# Second-order optimality conditions for the bilinear optimal control of a
degenerate parabolic equation

Cyrille Kenne  Laboratoire LAMIA, Université des Antilles, Campus Fouillole,
97159 Pointe-à-Pitre Guadeloupe (FWI)- Laboratoire L3MA, UFR STE et IUT,
Université des Antilles, 97275 Schoelcher, Martinique<EMAIL_ADDRESS>,
Landry Djomegne  University of Dschang, BP 67 Dschang, Cameroon, West region
<EMAIL_ADDRESS>and  Pascal Zongo  Laboratoire L3MA, UFR STE et IUT,
Université des Antilles, 97275 Schoelcher, Martinique<EMAIL_ADDRESS>
###### Abstract.

The main purpose of this paper is the study of second-order optimality
conditions for the bilinear control of a strongly degenerate parabolic
equation. The equation is degenerate at the boundary of the spatial domain.
The well-posedness of the state equation, as well as weak maximum principles
are established. We prove some differentiability properties of the control-to-
state operator and the existence of optimal solutions. Finally, we derive
first- and second-order optimality conditions for the system.

###### Key words and phrases:

Degenerate parabolic systems; optimal control; first- and second-order
optimality conditions; weak maximum principle.

###### 2010 Mathematics Subject Classification:

35B50; 35K65; 35Q93; 49J20; 49K20.

##  1\. Introduction

Bilinear systems are used for the description of several complex phenomena in
chemistry, physics, economics, biology and climate science [16, 19, 27]. They
have been also used to describe the complex interaction between large ice
masses and solar radiation on climate [9, 28]. In bilinear systems, there is a
high nonlinear dependence of the state function with respect to the control
function and so some results need more attention.

Up to date only a few papers concerning bilinear optimal control problems were
devoted to the derivation of second order necessary optimality conditions (see
e.g [2, 3, 5, 6, 7]). The authors in [4], considered the optimal control of an
infinite dimensional bilinear system governed by a strongly continuous
semigroup operator involving a $L^{1}$ time dependent control. Using the Goh
transform, they derived first and second optimality conditions.

Recently, M. Aronna et al. [8] studied a bilinear optimal control problem
subject to the Fokker-Planck equation. With a time dependent control, they
derived first and second order necessary and sufficient optimality conditions,
under quite mild regularity assumptions. In the above cited works, the control
were applied to non degenerate systems.

However, degenerate equations arise in the mathematical description of a wide
variety of phenomena such as sedimentation-consolidation processes [30] and
biological models. Many problems that are relevant for applications are
described by degenerate equations, with degeneracy occurring at the boundary
of the space domain [1]. Indeed, in the context of population genetics,
degenerate systems are used in the gene frequency models, which in a
population of a hypothetical species may be formulated as a Markov chain (see
e.g. [29]). We also refer to [16] for the Fleming-Viot model in population
genetics.

In this work, we study the bilinear optimal control of a strongly degenerate
parabolic equation. Here, the control given by the multiplicative term $v$ in
(1.1) depends on time and space and acts on an open subset of the whole
domain. We first establish a weak maximum principle for our system in order to
improve the regularity of solutions and we prove the existence of optimal
solutions. We also derive the first order optimality conditions. Next, using
the results of [12], we derive the second order necessary and sufficient
conditions for optimality. Importantly, we prove that for our problem, the
local solution satisfies the sufficient condition (6.12). This result is often
very difficult or impossible to prove in general. To the best of our
knowledge, such results have never been studied for degenerate systems
involving bilinear controls.

In this paper, we are interested in the bilinear control problem for the
following degenerate parabolic equation

$\left\\{\begin{array}[]{rllll}\displaystyle
y_{t}-\left(a(x)y_{x}\right)_{x}&=&v(t,x)\chi_{\omega}y&\mbox{in}&Q:=(0,T)\times\Omega,\\\
\displaystyle a(x)y_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
\displaystyle y(0,\cdot)&=&y^{0}&\mbox{in}&\Omega,\end{array}\right.$ (1.1)

where $\Omega=(-1,1)$, $\omega$ is a nonempty subset of $\Omega$ and
$\omega_{T}=(0,T)\times\omega$. In the system (1.1), $y=y(t,x)$ denotes the
state and $v(t,x)$ is the control function acting on the system via the subset
$\omega$. Here $\chi_{\omega}$ denotes the characteristic function of the
control set $\omega$ and $y^{0}\in L^{2}(\Omega)$ is the initial data. We
denote by $y_{t}$ and $y_{x}$ the partial derivative of $y$ with respect $t$
and $x$ respectively. The function $a:=a(\cdot)$ represents the diffusion
coefficient and is assumed to vanish at the extreme points of $[-1,1]$. The
system (1.1) is said to be strongly degenerate when
$a\in\mathcal{C}^{1}([-1,1])$ and so, $\displaystyle\frac{1}{a}\notin
L^{1}(-1,1)$. We make the following assumptions:

######  Assumption 1.1.

$\begin{array}[]{llll}\displaystyle a\in\mathcal{C}^{1}([-1,1]),\ \ a>0\
\mbox{in}\ (-1,1)\ \mbox{and}\ a(-1)=a(1)=0,\\\ a\;\;\text{is such
that}\,\,\,\displaystyle\int_{0}^{x}\frac{ds}{a(s)}\in
L^{1}(-1,1).\end{array}$ (1.2)

######  Remark 1.2.

An example of the diffusion coefficient $a$ is given by $a(x)=1-x^{2}$. In
that case, the principal part of the operator in (1.1) coincides with that of
the Budyko-Sellers model (see [9, 14, 28]). It is an energy balance model
studying the role played by continental and oceanic areas of ice on the
evolution of the climate.

The controllability results for bilinear control problems applied on
degenerate parabolic systems have been widely considered in recent papers (see
e.g. [11, 17, 18]). In [10], the authors studied the global approximate
multiplicative controllability for the linear degenerate parabolic Cauchy-
Neumann problem (1.1). The optimal control problems related to system (1.1)
remain in the general sense not discussed yet. However, those bilinear optimal
control problems remain very important from the perspective of their
application in several fields. The main results of this paper are summarized
as follows:

  1. (i)

We prove the existence and the uniqueness of the solution to the degenerate
problem (1.1);

  2. (ii)

We establish maximum principle results for the system (1.1);

  3. (iii)

We derive the first and second conditions for optimality.

The rest of this paper is organized as follows. In Section 2, we give some
notations and definitions of weighted functional spaces and their associated
norms for the need of this work; we prove some results on existence and
uniqueness of the weak solution of the system (1.1) and establish some weak
maximum principle results. In Section 3, we formulate the optimal control
problem and prove the existence of an optimal control solution to (3.1)-(3.3).
Properties of the control-to-state mapping is discussed in Section 4, and the
derivation of the first order necessary optimality conditions is derived in
Section 5. Finally, we present the second order necessary and sufficient
optimality conditions in Section 6.

##  2\. Well-posedness results

Here, we discuss the well-posedness of problem (1.1) by introducing the
weighted Sobolev spaces $H^{1}_{a}(\Omega)$ and $H^{2}_{a}(\Omega)$ defined as
follows (loc. abs. cont. means locally absolutely continuous).

$H^{1}_{a}(\Omega)=\\{u\in L^{2}(\Omega):u\ \mbox{is loc. abs. cont. in}\
\Omega:\sqrt{a}u_{x}\in L^{2}(\Omega)\\},$

endowed with the norm

$\|u\|^{2}_{H^{1}_{a}(\Omega)}=\|u\|^{2}_{L^{2}(\Omega)}+\|\sqrt{a}u_{x}\|^{2}_{L^{2}(\Omega)},\
\ \ u\in H^{1}_{a}(\Omega)$

and

$\displaystyle H^{2}_{a}(\Omega)=\\{u\in H^{1}_{a}(\Omega):a(x)u_{x}\in
H^{1}(\Omega)\\},$

endowed with the norm

$\|u\|^{2}_{H^{2}_{a}(\Omega)}=\|u\|^{2}_{H^{1}_{a}(\Omega)}+\|(a(x)u_{x})_{x}\|^{2}_{L^{2}(\Omega)}.$

Under the assumptions (1.2), we have the following result (see [11]).

######  Lemma 2.1.

We have

$H^{1}_{a}(\Omega)\hookrightarrow L^{2}(\Omega),$

with compact embedding. Moreover, there exists a positive constant $C$, such
that

$\|y\|_{L^{2}(\Omega)}\leq C\|y\|_{H^{1}_{a}(\Omega)},\;\;\;\forall y\in
H^{1}_{a}(\Omega).$

######  Remark 2.2.

Note that the functions of the Sobolev space $H^{1}_{a}(\Omega)$ are not
necessarily bounded when the operator is strongly degenerate.

Let $(H^{1}_{a}(\Omega))^{\prime}$ be the topological dual space of
$H^{1}_{a}(\Omega)$ and we denote by $\left\langle\cdot,\cdot\right\rangle$
the dual product between $H^{1}_{a}(\Omega)$ and
$(H^{1}_{a}(\Omega))^{\prime}$. If we set

$W_{a}(0,T)=\left\\{\rho\in L^{2}((0,T);H^{1}_{a}(\Omega));\rho_{t}\in
L^{2}\left((0,T);(H^{1}_{a}(\Omega))^{\prime}\right)\right\\},$ (2.1)

then $W_{a}(0,T)$ endowed with the norm

$\|\rho\|^{2}_{W_{a}(0,T)}=\|\rho\|^{2}_{L^{2}((0,T);H^{1}_{a}(\Omega))}+\|\rho_{t}\|^{2}_{L^{2}\left((0,T);(H^{1}_{a}(\Omega))^{\prime}\right)}$
(2.2)

is a Hilbert space. Moreover, we have the following embedding is continuous

$W_{a}(0,T)\subset C([0,T],L^{2}(\Omega)).$ (2.3)

###  2.1. Existence results

In this subsection, we prove the existence and uniqueness of a solution of the
system (1.1). We will simply denote by $\|\cdot\|_{\infty}$ the
$L^{\infty}$-norm in $\omega_{T}$. We first consider the following system

$\left\\{\begin{array}[]{rllll}\displaystyle
p_{t}-\left(a(x)p_{x}\right)_{x}&=&v(t,x)\chi_{\omega}p+f&\mbox{in}&Q,\\\
\displaystyle a(x)p_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
\displaystyle p(0,\cdot)&=&p^{0}&\mbox{in}&\Omega.\end{array}\right.$ (2.4)

Now, we define the weak solution to the system (2.4) as follows.

######  Definition 2.3.

Let $f\in L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})$, $v\in
L^{\infty}(\omega_{T})$ and $p^{0}\in L^{2}(\Omega)$. We say that a function
$p\in W_{a}(0,T)$ is a weak solution to (2.4), if the following equality

$\begin{array}[]{lll}\displaystyle-\int_{Q}\phi_{t}p\,\mathrm{d}x\,\mathrm{d}t+\int_{Q}a(x)p_{x}\phi_{x}\,\mathrm{d}x\,\mathrm{d}t-\int_{\omega_{T}}vp\,\phi\;\mathrm{d}x\,\mathrm{d}t=\displaystyle\int_{\Omega}p^{0}\,\phi(0,\cdot)\,\mathrm{d}x+\int_{0}^{T}\left\langle
f,\phi\right\rangle\,\mathrm{d}t,\end{array}$ (2.5)

holds, for every

$\phi\in H(Q)=\left\\{z\in L^{2}((0,T);H^{1}_{a}(\Omega)):\,z_{t}\in
L^{2}(Q)\hbox{ and }z(T,\cdot)=0\hbox{ in }\Omega\right\\}.$ (2.6)

######  Remark 2.4.

If $\varphi\in H(Q)$, then $\varphi_{t}\in L^{2}(Q)\hookrightarrow
L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})$. Consequently, $\varphi\in
W_{a}(0,T)$. Therefore, $\varphi(0,\cdot)$ and $\varphi(T,\cdot)$ exist and
belong to $L^{2}(\Omega)$.

We state the well posedness of the system (2.4) in the following theorem.

######  Theorem 2.5.

Let $f\in L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})$, $v\in
L^{\infty}(\omega_{T})$ and $p^{0}\in L^{2}(\Omega)$. Then, there exists a
unique weak solution $p\in W_{a}(0,T)$ to (2.4) in the sense of Definition
2.3. In addition, the following estimate holds

$\|p\|_{{\mathcal{C}}([0,T];L^{2}(\Omega))}+\|p\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
2e^{(\|v\|_{\infty}+1)T}\left(\|p^{0}\|_{L^{2}(\Omega)}+\|f\|_{L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})}\right).$
(2.7)

Before the proof of Theorem 2.5, we state and prove the following intermediate
result. We set $z(t,x)=e^{-rt}p(t,x)$ for a suitable $r>0$, which will be
chosen later. Then $p$ is solution to (2.4) if and only if $z$ is solution to

$\left\\{\begin{array}[]{rllll}\displaystyle
z_{t}-\left(a(x)z_{x}\right)_{x}+rz&=&v(t,x)\chi_{\omega}z+e^{-rt}f&\mbox{in}&Q,\\\
\displaystyle a(x)z_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
\displaystyle z(0,\cdot)&=&p^{0}&\mbox{in}&\Omega.\end{array}\right.$ (2.8)

We have the following definition.

######  Definition 2.6.

Let $f\in L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})$, $v\in
L^{\infty}(\omega_{T})$ and $p^{0}\in L^{2}(\Omega)$. We say that a function
$z\in W_{a}(0,T)$ is a weak solution to (2.8), if for every $\phi\in H(Q)$
defined by (2.6), the following equality holds

$\begin{array}[]{lll}\displaystyle-\int_{Q}\phi_{t}z\,\mathrm{d}x\,\mathrm{d}t+\int_{Q}a(x)z_{x}\phi_{x}\,\mathrm{d}x\,\mathrm{d}t+r\int_{Q}z\phi\,\mathrm{d}x\,\mathrm{d}t-\int_{\omega_{T}}vz\,\phi\;\mathrm{d}x\,\mathrm{d}t=\displaystyle\int_{\Omega}p^{0}\,\phi(0,\cdot)\mathrm{d}x+\int_{0}^{T}e^{-rt}\left\langle
f,\phi\right\rangle\,\mathrm{d}t.\end{array}$ (2.9)

On the basis of Theorem 1.1 [25], we can prove the well posed of (2.8).

######  Theorem 2.7.

Let $f\in L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})$, $v\in
L^{\infty}(\omega_{T})$ and $p^{0}\in L^{2}(\Omega)$. Then, the system (2.8)
admits a unique weak solution $z\in W_{a}(0,T)$ in the sense of Definition
2.6. In addition, the following estimate holds

$\|z\|_{{\mathcal{C}}([0,T];L^{2}(\Omega))}+\|z\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
2\left(\|p^{0}\|_{L^{2}(\Omega)}+\|f\|_{L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})}\right).$
(2.10)

###### Proof.

We make the proof in three steps.
Step 1. For the existence result, we apply [25, Theorem 1.1, Page 37 ]. For
the sake of completeness, we recall here the sketch of the proof. We set
$r=\|v\|_{\infty}+1$. We endow $H(Q)$ with the norm defined by

$\|\varphi\|^{2}_{H(Q)}:=\|\varphi\|^{2}_{L^{2}((0,T);H^{1}_{a}(\Omega))}+\|\varphi(0,\cdot)\|^{2}_{L^{2}(\Omega)},\,\forall\varphi\in
H(Q).$

Therefore, it is clear that for any $\rho\in H(Q),$ we have

$\|\rho\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq\|\rho\|_{H(Q)}.$

This shows that we have the continuous embedding $H(Q)\hookrightarrow
L^{2}((0,T);H^{1}_{a}(\Omega))$.

Now, let $\varphi\in H(Q)$ and consider the bilinear form
$\mathcal{E}(\cdot,\cdot)$ defined on $L^{2}((0,T);H^{1}_{a}(\Omega))\times
H(Q)$ by:

$\begin{array}[]{lll}\mathcal{E}(z,\phi)&=&\displaystyle-\int_{Q}\phi_{t}z\,\mathrm{d}x\,\mathrm{d}t+\int_{Q}a(x)z_{x}\phi_{x}\,\mathrm{d}x\,\mathrm{d}t+r\int_{Q}z\phi\,\mathrm{d}x\,\mathrm{d}t-\int_{\omega_{T}}vz\,\phi\;\mathrm{d}x\,\mathrm{d}t.\end{array}$
(2.11)

Using the Cauchy Schwarz’s inequality, we obtain

$\begin{array}[]{lll}|\mathcal{E}(z,\phi)|&\leq&\displaystyle\left[[(2\|v\|_{\infty}+1)\|\phi\|_{L^{2}(Q)}+\|\phi_{t}\|_{L^{2}(Q)}]^{2}+\|\sqrt{a(\cdot)}\phi_{x}\|^{2}_{L^{2}(Q)}\right]^{1/2}\|z\|_{L^{2}((0,T);H^{1}_{a}(\Omega)}.\end{array}$

Therefore, there exists a constant $C=C(\phi,\|v\|_{\infty})>0$ such that

$|\mathcal{E}(z,\phi)|\leq C\|z\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}.$

Consequently, for every fixed $\phi\in H(Q),$ the functional
$z\mapsto\mathcal{E}(z,\phi)$ is continuous on
$L^{2}((0,T);H^{1}_{a}(\Omega)).$

Next, for every $\phi\in H(Q)$ we have

$\begin{array}[]{lllll}\mathcal{E}(\phi,\phi)&=&-\displaystyle\int_{Q}\phi_{t}\,\phi\,\mathrm{d}x\,\mathrm{d}t+\int_{Q}a(x)\phi^{2}_{x}\,\mathrm{d}x\,\mathrm{d}t+r\int_{Q}\phi^{2}\,\mathrm{d}x\,\mathrm{d}t-\int_{\omega_{T}}v\phi^{2}\;\mathrm{d}x\,\mathrm{d}t\\\
&\geq&\displaystyle\frac{1}{2}\|\phi(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{Q}a(x)\phi^{2}_{x}\,\mathrm{d}x\,\mathrm{d}t+r\int_{Q}\phi^{2}\,\mathrm{d}x\,\mathrm{d}t-\|v\|_{\infty}\int_{Q}\phi^{2}\;\mathrm{d}x\,\mathrm{d}t\\\
&\geq&\displaystyle\frac{1}{2}\|\phi(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{Q}a(x)\phi^{2}_{x}\,\mathrm{d}x\,\mathrm{d}t+\int_{Q}\phi^{2}\,\mathrm{d}x\,\mathrm{d}t,\
\mbox{because}\ r=\|v\|_{\infty}+1\\\
&\geq&\displaystyle\frac{1}{2}\|\phi(0,\cdot)\|^{2}_{L^{2}(\Omega)}+\|\phi\|^{2}_{L^{2}((0,T);H^{1}_{a}(\Omega))}\\\
&\geq&\displaystyle\frac{1}{2}\|\phi\|^{2}_{H(Q)}.\end{array}$

Finally, we consider the linear functional $L(\cdot):H(Q)\to\mathbb{R}$
defined by

$L(\phi)=:\displaystyle\int_{0}^{T}e^{-rt}\left\langle
f,\phi\right\rangle\,\mathrm{d}t+\int_{\Omega}p^{0}\,\phi(0,\cdot)\
\mathrm{d}x.$

Then using Cauchy Schwarz’s inequality, we get

$|L(\phi)|\leq\left(\|f\|^{2}_{L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})}+\|y^{0}\|^{2}_{L^{2}(\Omega)}\right)^{1/2}\|\phi\|_{H(Q)}.$

Therefore, it follows from [25, Theorem 1.1, page 37 ] that there exists $y\in
L^{2}((0,T);H^{1}_{a}(\Omega))$ such that

$\mathcal{E}(y,\phi)=L(\phi),\quad\forall\phi\in H(Q).$ (2.12)

We have shown that the system (2.8) has a solution $y\in
L^{2}((0,T);H^{1}_{a}(\Omega))$ in the sense of Definition 2.6. But since
$z_{t}=\left(a(x)z_{x}\right)_{x}-rz+v(t,x)\chi_{\omega}z+e^{-rt}f\in
L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime}),$ we deduce that $z\in W_{a}(0,T)$.

Step 2. We prove the uniqueness of solution to (2.8).
Assume that there exist $z_{1}$ and $z_{2}$ solutions to (2.8) with the same
right hand side $f,\,v$ and initial datum $p^{0}$. Set
$\tilde{z}:=z_{1}-z_{2}$. Then $\tilde{z}$ satisfies

$\left\\{\begin{array}[]{rllll}\displaystyle\tilde{z}_{t}-\left(a(x)\tilde{z}_{x}\right)_{x}+r\tilde{z}&=&v(t,x)\chi_{\omega}\tilde{z}&\mbox{in}&Q,\\\
\displaystyle a(x)\tilde{z}_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
\displaystyle\tilde{z}(0,\cdot)&=&0&\mbox{in}&\Omega.\end{array}\right.$
(2.13)

So, if we multiply the first equation of (2.13) by $\tilde{z}$, then use the
integration by parts over $Q$, we obtain

$\frac{1}{2}\|\tilde{z}(T,\cdot)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{T}\|\sqrt{a(\cdot)}\tilde{z}(t)\|^{2}_{L^{2}(\Omega)}\,dt+r\|\tilde{z}\|^{2}_{L^{2}(Q)}\leq\|v\|_{L^{\infty}(Q)}\|\tilde{z}\|^{2}_{L^{2}(Q)}.$

Using the fact that $r=\|v\|_{\infty}+1$ in this latter inequality, we deduce
that

$\begin{array}[]{lll}\displaystyle\frac{1}{2}\|\tilde{z}(T,\cdot)\|^{2}_{L^{2}(\Omega)}+\|\tilde{z}\|^{2}_{L^{2}((0,T);H^{1}_{a}(\Omega))}&\leq&0.\\\
\end{array}$

Hence, $\tilde{z}=0$ in $\Omega$. Thus, $z_{1}=z_{2}$ in $\Omega$ and we have
shown the uniqueness.
Step 3. We show the estimates (2.10).
Multiplying the first equation in (2.8) by $z\in
L^{2}((0,T);H^{1}_{a}(\Omega))$, using the integration by parts over $\Omega$
and Cauchy-Schwarz and Young inequalities, we obtain

$\begin{array}[]{llll}\displaystyle\frac{1}{2}\frac{d}{dt}\|z(t)\|^{2}_{L^{2}(\Omega)}+\|\sqrt{a(\cdot)}z(t)\|^{2}_{L^{2}(\Omega)}+r\|z(t)\|^{2}_{L^{2}(\Omega)}\\\
=\displaystyle e^{-rt}\left\langle
f(t),z(t)\right\rangle\,\mathrm{d}x+\int_{\omega}vz^{2}(t)\,\mathrm{d}x\\\
\leq\displaystyle\frac{1}{2}\|f(t)\|^{2}_{(H^{1}_{a}(\Omega))^{\prime}}+\frac{1}{2}\|z(t)\|^{2}_{H^{1}_{a}(\Omega)}+\displaystyle\|v\|_{\infty}\|z(t)\|^{2}_{L^{2}(\Omega)}.\end{array}$

Hence, due to the fact that $r=\|v\|_{\infty}+1$, we obtain

$\begin{array}[]{llll}\displaystyle\frac{d}{dt}\|z(t)\|^{2}_{L^{2}(\Omega)}+\|z(t)\|^{2}_{H^{1}_{a}(\Omega)}&\leq&\|f(t)\|^{2}_{(H^{1}_{a}(\Omega))^{\prime}}.\end{array}$
(2.14)

Now, integrating this latter inequality on $(0,\tau)$, with $\tau\in[0,T]$, we
obtain

$\begin{array}[]{llll}\displaystyle\|z(\tau)\|^{2}_{L^{2}(\Omega)}+\int_{0}^{\tau}\|z(t)\|^{2}_{H^{1}_{a}(\Omega)}\,\mathrm{d}t&\leq&\displaystyle\int_{0}^{T}\|f(t,\cdot)\|^{2}_{(H^{1}_{a}(\Omega))^{\prime}}\,\mathrm{d}t+\|p^{0}\|^{2}_{L^{2}(\Omega)},\end{array}$

from which we deduce that

$\begin{array}[]{lll}\displaystyle\sup_{\tau\in[0,T]}\|z(\tau,\cdot)\|^{2}_{L^{2}(\Omega)}&\leq&\left[\|p^{0}\|^{2}_{L^{2}(\Omega)}+\|f\|^{2}_{L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})}\right],\\\
\displaystyle\int_{0}^{T}\left\|z(t,\cdot)\right\|^{2}_{H^{1}_{a}(\Omega)}\,\mathrm{d}t&\leq&\displaystyle\left[\|p^{0}\|^{2}_{L^{2}(\Omega)}+\|f\|^{2}_{L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})}\right].\end{array}$

Hence,

$\begin{array}[]{lll}\displaystyle\|z\|_{{\mathcal{C}}([0,T];L^{2}(\Omega))}+\left\|z\right\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
2\left(\|p^{0}\|_{L^{2}(\Omega)}+\|f\|_{L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})}\right).\end{array}$

∎

Now, we prove the Theorem 2.5.

###### Proof of Theorem 2.5.

The existence of a unique weak solution $p\in W_{a}(0,T)$ to (2.4) in the
sense of Definition 2.3 follows directly from the Theorem 2.7. To prove the
estimate (2.7), we replace $z$ by its value $e^{-(\|v\|_{\infty}+1)t}p$ in the
estimate (2.10) to obtain:

$\|p\|_{{\mathcal{C}}([0,T];L^{2}(\Omega))}+\|p\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
2e^{(\|v\|_{\infty}+1)T}\left(\|p^{0}\|_{L^{2}(\Omega)}+\|f\|_{L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})}\right).$

∎

By taking $f=0$ in (2.4), we obtain the following well posed result for the
system (1.1).

######  Theorem 2.8.

Let $v\in L^{\infty}(\Omega)$ and $y^{0}\in L^{2}(\Omega)$. Then, there exists
a unique weak solution $y\in W_{a}(0,T)$ to (1.1) in the sense of Definition
2.3 (with $f=0$). In addition, the following estimate holds

$\|y\|_{{\mathcal{C}}([0,T];L^{2}(\Omega))}+\|y\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
2e^{(\|v\|_{\infty}+1)T}\|y^{0}\|_{L^{2}(\Omega)}.$ (2.15)

######  Remark 2.9.

Notice that if $f\in L^{2}(Q)$, then the Theorem 2.5 remain true and the
estimates (2.15) holds with
$\|f\|_{L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})}$ replaced by
$\|f\|_{L^{2}(Q)}$.

###  2.2. Maximum principle results

The aim of this subsection is to establish a weak maximum principle for the
system (1.1) in order to improve the regularity of the solution.

We have the following result. The proof is inspired from [10].

######  Theorem 2.10.

Let $y^{0}\in L^{2}(\Omega)$ be such that $y^{0}(x)\geq 0$ almost everywhere
in $\Omega$ and $v\in L^{\infty}(\Omega)$. Then the weak solution of (1.1) is
positive almost everywhere in $Q$.

###### Proof.

We just recall here the sketch of the proof. For $(t,x)\in Q$, we write
$y(t,x)=y^{+}(t,x)-y^{-}(t,x)$, where $y^{+}(t,x)=\max(y(t,x),0)$ and
$y^{-}(t,x)=\max(0,-y(t,x))$. It is sufficient to show that $y^{-}(t,x)=0$.
Notice that we have

$\begin{array}[]{llllll}y^{+}y^{-}=0\hbox{ in }Q\;\;\text{and }\ \
y^{-}(0,x)=\max(0,-y^{0}(x))=0\hbox{ in }\Omega,\\\ \end{array}$

and $y^{-}\in
L^{2}((0,T);H^{1}_{a}(\Omega)\cap\mathcal{C}([0,T];L^{2}(\Omega)).$

Multiplying the first equation in (1.1) by $y^{-}(t,x)$ and integrating over
$\Omega$, we obtain

$\begin{array}[]{llll}\displaystyle\langle
y_{t}(t),y^{-}(t)\rangle+\int_{\Omega}(a(x)y_{x}(t))_{x}y^{-}(t)\,\mathrm{d}x&=&\displaystyle\int_{\omega}vy(t)y^{-}(t)\,\mathrm{d}x.\end{array}$

Observe that the previous equality can be rewritten as

$-\displaystyle\frac{1}{2}\frac{d}{dt}\|y^{-}(t)\|^{2}_{L^{2}(\Omega)}-\int_{\Omega}a(x)y^{2}_{x}(t)\,\mathrm{d}x=-\displaystyle\int_{\omega}v(t)\left|y^{-}(t)\right|^{2}\,\mathrm{d}x.$

From which we deduce that

$\displaystyle\frac{d}{dt}\|y^{-}(t)\|^{2}_{L^{2}(\Omega)}\leq
2\|v\|_{\infty}\int_{\Omega}\left|y^{-}(t)\right|^{2}\,\mathrm{d}x.$

By using the Gronwall’s lemma it follows that

$\|y^{-}(t)\|^{2}_{L^{2}(\Omega)}\leq
e^{2T\|v\|_{\infty}}\|y^{-}(0,\cdot)\|_{L^{2}(\Omega)}=0.$

Hence $y^{-}(t,x)=0$ for almost every $(t,x)\in Q$. Consequently, $y\geq 0$
almost everywhere in $Q$. ∎

The next result state a weak maximum principle for the system (2.4).

######  Theorem 2.11.

Let $p^{0}\in L^{\infty}(\Omega)$, $f\in L^{\infty}(Q)$ and $v\in
L^{\infty}(\omega_{T})$. Then the weak solution $p\in W_{a}(0,T)$ of (2.4)
belongs to $L^{\infty}(Q)$ and the following estimate holds

$\begin{array}[]{lllll}p(t,x)\leq
e^{(\|v\|_{\infty}+1)T}\left(\|p^{0}\|_{L^{\infty}(\Omega)}+\|f\|_{L^{\infty}(\Omega)}\right)\;\hbox{
a.e. in }\Omega\times[0,T].\end{array}$ (2.16)

Moreover,

$\begin{array}[]{lllll}\|p\|_{L^{\infty}(Q)}\leq
e^{(\|v\|_{\infty}+1)T}\left(\|p^{0}\|_{L^{\infty}(\Omega)}+\|f\|_{L^{\infty}(Q)}\right).\end{array}$
(2.17)

###### Proof.

For any $(t,x)\in Q$, we set $z(t,x)=e^{-(\|v\|_{\infty}+1)t}p(t,x)$, where
$p$ is the weak solution to (2.4). Then according to Theorem 2.7, $z\in
W_{a}(0,T)$ is the unique weak solution of (2.8), that we recall here

$\left\\{\begin{array}[]{rllll}\displaystyle
z_{t}-\left(a(x)z_{x}\right)_{x}+(\|v\|_{\infty}+1)z&=&v(t,x)\chi_{\omega}z+e^{-(\|v\|_{\infty}+1)t}f&\mbox{in}&Q,\\\
\displaystyle a(x)z_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
\displaystyle z(0,\cdot)&=&p^{0}&\mbox{in}&\Omega.\end{array}\right.$

We claim that

$z(t,x)\leq\|p^{0}\|_{L^{\infty}(\Omega)}+\|f\|_{L^{\infty}(Q)}\;\hbox{ a.e.
in }\Omega\times[0,T].$ (2.18)

We set $w(t,x)=\|p^{0}\|_{L^{\infty}(\Omega)}+\|f\|_{L^{\infty}(Q)}-z(t,x)$
for any $(x,t)\in\Omega\times(0,T)$. Then,

$\begin{array}[]{lll}w(0,x)&=&\|f\|_{L^{\infty}(Q)}+(\|p^{0}\|_{L^{\infty}(\Omega)}-p^{0}(x)),\\\
&\geq&0,\end{array}$

for any $x\in\Omega.$ Moreover, $w$ satisfies

$\left\\{\begin{array}[]{rllll}\displaystyle
w_{t}-(a(x)w_{x})_{x}+(\|v\|_{\infty}+1)w&=&vw\chi_{\omega}+(\|v\|_{\infty}-v\chi_{\omega})K+\left(K-e^{-(\|v\|_{\infty}+1)t}f\right)&\mbox{in}&Q,\\\
\displaystyle a(x)w_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
w(0,\cdot)&=&\|p^{0}\|_{L^{\infty}(\Omega)}-p^{0}&\mbox{in}&\Omega,\end{array}\right.$
(2.19)

with $K=\|p^{0}\|_{L^{\infty}(\Omega)}+\|f\|_{L^{\infty}(Q)}$. We write
$w(t,x)=w^{+}(t,x)-w^{-}(t,x)$, where $w^{+}(t,x)=\max(w(t,x),0)$ and
$w^{-}(t,x)=\max(0,-w(t,x))$. It is sufficient to show that $w^{-}(t,x)=0$ for
almost every $(t,x)\in Q$. We have that $w^{-}\in
L^{2}((0,T);H^{1}_{a}(\Omega)\cap\mathcal{C}([0,T];L^{2}(\Omega)).$ Notice
that $w^{+}w^{-}=0$ and since $w(0,x)\geq 0$ in $\Omega$,
$w^{-}(0,x)=\max(0,-w(0,\cdot))=0\hbox{ in }\Omega.$

If we multiply the first equation in (2.19) by $w^{-}(t)$ and integrate over
$\Omega$, we obtain

$\begin{array}[]{llll}\displaystyle\langle
w_{t}(t),w^{-}(t)\rangle-\int_{\Omega}(a(x)w_{x}(t))_{x}w^{-}(t)\,\mathrm{d}x+(\|v\|_{\infty}+1)\int_{\Omega}w(t)w^{-}(t)\,\mathrm{d}x\\\
=\displaystyle\int_{\omega}v(t)w(t)w^{-}(t)\,\mathrm{d}x+K\int_{\omega}(\|v\|_{\infty}-v(t))w^{-}(t)\,\mathrm{d}x+\int_{\Omega}\left(K-e^{-(\|v\|_{\infty}+1)t}f(t)\right)w^{-}(t)\,\mathrm{d}x.\end{array}$
(2.20)

Observe that

$\begin{array}[]{llll}\displaystyle\langle w_{t}(t),w^{-}(t)\rangle=-\langle
w^{-}_{t}(t),w^{-}(t)\rangle=\displaystyle-\frac{1}{2}\frac{d}{dt}\|w^{-}(t)\|^{2}_{L^{2}(\Omega)},\end{array}$

$\begin{array}[]{llll}\displaystyle-\int_{\Omega}(a(x)w_{x}(t))_{x}w^{-}(t)\,\mathrm{d}x=\int_{\Omega}a(x)w_{x}(t)(-w_{x}(t))\,\mathrm{d}x=-\int_{\Omega}a(x)w^{2}_{x}(t)\,\mathrm{d}x,\end{array}$

$\begin{array}[]{llll}\displaystyle(\|v\|_{\infty}+1)\int_{\Omega}w(t)w^{-}(t)\,\mathrm{d}x=-(\|v\|_{\infty}+1)\int_{\Omega}(w^{-}(t))^{2}\,\mathrm{d}x\end{array}$

and

$\begin{array}[]{llll}\displaystyle\int_{\Omega}v(t)w(t)w^{-}(t)\,\mathrm{d}x=-\int_{\Omega}v(t)(w^{-}(t))^{2}\,\mathrm{d}x.\end{array}$

Therefore, (2.20) can be rewritten as

$\begin{array}[]{llll}\displaystyle\frac{1}{2}\frac{d}{dt}\|w^{-}(t)\|^{2}_{L^{2}(\Omega)}+\int_{\Omega}a(x)w^{2}_{x}(t)\,\mathrm{d}x+(\|v\|_{\infty}+1)\int_{\Omega}(w^{-}(t))^{2}\,\mathrm{d}x\\\
=\displaystyle\int_{\Omega}v(t)(w^{-}(t))^{2}\,\mathrm{d}x-K\int_{\omega}(\|v\|_{\infty}-v(t))w^{-}(t)\,\mathrm{d}x-\int_{\Omega}\left(K-e^{-(\|v\|_{\infty}+1)t}f(t)\right)w^{-}(t)\,\mathrm{d}x.\end{array}$
(2.21)

On the one hand, since $a(\cdot)>0$ in $\Omega$, then the left hand side of
(2.21) can be lower bounded by its first term. On the other hand,
$\|v\|_{\infty}-v\geq 0$ and $\displaystyle
K-e^{-(\|v\|_{\infty}+1)t}f=\|p^{0}\|_{L^{\infty}(\Omega)}+\left(\|f\|_{L^{\infty}(Q)}-e^{-(\|v\|_{\infty}+1)t}f\right)\geq
0$ almost every where in $Q$ and so the right hand side of (2.21) can be upper
bounded by the term
$\displaystyle\|v\|_{\infty}\int_{\Omega}\left(w^{-}(t)\right)^{2}dx$. Hence
from (2.21), we deduce that

$\displaystyle\frac{d}{dt}\|w^{-}(t)\|^{2}_{L^{2}(\Omega)}\leq
2\|v\|_{\infty}\int_{\Omega}\left(w^{-}(t)\right)^{2}\,\mathrm{d}x.$

By using the Gronwall’s lemma it follows that

$\|w^{-}(t)\|^{2}_{L^{2}(\Omega)}\leq
e^{2t\|v\|_{\infty}}\|w^{-}(0,\cdot)\|_{L^{2}(\Omega)}=0.$

Hence $w^{-}(t,x)=0$ for almost every $(t,x)\in Q$, that is $w(t,x)\geq 0$ for
almost every $(t,x)\in Q$. Consequently,

$z(t,x)\leq\|y^{0}\|_{L^{\infty}(\Omega)}+\|f\|_{L^{\infty}(Q)}\;\hbox{ a.e.
in }[0,T]\times\Omega.$

Now by taking $z(t,x)=e^{-(\|v\|_{\infty}+1)t}p(t,x)$ in this latter
inequality, we deduce (2.16) and then (2.17). ∎

By taking $f=0$ in the Corollary 2.12, we obtain the following weak maximum
principle for the system (1.1).

######  Corollary 2.12.

Let $y^{0}\in L^{\infty}(\Omega)$ and $v\in L^{\infty}(\omega_{T})$. Then the
weak solution $y\in W_{a}(0,T)$ of (1.1) belongs to $L^{\infty}(Q)$. More
precisely, the following estimate holds

$\begin{array}[]{lllll}y(x,t)\leq
e^{(\|v\|_{\infty}+1)T}\|y^{0}\|_{L^{\infty}(\Omega)}\;\hbox{ a.e. in
}\Omega\times[0,T].\end{array}$ (2.22)

Moreover,

$\begin{array}[]{lllll}\|y\|_{L^{\infty}(Q)}\leq
e^{(\|v\|_{\infty}+1)T}\|y^{0}\|_{L^{\infty}(\Omega)}.\end{array}$ (2.23)

##  3\. Resolution of the optimization problem

We state the following definition.

######  Definition 3.1.

We define the control-to-state mapping

$G:L^{\infty}(\omega_{T})\to W_{a}(0,T),$

which associates to each $u\in L^{\infty}(\omega_{T})$ the unique weak
solution $y\in W_{a}(0,T)$ of (1.1). Sometimes, we may write $G(u)$ to denote
the state $y$ corresponding to the control $u$.

We consider the following optimal control problem:

$\inf_{v\in\mathcal{U}}J(G(v),v),$ (3.1)

where

$\displaystyle\mathcal{U}=\left\\{w\in L^{\infty}(\omega_{T}):m\leq w\leq
M\right\\}$ (3.2)

and

$J(G(v),v)=\frac{1}{2}\|G(v)(T)-y^{d}\|^{2}_{L^{2}(\Omega)}+\frac{\alpha}{2}\|v\|^{2}_{L^{2}(\omega_{T})},$
(3.3)

with $\alpha>0$, $m,M\in\mathbb{R}$ with $m<M$, $y^{d}\in
L^{\infty}({\Omega})$ and $G(v)\in W_{a}(0,T)\cap L^{\infty}(Q)$ is solution
to (1.1).

######  Definition 3.2.

We say that $u\in\mathcal{U}$ is a $L^{\infty}$-local solution (resp.,
$L^{2}$-local solution) of (3.1)-(3.3) if there exists $\varepsilon>0$ such
that $J(G(u),u)\leq J(G(v),v)$ holds for every $v\in\mathcal{U}\cap
B^{\infty}_{\varepsilon}(u)$ (resp. $v\in\mathcal{U}\cap
B^{2}_{\varepsilon}(u)$), where $B^{\infty}_{\varepsilon}(u)$ (resp.,
$B^{2}_{\varepsilon}(u)$) denote the open ball in $L^{\infty}(Q)$ (resp.,
$L^{2}(Q)$) of radius $\varepsilon$ centered at $u$.

###  3.1. Existence of optimal controls

In this Section we prove the existence of optimal controls.

######  Theorem 3.3.

Let $\alpha>0$, $v\in\mathcal{U}$ and $y^{0},y^{d}\in L^{\infty}(\Omega)$.
Then there exists a solution $u\in\mathcal{U}$ of the optimal control problem
(3.1)-(3.3).

###### Proof.

Notice that $J(G(v),v)\geq 0$ for all $v\in\mathcal{U}$. Let
$\\{(G(v^{n}),v_{n})\\}_{n}\subset W_{a}(0,T)\times\mathcal{U}$ be a
minimizing sequence such that

$\lim_{n\to\infty}J(G(v^{n}),v^{n})\to\min_{v\in\mathcal{U}}J(G(v),v).$

Since $y^{n}:=G(v^{n})$ is the state associated to the control $v^{n}$, there
is a positive constant $C$ independent of $n$ which may varies form one line
to another such that

$\|y^{n}(T)\|_{L^{2}(\Omega)}\leq C$ (3.4)

and

$\|v^{n}\|_{L^{2}(\omega_{T})}\leq C.$ (3.5)

Moreover, $y^{n}$ being solution of (1.1) with $v=v^{n}$, it follows from
(2.15) that

$\|y^{n}\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
2e^{(\|v^{n}\|_{\infty}+1)T}\|y^{0}\|_{L^{2}(\Omega)}\leq
2e^{(\beta+1)T}\|y^{0}\|_{L^{2}(\Omega)},$ (3.6)

with $\beta=\max\\{|m|,|M|\\}$. Consequently,

$\displaystyle\|v^{n}y^{n}\|_{L^{2}(\omega_{T})}\leq C\|y^{n}\|_{L^{2}(Q)}\leq
C\|y^{n}\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
2Ce^{(\beta+1)T}\|y^{0}\|_{L^{2}(\Omega)}.$ (3.7)

From (3.4)-(3.7), we have that there exist $\eta\in L^{2}(\Omega)$, $u\in
L^{2}(\omega_{T})$ and $y\in L^{2}((0,T);H^{1}_{a}(\Omega))$ such that (up to
a subsequence), as $n\to\infty$, we have

$y^{n}(T)\rightharpoonup\eta\text{ weakly in }L^{2}(\Omega),$ (3.8)

$v^{n}\rightharpoonup u\text{ weakly in }L^{2}(\omega_{T}),$ (3.9)

$y^{n}\rightharpoonup y\text{ weakly in }L^{2}(0,T;H^{1}_{a}(\Omega))$ (3.10)

and one obtain easily that

$v^{n}y^{n}\rightharpoonup uy\text{ weakly in }L^{2}(\omega_{T}).$ (3.11)

Since $\mathcal{U}$ is a closed convex subset of $L^{2}(\omega_{T})$, we have
that $\mathcal{U}$ is weakly closed and so

$u\in\mathcal{U}.$ (3.12)

Using the convergences (3.8)-(3.11) and classical arguments (see e.g. [15, 21,
23, 24, 26]), we show that $y=y(u)$ and

$y^{n}(T)\rightharpoonup y(T)\text{ weakly in }L^{2}(\Omega).$ (3.13)

Moreover, using (3.9), (3.12), (3.13) and the lower semi-continuity of the
functional cost $J$, it follows that

$\displaystyle J(G(u),u)$ $\displaystyle\leq$
$\displaystyle\liminf_{n\to\infty}J(G(v^{n}),v^{n})=\inf_{v\in\mathcal{U}}J(G(v),v).$

∎

##  4\. Regularity results of the control-to-state mapping

In the rest of the paper, we assume that $y^{0},y^{d}\in L^{\infty}(\Omega)$,
so the weak solution $y$ of (1.1) belongs to $W_{a}(0,T)\cap L^{\infty}(Q)$
(see Corollary 2.12).

Using the Implicit Function Theorem, we establish in this section some
regularity results on the control-to-state operator. The proofs are inspired
from [8]. We define the mapping

$(y,u)\mapsto\mathcal{G}(y,u):=(y_{t}-(a(x)y_{x})_{x}-u\chi_{\omega}y,y(0)-y^{0})$
(4.1)

from $W_{a}(0,T)\times L^{\infty}(\omega_{T})\to
L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})\times L^{2}(\Omega).$ Then, the
state equation $y$ solution of (1.1) can be viewed as the equation

$\mathcal{G}(y,u)=0.$ (4.2)

######  Proposition 4.1.

The mapping $\mathcal{G}$ is of class $\mathcal{C}^{\infty}$. Moreover, the
control-to-state mapping $G:L^{\infty}(\omega_{T})\to W_{a}(0,T),u\mapsto y$
is also of class $\mathcal{C}^{\infty}$.

###### Proof.

For all $\phi\in L^{2}((0,T);H^{1}_{a}(\Omega))$, we write the first component
$\mathcal{G}_{1}$ of $\mathcal{G}$

$\mathcal{G}_{1}(y,u)(\phi):=\left\langle
y_{t},\phi\right\rangle+\int_{Q}a(x)y_{x}\phi_{x}\,\mathrm{d}x\,\mathrm{d}t-\int_{\omega_{T}}uy\,\phi\;\mathrm{d}x\,\mathrm{d}t.$

On the one hand, the first two terms of $\mathcal{G}_{1}$ define linear and
continuous mapping from $W_{a}(0,T)$ to
$L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})$. Thus they are of class
$\mathcal{C}^{\infty}$. On the other hand the last term of $\mathcal{G}_{1}$
defines a bilinear and continuous mapping from $W_{a}(0,T)\times
L^{\infty}(\omega_{T})$ to $L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime}).$
Therefore $\mathcal{G}_{1}$ is of class $\mathcal{C}^{\infty}$. The second
component of $\mathcal{G}$ is clearly of class $\mathcal{C}^{\infty}$.

Moreover,

$\partial_{y}\mathcal{G}(y,u)\varphi=(\varphi_{t}-(a(x)\varphi_{x})_{x}-u\varphi\chi_{\omega},\varphi(0)).$

For $u\in L^{\infty}(\omega_{T})$, $\varphi^{0}\in L^{2}(\Omega)$ and $f\in
L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})$, then the Theorem 2.5 shows that
the problem

$\left\\{\begin{array}[]{rllll}\displaystyle\varphi_{t}-(a(x)\varphi_{x})_{x}&=&u\varphi\chi_{\omega}+f&\mbox{in}&Q,\\\
\displaystyle a(x)\varphi_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
\varphi(0,\cdot)&=&\varphi^{0}&\mbox{in}&\Omega\end{array}\right.$

has a unique weak solution $\varphi:=\varphi(\varphi^{0},f)$ in $W_{a}(0,T)$.
Moreover, $\varphi$ depends continuously on $\varphi^{0}\in L^{2}(\Omega)$ and
on $f\in L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})$. Hence
$\displaystyle\partial_{y}\mathcal{G}(y,u)$ define an isomorphism from
$W_{a}(0,T)$ to $L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})\times
L^{2}(\Omega)$. Using the Implicit Function Theorem, we deduce that
$\mathcal{G}(y,u)=0$ implicitly define the control-to-state operator
$G:u\mapsto y$ which is itself of class $\mathcal{C}^{\infty}$. ∎

The following result prove the Lipschitz continuity of the control-to-state
mapping $G$.

######  Proposition 4.2.

Let $v\in L^{\infty}(\omega_{T})$ and $y^{0}\in L^{\infty}(\Omega)$. Then the
control-to-state mapping $v\mapsto G(v)$ is Lipschitz continuous function from
$L^{2}(\omega_{T})$ onto $W_{a}(0,T)$. More precisely, for all $v_{1},v_{2}\in
L^{\infty}(\omega_{T})$, there is a constant
$C=C(\|v_{1}\|_{\infty},\|v_{2}\|_{\infty},T)>0$ such that the following
estimate holds

$\|G(v_{1})-G(v_{2})\|_{\mathcal{C}([0,T];L^{2}(\Omega))}+\|G(v_{1})-G(v_{2})\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
C\|y^{0}\|_{L^{\infty}(\Omega)}\|v_{1}-v_{2}\|_{L^{2}(\omega_{T})}.$ (4.3)

###### Proof.

Let $v_{1},v_{2}\in L^{\infty}(\omega_{T}).$ Set $z:=y(v_{1})-y(v_{2})$ with
$y(v_{1})$ and $y(v_{2})$ being solutions of (1.1) with $v=v_{1}$ and
$v=v_{2}$ respectively. Then, $z$ satisfies the following system

$\left\\{\begin{array}[]{lllll}\displaystyle
z_{t}-(a(x)z_{x})_{x}&=&(v_{1}z+(v_{1}-v_{2})y(v_{2}))\chi_{\omega}&\hbox{in}&Q,\\\
\displaystyle a(x)z_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
z(0,\cdot)&=&0&\hbox{in}&\Omega.\end{array}\right.$ (4.4)

Using the estimation (2.7) with $f=(v_{1}-v_{2})y(v_{2})\chi_{\omega}\in
L^{2}(Q)$, it follows that

$\|z\|_{\mathcal{C}([0,T];L^{2}(\Omega))}+\|z\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
2C\|G(v_{2})\|_{L^{\infty}(Q)}\|v_{1}-v_{2}\|_{L^{2}(\omega_{T})},$ (4.5)

for some $C=C(\|v_{1}\|_{\infty},T)>0$. Observing that $y^{0}\in
L^{\infty}(\Omega)$, then using Theorem 2.12, we have from (2.23) that

$\|G(v_{2})\|_{L^{\infty}(Q)}\leq
C(\|v_{2}\|_{\infty},T)\|y^{0}\|_{L^{\infty}(\Omega)}.$ (4.6)

Therefore, by combining (4.5) and (4.6), we deduce that

$\displaystyle\|z\|_{{\mathcal{C}}([0,T];L^{2}(\Omega))}+\|z\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
C(\|v_{1}\|_{\infty},\|v_{2}\|_{\infty},T)\|y^{0}\|_{L^{\infty}(\Omega)}\|v_{1}-v_{2}\|_{L^{2}(\omega_{T})},$

from which we deduce (4.3). ∎

##  5\. First-order necessary optimality conditions

Before going further, we define the reduced cost functional as follows

$\mathcal{J}(v):=J(G(v),v).$ (5.1)

Note that $\mathcal{J}$ is continuously Fréchet differentiable, since $J$ and
$G$ have this property. The main purpose of this section is to establish the
following result.

######  Theorem 5.1 (First order necessary optimality conditions).

Let $\alpha>0,$ $v\in\mathcal{U}$ and $y^{0},y^{d}\in L^{\infty}(\Omega)$ .
Let $u$ be an $L^{\infty}$-local minimum for the minimization problem (3.1).
Then the following holds

$\mathcal{J}^{\prime}(u)(v-u)\geq 0\;\;\;\text{for
every}\;\;\;v\in\mathcal{U}.$ (5.2)

Moreover, there exists $y,q\in W_{a}(0,T)\cap L^{\infty}(Q)$ such that the
triple $(y,u,q)$ satisfies

$\left\\{\begin{array}[]{lllll}\displaystyle
y_{t}-(a(x)y_{x})_{x}&=&uy\chi_{\omega}&in&Q,\\\ \displaystyle
a(x)y_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
y(0,\cdot)&=&y^{0}&in&\Omega,\end{array}\right.$ (5.3)
$\left\\{\begin{array}[]{lllll}\displaystyle-
q_{t}-(a(x)q_{x})_{x}&=&uq\chi_{\omega}&in&Q,\\\ \displaystyle
a(x)q_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
q(T,\cdot)&=&y(T;u)-y^{d}&in&\Omega,\end{array}\right.$ (5.4)

$\displaystyle\int_{\omega_{T}}(\alpha u+y(u)q)(v-u)\
\mathrm{d}x\,\mathrm{d}t\geq 0,\quad\forall v\in\mathcal{U},$ (5.5)

and equivalently

$u=\min\left(\max\left(m,-\frac{q}{\alpha}y(u)\right),M\right)\text{ in
}\omega_{T}.$ (5.6)

######  Remark 5.2.

We notice that any $L^{\infty}$-local solution of the variational inequality
(5.2) is also a $L^{2}$-local one.

We first prove some important results needed further. The proofs follow
similar arguments as in [20]. We recall here the sketch of the proofs.

######  Proposition 5.3.

Let $G:L^{\infty}(\omega_{T})\to W_{a}(0,T)\cap L^{\infty}(Q),v\mapsto y$ be
the control-to-state operator, where $y$ is the solution of (1.1). Then, the
directional derivate of $G$ in every direction $w\in L^{\infty}(\omega_{T})$
is given by

$G^{\prime}(u)w=\rho,$

where the state $y=G(u)$ correspond to $u$ and $\rho\in W_{a}(0,T)$ is the
unique weak solution of

$\left\\{\begin{array}[]{lllll}\displaystyle\rho_{t}-(a(x)\rho_{x})_{x}&=&(u\rho+wy)\chi_{\omega}&\hbox{in}&Q,\\\
\displaystyle a(x)\rho_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
\rho(0,\cdot)&=&0&\hbox{in}&\Omega.\end{array}\right.$ (5.7)

###### Proof.

Let $v,w\in L^{\infty}(\omega_{T})$. For the existence and uniqueness of
$\rho\in W_{a}(0,T)$ and the estimate, we can apply the Theorem 2.8, with
$f=wy\chi_{\omega}\in L^{2}(Q)\hookrightarrow
L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})$. Now, let $\lambda>0$ and set
$\displaystyle y_{\lambda}:=\frac{y(v+\lambda w)-y(v)}{\lambda}.$ Then
$y_{\lambda}$ is solution to the problem

$\left\\{\begin{array}[]{lllll}\displaystyle(y_{\lambda})_{t}-(a(x)(y_{\lambda})_{x})_{x}&=&(vy_{\lambda}+wy(v+\lambda
w))\chi_{\omega}&\hbox{in}&Q,\\\ \displaystyle
a(x)(y_{\lambda})_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
y_{\lambda}(0,\cdot)&=&0&\hbox{in}&\Omega.\end{array}\right.$

Define $p_{\lambda}:=e^{-rt}(y_{\lambda}-\rho)$, with $r>0$. Then
$p_{\lambda}$ is a solution to

$\left\\{\begin{array}[]{lllll}\displaystyle(p_{\lambda})_{t}-(a(x)(p_{\lambda})_{x})_{x}+rp_{\lambda}&=&[vp_{\lambda}+e^{-rt}w(y(v+\lambda
w)-y(v))]\chi_{\omega}&\hbox{in}&Q,\\\ \displaystyle
a(x)(p_{\lambda})_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
p_{\lambda}(0,\cdot)&=&0&\hbox{in}&\Omega.\end{array}\right.$ (5.8)

Using the estimation (2.10), we obtain that

$\displaystyle\|p_{\lambda}\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq\|w\|_{\infty}\|y(v+\lambda
w)-y(v)\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}.$

By taking the limit as $\lambda\to 0$ in this latter inequality and using the
Proposition 4.2, we obtain that $p_{\lambda}\to 0$ strongly in
$L^{2}((0,T);H^{1}_{a}(\Omega))$. Hence $y_{\lambda}\to\rho$ strongly in
$L^{2}((0,T);H^{1}_{a}(\Omega))$ as $\lambda\to 0$. This proves (5.7). ∎

######  Proposition 5.4 ( Fréchet differentiability of $\mathcal{J}$).

Let $u\in L^{\infty}(\omega_{T})$, $y=G(u)$ be the solution of (5.3) and $q$
be the solution of (5.4). Under the hypothesis of Propositions 6.3, the
functional $\mathcal{J}:L^{\infty}(\omega_{T})\to\mathbb{R}$ is continuously
Fréchet differentiable. Moreover for every $w\in L^{\infty}(\omega_{T})$, we
have

$\mathcal{J}^{\prime}(u)w=\int_{\omega_{T}}(\alpha
u+yq)w\,\mathrm{d}x\,\mathrm{d}t.$ (5.9)

###### Proof.

Let $u,w\in L^{\infty}(\omega_{T})$, we have after some straightforward
calculations and using Proposition 5.3,

$\lim_{\lambda\to 0}\frac{\mathcal{J}(u+\lambda
w)-\mathcal{J}(u)}{\lambda}=\int_{\Omega}\rho(T)(y(T,u)-y^{d})\,\mathrm{d}x+\alpha\int_{\omega_{T}}uw\
\mathrm{d}x\,\mathrm{d}t,$ (5.10)

where $\rho\in W_{a}(0,T)$ is the unique solution of (5.7), that we recall
here:

$\left\\{\begin{array}[]{lllll}\displaystyle\rho_{t}-(a(x)\rho_{x})_{x}&=&(u\rho+wy)\chi_{\omega}&\hbox{in}&Q,\\\
\displaystyle a(x)\rho_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
\rho(0,\cdot)&=&0&\hbox{in}&\Omega.\end{array}\right.$

To interpret (5.10), we use the adjoint state given by (5.4), that we recall
here

$\left\\{\begin{array}[]{lllll}\displaystyle-
q_{t}-(a(x)q_{x})_{x}&=&uq\chi_{\omega}&in&Q,\\\ \displaystyle
a(x)q_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
q(T,\cdot)&=&y(T;u)-y^{d}&in&\Omega.\end{array}\right.$

Make the change of variable $t\mapsto T-t$ in this latter system, we have that
$\varphi(x,t)=q(x,T-t)$ satisfies

$\left\\{\begin{array}[]{lllll}\displaystyle\varphi_{t}-(a(x)\varphi_{x})_{x}&=&\tilde{u}\varphi\chi_{\omega}&\hbox{in}&Q,\\\
\displaystyle\displaystyle a(x)\varphi_{x}(\cdot,x)|_{x=\pm
1}&=&0&\hbox{in}&(0,T),\\\
\varphi(0,\cdot)&=&y^{0}-y^{d}&\hbox{in}&\Omega,\end{array}\right.$ (5.11)

where $\tilde{u}=u(T-t)$. Since $y^{0}-y^{d}\in L^{\infty}(\Omega)$ and
$\tilde{u}\in L^{\infty}(\omega_{T})$, then applying Theorem 2.8 with $f=0$
and Corollary 2.12, we deduce that, there exists a unique adjoint state $q\in
W_{a}(0,T)\cap L^{\infty}(Q)$ solution to (5.4) in the sense of Definition
2.3.

So, if we multiply the first equation in (5.7) by $q$ solution of (5.4), and
integrate by parts over $Q$, we get

$\int_{\Omega}y(T)(y(T;u)-y^{d})\,\mathrm{d}x=\int_{\omega_{T}}wy(u)q\
\mathrm{d}x\,\mathrm{d}t,$ (5.12)

which combining with (5.10) gives

$\displaystyle\mathcal{J}^{\prime}(u)w=\int_{\omega_{T}}(\alpha u+y(u)q)w\
\mathrm{d}x\,\mathrm{d}t.$

∎

We now prove the main result of this section.

###### Proof of Theorem 5.1.

Let $v\in\mathcal{U}$ be arbitrary. Since $\mathcal{U}$ is convex, then
$u+\lambda(v-u)\in\mathcal{U}$ for all $\lambda\in(0,1]$. But $u$ is an
$L^{\infty}$-local minimum, so $\mathcal{J}(u+\lambda(v-u))\geq\mathcal{J}(u)$
and hence

$\frac{\mathcal{J}(u+\lambda(v-u))-\mathcal{J}(u)}{\lambda}\geq
0\qquad\forall\;\lambda\in(0,1].$

By letting $\lambda\to 0$ in this latter inequality, we obtain (5.2). We have
already shown (5.3) in Theorem 3.3. To complete the proof of the Theorem 5.1,
we write

$\displaystyle\lim_{\lambda\to 0}\frac{\mathcal{J}(u+\lambda
w)-\mathcal{J}(u)}{\lambda}\geq 0\quad\forall v\in\mathcal{U}.$

Using Proposition 5.4, with $w=v-u$, we obtain that

$\displaystyle\int_{\omega_{T}}(\alpha u+y(u)q)(v-u)\
\mathrm{d}x\,\mathrm{d}t\geq 0\quad\forall v\in\mathcal{U},$

where $q$ is solution to (5.4).

∎

######  Remark 5.5.

The relation (5.2) (equivalently (5.5)) is called first order necessary
conditions. We notice that from Theorem 5.1, a necessary and sufficient
condition for (5.5) to hold is that for a.e $(t,x)\in Q$,

$\left\\{\begin{array}[]{lllll}\displaystyle u=m&if&\alpha
u(t,x)+y(u)q(t,x)>0,\\\ \displaystyle u\in[m,M]&if&\alpha
u(t,x)+y(u)q(t,x)=0,\\\ u=M&if&\alpha u(t,x)+y(u)q(t,x)<0.\end{array}\right.$
(5.13)

Note that from (5.13), we have $|\alpha u+y(u)q|>0$ implies $u=m$ or $u=M$.

######  Remark 5.6.

By a change of variable $t\mapsto T-t$, one can show that the solution $q$ of
the adjoint state (5.4) which belongs to $W_{a}(0,T)\cap L^{\infty}(Q)$
satisfies the estimates:

$\|q\|_{{\mathcal{C}}([0,T];L^{2}(\Omega))}+\|q\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
C(\|u\|_{\infty},T)\left[\|y^{0}\|_{L^{\infty}(\Omega)}+\|y^{d}\|_{L^{\infty}(\Omega)}\right]$
(5.14)

and

$\|q\|_{L^{\infty}(Q)}\leq
e^{(\|u\|_{\infty}+1)T}\left[\|y^{0}\|_{L^{\infty}(\Omega)}+\|y^{d}\|_{L^{\infty}(\Omega)}\right].$
(5.15)

We end this section by the following important results.

######  Lemma 5.7.

For every $u\in L^{\infty}(\omega_{T})$, the linear mapping
$v\mapsto\mathcal{J}^{\prime}(u)v$ can be extended to a linear continuous
mapping $\mathcal{J}^{\prime}(u):L^{2}(\omega_{T})\to\mathbb{R}$ given by
(5.9).

###### Proof.

Let $u\in L^{\infty}(\omega_{T})$ and $v\in L^{2}(\omega_{T})$. From (5.9), we
have

$\mathcal{J}^{\prime}(u)v=\int_{\omega_{T}}(\alpha
u+yq)v\,\mathrm{d}x\,\mathrm{d}t,$

where $y$ and $q$ are respectively solutions to (5.3) and (5.4). Then using
(2.22) and (5.14), there exists a constant
$C=C\left(\alpha,\|u\|_{\infty},T,\|y^{0}\|_{L^{\infty}(\Omega)},\|y^{d}\|_{L^{\infty}(\Omega)}\right)$
independent of $v$ such that

$\begin{array}[]{lll}\displaystyle|\mathcal{J}^{\prime}(u)v|\leq
C\|v\|_{L^{2}(\Omega)}.\end{array}$

Hence, the mapping $v\mapsto\mathcal{J}^{\prime}(u)v$ is a linear continuous
mapping on $L^{2}(\omega_{T})$. ∎

Arguing as the same as in the proof of Proposition 4.2, we obtain the
following.

######  Proposition 5.8.

Let $u\in L^{\infty}(\omega_{T})$. Then the mapping $u\mapsto q(u)$ solution
of the adjoint problem (5.4) is a Lipschitz continuous function from
$L^{2}(\omega_{T})$ onto $W_{a}(0,T)$. More precisely for all $u_{1},u_{2}\in
L^{\infty}(\omega_{T})$, there exists a constant
$C=C(\|u_{1}\|_{\infty},\|u_{2}\|_{\infty},\|y^{d}\|_{L^{\infty}(\Omega)},\|y^{0}\|_{L^{\infty}(\Omega)},T)>0$
such that the following estimate

$\|q(u_{1})-q(u_{2})\|_{{\mathcal{C}}([0,T];L^{2}(\Omega))}+\|q(u_{1})-q(u_{2})\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
C\|u_{1}-u_{2}\|_{L^{2}(\omega_{T})}$ (5.16)

holds.

##  6\. Second-order necessary and sufficient optimality conditions

Since, the cost functional (3.3) associated to the optimization problem (3.1)
is non-convex, the first order optimality conditions developed in Theorem 5.1
are necessary but not sufficient for optimality. In this section, we develop
the second order optimality conditions which ensure the sufficiency. They also
ensure the stability of optimal solutions with respect to perturbations of the
problems such as finite element discretization [13].

Using the same arguments as in the proof of Proposition 6.3, we obtain the
following.

######  Proposition 6.1 (Twice Fréchet differentiability of $G$).

Under the hypothesis of Proposition 6.3, the control-to-state mapping
$G:v\mapsto y$, is twice continuously Fréchet differentiable from
$L^{\infty}(\omega_{T})$ into $W_{a}(0,T)$. Moreover, the second derivative of
$G$ at $u\in L^{\infty}(\omega_{T})$ is given by the expression

$G^{\prime\prime}(u)[w,h]=z,$

where $w,h\in L^{\infty}(\omega_{T})$ and with $z\in W_{a}(0,T)$ being the
uniquely determined weak solution of

$\left\\{\begin{array}[]{lllll}\displaystyle
z_{t}-(a(x)z_{x})_{x}&=&(uz+hG^{\prime}(u)w+wG^{\prime}(u)h)\chi_{\omega}&\hbox{in}&Q,\\\
\displaystyle a(x)z_{x}(\cdot,x)|_{x=\pm 1}&=&0&\mbox{in}&(0,T),\\\
z(0,\cdot)&=&0&\hbox{in}&\Omega.\end{array}\right.$ (6.1)

######  Proposition 6.2 (Twice Fréchet differentiability of $\mathcal{J}$).

Let $u\in L^{\infty}(\omega_{T})$ and $y=G(u)$ be the solution of (5.3) and
$q$ be the solution of (5.4). Under the hypothesis of Propositions 6.1 and
6.3, the functional $\mathcal{J}:L^{\infty}(\omega_{T})\to\mathbb{R}$ is twice
continuously Fréchet differentiable. Moreover for every $u,w,h\in
L^{\infty}(\omega_{T})$, we have

$\mathcal{J}^{\prime\prime}(u)[w,h]=\int_{\omega_{T}}[hG^{\prime}(u)w+wG^{\prime}(u)h]q\,\mathrm{d}x\,\mathrm{d}t+\int_{\Omega}(G^{\prime}(u)w)(T)(G^{\prime}(u)h)(T)\,\mathrm{d}x+\alpha\int_{\omega_{T}}hw\,\mathrm{d}x\,\mathrm{d}t.$
(6.2)

###### Proof.

It follows from Lemma 4.1, Proposition 6.1, straightforward computations and
Lebesgue dominated convergence theorem. ∎

We have the following important result.

######  Lemma 6.3.

Let $G:L^{\infty}(\omega_{T})\to W_{a}(0,T)\cap L^{\infty}(Q),u\mapsto y$ be
the control-to-state operator, where $y$ is solution to (1.1). Then for any
$u\in L^{\infty}(\omega_{T})$, the linear mapping $v\mapsto G^{\prime}(u)v$
can be extended to a linear continuous mapping from $L^{2}(\omega_{T})\to
W_{a}(0,T)$. In addition, the following estimate holds

$\|G^{\prime}(u)w\|_{{\mathcal{C}}([0,T];L^{2}(\Omega))}+\|G^{\prime}(u)w\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
2e^{2(\|u\|_{\infty}+1)T}\|w\|_{L^{2}(\omega_{T})},$ (6.3)

for all $v\in L^{2}(\omega_{T}).$

###### Proof.

For the extension, it is sufficient to prove that for any $v\in
L^{2}(\omega_{T})$, the system (5.7) has a unique solution $\rho\in
W_{a}(0,T)$. This follows directly from Theorem 2.8, with
$f=wy\chi_{\omega}\in L^{2}(Q)\hookrightarrow
L^{2}((0,T);(H^{1}_{a}(\Omega))^{\prime})$. Note that $f\in L^{2}(Q)$ because
$y\in L^{\infty}(Q)$. For the estimate, we take $y^{0}=0$ and
$f=wy\chi_{\omega}$ in (2.7), to obtain

$\|G^{\prime}(u)w\|_{{\mathcal{C}}([0,T];L^{2}(\Omega))}+\|G^{\prime}(u)w\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
2e^{(\|u\|_{\infty}+1)T}\|w\|_{L^{2}(\omega_{T})}\|y\|_{L^{\infty}(Q)},$

which using the estimate (2.22) gives

$\|G^{\prime}(u)w\|_{{\mathcal{C}}([0,T];L^{2}(\Omega))}+\|G^{\prime}(u)w\|_{L^{2}((0,T);H^{1}_{a}(\Omega))}\leq
2e^{2(\|u\|_{\infty}+1)T}\|w\|_{L^{2}(\omega_{T})}.$

∎

######  Lemma 6.4.

Let $u\in L^{\infty}(\omega_{T})$, then, the bilinear mapping
$(w,h)\mapsto\mathcal{J}^{\prime\prime}(u)[w,h]$ can be extended to a bilinear
continuous mapping on $\mathcal{J}^{\prime\prime}(u):L^{2}(\omega_{T})\times
L^{2}(\omega_{T})\to\mathbb{R}$ given by (6.2).

###### Proof.

$u\in L^{\infty}(\omega_{T})$ and $w,h\in L^{2}(\omega_{T})$. From (6.2), we
have

$\mathcal{J}^{\prime\prime}(u)[w,h]=\int_{\omega_{T}}[hG^{\prime}(u)w+wG^{\prime}(u)h]q\,\mathrm{d}x\,\mathrm{d}t+\int_{\Omega}(G^{\prime}(u)w)(T)(G^{\prime}(u)h)(T)\,\mathrm{d}x+\alpha\int_{\omega_{T}}hw\,\mathrm{d}x\,\mathrm{d}t,$

where $G^{\prime}(u)w$ is solution to (5.7). Using Cauchy Schwarz’s
inequality, (5.14) and (6.3), we obtain

$\begin{array}[]{lll}\displaystyle|\mathcal{J}^{\prime\prime}(u)[w,h]|&\leq&\|q\|_{L^{\infty}(Q)}\left[\|h\|_{L^{2}(\omega_{T})}\|G^{\prime}(u)w\|_{L^{2}(Q)}+\|w\|_{L^{2}(\omega_{T})}\|G^{\prime}(u)h\|_{L^{2}(Q)}\right]\\\
&+&\|(G^{\prime}(u)w)(T)\|_{L^{2}(\Omega)}\|(G^{\prime}(u)h)(T)\|_{L^{2}(\Omega)}+\alpha\|w\|_{L^{2}(\omega_{T})}\|h\|_{L^{2}(\omega_{T})}\\\
&\leq&C\|w\|_{L^{2}(\omega_{T})}\|h\|_{L^{2}(\omega_{T})}.\end{array}$

Therefore, there exists a constant
$C=C(\alpha,\|u\|_{\infty},\|y^{0}\|_{L^{\infty}(\Omega)},\|y^{d}\|_{L^{\infty}(\Omega)},T)>0$
independent of $v$ and $h$ such that

$\begin{array}[]{lll}\displaystyle|\mathcal{J}^{\prime\prime}(u)[w,h]|\leq
C\|w\|_{L^{2}(\omega_{T})}\|h\|_{L^{2}(\omega_{T})}.\end{array}$

Hence, the mapping $(w,h)\mapsto\mathcal{J}^{\prime\prime}(u)[w,h]$ is a
bilinear continuous mapping on $L^{2}(\omega_{T})\times L^{2}(\omega_{T})$. ∎

Before stating second order necessary and sufficient conditions, we first
introduce some preliminary concepts retrieved from [31].

######  Definition 6.5.

  1. -

The set of strongly active constraints is the set $A_{\tau}(u)$ defined by

$A_{\tau}(u):=\\{(t,x)\in\omega_{T}:|u(t,x)+y(u)q(t,x)|>\tau\\}.$

  2. -

We introduce the following for a.e $(t,x)\in\omega_{T}$,

$v(t,x)\left\\{\begin{array}[]{lllll}\displaystyle\geq 0&if&u(t,x)=m,\\\
\displaystyle\leq 0&if&u(t,x)=M,\\\ 0&if&(t,x)\in
A_{\tau}(u).\end{array}\right.$ (6.4)

The $\tau$-critical (see e.g [13, 31]) associated to a control $u$ is defined
by

$C_{\tau}(u)=\\{v\in
L^{\infty}(\omega_{T}):v\;\text{fulfills}\;\eqref{eq1}\\}.$ (6.5)

In the rest of the paper, we will adopt the following notation
$\mathcal{J}^{\prime\prime}(u)v^{2}:=\mathcal{J}^{\prime\prime}(u)[v,v]$. We
have the following important remark stated in [13].

######  Remark 6.6.

Thanks to Lemma 6.4, the second order conditions based on the $\tau$-critical
cone defined in (6.5) as a subset of $L^{\infty}(\omega_{T})$ can be
transferred to the extended to a closed and convex cone $L^{2}(\omega_{T})$.

######  Proposition 6.7 (Second order necessary optimality conditions).

Let $u\in\mathcal{U}$ be a $L^{\infty}$-local solution of problem (3.1). Then
$\mathcal{J}^{\prime\prime}(u)v^{2}\geq 0$, for all $v\in C_{0}(u)$.

###### Proof.

The proof follows similar arguments as [22, 31]. Let $v\in C_{0}(u)$ and
$\theta\in(0,1)$. For $n\in\mathbb{N}$, we define the set

$I_{n}=\left\\{(x,t)\in\omega_{T}:m+\frac{1}{n}\leq u(x,t)\leq
M-\frac{1}{n}\right\\}.$

Let $v_{n}:=\chi_{n}v$, where

$\chi_{n}(x,t)=\left\\{\begin{array}[]{lllll}\displaystyle 1&if&(x,t)\in
I_{n}\;\text{or}\;u(x,t)\in\\{m,M\\}\;\text{and}\;M-m\geq\displaystyle\frac{1}{n},\\\
0&if&u(x,t)\in\displaystyle\left(m,m+\frac{1}{n}\right)\cup\left(M-\frac{1}{n},M\right).\end{array}\right.$

Then, $\chi_{n}(x,t)=0$ also if $u(x,t)\in\\{m,M\\}$ and $\displaystyle
M-m<\displaystyle\frac{1}{n}$, and we have $\bar{u}=u+\theta
v_{n}\in\mathcal{U}$ for $\theta\in(0,1)$. Moreover using that $u$ is a
locally optimal control, we deduce that

$\displaystyle
0\leq\frac{\mathcal{J}(\bar{u})-\mathcal{J}(u)}{\theta}=\mathcal{J}^{\prime}(u)v_{n}+\frac{1}{2}\theta\mathcal{J}^{\prime\prime}(u)v_{n}^{2}+\theta^{-1}r^{2}(u,\theta
v_{n}),$

where $r^{2}(u,\theta v_{n})$ represents the second-order remainder. Since
$v\in C_{0}(u)$, then $v_{n}\in C_{0}(u)$ and then
$\displaystyle\mathcal{J}^{\prime}(u)v_{n}=\int_{\omega_{T}}(\alpha u-\rho
q)v_{n}\,\mathrm{d}x\,\mathrm{d}t=0$. Therefore, by dividing the both
inequalities of the previous identity by $\theta$, we obtain

$\displaystyle
0\leq\frac{1}{2}\mathcal{J}^{\prime\prime}(u)v_{n}^{2}+\theta^{-2}r^{2}(u,\theta
v_{n}).$

Taking the limit as $\theta\to 0$ in the last inequality yields

$\mathcal{J}^{\prime\prime}(u)v_{n}^{2}\geq 0.$ (6.6)

It remains to prove that as $n\to\infty$, $v_{n}\to v$ in $L^{2}(\omega_{T})$.
First we note that for a.e $(x,t)\in\omega_{T}$, $v_{n}(x,t)\to v(x,t)$
pointwise almost everywhere as $n\to\infty$. In addition
$|v_{n}(x,t)|^{2}\leq|v(x,t)|^{2}$ pointwise everywhere for all
$n\in\mathbb{N}$. Then, using the Lebesgue’s dominated convergence theorem, we
deduce that as $n\to\infty$, $v_{n}\to v$ in $L^{2}(\omega_{T})$. Hence, by
taking the limit as $n\to\infty$ in (6.6) and using the continuity of
$v\to\mathcal{J}^{\prime\prime}(u)v^{2}$ in $L^{2}(\omega_{T})$, we obtain
$\mathcal{J}^{\prime\prime}(u)v^{2}\geq 0$. ∎

######  Theorem 6.8.

Let $u\in\mathcal{U}$ be a control satisfying the first order optimality
conditions (5.2). Then the following hold:

  1. (1)

The functional $\mathcal{J}:L^{\infty}(\omega_{T})\to\mathbb{R}$ is of class
$\mathcal{C}^{2}$. Furthermore, for every $u\in\mathcal{U}$, there exist
continuous extensions

$\mathcal{J}^{\prime}(u)\in\mathcal{L}(L^{2}(\omega_{T}),\mathbb{R})\;\;\;\;\text{and}\;\;\;\mathcal{J}^{\prime\prime}(u)\in\mathcal{B}(L^{2}(\omega_{T}),\mathbb{R}).$
(6.7)

  2. (2)

For any sequence
$\left\\{(u_{k},v_{k})\right\\}_{k=1}^{\infty}\subset\mathcal{U}\times
L^{2}(\omega_{T})$ with $\|u_{k}-u\|_{L^{2}(\omega_{T})}\to 0$ and
$v_{k}\rightharpoonup v$ weakly in $L^{2}(\omega_{T})$,

$\displaystyle\mathcal{J}^{\prime}(u)v=\lim_{k\to\infty}\mathcal{J}^{\prime}(u_{k})v_{k},$
(6.8)

$\displaystyle\mathcal{J}^{\prime\prime}(u)v^{2}\leq\liminf_{k\to\infty}\mathcal{J}^{\prime\prime}(u_{k})v_{k}^{2},$
(6.9)

$\displaystyle\text{If}\;v=0,\;\;\text{then}\;\;\;\Lambda\liminf_{k\to\infty}\|v_{k}\|^{2}_{L^{2}(\omega_{T})}\leq\liminf_{k\to\infty}\mathcal{J}^{\prime\prime}(u_{k})v_{k}^{2},$
(6.10)

for some $\Lambda>0$.

###### Proof.

The proof follows similar arguments as in [22]. Indeed, using Proposition 5.4,
Lemma 5.7, Proposition 6.2 and Lemma 6.4, we obtain the first point (1) of
Theorem 6.8.
Now, we prove the point (2) in three steps.
Let $\left\\{(u_{k},v_{k})\right\\}_{k=1}^{\infty}$ be a sequence of
$\mathcal{U}\times L^{2}(\omega_{T})$ such that
$\|u_{k}-u\|_{L^{2}(\omega_{T})}\to 0$ and $v_{k}\rightharpoonup v$ weakly in
$L^{2}(\omega_{T})$.
Step 1. We prove (6.8).
Using the Lipschitz continuity property of $G$ given in Proposition 4.2, we
obtain that $\displaystyle G(u_{k})\to G(u)$ in
$L^{2}((0,T);H^{1}_{a}(\Omega))$ as $k\to\infty$. We also obtain due to
Proposition 5.8 that the solution of system (5.4) satisfies $\displaystyle
q(u_{k})\to q(u)$ in $L^{2}((0,T);H^{1}_{a}(\Omega))$ as $k\to\infty$. Since,
$G(u_{k})$ and $G(u)$ are respectively solutions to (1.1) with $v=u_{k}$ and
$v=u$. Therefore, using (2.23), we deduce that in particular the sequence
$G(u_{k})$ is bounded in $L^{2}(Q)$ and $G(u)\in L^{\infty}(Q)$. Hence
$G(u_{k})q(u_{k}),G(u)q(u)\in L^{2}(Q)$ and

$G(u_{k})q(u_{k})\to
G(u)q(u)\;\;\text{in}\;\;L^{2}(Q)\;\;\text{as}\;\;k\to\infty.$

From the expression of $\mathcal{J}^{\prime}$ given in (5.9), we deduce using
the latter convergence that:

$\displaystyle\displaystyle\lim_{k\to\infty}\mathcal{J}^{\prime}(u_{k})v_{k}=\lim_{k\to\infty}\int_{\omega_{T}}(\alpha
u_{k}+G(u_{k})q(u_{k}))v_{k}\,\mathrm{d}x\,\mathrm{d}t=\int_{\omega_{T}}(\alpha
u+G(u)q(u))v\,\mathrm{d}x\,\mathrm{d}t=\mathcal{J}^{\prime}(u)v.$

Thus (6.8) is proved.
Step 2. Let us show (6.9). We have

$\mathcal{J}^{\prime\prime}(u_{k})v_{k}^{2}=2\int_{\omega_{T}}v_{k}(G^{\prime}(u_{k})v_{k})q(u_{k})\mathrm{d}x\,\mathrm{d}t+\int_{\Omega}|(G^{\prime}(u_{k})v_{k})(T)|^{2}\,\mathrm{d}x+\alpha\int_{\omega_{T}}|v_{k}|^{2}\,\mathrm{d}x\,\mathrm{d}t.$
(6.11)

Note that $G^{\prime}(u_{k})v_{k}$ is the unique weak solution of (5.7) with
$u=u_{k}$ and $w=v_{k}$. Then, we claim that

$(G^{\prime}(u_{k})v_{k})q(u_{k})\;\;\text{converges strongly to
}(G^{\prime}(u)v)q(u)\;\;\text{in }L^{2}(Q)\;\;\text{ as }k\to\infty.$

The latter convergence follows from the boundedness of the sequences
$(G(u_{k}),q(u_{k}))$ and $v_{k}$ in $L^{\infty}(Q)\times L^{\infty}(Q)$ and
in $L^{2}(\omega_{T})$ respectively, the estimation (6.3), Lemma 2.1 and the
convergence $\displaystyle q(u_{k})\to q(u)$ in
$L^{2}((0,T);H^{1}_{a}(\Omega))$ as $k\to\infty$. Therefore, taking the limit
as $k\to\infty$ in (6.11) and using the lower-semi continuity of the
$L^{2}$-norm, we deduce that

$\displaystyle\displaystyle\lim_{k\to\infty}\mathcal{J}^{\prime\prime}(u_{k})v_{k}^{2}\\!\\!\\!$
$\displaystyle\geq$
$\displaystyle\\!\\!\\!2\lim_{k\to\infty}\int_{\omega_{T}}v_{k}(G^{\prime}(u_{k})v_{k})q(u_{k})\mathrm{d}x\,\mathrm{d}t+\liminf_{k\to\infty}\left[\int_{\Omega}|(G^{\prime}(u_{k})v_{k})(T)|^{2}\,\mathrm{d}x+\alpha\int_{\omega_{T}}|v_{k}|^{2}\,\mathrm{d}x\,\mathrm{d}t\right]$
$\displaystyle\geq$ $\displaystyle
2\int_{\omega_{T}}v(G^{\prime}(u)v)q(u)\mathrm{d}x\,\mathrm{d}t+\int_{\Omega}|(G^{\prime}(u)v)(T)|^{2}\,\mathrm{d}x+\alpha\int_{\omega_{T}}|v|^{2}\,\mathrm{d}x\,\mathrm{d}t$
$\displaystyle=$ $\displaystyle\mathcal{J}^{\prime\prime}(u)[v,v].$

Hence (6.9) is obtained.

Step 3. Finally, we show (6.10). If $v=0$, then in (6.11), the first and the
second terms tend to $0$, except the last one. Hence,

$\Lambda\liminf_{k\to\infty}\|v_{k}\|^{2}_{L^{2}(\omega_{T})}\leq\lim_{k\to\infty}\mathcal{J}^{\prime\prime}(u_{k})v_{k}^{2},$

with $\Lambda=\alpha$. ∎

Now, we state one of the main results of this paper.

######  Theorem 6.9 (Second order sufficient optimality conditions).

Let $y_{0},y_{d}\in L^{\infty}(\Omega)$, and $\beta=\max\\{|m|,|M|\\}$. Let
$u\in\mathcal{U}$ be a control satisfying the first order optimality
conditions (5.5). We assume that the penalization constant $\alpha$ satisfies

$\alpha>4e^{3(\beta+1)T}\left(\|y^{0}\|_{L^{\infty}(\Omega)}+\|y^{d}\|_{L^{\infty}(\Omega)}\right).$
(6.12)

Then, there are two constants $\varepsilon>0$ and $\gamma>0$ such that

$\mathcal{J}(v)\geq\mathcal{J}(u)+\frac{\gamma}{2}\|v-u\|^{2}_{L^{2}(\omega_{T})}\quad\forall
v\in\mathcal{U}\cap B^{2}_{\varepsilon}(u).$ (6.13)

###### Proof.

Let $\tau>0$ and $v\in\mathcal{C}_{\tau}(u)$, we first show that there exists
$\delta>0$ such that

$\mathcal{J}^{\prime\prime}(u)v^{2}\geq\delta\|v\|^{2}_{L^{2}(\omega_{T})}.$
(6.14)

We have

$\mathcal{J}^{\prime\prime}(u)v^{2}=2\int_{\omega_{T}}v(G^{\prime}(u)v)q(u)\mathrm{d}x\,\mathrm{d}t+\int_{\Omega}|(G^{\prime}(u)v)(T)|^{2}\,\mathrm{d}x+\alpha\int_{\omega_{T}}|v|^{2}\,\mathrm{d}x\,\mathrm{d}t.$
(6.15)

We note that since $v\in\mathcal{C}_{\tau}(u)$ (i.e $v(x,t)=0$ for $(x,t)\in
A_{\tau}(u)$ ), we can write the first term of (6.15) as

$\begin{array}[]{lll}\displaystyle
2\int_{\omega_{T}}v(G^{\prime}(u)v)q(u)\mathrm{d}x\,\mathrm{d}t&=&\displaystyle
2\int_{\\{(x,t)\in
A_{\tau}(u)\\}}v(G^{\prime}(u)v)q(u)\mathrm{d}x\,\mathrm{d}t+2\int_{\\{(x,t)\notin
A_{\tau}(u)\\}}v(G^{\prime}(u)v)q(u)\mathrm{d}x\,\mathrm{d}t\\\
&=&\displaystyle 2\int_{\\{(x,t)\notin
A_{\tau}(u)\\}}v(G^{\prime}(u)v)q(u)\mathrm{d}x\,\mathrm{d}t.\end{array}$

Therefore, using the Cauchy Schwartz inequality, (5.15) and (6.3) we obtain

$\begin{array}[]{lll}\displaystyle\left|2\int_{\omega_{T}}v(G^{\prime}(u)v)q(u)\mathrm{d}x\,\mathrm{d}t\right|&=&\displaystyle
2\left|\int_{\\{(x,t)\notin
A_{\tau}(u)\\}}v(G^{\prime}(u)v)q(u)\mathrm{d}x\,\mathrm{d}t\right|\\\
&\leq&\displaystyle
2\|q\|_{L^{\infty}(Q)}\|(G^{\prime}(u)v)\|_{L^{2}(Q)}\|v\|_{L^{2}(\omega_{T})}\\\
&\leq&4e^{3(\beta+1)T}\left(\|y^{0}\|_{L^{\infty}(\Omega)}+\|y^{d}\|_{L^{\infty}(\Omega)}\right)\|v\|^{2}_{L^{2}(\omega_{T})},\end{array}$

with $\beta=\max\\{|m|,|M|\\}$. Hence,

$\mathcal{J}^{\prime\prime}(u)v^{2}\geq\left(\alpha-4e^{3(\beta+1)T}\left(\|y^{0}\|_{L^{\infty}(\Omega)}+\|y^{d}\|_{L^{\infty}(\Omega)}\right)\right)\|v\|^{2}_{L^{2}(\omega_{T})}.$
(6.16)

Choosing $\alpha$ such that
$\alpha>4e^{3(\beta+1)T}\left(\|y^{0}\|_{L^{\infty}(\Omega)}+\|y^{d}\|_{L^{\infty}(\Omega)}\right)$,
we obtain (6.14) with
$\delta=\alpha-4e^{3(\beta+1)T}\left(\|y^{0}\|_{L^{\infty}(\Omega)}+\|y^{d}\|_{L^{\infty}(\Omega)}\right)>0$.
Moreover, from Theorem 6.8, the assumptions of [12, Theorem 2.3 ] are
fulfilled. Hence, (6.13) holds. ∎

## References

  * [1] F. Alabau-Boussouira, P. Carnassa, and G. Fragnelli.  Carleman estimates for degenerate parabolic operators with applications to null controllability.  Journal of Evolution Equations, 6:161–204, 2006.
  * [2] M. S. Aronna.  Singular solutions in optimal control: second order conditions and a shooting algorithm.  arXiv preprint arXiv:1210.7425, 2012.
  * [3] M. S. Aronna, J. F. Bonnans, and B. S. Goh.  Second order analysis of control-affine problems with scalar state constraint.  Mathematical Programming, 160(1):115–147, 2016.
  * [4] M. S. Aronna, J. F. Bonnans, and A. Kröner.  Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations.  Mathematical Programming, 168(1):717–757, 2018.
  * [5] M. S. Aronna, J. F. Bonnans, and A. Kroner.  Optimal control of pdes in a complex space setting: Application to the schroödinger equation.  SIAM Journal on Control and Optimization, 57(2):1390–1412, 2019\.
  * [6] M. S. Aronna, J. F. Bonnans, and A. Kröner.  State-constrained control-affine parabolic problems i: first and second order necessary optimality conditions.  Set-Valued and Variational Analysis, 29(2):383–408, 2021.
  * [7] M. S. Aronna, J. F. Bonnans, and A. Kroöner.  State constrained control-affine parabolic problems ii: Second order sufficient optimality conditions.  SIAM Journal on Control and Optimization, 59(2):1628–1655, 2021\.
  * [8] M. S. Aronna and F. Tröltzsch.  First and second order optimality conditions for the control of fokker-planck equations.  ESAIM: Control, Optimisation and Calculus of Variations, 27:15, 2021\.
  * [9] M. I. Budyko.  The effect of solar radiation variations on the climate of the earth.  tellus, 21(5):611–619, 1969.
  * [10] P. Cannarsa and G. Floridia.  Approximate controllability for linear degenerate parabolic problems with bilinear control.  arXiv preprint arXiv:1106.4232, 2011.
  * [11] P. Cannarsa and G. Floridia.  Approximate multiplicative controllability for degenerate parabolic problems with robin boundary conditions.  Communications in Applied and Industrial Mathematics, 2(2):1–16, 2011.
  * [12] E. Casas and F. Tröltzsch.  Second order analysis for optimal control problems: improving results expected from abstract theory.  SIAM Journal on Optimization, 22(1):261–279, 2012.
  * [13] E. Casas and F. Tröltzsch.  Second order optimality conditions and their role in pde control.  Jahresbericht der Deutschen Mathematiker-Vereinigung, 117(1):3–44, 2015.
  * [14] J. Diaz, G. Hetzer, and L. Tello.  An energy balance climate model with hysteresis.  Nonlinear Analysis: Theory, Methods & Applications, 64(9):2053–2074, 2006.
  * [15] R. Dorville.  Bilinear boundary optimal control with final observation for the heat equation.  Applicable Analysis, 2021.
  * [16] C. L. Epstein and R. Mazzeo.  Degenerate diffusion operators arising in population biology.  Number 185. Princeton University Press, 2013.
  * [17] G. Floridia.  Approximate controllability for nonlinear degenerate parabolic problems with bilinear control.  Journal of Differential Equations, 257(9):3382–3422, 2014.
  * [18] G. Floridia.  Nonnegative controllability for a class of nonlinear degenerate parabolic equations with application to climate science.  arXiv preprint arXiv:2003.04966, 2020.
  * [19] M. Ghil.  Climate stability for a sellers-type model.  Journal of Atmospheric Sciences, 33(1):3–20, 1976.
  * [20] C. Kenne.  Sur les modèles de dynamique de populations et l’émergence de la maladie dans les eaux douces.  PhD thesis, Université des Antilles, 2022.
  * [21] C. Kenne, G. Leugering, and G. Mophou.  Optimal control of a population dynamics model with missing birth rate.  SIAM Journal on Control and Optimization, 58:1289–1313, 2020.
  * [22] C. Kenne, G. Mophou, and M. Warma.  Bilinear optimal control for a fractional diffusive equation.  arXiv preprint arXiv:2210.17494, 2022.
  * [23] C. Kenne and B. Nkemzi.  Optimal control of averaged state of a population dynamics model.  In Studies in Evolution Equations and Related Topics, pages 113–127. Springer, 2021.
  * [24] C. Kenne, P. Zongo, R. Dorville, and G. Mophou.  Optimal control of a coupled degenerate population dynamics model with unknown birth rates.  Nonlinear Studies (NS), 28(4):1225–1252, 2021.
  * [25] J. L. Lions.  Equations différentielles opérationnelles: et problèmes aux limites, volume 111.  Springer-Verlag, 2013.
  * [26] G. Mophou, M. Kéré, and L. L. D. Njoukoué.  Robust hierarchic control for a population dynamics model with missing birth rate.  Mathematics of Control, Signals, and Systems, 32(2):209–239, 2020\.
  * [27] G. R. North and K.-Y. Kim.  Energy Balance Climate Models.  John Wiley & Sons, 2017.
  * [28] W. D. Sellers.  A global climatic model based on the energy balance of the earth-atmosphere system.  Journal of Applied Meteorology (1962-1982), pages 392–400, 1969\.
  * [29] N. Shimakura.  Partial differential operators of elliptic type.  Amer Mathematical Society, 1992.
  * [30] E. M. Tory, K. H. Karlsen, R. Bürger, and S. Berres.  Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression.  SIAM Journal on Applied Mathematics, 64(1):41–80, 2003.
  * [31] F. Tröltzsch.  Optimal control of partial differential equations: theory, methods, and applications, volume 112.  American Mathematical Soc., 2010.