which we know that the element
$y_{u}^{\xi}:=\mathcal{S}_{\xi}(u)\in W_{\bar{s}}^{2,1}$ uniquely solves (46).
Analogously with force-to-velocity map, $\mathcal{S}_{\xi}$ can be shown to be
of class $C^{\infty}$, specifically, for given $u\in\mathcal{U}$ and $v\in
L^{\bar{s}}(Q)^{2}$ with $u+v\in\mathcal{U}$ one gets an element
$\mathcal{S}_{\xi}^{\prime}(u)v\in W_{\bar{s}}^{2,1}$ that solves (18) with
$\overline{y}_{1}=\overline{y}_{2}=y_{u}^{\xi}$ and zero initial datum.
Furthermore, we get the adjoint of
$\mathcal{S}_{\xi}^{\prime}(u)\in\mathcal{L}(L^{\bar{s}}(Q)^{2};W^{2,1}_{\bar{s},0})$,
i.e., the element $\mathcal{S}_{\xi}^{\prime}(u)^{*}v\in W^{2,1}_{\bar{s}}$
solves (21) with $\overline{y}_{1}=\overline{y}_{2}=y_{u}^{\xi}$ and ${v}$ as
the right-hand side.

For $\eta\in L^{\bar{s}}(Q)$ and $\varepsilon\geq 0$, we consider the
perturbed objective function

$\displaystyle\mathcal{J}_{\xi,\eta}^{\varepsilon}(u):=\frac{1}{2}\int_{0}^{T}\int_{\Omega}|y_{u}^{\xi}(x,t)-\big{(}y_{d}(x,t)+\eta(x,t)\big{)}|^{2}\,dx\,dt+\frac{\varepsilon}{2}\int_{0}^{T}\int_{\Omega}|u|^{2}\,dx\,dt,$

where $y_{u}^{\xi}=\mathcal{S}_{\xi}(u)$. The perturbation $\eta$ represent
uncertainty in the tracking data and the parameter $\varepsilon$ is a weight
parameter for the Tikhonov regularization term.

The perturbed optimal control problem can now be written as

$\displaystyle\min_{u\in\mathcal{U}}J_{\xi,\eta}^{\varepsilon}(u)\quad\text{
subject to }\eqref{dynsysper}.$ (P${}_{\xi,\eta}^{\varepsilon}$)

Due to the convexity in the control variable, the functional
$\mathcal{J}_{\xi,\eta}^{\varepsilon}$ is lower semicontinuous. Therefore, the
existence of at least one global minimizer of problem
(P${}_{\xi,\eta}^{\varepsilon}$) is guaranteed. Furthermore, one can obtain a
first-order necessary condition for a minimizer $\widehat{u}\in\mathcal{U}$ of
(P${}_{\xi,\eta}^{\varepsilon}$) which we can write as
$0\in\varepsilon\widehat{u}+w_{\widehat{u}}^{\eta}+N_{\mathcal{U}}(\widehat{u})$
where
$w_{\widehat{u}}^{\eta}:=\mathcal{S}_{\xi}^{\prime}(\widehat{u})^{*}(y_{\widehat{u}}^{\xi}-y_{d}-\eta)\in
W_{\bar{s}}^{2,1}$, $y_{\widehat{u}}^{\xi}=\mathcal{S}_{\xi}(\widehat{u})$,
and $N_{\mathcal{U}}(\widehat{u})$ is the normal cone to $\mathcal{U}$ at
$\widehat{u}$, see $\eqref{normalcone}$. Let us introduce
$\mathfrak{W}:=W^{2-\frac{2}{\bar{s}}}_{0}(\Omega)^{2}\times
L^{\infty}(Q)^{2}\times[0,+\infty)$ as the set of perturbations.

Aside from the growth assumption imposed to obtain the second-order sufficient
condition, we shall rely on a slightly modified growth assumption from which
we can get the stability desired.

######  Assumption 5.1.

Let $\mu\in[1,2)$ and $\bar{u}\in\mathcal{U}$. Suppose that

$\displaystyle\mathcal{J}^{\prime}(\bar{u})(u-\bar{u})+\mathcal{J}^{\prime\prime}(\bar{u})(u-\bar{u})^{2}\geq\|u-\bar{u}\|_{L^{1}(Q)^{2}}^{\mu+1}\quad\text{for
all $u\in\mathcal{U}$ with $\|u-\bar{u}\|_{L^{1}(Q)^{2}}\leq\delta$.}$ (47)

We note that Assumption 5.1 is a weaker one compared to the growth assumption
(40), i.e., if $\bar{u}\in\mathcal{U}$ satisfies $0\in
w_{\bar{u}}+N_{\mathcal{U}}(\bar{u})$ then Assumption 5.1 implies (40). Both
assumptions imply that $\bar{u}$ is bang-bang, and both will give us the
stability we want. The advantage for assuming Assumption 5.1 is that we can
directly apply the results in the Appendix while (40) requires a more nuanced
proof to achieve the stability.

######  Theorem 5.2.

Let $\bar{u}$ be a local minimizer of problem (P), $\mu\in[1,2)$ and suppose
that either Assumption 5.1 or (40) holds. Then there exist $\delta>0$ and
$\kappa>0$ such that for all $\xi,\eta,\varepsilon\in\mathfrak{W}$ and all
local minimizers $\widehat{u}\in\mathcal{U}$ of problem
(P${}_{\xi,\eta}^{\varepsilon}$) with
$\|\widehat{u}-\bar{u}\|_{L^{1}(Q)^{2}}\leq\delta$, it holds that

$\displaystyle\|\widehat{u}-\bar{u}\|_{L^{1}(Q)^{2}}\leq\kappa\Big{(}\|\xi\|_{W^{2-\frac{2}{\bar{s}},s}_{0.\sigma}(\Omega)^{2}}+\|\eta\|_{L^{2}(Q)}+\varepsilon\Big{)}^{\frac{1}{\mu}}.$

To prove the main result above, we will rely on the following lemmata that set
up the application of the abstract results from the Appendix.

######  Lemma 5.3.

Let $u\in\mathcal{U}$ and $v\in L^{\bar{s}}(Q)^{2}$ such that
$u+v\in\mathcal{U}$. Let $y_{u}^{\xi}=\mathcal{S}_{\xi}(u)$,
$z_{u,v}^{\xi}:=\mathcal{S}_{\xi}^{\prime}(u)v\in W_{\bar{s}}^{2,1}$,
$y_{u}=\mathcal{S}(u)$ and $z_{u,v}:=\mathcal{S}^{\prime}(u)v\in
W_{\bar{s}}^{2,1}$. Then, there exists $c>0$ independent of $u$ such that

$\displaystyle\max\\{\|z_{u,v}^{\xi}-z_{u,v}\|_{L^{2}(Q)^{2}},\|\nabla(z_{u,v}^{\xi}-z_{u,v})\|_{L^{2}(Q)^{2}}\\}\leq
c\|y_{u}^{\xi}-y_{u}\|_{C(\overline{Q})^{2}}\|v\|_{L^{2}(Q)^{2}}.$

###### Proof.

We only show the inequality $\|z_{u,v}^{\xi}-z_{u,v}\|_{L^{2}(Q)^{2}}\leq
c\|y_{u}^{\xi}-y_{u}\|_{C(\overline{Q})^{2}}\|v\|_{L^{2}(Q)^{2}}$, as the
other one follows analogously as in the forecoming arguments and the proof of
[10, Lemma 3.9].

First, note that the element $\mathfrak{Z}=z_{u,v}^{\xi}-z_{u,v}\in
W^{2,1}_{\bar{s}}$ solves the linear system

$\displaystyle\begin{aligned} \partial_{t}\mathfrak{Z}+\nu
A\mathfrak{Z}+B^{\prime}(y_{u}^{\xi})\mathfrak{Z}+\nabla\mathfrak{q}&=B^{\prime}(y_{u}-y_{u}^{\xi})z_{u,v}&&\text{in
}L^{\bar{s}}(Q)^{2}\\\ \mathfrak{Z}(0)&=0&&\text{in
}W_{0}^{2-2/\bar{s},\bar{s}}(\Omega)^{2}.\end{aligned}$

Now, let $f\in L^{2}(I;H)$ and consider the adjoint system

$\displaystyle\begin{aligned} -\partial_{t}\mathfrak{w}+\nu
A\mathfrak{w}+B^{\prime}(y_{u}^{\xi})^{*}\mathfrak{w}+\nabla\mathfrak{r}&=f&&\text{in
}L^{2}(I;H)\\\ \mathfrak{w}(T)&=0&&\text{in }H.\end{aligned}$

From Theorem 3.6, we know that $\|\nabla\mathfrak{w}\|_{L^{2}(Q)^{2}}\leq
c\|f\|_{L^{2}(Q)^{2}}$ for some constant $c>0$. Hence, we get that

$\displaystyle\begin{aligned} \int_{0}^{T}\int_{\Omega}\mathfrak{Z}\cdot
f\operatorname*{d\\!}x\operatorname*{d\\!}t&=\int_{0}^{T}\int_{\Omega}\mathfrak{Z}\cdot(-\partial_{t}\mathfrak{w}+\nu
A\mathfrak{w}+B^{\prime}(y_{u}^{\xi})^{*}\mathfrak{w}+\nabla\mathfrak{r})\operatorname*{d\\!}x\operatorname*{d\\!}t\\\
&=\int_{0}^{T}\int_{\Omega}(\partial_{t}\mathfrak{Z}+\nu
A\mathfrak{Z}+B^{\prime}(y_{u}^{\xi})\mathfrak{Z}+\nabla\mathfrak{q})\cdot\mathfrak{w}\operatorname*{d\\!}x\operatorname*{d\\!}t\\\
&=\int_{0}^{T}\int_{\Omega}B^{\prime}(y_{u}-y_{u}^{\xi})z_{u,v}\cdot\mathfrak{w}\operatorname*{d\\!}x\operatorname*{d\\!}t\\\
&\leq
c\|y_{u}-y_{u}^{\xi}\|_{C(\overline{Q})^{2}}\|z_{u,v}\|_{L^{2}(Q)^{2}}\|\nabla\mathfrak{w}\|_{L^{2}(Q)^{2}}\\\
&\leq
c\|y_{u}-y_{u}^{\xi}\|_{C(\overline{Q})^{2}}\|z_{u,v}\|_{L^{2}(Q)^{2}}\|f\|_{L^{2}(Q)^{2}}.\end{aligned}$
(48)

The fact that $\|z_{u,v}\|_{L^{2}(Q)^{2}}\leq c\|v\|_{L^{2}(Q)^{2}}$ proves
the claim.

∎

######  Lemma 5.4.

Let $u\in\mathcal{U}$. Let $y_{u}^{\xi}=\mathcal{S}_{\xi}(u)$,
$w_{u}^{\eta}:=\mathcal{S}_{\xi}^{\prime}(u)^{*}(y_{u}^{\xi}-y_{d}-\eta)\in
W_{\bar{s}}^{2,1}$.Then, there exists $c>0$ (independent of $u$) such that

$\displaystyle\|w_{u}^{\eta}-w_{u}\|_{W_{\bar{s}}^{2,1}}\leq
c\Big{(}\|\xi\|_{W^{2-\frac{2}{\bar{s}},s}_{0.\sigma}(\Omega)^{2}}+\|\eta\|_{L^{\bar{s}}(Q)}\Big{)}$

for all $\xi\in W^{2-\frac{2}{\bar{s}},\bar{s}}_{0.\sigma}(\Omega)^{2}$ and
$\eta\in L^{\bar{s}}(Q)$, where
$w_{u}=\mathcal{S}^{\prime}(u)^{*}(y_{u}-y_{d})$ and $y_{u}={\mathcal{S}}(u)$.

###### Proof.

We begin by obtaining some estimates for the element
$z_{u}^{\xi}:=y_{u}^{\xi}-y_{u}\in W^{2,1}_{\bar{s}}$ which solves the system

$\displaystyle\left\\{\begin{aligned} \partial_{t}z_{u}^{\xi}+\nu
Az_{u}^{\xi}+\widetilde{B}(y_{u},y_{u}^{\xi})z_{u}^{\xi}+\nabla q&=0&&\text{in
}L^{s}(Q)^{2},\\\ z_{u}^{\xi}(0)&=\xi&&\text{in
}W_{0,\sigma}^{2-2/\bar{s},\bar{s}}(\Omega)^{2}.\end{aligned}\right.$

According to Theorem 3.5, this solution satisfies
$\|z_{u}^{\xi}\|_{W^{2,1}_{\bar{s}}}\leq
c\|\xi\|_{W_{0,\sigma}^{2-2/\bar{s},\bar{s}}(\Omega)^{2}}$.

Now the difference $\mathfrak{w}_{u}^{\eta}:=w_{u}^{\eta}-w_{u}$ solves the
system

$\displaystyle\begin{aligned} -\partial_{t}\mathfrak{w}_{u}^{\eta}+\nu
A\mathfrak{w}_{u}^{\eta}+B^{\prime}(y_{u}^{\xi})^{*}\mathfrak{w}_{u}^{\eta}+\nabla\mathfrak{r}&=B^{\prime}(z_{u}^{\xi})^{*}w_{u}+z_{u}^{\xi}+\eta&&\text{in
}L^{\bar{s}}(Q)^{2}\\\ \mathfrak{w}_{u}^{\eta}(T)&=0&&\text{in
}W_{0}^{2-2/{\bar{s}},\bar{s}}(\Omega)^{2}.\end{aligned}$ (49)

From $W^{2,1}_{s}\hookrightarrow C(\overline{I};C^{1}(\overline{\Omega})^{2})$
for $s>4$ and Theorem 3.7, supplemented with (14) and (32) we get

$\displaystyle\begin{aligned}
\|w_{u}\|_{C(\overline{I};C^{1}(\overline{\Omega})^{2})}\leq
c\left(\|y_{v}-y_{d}\|_{L^{s}(Q)^{2}}\right)\leq M_{p}.\end{aligned}$ (50)

Hence, the first term on the right-hand side of (49) can be estimated below as
follows:

$\displaystyle\begin{aligned}
\|B^{\prime}(z_{u}^{\xi})^{*}w_{u}\|_{L^{s}}&\leq\inf_{\|v\|_{L^{s^{\prime}}}=1}\left|\langle
B^{\prime}(z_{u}^{\xi})^{*}w_{u},v\rangle_{L^{2}(V)}\right|\\\
&=\inf_{\|v\|_{L^{s^{\prime}}}=1}\left|\langle
B(z_{u}^{\xi},w_{u}),v\rangle_{L^{2}(V)}\right|+\left|\langle
B(v,w_{u}),z_{u}^{\xi}\rangle_{L^{2}(V)}\right|\\\ &\leq
cM_{p}\|z_{u}^{\xi}\|_{L^{s}(Q)}.\end{aligned}$ (51)

From Theorem 3.7, we therefore get

$\displaystyle\begin{aligned}
\|w_{u}^{\eta}-w_{u}\|_{W_{\bar{s}}^{2,1}}&\leq\|B^{\prime}(z_{u}^{\xi})^{*}w_{u}\|_{L^{s}}+\|z_{u}^{\xi}\|_{L^{s}}+\|\eta\|_{L^{s}}\\\
&\leq(cM_{p}+1)\|z_{u}^{\xi}\|_{L^{s}}+\|\eta\|_{L^{s}}\\\ &\leq
c\Big{(}\|\xi\|_{W^{2-\frac{2}{\bar{s}},s}_{0.\sigma}(\Omega)^{2}}+\|\eta\|_{L^{\bar{s}}(Q)}\Big{)}.\end{aligned}$
(52)

∎

######  Lemma 5.5.

Let $\bar{u}\in\mathcal{U}$ and $\mu\in[1,2)$ satisfy (5.1). There exist
positive numbers $\delta$ and $c$ such that

$\displaystyle\|u-\bar{u}\|_{L^{1}(Q)^{2}}\leq
c\|\rho\|_{L^{\infty}(Q)^{2}}^{\frac{1}{{\mu}}}$ (53)

for all $\rho\in L^{\infty}(Q)$ and $u\in\mathcal{U}$ satisfying $\rho\in
w_{u}+N_{\mathcal{U}}(u)$ and $\|u-\bar{u}\|_{L^{1}(Q)^{2}}\leq\delta$.

###### Proof.

By Proposition A.11, there exists positive constants $c$ and $\alpha$ such
that $\mathcal{J}^{\prime}(u)(u-\bar{u})\geq
c\|u-\bar{u}\|_{L^{1}(Q)^{2}}^{\mu+1}$ for all $u\in\mathcal{U}$ with
$\|u-\bar{u}\|_{L^{1}(Q)^{2}}\leq\alpha$. Given $\rho\in L^{\infty}(Q)^{2}$
and $u\in\mathcal{U}$ satisfying $\rho\in w_{u}+N_{\mathcal{U}}(u)$ and
$\|u-\bar{u}\|_{L^{1}(Q)^{2}}\leq\alpha$, it holds

$\int_{0}^{T}\langle w_{u}-\rho,\bar{u}-u\rangle\,dt\geq 0.$ (54)

Now rearranging the terms in (54) and applying Proposition A.11, we find

$\displaystyle
c\|u-\bar{u}\|_{L^{1}(Q)^{2}}^{\mu+1}\leq\mathcal{J}^{\prime}(u)(u-\bar{u})=\int_{0}^{T}\langle
w_{u},u-\bar{u}\rangle\,dt\leq\|\rho\|_{L^{\infty}(Q)^{2}}\|\bar{u}-u\|_{L^{1}(Q)^{2}},$

which proves the claim. ∎

###### Proof of Theorem 5.2.

The proof will be divided into two cases, one with Assumption 5.1 and the
other case is where we use (40). Let us begin with the former case. By
defining
$\rho:=w_{\widehat{u}}-\varepsilon\widehat{u}-w_{\widehat{u}}^{\eta}\in
L^{\infty}(Q)^{2}$, where
$w_{\widehat{u}}=\mathcal{S}^{\prime}(\widehat{u})^{*}(y_{\widehat{u}}-y_{d})$
and
$w_{\widehat{u}}^{\eta}=\mathcal{S}_{\xi}^{\prime}(\widehat{u})^{*}(y_{\widehat{u}}-y_{d}-\eta)$,
and utilizing Lemma 5.4 we get

$\displaystyle\|\rho\|_{L^{\infty}(Q)}\leq\|w_{\widehat{u}}-w_{\widehat{u}}^{\eta}\|_{L^{\infty}(Q)^{2}}+\varepsilon\|\widehat{u}\|_{L^{\infty}(Q)^{2}}\leq
c\Big{(}\|\xi\|_{W^{2-\frac{2}{\bar{s}},s}_{0.\sigma}(\Omega)^{2}}+\|\eta\|_{L^{2}(Q)}+\varepsilon\Big{)},$

where $c>0$ is dependent on $M_{\mathcal{U}}$ and the constant in Lemma 5.4.
As $\widehat{u}\in\mathcal{U}$ is a local minimizer of problem
(P${}_{\xi,\eta}^{\varepsilon}$), we get
$0\in\varepsilon\widehat{u}+w_{\widehat{u}}^{\eta}+N_{\mathcal{U}}(\widehat{u})$
and thus $\rho\in w_{\widehat{u}}+N_{\mathcal{U}}(\widehat{u})$. Then by Lemma
5.5,

$\displaystyle\begin{aligned} \|\widehat{u}-\bar{u}\|_{L^{1}(Q)^{2}}\leq
c\|\rho\|_{L^{\infty}(Q)}^{\frac{1}{{\mu}}}&\leq
cc_{2}^{\frac{1}{{\mu}}}\Big{(}\|\xi\|_{W^{2-\frac{2}{\bar{s}},s}_{0.\sigma}(\Omega)^{2}}+\|\eta\|_{L^{2}(Q)}+\varepsilon\Big{)}^{\frac{1}{{\mu}}}\\\
&:=\kappa\Big{(}\|\xi\|_{W^{2-\frac{2}{\bar{s}},s}_{0.\sigma}(\Omega)^{2}}+\|\eta\|_{L^{2}(Q)}+\varepsilon\Big{)}^{\frac{1}{\mu}}.\end{aligned}$
(55)

For the case where we assume (40), we first note that since $\widehat{u}$ is a
minimizer of Problem (P${}_{\xi,\eta}^{\varepsilon}$) we have
$J_{\xi,\eta}^{\varepsilon}(\widehat{u})-\mathcal{J}_{\xi,\eta}^{\varepsilon}(\bar{u})\leq
0$ which is equivalent to writing

$\displaystyle\begin{aligned}
0&\leq-(\mathcal{G}(\widehat{u})-\mathcal{G}(\bar{u}))+\int_{0}^{T}\int_{\Omega}\langle\eta,y_{\widehat{u}}^{\xi}-y_{\bar{u}}^{\xi}\rangle\,dx\,dt+\frac{\varepsilon}{2}\int_{0}^{T}\int_{\Omega}|\bar{u}|^{2}-|\widehat{u}|^{2}\,dx\,dt\\\
&\leq-(\mathcal{G}(\widehat{u})-\mathcal{G}(\bar{u}))+\|\eta\|_{L^{2}(Q)^{2}}\|y_{\widehat{u}}^{\xi}-y_{\bar{u}}^{\xi}\|_{L^{2}(Q)^{2}}+M_{\mathcal{U}}\varepsilon\|\bar{u}-\widehat{u}\|_{L^{1}(Q)^{2}}\\\
&\leq-(\mathcal{G}(\widehat{u})-\mathcal{G}(\bar{u}))+c(\|\eta\|_{L^{\bar{s}}(Q)^{2}}+\varepsilon)\|\bar{u}-\widehat{u}\|_{L^{1}(Q)^{2}}\end{aligned}$
(56)

where for a fixed $\xi\in L^{\infty}(Q)^{2}$ we used the notation
$\mathcal{G}(u):=\frac{1}{2}\int_{0}^{T}\int_{\Omega}|y_{u}^{\xi}(x,t)-y_{d}(x,t)|^{2}\,dx\,dt.$
We note that to achieve the last line we used the fact that
$y_{\widehat{u}}^{\xi}-y_{\bar{u}}^{\xi}\in W^{2,1}_{s}$ solves an equation of
the form (18) with the right-hand side $\widehat{u}-\bar{u}$ and initial datum
equal to zero, which implies
$\|y_{\widehat{u}}^{\xi}-y_{\bar{u}}^{\xi}\|_{L^{2}(Q)^{2}}\leq
c\|\widehat{u}-\bar{u}\|_{L^{1}(Q)^{2}}$ for some constant $c>0$.

By Taylor’s theorem, for some $\theta\in[0,1]$ we find
$u_{\theta}=\bar{u}+\theta(\widehat{u}-\bar{u})\in\mathcal{U}$ such that

$\mathcal{G}(\widehat{u})-\mathcal{G}(\bar{u}):=\mathcal{G}^{\prime}(\bar{u})(\widehat{u}-\bar{u})+\frac{1}{2}\mathcal{G}^{\prime\prime}(u_{\theta})(\widehat{u}-\bar{u})^{2}.$

From this we rewrite (56) as

$\displaystyle\mathcal{J}^{\prime}(\bar{u})(\widehat{u}-\bar{u})+\frac{1}{2}\mathcal{J}^{\prime\prime}(u_{\theta})(\widehat{u}-\bar{u})^{2}\leq
G_{1}+\frac{1}{2}G_{2}+c(\|\eta\|_{L^{\bar{s}}(Q)^{2}}+\varepsilon)\|\bar{u}-\widehat{u}\|_{L^{1}(Q)^{2}}$

where
$G_{1}=\mathcal{J}^{\prime}(\bar{u})(\widehat{u}-\bar{u})-\mathcal{G}^{\prime}(\bar{u})(\widehat{u}-\bar{u})$
and
$G_{2}=\mathcal{J}^{\prime\prime}(u_{\theta})(\widehat{u}-\bar{u})^{2}-\mathcal{G}^{\prime\prime}(u_{\theta})(\widehat{u}-\bar{u})^{2}$.
Let us now get some estimates for $G_{1}$ and $G_{2}$, respectively.

For $G_{1}$, we note that we can write the derivative of $\mathcal{G}$ as
$\mathcal{G}^{\prime}(\bar{u})(\widehat{u}-\bar{u})=\int_{0}^{T}\int_{\Omega}w_{\bar{u}}^{\eta}\cdot(\widehat{u}-\bar{u})\operatorname*{d\\!}x\operatorname*{d\\!}t$
where
$w_{\bar{u}}^{\eta}=\mathcal{S}_{\xi}^{\prime}(\bar{u})^{*}(y_{\bar{u}}^{\xi}-y_{d}-\eta)$
so that, by additionally employing Lemma 5.4 and the embedding
$W^{2,1}_{\bar{s}}\hookrightarrow C(\overline{Q})^{2}$, we get

$\displaystyle|G_{1}|$
$\displaystyle\leq\|w_{\bar{u}}^{\eta}-w_{\bar{u}}\|_{C(\overline{Q})^{2}}\|\widehat{u}-\bar{u}\|_{L^{1}(Q)^{2}}\leq
c\Big{(}\|\xi\|_{W^{2-\frac{2}{\bar{s}},\bar{s}}_{0,\sigma}(\Omega)^{2}}+\|\eta\|_{L^{2}(Q)}\Big{)}\|\widehat{u}-\bar{u}\|_{L^{1}(Q)^{2}}.$
(57)

The estimate for $G_{2}$ will be divided into two parts, i.e., using the same
form as the second Fréchet derivative of $\mathcal{J}$ from Theorem 4.9 we
write $G_{2}=G_{2,1}+G_{2,2}$ where

$\displaystyle\begin{aligned}
&G_{2,1}=\int_{0}^{T}\int_{\Omega}|z_{u_{\theta},\widehat{u}-\bar{u}}|^{2}-|z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}|^{2}\operatorname*{d\\!}x\operatorname*{d\\!}t\\\
&G_{2,2}=2\int_{0}^{T}\int_{\Omega}[(z_{u_{\theta},\widehat{u}-\bar{u}}\cdot\nabla)z_{u_{\theta},\widehat{u}-\bar{u}}]\cdot
w_{u_{\theta}}-[(z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}\cdot\nabla)z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}]\cdot
w_{u_{\theta}}^{\eta}\operatorname*{d\\!}x\operatorname*{d\\!}t\end{aligned}$

Knowing that
$\max\\{\|z_{u_{\theta},\widehat{u}-\bar{u}}\|_{L^{2}(Q)^{2}},\|z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}\|_{L^{2}(Q)^{2}}\\}\leq
c\|\widehat{u}-\bar{u}\|_{L^{2}(Q)^{2}}$, and due to Lemma 5.3 we majorize
$G_{2,1}$ as follows:

$\displaystyle\begin{aligned}
|G_{2,1}|&\leq\|z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}+z_{u_{\theta},\widehat{u}-\bar{u}}\|_{L^{2}(Q)^{2}}\|z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}-z_{u_{\theta},\widehat{u}-\bar{u}}\|_{L^{2}(Q)^{2}}\\\
&\leq
c\|\widehat{u}-\bar{u}\|_{L^{1}(Q)^{2}}\|y^{\xi}_{u_{\theta}}-y_{u_{\theta}}\|_{C(\overline{Q})^{2}}.\end{aligned}$

Similarly, using additionally Lemma 5.4, we majorize the remaining term as

$\displaystyle\begin{aligned}
|G_{2,2}|\leq&\,\left|\int_{0}^{T}\int_{\Omega}[((z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}-z_{u_{\theta},\widehat{u}-\bar{u}})\cdot\nabla)z_{u_{\theta},\widehat{u}-\bar{u}}]\cdot
w_{u_{\theta}}\operatorname*{d\\!}x\operatorname*{d\\!}t\right|\\\
&+\left|\int_{0}^{T}\int_{\Omega}[(z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}\cdot\nabla)(z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}-z_{u_{\theta},\widehat{u}-\bar{u}})]\cdot
w_{u_{\theta}}\operatorname*{d\\!}x\operatorname*{d\\!}t\right|\\\
&+\left|\int_{0}^{T}\int_{\Omega}[(z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}\cdot\nabla)z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}]\cdot(w_{u_{\theta}}^{\eta}-w_{u_{\theta}})\operatorname*{d\\!}x\operatorname*{d\\!}t\right|\\\
\leq&\,\|z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}-z_{u_{\theta},\widehat{u}-\bar{u}}\|_{L^{2}(Q)^{2}}\|\nabla
z_{u_{\theta},\widehat{u}-\bar{u}}\|_{L^{2}(Q)^{2}}\|w_{u_{\theta}}\|_{C(\overline{Q})^{2}}\\\
&+\|z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}\|_{L^{2}(Q)^{2}}\|\nabla(z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}-z_{u_{\theta},\widehat{u}-\bar{u}})\|_{L^{2}(Q)^{2}}\|w_{u_{\theta}}\|_{C(\overline{Q})^{2}}\\\
&+\|z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}\|_{L^{2}(Q)^{2}}\|\nabla(z_{u_{\theta},\widehat{u}-\bar{u}}^{\xi}-z_{u_{\theta},\widehat{u}-\bar{u}})\|_{L^{2}(Q)^{2}}\|w_{u_{\theta}}^{\eta}-w_{u_{\theta}}\|_{C(\overline{Q})^{2}}\\\
\leq&\,c\|\widehat{u}-\bar{u}\|_{L^{1}(Q)^{2}}(\|y^{\xi}_{u_{\theta}}-y_{u_{\theta}}\|_{C(\overline{Q})^{2}}+\|\xi\|_{W^{2-\frac{2}{\bar{s}},\bar{s}}_{0,\sigma}(\Omega)^{2}}+\|\eta\|_{L^{\bar{s}}(Q)})\end{aligned}$
(58)

Following the proof in Lemma 5.4, we know that
$\|y^{\xi}_{u_{\theta}}-y_{u_{\theta}}\|_{C(\overline{Q})^{2}}\leq
c\|\xi\|_{W^{2-\frac{2}{\bar{s}},s}_{0,\sigma}}$ for some constant $c>0$. With
this relation, and the estimates (57) and (58) we get

$\displaystyle\mathcal{J}^{\prime}(\bar{u})(\widehat{u}-\bar{u})+\frac{1}{2}\mathcal{J}^{\prime\prime}(u_{\theta})(\widehat{u}-\bar{u})^{2}\leq
c(\|\xi\|_{W^{2-\frac{2}{\bar{s}},s}_{0,\sigma}(\Omega)^{2}}+\|\eta\|_{L^{\bar{s}}(Q)^{2}}+\varepsilon)\|\bar{u}-\widehat{u}\|_{L^{1}(Q)^{2}}.$
(59)

From Lemma 4.11, for any $\varepsilon>0$ we can find a $\delta_{1}>0$ such
that

$\displaystyle|\mathcal{J}^{\prime\prime}(\bar{u}_{\theta})(u-\bar{u})^{2}-\mathcal{J}^{\prime\prime}(\bar{u})(u-\bar{u})^{2}|\leq\rho\|u-\bar{u}\|_{L^{1}(Q)^{2}}^{\mu+1}$

whenever $u\in\mathcal{U}$ satisfies
$\|u-\bar{u}\|_{L^{1}(Q)^{2}}\leq\delta_{1}$. Combining this with (40) we
estimate the left-hand side of the inequality above as

$\displaystyle\begin{aligned}
&\mathcal{J}^{\prime}(\bar{u})(\widehat{u}-\bar{u})+\frac{1}{2}\mathcal{J}^{\prime\prime}(\bar{u}_{\theta})(\widehat{u}-\bar{u})^{2}\\\
&=\mathcal{J}^{\prime}(\bar{u})(\widehat{u}-\bar{u})+\frac{1}{2}\mathcal{J}^{\prime\prime}(\bar{u})(\widehat{u}-\bar{u})^{2}+\frac{1}{2}\left[\mathcal{J}^{\prime\prime}(u_{\theta})(\widehat{u}-\bar{u})^{2}-\mathcal{J}^{\prime\prime}(\bar{u})(\widehat{u}-\bar{u})^{2}\right]\\\
&\geq\left(1-\frac{\varepsilon}{2}\right)\|u-\bar{u}\|_{L^{1}(Q)^{2}}^{\mu+1}\end{aligned}$
(60)

whenever
$\|u-\bar{u}\|_{L^{1}(Q)^{2}}\leq\delta:=\min\\{\delta_{1},\delta_{2}\\}$,
where $\delta_{2}>0$ is as in the assumption (40). The arbitrariness of
$\varepsilon>0$ allows us to choose $\varepsilon<2$. Combining (59) and (60)
proves our claim.

∎

To finally end this section, we mention that according to Proposition A.11,
local minimizers $\bar{u}\in\mathcal{U}$ that satify (41) should also satisfy
(40). This implies that said local minimizers are bang-bang and should satisfy
the stability we just proved.

##  Appendix A Appendix

This section is devoted to collecting the stability results of the paper in an
abstract framework, as the same principles can be applied to other types of
optimization problems. We employ normed spaces since they constitute the right
setting in the context of control and optimization, and norms allow a
positively homogeneous measure for the notions of growth and convergence, in
contrast with metric spaces.

The results of this section focus mainly on necessary and sufficient
conditions for stability of the first-order necessary conditions in
optimization. For the convenience of the reader, this section is intended to
be absolutely self-contained and independent of other sections.

###  A.1 A first-order variant of Ekeland’s principle

The first subsection is of technical nature and is devoted to recalling a few
results of variational analysis that will be used later on. In particular, we
state a first-order variant of the seminal Ekeland’s variational principle. In
order to avoid unnecessary technical issues we work directly with convex
subsets $\mathcal{U}$ of a normed space $(U,\|\cdot\|_{U})$ as the domain of
functions, unless otherwise stated. Recall, that a real-valued function
$\mathcal{J}$ from a nonempty convex subset $\mathcal{U}$ of a normed space
$U$ is Gateaux differentiable a $u\in\mathcal{U}$ if there exists a linear
mapping $\mathcal{J}^{\prime}(u)\in U^{*}$ such that

$\displaystyle{\mathcal{J}}^{\prime}(\bar{u})v=\lim_{\varepsilon\to
0^{+}}\frac{{\mathcal{J}}(\bar{u}+\varepsilon
v)-{\mathcal{J}}(\bar{u})}{\varepsilon}\quad\forall v\in\mathcal{U}-u.$

Working with functions defined on convex domains has the advantage of simple
tangent and normal cone formulations, which in turn implies that the first-
order necessary condition also takes a simpler form.

######  Definition A.1.

Let $\bar{u}\in\mathcal{U}$, the set

$\displaystyle N_{\mathcal{U}}(\bar{u}):=\left\\{\rho\in U^{*}:\text{
}\rho(u-\bar{u})\leq 0\quad\text{for all $u\in\mathcal{U}$}\right\\}$ (61)

is called the normal cone to $\mathcal{U}$ at $\bar{u}$.

The first-order necessary condition is well-known for Gateaux differentiable
functions, and on it relies a lot of the work carried in optimization and
variational analysis; see [20, pp. 11-13].

######  Proposition A.2.

Let $\bar{u}\in\mathcal{U}$ be a local minimizer of ${\mathcal{J}}$. If
${\mathcal{J}}$ is Gateaux differentiable at $\bar{u}$, then
$0\in\mathcal{J}^{\prime}(\bar{u})+N_{\mathcal{U}}(\bar{u})$.

Second-order necessary conditions for optimality are also well known. See [5,
Lemma 3.44] and [29, Theorem 3.45] for more general second-order necessary
conditions.

We give now a technical lemma based on the celebrated Sion’s Minimax Theorem.

######  Lemma A.3.

Let $\psi:\mathcal{U}\to\mathbb{R}$ be a convex lower semicontinuous function.
Let $\hat{u}\in U$ and $\gamma>0$. There exists $\hat{\rho}\in U^{*}$ with
$\|\hat{\rho}\|_{U^{*}}\leq\gamma$ such that

$\displaystyle\inf_{u\in
M}\Big{\\{}\psi(u)+\gamma\|u-\hat{u}\|_{U}\Big{\\}}=\inf_{u\in\mathcal{U}}\left\\{\psi(u)-\hat{\rho}(u-\hat{u})\right\\}.$

###### Proof.

Let $\mathbb{B}^{*}$ be the unit ball of $U^{*}$. Define
$f:\mathcal{U}\times\mathbb{B}^{*}\to\mathbb{R}$ by
$f(u,\rho):=\psi(u)+\gamma\rho(u-\hat{u})$. Note that $U^{*}$ endowed with the
weak* topology is a linear topological space, and $\mathbb{B}^{*}$ is weak*
compact by Banach-Alaoglu Theorem. The function
$f(\cdot,\rho):\mathcal{U}\to\mathbb{R}$ is convex and lower semicontinuous
for each $\rho\in U^{*}$. The function
$f(u,\cdot):\mathbb{B}^{*}\to\mathbb{R}$ is weak* continuous and affine for
each $u\in\mathcal{U}$. The hypotheses of Sion’s Minimax Theorem ([30,
Corollary 3.3]) are then satisfied, and hence

$\displaystyle\inf_{u\in\mathcal{U}}\sup_{\rho\in\mathbb{B}^{*}}\left\\{\psi(u)+\gamma\rho(u-\hat{u})\right\\}=\sup_{\rho\in\mathbb{B}^{*}}\inf_{u\in\mathcal{U}}\\{\psi(u)+\gamma\rho(u-\hat{u})\\}.$
(62)

Let $h:\mathbb{B}^{*}\to\mathbb{R}$ be given by
$h(\rho):=\inf_{u\in\mathcal{U}}\\{\psi(u)+\gamma\rho(u-\hat{u})\\}$. Clearly,
$h$ is weak* upper semicontinuous as it is the infimum of weak* continuous
functions; and since $\mathbb{B}^{*}$ is weak* compact, there exists
$\rho^{*}\in\mathbb{B}^{*}$ such that
$\sup_{\rho\in\mathbb{B}^{*}}h(\rho)=h(\rho^{*})$. This implies

$\displaystyle\sup_{\rho\in\mathbb{B}^{*}}\inf_{u\in\mathcal{U}}\\{\psi(u)+\gamma\rho(u-\hat{u})\\}=\inf_{u\in\mathcal{U}}\\{\psi(u)+\gamma\rho^{*}(u-\hat{u})\\}$
(63)

Finally, by (62) and (63),

$\displaystyle\inf_{u\in\mathcal{U}}\Big{\\{}\psi(u)+\gamma\|u-\hat{u}\|_{U}\Big{\\}}$
$\displaystyle=\inf_{u\in\mathcal{U}}\sup_{\rho\in\mathbb{B}^{*}}\Big{\\{}\psi(u)+\gamma\rho(u-\hat{u})\Big{\\}}=\inf_{u\in\mathcal{U}}\Big{\\{}\psi(u)+\gamma\rho^{*}(u-\hat{u})\Big{\\}}.$

The results follow defining $\hat{\rho}:=-\gamma\rho^{*}$. ∎

We can now prove the following variant of Ekeland’s principle.

######  Lemma A.4.

Assume that $\mathcal{U}$ is a closed convex subset of $U$. Let
${\mathcal{J}}:\mathcal{U}\to\mathbb{R}$ be lower semicontinuous and Gateaux
differentiable. Let $\bar{u}\in U$ and $r>0$ such that

$\displaystyle{\mathcal{J}}(\bar{u})\leq{\mathcal{J}}(s)\quad\text{for all
$s\in\mathcal{U}$ with $\|s-\bar{u}\|_{U}\leq r$}.$

Let $u\in\mathcal{U}$ and $\varepsilon>0$ satisfy

$\displaystyle\|u-\bar{u}\|_{U}<r\quad\text{and}\quad{\mathcal{J}}(u)\leq{\mathcal{J}}(\bar{u})+\varepsilon.$

Then for every $\lambda\in\big{(}0,r-\|u-\bar{u}\|_{U}\big{)}$ there exist
$\hat{u}\in\mathcal{U}$ and $\hat{\rho}\in U^{*}$ such that

  * (i)

$\|u-\hat{u}\|_{U}\leq\lambda$;

  * (ii)

$\displaystyle\|\hat{\rho}\|_{U^{*}}\leq\frac{\varepsilon}{\lambda}$;

  * (iii)

$\hat{\rho}\in\mathcal{J}^{\prime}(\hat{u})+N_{\mathcal{U}}(\hat{u})$.

###### Proof.

Let $S:=\\{s\in\mathcal{U}:\|s-\bar{u}\|_{U}\leq r\\}$. Since $S$ is a closed
subset of $U$, $S$ is a complete metric space endowed with the metric induced
from the norm of $U$, and ${\mathcal{J}}|_{S}$ is lower semicontinuous. We can
apply Ekeland’s Principle ([22, Theorem 1.1]) to obtain that for every
$\lambda\in\big{(}0,r-\|u-\bar{u}\|_{U}\big{)}$ there exists $\hat{u}\in S$
such that

  * (a)

$\|u-\hat{u}\|_{U}\leq\displaystyle\lambda$;

  * (b)

${\mathcal{J}}(\hat{u})\leq{\mathcal{J}}(u)$;

  * (c)

$\hat{\mathcal{J}}(\hat{u})\leq\hat{\mathcal{J}}(s)$ for all $s\in S$, where
$\hat{\mathcal{J}}:\mathcal{U}\to\mathbb{R}$ is given by

$\displaystyle\hat{\mathcal{J}}(s):={\mathcal{J}}(s)+\frac{\varepsilon}{\lambda}\|s-\hat{u}\|_{U}.$

Let $r_{\lambda}:=r-\|u-\bar{u}\|_{U}-\lambda$; clearly $r_{\lambda}>0$. If
$s\in\mathcal{U}$ satisfies $\|s-\hat{u}\|_{U}\leq r_{\lambda}$, then

$\displaystyle\|s-\bar{u}\|_{U}\leq\|s-\hat{u}\|_{U}+\|\hat{u}-u\|_{U}+\|u-\bar{u}\|_{U}\leq
r_{\lambda}+\lambda+\|u-\bar{u}\|_{U}=r.$

Thus, from item $(c)$, $\hat{\mathcal{J}}(\hat{u})\leq\hat{\mathcal{J}}(s)$
for all $s\in\mathcal{U}$ with $\|s-\hat{u}\|_{U}\leq r_{\lambda}$. We
conclude that $\hat{u}$ is a local minimizer of $\hat{\mathcal{J}}$. From
this, we get

$\displaystyle 0\leq\liminf_{t\to 0^{+}}\hskip
1.42271pt\frac{\hat{\mathcal{J}}(\hat{u}+t(s-\hat{u}))-\hat{\mathcal{J}}(\hat{u})}{t}=\mathcal{J}^{\prime}(\hat{u})(s-\hat{u})+\frac{\varepsilon}{\lambda}\|s-\hat{u}\|_{U}\quad\forall
s\in\mathcal{U}.$

This can be rewritten as

$\displaystyle
0=\inf_{s\in\mathcal{U}}\Big{\\{}\mathcal{J}^{\prime}(\hat{u})(s-\hat{u})+\frac{\varepsilon}{\lambda}\|s-\hat{u}\|_{U}\Big{\\}}$

By Lemma A.3, there exists $\hat{\rho}\in U^{*}$ with
$\|\hat{\rho}\|_{U^{*}}\leq\varepsilon/\lambda$ such that

$\displaystyle
0=\inf_{s\in\mathcal{U}}\Big{\\{}\mathcal{J}^{\prime}(\hat{u})(s-\hat{u})+\frac{\varepsilon}{\lambda}\|s-\hat{u}\|_{U}\Big{\\}}=\inf_{s\in\mathcal{U}}\Big{\\{}\mathcal{J}^{\prime}(\hat{u})(s-\hat{u})-\hat{\rho}(s-\hat{u})\Big{\\}}\leq\mathcal{J}^{\prime}(\hat{u})(v-\hat{u})-\hat{\rho}(v-\hat{u})$

for all $v\in\mathcal{U}$. This implies
$\hat{\rho}\in\mathcal{J}^{\prime}(\hat{u})+N_{\mathcal{U}}(\hat{u})$.
Clearly, $\hat{u}$ and $\hat{\rho}$ satisfy items $(i)$-$(iii)$. ∎

###  A.2 Strong Hölder subregularity of the optimality mapping

This subsection is devoted to study the behavior of critical points under the
presence of perturbations. We derive necessary and sufficient conditions for
stability of the variational inequality describing the first-order necessary
condition at critical points. From this point on, we assume that
$\mathcal{J}:\mathcal{U}\to\mathbb{R}$ is Gateaux differentiable, unless we
specify otherwise.

Stability of the first-order necessary conditions. The following definition
gives a suitable notion of the correspondence between solutions of the
perturbed variational inequality and the perturbations.

######  Definition A.5.

The set-valued mapping $\Phi_{\mathcal{J}}:\mathcal{U}\twoheadrightarrow
U^{*}$ given by

$\displaystyle\Phi_{\mathcal{J}}(u):=\mathcal{J}^{\prime}(u)+N_{\mathcal{U}}(u)$

is called the optimality mapping.

We now give the definition of stability that we wish to analyze, i.e., the so-
called strong (metric) subregularty.

######  Definition A.6.

Let $\bar{u}\in\mathcal{U}$ satisfy $0\in\Phi_{\mathcal{J}}(\bar{u})$. We say
that the optimality mapping $\Phi_{\mathcal{J}}:\mathcal{U}\twoheadrightarrow
U^{*}$ is strongly (Hölder) subregular at $\bar{u}$ (with exponent
$\theta\in(0,\infty)$) if there exist positive numbers $\alpha$ and $\kappa$
such that the following property holds. For all $u\in\mathcal{U}$ and $\rho\in
U^{*}$,

$\displaystyle\|u-\bar{u}\|_{U}\leq\alpha\quad\text{and}\quad\rho\in\mathcal{J}^{\prime}(u)+N_{\mathcal{U}}(u)\quad\text{imply}\quad\|u-\bar{u}\|_{U}\leq\kappa\|\rho\|_{U^{*}}^{\theta}.$
(64)

We now proceed to state both sufficient and necessary conditions for this
notion of stability.

Sufficient conditions. The proof of the sufficient condition for stability
shown in the next theorem follows the arguments as in [19, Theorem 1] to the
letter, where it was previously proved in the context of optimal control.

######  Theorem A.7.

Let $\bar{u}\in\mathcal{U}$ such that $0\in\Phi_{\mathcal{J}}(\bar{u})$, and
$\mu\in(0,\infty)$. Suppose there exist positive numbers $\delta$ and $c$ such
that

$\displaystyle\mathcal{J}^{\prime}(u)(u-\bar{u})\geq
c\|u-\bar{u}\|_{U}^{\mu+1}\quad\text{for all $u\in\mathcal{U}$ with
$\|u-\bar{u}\|_{U}\leq\delta$.}$ (65)

Then the optimality mapping $\Phi_{\mathcal{J}}:\mathcal{U}\twoheadrightarrow
U^{*}$ is strongly Hölder subregular at $\bar{u}$ with exponent $1/\mu$.

###### Proof.

Let $u\in\mathcal{U}$ and $\rho\in U^{*}$ be arbitrary satisfying
$\|u-\bar{u}\|_{U}\leq\delta$ and $\rho\in\Phi_{\mathcal{J}}(u)$. Then, as
$\rho-\mathcal{J}^{\prime}(u)\in N_{\mathcal{U}}(u)$ and
$\bar{u}\in\mathcal{U}$, we have

$\displaystyle
0\geq\big{(}\rho-\mathcal{J}^{\prime}(u)\big{)}(\bar{u}-u)=\rho(\bar{u}-u)+\mathcal{J}^{\prime}(u)(u-\bar{u})\geq-\|\rho\|_{U^{*}}\|u-\bar{u}\|_{U}+c\|u-\bar{u}\|_{U}^{\mu+1}.$

Hence, $\|u-\bar{u}\|_{U}\leq c^{-1/\mu}\|\rho\|_{U^{*}}^{1/\mu}.$ The result
follows defining $\alpha:=\delta$ and $\kappa:=c^{-1/\mu}$. ∎

Growth assumption (65) appeared first in [19, Assumption 2] as a natural
hypothesis for an affine optimal control problem; see also [16, Proposition
4.3], where this kind of growth was proven for an elliptic optimal control
problem under a linearized growth hypothesis. A similar assumption of this
type appeared in [18, Assumption A2], where stability results for an affine
optimal control problem were studied. In Proposition A.12 below, we give
further details on growth (65) and its linearization.

Necessary conditions. In order to establish necessary conditions for stability
in the form of growth properties of functionals, we will use Ekeland’s
principle in the form of Lemma A.4, following the approach used in [3, 4]. In
those papers, the subregularity property of the subdifferential of convex
functions was characterized in terms of quadratic growth conditions; see also
[27], where a similar approach was used for the limiting subdifferential. In
all those three papers only Lipschitz stability and quadratic growth
conditions were considered. We make simple refinements in those arguments to
consider both Hölder stability and higher-order growth conditions.

In the next theorem, we argue similarly to the proof of [3, Theorem 3.3]; see
also the proofs of [4, Theorem 2.1] and [27, Theorem 3.1] for parallel
arguments.

######  Theorem A.8.

Let $(U,\|\cdot\|_{U})$ be a Banach space, $\mathcal{U}$ a closed convex
subset of $U$ and ${\mathcal{J}}:\mathcal{U}\to\mathbb{R}$ a lower
semicontinuous Gateaux differentiable function. Let $\bar{u}\in\mathcal{U}$ be
a local minimizer of ${\mathcal{J}}$ and $\mu\in(0,\infty)$. Suppose that the
optimality mapping $\Phi_{\mathcal{J}}:\mathcal{U}\twoheadrightarrow U^{*}$ is
strongly Hölder subregular at $\bar{u}$ with exponent $1/\mu$. Then there
exist positive numbers $\delta$ and $c$ such that

$\displaystyle{\mathcal{J}}(u)-{\mathcal{J}}(\bar{u})\geq
c\|u-\bar{u}\|_{U}^{\mu+1}\quad\text{for all $u\in\mathcal{U}$ with
$\|u-\bar{u}\|_{U}\leq\delta$.}$ (66)

###### Proof.

Let $\alpha$ and $\kappa$ be positive numbers such that property (64) holds.
Suppose that (66) does not hold. Then there would exist
$u\in\mathcal{U}\setminus\\{\bar{u}\\}$ satisfying
$\|u-\bar{u}\|_{U}<2\alpha/3$ such that

$\displaystyle{\mathcal{J}}(u)-{\mathcal{J}}(\bar{u})<\frac{1}{2^{2\mu+1}\kappa^{\mu}}\|u-\bar{u}\|_{U}^{\mu+1}.$
(67)

Let $\varepsilon:=2^{-(2\mu+1)}\kappa^{-\mu}\|u-\bar{u}\|_{U}^{\mu+1}$ and
$\lambda:=2^{-1}\|u-\bar{u}\|_{U}$. Note that

$\displaystyle\lambda=\frac{3}{2}\|u-\bar{u}\|_{U}-\|u-\bar{u}\|_{U}<\alpha-\|u-\bar{u}\|_{U}.$

From Lemma A.4, we conclude the existence of $\hat{u}\in\mathcal{U}$ and
$\hat{\rho}\in U^{*}$ such that

  * (i)

$\displaystyle\|u-\hat{u}\|_{U}\leq\frac{1}{2}\|u-\bar{u}\|_{U}$;

  * (ii)

$\displaystyle\|\hat{\rho}\|_{U^{*}}\leq\frac{1}{4^{\mu}\kappa^{\mu}}\|u-\bar{u}\|_{U}^{\mu}$;

  * (iii)

$\hat{\rho}\in\mathcal{J}^{\prime}(\hat{u})+N_{\mathcal{U}}(\hat{u})$.

Observe that
$\|\hat{u}-\bar{u}\|_{U}\leq\|\hat{u}-u\|_{U}+\|u-\bar{u}\|_{U}\leq
1/2\|u-\bar{u}\|_{U}+\|u-\bar{u}\|_{U}<\alpha$. By subregularity of the
optimality mapping at $\bar{u}$, we get

$\displaystyle\|\hat{u}-\bar{u}\|_{U}\leq\kappa\|\hat{\rho}\|_{U^{*}}^{1/\mu}\leq\frac{1}{4}\|u-\bar{u}\|_{U}.$
(68)

By item $(i)$, we have
$\|u-\bar{u}\|_{U}\leq\|u-\hat{u}\|_{U}+\|\hat{u}-\bar{u}\|_{U}\leq
1/2\|u-\bar{u}\|_{U}+\|\hat{u}-\bar{u}\|_{U}$. This implies
$\|u-\bar{u}\|_{U}\leq 2\|\hat{u}-\bar{u}\|_{U}$. Combining this with (68), we
get

$\displaystyle\|\hat{u}-\bar{u}\|_{U}\leq\frac{1}{4}\|u-\bar{u}\|_{U}\leq\frac{1}{2}\|\hat{u}-\bar{u}\|_{U},$

and hence $\hat{u}=u=\bar{u}$. A contradiction to (67). ∎

Growth condition (66) is well known in optimization. See, for example, [13,
Theorem 2.4] or [25, Theorem III] in affine optimal control; and [24, Theorem
1] or [26, Theorem I] in the quantitative study of eigenvalues stability for
the Schrödinger operator.

###  A.3 Hölder growth of real-valued functions

In this section, we study how to reduce growth conditions (65) and (66) to
linearized versions. This is to facilitate the understanding of their
feasibility. Recall that a Gateaux differentiable function
$\mathcal{J}:\mathcal{U}\to\mathbb{R}$ from a nonempty convex subset
$\mathcal{U}$ of a normed space $U$ is said to be twice Gateaux differentiable
a $u\in\mathcal{U}$ if there exists a bounded bilinear map
$\mathcal{J}^{\prime\prime}(u):U\times U\to\mathbb{R}$ such that

$\displaystyle{\mathcal{J}^{\prime\prime}}(\bar{u})(v,w)=\lim_{\varepsilon\to
0^{+}}\frac{{\mathcal{J}^{\prime}}(\bar{u}+\varepsilon
w)v-{\mathcal{J}^{\prime}}(\bar{u})v}{\varepsilon}\quad\forall
v,w\in\mathcal{U}-u.$

We will now assume, unless stated differently, that $\mathcal{J}$ is twice
Gateaux differentiable. A Hölder-type second order condition. In order to
transfer conditions (65) and (66) from being satisfied by a nonlinear function
to a second order polynomial, we will employ the following weakened version of
“twice continuously differentiable”.

######  Definition A.9.

Let $\bar{u}\in\mathcal{U}$ and $\mu\in[1,\infty)$. We say that
${\mathcal{J}}$ has changing curvature of order $\mu$ at $\bar{u}$ if for
every $\varepsilon>0$ there exists $\delta>0$ such that

$\displaystyle|\mathcal{J}^{\prime\prime}(\bar{u}+v)v^{2}-\mathcal{J}^{\prime\prime}(\bar{u})v^{2}|\leq\varepsilon\|v\|_{U}^{\mu+1}$
(69)

for all $v\in U$ with $\bar{u}+v\in\mathcal{U}$ and $\|v\|_{U}\leq\delta$.

As we shall utilize the assumption above for the remainder of this paper, we
shall assum from hereon that $\bar{u}\in\mathcal{U}$ and $\mu\in[1,+\infty)$.

Properties like (69) have appeared ubiquitously in the optimal control
literature of bang-bang controls. See, for example, [13, p. 4207], where it
appeared as a standard assumption in abstract optimal control; or [17, Lemma
11] where it appeared as natural property in the context of parabolic
problems.

######  Proposition A.10.

Suppose that ${\mathcal{J}}$ has changing curvature of order $\mu$ at
$\bar{u}$, then the following statements hold:

  * (i)

For all $\varepsilon>0$ there exists $\delta>0$ such that

$\displaystyle|{\mathcal{J}}(\bar{u}+v)-{\mathcal{J}}(\bar{u})-\mathcal{J}^{\prime}(\bar{u})v-\frac{1}{2}\mathcal{J}^{\prime\prime}(\bar{u})v^{2}|\leq\varepsilon\|v\|_{U}^{\mu+1}$

for all $v\in U$ satisfying $\bar{u}+v\in\mathcal{U}$ and
$\|v\|_{U}\leq\delta$.

  * (ii)

For all $\varepsilon>0$ there exists $\delta>0$ such that

$\displaystyle|\mathcal{J}^{\prime}(\bar{u}+v)v-\mathcal{J}^{\prime}(\bar{u})v-\mathcal{J}^{\prime\prime}(\bar{u})v^{2}|\leq\varepsilon\|v\|_{U}^{\mu+1}$

for all $v\in U$ satisfying $\bar{u}+v\in\mathcal{U}$ and
$\|v\|_{U}\leq\delta$.

###### Proof.

For each $v\in U$ with $\bar{u}+v\in\mathcal{U}$, define
$h_{v}:[0,1]\to\mathbb{R}$ by $h_{v}(t)={\mathcal{J}}(\bar{u}+tv)$. We can
apply Taylor’s Theorem to conclude that for each $v\in\mathcal{U}-\bar{u}$
there exists $t_{v}\in[0,1]$ such that
$h(1)-h(0)=h^{\prime}(0)+h^{\prime\prime}(t_{v})/2$. That is,

$\displaystyle{\mathcal{J}}(\bar{u}+v)-{\mathcal{J}}(\bar{u})=\mathcal{J}^{\prime}(\bar{u})v+\frac{1}{2}\mathcal{J}^{\prime\prime}(\bar{u}+t_{v}v)v^{2}\quad\forall
v\in\mathcal{U}-\bar{u}.$ (70)

Let $\varepsilon>0$ be given. By definition of changing curvature of order
$\mu$ at a point, we can find $\delta>0$ such that

$\displaystyle|\mathcal{J}^{\prime\prime}(\bar{u}+t_{v}v)(t_{v}v)^{2}-\mathcal{J}^{\prime\prime}(\bar{u})(t_{v}v)^{2}|\leq
2\,\varepsilon\|t_{v}v\|_{U}^{\mu+1}$ (71)

whenever $v\in\mathcal{U}-\bar{u}$ satisfies $\|v\|_{U}\leq\delta$. Then,
combining (70) and (71), we get

$\displaystyle
t_{v}^{2}|{\mathcal{J}}(\bar{u}+v)-{\mathcal{J}}(\bar{u})-\mathcal{J}^{\prime}(\bar{u})v-\frac{1}{2}\mathcal{J}^{\prime\prime}(\bar{u})v^{2}|$
$\displaystyle=\frac{1}{2}|\mathcal{J}^{\prime\prime}(\bar{u}+t_{v}v)v^{2}-\mathcal{J}^{\prime\prime}(\bar{u})v^{2}|\leq
t_{v}^{\mu+1}\varepsilon\|v\|_{U}^{\mu+1}$

for all $v\in U$ satisfying $\bar{u}+v\in\mathcal{U}$ and
$\|v\|_{U}\leq\delta$. Since $\mu\geq 1$, it follows that

$\displaystyle|{\mathcal{J}}(\bar{u}+v)-{\mathcal{J}}(\bar{u})-\mathcal{J}^{\prime}(\bar{u})v-\frac{1}{2}\mathcal{J}^{\prime\prime}(\bar{u})v^{2}|\leq
t_{v}^{\mu-1}\varepsilon\|v\|_{U}^{\mu+1}\leq\varepsilon\|v\|_{U}^{\mu+1}$

for all $v\in U$ satisfying $\bar{u}+v\in\mathcal{U}$ and
$\|v\|_{U}\leq\delta$. Thus, item $(i)$ holds.

The proof of item $(ii)$ is analogous; it follows defining
$k_{v}:[0,1]\to\mathbb{R}$ given by
$k_{v}(t):=\mathcal{J}^{\prime}(\bar{u}+tv)v$ for each
$v\in\mathcal{U}-\bar{u}$, and applying the Mean Value Theorem to each
function $k_{v}$. ∎

Growth of functionals and their differentials. One easy consequence of
Proposition A.10 is the following characterization of growth condition (66)
which follows directly from item $(i)$ of Proposition A.10.

######  Proposition A.11.

Suppose that ${\mathcal{J}}$ has changing curvature of order $\mu$ at
$\bar{u}$, then the following statements are equivalent:

  * (i)

There exist positive numbers $\alpha$ and $c$ such that

$\displaystyle{\mathcal{J}}(u)-{\mathcal{J}}(\bar{u})\geq
c\|u-\bar{u}\|_{U}^{\mu+1}\quad\text{for all $u\in\mathcal{U}$ with
$\|u-\bar{u}\|_{U}\leq\alpha$.}$

  * (ii)

There exist positive numbers $\alpha$ and $c$ such that

$\displaystyle\mathcal{J}^{\prime}(\bar{u})(u-\bar{u})+\frac{1}{2}\mathcal{J}^{\prime\prime}(\bar{u})(u-\bar{u})^{2}\geq
c\|u-\bar{u}\|_{U}^{\mu+1}\quad\text{for all $u\in\mathcal{U}$ with
$\|u-\bar{u}\|_{U}\leq\alpha$.}$

On the other hand, a trivial, but important, consequence of Proposition A.10
item $(ii)$ is the characterization of growth condition (65).

######  Proposition A.12.

Let $(U,\|\cdot\|_{U})$ be a normed space, $\mathcal{U}$ a convex subset of
$U$ and ${\mathcal{J}}:\mathcal{U}\to\mathbb{R}$ a twice Gâteaux
differentiable function. Let $\bar{u}\in\mathcal{U}$ and $\mu\in[1,\infty)$.
If ${\mathcal{J}}$ has changing curvature of order $\mu$ at $\bar{u}$, then
the following statements are equivalent.

  * (i)

There exist positive numbers $\alpha$ and $c$ such that

$\displaystyle\mathcal{J}^{\prime}(u)(u-\bar{u})\geq
c\|u-\bar{u}\|_{U}^{\mu+1}\quad\text{for all $u\in\mathcal{U}$ with
$\|u-\bar{u}\|_{U}\leq\alpha$.}$

  * (ii)

There exist positive numbers $\alpha$ and $c$ such that

$\displaystyle\mathcal{J}^{\prime}(\bar{u})v+\mathcal{J}^{\prime\prime}(\bar{u})v^{2}\geq
c\|u-\bar{u}\|_{U}^{\mu+1}\quad\text{for all $u\in\mathcal{U}$ with
$\|u-\bar{u}\|_{U}\leq\alpha$.}$

We mention that this characterization has appeared before in PDE-constrained
optimization; see [16, Prosposition 4.1] or [17, Lemma 12].

## References

  * [1] Robert A. Adams and John J.F. Fournier.  Sobolev Spaces, volume 140.  Elsevier, Langford Lane, Kidlington, 2003.
  * [2] Herbert Amann.  Linear parabolic problems involving measures.  Real Academia de Ciencias Exactas, Fisicas y Naturales. Revista. Serie A, Matematicas, 95(1):85–119, 2001.
  * [3] Francisco J. Aragón Artacho and Michel H. Geoffroy.  Characterization of metric regularity of subdifferentials.  J. Convex Anal., 15(2):365–380, 2008.
  * [4] Francisco J. Aragón Artacho and Michel H. Geoffroy.  Metric subregularity of the convex subdifferential in Banach spaces.  J. Nonlinear Convex Anal., 15(1):35–47, 2014.
  * [5] J. Frédéric Bonnans and Alexander Shapiro.  Perturbation analysis of optimization problems.  Springer Series in Operations Research. Springer-Verlag, New York, 2000\.
  * [6] Eduardo Casas.  Second order analysis for bang-bang control problems of pdes.  SIAM Journal on Control and Optimization, 50(4):2355–2372, 2023/05/25 2012.
  * [7] Eduardo Casas and Konstantinos Chrysafinos.  A discontinuous Galerkin time-stepping scheme for the velocity tracking problem.  SIAM J. Numer. Anal., 50(5):2281–2306, 2012.
  * [8] Eduardo Casas and Konstantinos Chrysafinos.  Error estimates for the discretization of the velocity tracking problem.  Numer. Math., 130(4):615–643, 2015.
  * [9] Eduardo Casas and Konstantinos Chrysafinos.  Error estimates for the approximation of the velocity tracking problem with bang-bang controls.  ESAIM Control Optim. Calc. Var., 23(4):1267–1291, 2017.
  * [10] Eduardo Casas and Konstantinos Chrysafinos.  Error estimates for the approximation of the velocity tracking problem with bang-bang controls.  ESAIM Control Optim. Calc. Var., 23(4):1267–1291, 2017.
  * [11] Eduardo Casas, Alberto Domínguez Corella, and Nicolai Jork.  New assumptions for stability analysis in elliptic optimal control problems, 2022.
  * [12] Eduardo Casas and Fredi Tröltzsch.  Stability for semilinear parabolic optimal control problems with respect to initial data.  Appl. Math. Optim., 86(2):Paper No. 16, 31, 2022.
  * [13] Eduardo Casas, Daniel Wachsmuth, and Gerd Wachsmuth.  Second-order analysis and numerical approximation for bang-bang bilinear control problems.  SIAM J. Control Optim., 56(6):4203–4227, 2018.
  * [14] R. Cibulka, A. L. Dontchev, and A. Y. Kruger.  Strong metric subregularity of mappings in variational analysis and optimization.  J. Math. Anal. Appl., 457(2):1247–1282, 2018.
  * [15] Klaus Deckelnick and Michael Hinze.  A note on the approximation of elliptic control problems with bang-bang controls.  Comput. Optim. Appl., 51(2):931–939, 2012.
  * [16] Alberto Domínguez Corella, Nicolai Jork, and Vladimir Veliov.  Stability in Affine Optimal Control Problems Constrained by Semilinear Elliptic Partial Differential Equations.  ESAIM Control Optim. Calc. Var., 28:Paper No. 79, 30, 2022.
  * [17] Alberto Domínguez Corella, Nicolai Jork, and Vladimir M. Veliov.  On the solution stability of parabolic optimal control problems.  Computational Optimization and Applications, 2023.
  * [18] Alberto Domínguez Corella, Marc Quincampoix, and Vladimir M. Veliov.  Strong bi-metric regularity in affine optimal control problems.  Pure Appl. Funct. Anal., 6(6):1119–1137, 2021.
  * [19] Alberto Domínguez Corella and Vladimir M. Veliov.  Hölder regularity in bang-bang type affine optimal control problems.  In Large-scale scientific computing, volume 13127 of Lecture Notes in Comput. Sci., pages 306–313. Springer, Cham, 2022.
  * [20] Asen L. Dontchev.  Lectures on variational analysis, volume 205 of Applied Mathematical Sciences.  Springer, Cham, [2021] ©2021.
  * [21] Asen L. Dontchev and R. Tyrrell Rockafellar.  Implicit functions and solution mappings.  Springer Series in Operations Research and Financial Engineering. Springer, New York, second edition, 2014.  A view from variational analysis.
  * [22] I. Ekeland.  On the variational principle.  J. Math. Anal. Appl., 47:324–353, 1974.
  * [23] M. D. Gunzburger and S. Manservisi.  The velocity tracking problem for Navier-Stokes flows with bounded distributed controls.  SIAM J. Control Optim., 37(6):1913–1945, 1999.
  * [24] Idriss Mazari.  Quantitative inequality for the eigenvalue of a Schrödinger operator in the ball.  J. Differential Equations, 269(11):10181–10238, 2020.
  * [25] Idriss Mazari.  Quantitative estimates for parabolic optimal control problems under $L^{\infty}$ and $L^{1}$ constraints in the ball: quantifying parabolic isoperimetric inequalities.  Nonlinear Anal., 215:Paper No. 112649, 48, 2022.
  * [26] Idriss Mazari and Domènec Ruiz-Balet.  Quantitative stability for eigenvalues of Schrödinger operator, quantitative bathtub principle, and application to the turnpike property for a bilinear optimal control problem.  SIAM J. Math. Anal., 54(3):3848–3883, 2022.
  * [27] B. S. Mordukhovich and T. T. A. Nghia.  Second-order variational analysis and characterizations of tilt-stable optimal solutions in infinite-dimensional spaces.  Nonlinear Anal., 86:159–180, 2013.
  * [28] M. Quincampoix and V. M. Veliov.  Metric regularity and stability of optimal control problems for linear systems.  SIAM J. Control Optim., 51(5):4118–4137, 2013.
  * [29] Andrzej Ruszczyński.  Nonlinear optimization.  Princeton University Press, Princeton, NJ, 2006.
  * [30] Maurice Sion.  On general minimax theorems.  Pacific J. Math., 8:171–176, 1958.
  * [31] Roger Temam.  Navier-Stokes equations. Theory and numerical analysis.  Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
  * [32] Fredi Tröltzsch and Daniel Wachsmuth.  Second-order sufficient optimality conditions for the optimal control of Navier-Stokes equations.  ESAIM Control Optim. Calc. Var., 12(1):93–119, 2006.
  * [33] Nikolaus von Daniels and Michael Hinze.  Variational discretization of a control-constrained parabolic bang-bang optimal control problem.  J. Comput. Math., 38(1):14–40, 2020.

Alberto Domínguez Corella

Friedrich-Alexander-Universität Erlangen-Nürnberg

Department of Data Science, Chair for Dynamics, Control and Numerics
(Alexander von Humboldt-Professorship)

Erlangen, Germany

E-mail<EMAIL_ADDRESS>
Nicolai Jork

Institute of Statistics and Mathematical Methods in Economics

Vienna University of Technology, Austria

E-mail<EMAIL_ADDRESS>
Šarka Nečasovà

Institute of Mathematics of the Academy of Sciences of the Czech Republic

Žitna 25, 115 67 Praha 1, Czech Republic

E-mail<EMAIL_ADDRESS>
John Sebastian H. Simon

Institute of Mathematics of the Academy of Sciences of the Czech Republic

Žitna 25, 115 67 Praha 1, Czech Republic

E-mail<EMAIL_ADDRESS>