\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 72, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/72\hfil Optimal control problem without initial conditions]
{Optimal control for systems governed by parabolic equations without
initial conditions with controls in the coefficients}

\author[M. Bokalo, A. Tsebenko \hfil EJDE-2017/72\hfilneg]
{Mykola Bokalo, Andrii Tsebenko}

\address{Mykola Bokalo \newline
Department of Differential Equations\\
Ivan Franko National University of Lviv\\
Lviv, Ukraine}
\email{mm.bokalo@gmail.com}

\address{Andrii Tsebenko \newline
Department of Differential Equations\\
Ivan Franko National University of Lviv\\
Lviv, Ukraine}
\email{amtseb@gmail.com}

\dedicatory{Communicated by Suzanne Lenhart}

\thanks{Submitted December 7, 2016. Published March 14, 2017.}
\subjclass[2010]{35K10, 49J20, 58D25}
\keywords{Optimal control; problems without initial conditions;
\hfill\break\indent evolution equation}

\begin{abstract}
 We consider an optimal control problem for systems described by a Fourier
 problem for parabolic equations. We prove the existence of solutions, and 
 obtain necessary conditions of the optimal control in the case of final 
 observation when the control functions occur in the coefficients.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

Optimal control of determined systems governed by partial 
differential equations (PDEs) is currently  of much interest.
Optimal control problems for PDEs are most completely studied for 
the case in which the control functions occur either
on the right-hand sides of the state equations, or the boundary or
initial conditions. So far, problems in which control functions
occur in the coefficients of the state equations are less studied.
A  simple model of such type problem is the following.

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with piecewise smooth
boundary $\Gamma$, $T>0$, $Q:=\Omega\times(0,T)$,
$\Sigma:=\partial\Omega\times(0,T)$.
A state of controlled system for given control $v\in U:=L^\infty(Q)$
is defined by a weak solution
$y=y(v)=y(x,t;v)$, $(x,t)\in Q$, from the space
$L^2(0,T;H^1_0(\Omega))\cap C([0,T];L^2(\Omega))$, of the problem
\begin{gather*}
y_t-\Delta y + vy=f \in L^2(Q),\quad
y\big|_{\Sigma}=0, \quad  y\big|_{t=0}=y_0 \in L^2(\Omega).
\end{gather*}
The cost functional is
\[
J(v):=\|y(\cdot,T;v)-z_0(\cdot)\|^2_{L^2(\Omega)}+\mu\|v\|^2_{L^\infty(Q)}\quad
 \forall v\in U,
\]
 where $\mu>0$, $z_0\in L^2(\Omega)$ are given.
An optimal control problem is to find a function
$u\in U_{\partial}:=\big\{ v \in U :  v\geq 0  \text{ a. e. on } Q \big\}$
such that
$$
J(u)= \inf _{v\in U_{\partial}}J(v).
$$
This problem is nonlinear, since the dependence
between the state and the control is nonlinear.

The direct generalization of this problem is given as only one among many other
problems which were considered in monograph \cite{Lions}.
Other various generalizations of this problem were investigated in many papers,
including
\cite{Akimenko,Lenhart1,Bokalo1,Bokalo2,Lenhart3,Farag2009,Fister,Khater,
Lou,Zuliang, Tagiev,Tagiev2013} where the state of controlled system is
described by the initial-boundary value  problems for parabolic equations.


In \cite{Akimenko,Zuliang,Tagiev,Tagiev2013} the state of controlled system is
 described by linear parabolic equations and systems, while in
\cite{Akimenko,Zuliang} control functions  appears as  coefficients at
lower derivatives, and in \cite{Tagiev,Tagiev2013} the control functions are
coefficients at higher derivatives.
In \cite{Zuliang} the existence and uniqueness of optimal control in the
case of final observation was shown and a necessary optimality condition
in the form of the generalized rule
of Lagrange multipliers was obtained.
In  \cite{Akimenko} the authors proved the existence of at
least one optimal control for system governed by  a system of general
parabolic equations  with degenerate discontinuous parabolicity coefficient.
In papers \cite{Tagiev,Tagiev2013} the authors consider cost
function in general form, and as special case it includes different
kinds of specific practical optimization problems. The
well-posedness of the problem statement is investigated and a necessary optimality
condition in the form of the generalized principle of Lagrange multiplies is
established in this papers.

In \cite{Lenhart1,Lenhart3,Farag2009,Fister,Khater,Lou} the authors investigate
optimal control of systems governed by  nonlinear PDEs.
In particular, in \cite{Lenhart1} the problem of allocating resources to maximize
the net benefit in the conservation of a single species is studied.
The population model is an equation with density dependent
growth and spatial-temporal resource control coefficient.
The existence of an optimal control and the uniqueness and
the characterization of the optimal control are established.
Numerical simulations illustrate several cases with Dirichlet
and Neumann boundary conditions.
In \cite{Lenhart3} the problem of optimal control of a Kirchhoff plate
is considered. A bilinear control is used as a force to make the plate close to a desired profile taking into the account, a quadratic cost of control. The authors prove the existence of an optimal control and characterize it uniquely through the solution of an optimality system.
In \cite{Farag2000} the optimal control
problem is converted to an optimization problem which is solved
using a penalty function technique.
The existence and uniqueness theorems are
investigated. The derivation of formula for the gradient of
the modified function is explained by solving the adjoint problem.
Paper \cite{Khater} presents analytical and numerical
solutions of an optimal control problem for quasilinear parabolic equations.
The existence and uniqueness of the solution are shown.
The derivation of formula for the gradient of
the modified cost function by solving the conjugated boundary value
problem is explained.
In \cite{Lenhart} the authors consider the optimal control of the degenerate
parabolic equation governing a diffusive population
with logistic growth terms. The optimal control is characterized in terms
of the solution of the optimality system, which is  the state  equation coupled
with the adjoint equation.
Uniqueness  for the solutions of the optimality system is valid for a sufficiently
small time interval due to the opposite time orientations of the two equations
involved.
In  \cite{Lou} optimal control for semilinear parabolic equations
without Cesari-type conditions is investigated.

In this article, we study an optimal control problem (see
\eqref{equation}, \eqref{BoundaryCondition}, \eqref{incondition},
\eqref{funcvart}, \eqref{varinf} below) for systems whose states are  described
by  problems without initial conditions or, other words, Fourier problems
for parabolic equations. The model example of  considered optimal control
problem is a problem which differs from the previous one
(see beginning of this section) by the following facts: the initial moment is
$-\infty$ and, correspondingly,  the state equation and control functions
are considered in the domain $Q=\Omega\times(-\infty,T)$, a boundary condition
is given on the surface
$\Sigma=\partial\Omega\times(-\infty,T)$, while the initial condition is
replaced by the condition
\begin{gather*}\label{poch1}
\lim_{t\to-\infty}\|y(\cdot,t)\|_{L^2(\Omega)}=0.
\end{gather*}

The problem without initial conditions for evolution equations  describes
processes that started a long time ago
and initial conditions do not affect on them in the actual time moment.
Such problem  were  investigated in the works of many mathematicians
(see \cite{Bokalo3, Bokalo, Showalter97} and bibliography there).

As we know among numerous works devoted to the optimal control problems
for PDEs, only in  \cite{Bokalo1,Bokalo2} the state of controlled
system is described by the solution of Fourier problem for parabolic equations.
In the current paper, unlike the above two, we consider optimal
control problem in case when the control functions
occur in the coefficients of the state equation.
The main result of this paper is existence of the solution of  this problem.


The outline of this article is as follows.
In Section \ref{Sect1}, we give notation, definitions of function spaces
and auxiliary results.
In Section \ref{Sect2}, we prove existence and uniqueness of the solutions
for the state equations. Furthermore, we construct a priori estimates
for the weak solutions of the state equations.
In Section \ref{Sect3}, we formulate the optimal control problem.
Finally, the existence and necessary conditions of the optimal control are
presented in Section \ref{Sect4}.



\section{Preliminaries} \label{Sect1}


Let  $n$ be a  natural number, $\mathbb{R}^n$ be the linear space of ordered
collections $x=(x_1,\ldots,x_n)$  of real numbers with the norm
$|x|:=(|x_1|^2+\ldots+|x_n|^2)^{1/2}$.
 Suppose that $\Omega$ is a bounded domain in $\mathbb{R}^n$ with piecewise
smooth boundary  $\Gamma$. Set  $S:=(-\infty,0]$,
$Q:=\Omega\times S$, $ \Sigma:=\Gamma\times S$.

Denote by $L^{\infty}_{\rm loc}(\overline{Q})$ the linear space of measurable
functions on $Q$ such that their restrictions to any bounded measurable set
$Q'\subset Q$ belong to the space $L^\infty(Q')$.


Let $X$ be an arbitrary Hilbert space with the scalar product $(\cdot,\cdot)_X$
and the norm $\|\cdot\|_X$. Denote by $L^2_{\rm loc}(S;X)$
the linear space of measurable functions defined on $S$ with values in $X$,
 whose restrictions to any segment $[a,b]\subset S$ belong to the space
$L^2(a,b;X)$.

Let $\omega\in \mathbb{R}$, $\alpha \in C(S)$ be such that $\alpha(t)>0$
for all $t\in S$, $\gamma=\alpha$ or $\gamma=1/\alpha$, and let $X$ be as above.
Put by definition
\begin{equation*}
L_{\omega,\gamma}^2(S;X):=\Big\{f\in L^2_{\rm loc}(S;X):
\int_{S}\gamma(t)e^{2\omega\int_{0}^{t}\alpha(s)ds}\|f(t)\|_X^2dt<\infty\Big\}.
\end{equation*}
This space is a Hilbert space with respect to the scalar product
\begin{equation*}\label{skal}
(f,g)_{L_{\omega,\gamma}^2(S;X)}=
\int_S\gamma(t)\,e^{2\omega\int_0^t\alpha(s)\,ds}(f(t),g(t))_X\,dt
\end{equation*}
and the norm
$$
\|f\|_{L_{\omega,\gamma}^2(S;X)}:=\Big(
\int_S\gamma(t)\,e^{2\omega\int_0^t\alpha(s)\,ds}\|f(t)\|^2_{X}\,dt\Big)^{1/2}.
$$


For an interval  $I$, we denote by $C^1_c(I)$ the linear space of
continuously differentiable functions on $I$ with compact supports (if
$I=(t_1,t_2)$, then we will write $C_c^1(t_1,t_2)$ instead of
 $C_c^1((t_1,t_2))$).


Let ${H}^1(\Omega):=\{v\in L^2(\Omega): v_{x_i} \in L^2(\Omega)\ (i=\overline{1,n})\}$
  be a Sobolev space, which is a Hilbert space with respect to the scalar product
$(v,w)_{H^1(\Omega)}:=\int_\Omega\big\{\nabla v \nabla w+
vw\big\}\,dx$
and the corresponding norm
$\|v\|_{H^1(\Omega)}:=\big(\int_{\Omega}\big\{|\nabla v|^2+
|v|^2\big\}\,dx\big)^{1/2}$, where
$\nabla v=(v_{x_1},\dots,v_{x_n}),\ |\nabla v|^2=\sum_{i=1}^n |v_{x_i}|^2$.
Under $H_0^1(\Omega)$ we mean the closure in ${H^1(\Omega)}$ of the space
$C_c^\infty(\Omega)$ consisting of infinitely differentiable functions on $\Omega$
with compact supports.
Denote by $H^{-1}(\Omega)$ the dual space of $H_0^1(\Omega)$, that is,
the space of all continuous linear functionals on $H_0^1(\Omega)$.

We suppose (after appropriate identification of functionals), that the space
$L^2(\Omega)$ is a subspace of $H^{-1}(\Omega)$. Identifying
spaces $L^2(\Omega)$ and $\big(L^2(\Omega)\big)'$, we obtain  continuous and
dense embeddings
\begin{equation}\label{embedding}
H^1_0(\Omega)\subset L^2(\Omega)\subset H^{-1}(\Omega).
\end{equation}
Note, that in this case $\langle g,v\rangle_{H^1_0(\Omega)}=(g,v)$ for every
$v\in H^1_0(\Omega), g\in L^2(\Omega)$, where
$(\cdot,\cdot)$  is the scalar product on $L^2(\Omega)$ and
$\langle\cdot,\cdot\rangle_{H^1_0(\Omega)}$   is the scalar product
for the duality $H^{-1}(\Omega)$, $H_0^1(\Omega)$.
Therefore, further we use the notation $(\cdot,\cdot)$ instead of
$\langle\cdot,\cdot\rangle_{H^1_0(\Omega)}$.

We define
\begin{equation}\label{Kfridrix}
K:=\inf_{v\in H^1_0(\Omega),\  v\neq 0}\frac{\int_{\Omega}|\nabla v|^2\,dx}
{\int_{\Omega}|v|^2\,dx}.
\end{equation}
It is well known that the constant $K$ is finite and coincides with the first
eigenvalue of the  eigenvalue problem
\begin{equation}\label{eigenvalue}
    -\Delta v=\lambda v, \quad
    v|_{\partial \Omega}=0.
\end{equation}
From \eqref{Kfridrix} it clearly follows the Friedrichs inequality
\begin{equation}\label{Fridrix}
\int_{\Omega} |\nabla v|^2\,dx\geq K \int_{\Omega} |v|^2 \,dx \quad
\forall  v \in H_0^1(\Omega).
\end{equation}

Further, an important role will be played by the following statement,
which is a well-known result (see, e.g. \cite[Theorem 3, p. 287]{Evans}),
but we reformulate it according to our needs.

\begin{lemma}\label{lem1}
Suppose that a function $z \in L^2(t_1,t_2;H_0^1(\Omega))$, with
$t_1<t_2$, satisfies
\begin{equation}\label{Lemma11}
\int_{t_1}^{t_2} \int_\Omega \big\{  -z\psi\varphi'
+(g_0\psi+\sum_{i=1}^n g_i\psi_{x_i})\varphi \big\}\,dx\,dt=0,
\end{equation}
for $\psi \in H_0^1(\Omega)$, $\varphi \in C_c^1(t_1,t_2)$, where
$g_i \in L^2(\Omega\times(t_1,t_2))$ $(i=\overline{0,n})$.
Then

(1) the derivative $z_t$ of the function $z$  in the sense
$D'(t_1,t_2;H^{-1}(\Omega))$ (the distributions space) belongs to
$L^2(t_1,t_2;H^{-1}(\Omega))$, furthermore for a.e.
$t\in (t_1,t_2)$,
\begin{gather} \label{lemma111}
z_t(\cdot,t)=-g_0(\cdot,t)+\sum_{i=1}^n\big(g_i(\cdot,t)\big)_{x_i}\quad
\text{in } H^{-1}(\Omega), \\
\label{lemma112}
\frac{1}{2}\frac{d}{dt}\|z(\cdot,t)\|^2_{L^2(\Omega)}=(z_t(\cdot,t),z(\cdot,t)),\\
\label{Lemma13}
\int_{t_1}^{t_2}\|z_t(\cdot,t)\|^2_{H^{-1}(\Omega)}\,dt
\leq  \sum_{i=0}^n \|g_i\|^2_{L^2(\Omega\times(t_1,t_2))};
\end{gather}

(2) the function $z$ belongs to the space $C([t_1,t_2];L^2(\Omega))$ and for all
$\tau_1,\tau_2 \in [t_1,t_2]\ (\tau_1<\tau_2)$ and for every
$\theta \in C^1([t_1,t_2])$, $q\in L^2(t_1,t_2;H^1_0(\Omega))$  we have
\begin{gather}\label{Lemma12}
\begin{aligned}
&\frac12\theta(t)\int_\Omega |z(x,t)|^2\,dx \Big|_{t=\tau_1}^{t=\tau_2}
 - \frac12\int_{\tau_1}^{\tau_2}\int_\Omega |z|^2\theta'\,dx\,dt \\
&+   \int_{\tau_1}^{\tau_2}\int_\Omega \big\{  g_0z
 +\sum_{i=1}^ng_iz_{x_i} \big\}\theta\,dx\,dt=0,
\end{aligned} \\
\label{Lemma121}
\int_{\tau_1}^{\tau_2} \big(z_t(\cdot,t),q(\cdot,t)\big)\,dt
+\int_{\tau_1}^{\tau_2}\int_\Omega  g_0q\,dx\,dt
+\sum_{i=1}^n\int_{\tau_1}^{\tau_2}\int_\Omega g_iq_{x_i}\,dx\,dt=0.
\end{gather}
\end{lemma}

\begin{proof}
As it has already been mentioned, this lemma  follows directly from the
 well-known result. But for clarity we re-present schematically some
points of the proof.
The first statement is:
Since the spaces $L^2(t_1,t_2;H^1_0(\Omega))$,
$L^2(t_1,t_2;H^{-1}(\Omega))$ can be identified with subspaces of the
space of distributions $D'(t_1,t_2;H^{-1}(\Omega))$, then it allows us
to speak about derivatives of functions from $L^2(t_1,t_2;H^1_0(\Omega))$
in the sense  $D'(t_1,t_2;H^{-1}(\Omega))$ and their belonging to the space
$ L^2(t_1,t_2;H^{-1}(\Omega))$.

Let us rewrite equality \eqref{Lemma11}  in the form
\begin{equation}\label{Lemma11'}
-\int_{t_1}^{t_2}\int_\Omega   z\psi\varphi'\,dx\,dt
=-\int_{t_1}^{t_2}\int_\Omega(g_0\psi+\sum_{i=1}^n g_i\psi_{x_i})\varphi\,dx\,dt,
\end{equation}
for $\psi \in H_0^1(\Omega)$, $\varphi \in C_c^1(t_1,t_2)$.
According to the definition of the derivative of distributions from
$D'(t_1,t_2;H^{-1}(\Omega))$, \eqref{Lemma11'} implies  existence of $z_t$
and its belonging to the space $L^2(t_1,t_2;H^{-1}(\Omega))$, then according
to \cite[Theorem 3, p. 287]{Evans} identity \eqref{lemma112} holds.
From \eqref{Lemma11'} for almost all $t\in (t_1,t_2)$ we have
\begin{equation}\label{lemma1111}
\big(z_t(\cdot,t),\psi(\cdot)\big)=-\int_\Omega\big[ g_0(x,t)\psi(x) +
\sum_{i=1}^ng_i(x,t)\psi_{x_i}(x) \big]\,dx,
\end{equation}
that is, \eqref{lemma111}  holds.

From \eqref{lemma1111}, using the Cauchy-Schwarz inequality,
for almost all $t\in (t_1,t_2)$ we obtain
\begin{equation}\label{Lemma14}
\begin{aligned}
&\big|\big(z_t(\cdot,t),\psi(\cdot)\big)\big| \\
&\leq \|g_0(\cdot,t)\|_{L^2(\Omega)} \|\psi(\cdot)\|_{L^2(\Omega)}
+ \sum_{i=1}^n \|g_i(\cdot,t)\|_{L^2(\Omega)} \|\psi_{x_i}(\cdot)\|_{L^2(\Omega)}\\
&\leq \Big( \sum_{i=0}^n \|g_i(\cdot,t)\|^2_{L^2(\Omega)} \Big)^{1/2}
 \|\psi(\cdot)\|_{H^1(\Omega)}.
\end{aligned}
\end{equation}
From \eqref{Lemma14} it follows that for almost all $t\in (t_1,t_2)$
the following estimate is valid
$$
\|z_t(\cdot,t)\|^2_{H^{-1}(\Omega)}
\leq  \sum_{i=0}^n \|g_i(\cdot,t)\|^2_{L^2(\Omega)},
$$
which easily implies \eqref{Lemma13}.

Let us prove the second statement of Lemma \ref{lem1}.
The fact that the function $z$ belongs to the space $C([t_1,t_2];L^2(\Omega))$
follows directly from  \cite[Theorem 3, p. 287]{Evans} according to the first
statement.

Since for a.e. $t\in S$ the function $q(\cdot,t) \in H^1_0(\Omega)$,
we can take $\psi(\cdot)=q(\cdot,t)$ in \eqref{lemma1111} and obtain
\begin{equation}\label{lemma122}
\big(z_t(\cdot,t),q(\cdot,t)\big)
=-\int_\Omega\big[ g_0(x,t)q(x,t) + \sum_{i=1}^ng_i(x,t)q_{x_i}(x,t) \big]\,dx,
\quad t\in S.
\end{equation}
Integrating this inequality by $t$ over $(\tau_1,\tau_2)$ for arbitrary
$\tau_1, \tau_2 \in S$, we obtain \eqref{Lemma121}.

Taking $q(\cdot,t)=\theta(t) z(\cdot,t)$, $t\in S$, in \eqref{Lemma121}
and integrating over $(\tau_1,\tau_2)$, we obtain
\begin{equation}\label{lemma15}
\int_{\tau_1}^{\tau_2} \theta(t) \big(z_t(\cdot,t),z(\cdot,t)\big)\,dt
+\int_{\tau_1}^{\tau_2}\int_\Omega \big\{ g_0z
+\sum_{i=1}^n g_iz_{x_i}\big\}\theta\,dx\,dt=0.
\end{equation}
Using  \eqref{lemma112} and integration by parts, we have
\begin{align*}
\int_{\tau_1}^{\tau_2} \theta(t) \big(z_t(\cdot,t),z(\cdot,t)\big)\,dt
&=\frac{1}{2}\int_{\tau_1}^{\tau_2} \theta(t)
\frac{d}{dt}\|z(\cdot,t),z(\cdot,t\|^2_{L^2(\Omega)}\,dt \\
&=\frac{1}{2}\theta(t)\|z(\cdot,t)\|^2_{L^2(\Omega)}\Big|_{t=\tau_1}^{t=\tau_2}
-\frac{1}{2} \int_{\tau_1}^{\tau_2} \theta'\|z(\cdot,t)\|_{L^2(\Omega)}\,dt,
\end{align*}
which, together with \eqref{lemma15}, gives  \eqref{Lemma12}.
\end{proof}

\section{Well-posedness of the problem without initial conditions for
linear parabolic equations}\label{Sect2}

Consider the equation
\begin{equation}\label{equation1}
y_t  -\sum_{i,j=1}^n\big(a_{ij}(x,t) y_{x_i}\big)_{x_j}+a_0(x,t)y
= f(x,t), \quad (x,t)\in Q,
\end{equation}
where $y:\overline{Q}\to \mathbb{R}$ is an unknown function and
data-in satisfies conditions:
\begin{itemize}
\item[(A1)]  $ a_0, a_{ij} \in L^{\infty}_{\rm loc}(\overline{Q})$,
$a_{ij}=a_{ji}$ $(i,j=\overline{1,n})$,
  $a_0(x,t)\geq 0$ for a. e. $ (x,t) \in Q$,
 there exists a function $\alpha \in C(S)$ such that $\alpha(t)>0$
 for all $t\in S$  and
 $\sum_{i,j=1}^na_{ij}(x,t)\xi_i\xi_j \geq \alpha(t)|\xi|^2$
  for every $\xi \in \mathbb{R}^n$ and for a. e. $(x,t) \in Q$;

\item[(A2)] $f\in L_{\rm loc}^2(S;L^2(\Omega))$.

\end{itemize}
Additionally, we impose the boundary condition
\begin{equation}\label{BoundaryCondition}
y\big|_{\Sigma}=0
\end{equation}
on a solution of equation \eqref{equation1}.


\begin{definition}\label{definition} \rm
A weak solution of problem \eqref{equation1}, \eqref{BoundaryCondition}
is a function $y$ which belongs to
$L_{\rm loc}^2(S;{H}^1_0(\Omega))\cap C(S;L^2(\Omega))$ and satisfies
\begin{equation}\label{equationFull1}
\begin{aligned}
&\iint_Q\Big\{-y \psi \varphi' +
 \sum_{i,j=1}^n a_{ij} y_{x_i} \psi_{x_j}\varphi+
a_0y\psi\varphi\Big\}\,dx\,dt \\
&=\iint_Q f\psi\varphi\,dx\,dt,
\quad\psi \in H_0^1(\Omega),\; \varphi\in C_c^1(-\infty,0).
\end{aligned}
\end{equation}
In other words: a weak solution of problem \eqref{equation1},
\eqref{BoundaryCondition} is the function $y$ which belongs to
$L_{\rm loc}^2(S;{H}^1_0(\Omega))\cap C(S;L^2(\Omega))$
with $y_t\in L_{\rm loc}^2(S;H^{-1}(\Omega))$, and satisfies
$$
y_t-\sum_{i,j=1}^n(a_{ij}y_{x_i})_{x_j} + a_0y=f\quad \text{in }
 L_{\rm loc}^2(S;{H}^{-1}(\Omega)).
$$
\end{definition}

\begin{remark}\label{remark1} \rm
There may exist many weak solutions of problem
\eqref{equation1}, \eqref{BoundaryCondition}. E.g.,
the functions $y_c(x,t)=cv(x)e^{-Kt}$, $(x,t) \in \overline{Q}\ (c\in \mathbb{R})$,
where $v$ is
an eigenfunction of problem \eqref{eigenvalue} corresponding to the first
eigenvalue, are weak solutions of problem \eqref{equation1},
\eqref{BoundaryCondition} when $a_{ij}=\delta_{ij},\ a_0=0$ and
$f=0$, where $\delta_{ij}$ is Kronecker's delta $(i,j=\overline{1,n})$. Therefore,
to ensure uniqueness of the weak
solution of \eqref{equation1} satisfying condition \eqref{BoundaryCondition},
we have to impose some additional conditions on solutions, for instance,
some restrictions on their behavior as $t\to-\infty$.
\end{remark}

We will consider the problem of finding the weak solution of \eqref{equation1},
\eqref{BoundaryCondition} satisfying the analogue of the initial condition
\begin{equation}\label{incondition}
\lim_{t\to-\infty} e^{\omega\int_{0}^{t}\alpha(s)ds}\|y(\cdot,t)\|_{L^2(\Omega)}=0,
\end{equation}
where $\omega\in \mathbb{R}$ is given.


We will briefly call this problem by problem  \eqref{equation1},
\eqref{BoundaryCondition}, \eqref{incondition}, and the function $y$
is called the solution of problem \eqref{equation1}, \eqref{BoundaryCondition},
\eqref{incondition}.

\begin{theorem}\label{thm1}
Suppose that condition {\rm (A1)} holds, $K$ is a constant defined
by \eqref{Kfridrix}. The following two statements hold:

(1) If $\omega\leq K$
then  \eqref{equation1}, \eqref{BoundaryCondition}, \eqref{incondition}
 has at most one weak solution.

(2)  If $\omega< K$ and
\begin{equation}\label{f}
f\in L^2_{\omega,1/\alpha}(S;L^2(\Omega)),
\end{equation}
then there exists a unique weak solution of
 \eqref{equation1}, \eqref{BoundaryCondition}, \eqref{incondition},
it belongs to the space $L_{\omega,\alpha}^2(S;{H}^1_0(\Omega))$ and the
following estimates are satisifed
\begin{gather} \label{estimate7}
e^{\omega\int_0^{\tau} \alpha(s)  \,ds} \|y(\cdot,\tau)\|_{L^2(\Omega)}\leq
C_1\|f\|_{L_{\omega,1/\alpha}^2(S_\tau;L^2(\Omega))}, \quad \tau \in S,
\\ \label{estimate2}
\|y\|_{L_{\omega,\alpha}^2(S_\tau;{H}^1_0(\Omega))}
\leq C_2\|f\|_{L_{\omega,1/\alpha}^2(S_\tau;L^2(\Omega))}, \quad \tau \in S,
\end{gather}
where $S_\tau:=(-\infty,\tau]$ $(\tau\in (-\infty,0]$, $S_0=S)$, $C_1, C_2$
are positive constants depending only on $K$ and $\omega$.
\end{theorem}

\begin{remark}\label{remark2} \rm
In the particular case of equation \eqref{equation1}, which was considered
in Remark \ref{remark1}, we have $\alpha(t)=1$, therefore condition
\eqref{incondition} takes on the form:
\[
e^{\omega t} \|y(\cdot,t)\|_{L^2(\Omega)} \to 0 \quad\text{as }
t\to -\infty.
\]
 Obviously in this case for the nonzero solutions of
 \eqref{equation1}, \eqref{BoundaryCondition}, \eqref{incondition},
indicated in Remark \ref{remark1}, we have
$\lim_{t\to -\infty} e^{K t}\|y_c(\cdot,t)\|_{L^2(\Omega)} =C $, where $C$ is
a nonzero constant;
$\lim_{t\to -\infty} e^{\omega t}\|y_c(\cdot,t)\|_{L^2(\Omega)} = +\infty$,
if $\omega<K$;
 $\lim_{t\to -\infty}e^{\omega t}\|y_c(\cdot,t)\|_{L^2(\Omega)} =0$,
 if $\omega>K$.
 This means that the condition $\omega\leq K$ is essential for ensuring
 the uniqueness of the weak solution of
 \eqref{equation1}, \eqref{BoundaryCondition}, \eqref{incondition},
 i.e., it cannot be simplified.
\end{remark}


\begin{proof}[Proof of Theorem \ref{thm1}]
In the proof  we use the same technique as in the proofs of corresponding
results in [4,5]. Nevertheless, we present the proof, because it is
important for us to obtain more precise estimates of the solution of
 \eqref{equation1}, \eqref{BoundaryCondition}, \eqref{incondition}
and to track how this solution  depends on the coefficient
(which serves as a control in the following sections).

Let us prove the first statement of Theorem \ref{thm1}.
Assume the opposite. Let $y_1,y_2$ be two  weak solutions of
 \eqref{equation1}, \eqref{BoundaryCondition}, \eqref{incondition}.
Substituting them one by one into integral identity \eqref{equationFull1}
and subtracting the
obtained equalities, for the  difference $z:=y_1-y_2$ we obtain
\begin{equation}\label{equationFull12}
-\iint_Q  z \psi \varphi'\,dx\,dt +
\iint_Q \Big(\sum_{i,j=1}^n a_{ij} z_{x_i} \psi_{x_j} +
a_0z\psi\Big)\varphi\,dx\,dt=0,
\end{equation}
for all $ \psi \in H_0^1(\Omega)$, $\varphi\in C_c^1(-\infty,0)$.

From \eqref{incondition} it follows that
\begin{equation}\label{uniq}
e^{2\omega\int_{0}^{t}\alpha(s)ds}\int_\Omega|z(x,t)|^2\,dx \to 0
\quad \text{as }  t\to -\infty.
\end{equation}
According to Lemma \ref{lem1} with
$\theta(t)=2e^{2\omega\int_0^t \alpha(s)\,ds}$, $t \in \mathbb{R}$,
\eqref{equationFull12} implies that
\begin{align*}
&e^{2\omega\int_0^{\tau_2} \alpha(s)\,ds}\int_\Omega |z(x,\tau_2)|^2\,dx
 -  e^{2\omega\int_0^{\tau_1} \alpha(s)\,ds}\int_\Omega |z(x,\tau_1)|^2\,dx
\\
&- 2\omega\int_{\tau_1}^{\tau_2}\int_\Omega \alpha(t)
e^{2\omega\int_0^{t} \alpha(s)\,ds} |z|^2  \,dx\,dt
\\
&+2\int_{\tau_1}^{\tau_2}\int_\Omega e^{2\omega\int_0^{t} \alpha(s)\,ds}
\Big[ \sum_{i,j=1}^n a_{ij} z_{x_i}z_{x_j} + a_0|z|^2 \Big] \,dx\,dt=0,
\end{align*}
where $\tau_1,\tau_2 \in S$ $(\tau_1<\tau_2)$ are arbitrary numbers.

Taking into account condition (A1) and inequality \eqref{Fridrix}, we obtain
\begin{equation} \label{001}
\begin{aligned}
& e^{2\omega\int_0^{\tau_2} \alpha(s)\,ds}\int_\Omega |z(x,\tau_2)|^2\,dx
 -  e^{2\omega\int_0^{\tau_1} \alpha(s)\,ds}\int_\Omega |z(x,\tau_1)|^2\,dx\\
&+ 2(K- \omega)\int_{\tau_1}^{\tau_2}\int_\Omega \alpha(t)
 e^{2\omega\int_0^{t} \alpha(s)  \,ds}|z|^2\,dx\,dt
\leq 0.
\end{aligned}
\end{equation}

Since $\omega \leq K$, from \eqref{001} we obtain
\begin{equation}\label{eq1}
e^{2\omega\int_0^{\tau_2} \alpha(s)  \,ds}\int_\Omega  |z(x,\tau_2)|^2\,dx \leq
e^{2\omega\int_0^{\tau_1} \alpha(s)  \,ds}\int_\Omega  |z(x,\tau_1)|^2\,dx.
\end{equation}
In \eqref{eq1} fix $\tau_2$ and let $\tau_1$ to $-\infty$.
According to condition \eqref{uniq} we obtain the equality
\[
 e^{2\omega \int_0^{\tau_2} \alpha(s)  \,ds}\int_\Omega |z(x,\tau_2)|^2\,dx=0.
\]
 Since $\tau_2 \in S$ is an arbitrary number, we have  $z(x,t)=0$ for a. e.
 $(x,t) \in Q$, that is,  $y_1(x,t)=y_2(x,t)=0$ for a. e.  $(x,t) \in Q$.
The resulting contradiction proves the first statement.

Let us prove the second statement. First we determine a priori estimates
of a weak solution of  \eqref{equation1}, \eqref{BoundaryCondition}, \eqref{incondition}.
According to Lemma \ref{lem1}, condition \eqref{equationFull1} implies
\begin{equation} \label{1.291}
\begin{aligned}
&\frac12\theta(\tau_2)\int_\Omega |y(x,\tau_2)|^2\,dx
 -\frac12\theta(\tau_1)\int_\Omega |y(x,\tau_1)|^2\,dx \\
&-\frac12 \int_{\tau_1}^{\tau_2}\int_\Omega |y|^2\theta'\,dx\,dt
+\int_{\tau_1}^{\tau_2}\int_\Omega
\Big[ \sum_{i,j=1}^na_{ij} y_{{x_i}}y_{{x_j}} + a_0|y|^2 \Big]\theta\,dx\,dt \\
&=\int_{\tau_1}^{\tau_2}\int_\Omega fy\theta\,dx\,dt,
\end{aligned}
\end{equation}
where $ \theta \in C^1(S)$ is an arbitrary function,
$\tau_1,\tau_2 \in S \ (\tau_1<\tau_2)$ are arbitrary numbers.
Further assume that $\theta(t)\geq0$ for all $t\in S$.

Using the Cauchy inequality with ``$\varepsilon$``:
$$
ab\leq\frac\varepsilon2 a^2+\frac1{2\varepsilon}b^2,\quad
 a,b\in \mathbb{R},\; \varepsilon>0,
$$
we estimate the right side of \eqref{1.291} as follows:
\begin{equation} \label{1.303}
\big|\int_{\tau_1}^{\tau_2}\int_\Omega  f y\theta\,dx\,dt \big|
\leq  \frac{\varepsilon}{2}\int_{\tau_1}^{\tau_2}
 \int_\Omega \alpha|y|^2\theta\,dx\,dt +
\frac{1}{2\varepsilon}\int_{\tau_1}^{\tau_2}\int_\Omega
 [\alpha]^{-1}|f|^2\theta\,dx\,dt,
\end{equation}
where $\varepsilon>0$ is arbitrary.

From condition (A1) we obtain
\begin{equation}\label{130}
\int_{\tau_1}^{\tau_2}\int_\Omega \big[ \sum_{i,j=1}^na_{ij} y_{{x_i}}y_{{x_j}}
+a_0|y|^2\big]\theta\,dx\,dt
\geq \int_{\tau_1}^{\tau_2}\int_\Omega \alpha |\nabla y|^2\theta\,dx\,dt,
\end{equation}
where $\nabla y:=(y_{x_1},\dots,y_{x_n})$ is the gradient of $y$.

According to \eqref{1.303} and \eqref{130}, equality \eqref{1.291} implies
\begin{align*}
&\frac12\theta(\tau_2)\int_\Omega |y(x,\tau_2)|^2\,dx
 -\frac12\theta(\tau_1)\int_\Omega |y(x,\tau_1)|^2\,dx \\
&- \frac12 \int_{\tau_1}^{\tau_2}\int_\Omega |y|^2\theta'\,dx\,dt
+\int_{\tau_1}^{\tau_2}\int_\Omega \alpha |\nabla y|^2\theta\,dx\,dt \\
&\leq \frac{\varepsilon}{2}\int_{\tau_1}^{\tau_2}
 \int_\Omega \alpha |y|^2\theta \,dx\,dt
 +\frac{1}{2\varepsilon}\int_{\tau_1}^{\tau_2}\int_\Omega [\alpha]^{-1}
 |f|^2\theta\,dx\,dt,
\end{align*}
where $\varepsilon>0$ is arbitrary.

Taking $\theta(t)=2e^{2\omega\int_0^{t} \alpha(s)  \,ds}$ with $t \in S$, we obtain
 % \label{1.303'}
\begin{align*}
&e^{2\omega\int_0^{\tau_2} \alpha(s)  \,ds}\int_\Omega |y(x,\tau_2)|^2dx
- e^{2\omega\int_0^{\tau_1} \alpha(s)  \,ds}\int_\Omega |y(x,\tau_1)|^2dx
\\
&- 2\omega \int_{\tau_1}^{\tau_2}\int_\Omega \alpha(t) e^{2\omega\int_0^{t}
\alpha(s)  \,ds} |y|^2dx\,dt +
2\int_{\tau_1}^{\tau_2}\int_\Omega \alpha(t)e^{2\omega\int_0^{t} \alpha(s)  \,ds}
|\nabla y|^2\,dx\,dt
\\
&\leq \varepsilon \int_{\tau_1}^{\tau_2}\int_\Omega \alpha(t)
 e^{2\omega\int_0^{t} \alpha(s)  \,ds} |y|^2\,dx\,dt+
\frac 1{\varepsilon}\int_{\tau_1}^{\tau_2}\int_\Omega [\alpha(t)]^{-1}
 e^{2\omega\int_0^{t} \alpha(s)  \,ds} |f|^2\,dx\,dt.
\end{align*}
By the above inequality and using \eqref{Fridrix}, we obtain
\begin{equation} \label{1.321}
\begin{aligned}
&e^{2\omega\int_0^{\tau_2} \alpha(s)  \,ds}\int_\Omega |y(x,\tau_2)|^2\,dx
- e^{2\omega\int_0^{\tau_1} \alpha(s)  \,ds}\int_\Omega |y(x,\tau_1)|^2\,dx\\
&+ \chi(K,\omega,\varepsilon)\int_{\tau_1}^{\tau_2}
\int_\Omega \alpha(t)e^{2\omega\int_0^{t} \alpha(s)  \,ds}|\nabla y|^2\,dx\,dt \\
&\leq \frac1{\varepsilon}\int_{\tau_1}^{\tau_2}
 \int_\Omega [\alpha(t)]^{-1} e^{2\omega\int_0^{t} \alpha(s)  \,ds}|f|^2\,dx\,dt,
\end{aligned}
\end{equation}
where
\[
\chi (K,\omega,\varepsilon):=\begin{cases}
\frac{2(K-\omega)-\varepsilon}{K} &\text{if } 0<\omega<K,\\
\frac{2K-\varepsilon}{K}&\text{if } \omega\leq 0.
\end{cases}
\]
Taking $\varepsilon=K$ if $\omega \leq 0$, and $\varepsilon=K-\omega$ if
$0<\omega<K$ in \eqref{1.321}, we obtain
\begin{equation} \label{1.341}
\begin{aligned}
&e^{2\omega\int_0^{\tau_2} \alpha(s)  \,ds}\int_\Omega |y(x,\tau_2)|^2\,dx
-e^{2\omega\int_0^{\tau_1} \alpha(s)  \,ds}\int_\Omega |y(x,\tau_1)|^2\,dx
\\
&+C_3\int_{\tau_1}^{\tau_2}\int_\Omega \alpha(t)e^{2\omega\int_0^{t} \alpha(s)
\,ds}|\nabla y|^2\,dx\,dt \\
&\leq C_4\int_{\tau_1}^{\tau_2}\int_\Omega [\alpha(t)]^{-1} e^{2\omega\int_0^{t}
\alpha(s)  \,ds}|f|^2\,dx\,dt,
\end{aligned}
\end{equation}
where $C_3>0$, $C_4>0$ are constants depending only on $K$ and $\omega$.

Taking into account \eqref{incondition} and \eqref{f}, we let $\tau_1\to-\infty$
in \eqref{1.341}. As a result, adopting  $\tau_2=\tau \in S$, we obtain
\begin{equation} \label{1.351}
\begin{aligned}
&e^{2\omega\int_0^{\tau} \alpha(s)  \,ds}\int_\Omega |y(x,\tau)|^2\,dx
 +C_3\int_{-\infty}^{\tau}\int_\Omega \alpha(t)e^{2\omega\int_0^{t} \alpha(s)  \,ds}
 |\nabla y|^2\,dx\,dt\\
&\leq C_4\int_{-\infty}^{\tau}\int_\Omega [\alpha(t)]^{-1}
 e^{2\omega\int_0^{t} \alpha(s)  \,ds}|f|^2\,dx\,dt.
\end{aligned}
\end{equation}
Hence, using inequality \eqref{Fridrix}, we easily obtain estimates
\eqref{estimate7} and \eqref{estimate2}.

Now let us prove the existence of a weak solution of problem
\eqref{equation1}, \eqref{BoundaryCondition}, \eqref{incondition}.
First, for each $m\in N$ we define $Q_m:=\Omega\times(-m,0]$,
$f_{m}(\cdot,t):= f(\cdot,t)$, if  $-m<t\leq 0$, and
$f_{m}(\cdot,t):= 0$, if $t\leq -m$,
and consider the problem of finding a function
$y_m \in L^2(-m,0;H^1_0(\Omega))\cap$ $ C([-m,0];L^2(\Omega))$ satisfying
the initial condition
\begin{equation}\label{inconditionm}
y_m(x,-m)=0, \quad x\in \Omega,
\end{equation}
(as an element of space $C([-m,0];L^2(\Omega))$) and equation \eqref{equation1}
in $Q_m$ in the sense of integral identity; that is,
\[
\iint_{Q_m}  \Big\{ -y_m \psi \varphi' +
 \sum_{i,j=1}^n  a_{ij} y_{m,{x_i}} \psi_{x_j} \varphi +
a_0y_m\psi \varphi \Big\}\,dx\,dt
=\iint_{Q_m}  f_m\psi\varphi\,dx\,dt,
\]  %\label{equationFullm1}
for $\psi \in H^1_0(\Omega)$, $\varphi\in C_c^1(-m,0)$.

The existence and uniqueness of the solution of this problem easily follows
from the known results  (see, for example, \cite{Gaevskyy}).
For every $m\in \mathbb{N}$ we extend $y_m$ by zero for the entire set $Q$
and keep the same notation  $y_m$ for this extension.
Note that for each $m\in N$, the function $y_m$ belongs to
$L^2(S;H^1_0(\Omega))\cap C(S;L^2(\Omega))$ and satisfies integral
identity \eqref{equationFull1} with $f_{m}$ substituted for $f$, i.e.,
\begin{equation}
\iint_{Q} \Big\{-y_m \psi \varphi' +
\sum_{i,j=1}^n a_{ij} y_{m,{x_i}} \psi_{x_j}\varphi+
a_0y_m\psi\varphi\Big\}\,dx\,dt
=\iint_{Q} f_m\psi\varphi\,dx\,dt, \label{equationFullm1}
\end{equation}
for $\psi \in H_0^1(\Omega)$, $\varphi\in C_c^1(-\infty,0)$.
Consequently, we have shown that $y_m$ is a weak solution of problem
\eqref{equation1}, \eqref{BoundaryCondition}, \eqref{incondition}
with $f_m$ substituted for $f$.
Therefore, for $y_{m}$ we obtain estimates similar to
 \eqref{estimate7}, \eqref{estimate2}, in particular, for
$\tau \in S$,
\begin{equation} \label{estimate17}
e^{2\omega\int_0^{\tau} \alpha(s)  \,ds} \|y_m(\cdot,\tau)\|^2_{L^2(\Omega)}\leq
C_1\int_{-\infty}^\tau [\alpha(t)]^{-1} e^{2\omega\int_0^{t} \alpha(s)
\,ds}\|f(\cdot,t)\|^2_{L^2(\Omega)}\,dt,
\end{equation}
Let us take identity \eqref{equationFullm1} with  alternating $m=k$ and $m=l$,
where $k,l$ are arbitrary positive integers, $l>k$, and then subtract the
obtained identities.
As a result, we obtain the same identity as $\eqref{equationFullm1}$
with $z_{k,l}:=y_k-y_l,\ f_{k,l}:=f_k-f_l$ instead of $y_m$ and $f_m$,  respectively.
Finally taking into account that the function $z_{k,l}$ satisfies conditions
\eqref{BoundaryCondition} and \eqref{incondition}, replacing $y$ with $z_{k,l}$,
we see that the function $z_{k,l}$ is a weak solution of the problem,
which differs from problem
 \eqref{equation1}, \eqref{BoundaryCondition}, \eqref{incondition}
only in that instead of $y$ and $f$, there are $z_{k,l}$ and $f_{k,l}$, respectively.
Thus, for $z_{k,l}$ we have estimates similar to
\eqref{estimate7}, \eqref{estimate2}, i.e.
\begin{gather} \label{estimate71}
\begin{aligned}
&e^{2\omega\int_0^{\tau} \alpha(s)  \,ds} \|y_k(\cdot,\tau)
 -y_l(\cdot,\tau)\|^2_{L^2(\Omega)}  \\
&\leq C_1\int_{-l}^{-k} [\alpha(t)]^{-1}
e^{2\omega\int_0^{t} \alpha(s)  \,ds}\|f(\cdot,t)\|^2_{L^2(\Omega)}\,dt,
\quad \tau \in S,
\end{aligned}\\
\label{estimate21}
\|y_k-y_l\|_{L_{\omega,\alpha}^2(S;{H}^1_0(\Omega))}
\leq C_2\int_{-l}^{-k}[\alpha(t)]^{-1}e^{2\omega\int_0^t \alpha(s)\,ds}
\|f(\cdot,t)\|^2_{L^2(\Omega)}\,dt.
\end{gather}
Condition \eqref{f} implies that the right-hand sides of inequalities
\eqref{estimate71} and \eqref{estimate21} tend to zero when $k$ and $l$
tend to $+\infty$.
This means that the sequence $\{y_m\}_{m=1}^\infty$ is a Cauchy sequence
in the space $L^2_{\omega,\alpha}(S;H^1_0(\Omega))$ and $C(S;L^2(\Omega))$.
Consequently, we obtain the existence of the function
$y\in L^2_{\omega,\alpha}(S;H_0^1(\Omega))\cap C(S;L^2(\Omega))$ such that
\begin{equation} \label{1.40}
 y_{m} \underset{m\to\infty}{\longrightarrow}
  y \quad \text{strongly in }
 L^2_{\omega,\alpha}(S;H_0^1(\Omega))\text{ and } C(S;L^2(\Omega)).
\end{equation}
Note that \eqref{1.40} implies
\begin{equation}\label{1.41}
y_{m}\ {\underset{m\to\infty}{\longrightarrow}}
  y, \quad
  y_{m,{x_i}} {\underset{m\to\infty}{\longrightarrow}}
  y_{x_i}\ (i=\overline{1,n})\quad  \text{strongly in }
 L^2_{\rm loc}(S;L^2(\Omega)).
\end{equation}

Let us show that the function $y$ is a weak solution of
\eqref{equation1}, \eqref{BoundaryCondition}, \eqref{incondition}.
To do this, first  we let $m\to \infty$ in identity \eqref{equationFullm1},
 taking into account
\eqref{1.41} and the definition of the function $f_m$. Consequently, we obtain
identity \eqref{equationFull1}.
Now, taking into account \eqref{1.40}, we let $m\to+\infty$ in \eqref{estimate17}.
From the resulting inequality and condition \eqref{f}, we obtain condition
\eqref{incondition}. Hence, we have proven that  $y$ is a  weak solution of
problem \eqref{equation1}, \eqref{BoundaryCondition}, \eqref{incondition}.
\end{proof}

\section{Formulation of the optimal control problem and  main result}\label{Sect3}

Let $U:=L^{\infty}(Q)$ be a space of controls and $U_{\partial}$ be a convex
and closed subset of $\{v \in U : v\geq 0  \text{ a. e. in }  Q \}$.
We suppose that $U_\partial$ is the set of admissible controls.

We assume that the state of the investigated evolutionary system for a given
control $v\in U_\partial$ is described by a weak solution of
 \eqref{equation1}, \eqref{BoundaryCondition}, \eqref{incondition} when
$a_0=\widetilde{a}_0+v$, where $\widetilde{a}_0\in L^{\infty}_{\rm loc}(\overline{Q})$
 is a given function such that $\widetilde{a}_0\geq0$ a. e. in $Q$.
Then, equation \eqref{equation1}  has the form
\begin{equation}\label{equation}
y_t  -\sum_{i,j=1}^n\big(a_{ij}(x,t) y_{x_i}\big)_{x_j}+(\widetilde{a}_0(x,t)
+ v(x,t))y = f(x,t), \quad (x,t)\in Q.
\end{equation}
The specified problem will be called problem
\eqref{equation}, \eqref{BoundaryCondition}, \eqref{incondition}.
The weak solution $y$ of
\eqref{equation}, \eqref{BoundaryCondition}, \eqref{incondition}
for a given control $v$, denoted by $y$, or $y(v)$, or
$y(x,t)$, $(x,t) \in Q$, or $y(x,t;v)$, $(x,t)\in Q$.
Further, we assume that conditions (A1), \eqref{f} and the inequality
$\omega<K$ hold. From the previous section
(see Theorem \ref{thm1}), we immediately obtain the existence and
uniqueness of the weak solution of problem
 \eqref{equation}, \eqref{BoundaryCondition}, \eqref{incondition}
 and its estimates \eqref{estimate7}, \eqref{estimate2}.

We assume that the cost functional has the form
\begin{equation}\label{funcvart}
  J(v)=\|y(\cdot,0;v)-z_{0}(\cdot)\|_{L^2(\Omega)}^2+\mu\|v\|_{L^{\infty}(Q)},
  \quad v\in U_\partial,
 \end{equation}
where $z_{0}\in L^2(\Omega)$, $\mu\geq0$ if $U_\partial$ is bounded, and
$\mu>0$ otherwise.

We consider the following optimal control problem:
find a control $u\in U_{\partial}$ such that
\begin{equation}\label{varinf}
J(u)= \inf _{v\in U_{\partial}}J(v).
\end{equation}
We call this problem \eqref{varinf}, and its solutions will be
called \emph{optimal controls}.

The main results of this paper are the following.

\begin{theorem}[Existence of an optimal control] \label{thm2}
With the above assumptions in this section, a set of optimal controls of
 problem \eqref{varinf} is nonempty and $*$-weakly closed in $L^\infty(Q)$.
\end{theorem}


\begin{theorem}[Necessary conditions of an optimal control]\label{thm3}
Let $U_\partial$ be bounded, $\mu=0$, and
\begin{equation}\label{alpha}
\alpha(t)\geq\alpha_0={\rm const.}>0 \quad \text{for a.e.}\quad t\in S.
\end{equation}
Then an optimal control of problem \eqref{varinf} satisfies the relations
\begin{equation} \label{y_problem}
\begin{gathered}
    y\in L^2_{\omega,\alpha}(S;H^1_0(\Omega)),\quad
    y_t\in L^2_{\rm loc}(S;H^{-1}(\Omega)),
    \\
    y_t  -\sum_{i,j=1}^n(a_{ij} y_{x_i})_{x_j}+(\widetilde{a}_0 + u)y = f \quad
\text{in } L^2_{\rm loc}(S;H^{-1}(\Omega)),
    \\
    y\big|_{\Sigma}=0, \quad
    \lim_{t\to-\infty} e^{\omega\int_{0}^{t}\alpha(s)ds}
\|y(\cdot,t)\|_{L^2(\Omega)}=0,
    \end{gathered}
\end{equation}
\begin{equation} \label{p_equation}
  \begin{gathered}
    p\in L^2_{-\omega,1/\alpha}(S;H^1_0(\Omega)),\quad
    p_t\in L^2_{\rm loc}(S;H^{-1}(\Omega)),
    \\
  -p_t-\sum_{i,j=1}^n (a_{i j}p_{x_i})_{x_j}+(\widetilde{a}_0+u)p=0 \quad \text{\rm in} \quad L^2_{\rm loc}(S;H^{-1}(\Omega)),
    \\
    p\big|_\Sigma=0,\quad p(\cdot,0)=y(\cdot,0)-z_0(\cdot),
   \end{gathered}
\end{equation}
\begin{equation}  \label{Jdiff5}
    \iint_Q yp(v-u)\,dx\,dt \leq 0 \quad \forall\, v\in U_\partial.
\end{equation}
\end{theorem}

Since  $y$ belongs to $L^2_{\omega,\alpha}(S;H^1_0(\Omega))$, and
$p$ belongs to
$L^2_{-\omega,1/\alpha}(S;H^1_0(\Omega))$,  the product $py$ belongs to
 $L^1(Q)$, and thus the left-hand side of inequality \eqref{Jdiff5} is well-defined.

Problem \eqref{p_equation} is called an adjoint problem, its solution is
called an adjoint state and is introduced in order to characterize an optimal
 control.


\section{Proof of main results}\label{Sect4}

\begin{proof}[Proof of Theorem \ref{thm2}]
 Since the cost functional $J$ is bounded below, there exists a minimizing sequence
$\{v_k\}$ in $ U_\partial$: $\lim_{k\to \infty} J(v_k)= \inf_{v\in U_\partial}J(v)$.
This and \eqref{funcvart} imply that the sequence $\{v_k\}$ is bounded in the
space $L^{\infty}(Q)$, that is
\begin{equation}\label{esssup}
\operatorname{ess\,sup}_{(x,t)\in Q}|v_k(x,t)|\leq C_5,
\end{equation}
where $C_5$ is a constant, which does not depend on $k$.

Since for each $k\in \mathbb{N}$ the function $y_k:=y(v_k)\ (k\in \mathbb{N})$
is a weak solution of  \eqref{equation}, \eqref{BoundaryCondition},
\eqref{incondition} for $v=v_k$, the following identity holds:
\begin{equation}\label{equationFullm}
\begin{aligned}
&\iint_Q\Big\{-y_k \psi \varphi' +
 \sum_{i,j=1}^n a_{ij} y_{k,{x_i}} \psi_{x_j}\varphi+
(\widetilde{a}_0+v_k)y_k\psi\varphi\Big\}\,dx\,dt \\
&=\iint_Q f\psi\varphi\,dx\,dt,
\quad\psi \in H_0^1(\Omega),\; \varphi\in C_c^1(-\infty,0).
\end{aligned}
\end{equation}
According to Theorem \ref{thm1} we have the estimates
\begin{gather} \label{estim31}
e^{2\omega\int_0^{\tau} \alpha(s)  \,ds} \|y_k(\cdot,\tau)\|^2_{L^2(\Omega)}
\leq C_1 \|f\|_{L^2_{\omega,1/\alpha}(S_\tau;L^2(\Omega))}\,,   \quad \tau \in S,
\\ \label{estim32}
\|y_k\|_{L^2_{\omega,\alpha}(S_\tau;H_0^1(\Omega))}
\leq C_2\|f\|_{L^2_{\omega,1/\alpha}(S_\tau;L^2(\Omega))}\ .
\end{gather}
Taking into account the first statement of Lemma \ref{lem1},
from \eqref{equationFullm} for arbitrary  $\tau_1, \tau_2 \in S\ (\tau_1<\tau_2)$
we obtain
\begin{equation}\label{derivY}
\int_{\tau_1}^{\tau_2}\|y_{k,t}\|^2_{H^{-1}(\Omega)}\,dt
\leq \int_{\tau_1}^{\tau_2}\int_\Omega\Big( \sum_{j=1}^n\big|
\sum_{i=1}^n a_{ij}y_{k,x_i}\big|^2 + |(\widetilde{a}_0+v_k)y_k-f|^2 \Big)dx\,dt.
\end{equation}

By condition (A1), \eqref{f}, \eqref{esssup}, and \eqref{estim32},
estimate \eqref{derivY} implies
\begin{equation}\label{derivYestim}
\int_{\tau_1}^{\tau_2}\|y_{k,t}(\cdot,t)\|^2_{H^{-1}(\Omega)}\,dt \leq C_6,
\end{equation}
where $\tau_1,\tau_2 \in S\ (\tau_1<\tau_2)$  are arbitrary, $C_6>0$
is a constant which depends on  $\tau_1$ and $\tau_2$, but does not depend on $k$.

By the Compactness Lemma (see \cite[Proposition 4.2]{Lions2}), and the compactness
of the embedding $H_0^1(\Omega)\subset L^2(\Omega)$ (see \cite[p. 245]{Lions}),
estimates  \eqref{esssup}, \eqref{estim32}, \eqref{derivYestim}  yield that there
exist a subsequence of the sequence $\{v_k,y_k\}$ (which is also denoted by
 $\{v_k,y_k\}$) and functions
$u\in  U_\partial$, and $y\in L_{\omega,\alpha}^2(S;H_0^1(\Omega))$ such that
\begin{gather}
\label{ConvergenceV}
v_k\underset{k\to\infty}{\longrightarrow}u \quad *\text{-weakly in} \quad L^\infty(Q),
\\ \label{weakLimit}
 y_k\underset{k\to\infty}{\longrightarrow}y \quad \text{weakly in} \quad L^2_{\omega,\alpha}(S;H_0^1(\Omega)),
\\ \label{ConvergenceStrongY}
y_k\underset{k\to\infty}{\longrightarrow}y \quad \text{strongly in} \quad L^2_{\rm loc}(S;L^2(\Omega)).
\end{gather}
Note that \eqref{weakLimit} implies
\begin{gather}
 \label{ConvergenceWeakY}
y_{k}\underset{k\to\infty}{\longrightarrow}y, \quad
y_{k,{x_i}}
\underset{k\to\infty}{\longrightarrow}
y_{x_i}\ (i=\overline{1,n})
\quad \text{weakly in} \quad L^2_{\rm loc}(S;L^2(\Omega)).
\end{gather}
Let us show that \eqref{ConvergenceV} and \eqref{ConvergenceStrongY} yield
\begin{equation}\label{convergenceYV}
\iint_Q y_kv_k\psi\varphi\,dx\,dt
\underset{k\to\infty}{\longrightarrow}\iint_Q yu\psi\varphi\,dx\,dt \quad\forall
  \psi \in H_0^1(\Omega),\forall \ \varphi\in C_c^1(-\infty,0).
\end{equation}
Indeed, let $g:=\psi\varphi$, and $t_1, t_2 \in S$ be such that
$\operatorname{supp}\varphi\subset [t_1,t_2]$. Then we have
\begin{equation}
\begin{aligned}
\iint _{Q} y_kv_kg\,dx\,dt
&=\int _{t_{1}}^{t_{2}}\int_\Omega ( y_kv_k - yv_k + yv_k)g\,dx\,dt
\\ \label{conv}
&=\int _{t_{1}}^{t_{2}}\int_\Omega yv_kg\,dx\,dt
 +\int _{t_{1}}^{t_{2}}\int_\Omega (y_k-y)v_kg\,dx\,dt.
\end{aligned}
\end{equation}
From \eqref{esssup} and \eqref{ConvergenceStrongY} it follows that
\begin{equation}\label{conv1}
\begin{aligned}
&\Big| \int _{t_{1}}^{t_{2}}\int_\Omega (y_k-y)v_kg\,dx\,dt \Big|\\
&\leq \Big(\int _{t_{1}}^{t_{2}}\int_\Omega |v_kg|^2\,dx\,dt\Big)^{1/2}
\Big(\int _{t_{1}}^{t_{2}}\int_\Omega |y_k-y|^2\,dx\,dt\Big)^{1/2} 
\to 0\quad \text{as } k\to \infty.
\end{aligned}
\end{equation}
Thus, using \eqref{ConvergenceV} and \eqref{conv1}, \eqref{conv}
implies \eqref{convergenceYV}.

Using  \eqref{ConvergenceWeakY} and \eqref{convergenceYV}, and letting
$k\to\infty$ in \eqref{equationFullm}, we obtain
\begin{equation}\label{equationFull}
\begin{aligned}
&\iint_Q\Big\{-y \psi \varphi' +
 \sum_{i,j=1}^n a_{ij} y_{{x_i}} \psi_{x_j}\varphi+
(\widetilde{a}_0+u)y\psi\varphi\Big\}\,dx\,dt \\
&=\iint_Q f\psi\varphi\,dx\,dt,
\quad\psi \in H_0^1(\Omega),\; \varphi\in C_c^1(-\infty,0).
\end{aligned}
\end{equation}
According to Lemma \ref{lem1}, identity \eqref{equationFull}
implies that $y\in C(S;L^2(\Omega))$ and $y_t \in L^2_{\rm loc}(S;H^{-1}(\Omega))$.
Hence, the function $y=y(u)$ is a weak solution of problem
\eqref{equation}, \eqref{BoundaryCondition}.
Let us show that $y$ satisfies condition \eqref{incondition}.
First, we prove the  convergence
\begin{equation}\label{ConvergenceYInL2Om}
\forall \tau \in S:\quad y_k(\cdot,\tau)\underset{k\to\infty}{\longrightarrow}
y(\cdot,\tau) \quad \text{strongly in }  L^2(\Omega).
\end{equation}
For this purpose, we subtract  \eqref{equationFullm} from
 \eqref{equationFull}. To the resulting identity, we apply
Lemma \ref{lem1} with
$z=y-y_k$, $g_0=(\widetilde{a}_0+u)y-(\widetilde{a}_0+v_k)y_k$,
$g_i=\sum_{j=1}^n a_{ij}(y_{x_j}-y_{k,x_j})\ (i=\overline{1,n})$,
$\theta(t)=2(t-\tau+1)$, $\tau_1=\tau-1$, $\tau_2=\tau$,  where
$\tau \in S$ is arbitrary.
Consequently,
\begin{equation} \label{50}
\begin{aligned}
&\int_\Omega |y(x,\tau)-y_k(x,\tau)|^2\,dx -
 \int_{\tau-1}^{\tau}\int_\Omega |y-y_k|^2\,dx\,dt\\
&+\int_{\tau-1}^{\tau}\int_\Omega \Big[ \sum_{i,j=1}^na_{ij} (y_{x_i}
 - y_{k,x_i})(y_{x_j}-y_{k,x_j})\Big]\theta\,dx\,dt\\
&+\int_{\tau-1}^{\tau}\int_\Omega\big((\widetilde{a}_0+u)y
 - (\widetilde{a}_0+v_k)y_k\big)\big( y -y_k \big)\theta\,dx\,dt=0.
\end{aligned}
\end{equation}
Let us transform the last term on the left side of \eqref{50} as follows:
\begin{equation} \label{50'}
\begin{aligned}
&\int_{\tau-1}^{\tau}\int_\Omega\big((\widetilde{a}_0+u)y
 - (\widetilde{a}_0+v_k)y_k\big)\big( y -y_k \big)\theta\,dx\,dt\\
&=\int_{\tau-1}^{\tau}\int_\Omega
\big((\widetilde{a}_0+u)y - (\widetilde{a}_0+v_k)(y_k-y+y)\big)
\big( y -y_k \big) \theta\,dx\,dt \\
&=\int_{\tau_1}^{\tau_2}\int_\Omega
\big[(\widetilde{a}_0+v_k)|y-y_k|^2+(u-v_k)y(y-y_k)\big] \theta\,dx\,dt.
\end{aligned}
\end{equation}
From \eqref{50}, taking into account (A1) and \eqref{50'}, we obtain
\begin{equation}  \label{51}
\begin{aligned}
&\int_\Omega |y(x,\tau)-y_k(x,\tau)|^2\,dx+
2\int_{\tau-1}^{\tau}\int_\Omega (\widetilde{a}_0 + v_k)|y - y_k|^2\,dx\,dt \\
&\leq  \int_{\tau-1}^{\tau}\int_\Omega |y( y - y_k) ||u -v_k|\,dx\,dt
+ \int_{\tau-1}^{\tau}\int_\Omega |y-y_k|^2\,dx\,dt.
\end{aligned}
\end{equation}
Using \eqref{esssup} and Cauchy-Schwarz inequality, \eqref{51} yields
\begin{equation} \label{511}
\begin{aligned}
&\int_\Omega |y(x,\tau)-y_k(x,\tau)|^2\,dx\\
&\leq C_7\Big(\int_{\tau-1}^{\tau}\int_\Omega
|y - y_k|^2\,dx\,dt \Big)^{1/2}
+ \int_{\tau-1}^{\tau}\int_\Omega |y-y_k|^2\,dx\,dt,
\end{aligned}
\end{equation}
where $C_7>0$ is a constant which does not depend on $k$.

From \eqref{ConvergenceStrongY}, according to \eqref{511},
 we obtain \eqref{ConvergenceYInL2Om}.
Taking into account \eqref{ConvergenceYInL2Om}, letting $k\to \infty$
in \eqref{estim31}, the resulting inequality, according to condition \eqref{f},
implies
\begin{equation}
\lim_{\tau\to -\infty}  e^{2\omega\int_0^{\tau} \alpha(s)  \,ds}
\int_\Omega  |y(x,\tau)|^2\,dx =0.
\end{equation}
Hence, we have shown that  $y=y(u)= y(x,t;u)$, $(x,t)\in Q$, is the state
of the controlled system for the control $u$.

It remains to prove that $u$ is a minimizing element of the functional $J$.
Indeed, \eqref{ConvergenceYInL2Om} implies
\begin{equation}\label{oneBeforeLast}
\|y_k(\cdot,0)-z_{0}(\cdot)\|_{L^2(\Omega)}^2
\underset{k\to\infty}{\longrightarrow} \|y(\cdot,0)-z_{0}(\cdot)\|_{L^2(\Omega)}^2.
\end{equation}
Also, \eqref{ConvergenceV} and properties of $*$-weakly convergent sequences yield
\begin{equation}\label{TheLast}
\liminf_{k\to\infty}\|v_k\|_{L^{\infty}(Q)}\geq  \|u\|_{L^{\infty}(Q)}.
\end{equation}

From \eqref{funcvart}, \eqref{oneBeforeLast} and \eqref{TheLast}, it easily
follows that
$\lim_{k\to \infty}J(v_k)\geq J(u)$. Thus, we have shown that $u$ is a
solution of  problem \eqref{varinf}.

Now let us show that the set of optimal controls of problem \eqref{varinf}
is $*$-weakly closed. Indeed, let $\{u_k\}$ is a sequence of optimal controls
such that $u_k\to u$ $*$-weakly in $L^\infty(Q)$. Similarly as above we
show that $\liminf _{k\to\infty} J(u_k)\geq J(u)$.
 But $J(u_k)=\inf_{v\in U_\partial}J(v)$ $\forall k \in \mathbb{N}$.
Then $u$ is an optimal control of \eqref{varinf}.
\end{proof}

We now turn to the proof of Theorem \ref{thm3}.
To do this we need some extra statements.

\begin{lemma}\label{remark4}
Under condition \eqref{alpha} the following continuous embeddings hold
$$
L^2_{\omega,\alpha}(S_\tau;H_0^1(\Omega))
\subset L^2_{\omega,1/\alpha}(S_\tau;H^1_0(\Omega))
\subset L^2_{\omega,1/\alpha}(S_\tau;L^2(\Omega)) \quad \forall \tau \in S,
$$
so, there exist positive constants $C_8,C_9$ such that for arbitrary
$z\in L^2_{\omega,\alpha}(S;H_0^1(\Omega))$ and $\tau \in S$ we have
\begin{equation}\label{embeddings}
\|z\|_{L^2_{\omega,1/\alpha}(S_\tau;L^2(\Omega))}
\leq C_8 \|z\|_{L^2_{\omega,1/\alpha}(S_\tau;H_0^1(\Omega))}
\leq C_9 \|z\|_{L^2_{\omega,\alpha}(S_\tau;H_0^1(\Omega))}.
\end{equation}
\end{lemma}

\begin{proof}
The first inequality of \eqref{embeddings} follows easily from \eqref{Kfridrix}.
According to  \eqref{alpha} we have
$1/{\alpha(t)}\leq {1}/{\alpha_0}\leq{\alpha(t)}/{(\alpha_0)^2}$
for a.e. $t\in S$.
This yields
$$
\int_{-\infty}^\tau\int_\Omega [\alpha(t)]^{-1}e^{2\omega \int_{0}^{t}{\alpha }(s)ds}
|\nabla z|^2\,dx\,dt
\leq [\alpha_0]^{-2}\int_{-\infty}^\tau\int_\Omega \alpha(t)
e^{2\omega \int_{0}^{t}{\alpha }(s)ds} |\nabla z|^2\,dx\,dt.
$$
So, we obtain \eqref{embeddings} with $C_{9}=C_8[\alpha_0]^{-2}$.
\end{proof}

To proof Theorem \ref{thm3}, we need to differentiate the map
$v\mapsto J(v)$ with respect to the control $v$. Since $y=y(v)$ appears
in $J(v)$, we first prove the appropriate differentiability  of the map
$v\mapsto y(v)$ whose derivative is called {\it sensitivity}
(see \cite[Section 5]{Lenhart1}).

\begin{lemma}\label{lem3}
For every $u,v\in U_\partial$ there exists function
$\chi=\chi(u,v)=\chi(x,t;u,v)=\chi(x,\tau)$, $(x,t) \in Q$,
from $L^2_{\omega,\alpha}(S;H^1_0(\Omega))$ such that
$\chi_t\in L^2_{\rm loc}(S;H^{-1}(\Omega))$ (so
$\chi\in C(S;L^2(\Omega))$), and
\begin{gather}
\label{chi_convergence1}
\chi^\varepsilon(u,v):=\frac{y(u+\varepsilon (v-u))-y(u)}{\varepsilon}
\underset{\varepsilon\to 0+}{\longrightarrow} \chi(u,v) \quad\text{weakly in }
  L^2_{\omega,\alpha}(S;H^1_0(\Omega)),
\\ \label{chi_strongconvergence}
\chi^\varepsilon(u,v)\underset{\varepsilon\to 0+}{\longrightarrow} \chi(u,v)
\quad\text{strongly in } L^2_{\rm loc}(S;L^2(\Omega)),
\\ \label{chi_convergence2}
\forall \tau\in S:\quad \chi^\varepsilon(\cdot,\tau)
 \underset{\varepsilon\to 0+}{\longrightarrow} \chi(\cdot,\tau) \quad\text{strongly in }
 L^2(\Omega)\; ( \varepsilon\in (0,1)).
\end{gather}
Moreover, sensitivity $\chi$ is a weak solution of the problem
\begin{gather}
    \label{chi_equation}
    \chi_t-\sum_{i,j=1}^n (a_{i j}\chi_{x_i})_{x_j}+(\widetilde{a}_0+u)\chi=(u-v)y,
    \\ \label{chi_boundarycondition}
    \chi\big|_\Sigma=0,
    \\ \label{chi_incondition}
    \lim_{t\to-\infty} e^{\omega\int_{0}^{t}\alpha(s)ds}\|\chi(\cdot,t)\|_{L^2(\Omega)}=0.
\end{gather}
\end{lemma}


\begin{proof}
First we denote $w:=v-u$, $v^\varepsilon:=u+\varepsilon w$,
 $\varepsilon \in (0,1)$. Since the set $U_\partial$ is convex then for each
$\varepsilon\in (0,1)$ an element $v^\varepsilon$ belongs to $U_\partial$ for
all $u,v \in U_\partial$.
It is clear that
\begin{equation}\label{lemma3_convergence_ve}
v^\varepsilon \underset{\varepsilon\to 0}{\longrightarrow}u \quad \text{strongly in }
 L^\infty(Q).
\end{equation}
Let the function $y^\varepsilon:=y(v^\varepsilon)$ be a weak solution of problem
\eqref{equation}, \eqref{BoundaryCondition}, \eqref{incondition} for
$v=v^\varepsilon$, where $\varepsilon \in (0,1)$. Theorem \ref{thm1}
imply that $y^\varepsilon$ exists, it is unique, belongs to
$L^2_{\omega,\alpha}(S;H_0^1(\Omega))$ and the following estimates hold
\begin{gather}
\label{estim31_y_e}
e^{\omega\int_0^{\tau} \alpha(s)  \,ds} \|y^\varepsilon(\cdot,\tau)
\|_{L^2(\Omega)}\leq C_1\|f\|_{L^2_{\omega,1/\alpha}(S_\tau;L^2(\Omega))},
\quad \tau \in S,
\\ \label{estim32_y_e}
\|y^\varepsilon\|_{L^2_{\omega,\alpha}(S_\tau;H_0^1(\Omega))}
\leq C_2\|f\|_{L^2_{\omega,1/\alpha}(S_\tau;L^2(\Omega))}\, ,
\quad \tau \in S.
\end{gather}
Also by Lemma \ref{remark4} and \eqref{estim32_y_e} we have
\begin{equation}\label{estim32_y_e1}
\begin{aligned}
\|y^\varepsilon\|_{L^2_{\omega,1/\alpha}(S_\tau;L^2(\Omega))}
&\leq C_9\|y^\varepsilon\|_{L^2_{\omega,\alpha}(S_\tau;H_0^1(\Omega))} \\
&\leq C_2C_9\|f\|_{L^2_{\omega,1/\alpha}(S_\tau;L^2(\Omega))},\quad
    \tau \in S.
\end{aligned}
\end{equation}

  Repeating the proof of Theorem \ref{thm2}
with $v_k$ being replaced by $v^\varepsilon$ and $y_k$  replaced by
$y^\varepsilon$ we easily obtain convergence similar to
\eqref{weakLimit}, \eqref{ConvergenceStrongY}, \eqref{ConvergenceYInL2Om}, i.e.,
\begin{gather}
  y^\varepsilon\underset{\varepsilon\to 0}{\longrightarrow}y \quad \text{weakly in }
L^2_{\omega,\alpha}(S;H_0^1(\Omega)),
\\ \label{lemma3_ConvergenceStrong_ye}
y^\varepsilon\underset{\varepsilon\to 0}{\longrightarrow}y \quad \text{strongly in }
 L^2_{\rm loc}(S;L^2(\Omega)),
\\ \label{lemma3_Convergence_ye_L2Om}
\forall \tau \in S:\quad y^\varepsilon(\cdot,\tau)
\underset{\varepsilon\to 0}{\longrightarrow} y(\cdot,\tau) \quad \text{strongly in }
 L^2(\Omega),
\end{gather}
where $y:=y(u)$ is a solution of  \eqref{equation}, \eqref{BoundaryCondition},
\eqref{incondition} for $v=u$, that is, problem \eqref{y_problem}.

Obviously, by the definition of $\chi^\varepsilon$, we obtain that $\chi^\varepsilon$
is the weak solution of the problem
\begin{gather}
    \label{chi_e_equation}
    \chi^\varepsilon_t-\sum_{i,j=1}^n (a_{i j}\chi^\varepsilon_{x_i})_{x_j}
+(\widetilde{a}_0+u)\chi^\varepsilon=-wy^\varepsilon,
   \\ \label{chi_e_boundarycondition}
    \chi^\varepsilon\big|_\Sigma=0,
    \\ \label{chi_e_incondition}
    \lim_{t\to-\infty} e^{\omega\int_{0}^{t}\alpha(s)ds}\|\chi^\varepsilon
(\cdot,t)\|_{L^2(\Omega)}=0.
\end{gather}
In particular, we have
\begin{equation}\label{equationFullchi}
\begin{aligned}
&\iint_Q\Big\{-\chi^\varepsilon\psi \varphi' +
 \sum_{i,j=1}^n a_{ij} \chi^\varepsilon_{x_i} \psi_{x_j}\varphi+
(\widetilde{a}_0+u)\chi^\varepsilon\psi\varphi\Big\}\,dx\,dt \\
&=-\iint_Q wy^\varepsilon \psi\varphi\,dx\,dt,
\quad\psi \in H_0^1(\Omega),\,\, \varphi\in C_c^1(-\infty,0).
\end{aligned}
\end{equation}

Clearly, problem \eqref{chi_e_equation}-\eqref{chi_e_incondition} coincides
with problem \eqref{equation}, \eqref{BoundaryCondition}, \eqref{incondition}
when $v=u$ and $f=-wy^\varepsilon$.
Hence, taking into account Theorem \ref{thm1} we obtain that
$\chi^\varepsilon$ belongs to $L^2_{\omega,\alpha}(S;H_0^1(\Omega))$,
$\chi^\varepsilon_t$ belongs to $L^2_{\rm loc}(S;H^{-1}(\Omega))$,
and satisfies the following estimates
\begin{gather*}
e^{\omega\int_0^{\tau} \alpha(s)  \,ds}
\|\chi^\varepsilon(\cdot,\tau)\|_{L^2(\Omega)}
\leq C_1 \|wy^\varepsilon\|_{L^2_{\omega,1/\alpha}(S_\tau;L^2(\Omega))},\quad
\tau \in S,
\\
\|\chi^\varepsilon\|_{L^2_{\omega,\alpha}(S_\tau;H_0^1(\Omega))}
\leq C_2\|wy^\varepsilon\|_{L^2_{\omega,1/\alpha}(S_\tau;L^2(\Omega))}, \quad
\tau \in S.
\end{gather*}
Estimate \eqref{estim32_y_e1} implies that
\begin{equation}\label{estim31_chi_e}
\|wy^\varepsilon\|_{L^2_{\omega,1/\alpha}(S_\tau;L^2(\Omega))}
\leq  C_2C_9\|w\|_{L^\infty(Q)} \|f\|_{L^2_{\omega,1/\alpha}(S_\tau;L^2(\Omega))},
\end{equation}
which yields
\begin{gather}
\label{estim31_chi_e_f}
e^{\omega\int_0^{\tau} \alpha(s)  \,ds} \|\chi^\varepsilon(\cdot,\tau)
\|_{L^2(\Omega)}
\leq C_{10}\|f\|_{L^2_{\omega,1/\alpha}(S_\tau;L^2(\Omega))},   \quad \tau \in S,
\\ \label{estim32_chi_e_f}
\|\chi^\varepsilon\|_{L^2_{\omega,\alpha}(S_\tau;H_0^1(\Omega))}
\leq C_{11} \|f\|_{L^2_{\omega,1/\alpha}(S_\tau;L^2(\Omega))}, \quad \tau \in S,
\end{gather}
where $C_{10}, C_{11}$ are positive constants which do not depend on $\varepsilon$.

Since $L^2_{\omega,\alpha}(S;H_0^1(\Omega))$ is a Hilbert space,
then estimate \eqref{estim32_chi_e_f} yield the existence of function
$\chi\in L^2_{\omega,\alpha}(S;H_0^1(\Omega))$ such that
convergence \eqref{chi_convergence1} holds.

Convergence \eqref{chi_convergence1}, \eqref{lemma3_ConvergenceStrong_ye}
imply that we can pass to the limit in \eqref{equationFullchi} as $\varepsilon\to0$
and we obtain that function $\chi$ satisfies
\eqref{chi_equation}, \eqref{chi_boundarycondition}, so it suffices to prove that
function $\chi$ satisfies condition \eqref{chi_incondition} and convergence
\eqref{chi_strongconvergence}, \eqref{chi_convergence2}.

From \eqref{equationFullchi} and the first statement of Lemma \ref{lem1},
for arbitrary  $\tau_1, \tau_2 \in S\ (\tau_1<\tau_2)$ we obtain
\begin{equation}\label{deriv_chi}
\int_{\tau_1}^{\tau_2}\|\chi^\varepsilon_t\|^2_{H^{-1}(\Omega)}\,dt
\leq \int_{\tau_1}^{\tau_2}\int_\Omega\Big( \sum_{j=1}^n
\big|\sum_{i=1}^n a_{ij}\chi^\varepsilon_{x_i}\big|^2
+ |(\widetilde{a}_0+u)\chi^\varepsilon+wy^\varepsilon|^2 \Big)dx\,dt.
\end{equation}

By condition (A1), and \eqref{estim32_chi_e_f},
estimate \eqref{deriv_chi} implies that
\begin{equation}\label{deriv_chi_estim}
\int_{\tau_1}^{\tau_2}\|\chi^\varepsilon_t(\cdot,t)\|^2_{H^{-1}(\Omega)}\,dt
 \leq C_{12},
\end{equation}
where $\tau_1,\tau_2 \in S\ (\tau_1<\tau_2)$  are arbitrary, $C_{12}>0$
is a constant which depends on  $\tau_1$ and $\tau_2$, but does not depend on $k$.

Having estimates \eqref{estim32_chi_e_f}, \eqref{deriv_chi_estim}, we can
conclude (similarly as it was done for \eqref{ConvergenceStrongY})
that there exists a subsequence of  $\{v^\varepsilon,y^\varepsilon\}$
(which is also denoted by $\{v^\varepsilon,y^\varepsilon\}$) such that
\begin{equation}
\label{ConvergenceStrong_chi}
\chi^\varepsilon\underset{\varepsilon\to 0}{\longrightarrow}\chi \quad
\text{strongly in }  L^2_{\rm loc}(S;L^2(\Omega)).
\end{equation}
Let us prove the following convergence:
\begin{equation}\label{Convergence_chi_L2Om}
\forall \tau \in S:\quad \chi^\varepsilon(\cdot,\tau)
\underset{\varepsilon\to 0}{\longrightarrow} \chi(\cdot,\tau) \quad
\text{strongly in} \quad L^2(\Omega).
\end{equation}
For this purpose, we subtract identity \eqref{chi_e_equation} from identity
\eqref{chi_equation}. To the resulting identity, we apply Lemma \ref{lem1} with
$z=\chi-\chi^\varepsilon$, $g_0=(\widetilde{a}_0+u)(\chi-\chi^\varepsilon)
-wy^\varepsilon$, $g_i=\sum_{j=1}^n a_{ij}(\chi_{x_j}
-\chi^\varepsilon_{x_j})\ (i=\overline{1,n})$,
$\theta(t)=2(t-\tau+1)$, $\tau_1=\tau-1$, $\tau_2=\tau$, where
$ \tau \in S$ is arbitrary.
Consequently, we obtain
\begin{equation} \label{lemma3_lemma1}
\begin{aligned}
&\int_\Omega |\chi(x,\tau)-\chi^\varepsilon(x,\tau)|^2\,dx -
 \int_{\tau-1}^{\tau}\int_\Omega |\chi-\chi^\varepsilon|^2\,dx\,dt\\
&+\int_{\tau-1}^{\tau}\int_\Omega \Big[ \sum_{i,j=1}^n a_{ij} (\chi_{x_i}
  - \chi^\varepsilon_{x_i})(\chi_{x_j}-\chi^\varepsilon_{x_j})\Big]\theta\,dx\,dt\\
&+\int_{\tau-1}^{\tau}\int_\Omega\big(
(\widetilde{a}_0+u)(\chi - \chi^\varepsilon)-wy^\varepsilon\big)
\big( \chi -\chi^\varepsilon \big)\theta\,dx\,dt=0.
\end{aligned}
\end{equation}
Taking into account  (A1), we obtain
\begin{equation} \label{lemma3_lemma1_1}
\begin{aligned}
&\int_\Omega |\chi(x,\tau)-\chi^\varepsilon(x,\tau)|^2\,dx \\
&\leq  \int_{\tau-1}^{\tau}\int_\Omega |\chi-\chi^\varepsilon|^2\,dx\,dt
  + \int_{\tau-1}^{\tau}\int_\Omega
|w||y^\varepsilon||\chi - \chi^\varepsilon|\,dx\,dt.
\end{aligned}
\end{equation}
Using Cauchy-Schwarz inequality, the above inequality yields
\begin{align*}
\int_\Omega |\chi(x,\tau)-\chi^\varepsilon(x,\tau)|^2\,dx 
&\leq C_{13}\Big[ \int_{\tau-1}^{\tau}\int_\Omega
|\chi - \chi^\varepsilon|^2\,dx\,dt \\
&\quad +\Big(\int_{\tau-1}^{\tau}\int_\Omega
|\chi - \chi^\varepsilon|^2\,dx\,dt \Big)^{1/2}\Big(\int_{\tau-1}^{\tau}\int_\Omega
|y^\varepsilon|^2\,dx\,dt \Big)^{1/2}\Big],
\end{align*}
where $C_{13}>0$ is a constant depending on $\|w\|_{L^\infty(Q)}$ only.

By \eqref{lemma3_ConvergenceStrong_ye} and \eqref{ConvergenceStrong_chi},
we obtain \eqref{Convergence_chi_L2Om}.
Taking into account \eqref{Convergence_chi_L2Om}, and letting $\varepsilon\to 0$
in \eqref{estim31_chi_e_f}, the resulting inequality, according to condition
\eqref{f}, implies \eqref{chi_incondition}.
\end{proof}

\begin{lemma}\label{lem4}
There exists a unique weak solution of \eqref{p_equation},
and if $\omega<K$, then it belongs to
$L^2_{-\omega,1/\alpha}(S;H^1_0(\Omega))$ and satisfies the following estimates:
\begin{gather} \label{estim}
e^{-\omega \int_{0}^{\tau}{\alpha }(s)ds}\|p(\cdot,\tau)\|_{L^2(\Omega)}\,dx
\leq C_{14} \|p(\cdot,0)\|_{L^2(\Omega)},\quad \tau \in S,\\
\label{pestim1}
\|p\|_{L^2_{-\omega,1/\alpha}(S;H^1_0(\Omega))}\leq C_{14}\|p(\cdot,0)\|_{L^2(\Omega)},
\end{gather}
where $C_{14}>0$ is a  constant independent of $p$.
\end{lemma}

\begin{proof}
The existence of a unique weak solution $p$ of \eqref{p_equation}
is a well-known fact.
Lemma \ref{lem1} yield $p_t\in L^2_{\rm loc}(S;H^{-1}(\Omega))$.
To conclude, it suffices to prove estimates \eqref{estim} and \eqref{pestim1}.

According to Lemma \ref{lem1} when $\tau_1=\tau<0, \tau_2=0$, $z=-p$,
$g_0=(\widetilde{a}_0+u)p$, $g_i=\sum_{j=1}^na_{ij} p_{x_j}$ ($i=\overline{1,n}$),
while $\theta\in C^1(S)$ is arbitrary function, we obtain
\begin{align*} %\label{estim0}
&\frac12\theta(0)\int_\Omega |p(x,0)|^2\,dx
 -\frac12\theta(\tau)\int_\Omega |p(x,\tau)|^2\,dx
 -\frac12 \int_{\tau}^{0}\int_\Omega |p|^2\theta'\,dx\,dt\\
&-\int_{\tau}^{0}\int_\Omega \big[ \sum_{i,j=1}^na_{ij} p_{x_i}p_{x_j}
+(\widetilde{a}_0+u)|p|^2 \big]\theta\,dx\,dt=0.
\end{align*}
Taking $\theta(t)=e^{-2\omega \int_{0}^{t}{\alpha }(s)ds}$, $t\in S$, we obtain
\begin{align*}
&\frac12e^{-2\omega \int_{0}^{\tau}{\alpha }(s)ds}\int_\Omega |p(x,\tau)|^2\,dx
- \omega \int_{\tau}^{0}\int_\Omega \alpha(t)e^{-2\omega \int_{0}^{t}{\alpha }(s)ds}
 |p|^2\,dx\,dt\\
&+\int_{\tau}^{0}\int_\Omega e^{-2\omega \int_{0}^{t}{\alpha }(s)ds}
\big[ \sum_{i,j=1}^na_{ij} p_{x_i}p_{x_j} + (\widetilde{a}_0+u)|p|^2 \big]\,dx\,dt\\
&=\frac12\int_\Omega |p(x,0)|^2\,dx.
\end{align*}
From this, using condition (A1) we have
\begin{align*}
&e^{-2\omega\int_0^{\tau} \alpha(s)  \,ds}\int_\Omega |p(x,\tau)|^2dx
- 2\omega \int_{\tau}^{0}\int_\Omega \alpha(t)
e^{-2\omega\int_0^{t} \alpha(s)  \,ds} |p|^2dx\,dt
\\
&+ 2(\delta+1-\delta)\int_{\tau}^{0}\int_\Omega \alpha(t)
e^{-2\omega\int_0^{t} \alpha(s)  \,ds}|\nabla p|^2\,dx\,dt\\
&\leq \int_\Omega |p(x,0)|^2\,dx.
\end{align*}
According to \eqref{Fridrix}, for arbitrary $\delta \in (0,1)$ we obtain
\begin{align*}
&e^{-2\omega\int_0^{\tau} \alpha(s)  \,ds}\int_\Omega |p(x,\tau)|^2dx
+2(\delta K- \omega) \int_{\tau}^{0}\int_\Omega \alpha(t) e^{-2\omega\int_0^{t} \alpha(s)  \,ds} |p|^2dx\,dt
\\
&+ 2(1-\delta)\int_{\tau}^{0}\int_\Omega \alpha(t)e^{-2\omega\int_0^{t}
 \alpha(s)  \,ds}|\nabla p|^2\,dx\,dt \\
&\leq \int_\Omega |p(x,0)|^2\,dx.
\end{align*}
Since $\omega<K$ we choose $\delta \in [0,1)$ such that $\delta K-\omega>0$
and obtain
\begin{equation} \label{pestim2}
\begin{aligned}
&e^{-2\omega \int_{0}^{\tau}{\alpha }(s)ds}\int_\Omega |p(x,\tau)|^2\,dx + \int_{\tau}^{0}\int_\Omega \alpha(t)e^{-2\omega \int_{0}^{t}{\alpha }(s)ds}  |\nabla p|^2 \,dx\,dt
\\
&\leq C_{11}\int_\Omega |p(x,0)|^2\,dx,
\end{aligned}
\end{equation}
where $C_{11}>0$ is a  constant depending on $\omega$ and $K$ only.
From \eqref{pestim2} according to Lemma \ref{remark4}  we easily
obtain \eqref{estim} and \eqref{pestim1}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3}]
Let $u$ be an optimal control of problem \eqref{varinf}, $v\in U_\partial$ 
be an arbitrary, then using the same notations as in the proof of 
Lemma \ref{lem3}, for all $\varepsilon\in(0,1)$ we have
\begin{equation}\label{Jdiff1}
J(v^\varepsilon)-J(u)\geq 0.
\end{equation}
Multiplying variational inequality \eqref{Jdiff1} by $1/\varepsilon$ and
denoting $w=v-u$, we obtain
%\label{Jdiff2}
\begin{align*}
0&\leq \frac{1}{\varepsilon}\big(J(v^\varepsilon)-J(u)\big) \\
&=\frac{1}{\varepsilon}\Big[\int_\Omega\big( y(x,0;v^\varepsilon)
 -z_{0}(x)\big)^2dx -\int_\Omega\big( y(x,0;u)-z_{0}(x)\big)^2dx\Big]\\
&= \frac{1}{\varepsilon}\int_\Omega\big( y^2(x,0;v^\varepsilon)-y^2(x,0;u)
- 2z_{0}(x)y^2(x,0;v^\varepsilon) + 2z_0(x)y(x,0;u)\big)\,dx\\
&=\int_\Omega \frac{y(x,0;v^\varepsilon)-y(x,0;u)}{\varepsilon}
\Big[y(x,0;v^\varepsilon)+y(x,0;u) -2z_0(x)\Big]dx.
\end{align*}
We rewrite  the above inequality  as 
\begin{gather}\label{Jdiff3}
\int_\Omega\Big(\chi^\varepsilon(x,0)\big(y^\varepsilon(x,0)+y(x,0)\big)
-2\chi^\varepsilon(x,0)z_0(x)\Big)dx \geq 0.
\end{gather}


According to  \eqref{chi_convergence2} and 
\eqref{lemma3_Convergence_ye_L2Om}, we pass to the limit in \eqref{Jdiff3} 
as $\varepsilon\to 0+$. As a result we obtain the following variational inequality
\begin{gather} \label{Jdiff4}
\int_\Omega\chi(x,0)\big(y(x,0)-z_0(x)\big)dx \geq 0.
\end{gather}
Applying  formula \ref{Lemma121} of Lemma \eqref{lem1} for the adjoint 
problem \eqref{p_equation}
with test function $\chi$, i.e. $q=\chi$, for any $\tau \in S$, we obtain
\begin{align*}
0&=\int_\tau^0\big(-p_t-\sum_{i,j=1}^n (a_{i j}p_{x_i})_{x_j}
+(\widetilde{a}_0+u)p,\chi\big)\,dt \\
&=-\int_\tau^0(p_t,\chi)\,dt
 +\int_\tau^0\int_\Omega\Big[\sum_{i,j=1}^n a_{i j}p_{x_i}\chi_{x_j} 
 +(\widetilde{a}_0+u)p\chi\Big]dx\,dt=0.
\end{align*}
Using integration by parts  and condition (A1) 
(the symmetry of coefficients $a_{ij}$), we obtain
\begin{equation} \label{p_chi_equation2}
\begin{aligned}
0&=-\int_\Omega p\chi\,dx\Big|_{t=\tau}^{t=0}
    +\int_\tau^0(p,\chi_t)\,dt \\
&\quad    +\int_\tau^0\int_\Omega\Big[\sum_{i,j=1}^n a_{i j}p_{x_i}\chi_{x_j}
 +(\widetilde{a}_0+u)p\chi\Big]\,dx\,dt    \\
&=-\int_\Omega \chi(x,0) p(x,0)\,dx
    +\int_\Omega \chi(x,\tau) p(x,\tau)\,dx     \\
&\quad + \int_\tau^0 (\chi_t, p)\,dt
    +\int_\tau^0\int_\Omega\Big[
    \sum_{i,j=1}^n a_{ij}\chi_{x_i}p_{x_j}+(\widetilde{a}_0+u)\chi p\Big]dx\,dt
    \\
&=-\int_\Omega \chi(x,0) p(x,0)\,dx
    +\int_\Omega \chi(x,\tau) p(x,\tau)\,dx \\
&\quad +\int_\tau^0 \big(\chi_t-\sum_{i,j=1}^n (a_{i j}\chi_{x_i})_{x_j}
 +(\widetilde{a}_0+u)\chi,p\big)
 -\int_\Omega \chi(x,0) p(x,0)\,dx \\
&\quad +\int_\Omega \chi(x,\tau) p(x,\tau)\,dx
 -\int_\tau^0\int_\Omega wyp\,dx\,dt.
\end{aligned}
\end{equation}
By the weak solution formulation for problem
\eqref{chi_equation}--\eqref{chi_incondition}, from  \eqref{p_chi_equation2} we obtain
\begin{gather} \label{p_chi_equation3}
    \int_\Omega \big(y(x,0)-z_0(x)\big)\chi(x,0)\,dx
    =\int_\Omega p(x,\tau)\chi(x,\tau)\,dx
    -\int_\tau^0\int_\Omega pyw\,dx\,dt.
\end{gather}
Let us show that we can pass  to the limit in \eqref{p_chi_equation3}
as $t\to-\infty$. Indeed, according to \eqref{estim} and \eqref{chi_incondition},
we have
\begin{equation}\label{351}
\begin{aligned}
\int_\Omega|p(x,\tau)\chi(x,\tau)|\,dx
&\leq \|p(\cdot,\tau)|_{L^2(\Omega)}
\|\chi(\cdot,\tau)\|_{L^2(\Omega)} \\
&\leq e^{\omega\int _{0}^{\tau}\alpha(s)ds}\|p(\cdot,0)\|_{L^2(\Omega)}
\|\chi(\cdot,\tau)\|_{L^2(\Omega)} \\
&= \|p(\cdot,0)\|_{L^2(\Omega)}\gamma(\tau),
\end{aligned}
\end{equation}
where because of condition \eqref{chi_incondition}, the function
$\gamma(t):=e^{\omega\int _{0}^{\tau}\alpha(s)ds}\|\chi(\cdot,\tau)\|_{L^2(\Omega)}$,
$t\in S$, is such that
$\gamma(t)\to  0$ as $t\to -\infty$.


Condition (A2), Theorem \ref{thm1} (estimate \eqref{estimate2})  
and estimate \eqref{pestim1}, by the Cauchy-Schwarz inequality, imply 
\begin{gather*}
\int_\tau^0\int_\Omega |pyw|\,dx\,dt
\leq \Big(\int_\tau^0\int_\Omega [\alpha]^{-1}e^{-2\omega\int_{0}^{t}
\alpha(s)ds}|p(x,t)|^2dx\,dt\Big)^{1/2}, \\
\Big(\int_\tau^0\int_\Omega
\alpha e^{2\omega\int_{0}^{t}\alpha(s)ds}|y|^2dx\,dt\Big)^{1/2}
\leq C_2 C_{10}\|p(\cdot,0)\|_{L^2(\Omega)}^2 
 \|f\|_{L_{\omega,1/\alpha}^2(S;L^2(\Omega))},
\end{gather*}
which yields $wpy\in L^1(Q)$.

According to this and \eqref{351}, we pass to the limit in 
\eqref{p_chi_equation3} as $\tau\to-\infty$, and obtain
\begin{gather}  \label{p_chi_equation4}
    \int_\Omega \big(y(x,0)-z_0(x)\big)\chi(x,0)\,dx
    =-\iint_Q pyw\,dx\,dt.
\end{gather}
From \eqref{Jdiff4} taking into account \eqref{p_chi_equation4} 
we obtain \eqref{Jdiff5}.
\end{proof}


\subsection*{Acknowledgments} 
We want to thank the anonymous referees for the careful reading and 
their helpful suggestions.


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\end{document}