# Approximate control of parabolic equations with on-off shape controls by
Fenchel duality

Camille Pouchol  Laboratoire MAP5 UMR 8145, Université Paris Cité, 75006
Paris, France. Email address<EMAIL_ADDRESS>Emmanuel Trélat
Sorbonne Université, Université de Paris, CNRS, Laboratoire Jacques-Louis
Lions, 75005 Paris, France. Email address: emmanuel.trelat@sorbonne-
universite.fr  CAGE, INRIA, Paris, France.  Christophe Zhang  SPHINX, INRIA,
Faculté des Sciences et Technologies Campus, Boulevard des Aiguillettes 54506
Vandœuvre-lès-Nancy, France. Email address<EMAIL_ADDRESS>
###### Abstract

We consider the internal control of linear parabolic equations through on-off
shape controls, i.e., controls of the form $\displaystyle
M(t)\chi_{\omega(t)}$ with $\displaystyle M(t)\geq 0$ and
$\displaystyle\omega(t)$ with a prescribed maximal measure.

We establish small-time approximate controllability towards all possible final
states allowed by the comparison principle with nonnegative controls. We
manage to build controls with constant amplitude $\displaystyle
M(t)\equiv\overline{M}$. In contrast, if the moving control set
$\displaystyle\omega(t)$ is confined to evolve in some region of the whole
domain, we prove that approximate controllability fails to hold for small
times.

The method of proof is constructive. Using Fenchel-Rockafellar duality and the
bathtub principle, the on-off shape control is obtained as the bang-bang
solution of an optimal control problem, which we design by relaxing the
constraints.

Our optimal control approach is outlined in a rather general form for linear
constrained control problems, paving the way for generalisations and
applications to other PDEs and constraints.

###### Contents

  1. 1 Introduction
    1. 1.1 Constrained internal control
    2. 1.2 Main results
    3. 1.3 General results
    4. 1.4 Proof strategy and related works
    5. 1.5 Extensions and perspectives
  2. 2 Building the optimal control problem
    1. 2.1 Convex analytic framework
    2. 2.2 Approximate controllability by Fenchel duality ([24])
    3. 2.3 Convex analytic interpretation of the bathtub principle
    4. 2.4 From the static bathtub principle to the dual problem and its corresponding cost
  3. 3 Approximate controllability results
    1. 3.1 Strong duality
    2. 3.2 Coercivity of $\displaystyle J_{T,\varepsilon}$, nonnegative approximate controllability
    3. 3.3 Characterisation of the minimisers
    4. 3.4 Uniqueness
  4. 4 Obstructions to controllability
  5. 5 Further comments
    1. 5.1 Properties of the value function in the general case
    2. 5.2 Obstructions
      1. 5.2.1 Obstruction to reachability and small-time controllability
      2. 5.2.2 Characterisation of minimal time controls
  6. A Convex analysis
    1. A.1 Core properties of Fenchel conjugation
    2. A.2 Some properties of indicator and support functions
    3. A.3 Technical lemmas
    4. A.4 Fenchel-Rockafellar duality
    5. A.5 Parametric convex optimisation
  7. B The classical bathtub principle

##  1 Introduction

###  1.1 Constrained internal control

This article is devoted to the internal approximate controllability problem at
time $\displaystyle T>0$ for linear parabolic equations on a domain
$\displaystyle\Omega$ by means of on-off shape controls, i.e., internal
controls taking the form

$\displaystyle\forall t\in(0,T),\,\forall x\in\Omega,~\quad
u(t,x)=M(t)\chi_{\omega(t)}(x),$

where, at a given time $\displaystyle t\in(0,T)$,

  * •

$\displaystyle M(t)>0$ is the nonnegative amplitude of the control

  * •

$\displaystyle\chi_{\omega(t)}$ is the characteristic function of the set
$\displaystyle\omega(t)\subset\Omega$, i.e.,
$\displaystyle\chi_{\omega(t)}(x):=\begin{cases}1&\text{ if }x\in\omega(t),\\\
0&\text{ otherwise}\end{cases}.$

Both the amplitude and location may be subject to constraints. This problem is
a paradigmatic simplification of many practical situations where one can act
on a complex system with on-off devices that can be moved in time, while their
shape can also be modified.

Along the introduction, we expose our results for general operators
$\displaystyle A$, while first illustrating them in the case of the controlled
linear heat equation with Dirichlet boundary conditions

$\left\\{\begin{aligned} y_{t}-\Delta y&=u\ \textrm{in}\ \Omega,\\\ y&=0\
\textrm{on}\ \partial\Omega,\\\ y(0)&=y_{0}\ \textrm{in}\
\Omega.\end{aligned}\right.$ (1)

In this setting, $\displaystyle\Omega$ is an open connected bounded subset of
$\displaystyle\mathbb{R}^{d}$, with $\displaystyle C^{2}$ boundary, and
$\displaystyle y_{0}\in L^{2}(\Omega)$.

##### Control without constraints.

When constraints are removed, generic parabolic equations are well-known to be
approximately controllable [2, 42], and even null-controllable [13, 22] in
arbitrarily small time by means of internal controls, acting only on an
arbitrary fixed measurable subset $\displaystyle\omega\subset\Omega$ of
positive measure.

This more precisely means that for any time $\displaystyle T>0$, any
measurable set $\displaystyle\omega\subset\Omega$ of positive measure, any
$\displaystyle\varepsilon>0$, any $\displaystyle y_{0}\in L^{2}(\Omega)$ and
target $\displaystyle y_{f}\in L^{2}(\Omega)$, there holds

$\displaystyle\exists u\in L^{2}((0,T)\times\Omega),\,\text{ such that
}\;\forall t\in(0,T),\;\mathrm{supp}(u(t,\cdot))\subset\omega\text{ and
}\|y(T)-y_{f}\|_{L^{2}(\Omega)}\leq\varepsilon,$

where $\displaystyle\mathrm{supp}(u)$ refers to the essential support of a
function $\displaystyle u\in L^{2}(\Omega)$.

##### Constrained control.

In view of applications where unilateral or bilateral or $\displaystyle
L^{\infty}$ constraints naturally appear, constrained controllability has been
an active area of research [8, 1, 34], whether in finite or infinite
dimension.

In various contexts, control constraints have been shown to lead to
controllability obstructions, even for unilateral constraints. Some states are
out of reach, regardless of how large $\displaystyle T>0$ may be [37, 41]. On
the other hand, some states are reachable but only for $\displaystyle T$ large
enough: constraints may lead to the appearance of a minimal time of
controllability [25, 36, 26, 27].

In the case of unilateral constraints for linear control problems in finite
dimension, these obstructions can be categorised thanks to Brunovsky’s normal
form as done in [26], leading to the existence of a positive minimal time. In
infinite dimension, however, we are only aware of obstructions based on the
comparison principle (see [25] and [37]). The present work uncovers another
type of obstruction, already hinted at in [36].

###  1.2 Main results

As our results require different sets of hypotheses and in order to give a
quick glance at the main ideas, we first present them in the simplified
context of the heat equation (1).

Given a constraint set $\displaystyle\mathcal{U}_{+}\subset L^{2}(\Omega)$, we
will be considering control constraints of the form

$\displaystyle\forall t\in(0,T),\quad u(t)\in\mathcal{U}_{+}.$

Here, the notation $\displaystyle\mathcal{U}_{+}$ emphasises that we will
always deal with constraints that include the nonnegativity constraint, i.e.,
sets $\displaystyle\mathcal{U}_{+}$ such that
$\displaystyle\mathcal{U}_{+}\subset\\{u\in L^{2}(\Omega),\,u\geq 0\\}$.

Now, when the control $\displaystyle u$ satisfies $\displaystyle u\geq 0$, if
follows from _the parabolic comparison principle_ satisfied by the Dirichlet
Laplacian [16] that

$\forall t\geq 0,\quad y(t)\geq e^{t\Delta}y_{0},$ (2)

where $\displaystyle(e^{t\Delta})_{t\geq 0}$ denotes the heat semigroup with
Dirichlet boundary conditions. Hence, targets $\displaystyle y_{f}$ which do
not satisfy $\displaystyle y_{f}\geq e^{T\Delta}y_{0}$ cannot be reached with
nonnegative controls, let alone on-off shape controls.

Taking into account the obstruction to controllability given by the inequality
(2), we adapt the usual definition of approximate controllability to the
context of nonnegative controls.

More precisely, we say that system (1) is nonnegatively approximately
controllable with controls in $\displaystyle\mathcal{U}_{+}$ in time
$\displaystyle T>0$, if for all $\displaystyle\varepsilon>0$, and all
$\displaystyle y_{0},y_{f}\in L^{2}(\Omega)$ such that $\displaystyle
y_{f}\geq e^{T\Delta}y_{0}$, there exists a control $\displaystyle u\in
L^{2}((0,T)\times\Omega)$ with values in $\displaystyle\mathcal{U}_{+}$ such
that the corresponding solution to (1) satisfies
$\displaystyle\|y(T)-y_{f}\|_{L^{2}(\Omega)}\leq\varepsilon$.

##### On-off shape control.

For our first main result, we focus on nonnegative approximate controllability
with on-off shape controls: for a fixed $\displaystyle L\in(0,1)$, we consider
the constraint set

$\displaystyle\mathcal{U}_{L}^{\mathrm{shape}}:=\\{M\chi_{\omega},\quad\omega\subset\Omega,\quad|\omega|\leq
L|\Omega|,\,M>0\\}\subset L^{2}(\Omega).$

Within the above class of on-off shape controls, we establish nonnegative
approximate controllability in arbitrary time (see Theorem 3.1 for the precise
and general statement), whatever the value of $\displaystyle L\in(0,1)$.

######  Theorem A.

For any $\displaystyle L\in(0,1)$, $\displaystyle T>0$, system (1) is
nonnegatively approximately controllable with controls in
$\displaystyle\mathcal{U}_{\mathrm{shape}}^{L}$ in time $\displaystyle T$.

To establish this result, we draw from Lions’s strategy in [24], which
develops a constructive approach in studying the approximate controllability
of a linear wave equation. The idea is to consider the requirement
$\displaystyle\|y(T)-y_{f}\|_{L^{2}(\Omega)}\leq\varepsilon$ as a constraint.
With $\displaystyle L_{T}u:=\textstyle\int_{0}^{T}e^{(T-t)\Delta}u(t)\,dt$ and
since $\displaystyle y(T)=L_{T}u+S_{T}y_{0}$, Lions considers the constrained
optimal control problem

$\displaystyle\pi:=\inf\left\\{\frac{1}{2}\|u\|_{L^{2}((0,T)\times\Omega)}^{2},\
\|e^{T\Delta}y_{0}+L_{T}u-y_{f}\|_{L^{2}(\Omega)}\leq\varepsilon\right\\}.$

The infimum satisfies $\displaystyle\pi<+\infty$ if and only if there exists
$\displaystyle u\in L^{2}((0,T)\times\Omega)$ steering $\displaystyle y_{0}$
to a closed $\displaystyle\varepsilon$-ball around $\displaystyle y_{f}$. To
find minimisers, i.e., to build controls, note that

$\displaystyle\pi=\inf_{u\in
L^{2}((0,T)\times\Omega)}\textstyle\frac{1}{2}\|u\|_{L^{2}((0,T)\times\Omega)}^{2}+G_{T,\varepsilon}(L_{T}u)=\displaystyle\inf_{u\in
L^{2}((0,T)\times\Omega)}F_{T}(u)+G_{T,\varepsilon}(L_{T}u),$

with $\displaystyle
F_{T}(u)=\textstyle\frac{1}{2}\|u\|_{L^{2}((0,T)\times\Omega)}^{2}$ and

$\displaystyle G_{T,\varepsilon}(y)=\left\\{\begin{aligned}
0\quad&\text{if}\quad\|e^{T\Delta}y_{0}+y-y_{f}\|_{L^{2}(\Omega)}\leq\varepsilon,\\\
+\infty\quad&\textrm{otherwise}.\end{aligned}\right.$

From this optimisation problem, one computes its Fenchel dual optimisation
problem, which reads

$\displaystyle d:=-\inf_{p_{f}\in
L^{2}(\Omega)}F_{T}^{\ast}(L_{T}^{\ast}p_{f})+G_{T,\varepsilon}^{\ast}(-p_{f})=-\inf_{p_{f}\in
L^{2}(\Omega)}\textstyle\frac{1}{2}\|L_{T}^{\ast}p_{f}\|_{L^{2}((0,T)\times\Omega)}^{2}+G_{T,\varepsilon}^{\ast}(-p_{f}),$

where $\displaystyle F_{T}^{\ast}$($\displaystyle=F_{T}$) and $\displaystyle
G_{T,\varepsilon}^{\ast}$ are the Fenchel conjugates of $\displaystyle F_{T}$
and $\displaystyle G_{T,\varepsilon}$, respectively, and $\displaystyle
L_{T}^{\ast}$ is the adjoint of the linear bounded operator $\displaystyle
L_{T}:L^{2}((0,T)\times\Omega)\rightarrow L^{2}(\Omega)$. Recall that for a
given $\displaystyle p_{f}\in L^{2}(\Omega)$, $\displaystyle
p=L_{T}^{\ast}p_{f}$ is the solution to the adjoint equation ending at
$\displaystyle p_{f}$, i.e., it solves

$\left\\{\begin{aligned} p_{t}+\Delta p&=0,\\\ p=0\ &\textrm{on}\
\partial\Omega,\\\ p(T)=p_{f}\ &\textrm{on}\ \Omega.\end{aligned}\right.$ (3)

Under suitable hypotheses, the Fenchel-Rockafellar theorem [40] ensures that
$\displaystyle\pi=d$. As a result, one can then study the dual functional to
establish that $\displaystyle\pi=d<+\infty$, and that its minimum is attained.
Furthermore, the cost function $\displaystyle F_{T}$ is differentiable in this
case and the first order optimality condition for the (unique) variable
$\displaystyle p_{f}^{\star}$ minimising the dual functional then reads
$\displaystyle
L_{T}L_{T}^{\ast}p_{f}^{\star}=y_{f}-\varepsilon\textstyle\frac{p_{f}^{\star}}{\|p_{f}^{\star}\|_{L^{2}(\Omega)}}$.
The optimal control $\displaystyle u^{\star}:=L_{T}^{\ast}p_{f}^{\star}$ is
thus constructed from the minimiser of the dual problem $\displaystyle
p_{f}^{\star}$.

Accordingly, in this paper we reframe constrained approximate controllability
as an optimal control problem, replacing
$\displaystyle\textstyle\frac{1}{2}\|u\|_{L^{2}((0,T)\times\Omega)}^{2}$ of
[24] with a suitable cost functional $\displaystyle F_{T}$. This constitutes a
novel generalisation of Lions’s method.

As the detailed statements in Theorem 3.1 and Proposition 3.7 show:

  * •

Instead of using the $\displaystyle L^{2}$ norm as in the optimal control
problem studied in [24], we will consider the cost functional:

$F_{T}(u):=\frac{1}{2}\sup_{t\in[0,T]}\max\left(\|u(t,\cdot)\|_{L^{\infty}(\Omega)},\frac{\|u(t,\cdot)\|_{L^{1}(\Omega)}}{L|\Omega|}\right)^{2}+\delta_{\\{u\geq
0\\}}(u),$ (4)

where $\displaystyle\delta_{\\{u\geq 0\\}}(u)=0$ if $\displaystyle u\geq 0$
and $\displaystyle+\infty$ otherwise. The rather unusual form of the
minimisation criterion (4) is finely designed so as to handle nonnegativity
and the other (bound, volume) constraints we are dealing with.

  * •

The optimal controls have constant amplitude in time, i.e., $\displaystyle
M(t)\equiv\overline{M}$.

  * •

The proof is constructive: the optimal control $\displaystyle u^{\star}$ can
be computed from a unique dual optimal variable $\displaystyle p_{f}^{\star}$
solving the corresponding Fenchel dual problem. This computation generalises
what is done in [24] to the broader case of costs that are not differentiable
but still convex. More precisely, $\displaystyle u^{\star}$ is given by

$u^{\star}(t,\cdot)=\overline{M}\,\chi_{\\{p^{\star}(t,\cdot)>h(p^{\star}(t,\cdot))\\}},\quad\overline{M}=\int_{0}^{T}\int_{\\{p^{\star}(t,\cdot)>h(p^{\star}(t,\cdot))\\}}p^{\star}(t,x)\,dx\,dt,$

where $\displaystyle h:L^{2}(\Omega)\rightarrow\mathbb{R}$ is a function that
will be defined in Section 2.3, and $\displaystyle p^{\star}$ solves the
adjoint equation (3) with $\displaystyle p^{\star}(T)=p_{f}^{\star}$.

##### Obstructions to nonnegative controllability.

In the spirit of the unconstrained case, one may wonder whether nonnegative
approximate controllability can be achieved with controls acting only in some
prescribed time-independent subdomain $\displaystyle\omega$. We emphasise that
our first result does not a priori prevent the control from visiting the whole
domain $\displaystyle\Omega$.

Our second result proves that visiting the whole $\displaystyle\Omega$ is
necessary in the following sense: if the sets
$\displaystyle\omega(t),\,t\in(0,T)$ do not intersect some fixed open subset
of $\displaystyle\Omega$, nonnegative approximate controllability is lost for
small times.

######  Theorem B.

Assume that the constraint set $\displaystyle\mathcal{U}_{+}$ satisfies the
following property: there exists a ball $\displaystyle B(x,r)\subset\Omega$
with $\displaystyle x\in\Omega$ and $\displaystyle r>0$ such that

$\displaystyle\forall u\in\mathcal{U}_{+},\quad\mathrm{supp}(u)\cap
B(x,r)=\emptyset.$

Then, there exists $\displaystyle T^{\star}>0$ such that the control system
(1) is not nonnegatively approximately controllable with controls in
$\displaystyle\mathcal{U}_{+}$ in time $\displaystyle T\leq T^{\star}$.

We refer to Theorem 4.1 for the complete statement. Let us mention that
obstructions of this type have been reported for similar problems in [36].

##### Amplitude and time optimal control.

In Section 5, we gather several further results regarding the dependence of
the amplitude
$\displaystyle\overline{M}=\overline{M}(T,y_{0},y_{f},\varepsilon)$ with
respect to its arguments. Using duality once more, we study its dependence on
the final time $\displaystyle T$.

Focusing on the case $\displaystyle y_{0}=0$, we then establish an equivalence
between the optimal control problem and the related minimal time problem

$\inf\\{T>0,\quad\exists u\in
L^{2}((0,T)\times\Omega),\quad\|L_{T}u-y_{f}\|_{L^{2}(\Omega)}\leq\varepsilon,\quad
F_{T}(u)\leq\lambda\\},\quad\lambda>0.$

###  1.3 General results

Theorems A and B above have been stated for the heat equation with Dirichlet
boundary conditions, in order to provide the reader with a quick overview of
our main results. In fact, they all hold for more general semigroups under
suitable hypotheses presented hereafter.

The underlying general setting is that of linear control problems of the form

$\left\\{\begin{aligned} y_{t}-Ay&=u,\\\ y(0)&=y_{0}\ \textrm{in}\
\Omega\end{aligned}\right.$ (5)

where $\displaystyle\Omega$ is an open subset of
$\displaystyle\mathbb{R}^{d}$, and $\displaystyle A:D(A)\rightarrow
L^{2}(\Omega)$ is an operator generating a $\displaystyle C_{0}$ semigroup
$\displaystyle(S_{t})_{t\geq 0}$ on $\displaystyle L^{2}(\Omega)$ [14, 35].

In this more general context, we define nonnegative approximate
controllability as follows.

######  Definition 1.1.

Given a constraint set of nonnegative controls
$\displaystyle\mathcal{U}_{+}\subset L^{2}(\Omega)$, we say that system (5) is
nonnegatively approximately controllable with controls in
$\displaystyle\mathcal{U}_{+}$ in time $\displaystyle T$ if for all
$\displaystyle\varepsilon>0$, and all $\displaystyle y_{0},y_{f}\in
L^{2}(\Omega)$ such that $\displaystyle y_{f}\geq S_{T}y_{0}$, there exists a
control $\displaystyle u\in L^{2}((0,T)\times\Omega)$ with values in
$\displaystyle\mathcal{U}_{+}$ such that the corresponding solution to (5)
satisfies $\displaystyle\|y(T)-y_{f}\|_{L^{2}(\Omega)}\leq\varepsilon$.

##### General hypotheses for Theorem A.

We have previously presented Theorem A for the heat equation as a paradigmatic
example. Nevertheless, the underlying hypotheses on which some of our proofs
rely are much more general in nature; we review them below.

  * •

First, we consider the (unusual) unique-continuation like property

$\forall y\in L^{2}(\Omega),\quad\exists t\in(0,T),\;S_{t}y\,\text{ is
constant over }\,\Omega\quad\implies\quad y=0.$ (GUC)

This property is satisfied as soon as the three assumptions below hold:

    * –

for $\displaystyle y\in L^{2}(\Omega)$, $\displaystyle S_{t}y\in D(A)$ for all
$\displaystyle t>0$ (for instance, this is true if
$\displaystyle(S_{t})_{t\geq 0}$ is analytic [35]),

    * –

the only constant function in $\displaystyle D(A)$ is the zero
function,111This is the case for the Dirichlet Laplacian with domain
$\displaystyle D(A)=H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ if
$\displaystyle\Omega$ has a $\displaystyle C^{2}$ boundary.

    * –

$\displaystyle S_{t}$ is injective for all $\displaystyle t>0$.222This is the
case for groups, such as the wave equation, and for parabolic equations thanks
to the parabolic maximum principle. This is also true for analytic semigroups:
if $\displaystyle S_{t}y=0$ for some $\displaystyle t>0$, then $\displaystyle
S_{s}y=0$ for all $\displaystyle s\geq t$ and by analyticity $\displaystyle
S_{s}y=0$ for all $\displaystyle s\geq 0$, which for $\displaystyle s=0$
yields $\displaystyle y=0$.

  * •

Second, we will be interested in analytic-hypoellipticity:
$\displaystyle\partial_{t}-A$ is said to be analytic-hypoelliptic if any
distributional solution $\displaystyle y$ to $\displaystyle\partial_{t}y-Ay=f$
on $\displaystyle\Omega\times(0,T)$ with $\displaystyle f$ analytic on
$\displaystyle\Omega$ is analytic on $\displaystyle\Omega$, where analyticity
refers to real-analyticity.

  * •

Third, we will say that $\displaystyle(S_{t})_{t\geq 0}$ satisfies the
comparison principle if

$\forall y\in L^{2}(\Omega),\quad y\geq 0\implies\forall t>0,\;S_{t}y\geq 0.$
(6)

The first two properties are sufficient for the generalisation of Theorem A,
see Theorem 3.1. The third will play an important role when it comes to
minimal controllability times, and is in line with our definition of
nonnegative approximate controllability.

##### Elliptic operators.

As a generalisation of the Dirichlet Laplacian, let us discuss a large class
of uniformly elliptic operators that do satisfy these properties and to which
our obstruction result Theorem B generalises (see Theorem 4.1).

Let us assume that $\displaystyle\Omega$ is a bounded, open, connected subset
of $\displaystyle\mathbb{R}^{d}$, with $\displaystyle C^{2}$ boundary.
Defining $\displaystyle D(A):=H^{1}_{0}(\Omega)\cap H^{2}(\Omega)$, we
introduce operators of the form

$\forall y\in D(A),\quad Ay:=\sum_{1\leq i,j\leq
d}\partial_{x_{j}}(a_{ij}(x)\partial_{x_{i}}y)-\sum_{i=1}^{d}b_{i}(x)\partial_{x_{i}}y+c(x)y.$
(7)

When referring to operators of the form (7), we will always assume that the
functions $\displaystyle a_{ij}=a_{ji}$, $\displaystyle b_{i}$ are in
$\displaystyle W^{1,\infty}(\Omega)$, $\displaystyle c$ is in $\displaystyle
L^{\infty}(\Omega)$, and that the operator is uniformly elliptic, i.e., there
exists $\displaystyle\theta>0$ such that

$\displaystyle\forall
x\in\Omega,\;\forall\xi\in\mathbb{R}^{d},\quad\sum_{1\leq i,j\leq
d}a_{ij}(x)\xi_{i}\xi_{j}\geq\theta|\xi|^{2}.$

The adjoint of $\displaystyle A$ is given by

$\displaystyle\forall p\in D(A^{\ast}),\quad A^{\ast}p=\sum_{1\leq i,j\leq
d}\partial_{x_{i}}(a_{ij}(x)\partial_{x_{j}}p)+\sum_{i=1}^{d}b_{i}(x)\partial_{x_{i}}p+\left(c(x)-\sum_{i=1}^{d}\partial_{x_{i}}(b_{i}(x))\right)p,$

and we have $\displaystyle D(A^{\ast})=D(A)$.

Both $\displaystyle A$ and $\displaystyle A^{\ast}$ satisfy the parabolic
comparison principle [16], hence they satisfy the comparison principle (6).
They also satisfy the three conditions sufficient for the (GUC) property to
hold. 333The analyticity of the semigroup is well known for this class of
elliptic operators on open domains with $\displaystyle C^{2}$ boundary. There
are clearly no nonzero constant functions in $\displaystyle H^{2}\cap
H^{1}_{0}$. Finally, injectivity follows from the comparison principle (see
above footnote). Furthermore, both $\displaystyle\partial_{t}-A$ and
$\displaystyle\partial_{t}-A^{\ast}$ are analytic-hypoelliptic as soon as all
functions $\displaystyle a_{ij}$, $\displaystyle b_{i}$ and $\displaystyle c$
are analytic [33].

###  1.4 Proof strategy and related works

In the unconstrained case, approximate controllability of the heat equation is
a consequence of the unique continuation property, thanks to a general
property of linear control problems (see for example [12, Section 2.3]). In
the case of heat equations, the latter property can be obtained by the
Holmgren Uniqueness Theorem [2]. In contrast to these existence results, the
variational approach developed in [24] (see Section 1.2), handles approximate
controllability in a _constructive manner_.

Our strategy consists in extending this approach to the constrained case: the
main idea is to find a suitable cost function $\displaystyle F_{T}$ such that
optimal controls must satisfy the constraint $\displaystyle
u\in\mathcal{U}_{L}^{\mathrm{shape}}$. A remarkable feature of our strategy
lies in how we design the cost function: we do so by building an adequate
Fenchel dual function, instead of trying to find the cost function directly.

##### Constrained controllability.

Constrained control problems in infinite dimension have been studied in papers
such as [19, 3, 4, 5, 15]. In [15], sufficient conditions (in the form of
unique continuation properties) for controllability results are derived when
the control and states are constrained to some prescribed subspaces, but at
the expense of controlling only a finite-dimensional subpart of the final
state. In [19], the authors deal with a form of approximate controllability of
the heat equation akin to ours, focusing on minimal time problems. They derive
bang-bang type necessary optimality conditions for minimal time controls, and
then build such controls using an auxiliary optimisation problem.

The papers [3, 4, 5] address constrained exact controllability through
modified observability inequalities, thus giving abstract necessary and/or
sufficient conditions. One key difference with our work is that constraint
sets are assumed to be convex. In fact, all examples handled by [3, 4, 5]
feature isotropic constraints.

It is noteworthy that all the above references introduce so-called dual
functionals, drawing from the variational formulation of the Hilbert
Uniqueness Method. However, the formalism of Fenchel-Rockafellar duality in
itself, as developed in [24], has increasingly been abandoned in the
literature. Some notable exceptions are [44] in the context of stabilisation,
and, to some extent, [4], which uses Fenchel duality to study null-
controllability under some hypotheses.

We fully exploit the ideas hinted at in the latter paper by choosing a
different type of functional, which allows us to handle anisotropic, non-
convex constraints. In contrast with the aforementioned trend in the
literature, we work with Fenchel duality, but in a rather unusual way, in that
we will focus mainly on the dual problem. The nature of the actual primal
problem (optimal control problem) being solved follows effortlessly. To
perform the necessary computations, we will make extensive use of convex
analysis. Doing so bypasses many technical difficulties thanks to properties
of subdifferentials and Fenchel conjugates, among others, and allows for the
use of costs which are not differentiable but still smooth in the convex
analytic sense.

##### Bathtub principle for appropriate costs.

The second main idea is what underlies our choice of cost function
$\displaystyle F_{T}$, forcing optimal controls to satisfy the required the
on-off shape constraint. As the set of on-off shape controls is a non-convex
cone, we are led to relaxation, i.e., to consider the closure of its convex
hull. In order to build relevant costs, we then rely on the so-called bathtub
principle (actually, a relaxed version of it) [23].

For a given function $\displaystyle v\in L^{2}(\Omega)$, the latter principle
solves

$\sup_{u\in\overline{\mathcal{U}}_{L}}\int_{\Omega}u(x)v(x)\,dx,\qquad\overline{\mathcal{U}}_{L}:=\left\\{u\in
L^{2}(\Omega),\;0\leq u\leq 1\text{ and }\int_{\Omega}u\leq
L|\Omega|\right\\}.$

This optimisation problem comes up naturally in some control problems similar
to ours [21, 28], or in shape optimisation problems [38].

Interpreting the bathtub principle as a Fenchel conjugate leads us to design
the unusual cost functional (4). This allows us to design dual problems such
that optimal controls exist, and are characterised as maximisers of some
bathtub principle. Then, using analyticity properties for solutions of the
dual problem, we prove their uniqueness and hence their extremality, thereby
uncovering that they are on-off shape controls.

##### Bang-bang property of optimal controls.

Bang-bang controls (i.e., controls that saturate their constraints) are a
common feature in time optimal control problems. A growing literature on the
heat equation alone [43, 31, 29, 45, 47] shows that this property extends well
to some infinite-dimensional systems. In our case, we will see that the on-off
shape controls we have constructed can be understood as time-optimal controls.
As these controls are bang-bang, this yields another occurrence of the bang-
bang property in the time-optimal control of the heat equation.

Note, however, that in the references cited above, the controls are
constrained to lie in balls of specific function spaces, whereas we consider
non-negative constraints on the controls, which is an anisotropic constraint.
Moreover, the bang-bang property is usually established separately using
optimality conditions, having established controllability at the onset. In our
case, the Fenchel-Rockafellar duality approach allows to do all those things
simultaneously.

###  1.5 Extensions and perspectives

##### Operator, boundary conditions.

The (GUC) property and analytic-hypoellipticity are two key sufficient
properties for approximate controllability by on-off shape controls. We have
highlighted second-order elliptic operators with analytic coefficients
Dirichlet boundary conditions as an example. Our results apply to such
operators with Robin boundary conditions of the form $\displaystyle
a(x)y+b(x)\partial_{\nu}y=0$ over $\displaystyle\partial\Omega$ (with
$\displaystyle a,b$ analytic) as soon as the function $\displaystyle a$ does
not vanish on the whole of $\displaystyle\partial\Omega$ (more generally, as
soon as $\displaystyle a$ is nontrivial on any connected component of
$\displaystyle\partial\Omega$). This excludes the important case of Neumann
boundary conditions, which remains open.

Our approach also accommodates subelliptic operators. This includes a large
class of Hörmander operators, i.e., operators of the form $\displaystyle
A=\textstyle\sum_{i=1}^{m}X_{i}^{2}+X_{0}+V\mathrm{Id}$ with vector fields
$\displaystyle X_{1},\ldots,X_{m}$ generating a Lie algebra that equals
$\displaystyle\mathbb{R}^{d}$ on the whole of $\displaystyle\Omega$. Under
general regularity assumptions and boundary conditions, such an operator and
its adjoint generate a strongly continuous semigroup on $\displaystyle
L^{2}(\Omega)$, satisfy the comparison principle [6], all three conditions
sufficient for the (GUC) property, and are analytic-hypoelliptic for instance
if the characteristic manifold is an analytic symplectic manifold (see [32]).

##### Control operator.

Our results have been stated with the identity control operator. They extend
to the nonnegative control of

$\left\\{\begin{aligned} y_{t}-Ay&=\varphi u,\\\ y(0)=&y_{0}\ \textrm{in}\
\Omega\end{aligned}\right.$

where $\displaystyle\varphi\in L^{\infty}(\Omega)$ is positive, analytic.

An interesting perspective is to follow our proof strategy with boundary
control operators, where on-off shape controls now refer to characteristic
functions over the boundary $\displaystyle\partial\Omega$.

##### Exact nonnegative controllability.

In the case of unconstrained controllability with a control acting in some
fixed subset $\displaystyle\omega$, any function that can be reached exactly
is (at least) analytic in $\displaystyle\Omega\setminus\omega$, preventing
exact controllability to hold true.

On the one hand, this argument for (non)-exact controllability by on-off shape
controls fails since the control may act everywhere. On the other hand, our
approach heavily relies on targeting a ball
$\displaystyle\overline{B}(y_{f},\varepsilon)$ with
$\displaystyle\varepsilon>0$. As a result, exact nonnegative controllability
by on-off shape controls is an open and seemingly difficult question.

A related matter is that of the cost of approximate controllability as a
function of $\displaystyle\varepsilon\rightarrow 0$.

##### Abstract constrained control.

The strategy of proof developed in this article hints at generalisations,
where the method is applied to abstract linear control problems with abstract
constraint sets $\displaystyle\mathcal{U}$.

In particular, we expect it to lead to necessary and sufficient conditions for
controllability when $\displaystyle\mathcal{U}$ is convex. When
$\displaystyle\mathcal{U}$ is not convex as is the case for on-off shape
controls, this requires to study the convex hull of
$\displaystyle\mathcal{U}$, following the relaxation approach. This abstract
setting should allow us to discern how one can design a cost function
$\displaystyle F_{T}$, analogous to (4), tailored to a given
$\displaystyle\mathcal{U}$.

Further sufficient conditions should be derived to ensure that optimal
controls in the convex hull of $\displaystyle\mathcal{U}$ actually are in the
original constraint set $\displaystyle\mathcal{U}$. In the present work,
analytic-hypoellipticity and the (GUC) property play that role in the case of
on-off shape controls.

This will be the subject of an ulterior article.

##### Regularity of the sets $\displaystyle\omega(t)$.

Another problem is to analyse the complexity of the sets
$\displaystyle\omega(t)$ occupied by optimal controls over time. For instance,
how smooth ($\displaystyle BV$ regularity, number of connected components,
etc) are the sets $\displaystyle\omega(t)$ achieving approximate
controllability?

In view of applications, these are important issues for the controls to be
implementable in practice. For example, if the sets $\displaystyle\omega(t)$
are constrained to depend on a few parameters, or if they are restricted to
rigid movements, controllability is a totally open question.

##### Numerical approximation of optimal controls.

Optimal controls are given explicitly in terms of optimisers of the dual
problem: the constructive nature of our approach means that optimal controls
may be numerically computed, at least on paper.

Providing reliable and efficient methods to compute optimal controls is a
difficult issue which has been studied in the case of Lions’s cost functional
with $\displaystyle\varepsilon=0$ (i.e., exact controllability) [20, 7].
Similar results in a generalised setting with our Fenchel-Rockafellar-based
approach would be valuable.

Contrary to Lions’s cost functional, we note that ad hoc algorithms are
required in order to cope with functions that are not necessarily
differentiable, as is the case in the present paper. Recent primal-dual
algorithms designed for optimisation problems with objective functions of the
form $\displaystyle F(u)+G(L_{T}u)$ are likely to be good candidates [10].

##### Outline of the paper.

First, Section 2 lays out the convex analytic framework, that of Fenchel-
Rockafellar duality, and how it may be applied to constrained approximate
controllability. We then introduce the bathtub principle and interpret it in
terms of Fenchel conjugation in order to design a relevant optimal control
problem for our purposes. Section 3 is dedicated to the proof of our
nonnegative approximate controllability result given by Theorem 3.1, and
Section 4 to that of the obstruction result, Theorem 4.1. Finally, Section 5
gathers our results about further obstructions when the control amplitude is
bounded, along with our analysis of the corresponding minimal time control
problem.

##  2 Building the optimal control problem

###  2.1 Convex analytic framework

Let $\displaystyle H$ be a Hilbert space. We let $\displaystyle\Gamma_{0}(H)$
be the set of functions from $\displaystyle H$ to
$\displaystyle\mathopen{]}-\infty,+\infty]$ that are convex, lower
semicontinuous (abbreviated lsc) and proper (i.e., not identically
$\displaystyle+\infty$). For $\displaystyle f\in\Gamma_{0}(H)$, we let

$\displaystyle\mathrm{dom}(f)=\\{x\in H,\,f(x)<+\infty\\}$

be its domain.

##### Fenchel conjugate.

For a proper function $\displaystyle
f:H\rightarrow\mathopen{]}-\infty,+\infty]$, we denote $\displaystyle
f^{\ast}:H\rightarrow\mathopen{]}-\infty,+\infty]$ its convex conjugate, given
by the convex lsc function

$\displaystyle f^{\ast}(y):=\sup_{x\in H}\;\big{(}\langle
y,x\rangle-f(x)\big{)},\quad\forall y\in H.$

##### Support and indicator functions.

Given a subset $\displaystyle C\subset H$, the indicator function of
$\displaystyle C$ is the function defined by

$\delta_{C}(x):=\begin{cases}0&\text{if }x\in C\\\ +\infty&\text{if }x\notin
C\end{cases}\;,\quad\forall x\in H,$

and the support function of $\displaystyle C$ is defined by

$\displaystyle\sigma_{C}(p):=\sup_{x\in C}\langle
p,x\rangle=\delta_{C}^{\ast}(p),\quad\forall p\in H,$

i.e., the Fenchel conjugate function of the indicator function of
$\displaystyle C$.

##### Subdifferentials.

For $\displaystyle f\in\Gamma_{0}(H)$, we let

$\displaystyle\partial f(x):=\\{p\in H,\,\forall y\in H,f(y)\geq f(x)+\langle
p,y-x\rangle\\},$

be its subdifferential at a point $\displaystyle x\in H$.

Various common properties of Fenchel conjugates, support functions and
subdifferentials are used throughout the article. These are all recalled in
Appendix A, where a few additional lemmas are proved.

###  2.2 Approximate controllability by Fenchel duality ([24])

Let us explain how the approximate controllability problem is reformulated in
the context of Fenchel-Rockafellar duality [40] (see A.4 for a general
presentation), following the strategy introduced by Lions in [24]. We work
with the control problem (5), with the control space $\displaystyle
E:=L^{2}((0,T)\times\Omega)$ and the state space $\displaystyle
L^{2}(\Omega)$.

By Duhamel’s formula $\displaystyle y(T)=S_{T}y_{0}+L_{T}u$, the inclusion
$\displaystyle y(T)\in\overline{B}(y_{f},\varepsilon)$ (where the closed ball
of center $\displaystyle y_{f}$ and radius $\displaystyle\varepsilon$ is with
respect to the $\displaystyle L^{2}(\Omega)$-norm) can equivalently be written
as $\displaystyle L_{T}u\in\overline{B}(y_{f}-S_{T}y_{0},\varepsilon)$.

Given some cost functional $\displaystyle
F_{T}:E\rightarrow[0,+\infty]\in\Gamma_{0}(E)$, consider the optimal control
problem (which we will refer to as the primal problem)

$\pi:=\inf_{u\in E}F_{T}(u)+G_{T,\varepsilon}(L_{T}u).$

where

$\displaystyle
G_{T,\varepsilon}:=\delta_{\overline{B}(y_{f}-S_{T}y_{0},\varepsilon)}\in\Gamma_{0}(L^{2}(\Omega)).$

Now consider the Fenchel dual to the above problem, which writes

$d=-\inf_{p_{f}\in L^{2}(\Omega)}J_{T,\varepsilon}(p_{f}),\quad
J_{T,\varepsilon}(p_{f}):=F_{T}^{\ast}(L_{T}^{\ast}p_{f})+G_{T,\varepsilon}^{\ast}(-p_{f}).$
(8)

Thanks to the formulae for conjugates, we find

$\displaystyle G_{T,\varepsilon}^{\ast}(z)=\langle
y_{f}-S_{T}y_{0},z\rangle_{L^{2}}+\varepsilon\|z\|_{L^{2}},$

leading to

$\displaystyle
J_{T,\varepsilon}(p_{f})=F_{T}^{\ast}(L_{T}^{\ast}p_{f})-\langle
y_{f}-S_{T}y_{0},p_{f}\rangle_{L^{2}}+\varepsilon\|p_{f}\|_{L^{2}}.$

We recall that $\displaystyle p=L_{T}^{\ast}p_{f}$ solves the adjoint equation

$\left\\{\begin{aligned} p_{t}+A^{\ast}p&=0,\\\ p(T)=p_{f}\ &\textrm{in}\
\Omega.\end{aligned}\right.$ (9)

##### Strong duality.

Weak duality $\displaystyle\pi\geq d$ always holds. According to the Fenchel-
Rockafellar duality theorem recalled in Appendix A.4, the existence of
$\displaystyle p_{f}\in\mathrm{dom}(G_{T,\varepsilon}^{\ast})$ such that
$\displaystyle F_{T}^{\ast}$ is continuous at $\displaystyle
L_{T}^{\ast}p_{f}$ is a sufficient condition for strong duality
$\displaystyle\pi=d$ to hold. Since
$\displaystyle\mathrm{dom}(G_{T,\varepsilon}^{\ast})=L^{2}(\Omega)$, this
condition reduces to the existence of a point of continuity of the form
$\displaystyle L_{T}^{\ast}p_{f}$ for $\displaystyle F_{T}^{\ast}$. In the
cases covered here, we shall check that the chosen $\displaystyle
F_{T}^{\ast}$ is continuous at $\displaystyle 0$. When strong duality obtains,
it is therefore equivalent to work with the dual problem, which is easier to
handle especially when it has full domain, i.e., its objective function is
finite everywhere.

##### Non-trivial strong duality.

Furthermore, the primal value $\displaystyle\pi$ is attained if finite, i.e.,
if this equality is not the trivial $\displaystyle+\infty=+\infty$ (the
uncontrollable case). Thus, if $\displaystyle d$ is finite, $\displaystyle\pi$
is finite as well and attained: we may speak of optimal controls.

This requirement that $\displaystyle d$ be finite is by far the subtlest one.
It may be tackled by proving that the functional $\displaystyle
J_{T,\varepsilon}$ underlying the dual problem (written in infimum form
$\displaystyle\textstyle\inf_{p_{f}\in
L^{2}(\Omega)}J_{T,\varepsilon}(p_{f})$) has a minimum. In practice, we will
always find this to be the case, as the dual problem is usually unconstrained
(depending on the choice of $\displaystyle F_{T}$), unlike the primal problem.
Hence, both $\displaystyle\pi$ and $\displaystyle d$ will be attained and,
from Proposition A.8, any optimal dual variable $\displaystyle p_{f}^{\star}$
is such that any optimal control $\displaystyle u^{\star}$ satisfies

$u^{\star}\in\partial F_{T}^{\ast}(L_{T}^{\ast}p_{f}^{\star}).$ (10)

######  Proposition 2.1.

Assume that, for any $\displaystyle y_{0},y_{f}\in L^{2}(\Omega)$ such that
$\displaystyle y_{f}\geq S_{T}y_{0}$ and any $\displaystyle\varepsilon>0$,

  * •

there exists $\displaystyle p_{f}\in L^{2}(\Omega)$ such that $\displaystyle
F_{T}^{\ast}$ is continuous at $\displaystyle L_{T}^{\ast}p_{f}$,

  * •

$\displaystyle d\neq+\infty$.

If for any dual optimal variable $\displaystyle p_{f}^{\star}$, the controls
characterised by (10) are in $\displaystyle\mathcal{U}_{+}$, then the control
system (5) is nonnegatively approximately controllable with controls in
$\displaystyle\mathcal{U}_{+}$ in time $\displaystyle T$.

This shows how the choice of the cost $\displaystyle F_{T}$ impacts the
existence and properties of optimal controls. More precisely, it must be
pointed out that all the hypotheses of Proposition 2.1 are formulated with
respect to the dual problem. Accordingly, from the next section onwards, our
strategy will be to determine an adequate optimal control problem by designing
its dual problem.

Finally, we emphasise that (10) is only a necessary condition for the
optimality of $\displaystyle u^{\star}$. It becomes sufficient only when
$\displaystyle\partial F_{T}^{\ast}(L_{T}^{\ast}p_{f}^{\star})$ is reduced to
a singleton, which will occur in our case.

###  2.3 Convex analytic interpretation of the bathtub principle

Starting from the set of on-off shape controls of amplitude $\displaystyle 1$,

$\mathcal{U}_{L}:=\\{\chi_{\omega},\quad\omega\subset\Omega,\quad|\omega|\leq
L|\Omega|\\},$ (11)

where $\displaystyle|\cdot|$ denotes the Lebesgue measure, we define the
closure of its convex hull (which is also its weak-$\displaystyle\ast$ closure
for the $\displaystyle L^{\infty}(\Omega)$-topology)

$\overline{\mathcal{U}}_{L}:=\left\\{u\in L^{2}(\Omega),\;0\leq u\leq 1\text{
and }\int_{\Omega}u\leq L|\Omega|\right\\}.$ (12)

Given a fixed $\displaystyle v\in L^{2}(\Omega)$, we consider the (static)
maximisation problem

$\sup_{u\in\overline{\mathcal{U}}_{L}}\int_{\Omega}u(x)v(x)\,dx.$ (13)

This a relaxed version of the so-called bathtub principle, which gives the
maximum value as well as a characterisation of maximisers. For the sake of
readability, we introduce the necessary results for what follows, but refer to
Appendix B for a more detailed statement. For a given $\displaystyle v\in
L^{2}(\Omega)$, we let

$\Phi_{v}(r):=\left|\\{v>r\\}\right|.$ (14)

and its pseudo-inverse function

$\Phi^{-1}_{v}(s):=\inf_{r\in\mathbb{R}}\left\\{\Phi_{v}(r)\leq
s\right\\}=\inf_{r\in\mathbb{R}}\left\\{|\\{v>r\\}|\leq s\right\\}.$ (15)

Finally, we set

$h(v):=\max(0,\Phi^{-1}_{v}(L|\Omega|)).$ (16)

######  Remark 2.2.

The function $\displaystyle\Phi_{v}^{-1}$ is the Schwarz radial rearrangement
of $\displaystyle v$, see [18].

######  Lemma 2.3 (relaxed bathtub principle).

Let $\displaystyle v\in L^{2}(\Omega)$. The maximum in (13) equals

$\int_{0}^{\min(\Phi_{v}(0),L|\Omega|)}\Phi_{v}^{-1}.$

Furthermore, if all the level sets of the function $\displaystyle v$ have
measure zero, the maximum equals $\displaystyle\textstyle\int_{\\{v>h(v)\\}}v$
and is uniquely attained by

$u^{\star}:=\chi_{\\{v>h(v)\\}},$

We refer to Lemma B.2 for the comprehensive statement of the relaxed bathtub
principle. We may interpret the above results as a formula for the support
function of $\displaystyle\overline{\mathcal{U}}_{L}$ in $\displaystyle
L^{2}(\Omega)$:

$\sigma_{\overline{\mathcal{U}}_{L}}(v)=\sup_{u\in\overline{\mathcal{U}}_{L}}\left(\langle
u,v\rangle_{L^{2}}-\delta_{\overline{\mathcal{U}}_{L}}(u)\right))=\sup_{u\in\overline{\mathcal{U}}_{L}}\int_{\Omega}u(x)v(x)\,dx=\int_{0}^{\min(\Phi_{v}(0),L|\Omega|)}\Phi_{v}^{-1}.$
(17)

First, using the characterisation of the subdifferential given in Appendix A,
we arrive at the following characterisation for the solutions to the
maximisation problem given in Lemma 2.3:

######  Proposition 2.4.

Let $\displaystyle v\in L^{2}(\Omega)$. The maximisers of the relaxed bathtub
problem are given by the elements of
$\displaystyle\partial\sigma_{\overline{\mathcal{U}}_{L}}(v)$.

###### Proof.

We have, for $\displaystyle v\in L^{2}(\Omega)$,

$\operatorname*{arg\,max}_{u\in\overline{\mathcal{U}}_{L}}\langle
u,v\rangle_{L^{2}}=\operatorname*{arg\,max}_{u\in L^{2}}\langle
u,v\rangle_{L^{2}}-\delta_{\overline{\mathcal{U}}_{L}}(u)=\left\\{u\in
L^{2},\quad\langle
u,v\rangle_{L^{2}}-\delta_{\overline{\mathcal{U}}_{L}}(u)=\sigma_{\overline{\mathcal{U}}_{L}}(v)\right\\}=\partial\sigma_{\overline{\mathcal{U}}_{L}}(v),$

where we have used
$\displaystyle(\delta_{\overline{\mathcal{U}}_{L}})^{\ast}=\sigma_{\overline{\mathcal{U}}_{L}}$
along with (46) given in Appendix A.1. ∎

######  Remark 2.5.

Proposition 2.4 implies that for any maximiser $\displaystyle u$ of the
relaxed bathtub problem,

$\displaystyle v\in\partial\delta_{\overline{\mathcal{U}}_{L}}(u).$

Proposition A.4 in Appendix A.2 shows that this implies $\displaystyle
u\in\partial\overline{\mathcal{U}}_{L}$. Propositions 2.3 and 2.4 characterise
exactly which elements of the boundary
$\displaystyle\partial\overline{\mathcal{U}}_{L}$ are involved.

###  2.4 From the static bathtub principle to the dual problem and its
corresponding cost

Following Section 2.2 and recalling Proposition 2.1 and (10), we are looking
for a cost function $\displaystyle F_{T}$ such that the corresponding optimal
controls are on-off shape controls, and we have established that it suffices
to find a conjugate functional $\displaystyle F_{T}^{\ast}$ satisfying two key
properties. First, if there exists $\displaystyle p_{f}\in L^{2}(\Omega)$ such
that $\displaystyle F_{T}^{\ast}$ is continuous at $\displaystyle
L_{T}^{\ast}p_{f}$, and if we can provide the existence of a minimiser
$\displaystyle p_{f}^{\star}$ of $\displaystyle J_{T,\varepsilon}$, then
$\displaystyle\pi$ is attained and there exists at least one optimal control.
Second, any optimal control $\displaystyle u^{\star}$ should satisfy (10) so
$\displaystyle F_{T}^{\ast}$ should be chosen so that the subdifferential
$\displaystyle\partial F_{T}^{\ast}(L_{T}^{\ast}p_{f}^{\star})$ contains only
characteristic functions. Given Proposition A.4 and Section 2.3, elements of

$\displaystyle\partial\sigma_{\overline{\mathcal{U}}_{L}}(v),\quad v\in
L^{2}(\Omega)$

are bang-bang, in the sense that they are characteristic functions, under some
mild conditions that $\displaystyle v$ must satisfy.

To go from the static optimisation problem to the adequate dual problem, we
add a time dependency. Moreover, to ensure coercivity of the dual problem, we
add a quadratic exponent. All in all, we choose the following conjugate:

$F_{T}^{\ast}(p):=\frac{1}{2}\left(\int_{0}^{T}\sigma_{\overline{\mathcal{U}}_{L}}(p(t))\,dt\right)^{2}=\frac{1}{2}\left(\int_{0}^{T}\int_{0}^{\min(\Phi_{p(t)}(0),L|\Omega|)}\Phi_{p(t)}^{-1}(s)\,ds\,dt\right)^{2},\quad\forall
p\in E.$ (18)

Since the controllability problem corresponds to $\displaystyle
G_{T,\varepsilon}^{\ast}:=\sigma_{\overline{B}(y_{f}-S_{T}y_{0},\varepsilon)}$,
this defines a dual problem of the form (8). As pointed out in Section 2.2, we
are now dealing with an unconstrained optimisation problem (i.e., the domain
of the functions involved is the whole space $\displaystyle L^{2}(\Omega$)).

We can now derive the corresponding constrained optimisation problem, by
compute the actual cost $\displaystyle F_{T}$ associated to the choice (18)
for $\displaystyle F_{T}^{\ast}$ . We find, as announced by (4) in the
introduction:

######  Lemma 2.6.

The function $\displaystyle F_{T}^{\ast}$ defined by (18) satisfies
$\displaystyle F_{T}^{\star}\in\Gamma_{0}(E)$. Defining

$\mathcal{M}(u):=\max\left(\|u\|_{L^{\infty}},\frac{\|u\|_{L^{1}}}{L|\Omega|}\right),\quad\forall
u\in L^{2}(\Omega),$

its Fenchel conjugate $\displaystyle(F_{T}^{\ast})^{\ast}=F_{T}$ is given for
$\displaystyle u\in E$ by

$\displaystyle
F_{T}(u)=\frac{1}{2}\bigg{(}\sup_{t\in[0,T]}\mathcal{M}^{2}(u(t,\cdot))\bigg{)}+\delta_{\\{u\geq
0\\}}(u)=\frac{1}{2}\bigg{(}\sup_{t\in[0,T]}\max\Big{(}\|u(t,\cdot)\|_{L^{\infty}},\frac{\|u(t,\cdot)\|_{L^{1}}}{L|\Omega|}\Big{)}^{2}\bigg{)}+\delta_{\\{u\geq
0\\}}(u).$

###### Proof.

Lemma A.5 in Appendix A.3 shows that $\displaystyle
F_{T}^{\ast}\in\Gamma_{0}(E)$. We proceed by computing
$\displaystyle(F_{T}^{\ast})^{\ast}$. We have $\displaystyle
F_{T}^{\ast}={\textstyle\frac{1}{2}}H^{2}$, with $\displaystyle
H(p):=\textstyle\int_{0}^{T}\sigma_{\overline{\mathcal{U}}_{L}}(p(t,\cdot))\,dt$.
Since
$\displaystyle\sigma_{\overline{\mathcal{U}}_{L}}\in\Gamma_{0}(L^{2}(\Omega))$,
the definition of the support function together with Lemma A.5 in Appendix
show that $\displaystyle H\in\Gamma_{0}(E)$ with

$\displaystyle
H^{\ast}(u)=\int_{0}^{T}\sigma_{\overline{\mathcal{U}}_{L}}^{\ast}(u(t,\cdot))\,dt=\int_{0}^{T}\delta_{\overline{\mathcal{U}}_{L}}(u(t,\cdot))\,dt.$

Furthermore, we find the conjugate of
$\displaystyle\textstyle\frac{1}{2}H^{2}$ by using (45) in Appendix A.1, which
leads to

$\displaystyle\left(\frac{1}{2}H^{2}\right)^{\ast}(u)=\min_{\alpha>0}\left(\frac{1}{2}\alpha^{2}+\alpha
H^{\ast}\left(\frac{u}{\alpha}\right)\right),$

where we used that $\displaystyle\mathrm{dom}(H)=E$. Clearly,

$\displaystyle H^{\ast}\left(\frac{u}{\alpha}\right)=0\quad\text{if}\quad
u\geq 0\;\text{ and }\;\sup_{t\in[0,T]}\mathcal{M}(u(t,\cdot))\leq\alpha,$

and is $\displaystyle+\infty$ otherwise.

We end up with

$\displaystyle\left(\frac{1}{2}H^{2}\right)^{\ast}(u)=\min_{\alpha>0}\left(\frac{1}{2}\alpha^{2}+\delta_{\left\\{\sup_{t\in[0,T]}\mathcal{M}(u(t,\cdot))\leq\alpha\right\\}}(u)\right)+\delta_{\\{u\geq
0\\}}(u)=F_{T}(u).$

The lemma is proved. ∎

Note that $\displaystyle F_{T}^{\ast}$ is (positively)-homogeneous of degree
$\displaystyle 2$. Indeed, $\displaystyle
v\mapsto\sigma_{\overline{\mathcal{U}}_{L}}$ is positively-homogeneous of
degree $\displaystyle 1$, i.e.,
$\displaystyle\sigma_{\overline{\mathcal{U}}_{L}}(\lambda
v)=\lambda\sigma_{\overline{\mathcal{U}}_{L}}(v)$ for all
$\displaystyle\lambda>0$, $\displaystyle v\in L^{2}(\Omega)$.

We end this subsection by establishing a crucial property satisfied by
$\displaystyle F_{T}^{\ast}$. It will play a role similar to that of the
unique continuation property in proving that the dual functional is coercive.

######  Lemma 2.7.

For all $\displaystyle p_{f}\in L^{2}(\Omega)$, if $\displaystyle
F_{T}^{\ast}(L_{T}^{\ast}p_{f})=0$, then $\displaystyle p_{f}\leq 0$.

###### Proof.

By definition of $\displaystyle F_{T}^{\ast}$ and using (17), the equality
$\displaystyle F_{T}^{\ast}(L_{T}^{\ast}p_{f})=0$ entails

$\sup_{u\in\overline{\mathcal{U}}_{L}}\langle
u,p(t,\cdot)\rangle_{L^{2}}=\int_{0}^{\min(\Phi_{p(t,\cdot)}(0),L|\Omega|)}\Phi^{-1}_{p(t,\cdot)}=0,\quad\text{for
a.e. }t\in(0,T),$

where $\displaystyle p$ is the solution to the adjoint equation (9) such that
$\displaystyle p(T)=p_{f}$.

For a fixed $\displaystyle t\in(0,T)$, it is easily seen that the supremum on
the left-hand side is positive as soon as $\displaystyle p(t,\cdot)>0$ on a
set of positive measure, by appropriately choosing $\displaystyle u$ supported
in this set. We have proved that $\displaystyle p(t,\cdot)\leq 0$ for
$\displaystyle a.e.\,t\in(0,T)$ and in particular that $\displaystyle
p_{f}\leq 0$ since $\displaystyle p\in C([0,T];L^{2}(\Omega))$. ∎

##  3 Approximate controllability results

In this section, we state and prove our main result on approximate
controllability. The full statement for our Theorem A is given with more
details below, for general linear operators, satisfying the properties given
in Section 1.3.

We are considering the following optimal control problem:

$\pi=\inf_{u\in E}F_{T}(u)+G_{T,\varepsilon}(L_{T}u)=\inf_{u\in
E}\left\\{\frac{1}{2}\sup_{t\in[0,T]}\max\left(\|u(t)\|_{L^{\infty}},\frac{\|u(t)\|_{L^{1}}}{L|\Omega|}\right)^{2}+\delta_{\overline{B}(y_{f}-S_{T}y_{0},\varepsilon)}(L_{T}u)\right\\},$
(19)

whose dual problem is

$d=-\inf_{p_{f}\in L^{2}(\Omega)}J_{T,\varepsilon}(p_{f})=-\inf_{p_{f}\in
L^{2}(\Omega)}\left\\{\frac{1}{2}\left(\int_{0}^{T}\sigma_{\overline{\mathcal{U}}_{L}}(L_{T}^{\ast}p_{f}(t))\,dt\right)^{2}-\langle
y_{f}-S_{T}y_{0},p_{f}\rangle_{L^{2}}+\varepsilon\|p_{f}\|_{L^{2}}\right\\}.$
(20)

######  Theorem 3.1.

Assume that $\displaystyle A^{\ast}$ satisfies the (GUC) property and that
$\displaystyle\partial_{t}-A^{\ast}$ is analytic-hypoelliptic. Then for the
cost function $\displaystyle F_{T}$ defined by (4),

  * •

strong duality $\displaystyle\pi=d$ holds,

  * •

the dual problem (20) is attained at a unique minimiser $\displaystyle
p_{f}^{\star}\in L^{2}(\Omega)$,

  * •

there exists a unique optimal control $\displaystyle u^{\star}\in E$ for the
primal problem (19).

Furthermore, if the target is not reached by the trivial control
$\displaystyle u\equiv 0$, i.e., if

$y_{f}\notin\overline{B}(S_{T}y_{0},\varepsilon),$ (21)

then the optimal control is given by

$u^{\star}(t,\cdot)=\overline{M}\,\chi_{\\{p^{\star}(t,\cdot)>h(p^{\star}(t,\cdot))\\}},\quad\overline{M}=\int_{0}^{T}\int_{\\{p^{\star}(t,\cdot)>h(p^{\star}(t,\cdot))\\}}p^{\star}(t,x)\,dt\,dx,$
(22)

where $\displaystyle h$ is defined by (16), and where $\displaystyle
p^{\star}=L_{T}^{\star}p_{f}^{\star}$ is the solution of the adjoint equation
(9) satisfying $\displaystyle p^{\star}(T)=p_{f}^{\star}$.

######  Remark 3.2.

As mentioned in the introduction, Theorem 3.1 holds for uniformly elliptic
operators of the form (7) with analytic coefficients, and in particular the
classical heat equation with Dirichlet boundary conditions, on a bounded,
open, connected domain with $\displaystyle C^{2}$ boundary.

Throughout this section, we assume the hypotheses sufficient for Theorem 3.1,
i.e., that $\displaystyle A^{\ast}$ satisfies the (GUC) property and that
$\displaystyle\partial_{t}-A^{\ast}$ is analytic-hypoelliptic. The proof is
then scattered into the section as follows:

  * •

First, we establish that strong duality holds.

  * •

Second, we prove that the corresponding dual functional is coercive: hence,
the dual functional attains its minimum (the dual problem attains its
maximum).

  * •

Third we prove (22).

  * •

Finally, we investigate the uniqueness of optimal variables.

######  Remark 3.3.

As the proofs show, the first two steps and the uniqueness of dual optimal
variables are valid for any operator $\displaystyle A$. In particular, they do
not require that $\displaystyle A^{\ast}$ satisfy the (GUC) property and that
$\displaystyle\partial_{t}-A^{\ast}$ be analytic-hypoelliptic. Hence, strong
duality and existence of optimal controls does not require any specific
assumption the semigroup must satisfy. This remark will be of importance in
the next subsection where we manipulate optimal controls without making these
two hypotheses.

###  3.1 Strong duality

######  Lemma 3.4.

$\displaystyle F_{T}^{\ast}$ is continuous at $\displaystyle 0=L_{T}^{\ast}0$.

###### Proof.

By the Cauchy-Schwarz inequality,

$\displaystyle\forall u\in\overline{\mathcal{U}}_{L},\quad\langle
u,v\rangle_{L^{2}}\leq\|u\|_{L^{2}}\|v\|_{L^{2}}\leq|\Omega|^{1/2}\|v\|_{L^{2}},$

which leads to

$\displaystyle\sigma_{\overline{\mathcal{U}}_{L}}(v)=\int_{0}^{\min(\Phi_{v}(0),L|\Omega|)}\Phi_{v}^{-1}\leq|\Omega|^{1/2}\|v\|_{L^{2}}.$

As a result, we may bound with the Cauchy-Schwarz inequality again

$\displaystyle 0\leq
F_{T}^{\ast}(p)\leq\frac{1}{2}|\Omega|\left(\int_{0}^{T}\|p(t,\cdot)\|_{L^{2}}\,dt\right)^{2}\leq\frac{1}{2}T|\Omega|\,\|p\|_{E}^{2},$

hence the continuity of $\displaystyle F_{T}^{\ast}$ at $\displaystyle
0=L_{T}^{\ast}0$. ∎

The above lemma shows that the first condition of Proposition 2.1 is
satisfied, i.e., strong duality holds.

###  3.2 Coercivity of $\displaystyle J_{T,\varepsilon}$, nonnegative
approximate controllability

######  Proposition 3.5.

The functional $\displaystyle J_{T,\varepsilon}$ defined by

$J_{T,\varepsilon}(p_{f})=\left(\frac{1}{2}\int_{0}^{T}\sigma_{\overline{\mathcal{U}_{L}}}(L_{T}^{\ast}p_{f})dt\right)^{2}-\langle
y_{f}-S_{T}y_{0},p_{f}\rangle_{L^{2}}+\varepsilon\|p_{f}\|_{L^{2}}.$ (23)

is coercive on $\displaystyle L^{2}(\Omega)$, i.e.,

$\displaystyle
J_{T,\varepsilon}(p_{f})\xrightarrow[\|p_{f}\|_{L^{2}}\to\infty]{}\infty,$

and thus attains its minimum.

###### Proof.

Since we know that $\displaystyle J_{T,\varepsilon}$ is convex, proper,
strongly lsc, if $\displaystyle J_{T,\varepsilon}$ is coercive then
$\displaystyle\textstyle\inf_{p_{f}\in
L^{2}(\Omega)}J_{T,\varepsilon}(p_{f})\neq-\infty$, and that it is actually
attained.

We will actually prove a stronger condition than coercivity, namely

$\liminf_{\|p_{f}\|_{L^{2}}\to\infty}\frac{J_{T,\varepsilon}(p_{f})}{\|p_{f}\|_{L^{2}}}>0.$

Our method of proof follows that of [30, 19]. Take a sequence
$\displaystyle\|p_{f}^{n}\|_{L^{2}}\to\infty$. We denote
$\displaystyle\textstyle q_{f}^{n}:=\frac{p_{f}^{n}}{\|p_{f}^{n}\|_{L^{2}}}$,
and $\displaystyle q^{n}\in E$ the corresponding solution of the adjoint
equation (9), i.e., such that $\displaystyle q^{n}(T)=q_{f}^{n}$, which by
linearity is $\displaystyle\textstyle\frac{p^{n}}{\|p_{f}^{n}\|_{L^{2}}}$,
where $\displaystyle p^{n}=L_{T}^{\ast}p_{f}^{n}$ is the solution of (9) such
that $\displaystyle p^{n}(T)=p_{f}^{n}$. By positive homogeneity of
$\displaystyle F_{T}^{\ast}$ (of degree $\displaystyle 2$), we have

$\frac{J_{T,\varepsilon}(p^{n}_{f})}{\|p^{n}_{f}\|_{L^{2}}}=\|p^{n}_{f}\|_{L^{2}}F_{T}^{\ast}(L_{T}^{\ast}q_{f}^{n})-\left\langle
y_{f}-S_{T}y_{0},q_{f}^{n}\right\rangle_{L^{2}}+\varepsilon$

and hence if
$\displaystyle\liminf_{n\to\infty}F_{T}^{\ast}(L_{T}^{\ast}q_{f}^{n})>0$, then

$\liminf_{n\to\infty}\frac{J_{T,\varepsilon}(p^{n}_{f})}{\|p^{n}_{f}\|{L^{2}}}=+\infty.$

Let us now treat the remaining case where
$\displaystyle\liminf_{n\to\infty}F_{T}^{\ast}(L_{T}^{\ast}q_{f}^{n})=0$.
Since $\displaystyle\|q_{f}^{n}\|_{L^{2}}=1$, upon extraction of a
subsequence, we have $\displaystyle q_{f}^{n}\rightharpoonup q_{f}$ weakly in
$\displaystyle L^{2}(\Omega)$ for some $\displaystyle q_{f}\in L^{2}(\Omega)$.
Since $\displaystyle L_{T}^{\ast}\in L(L^{2}(\Omega),E)$, we have
$\displaystyle L_{T}^{\ast}q_{f}^{n}\rightharpoonup L_{T}^{\ast}q_{f}$ weakly
in $\displaystyle E$.

Now, since $\displaystyle F_{T}^{\ast}$ is convex and strongly lsc on
$\displaystyle E$, it is (sequentially) weakly lsc and taking the limit we
obtain $\displaystyle F_{T}^{\ast}(L_{T}^{\ast}q_{f})=0$. By Lemma 2.7, we
infer that $\displaystyle q_{f}\leq 0$.

Then, recalling that the target satisfies $\displaystyle y_{f}-S_{T}y_{0}\geq
0\Leftrightarrow y_{f}\geq S_{T}y_{0}$, we end up with

$\liminf_{n\to\infty}\frac{J_{T,\varepsilon}(p^{n}_{f})}{\|p^{n}_{f}\|_{L^{2}}}\geq-\langle
y_{f}-S_{T}y_{0},q_{f}\rangle_{L^{2}}+\varepsilon\geq\varepsilon>0,$

which concludes the proof. ∎

###  3.3 Characterisation of the minimisers

In this section, we assume that the target is not reached with the trivial
control $\displaystyle u=0$, i.e.,

$\displaystyle y_{f}\notin\overline{B}(S_{T}y_{0},\varepsilon)$

Note that, if (21) is not satisfied, the control $\displaystyle u=0$ steers
$\displaystyle y_{0}$ to the target, and is indeed a control in
$\displaystyle\mathcal{U}_{\textrm{shape}}^{L}$.

We first remark the following fact:

######  Lemma 3.6.

Under Assumption (21), any minimiser $\displaystyle p_{f}^{\star}$ of (20)
satisfies $\displaystyle p_{f}^{\star}\neq 0$.

###### Proof.

Suppose $\displaystyle p_{f}^{\star}=0$. Then, $\displaystyle d=0$ and by
strong duality, $\displaystyle\pi=0$. By Proposition 3.5, this value is
attained: there exists some optimal control $\displaystyle u^{\star}$ such
that

$\displaystyle F_{T}(u^{\star})+G_{T,\varepsilon}(L_{T}u^{\star})=0.$

This implies that $\displaystyle
F_{T}(u^{\star})=G_{T,\varepsilon}(L_{T}u^{\star})=0$. On the one hand, this
leads to $\displaystyle u^{\star}(t,\cdot)=0$ for a.e $\displaystyle
t\in(0,T)$, i.e., $\displaystyle u=0$, and on the other hand $\displaystyle
G_{T,\varepsilon}(L_{T}u^{\star})=G_{T,\varepsilon}(0)=0$, which is equivalent
to $\displaystyle 0\in\overline{B}(y_{f}-S_{T}y_{0},\varepsilon)$. This
contradicts Assumption (21). ∎

######  Proposition 3.7.

Under Assumption (21), any optimal control for (19) is of the form (22), with
$\displaystyle p^{\star}$ the solution of the adjoint equation (9) such that
$\displaystyle p^{\star}(T)=p_{f}^{\star}$, where $\displaystyle
p_{f}^{\star}$ is any dual optimal variable.

###### Proof.

Thanks to Proposition 3.5, we know that $\displaystyle J_{T,\varepsilon}$
defined by (20) attains its minimum. Let $\displaystyle p_{f}^{\star}$ be a
minimiser for $\displaystyle J_{T,\varepsilon}$, i.e., an optimal dual
variable. From Lemma 3.6, we have $\displaystyle p_{f}^{\star}\neq 0$. We
denote $\displaystyle p^{\star}$ the solution of the adjoint equation (9) such
that $\displaystyle p^{\star}(T)=p_{f}^{\star}$.

Let $\displaystyle u^{\star}$ be an optimal control. Thanks to Lemma 3.4, we
can apply the first identity of (50) in Proposition A.8 (see Appendix A.4) to
obtain $\displaystyle u^{\star}\in\partial
F_{T}^{\ast}(L_{T}^{\ast}p_{f}^{\star})=\partial F_{T}^{\ast}(p^{\star})$.
Using again the notation $\displaystyle
H(p):=\textstyle\int_{0}^{T}\sigma_{\overline{\mathcal{U}}_{L}}(p(t))\,dt$, so
that $\displaystyle F_{T}^{\ast}=\textstyle\frac{1}{2}H^{2}$, we have
$\displaystyle H(p^{\star})\geq 0$ and
$\displaystyle\operatorname{dom}(H)=L^{2}(\Omega)$.

Then, applying the generalised chain rule (see [11, Theorem 2.3.9, point
(ii)]) with the functions $\displaystyle x\mapsto\textstyle\frac{1}{2}x^{2}$
and $\displaystyle H$, we compute the subdifferential of the convex functional
$\displaystyle F_{T}^{\ast}$: $\displaystyle u^{\star}\in
H(p^{\star})\,\partial H(p^{\star}).$ Applying Lemma A.5 to $\displaystyle H$,
we find $\displaystyle
u^{\star}(t,\cdot)\in\overline{M}\partial\sigma_{\overline{\mathcal{U}}_{L}}(p^{\star}(t,\cdot))$
for a.e. $\displaystyle t\in(0,T)$, with
$\displaystyle\overline{M}:=H(p^{\star})$.

Now, let $\displaystyle t\in(0,T)$ be fixed and let us justify that all level
sets of $\displaystyle p^{\star}(t,\cdot)$ are of measure zero, i.e.,

$\displaystyle|\\{p^{\star}(t,\cdot)=\lambda\\}|=0,\quad\forall\lambda\in\mathbb{R},$

Indeed, since the operator $\displaystyle\partial_{t}-A^{\ast}$ is analytic-
hypoelliptic, we know that $\displaystyle p^{\star}(t,\cdot)$ is analytic on
$\displaystyle\Omega$. Hence, its level sets are of measure zero unless
$\displaystyle p^{\star}(t,\cdot)=S_{T-t}^{\ast}p_{f}^{\star}$ is constant.
Using the (GUC) property, this leads to $\displaystyle p_{f}^{\star}=0$,
contradicting (21).

Applying Propositions 2.3 and 2.4, and recalling that
$\displaystyle\partial\sigma_{\overline{\mathcal{U}}_{L}}(p^{\star}(t,\cdot))=\\{\chi_{\\{p^{\star}(t,\cdot)>h(p^{\star}(t,\cdot))\\}}\\}$,
we obtain the result. ∎

######  Remark 3.8.

As evidenced by the proof, a weaker (but less workable) property than
analytic-hypoellipticity is sufficient to infer that optimal controls are on-
off shape controls. Indeed, it suffices to require either one of the following
conditions (in decreasing order of strength):

  1. (i)

All solutions $\displaystyle t\mapsto p(t)$ of the adjoint equation such that
$\displaystyle p(T)\neq 0$ have zero-measure level sets.

  2. (ii)

For all solutions $\displaystyle t\mapsto p(t)$ of the adjoint equation such
that $\displaystyle p(T)\neq 0$, the level sets
$\displaystyle\\{p(t,\cdot)=h(p(t,\cdot))\\}$ (see (16) for the definition of
$\displaystyle h(p)$) have measure $\displaystyle 0$.

  3. (iii)

For all solutions $\displaystyle t\mapsto p(t)$ of the adjoint equation such
that $\displaystyle p(T)\neq 0$,

$\displaystyle\left\\{\begin{aligned}
&|\\{p(t,\cdot)=h(p(t,\cdot))\\}|=L|\Omega|-|\\{p(t,\cdot)>h(p(t,\cdot))\\}|,\quad&\text{if}\
h(p(t,\cdot))\neq 0\\\ &|\\{p(t,\cdot)=h(p(t,\cdot))\\}|=0,\quad&\text{if}\
h(p(t,\cdot))=0\end{aligned}\quad\mathrm{for\ a.e.}\ t\in[0,T].\right.$

Note that requirement $\displaystyle(iii)$ is minimal (see Lemma B.2 and
Remark B.3).

Finally, an even weaker requirement would be to restrict any of the above
$\displaystyle(i)$, $\displaystyle(ii)$ or $\displaystyle(iii)$ to a single
solution $\displaystyle t\mapsto p^{\star}(t)$ of the adjoint equation, namely
that with $\displaystyle p^{\star}(T)=p_{f}^{\star}$ where $\displaystyle
p_{f}^{\star}$ is the unique dual optimal variable (see below for the
uniqueness of optimal variables).

###  3.4 Uniqueness

Our first uniqueness statement below (i.e., that of the dual optimal variable)
is a consequence of Fenchel-Rockafellar duality, and the fact that we work
with a Hilbert space, rather than specific properties of the evolution
equation under consideration.

######  Remark 3.9.

Still applying Proposition A.8, we get

$L_{T}u^{\star}\in\partial
G_{T,\varepsilon}^{\ast}(-p_{f}^{\star})=\partial\sigma_{\overline{B}(y_{f}-S_{T}y_{0},\varepsilon)}(-p_{f}^{\star}).$

Flipping the subdifferentials, we get $\displaystyle-
p_{f}^{\star}\in\partial\delta_{\overline{B}(y_{f}-S_{T}y_{0},\varepsilon)}(L_{T}u^{\star})$.
Thanks to Proposition A.4, this means that $\displaystyle L_{T}u^{\star}$ lies
at the boundary of the closed ball
$\displaystyle\overline{B}(y_{f}-S_{T}y_{0},\varepsilon)$.

######  Proposition 3.10.

Under the assumptions of Theorem 3.1, the primal-dual optimal pairs
$\displaystyle(u^{\star},p_{f}^{\star})$ are unique.

###### Proof.

Uniqueness of the dual optimal variable. First note that if Assumption (21)
does not hold, then $\displaystyle 0$ is the unique optimal control, i.e.,

$\\{L_{T}u^{\star},\,u^{\star}\text{ is optimal}\\}=\\{0\\}.$ (24)

On the other hand, if Assumption (21) holds, according to Remark 3.9,and since
the set of minimisers of a convex function is convex, the set
$\displaystyle\\{L_{T}u^{\star},\,u^{\star}\text{ is optimal}\\}$ is a convex
subset of the sphere $\displaystyle S(y_{f}-S_{T}y_{0},\varepsilon)$. The
closed ball being strictly convex since we are working in the Hilbert space
$\displaystyle L^{2}(\Omega)$, there exists some $\displaystyle
y^{\star}\in\overline{B}(y_{f}-S_{T}y_{0},\varepsilon)$ with
$\displaystyle\|y^{\star}-(y_{f}-S_{T}y_{0})\|_{L^{2}}=\varepsilon$ such that

$\\{L_{T}u^{\star},\,u^{\star}\text{ is optimal}\\}=\\{y^{\star}\\}.$ (25)

Thus, in any case, the set of targets reached by optimal controls is always
reduced to a single point.

Now, let $\displaystyle p_{f}^{\star}$ be a dual optimal variable, and
$\displaystyle u^{\star}$ an optimal control. Then, as strong duality holds,
Proposition A.7 implies that the pair $\displaystyle(u^{\star},p_{f}^{\star})$
satisfies the two optimality conditions from (25). We then have

$p_{f}^{\star}\in-\partial
G_{T,\varepsilon}(L_{T}u^{\star})=-\partial\delta_{\overline{B}(y_{f}-S_{T}y_{0},\varepsilon)}(L_{T}u^{\star}).$
(26)

If Assumption (21) does not hold, then (26) and (24) imply $\displaystyle
p_{f}^{\star}\in-\partial\delta_{\overline{B}(y_{f}-S_{T}y_{0},\varepsilon)}(0)$.
If $\displaystyle\|y_{f}-S_{T}y_{0}\|_{L^{2}}<\varepsilon$, then
$\displaystyle 0\in B(y_{f}-S_{T}y_{0},\varepsilon)$ and

$p_{f}^{\star}\in-\partial\delta_{\overline{B}(y_{f}-S_{T}y_{0},\varepsilon)}(0)=\\{0\\}.$
(27)

Otherwise, $\displaystyle 0\in\partial B(y_{f}-S_{T}y_{0},\varepsilon)$ and
(47) yield

$\displaystyle
p_{f}^{\star}\in\left\\{\lambda\frac{y_{f}-S_{T}y_{0}}{\varepsilon},\lambda\geq
0\right\\}=\\{\lambda(y_{f}-S_{T}y_{0}),\lambda\geq 0\\}.$

Restricting the function $\displaystyle J_{T,\varepsilon}$ defining the primal
problem (19) to the above half-line, using the homogeneities of each of its
terms, and the fact that
$\displaystyle\|y_{f}-S_{T}y_{0}\|_{L^{2}}=\varepsilon$, we get

$\gamma_{0}(\lambda):=J_{T,\varepsilon}(\lambda(y_{f}-S_{T}y_{0}))=a_{0}\lambda^{2},\quad\lambda\geq
0.$ (28)

It is clear that $\displaystyle 0$ is the unique minimiser of
$\displaystyle\gamma_{0}$. From (27) and (28), $\displaystyle 0$ is the unique
dual optimal variable if (21) does not hold.

If Assumption (21) holds, then (26) and (25) imply

$\displaystyle
p_{f}^{\star}\in-\partial\delta_{\overline{B}(y_{f}-S_{T}y_{0},\varepsilon)}(y^{\star})=-\partial\delta_{\overline{B}(0,1)}\left(\frac{y^{\star}-(y_{f}-S_{T}y_{0})}{\varepsilon}\right).$

Since $\displaystyle y^{\star}$ lies at the boundary of
$\displaystyle\overline{B}(y_{f}-S_{T}y_{0},\varepsilon)$, formula (47) yields

$\displaystyle
p_{f}^{\star}\in\left\\{\lambda\left(\frac{y_{f}-S_{T}y_{0}-y^{\star}}{\varepsilon}\right),\lambda\geq
0\right\\}=\\{\lambda\left(y_{f}-S_{T}y_{0}-y^{\star}\right),\lambda\geq
0\\}.$

Restricting $\displaystyle J_{T,\varepsilon}$ to the above half-line as
previously, we find

$\displaystyle\gamma(\lambda):=J_{T,\varepsilon}(\lambda(y_{f}-S_{T}y_{0}-y^{\star}))=a\lambda^{2}+b\lambda,\quad\lambda\geq
0,$

where, using $\displaystyle\|y_{f}-S_{T}y_{0}-y^{\star}\|_{L^{2}}=\varepsilon$
and the homogeneities involved $\displaystyle
a=F_{T}^{\ast}(L_{T}^{\ast}(y_{f}-S_{T}y_{0}-y^{\star}))$ and $\displaystyle
b=-\langle
y_{f}-S_{T}y_{0},y_{f}-S_{T}y_{0}-y^{\star}\rangle_{L^{2}}+\varepsilon^{2}$.
By coercivity, $\displaystyle a>0$, and given Lemma 3.6, we have
$\displaystyle b<0$.

Thus, $\displaystyle\gamma$ has a unique minimiser
$\displaystyle\lambda^{\star}:=-b/2a>0$. Hence, $\displaystyle
p_{f}^{\star}=\lambda^{\star}(y_{f}-S_{T}y_{0}-y^{\star})$, and the dual
optimal variable is unique.

##### Uniqueness of the optimal control.

If Assumption (21) does not hold, then $\displaystyle 0$ is the unique optimal
control.

Now, suppose that Assumption (21) holds. We know from the proof of Proposition
3.7 that a given dual optimal variable uniquely determines one optimal
control. Moreover, as we have proved that strong duality holds, we can apply
Proposition A.7: for any pair of primal and dual optimal variables, the
relations (48) are satisfied. That is, any optimal control $\displaystyle
u^{\star}$ is uniquely determined by the unique dual optimal variable
$\displaystyle p_{f}^{\star}$ through the identity $\displaystyle
u^{\star}\in\partial F_{T}^{\ast}(L_{T}^{\ast}p_{f}^{\star})$. ∎

##  4 Obstructions to controllability

We here prove Theorem B, through the more general result below in the case of
second-order uniformly elliptic operators of the form (7). We use the notation
$\displaystyle A\subset\subset B$ to mean that there exists a compact set
$\displaystyle K$ such that $\displaystyle A\subset K\subset B$.

######  Theorem 4.1.

Let $\displaystyle\mathcal{U}_{+}\subset L^{2}(\Omega)$ be a constraint set of
nonnegative controls. Assume that there exists a ball $\displaystyle
B(x,r)\subset\Omega$ such that

$\displaystyle\forall u\in\mathcal{U}_{+},\quad\mathrm{supp}(u)\cap
B(x,r)=\emptyset.$

Let $\displaystyle A$ be a second-order uniformly elliptic operator of the
form (7). Let $\displaystyle y_{0}=0$ and $\displaystyle y_{f}\in
L^{2}(\Omega)$ be any target such that $\displaystyle y_{f}\geq S_{T}y_{0}=0$,
$\displaystyle y_{f}\neq 0$ and $\displaystyle\mathrm{supp}(y_{f})\subset
B(x,r)$. Then there exist $\displaystyle T^{\star}>0$ and
$\displaystyle\varepsilon>0$ such that for any time $\displaystyle T\leq
T^{\star}$, no control with values in $\displaystyle\mathcal{U}_{+}$ can steer
$\displaystyle 0$ to $\displaystyle\overline{B}(y_{f},\varepsilon)$.

The proof relies on the following lemma, inspired by [36].

######  Lemma 4.2.

Let $\displaystyle B(x,r)\subset\subset\Omega$. Under the assumptions of
Theorem 4.1, for any $\displaystyle K\subset B(x,r)$ compact, there exists
$\displaystyle p_{f}\in L^{2}(\Omega)$ such that

  * (i)

$\displaystyle p_{f}<0$ on $\displaystyle K$,

  * (ii)

$\displaystyle\exists\,T^{\star}>0$ such that for all $\displaystyle
t\in(0,T^{\star})$, $\displaystyle p(t,\cdot)\geq 0$ on
$\displaystyle\Omega\setminus B(x,r)$, where $\displaystyle p$ solves the
adjoint equation (9) with $\displaystyle p(T)=p_{f}$.

###### Proof.

Let us build $\displaystyle p_{f}$ such that for all $\displaystyle
1<r<+\infty$, $\displaystyle p_{f}\in W^{2,r}(\Omega)\cap
W^{1,r}_{0}(\Omega)$, with $\displaystyle p_{f}<0$ on $\displaystyle K$,
$\displaystyle p_{f}>0$ on $\displaystyle\Omega\setminus B(x,r)$,
$\displaystyle p_{f}=0$ on $\displaystyle\partial\Omega$, and
$\displaystyle\partial_{\nu}p_{f}<0$ on $\displaystyle\partial\Omega$.

To that end, we denote $\displaystyle\varphi_{1}$ the first eigenfunction of
the Dirichlet Laplacian on $\displaystyle\Omega$, which satisfies
$\displaystyle\varphi_{1}>0$ on $\displaystyle\Omega$ and
$\displaystyle\partial_{\nu}\varphi_{1}<0$ on $\displaystyle\partial\Omega$
and since $\displaystyle\Omega$ is of class $\displaystyle C^{2}$,
$\displaystyle\varphi_{1}\in W^{2,r}(\Omega)\cap W^{1,r}_{0}(\Omega)$ for all
$\displaystyle 1<r<+\infty$ [9][Theorem 9.32]. We then set $\displaystyle
p_{f}=\xi\varphi_{1}$ where $\displaystyle\xi\in C^{\infty}(\Omega)$ is chosen
to satisfy $\displaystyle\xi=1$ on $\displaystyle\Omega\setminus B(x,r)$ and
$\displaystyle\xi=-1$ on $\displaystyle K$. The function $\displaystyle p_{f}$
satisfies all the required properties (note that $\displaystyle
p_{f}=\varphi_{1}$ locally around $\displaystyle\partial\Omega$ since
$\displaystyle B(x,r)\subset\subset\Omega$, hence
$\displaystyle\partial_{\nu}p_{f}=\partial_{\nu}\varphi_{1}<0$ on
$\displaystyle\partial\Omega$).

We now set $\displaystyle q(t)=p(T-t)$, so that $\displaystyle q$ solves the
(forward) adjoint equation (9) with $\displaystyle q(0)=p_{f}$. Then, by
parabolic regularity, we both have $\displaystyle q\in
C([0,T]\times\overline{\Omega})$ and $\displaystyle\partial_{\nu}q\in
C([0,T]\times\partial\Omega)$ [36][Theorem 8.1]. As a result, by continuity
there exists $\displaystyle T^{\star}$ such that
$\displaystyle\partial_{\nu}q<0$ over
$\displaystyle[0,T^{\star}]\times\partial\Omega$, there exists some compact
set $\displaystyle K_{1}$ containing $\displaystyle B(x,r)$ such that
$\displaystyle q\geq 0$ on $\displaystyle[0,T^{\star}]\times(\Omega\setminus
K_{1})$.

Then, upon reducing $\displaystyle T^{\star}$ if necessary and by continuity
again, we have $\displaystyle q\geq 0$ over
$\displaystyle[0,T^{\star}]\times(K_{1}\setminus B(x,r))$, which concludes the
proof. ∎

###### Proof of Theorem 4.1.

Upon reducing $\displaystyle r$, we may without loss of generality assume that
$\displaystyle B(x,r)\subset\subset\Omega$. Letting $\displaystyle
K:=\mathrm{supp}(y_{f})$, we consider $\displaystyle p_{f}$ as given by Lemma
4.2, and the corresponding $\displaystyle T^{\star}$.

Let $\displaystyle T\leq T^{\star}$ be fixed. For any control $\displaystyle
u\in E$, any $\displaystyle y_{0},y_{f}\in L^{2}(\Omega)$, any solution to the
adjoint equation (9) such that $\displaystyle p(T)=p_{f}$, we have
$\displaystyle\textstyle\frac{d}{dt}\langle y(t),p(t)\rangle_{L^{2}}=\langle
p(t),u(t)\rangle_{L^{2}}$. As a result and owing to $\displaystyle y_{0}=0$,

$\langle y(T),p_{f}\rangle_{L^{2}}=\int_{0}^{T}\langle
p(t),u(t)\rangle_{L^{2}}\,dt.$ (29)

We now assume by contradiction that, for any $\displaystyle\varepsilon>0$
there exists a nonnegative control $\displaystyle u_{\varepsilon}\in E$
satisfying $\displaystyle\forall
t\in(0,T),\,\mathrm{supp}(u_{\varepsilon}(t))\cap B(x,r)=\emptyset$ and
steering $\displaystyle y_{0}=0$ to the ball
$\displaystyle\overline{B}(y_{f},\varepsilon)$ in time $\displaystyle T$. We
inspect the sign of the equality (29) along the controls $\displaystyle
u_{\varepsilon}$, $\displaystyle\varepsilon>0$.

On the one hand, because of the condition (ii) in Lemma 4.2 satisfied by
$\displaystyle p$, and owing to $\displaystyle u_{\varepsilon}\geq 0$, the
right-hand side of (29) is nonnegative, i.e.,

$\langle y(T),p_{f}\rangle_{L^{2}}\geq 0.$ (30)

On the other hand, the left-hand side of (29) satisfies

$\displaystyle\langle y(T),p_{f}\rangle_{L^{2}}=\langle
y_{f},p_{f}\rangle_{L^{2}}+\langle y(T)-y_{f},p_{f}\rangle_{L^{2}}\leq\langle
y_{f},p_{f}\rangle_{L^{2}}+\varepsilon\|p_{f}\|_{L^{2}}$

Now, $\displaystyle\langle y_{f},p_{f}\rangle_{L^{2}}<0$, because of (i) in
Lemma 4.2. As a result, there exists $\displaystyle\alpha>0$ such that
$\displaystyle p_{f}\leq-\alpha$ on $\displaystyle K$, so that

$\displaystyle\langle y_{f},p_{f}\rangle_{L^{2}}\leq-\alpha\int_{K}y_{f}<0,$

because $\displaystyle y_{f}$ is nonnegative and nontrivial on $\displaystyle
K$ by assumption.

Hence, for $\displaystyle\varepsilon>0$ small enough, $\displaystyle\langle
y(T),p_{f}\rangle_{L^{2}}<0$, which contradicts (30). ∎

######  Remark 4.3.

As the proof shows, the obstruction to nonnegative approximate controllability
in $\displaystyle\mathcal{U}_{+}$ does not rely on the comparison principle,
but is of dual nature as evidenced by the core idea behind it, i.e., to
construct $\displaystyle p_{f}$ and $\displaystyle y_{f}$ violating equality
(29). The proof of Theorem 4.1 follows directly from the existence of
$\displaystyle p_{f}$ satisfying the assumptions of Lemma 4.2. Hence, this
obstruction to nonnegative approximate controllability is rather general and
will be satisfied by any operator (including uniformly second-order elliptic
operator of the form (7)) for which such an element $\displaystyle p_{f}$ can
be built.

##  5 Further comments

###  5.1 Properties of the value function in the general case

For general linear operators generating a $\displaystyle C_{0}$ semigroup,
fixing $\displaystyle\Omega$, $\displaystyle L$, $\displaystyle\varepsilon$,
$\displaystyle y_{0}$ and $\displaystyle y_{f}$, we analyse the dependence
with respect to the final time $\displaystyle T$, for the optimal control
problem (19) studied in Section 3 for system (5).

By Lemma 3.4 and Proposition 3.5, the optimal control problem (19) is well-
posed, i.e., optimal controls exist (see also Remark 3.3), hence we may
consider

$\Pi(T):=\frac{1}{2}(\overline{M}(T))^{2}:=\inf\\{F_{T}(u),\quad u\in
E,\quad\|L_{T}u-(y_{f}-S_{T}y_{0})\|_{L^{2}}\leq\varepsilon\\},\quad T>0.$
(31)

When $\displaystyle A^{\ast}$ satisfies the (GUC) property and
$\displaystyle\partial_{t}-A^{\ast}$ is analytic-hypoelliptic,
$\displaystyle\overline{M}(T)$ is the amplitude of the unique optimal control
in Proposition 3.7.

Recall that by strong duality, we have

$\Pi(T)=\frac{1}{2}(\overline{M}(T))^{2}=-J_{T,\varepsilon}(p^{\star}_{T}),\quad\forall
T\geq 0,$ (32)

where $\displaystyle p_{T}^{\star}$ is the unique minimiser of $\displaystyle
J_{T,\varepsilon}$. This is exactly the identity obtained for the HUM method
where the cost functional $\displaystyle F_{T}$ is just
$\displaystyle\textstyle\frac{1}{2}\|\cdot\|_{E}^{2}$.

We first establish the continuity of $\displaystyle T\mapsto\overline{M}(T)$.

######  Proposition 5.1.

$\displaystyle\overline{M}$ (and thus $\displaystyle\Pi$) are continuous on
$\displaystyle(0,+\infty)$.

###### Proof.

Using (32), we prove the continuity by showing that
$\displaystyle(p_{f},T)\mapsto J_{T,\varepsilon}(p_{f})$ (given by (23))
satisfies the assumptions of Lemma A.9 with $\displaystyle H=L^{2}(\Omega)$
and $\displaystyle Z=(0,+\infty)$. Clearly, the first, second and fourth
assumptions are satisfied, hence we are left with proving that
$\displaystyle(p_{f},T)\mapsto J_{T,\varepsilon}(p_{f})$ is weak-strong lower
semicontinuous over $\displaystyle L^{2}(\Omega)\times(0,+\infty)$. The last
two terms of (23) are easily seen to be weak-strong lower semicontinuous over
$\displaystyle L^{2}(\Omega)\times(0,+\infty)$, hence we investigate the
property for the remaining term $\displaystyle
F_{T}^{\ast}(L_{T}^{\ast}p_{f})$.

Given $\displaystyle p_{f}\in L^{2}(\Omega)$ and $\displaystyle T>0$, let
$\displaystyle(p_{f}^{n})$ and $\displaystyle(T_{n})$ be two sequences such
that $\displaystyle p_{f}^{n}\rightharpoonup p_{f}$, $\displaystyle
T_{n}\rightarrow T$. We decompose

$\displaystyle\displaystyle F_{T_{n}}^{\ast}(L_{T_{n}}^{\ast}p_{f}^{n})$
$\displaystyle\displaystyle=F_{T}^{\ast}(L_{T}^{\ast}p_{f}^{n})+\left(F_{T_{n}}^{\ast}(L_{T_{n}}^{\ast}p_{f}^{n})-F_{T}^{\ast}(L_{T}^{\ast}p_{f}^{n})\right).$

By weak (sequential) lower semicontinuity of $\displaystyle F_{T}^{\ast}$ over
$\displaystyle L^{2}(0,T;L^{2}(\Omega))$, we find that the first term
satisfies

$\displaystyle
F_{T}^{\ast}(L_{T}^{\ast}p_{f})\leq\liminf_{n\rightarrow+\infty}F_{T}^{\ast}(L_{T}^{\ast}p_{f}^{n}).$

To conclude, we only need to prove that the second term tends to
$\displaystyle 0$ as $\displaystyle n\rightarrow+\infty$.

Using the notation $\displaystyle q_{n}$ for the solution to the forward
adjoint problem such that $\displaystyle q_{n}(0)=p_{f}^{n}$, i.e.,
$\displaystyle q_{n}(t)=S_{t}^{\ast}p_{f}^{n}$, we have

$\displaystyle\displaystyle
F_{T_{n}}^{\ast}(L_{T_{n}}^{\ast}p_{f}^{n})-F_{T}^{\ast}(L_{T}^{\ast}p_{f}^{n})$
$\displaystyle\displaystyle=\frac{1}{2}\left(\int_{0}^{T_{n}}\sigma_{\overline{\mathcal{U}}_{L}}(q_{n}(T_{n}-t))\,dt\right)^{2}-\frac{1}{2}\left(\int_{0}^{T}\sigma_{\overline{\mathcal{U}}_{L}}(q_{n}(T-t))\right)^{2}$
$\displaystyle\displaystyle=\frac{1}{2}\left(\int_{T}^{T_{n}}\sigma_{\overline{\mathcal{U}}_{L}}(q_{n}(t))\,dt\right)\left(\int_{0}^{T_{n}}\sigma_{\overline{\mathcal{U}}_{L}}(q_{n}(t))\,dt+\int_{0}^{T}\sigma_{\overline{\mathcal{U}}_{L}}(q_{n}(t))\,dt\right)$

Using the bound $\displaystyle
0\leq\sigma_{\overline{\mathcal{U}}_{L}}(p)\leq|\Omega|^{1/2}\|p\|_{L^{2}}$
(see the proof of Lemma 3.4) and the estimate
$\displaystyle\|S_{t}\|_{L(L^{2}(\Omega))}\leq C$ valid for all $\displaystyle
t\in[0,T+1]$ with $\displaystyle C>0$ some constant independent of
$\displaystyle n$, we have

$\displaystyle\displaystyle\left|\int_{0}^{T_{n}}\sigma_{\overline{\mathcal{U}}_{L}}(q_{n}(t))\,dt+\int_{0}^{T}\sigma_{\overline{\mathcal{U}}_{L}}(q_{n}(t))\,dt\right|\leq
C|\Omega|^{1/2}(T+T_{n})\,\|p_{f}^{n}\|_{L^{2}},$

a bounded quantity, and

$\displaystyle\displaystyle\left|\int_{T}^{T_{n}}\sigma_{\overline{\mathcal{U}}_{L}}(q_{n}(t))\,dt\right|\leq
C|\Omega|^{1/2}|T-T_{n}|\,\|p_{f}^{n}\|_{L^{2}},$

which tends to $\displaystyle 0$ as $\displaystyle n\rightarrow+\infty$. ∎

We now study the behaviour of $\displaystyle\overline{M}(T)$ near
$\displaystyle T=0$ and $\displaystyle T=+\infty$. We recall that
$\displaystyle\overline{M}(T)$ also depends on all other parameters
$\displaystyle y_{0}$, $\displaystyle y_{f}$, $\displaystyle\varepsilon$ and
$\displaystyle L$.

We now recall (see [35]) that there exist $\displaystyle C_{s}>0$,
$\displaystyle\alpha\in\mathbb{R}$ such that forall $\displaystyle t\geq 0$,
$\displaystyle\|S_{t}\|_{L(L^{2}(\Omega))}\leq C_{s}e^{\alpha t}$, and the
semi-group generated by $\displaystyle(A,D(A))$ is said to be exponentially
stable if $\displaystyle\alpha<0$.

######  Proposition 5.2.

We have

$\forall
T>0,\quad\overline{M}(T)\geq|\alpha|\frac{\|y_{f}-S_{T}y_{0}\|_{L^{2}}-\varepsilon}{\sqrt{L|\Omega|}(1-e^{\alpha
T})}.$ (33)

###### Proof.

Let $\displaystyle u^{\star}_{T}$ be an optimal control in time $\displaystyle
T$ for the optimal control problem (31), then

${\begin{array}[]{rl}\|L_{T}u^{\star}_{T}\|_{L^{2}}&=\left\|\int_{0}^{T}S_{T-t}u^{\star}_{T}(t,\cdot)dt\right\|_{L^{2}}\leq\int_{0}^{T}\|S_{T-t}u^{\star}_{T}(t,\cdot)\|_{L^{2}}dt\\\
&\leq\int_{0}^{T}e^{\alpha(T-t)}\|u^{\star}_{T}(t,\cdot)\|_{L^{2}}dt\leq\frac{1}{|\alpha|}(1-e^{\alpha
T})\overline{M}(T)\sqrt{L|\Omega|}.\end{array}}$

Now, by definition of our control problem, for all $\displaystyle T>0$,
$\displaystyle\|y_{f}-S_{T}y_{0}\|_{L^{2}}-\varepsilon\leq\|L_{T}u^{\star}_{T}\|_{L^{2}}$,
and the result follows. ∎

######  Corollary 5.3.

Assume that $\displaystyle y_{f}\notin\overline{B}(y_{0},\varepsilon)$. Then:

$\frac{1}{T}=\underset{T\rightarrow 0}{O}(\overline{M}(T)).$ (34)

In particular, $\displaystyle\overline{M}(T)\xrightarrow[T\to 0]{}+\infty$.

Assume that $\displaystyle y_{f}\notin\overline{B}(0,\varepsilon)$. If,
additionally, $\displaystyle(S_{t})_{t\geq 0}$ is exponentially stable, then

$\liminf_{T\rightarrow+\infty}\overline{M}(T)>0.$ (35)

###### Proof.

The estimate (34) is obtained by passing to the limit in (33), using that
$\displaystyle S_{T}y_{0}\xrightarrow[T\to 0]{}y_{0}$: the lower bound behaves
as
$\displaystyle\textstyle\frac{\|y_{f}-y_{0}\|_{L^{2}}-\varepsilon}{\sqrt{L|\Omega|}}\;\frac{1}{T}$.
The inequality (35) is obtained by passing to the limit $\displaystyle
T\rightarrow+\infty$ in (33), using that $\displaystyle
S_{T}y_{0}\xrightarrow[T\to\infty]{}0$:

$\displaystyle\liminf_{T\rightarrow+\infty}\overline{M}(T)\geq|\alpha|\frac{\|y_{f}\|_{L^{2}}-\varepsilon}{\sqrt{L|\Omega|}}>0.$

∎

###  5.2 Obstructions

We further investigate the behaviour of $\displaystyle\overline{M}$, and
establish results on the corresponding minimal time problem (37). The
comparison principle formulated in (6) will be a key ingredient in our study.

####  5.2.1 Obstruction to reachability and small-time controllability

Given the controllability result of Theorem 3.1, in order to study possible
obstructions, we introduce a new bound on the amplitude of the control, of the
form:

$M(u):=2\sqrt{F_{T}(u)}\leq M_{\max},\quad u\in E,$ (36)

for some $\displaystyle M_{\max}>0$. Note that such a constraint imposes
nonnegativity of the control. With this new constraint on the controls, we
illustrate a general property that is well known for finite-dimensional
systems: exponential stability prevents reachability.

In particular, the result below holds for uniformly elliptic operators of the
form (7) with $\displaystyle 0$th order coefficient satisfying $\displaystyle
c\leq 0$.

######  Proposition 5.4.

Assume that $\displaystyle(S_{t})_{t\geq 0}$ is exponentially stable. Let
$\displaystyle(y_{0},y_{f})$ be such that for all $\displaystyle T\geq 0$,
$\displaystyle y_{f}\geq S_{T}y_{0}$ and
$\displaystyle\|S_{T}y_{0}-y_{f}\|_{L^{2}}\geq\delta$ for some
$\displaystyle\delta>0$. Then, for all $\displaystyle 0<\varepsilon<\delta$
there exists $\displaystyle M_{\max}{M}(y_{0},y_{f},\varepsilon)>0$ satisfying

  * •

if $\displaystyle M_{\max}>M_{\max}(y_{0},y_{f},\varepsilon)$, there exists a
time $\displaystyle T>0$ and a control $\displaystyle u\in E$ satisfying (36),
steering $\displaystyle y_{0}$ to
$\displaystyle\overline{B}(y_{f},\varepsilon)$ in time $\displaystyle T$. If
$\displaystyle A^{\ast}$ satisfies the (GUC) property and
$\displaystyle\partial_{t}-A^{\ast}$ is analytic-hypoelliptic, the control may
be chosen to be in $\displaystyle\mathcal{U}^{L}_{\textrm{shape}}$.

  * •

if $\displaystyle M_{\max}<M_{\max}(y_{0},y_{f},\varepsilon)$, no such control
exists.

Moreover, for all $\displaystyle M_{\max}>0$, the control system (5) is not
nonnegatively approximately controllable with controls in
$\displaystyle\\{M(u)\leq M_{\max}\\}$ in any time $\displaystyle T>0$.

###### Proof.

Given Corollary 5.3, the function $\displaystyle\overline{M}(T)$ goes to
$\displaystyle+\infty$ as $\displaystyle T\rightarrow 0$, is bounded away from
$\displaystyle 0$ at infinity, and does not vanish over the interval
$\displaystyle(0,+\infty)$. Since it is continuous, we define

$\displaystyle
M_{\max}(y_{0},y_{f},\varepsilon):=\inf_{T>0}\overline{M}(T)>0,$

and the first two claims follow. When $\displaystyle A^{\ast}$ satisfies the
(GUC) property and $\displaystyle\partial_{t}-A^{\ast}$ is analytic-
hypoelliptic, the control may be chosen to be in
$\displaystyle\mathcal{U}^{L}_{\textrm{shape}}$ by Theorem 3.1.

Then, let $\displaystyle M_{\max}>0$. Taking $\displaystyle y_{f}\in
L^{2}(\Omega)$ such that
$\displaystyle\|y_{f}\|_{L^{2}}>\textstyle\frac{\sqrt{L|\Omega|}}{|\alpha|}M_{\max}+\varepsilon$
and $\displaystyle y_{0}\in L^{2}(\Omega)$ such that $\displaystyle y_{f}\geq
S_{T}y_{0}$ and $\displaystyle\|S_{T}y_{0}-y_{f}\|_{L^{2}}\geq\delta>0$.
Thanks to the proof of Corollary 5.3, we infer $\displaystyle
M_{\max}(y_{0},y_{f},\varepsilon)\geq|\alpha|\textstyle\frac{\|y_{f}\|_{L^{2}}-\varepsilon}{\sqrt{L|\Omega|}}>M_{\max}$.
It follows from the second claim that $\displaystyle y_{0}$ cannot be steered
to $\displaystyle y_{f}$ in any time $\displaystyle T>0$ with a control
$\displaystyle u$ such that $\displaystyle M(u)\leq M_{\max}$. Thus, system
(5) is not nonnegatively approximately controllable with such controls in any
time $\displaystyle T>0$.

∎

####  5.2.2 Characterisation of minimal time controls

Throughout this section, we let $\displaystyle\varepsilon>0$, $\displaystyle
y_{f}\in L^{2}(\Omega)$, we assume that (21) holds, and let $\displaystyle
y_{0}=0$. Hence we must have $\displaystyle\|y_{f}\|_{L^{2}}>\varepsilon$ and
the condition (21) is independent of $\displaystyle T$. Finally,
$\displaystyle y_{f}\geq S_{T}y_{0}$ here simply amounts to $\displaystyle
y_{f}\geq 0$.

Given the obstruction result of Proposition 5.4, we consider the minimal time
control problem:

$T^{\star}(\lambda)=\inf\\{T>0,\quad\exists u\in
E,\quad\|L_{T}u-y_{f}\|_{L^{2}}\leq\varepsilon,\quad
F_{T}(u)\leq\lambda\\},\quad\lambda>0.$ (37)

From our study of the optimal control problem (19), we know that this minimal
time is well defined for $\displaystyle\lambda\in\overline{M}((0,+\infty))$.
Under appropriate assumptions, we will show that it is reached, and
characterise the minimal time controls, by establishing a form of equivalence
between the optimal control problem and the corresponding minimal time
problem. This is now a well-known feature for parabolic equations (see [46,
19, 39]).

##### Further study of the value function $\displaystyle\overline{M}$.

Using strong duality again, we will establish that $\displaystyle\overline{M}$
is a non-increasing function under the assumption that $\displaystyle
A^{\ast}$ satisfies the comparison principle (6). We start with the following
general lemma:

######  Lemma 5.5.

Given any $\displaystyle 0<T_{1}<T_{2}$, and $\displaystyle y_{0}=0$, for a
general unbounded linear operator $\displaystyle A$, the dual functional
defined by (23) satisfies:

$J_{T_{1},\varepsilon}(p_{f})\leq J_{T_{2},\varepsilon}(p_{f}),\quad\forall
p_{f}\in L^{2}(\Omega),$ (38)

with equality if and only if

$L_{T_{2}}^{\ast}p_{f}(t)\leq 0,\quad\forall t\in[0,T_{2}-T_{1}].$ (39)

###### Proof.

Since $\displaystyle y_{0}=0$, inequality (38) follows immediately from the
comparison of the integral terms in the expression of the $\displaystyle
J_{T_{i},\varepsilon},i\in\\{1,2\\}.$ Moreover, for $\displaystyle p_{f}\in
L^{2}(\Omega)$, one has $\displaystyle
J_{T_{1},\varepsilon}(p_{f})=J_{T_{2},\varepsilon}(p_{f})$ if and only if

$\displaystyle\int_{0}^{T_{1}}\sigma_{\overline{\mathcal{U}}_{L}}\left(L_{T_{1}}^{\ast}p_{f}(t)\right)dt=\int_{0}^{T_{2}}\sigma_{\overline{\mathcal{U}}_{L}}\left(L_{T_{2}}^{\ast}p_{f}(t)\right)dt,$

that is, by definition of the operators $\displaystyle L_{T_{i}}^{\ast}$ (see
(9) which are obviously related by $\displaystyle
L_{T_{1},\varepsilon}^{\ast}p_{f}(t)=L_{T_{2},\varepsilon}^{\ast}p_{f}(T_{2}-T_{1}+t)$
for all $\displaystyle t\in(0,T_{1})$,

$\displaystyle\int_{0}^{T_{2}-T_{1}}\sigma_{\overline{\mathcal{U}}_{L}}\left(L_{T_{2}}^{\ast}p_{f}(t)\right)dt=0.$

Using the definition of the support function
$\displaystyle\sigma_{\overline{\mathcal{U}}_{L}}$ (see the proof of Lemma
2.7), this is equivalent to (39). ∎

######  Corollary 5.6.

The function $\displaystyle\overline{M}$ (and hence $\displaystyle\Pi$) are
non-increasing on $\displaystyle(0,+\infty)$.

We now denote
$\displaystyle\mu_{-}=\mu_{-}(y_{f}):=\lim_{T\rightarrow+\infty}\Pi(T)=\lim_{T\rightarrow+\infty}\textstyle\frac{1}{2}\overline{M}(T)^{2}$.
Note that $\displaystyle\mu_{-}\in[0,+\infty)$, and if the semi-group
generated by $\displaystyle A$ is exponentially stable,
$\displaystyle\mu_{-}>0$ as established by (35) in Corollary 5.3.

######  Proposition 5.7.

Assume $\displaystyle A^{\ast}$ satisfies the comparison principle (6). Then,
there exists $\displaystyle T_{\ell}=T_{\ell}(y_{f})\in(0,+\infty]$ such that
$\displaystyle\overline{M}$ is decreasing on $\displaystyle[0,T_{\ell})$, and
constant on $\displaystyle[T_{\ell},+\infty)$.

######  Remark 5.8.

The proposition above implies in particular that $\displaystyle\overline{M}$
either decreases on the whole of $\displaystyle(0,+\infty)$ to its limit
$\displaystyle\mu_{-}$ (if $\displaystyle T_{\ell}=+\infty)$, or reaches it at
$\displaystyle T_{\ell}<+\infty$ and then remains constant.

###### Proof.

By strong duality, Lemma 5.5 implies that $\displaystyle\overline{M}$ is non-
increasing. Let $\displaystyle T_{2}>T_{1}>0$, and denote $\displaystyle
p^{\star}_{T_{1}},p^{\star}_{T_{2}}$ the associated dual minimisers. Assume
that

$\overline{M}(T_{1})=\overline{M}(T_{2}).$ (40)

From Lemma 5.5, and by definition of $\displaystyle p_{T_{1}}^{\star}$, we
know that

$J_{T_{1},\varepsilon}(p_{T_{1}}^{\star})\leq
J_{T_{1},\varepsilon}(p_{T_{2}}^{\star})\leq
J_{T_{2},\varepsilon}(p_{T_{2}}^{\star}).$ (41)

From (32), (40) implies that $\displaystyle
J_{T_{1},\varepsilon}(p_{T_{1}}^{\star})=J_{T_{2},\varepsilon}(p_{T_{2}}^{\star}),$
so that all the inequalities in (41) actually are equalities.

By uniqueness of the dual optimal variable (Proposition 3.10), the first
equality implies that

$p_{T_{1}}^{\star}=p_{T_{2}}^{\star}=:p_{f}^{\star}.$ (42)

From Lemma 5.5, the second equality implies that

$L_{T_{2}}^{\ast}p_{f}^{\star}(t)\leq 0,\quad\forall t\in[0,T_{2}-T_{1}].$
(43)

From (42) and (43), we get $\displaystyle p_{T}^{\star}=p_{f}^{\star}$ for all
$\displaystyle T\in[T_{1},T_{2}]$. Now, for $\displaystyle T>T_{2}$, the
comparison principle (6) and inequality (43) imply that $\displaystyle
L_{T}^{\ast}p_{f}^{\star}(t)\leq 0$ for all $\displaystyle t\in[0,T-T_{1}].$
From Lemma 5.5, we then get $\displaystyle
J_{T,\varepsilon}(p_{f}^{\star})=J_{T_{1},\varepsilon}(p_{f}^{\star}),$ which
implies $\displaystyle
J_{T,\varepsilon}(p_{f}^{\star})=J_{T_{1},\varepsilon}(p_{f}^{\star})\leq
J_{T,\varepsilon}(p_{T}^{\star}).$ By definition of the dual minimiser
$\displaystyle p_{T}^{\star}$ of $\displaystyle J_{T,\varepsilon}$, we also
have $\displaystyle J_{T,\varepsilon}(p_{T}^{\star})\leq
J_{T,\varepsilon}(p_{f}^{\star}),$ and then finally, $\displaystyle
J_{T,\varepsilon}(p_{T}^{\star})=J_{T,\varepsilon}(p_{f}^{\star}),$ i.e.,
$\displaystyle p_{T}^{\star}=p_{f}^{\star}.$ This implies, thanks to (32),
that $\displaystyle\overline{M}(T)=\overline{M}(T_{1})=\overline{M}(T_{2}),$
which proves the proposition. ∎

######  Remark 5.9.

It follows from all the above and (43) that, when $\displaystyle A^{\ast}$
satisfies the comparison principle (6), if $\displaystyle T_{\ell}<+\infty$,
then

$L_{T}^{\ast}p_{T_{\ell}}^{\star}(t)\leq 0,\quad\forall T\geq
T_{\ell},\quad\forall t\in[0,T-T_{\ell}],$

and

$u_{T}^{\star}(t)=\left\\{\begin{aligned} &0&\quad\text{if}\quad
t\in(0,T-T_{\ell}),\\\ &u_{T_{\ell}}^{\star}(t-T+T_{\ell})&\quad\text{if}\quad
t\in(T-T_{\ell},T),\end{aligned}\right.,\quad\forall T\geq T_{\ell}$

is an optimal control on $\displaystyle[0,T]$ whenever $\displaystyle
u_{T_{\ell}}$ is an optimal control on $\displaystyle[0,T_{\ell}]$.

We now establish the relationship between the optimal control problem (31) and
the minimal time control problem.

######  Proposition 5.10.

Assume that $\displaystyle A^{\ast}$ satisfies the comparison principle (6).
Then, for all $\displaystyle T\in(0,T_{\ell})$, any optimal control for (31)
on $\displaystyle[0,T]$ is a minimal time control, that is,

$T^{\star}(\Pi(T))=T.$

Moreover, for any $\displaystyle\lambda>\mu_{-}$,

$\Pi(T^{\star}(\lambda))=\lambda.$

###### Proof.

We proceed by contradiction. Assume that $\displaystyle T^{\star}(\Pi(T))<T.$
Then, there exists $\displaystyle\delta>0$ and a control $\displaystyle
u_{\delta}\in L^{2}(0,T-\delta;L^{2}(\Omega))$ such that $\displaystyle
F_{T}(u_{\delta})\leq\Pi(T).$ Now, any optimal control $\displaystyle
u^{\star}_{\delta}$ (in the sense of optimal control problem (31) in time
$\displaystyle T-\delta$) satisfies $\displaystyle
F_{T}(u^{\star}_{\delta})\leq F_{T}(u_{\delta})$ (the inequality is not
necessarily strict, as $\displaystyle u_{\delta}$ could be an optimal
control), i.e., $\displaystyle\Pi(T-\delta)=F_{T}(u^{\star}_{\delta})\leq
F_{T}(u_{\delta})\leq\Pi(T),$ which contradicts the fact that $\displaystyle
T\mapsto\Pi(T)$ is a decreasing function on $\displaystyle(0,T_{\ell})$. Thus,
(5.10) holds.

Now, let $\displaystyle\lambda>\mu_{-}$. From Corollaries 5.3, 5.6 and
Proposition 5.1, there exists $\displaystyle T\in(0,T_{\ell})$ such that
$\displaystyle\Pi(T)=\lambda.$ Applying $\displaystyle T^{\star}$ to the above
and using (5.10), we get $\displaystyle
T^{\star}(\lambda)=T^{\star}(\Pi(T))=T.$ Then, applying $\displaystyle\Pi$ to
the above yields $\displaystyle\Pi(T^{\star}(\lambda))=\Pi(T)=\lambda.$ ∎

We can also formulate the above result in the following way: for all
$\displaystyle\lambda>\mu_{-}$,

$\displaystyle T^{\star}(\lambda)=\inf\\{T>0,\quad\Pi(T)\leq\lambda\\},$

that is, $\displaystyle T^{\star}$ is the pseudo-inverse of $\displaystyle\Pi$
on $\displaystyle(\mu_{-},+\infty)$.

In terms of the time optimal control problem, we now have a complete
characterisation of time optimal controls for (37):

######  Theorem 5.11.

Assume that $\displaystyle A^{\ast}$ satisfies the comparison principle (6).
For any $\displaystyle\lambda>\mu_{-}$, $\displaystyle
T^{\star}(\lambda)<+\infty$, and $\displaystyle
T^{\star}(\lambda)\xrightarrow[\lambda\to\infty]{}0$, $\displaystyle
T^{\star}(\lambda)\xrightarrow[\lambda\to\mu_{-}]{}+\infty$. As a consequence,
the domain of definition of $\displaystyle T^{\star}$ is
$\displaystyle(\mu_{-},+\infty)$, and on its domain of definition,
$\displaystyle T^{\star}$ is continuous and decreasing.

Moreover, if $\displaystyle A^{\ast}$ satisfies the (GUC) property and
$\displaystyle\partial_{t}-A^{\ast}$ is analytic-hypoelliptic, there exists a
unique minimal time control for (37), given by the optimal control problem
(19), and it lies in $\displaystyle\mathcal{U}^{L}_{\textrm{shape}}$ .

##### Acknowledgments.

The authors are grateful to Rémy Abergel for enlightening discussions about
Fenchel duality. All three authors acknowledge the support of the ANR project
TRECOS, grant number ANR-20-CE40-0009.

##  Appendix A Convex analysis

###  A.1 Core properties of Fenchel conjugation

A fundamental property of conjugation is involution (over
$\displaystyle\Gamma_{0}(H))$:

######  Theorem A.1 (Fenchel-Moreau).

Given any $\displaystyle f\in\Gamma_{0}(H)$, there holds $\displaystyle
f^{\ast}\in\Gamma_{0}(H)$ and $\displaystyle f^{\ast\ast}=f$.

Analogously to the classical gradient, the subdifferential can be used to
study optimality:

######  Proposition A.2 (Fermat’s rule).

Let $\displaystyle f\in\Gamma_{0}(H)$. $\displaystyle f$ attains a finite
global minimum over $\displaystyle H$ in $\displaystyle x^{\star}$ if and only
if

$0\in\partial f(x^{\star}).$

We now list further useful properties of the Fenchel conjugate:

  * •

multiplication by a real number: for $\displaystyle\alpha\in\mathbb{R}$,

$(\alpha f)^{\ast}(y)=\left\\{\begin{aligned} \alpha
f^{\ast}\left(\frac{y}{\alpha}\right)\qquad&\textrm{if}\quad\alpha\neq 0,\\\
\sigma_{\operatorname{dom}(f)}(y)\qquad&\textrm{if}\quad\alpha=0.\end{aligned}\right.$
(44)

  * •

the (suitably normalised) squared norm is its own conjugate:

$\left(\frac{1}{2}\|\cdot\|_{H}^{2}\right)^{\ast}=\frac{1}{2}\|\cdot\|_{H}^{2}.$
(45)

Let us also mention a result about composition [17]. First, let $\displaystyle
f\in\Gamma_{0}(H)$ and $\displaystyle g\in\Gamma_{0}(\mathbb{R})$ be non-
decreasing. Then,

$(g\circ f)^{\ast}(y)=\min_{\alpha\geq 0}\left(g^{\ast}(\alpha)+\alpha
f^{\ast}\Big{(}\frac{y}{\alpha}\Big{)}\right).$

Following (44), the convention for $\displaystyle\alpha=0$ is $\displaystyle
0\,f^{\ast}\Big{(}\frac{y}{0}\Big{)}=\sigma_{\mathrm{dom}(f)}(y).$

##### Link with the subdifferential.

We now give another characterisation of the subdifferential set, which
illustrates the link with convex conjugation: for $\displaystyle
f\in\Gamma_{0}(H)$,

$\displaystyle\partial f(x)=\\{p\in H,\,\langle
p,x\rangle_{H}-f(x)=f^{\ast}(p)\\}=\\{p\in H,\,\langle
x,p\rangle_{H}-f^{\ast}(p)=f(x)\\}$ (46)

Essentially, the subdifferential is the set of linear forms on which the
convex conjugate is attained.

Using this characterisation, we then get the Legendre-Fenchel identity, which
allows us to “flip” subdifferentials:

$p\in\partial f(x)\iff x\in\partial f^{\ast}(p),\quad
f\in\Gamma_{0}(H),\;\forall x,p\in H.$

###  A.2 Some properties of indicator and support functions

Indicator functions are a crucial tool to encode constraints in convex
optimisation problems. Their properties are closely linked to topological
properties of their indicated sets:

######  Proposition A.3.

We have $\displaystyle\delta_{C},\sigma_{C}\in\Gamma_{0}(H)$ as soon as
$\displaystyle C$ is non-empty, convex and closed.

The characterisation (46) of the subdifferential yields a useful result on
indicator functions:

######  Proposition A.4.

Let $\displaystyle C\subset H$ be a closed convex set with nonempty interior.
Then, for $\displaystyle x\in H$ we have the following:

$\displaystyle x\in\partial C\quad\iff\quad\partial\delta_{C}(x)\ \textrm{is a
nontrivial cone}.$

Equivalently, by convex conjugation,

$\displaystyle\exists p\neq 0,\;x\in\operatorname*{arg\,max}_{v\in C}\langle
v,p\rangle\ \iff\exists p\neq 0,\;\ x\in\partial\sigma_{C}(p)\ \iff\
x\in\partial C.$

##### Indicator function of a ball in a Hilbert space.

Consider the closed unit ball $\displaystyle\overline{B}(0,1)$ of
$\displaystyle H$. We have seen before that

$\displaystyle\sigma_{\overline{B}(0,1)}(y)=\big{(}\delta_{\overline{B}(0,1)}\big{)}^{\ast}(y)=\|y\|_{H}.$

Using (46), we get the following: for $\displaystyle x\in\overline{B}(0,1)$,

$\partial\delta_{\overline{B}(0,1)}(x)=\\{p\in H,\quad\langle
p,x\rangle_{H}=\sigma_{\overline{B}(0,1)}(p)\\}=\\{p\in H,\quad\langle
p,x\rangle_{H}=\|p\|_{H}\\}.$

From the Cauchy-Schwarz inequality we know that $\displaystyle\langle
p,x\rangle_{H}\leq\|p\|_{H}\|x\|_{H},$ it follows that $\displaystyle\langle
p,x\rangle_{H}=\|p\|_{H}$ if and only if $\displaystyle
x=\textstyle{\frac{p}{\|p\|_{H}}}$. This implies that

$\partial\delta_{\overline{B}(0,1)}(x)=\left\\{\begin{array}[]{cl}\\{0\\}&\qquad\textrm{if}\quad\|x\|_{H}<1,\\\
\\{\lambda x,\quad\lambda\geq
0\\}&\qquad\textrm{if}\quad\|x\|_{H}=1.\end{array}\right.$ (47)

###  A.3 Technical lemmas

######  Lemma A.5.

Let $\displaystyle f\in\Gamma_{0}(H)$ be such that

$\displaystyle F:u\in L^{2}(0,T;H)\longmapsto\int_{0}^{T}f(u(t))\,dt,$

is well-defined and proper. Then $\displaystyle F\in\Gamma_{0}(L^{2}(0,T;H))$,
and its Fenchel conjugate and subdifferential are given by

$\displaystyle\forall p\in L^{2}(0,T;H),\quad
F^{\ast}(p)=\int_{0}^{T}f^{\ast}(p(t))\,dt,$ $\displaystyle\partial
F(u)=\left\\{p\in L^{2}(0,T;H),\quad p(t)\in\partial f(u(t)),\ \text{for a.e.
}t\in(0,T)\right\\},\quad\forall u\in L^{2}(0,T;H).$

###### Proof.

Since $\displaystyle F$ is obviously convex, we only need to justify that
$\displaystyle F$ is lsc to infer $\displaystyle
F\in\Gamma_{0}(L^{2}(0,T;H))$. We let $\displaystyle u_{n}\rightarrow u$ be in
$\displaystyle L^{2}(0,T;H)$ and must show that $\displaystyle F(u)\leq\liminf
F(u_{n})$. Upon extraction of a subsequence, we may assume that $\displaystyle
F(u_{n})\rightarrow\liminf F(u_{n})$, and that $\displaystyle
u_{n}(t)\rightarrow u(t)$ in $\displaystyle H$ for a.e. $\displaystyle
t\in(0,T)$. Then, using successively the lsc of $\displaystyle f$ and Fatou’s
lemma, we find

$\displaystyle F(u)=\int_{0}^{T}f(u(t))\,dt\leq\int_{0}^{T}\liminf
f(u_{n}(t))\,dt\leq\liminf\int_{0}^{T}f(u_{n}(t))\,dt=\liminf F(u_{n}).$

For $\displaystyle p\in L^{2}(0,T;H)$, we compute

$\displaystyle F^{\ast}(p)=\sup_{u\in L^{2}(0,T;H)}\langle
p,u\rangle_{L^{2}(0,T;H)}-\int_{0}^{T}f(u(t))\,dt=\int_{0}^{T}\left(\sup_{u\in
H}\langle
p(t),u\rangle_{H}-f(u(t))\right)\,dt=\int_{0}^{T}f^{\ast}(p(t))\,dt.$

Using the characterisation given in (46), and Lemma A.5, we have the
following:

$\displaystyle\partial F(u)$ $\displaystyle=\operatorname*{arg\,max}_{p\in
L^{2}(0,T;H)}\\{\langle p,u\rangle-F^{\ast}(p)\\}$
$\displaystyle=\operatorname*{arg\,max}_{p\in
L^{2}(0,T;H)}\left\\{\int_{0}^{T}\langle p(t),u(t)\rangle
dt-\int_{0}^{T}f^{\ast}(p(t))dt\right\\}$
$\displaystyle=\operatorname*{arg\,max}_{p\in
L^{2}(0,T;H)}\left\\{\int_{0}^{T}\left(\langle
p(t),u(t)\rangle-f^{\ast}(p(t))\right)dt\right\\}$ $\displaystyle=\left\\{p\in
L^{2}(0,T;H),\quad p(t)\in\operatorname*{arg\,max}_{p\in H}\\{\langle
p,u(t)\rangle-f^{\ast}(p)\\}\right\\},$

and the result follows by the same characterisation of the subdifferential set
$\displaystyle\partial f(u(t))$. ∎

###  A.4 Fenchel-Rockafellar duality

Let $\displaystyle E$ and $\displaystyle F$ be two Hilbert spaces. Let
$\displaystyle f$ and $\displaystyle g$ be functions in
$\displaystyle\Gamma_{0}(E)$ and $\displaystyle\Gamma_{0}(F)$, respectively,
and $\displaystyle A:E\rightarrow F$ be a bounded operator. Consider the
(primal) optimisation problem

$\pi=\inf_{x\in E}\left(f(x)+g(Ax)\right).$ ($\displaystyle\mathcal{C}$)

and its dual problem

$d=\sup_{z\in F}\left(-f^{\ast}(A^{\ast}z)-g^{\ast}(-z)\right)=-\inf_{z\in
F}\left(f^{\ast}(A^{\ast}z)+g^{\ast}(-z)\right)$ ($\displaystyle\mathcal{D}$)

With the above notations, weak duality always holds, i.e., we always have
$\displaystyle\pi\geq d$. The Fenchel-Rockafellar theorem states when and how
strong duality holds, i.e., when $\displaystyle d=\pi$ [40].

######  Theorem A.6.

If there exists $\displaystyle\bar{x}\in E$ such that $\displaystyle g$ is
continuous at $\displaystyle A\bar{x}$ and $\displaystyle f(\bar{x})<+\infty$,
then

$\displaystyle\pi=d\quad\text{and}\quad d\text{ is attained {if finite}}.$

Symmetrically, if there exists $\displaystyle\bar{z}\in F$ such that
$\displaystyle f^{\ast}$ is continuous at $\displaystyle A^{\ast}\bar{z}$ and
$\displaystyle g^{\ast}(-\bar{z})<+\infty$, then

$\displaystyle d=\pi\quad\text{and}\quad\pi\text{ is attained {if finite}}.$

The second part of the theorem is obtained by applying the first part to
($\displaystyle\mathcal{D}$), $\displaystyle\inf_{z\in
F}\left(f^{\ast}(A^{\ast}z)+g^{\ast}(-z)\right),$ seen as a primal problem,
and ($\displaystyle\mathcal{C}$), rewritten as $\displaystyle\sup_{x\in
E}\left(-f(x)-g(Ax)\right),$ seen as its dual problem. This yields
$\displaystyle-d\geq-\pi,$ with equality under the corresponding assumptions.

##### Lagrangian and saddle-point interpretation.

Let us now define the Lagrangian for $\displaystyle(x,y)\in E\times F$ by

$\displaystyle\mathcal{L}(x,y):=\langle y,Ax\rangle+f(x)-g^{\ast}(y).$

If $\displaystyle(x^{\star},y^{\star})$ is a saddle point of the Lagrangian,
i.e.,

$x^{\star}\in\operatorname*{arg\,min}_{x\in E}\mathcal{L}(x,y^{\star})\;\text{
and }\;y^{\star}\in\operatorname*{arg\,max}_{y\in F}\mathcal{L}(x^{\star},y),$

then $\displaystyle(x^{\star},z^{\star})$ (with $\displaystyle
z^{\star}=-y^{\star}$) is a pair of primal and dual optimal variables, and
strong duality holds.

What matters is the converse: if $\displaystyle(x^{\star},z^{\star})$ is a
pair of primal and dual optimal variables and if strong duality holds, then
$\displaystyle(x^{\star},y^{\star})$ (with $\displaystyle
y^{\star}=-z^{\star}$) is a saddle point of $\displaystyle\mathcal{L}$.

Whenever $\displaystyle(x^{\star},y^{\star})$ is a primal-dual optimal pair,
Fermat’s rule and the Legendre-Fenchel identity yield

$\displaystyle x^{\star}\in\operatorname*{arg\,min}_{x\in
E}\mathcal{L}(x,y^{\star})\quad\iff\quad-A^{\ast}y^{\star}\in\partial
f(x^{\star})\quad\iff\quad x^{\star}\in\partial f^{\ast}(-A^{\ast}y^{\star}),$

as well as

$\displaystyle y^{\star}\in\operatorname*{arg\,max}_{y\in
F}\mathcal{L}(x^{\star},y)\quad\iff\quad Ax^{\star}\in\partial
g^{\ast}(y^{\star})\quad\iff\quad y^{\star}\in\partial g(Ax^{\star}),$

Summing up, we have the following proposition:

######  Proposition A.7.

Let $\displaystyle(x^{\star},z^{\star})$ be a pair of primal and dual optimal
variables. If strong duality holds, then

$x^{\star}\in\partial f^{\ast}(A^{\ast}z^{\star}),\quad Ax^{\star}\in\partial
g^{\ast}(-z^{\star}),$ (48) $z^{\star}\in-\partial g(Ax^{\star}),\quad
A^{\ast}z^{\star}\in\partial f(x^{\star}).$

Reinterpreting the Fenchel-Rockafellar theorem with the above and in a way
that is useful for controllability issues, we end up with

######  Proposition A.8.

Under the assumption that there exists $\displaystyle\bar{x}\in E$ such that
$\displaystyle g$ is continuous at $\displaystyle A\bar{x}$ and $\displaystyle
f(\bar{x})<+\infty$, if $\displaystyle\pi$ is finite, and attained at
$\displaystyle x^{\star}\in E$, then $\displaystyle d$ is attained at
$\displaystyle z^{\star}\in F$ satisfying

$z^{\star}\in-\partial g(Ax^{\star}),\quad A^{\ast}z^{\star}\in\partial
f(x^{\star}).$ (49)

Conversely, if $\displaystyle(x^{\star},z^{\star})$ satisfies
(49)$\displaystyle,(x^{\star},z^{\star})$ is a pair of primal and dual optimal
variables.

Similarly, under the assumption that there exists $\displaystyle\bar{z}\in E$
such that $\displaystyle f^{\ast}$ is continuous at $\displaystyle
A^{\ast}\bar{z}$ and $\displaystyle g^{\ast}(-\bar{z})<+\infty$, if
$\displaystyle d$ is finite, and attained at $\displaystyle z^{\star}\in F$,
then $\displaystyle\pi$ is attained at $\displaystyle x^{\star}\in E$
satisfying

$x^{\star}\in\partial f^{\ast}(A^{\ast}z^{\star}),\quad Ax^{\star}\in\partial
g^{\ast}(-z^{\star}).$ (50)

Conversely, if $\displaystyle(x^{\star},z^{\star})$ satisfies
(50)$\displaystyle,(x^{\star},z^{\star})$ is a pair of primal and dual optimal
variables.

###  A.5 Parametric convex optimisation

######  Lemma A.9.

Let $\displaystyle H$ be a Hilbert space, $\displaystyle Z$ be a metric space,
$\displaystyle f:H\times Z\rightarrow\mathbb{R}\cup{\\{+\infty\\}}$. Assume
that

  * •

$\displaystyle\forall\alpha\in Z$, $\displaystyle f(\cdot,\alpha)$ is convex
on $\displaystyle H$,

  * •

$\displaystyle\forall x\in H$, $\displaystyle f(x,\cdot)$ is continuous on
$\displaystyle Z$,

  * •

$\displaystyle f$ is sequentially weak-strong lower semicontinuous on
$\displaystyle H\times Z$, i.e.,

$\displaystyle\forall x_{n}\rightharpoonup
x,\;\forall\alpha_{n}\rightarrow\alpha,\quad
f(x,\alpha)\leq\liminf_{n\rightarrow+\infty}f(x_{n},\alpha_{n}),$

  * •

there exists a unique $\displaystyle x_{\alpha}\in H$ such that
$\displaystyle\inf_{x\in H}f(x,\alpha)=f(x_{\alpha},\alpha)$.

Then the mapping

$\displaystyle\alpha\in Z\longmapsto\inf_{x\in H}f(x,\alpha)$

is continuous on $\displaystyle Z$.

###### Proof.

Let $\displaystyle\alpha_{n}\rightarrow\alpha$. Denoting $\displaystyle
m(\alpha)=\inf_{x\in H}f(x,\alpha)=f(x_{\alpha},\alpha)$, let us show that
$\displaystyle m(\alpha_{n})$ converges to $\displaystyle m(\alpha)$.

Upper semicontinuity. For $\displaystyle x\in H$ fixed, thanks to the
continuity of $\displaystyle f(x,\cdot)$, we pass to the limit in
$\displaystyle f(x,\alpha_{n})\geq m(\alpha_{n})$ and find

$\displaystyle m(\alpha)=\inf_{x\in
H}f(x,\alpha)\geq\limsup_{n\rightarrow+\infty}m(\alpha_{n}).$

Lower semicontinuity. We denote $\displaystyle x_{n}=x_{\alpha_{n}}$. Let us
for the moment admit that $\displaystyle(x_{n})$ is bounded. Upon extraction,
we may assume that $\displaystyle x_{n}\rightharpoonup\bar{x}$ for some
$\displaystyle x\in H$. By sequential weak-strong lower semicontinuity,

$\displaystyle
f(\bar{x},\alpha)\leq\liminf_{n\rightarrow+\infty}f(x_{n},\alpha_{n})=\liminf_{n\rightarrow+\infty}m(\alpha_{n}).$

Since the left-hand side is bounded from below by $\displaystyle m(\alpha)$,
we have proved lower semicontinuity (and in fact
$\displaystyle\bar{x}=x_{\alpha}$).

We are left to proving the boundedness of $\displaystyle(x_{n})$ to conclude
the proof. Assume that $\displaystyle(x_{n})$ is not bounded. Upon extraction,
we may assume that

$\displaystyle y_{n}:=\frac{x_{n}}{\|x_{n}\|_{H}}\rightharpoonup y,$

for some $\displaystyle y\in H$. For any fixed $\displaystyle\lambda>0$, we
shall prove that $\displaystyle x_{\alpha}+\lambda y$ minimises $\displaystyle
f(\cdot,\alpha)$, which contradicts the fourth assumption that there exists a
single minimum point.

Indeed, we notice that

$\displaystyle\Big{(}1-\frac{\lambda}{\|x_{n}\|_{H}}\Big{)}x_{\alpha}+\frac{\lambda}{\|x_{n}\|_{H}}x_{n}\rightharpoonup
x_{\alpha}+\lambda y.$

Hence, by weak-strong lower semicontinuity, convexity, the fact $\displaystyle
x_{n}$ minimises $\displaystyle f(\cdot,\alpha_{n})$ and continuity,

$\displaystyle\displaystyle f(x_{\alpha}+\lambda y,\alpha)$
$\displaystyle\displaystyle\leq\liminf_{n\rightarrow+\infty}f\Big{(}\Big{(}1-\frac{\lambda}{\|x_{n}\|_{H}}\Big{)}x_{\alpha}++\frac{\lambda}{\|x_{n}\|_{H}}x_{n},\alpha_{n}\Big{)}$
$\displaystyle\displaystyle\leq\liminf_{n\rightarrow+\infty}\Big{(}1-\frac{\lambda}{\|x_{n}\|_{H}}\Big{)}f(x_{\alpha},\alpha_{n})+\frac{\lambda}{\|x_{n}\|_{H}}f(x_{n},\alpha_{n})$
$\displaystyle\displaystyle\leq\liminf_{n\rightarrow+\infty}\Big{(}1-\frac{\lambda}{\|x_{n}\|_{H}}\Big{)}f(x_{\alpha},\alpha_{n})+\frac{\lambda}{\|x_{n}\|_{H}}f(x_{\alpha},\alpha_{n})$
$\displaystyle\displaystyle=\liminf_{n\rightarrow+\infty}f(x_{\alpha},\alpha_{n})=f(x_{\alpha},\alpha).$

∎

##  Appendix B The classical bathtub principle

The classical bathtub principle characterises the maximisers, and gives the
maximum value, of the constrained scalar product maximisation:

$\sup_{u\in\widetilde{\mathcal{U}}^{\ast}_{L}}\int_{\Omega}u(x)v(x)\,dx,$ (51)

where $\displaystyle v\in L^{2}(\Omega)$ is arbitrary, and

$\widetilde{\mathcal{U}}_{L}^{\ast}:=\left\\{u\in L^{2}(\Omega),\;0\leq u\leq
1\text{ and }\int_{\Omega}u=L|\Omega|\right\\},$

is the convex hull (and $\displaystyle L^{\infty}$ weak-$\displaystyle\ast$
closure) of the set of characteristic functions whose support has the
corresponding fixed measure:

$\displaystyle\widetilde{\mathcal{U}}_{L}:=\\{\chi_{\omega},\quad\omega\subset\Omega,\quad|\omega|=L|\Omega|\\}.$

Recalling the notations (14) and (15) introduced in Section 2.3, the classical
bathtub principle reads:

######  Lemma B.1 (classical bathtub principle).

Let $\displaystyle v\in L^{2}(\Omega)$. Denote
$\displaystyle\rho(v):=\Phi^{-1}_{v}(L|\Omega|)$. The maximum in (51) equals

$\left(\int_{v>\rho(v)}v\right)+\rho(v)(L|\Omega|-|\\{v>\rho(v)\\}|)=\int_{0}^{L|\Omega|}\Phi_{v}^{-1},$

and the maximisers are given by

$u^{\star}:=\chi_{\\{v>\rho(v)\\}}+c(x)\chi_{\\{v=\rho(v)\\}},$

where $\displaystyle c$ is any measurable function such that $\displaystyle
0\leq c\leq 1$ and

$\displaystyle\int_{\\{v=\rho(v)\\}}c=L|\Omega|-|\\{v>\rho(v)\\}|.$

In particular, if all the level sets of the function $\displaystyle v$ have
zero measure, then the maximum is uniquely attained by

$u^{\star}:=\chi_{\\{v>\rho(v)\\}},$

and the maximum hence equals
$\displaystyle\displaystyle\int_{\\{v>\rho(v)\\}}v$.

Now, recall that we defined $\displaystyle\mathcal{U}_{L}$ by (11) and its
convex hull $\displaystyle\overline{\mathcal{U}}_{L}$ by (12) in Section 2.3.
They are respective relaxations of $\displaystyle\tilde{\mathcal{U}}_{L}$, and
its convex hull $\displaystyle\tilde{\mathcal{U}}^{\ast}_{L}$.

For $\displaystyle v\in L^{2}(\Omega)$, we consider the relaxed version of
(51):

$\sup_{u\in\overline{\mathcal{U}}_{L}}\int_{\Omega}u(x)v(x)\,dx.$ (52)

Then, the complete solution of Lemma 2.3 is given by the following:

######  Lemma B.2 (relaxed bathtub principle).

Let $\displaystyle v\in L^{2}(\Omega)$. Denote $\displaystyle
h(v)=\max(0,\Phi_{v}^{-1}(L|\Omega|))=\max(0,\rho(v))$. Then, the maximum in
(52) equals

$\displaystyle\int_{0}^{\min(\Phi_{v}(0),L|\Omega|)}\Phi_{v}^{-1},$

and the maximisers are given by

$u^{\star}:=\chi_{\\{v>h(v)\\}}+c(x)\chi_{\\{v=h(v)\\}},$

where $\displaystyle c$ is any measurable function such that $\displaystyle
0\leq c\leq 1$ and

$\displaystyle\left\\{\begin{aligned}
\int_{\\{v=h(v)\\}}c=L|\Omega|-|\\{v>h(v)\\}|\ &\textrm{ if }\ h(v)>0\\\
\int_{\\{v=h(v)\\}}c\leq L|\Omega|-|\\{v>h(v)\\}|\ &\textrm{ if }\
h(v)=0\end{aligned}\right.$

######  Remark B.3.

In particular, if $\displaystyle h(v)>0$, there is a unique maximiser in (52)
if and only if

$\displaystyle|\\{v>h(v)\\}|+|\\{v=h(v)\\}|=L|\Omega|.$

Indeed, when the above does not hold, $\displaystyle c$ can be chosen to have
values in $\displaystyle(0,1)$ on a set of nonzero measure, and the maximisers
is no longer unique.

On the other hand, if $\displaystyle h(v)=0$, as soon as
$\displaystyle|\\{v=h(v)\\}|>0$ the maximisers are no longer unique.

## References

  * [1] N. U. Ahmed.  Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints.  Journal of optimization Theory and Applications, 47(2):129–158, 1985.  Publisher: Springer.
  * [2] Nachman Aronszajn.  A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order.  Technical report, KANSAS UNIV LAWRENCE, 1956.
  * [3] Larbi Berrahmoune.  A variational approach to constrained controllability for distributed systems.  Journal of Mathematical Analysis and Applications, 416(2):805–823, 2014.
  * [4] Larbi Berrahmoune.  Constrained null controllability for distributed systems and applications to hyperbolic-like equations.  ESAIM: Control, Optimisation and Calculus of Variations, 25:32, 2019\.
  * [5] Larbi Berrahmoune.  A variational approach to constrained null controllability for the heat equation.  European Journal of Control, 52:42 – 48, 2020.
  * [6] Jean-Michel Bony.  Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés.  Annales de l’Institut Fourier, 19(1):277–304, 1969.
  * [7] Franck Boyer.  On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems.  ESAIM: Proc., 41:15–58, 2013.
  * [8] Robert F Brammer.  Controllability in linear autonomous systems with positive controllers.  SIAM Journal on Control, 10(2):339–353, 1972.
  * [9] Haïm Brézis.  Functional analysis, Sobolev spaces and partial differential equations, volume 2.  Springer, 2011.
  * [10] Antonin Chambolle and Thomas Pock.  A first-order primal-dual algorithm for convex problems with applications to imaging.  Journal of Mathematical Imaging and Vision, 40, 05 2011.
  * [11] Frank H Clarke.  Optimization and nonsmooth analysis.  SIAM, 1990.
  * [12] J.-M. Coron.  Control and nonlinearity, volume 136 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 2007.
  * [13] O Yu Emanuilov.  Controllability of parabolic equations.  Sbornik: Mathematics, 186(6):879–900, jun 1995.
  * [14] Klaus-Jochen Engel and Rainer Nagel.  One-parameter semigroups for linear evolution equations.  Semigroup Forum, 63:278–280, 06 2001.
  * [15] Sylvain Ervedoza.  Control issues and linear projection constraints on the control and on the controlled trajectory.  North-W. Eur. J. of Math., 6:165–197, 2020.
  * [16] Lawrence C Evans.  Partial differential equations.  Graduate studies in mathematics, 19(4):7, 1998.
  * [17] Jean-Baptiste Hiriart-Urruty.  A note on the Legendre-Fenchel transform of convex composite functions.  In Nonsmooth Mechanics and Analysis, pages 35–46. Springer, 2006\.
  * [18] Bernhard Kawohl.  Rearrangements and convexity of level sets in PDE, volume 1150.  Springer, 2006.
  * [19] Karl Kunisch and Lijuan Wang.  Time optimal control of the heat equation with pointwise control constraints.  ESAIM: Control, Optimisation and Calculus of Variations, 19(2):460–485, 2013.
  * [20] Stéphane Labbé and Emmanuel Trélat.  Uniform controllability of semidiscrete approximations of parabolic control systems.  Systems & Control Letters, 55(7):597–609, 2006.
  * [21] Gontran Lance, Emmanuel Trélat, and Enrique Zuazua.  Shape turnpike for linear parabolic PDE models.  Systems & Control Letters, 142:104733, 2020.
  * [22] Gilles Lebeau and Luc Robbiano.  Contrôle exact de l’équation de la chaleur.  Communications in Partial Differential Equations, 20(1-2):335–356, 1995.
  * [23] Elliott H. Lieb and Michael Loss.  Analysis.  Crm Proceedings & Lecture Notes. American Mathematical Society, 2001\.
  * [24] Jacques-Louis Lions.  Remarks on approximate controllability.  Journal d’Analyse Mathématique, 59(1):103, 1992.
  * [25] Jérôme Lohéac, Emmanuel Trélat, and Enrique Zuazua.  Minimal controllability time for the heat equation under unilateral state or control constraints.  Mathematical Models and Methods in Applied Sciences, 27(09):1587–1644, 2017.
  * [26] Jérôme Lohéac, Emmanuel Trélat, and Enrique Zuazua.  Minimal controllability time for finite-dimensional control systems under state constraints.  Automatica, 96:380–392, 2018.
  * [27] Jérôme Lohéac, Emmanuel Trélat, and Enrique Zuazua.  Nonnegative control of finite-dimensional linear systems.  Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 38(2):301–346, 2021.
  * [28] Idriss Mazari.  Quantitative estimates for parabolic optimal control problems under $\displaystyle{L}^{\infty}$ and $\displaystyle{L}^{1}$ constraints in the ball: Quantifying parabolic isoperimetric inequalities.  Nonlinear Analysis, 215:112649, 2022.
  * [29] Sorin Micu, Ionel Roventa, and Marius Tucsnak.  Time optimal boundary controls for the heat equation.  Journal of Functional Analysis, 263(1):25–49, 2012.  Publisher: Elsevier.
  * [30] Sorin Micu and Enrique Zuazua.  An introduction to the controllability of partial differential equations.  Quelques questions de théorie du contrôle. Sari, T., ed., Collection Travaux en Cours Hermann, to appear, 2004.
  * [31] Victor J Mizel and Thomas I Seidman.  An abstract bang-bang principle and time-optimal boundary control of the heat equation.  SIAM Journal on Control and Optimization, 35(4):1204–1216, 1997\.
  * [32] G. Métivier.  Hypoellipticité d’opérateurs à caractéristiques multiples.  Séminaire Équations aux dérivées partielles (Polytechnique), pages 1–13, 1980-1981.
  * [33] Edward Nelson.  Analytic vectors.  Annals of Mathematics, pages 572–615, 1959.
  * [34] Khoa Nguyen Son.  A unified approach to constrained approximate controllability for the heat equations and the retarded equations.  Journal of Mathematical Analysis and Applications, 150(1):1–19, 1990.  Publisher: Elsevier.
  * [35] Amnon Pazy.  Semigroups of linear operators and applications to partial differential equations, volume 44.  Springer Science & Business Media, 2012.
  * [36] Dario Pighin and Enrique Zuazua.  Controllability under positivity constraints of semilinear heat equations.  Mathematical Control & Related Fields, 8(3&4):935–964, 2018.
  * [37] Camille Pouchol, Emmanuel Trélat, and Enrique Zuazua.  Phase portrait control for 1D monostable and bistable reaction–diffusion equations.  Nonlinearity, 32(3):884, 2019.
  * [38] Yannick Privat, Emmanuel Trélat, and Enrique Zuazua.  Optimal shape and location of sensors for parabolic equations with random initial data.  Archive for Rational Mechanics and Analysis, 216(3):921–981, 2015\.  Publisher: Springer.
  * [39] Shulin Qin and Gengsheng Wang.  Equivalence between minimal time and minimal norm control problems for the heat equation.  SIAM Journal on Control and Optimization, 56(2):981–1010, 2018\.
  * [40] Ralph Rockafellar.  Duality and stability in extremum problems involving convex functions.  Pacific Journal of Mathematics, 21(1):167–187, 1967.
  * [41] Domènec Ruiz-Balet and Enrique Zuazua.  Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations.  Journal de Mathématiques Pures et Appliquées, 143:345–375, 2020.
  * [42] David L. Russell.  Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions.  SIAM Review, 20(4):639–739, 1978.
  * [43] E.J.P. Georg Schmidt.  The “bang-bang” principle for the time-optimal problem in boundary control of the heat equation.  SIAM Journal on Control and Optimization, 18(2):101–107, 1980.
  * [44] Emmanuel Trélat, Gengsheng Wang, and Yashan Xu.  Characterization by observability inequalities of controllability and stabilization properties.  Pure and Applied Analysis, 2(1):93–122, 2020.
  * [45] Gengsheng Wang, Yashan Xu, and Yubiao Zhang.  Attainable Subspaces and the Bang-Bang Property of Time Optimal Controls for Heat Equations.  SIAM Journal on Control and Optimization, 53(2):592–621, 2015.
  * [46] Gengsheng Wang and Enrique Zuazua.  On the Equivalence of Minimal Time and Minimal Norm Controls for Internally Controlled Heat Equations.  SIAM Journal on Control and Optimization, 50(5):2938–2958, 2012\.
  * [47] Dong-Hui Yang, Bao-Zhu Guo, Weihua Gui, and Chunhua Yang.  The Bang–Bang Property of Time-Varying Optimal Time Control for Null Controllable Heat Equation.  Journal of Optimization Theory and Applications, 182(2):588–605, 2019.  Publisher: Springer.