# Analysis and approximation to parabolic optimal control problems with
measure-valued controls in time

Wei Gong The State Key Laboratory of Scientific and Engineering Computing,
Institute of Computational Mathematics & National Center for Mathematics and
Interdisciplinary Sciences, Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, 100190 Beijing, China. Email address:
<EMAIL_ADDRESS>Dongdong Liang The Hong Kong Polytechnic University
Shenzhen Research Institute, Shenzhen 518057, China. Email address:
<EMAIL_ADDRESS>
###### Abstract

In this paper, we investigate an optimal control problem governed by parabolic
equations with measure-valued controls over time. We establish the well-
posedness of the optimal control problem and derive the first-order optimality
condition using Clarke’s subgradients, revealing a sparsity structure in time
for the optimal control. Consequently, these optimal control problems
represent a generalization of impulse control for evolution equations. To
discretize the optimal control problem, we employ the space-time finite
element method. Here, the state equation is approximated using piecewise
linear and continuous finite elements in space, alongside a Petrov-Galerkin
method utilizing piecewise constant trial functions and piecewise linear and
continuous test functions in time. The control variable is discretized using
the variational discretization concept. For error estimation, we initially
derive a priori error estimates and stabilities for the finite element
discretizations of the state and adjoint equations. Subsequently, we establish
weak-* convergence for the control under the norm
$\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$, with a convergence order of
$O(h^{\frac{1}{2}}+\tau^{\frac{1}{4}})$ for the state.

Keywords: Optimal control, parabolic equation, measure valued control, finite
element, error estimate

##  1 Introduction

Let $\Omega\subset{\mathbb{R}}^{d}$ ($d=2,3$) be a convex polyhedron with
boundary $\Gamma:=\partial\Omega$, and $I:=(0,T)$ with $T>0$. In this paper we
consider the following optimal control problem:

(1) $\displaystyle\min_{(u,q)\in
X\times\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}J(u,q)=\frac{1}{2}\|u-u_{d}\|_{L^{2}(I;L^{2}(\Omega))}^{2}+\frac{\beta}{2}\|u(T)-u_{T}\|_{L^{2}(\Omega)}^{2}+\alpha\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))},$

where $\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ is the control space of vector
measures that will be defined in the subsequent section, $X:=\\{v\in
L^{2}(I;L^{2}(\Omega)),\ v(T)\in L^{2}(\Omega)\\}$ is the observation space,
$u_{d}\in L^{2}(I;L^{2}(\Omega))$ and $u_{T}\in L^{2}(\Omega)$ are given
observations or target states, $\alpha>0$ is a regularization parameter,
$\beta\geq 0$ is a weight parameter. The state $u$ and the control
$q\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ in (1) are constrained by the
following parabolic equation with initial data $u_{0}\in L^{2}(\Omega)$ and
source $f\in L^{2}(I;L^{2}(\Omega))$:

(2) $\displaystyle\left\\{\begin{aligned} \partial_{t}u-\Delta
u&=f+\chi_{I_{c}\times\omega}q&&\mbox{in}\,\,\,\Omega\times I,\\\
u&=0&&\mbox{on}\,\,\Gamma\times I,\\\
u(0)&=u_{0}&&\mbox{in}\,\,\Omega,\end{aligned}\right.$

where $\omega\subset\Omega$ and $I_{c}\subset\subset I$ (relatively compact)
denote the spatial and temporal control domains, respectively. Here
$\chi_{I_{c}\times\omega}$ is the characteristic function of
$I_{c}\times\omega$ taking values $1$ in $I_{c}\times\omega$ and $0$
otherwise, which can be viewed as a zero extension operator.

In recent years, sparse controls of partial differential equations have
garnered significant attention. Initially motivated by actuator placement,
their application scope has since expanded to practical domains. Two main
approaches have emerged to achieve sparsity: one involves $L^{1}$-norm
regularization in the objective functional, while the other relies on measure-
valued controls. The pioneering work in this direction is [49], which
investigated $L^{1}(\Omega)$ control for linear elliptic equations.
Subsequently, [6] addressed the spatio-temporally sparse optimal control
problem of semilinear parabolic equations, introducing three different
sparsity-promoting terms in the objective functional: $L^{1}(\Omega\times I)$,
$L^{2}(I;L^{1}(\Omega))$, or $L^{1}(\Omega;L^{2}(I))$. For the latter case,
[13] provided an error estimate for its fully discrete finite element
approximation, with further refinement seen in [12]. Additionally, [32]
explored directional sparse control for parabolic equations, where controls
exhibit sparsity in space but not necessarily in time. Notably, the sparsity
pattern remains constant over time.

For measure-valued control problems, [4] investigates elliptic equations with
control space $\mathcal{M}(\Omega)$, while [5] addresses parabolic equations
in $L^{2}(I;\mathcal{M}(\Omega))$, providing error estimates for finite
element approximations. For parabolic control problems in space-time measure
$\mathcal{M}(I\times\Omega)$, [8] is a relevant reference. Extending the
directional sparsity concept ([32]) to measure spaces, [35] examines measure-
valued directional sparsity for parabolic control problems with control space
$\mathcal{M}(\Omega;L^{2}(I))$, deriving an a priori error estimate. Optimal
control of the linear second-order wave equation with measure-valued controls
in $\mathcal{M}(\Omega;L^{2}(I))$ is discussed in [36]. In [51], the authors
explore measure-valued optimal control problems for 1D wave equations with
control spaces of either measure-valued functions
$L_{w^{*}}^{2}(I;\mathcal{M}(\Omega))$ or vector measures
$\mathcal{M}(\Omega,L^{2}(I))$, deriving error estimates for the optimal state
variable and the error measured in the cost functional. Additionally, [31]
investigates a variational discretization of a parabolic optimal control
problem with space-time measure controls, employing a Petrov-Galerkin method
with piecewise constant states and piecewise linear and continuous test
functions in time for temporal discretization of the state equation.
References [14, 10, 53, 30] provide further insights into initial value
identification of parabolic equations in measure spaces.

For time-dependent systems, the control problem (1)-(2) posed in
$\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ yields controls with compact support
in time. This characteristic allows for determining the optimal moments for
control device actions, akin to a generalization of impulse control [1, 15,
21, 22, 46, 47, 52, 55, 56, 57, 58]. Recall that in impulse control problems,
the control $q$ in (1)-(2) is replaced by

$\displaystyle
q(x,t)=\sum\limits_{i=1}^{m}q_{i}(x)\otimes\delta_{\tau_{i}}(t),$

where $\delta_{\tau_{i}}$ denotes the Dirac delta measure concentrated at
$\tau_{i}\in I_{c}\subset\subset I$, and $q_{i}{\otimes}\delta_{\tau_{i}}$ are
linear functionals on space $C(\bar{I}_{c};L^{2}(\omega))$, with $q_{i}\in
L^{2}(\omega),i=1,2,\cdots,m$. Here, the impulse strengths
$q_{i},,i=1,2,\cdots,m,$ are optimized at prescribed time nodes
$\tau_{i}\in(0,T)$ [26]. However, in many cases, the interest lies in
optimizing both the time nodes and the impulse strengths. This motivation
leads to the formulation of the generalized impulse control problem as
described by (1)-(2). We remark that impulse control belongs to a class of
important control and has wide applications (see, for instance, [1, 56]). In
many cases impulse control is an interesting alternative to deal with systems
that cannot be acted on by means of continuous control inputs.

The contributions of this article are threefold. Firstly, we investigate the
well-posedness of both the state equation and the optimal control problem.
Additionally, we derive the first-order optimality condition, revealing that
the optimal control exhibits a sparsity structure independent of space.
Secondly, for the state equation approximation, we utilize piecewise linear
and continuous functions in space and a Petrov-Galerkin scheme from [20] in
time. Specifically, we employ piecewise constant trial functions and piecewise
linear and continuous test functions. We also employ a variational
discretization concept for the control. Lastly, we provide an a priori error
estimate for the finite element approximation of the control problem. Building
upon the a priori error and stability estimates for finite element
discretizations of the state and adjoint equations, we establish a convergence
order of $O(h^{\frac{1}{2}}+\tau^{\frac{1}{4}})$ for the approximation of the
state. Furthermore, we demonstrate weak-* convergence for the control under
the $\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ norm, with a convergence order of
$O(h+\tau^{\frac{1}{2}})$ for the discrete cost functional.

The remainder of this paper is organized as follows. Section 2 presents some
preliminary results, including the definition of very weak solutions to the
state equation, as well as the global and local regularity and weak-*
continuity of the state variable. In Section 3, we derive the first-order
optimality system and investigate the sparse structure and regularity of the
optimal control. Two discrete optimal control problems and their associated
optimality systems are provided in Section 4. Section 5 primarily focuses on
the error analysis for the optimal control problem.

##  2 Preliminaries

###  2.1 Notations for function spaces

Let $W^{k,p}(\Omega)$ $(k\in N_{+}\cup\\{0\\},\,1\leq p\leq\infty)$ be the
usual Sobolev space defined in $\Omega$ with the norm
$\|\cdot\|_{W^{k,p}(\Omega)}$. Note that $W_{0}^{k,p}(\Omega)$
$(H_{0}^{k}(\Omega))$ is the closed subspace of $W^{k,p}(\Omega)$
$(H^{k}(\Omega))$ with null-traces on the boundary $\Gamma$. We abbreviate it
by $H^{k}(\Omega):=W^{k,2}(\Omega)$ (resp.
$H_{0}^{k}(\Omega):=W_{0}^{k,2}(\Omega)$) $(k\geq 1)$ with norms
$\|\cdot\|_{H^{k}(\Omega)}$, and $L^{p}(\Omega):=W^{0,p}(\Omega)$ that is the
$p$-integrable function space in $\Omega$ with norms
$\|\cdot\|_{L^{p}(\Omega)}$. Particularly, $L^{2}(D)$ $(D=\Omega,\omega)$ is a
Hilbert space with inner products $(\cdot,\cdot)$ and norms $\|\cdot\|$. Let
${C}(\bar{I}_{c})$ be the Banach space consisting of continuous functions on
$\bar{I}_{c}$ equipped with the supremum norm $\|\cdot\|_{C(\bar{I}_{c})}$.
Let $\mathcal{M}(\bar{I}_{c})$ be the dual space of ${C}(\bar{I}_{c})$ that is
a Banach space under the norm

$\displaystyle\|v\|_{\mathcal{M}(\bar{I}_{c})}:=\sup\left\\{\int_{\bar{I}_{c}}wdv,\,\forall\omega\in{C}(\bar{I}_{c})\,,\,\|w\|_{{C}(\bar{I}_{c})}\leq
1\right\\}\quad\forall v\in\mathcal{M}(\bar{I}_{c}),$

which can be identified with the space of regular Borel measures in $I_{c}$.

For a given positive measure $\mu\in\mathcal{M}(\bar{I}_{b})$ the notation
$L^{p}(I_{b},\mu;L^{2}(\Omega))$ $(1\leq p)$ denotes the set of all functions
defined on a subset $I_{b}\subset I$ and valued in $L^{2}(\Omega)$, which is a
Banach space endowed with the norm

$\|v\|_{L^{p}(I_{b},\mu;L^{2}(\Omega))}:=\Big{(}\int_{I_{b}}\left\|v(t)\right\|^{p}d\mu(t)\Big{)}^{\frac{1}{p}}$

for $p<\infty$, and

$\|v\|_{L^{\infty}(I_{b},\mu;L^{2}(\Omega))}:=\mathop{\rm ess\ sup}_{t\in
I_{b}}\left\|v(t)\right\|$

for $p=\infty$. If $\mu$ is a Lebesgue measure, we abbreviate
$L^{p}(I_{b},\mu;L^{2}(\Omega))$ as $L^{p}(I_{b};L^{2}(\Omega))$.

The space $L^{2}(\omega;\mathcal{M}(\bar{I}_{c}))$, consisting of all weakly-*
measurable functions $q:\omega\rightarrow\mathcal{M}(\bar{I}_{c})$, is a
Banach space endowed with the norm

$\displaystyle\|q\|_{L^{2}(\omega;\mathcal{M}(\bar{I}_{c}))}:=\Big{(}\int_{\omega}\|q(x)\|^{2}_{\mathcal{M}(\bar{I}_{c})}dx\Big{)}^{1\over
2}\quad\forall q\in L^{2}(\omega;\mathcal{M}(\bar{I}_{c}))$

which can be identified with the dual of $L^{2}(\omega;{C}(\bar{I}_{c}))$,
where $L^{2}(\omega;{C}(\bar{I}_{c}))$ denotes the Banach space consisting of
all functions defined on $\omega$ and valued in ${C}(\bar{I}_{c})$ with the
norm

$\|v\|_{L^{2}(\omega;{C}(\bar{I}_{c}))}:=\Big{(}\int_{\omega}\|v(x)\|_{{C}(\bar{I}_{c})}^{2}dx\Big{)}^{\frac{1}{2}}\quad\forall
v\in L^{2}(\omega;{C}(\bar{I}_{c})).$

For any given Banach space $X$, e.g., $H^{1}_{0}(\Omega)$, $L^{2}(\Omega)$,
etc., the notation $C(\bar{I}_{b};X)$ denotes the set of all continuous
functions on $\bar{I}_{b}$ and valued in $X$, which is a Banach space under
the supremum norm

$\|v\|_{C(\bar{I}_{b};X)}:=\mathop{\rm
sup}\limits_{t\in\bar{I}_{b}}\left\|v(t)\right\|_{X}\quad\forall v\in
C(\bar{I}_{b};X).$

Then, we define $\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ as the space
containing all countably additive measures with bounded total variations
defined on the Borel sets $\mathcal{B}(\bar{I}_{c})$ and valued in
$L^{2}(\omega)$. For any $\mu\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$, the
variation measure $|\mu|\in\mathcal{M}(\bar{I}_{c})$ is defined as

$\displaystyle|\mu|(B):=\sup\Big{\\{}\sum\limits_{n=1}^{\infty}\|\mu(B_{n})\|_{L^{2}(\omega)}:\
\\{B_{n}\\}_{n=1}^{\infty}\subset\mathcal{B}(\bar{I}_{c})\ \mbox{is\ the\
disjoint\ partition\ of}\ B\Big{\\}}$

for any $B\in\mathcal{B}(\bar{I}_{c})$, where $\mathcal{B}(\bar{I}_{c})$
denotes the Borel set on $\bar{I}_{c}$. We denote by $|\mu|(\bar{I}_{c})$ the
total variation of $\mu$. The space $\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$
endowed with the norm
$\|\mu\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}=\||\mu|\|_{\mathcal{M}(\bar{I}_{c})}=|\mu|(\bar{I}_{c})$
is a Banach space (cf. [35] and [37, Chap 12, Sec. 3]), and that can be
identified to the dual of ${C}(\bar{I}_{c};L^{2}(\omega))$. In the following
we denote by $\langle\cdot,\cdot\rangle$ the duality pairing between
$\mathcal{M}(\bar{I}_{c})$ and ${C}(\bar{I}_{c})$,
$\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ and ${C}(\bar{I}_{c};L^{2}(\omega))$,
respectively.

For each $\mu\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$, the polar
decomposition of $\mu$ consisting of the variation measure
$|\mu|\in\mathcal{M}(\bar{I}_{c})$ and a space-time function $\mu^{\prime}\in
L^{1}(I_{c},|\mu|;L^{2}(\omega))$, where the temporal support of
$\mu^{\prime}$ is included in the support of $|\mu|$ (cf. [35, 37]), such that

$\displaystyle d\mu=\mu^{\prime}d|\mu|(t)\quad\mbox{\rm
and}\quad\langle\mu,w\rangle_{\bar{I}_{c}\times\omega}=\int_{I_{c}}(\mu^{\prime}(t),w(t))d|\mu|(t)$

for any $\mu\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ and
$w\in{C}(\bar{I}_{c};L^{2}(\omega))$. Furthermore, similar to equation (2.3)
in [35] we can show that $\mu^{\prime}\in
L^{\infty}(I_{c},|\mu|;L^{2}(\omega))$ with
$\|\mu^{\prime}\|_{L^{\infty}(I_{c},|\mu|;L^{2}(\omega))}\leq 1$ and (cf. [35,
37])

(3) $\displaystyle\|\mu^{\prime}(t)\|_{L^{2}(\omega)}=1\quad\mbox{for}\
|\mu|-\mbox{almost\ all}\ t\in\bar{I}_{c}.$

######  Remark 2.1.

Since
$L^{2}(\omega;{C}(\bar{I}_{c}))\hookrightarrow{C}(\bar{I}_{c};L^{2}(\omega))$
by Minkowski’s inequality, we have
$\mathcal{M}(\bar{I}_{c};L^{2}(\omega))\hookrightarrow
L^{2}(\omega;\mathcal{M}(\bar{I}_{c}))$. The difference is that
$L^{2}(\omega;\mathcal{M}(\bar{I}_{c}))$ yields measure valued functions whose
temporal supports are spatial dependent, while the ones for the former are
spatial independent (cf. [35]). Based on the above embedding, we see that for
each $\mu\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$, the representation
$\mu(x)\in\mathcal{M}(\bar{I}_{c})$ is well-defined for almost all
$x\in\omega$. Since $\mu^{\prime}\in L^{\infty}(I_{c},|\mu|;L^{2}(\omega))$,
it thus also belongs to $L^{2}(I_{c},|\mu|;L^{2}(\omega))$, or equivalently,
$L^{2}(\omega;L^{2}(I_{c},|\mu|))$. Therefore, $\mu^{\prime}(x)\in
L^{2}(I_{c},|\mu|)$ for a.e. $x\in\omega$. We can now write (cf. [35, eq.
(2.5)])

(4) $\displaystyle d\mu(x)=\mu^{\prime}(x)d|\mu|\quad\mbox{ a.e.}\
x\in\omega.$

######  Lemma 1.

For given $g\in L^{2}(I;H^{-1}(\Omega))$ and $z_{T}\in L^{2}(\Omega)$, there
exists a unique solution $z\in L^{2}(I;H^{1}_{0}(\Omega))\cap
H^{1}(I;H^{-1}(\Omega))$ to the following problem

(5) $\displaystyle\left\\{\begin{aligned} -\partial_{t}z-\Delta
z&=g&&\mbox{in}\,\,\,\Omega\times(0,T),\\\
z&=0&&\mbox{on}\,\,\Gamma\times(0,T),\\\
z(T)&=z_{T}&&\mbox{in}\,\,\Omega.\end{aligned}\right.$

Moreover, the following estimate holds

(6)
$\|z\|_{C(\bar{I};L^{2}(\Omega))}+\|\partial_{t}z\|_{L^{2}(I;H^{-1}(\Omega))}+\|z\|_{L^{2}(I;H^{1}_{0}(\Omega))}\leq
C\left(\|g\|_{L^{2}(I;H^{-1}(\Omega))}+\|z_{T}\|_{L^{2}(\Omega)}\right).$

If, in addition, $g\in L^{2}(I;L^{2}(\Omega))\ \mbox{\rm and}\ z_{T}\in
H^{1}_{0}(\Omega)$, then $z\in L^{2}(I;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))\cap\,H^{1}(I;L^{2}(\Omega))\cap\,C(\bar{I};H^{1}_{0}(\Omega))$
and there holds the stability estimate

(7)
$\|z\|_{C(\bar{I};H^{1}_{0}(\Omega))}+\|\partial_{t}z\|_{L^{2}(I;L^{2}(\Omega))}+\|z\|_{L^{2}(I;H^{2}(\Omega))}\leq
C\left(\|g\|_{L^{2}(I;L^{2}(\Omega))}+\|z_{T}\|_{H^{1}(\Omega)}\right).$

The constants $C>0$ is independent of $g$ and $z_{T}$ in the above two
stability estimates.

###### Proof.

The proof of the existence, uniqueness and stability estimate (6) of solutions
can be found in [24, Theorem 5.1]. The improved regularity and stability
estimate are classical; see, e.g., [25]. The regularity $z\in
C(\bar{I};H^{1}_{0}(\Omega))$ can be obtained from the fact that
$L^{2}(I;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\cap
H^{1}(I;L^{2}(\Omega))\hookrightarrow C(\bar{I};H^{1}_{0}(\Omega))$. ∎

###  2.2 Well-posedness of the state equation

To begin with, we first investigate the well-posedness of the state equation
(2). The very weak solution of equation (2) can be defined by transposition
techniques (cf. [41]), which will be given in the following.

######  Definition 2.

For any given $f\in L^{2}(I;L^{2}(\Omega))$,
$q\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ and $u_{0}\in L^{2}(\Omega)$, a
function $u\in L^{2}(I;L^{2}(\Omega))$ is called the very weak solution of
equation (2) if it satisfies

(8)
$(u,g)_{L^{2}(I;L^{2}(\Omega))}=\int_{I}(f,z_{g})_{L^{2}(\Omega)}dt+\langle
q,z_{g}\rangle_{\bar{I}_{c}\times\omega}+(u_{0},z_{g}(0))_{L^{2}(\Omega)}\quad\forall
g\in L^{2}(I;L^{2}(\Omega)),$

where $z_{g}\in C(\bar{I}_{c};L^{2}(\omega))$ is the solution to equation (5)
with the right-hand side $g$ and $z_{T}=0$, and
$\langle\cdot,\cdot\rangle_{\bar{I}_{c}\times\omega}$ denotes the duality
pairing between $\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ and
$C(\bar{I}_{c};L^{2}(\omega))$.

Since $z_{T}=0$ and $\Omega$ is convex, the solution of equation (5) satisfies
$z\in L^{2}(I;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\cap
H^{1}(I;L^{2}(\Omega))\hookrightarrow C(\bar{I};H_{0}^{1}(\Omega))$ by Lemma
1. Therefore, the above definition is well-defined.

######  Theorem 3.

Assume that $f\in L^{2}(I;L^{2}(\Omega))$, $u_{0}\in L^{2}(\Omega)$ and
$q\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$, then the parabolic equation (2)
admits a unique very weak solution $u\in L^{2}(I;L^{2}(\Omega))$. Furthermore,
$u\in L^{2}(I;H_{0}^{1}(\Omega))\cap L^{\infty}(I;L^{2}(\Omega))$ and there
holds the following estimate:

(9)
$\displaystyle\|u\|_{L^{2}(I;H_{0}^{1}(\Omega))}+\|u\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}+\|u_{0}\|_{L^{2}(\Omega)}),$

where $C>0$ is a constant independent of $f$, $q$ and $u_{0}$.

In addition, assume that $I_{c}=(t_{1},t_{2})$ with $0<t_{1}<t_{2}<T$, then
there exist $\hat{t}$ and $\tilde{t}$ satisfying $t_{2}<\hat{t}<\tilde{t}<T$,
i.e., $(\tilde{t},T)\subseteq(\hat{t},T)\subseteq(t_{2},T)$, such that
$u|_{(\tilde{t},T)}\in H^{1}((\tilde{t},T);L^{2}(\Omega))\cap
L^{2}((\tilde{t},T);H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\hookrightarrow
C([\tilde{t},T];H^{1}_{0}(\Omega))$ and

(10) $\begin{split}&\|{u}\|_{L^{2}((\tilde{t},T);H^{2}(\Omega)\cap
H_{0}^{1}(\Omega))}+\|{u}\|_{H^{1}((\tilde{t},T);L^{2}(\Omega))}+\|u\|_{C([\tilde{t},T];H^{1}_{0}(\Omega))}\\\
\leq&C(\|f\|_{L^{2}((\hat{t},T);L^{2}(\Omega))}+\|u\|_{L^{2}((\hat{t},T);L^{2}(\Omega))})\\\
\leq&C(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}+\|u_{0}\|_{L^{2}(\Omega)}),\end{split}$

where $C>0$ is a constant independent of $f$, $q$ and $u_{0}$.

###### Proof.

The proof of the existence of a unique very weak solution $u\in
L^{2}(I;H_{0}^{1}(\Omega))\cap L^{\infty}(I;L^{2}(\Omega))$ can be found in,
e.g., [27], [26, Theorem 2.2], [29, Theorem 2.4] or [42, Theorem 2.4] for
measure data in $\mathcal{M}([0,T])$. Here we include a brief proof for
completeness.

Since the state equation is linear, it suffices to consider the case either
$u_{0}=0$, $f=0$ or $q=0$. If $q=0$, $u_{0}\in L^{2}(\Omega)$ and $f\in
L^{2}(I;L^{2}(\Omega))$, it is obvious that problem (2) admits a unique weak
solution $u\in L^{2}(I;H_{0}^{1}(\Omega))\cap L^{\infty}(I;L^{2}(\Omega))$
satisfying (cf. [40, 50])

$\displaystyle\|u\|_{L^{2}(I;H_{0}^{1}(\Omega))}+\|u\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{0}\|_{L^{2}(\Omega)}).$

Now we consider the case $u_{0}=0$ and $f=0$. Let
$\\{q_{n}\\}_{n}\subset{C}(\bar{I}_{c}\times\bar{\omega})$ be the sequence
converging weakly to $q$ in $\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ and
satisfy

$\displaystyle\|q_{n}\|_{L^{1}(I_{c};L^{2}(\omega))}\leq\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}.$

Let $u_{n}$ be the solution of

(11) $\displaystyle\left\\{\begin{aligned} \partial_{t}u_{n}-\Delta
u_{n}&=\chi_{I_{c}\times\omega}q_{n}&&\mbox{in}\,\,\,\Omega\times(0,T],\\\
u_{n}&=0&&\mbox{on}\,\,\partial\Omega\times(0,T],\\\
u_{n}|_{t=0}&=0&&\mbox{in}\,\,\Omega,\end{aligned}\right.$

then one has $u_{n}\in L^{2}(I;H_{0}^{1}(\Omega))\cap
H^{1}(I;H^{-1}(\Omega))$. Let $z$ be the solution of problem (5) for given
$g\in\mathcal{D}(I\times\Omega)$ and $z_{T}=0$, it follows from Lemma 1 that
$z\in{C}(\bar{I};H_{0}^{1}(\Omega))$. Then, using integration by parts we
obtain

(12) $\displaystyle\int_{I}\int_{\Omega}gu_{n}dxdt$ $\displaystyle=$
$\displaystyle\int_{I}\int_{\Omega}(-\partial_{t}z-\Delta z)u_{n}dxdt$
$\displaystyle=$ $\displaystyle\int_{I_{c}}(q_{n},z)_{L^{2}(\omega)}dt$
$\displaystyle\leq$
$\displaystyle\|q_{n}\|_{L^{1}(I_{c};L^{2}(\omega))}\|z\|_{L^{\infty}(I_{c};L^{2}(\omega))}$
$\displaystyle\leq$
$\displaystyle\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\|z\|_{L^{\infty}(I_{c};L^{2}(\omega))}.$

Combining the following standard estimates (cf. [3, 26]):

$\displaystyle\|z\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C\|g\|_{L^{1}(I;L^{2}(\Omega))},\quad\|z\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C\|g\|_{L^{2}(I;H^{-1}(\Omega))},$

we conclude that $\\{u_{n}\\}_{n}$ is bounded in the space
$L^{\infty}(I;L^{2}(\Omega))$ by setting
$g:=\psi_{0}\in\mathcal{D}(I\times\Omega)$ and using the density of
$\mathcal{D}(I\times\Omega)$ in $L^{1}(I;L^{2}(\Omega))$, and also bounded in
$L^{2}(I;H_{0}^{1}(\Omega))$ by setting
$g:=\psi_{0}-\frac{\partial\psi_{j}}{\partial x_{j}}$,
$\psi_{j}\in\mathcal{D}(I\times\Omega)$, $j=1,\dots,d$ and using the density
of $\mathcal{D}(I\times\Omega)$ in $L^{2}(I;H^{-1}(\Omega))$ (cf. [5, 8]),
respectively. Thus, we can extract a subsequence, still denoted by
$\\{u_{n}\\}_{n}$, such that $u_{n}\rightarrow u$ weakly in
$L^{2}(I;H_{0}^{1}(\Omega))$ and $L^{\infty}(I;L^{2}(\Omega))$.

For any $g\in L^{2}(I;L^{2}(\Omega))$, let $z_{g}\in
H^{1}(I;L^{2}(\Omega))\cap L^{2}(I;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))$ be
the solution of equation (5) with $z_{T}=0$. Multiplying by $z_{g}$ in both
sides of equation (11) and integrating by parts give

$\displaystyle(u_{n},g)_{L^{2}(I;L^{2}(\Omega))}=\int_{I_{c}}\int_{\omega}z_{g}(x,t)dxdq_{n}(t),$

which yields the identity (8) with $f=0,\,u_{0}=0$ by passing to the limit in
the above identity. Therefore, $u$ is the very weak solution of equation (2).
By the weak lower semicontinuity of $\|\cdot\|_{L^{2}(I;H^{1}_{0}(\Omega))}$
and $\|\cdot\|_{L^{\infty}(I;L^{2}(\Omega))}$, we can obtain the estimate (9)
of $u$.

Since $I_{c}=(t_{1},t_{2})\subseteq(0,T)$, there exist $\hat{t}$ and
$\tilde{t}$ satisfying $t_{2}<\hat{t}<\tilde{t}<T$, such that
$(\tilde{t},T)\subseteq(\hat{t},T)\subseteq(t_{2},T)$. Therefore, we consider
a smooth cut-off function $\tilde{\omega}$ with the following properties:

$\displaystyle\tilde{\omega}(t)\in[0,1]\quad\forall
t\in[0,T];\quad\tilde{\omega}(t)=1\quad\forall
t\in(\tilde{t},T);\quad\tilde{\omega}(t)=0\quad\forall t\in(0,\hat{t}].$

Let $\tilde{u}:=\tilde{\omega}u$. Since $\bar{I}_{c}\cap{\rm
supp}\,\tilde{u}=\emptyset$, $\tilde{u}$ satisfies the following equation:

(13) $\displaystyle\left\\{\begin{aligned}
\partial_{t}\tilde{u}-\Delta\tilde{u}&=F&&\mbox{in}\,\,\,\Omega\times(\hat{t},T),\\\
\tilde{u}&=0&&\mbox{on}\,\,\Gamma\times(\hat{t},T),\\\
\tilde{u}(\hat{t})&=0&&\mbox{in}\,\,\Omega,\end{aligned}\right.$

where $F:=\partial_{t}\tilde{\omega}u+\tilde{\omega}f$. Since $F\in
L^{2}((\hat{t},T);L^{2}(\Omega))$, we can obtain $\tilde{u}\in
L^{2}((\hat{t},T);H^{2}(\Omega)\cap H_{0}^{1}(\Omega))\cap
H^{1}((\hat{t},T);L^{2}(\Omega))$ and there holds the following estimate:

$\begin{split}\|\tilde{u}\|_{L^{2}((\hat{t},T);H^{2}(\Omega)\cap
H_{0}^{1}(\Omega))}+\|\tilde{u}\|_{H^{1}((\hat{t},T);L^{2}(\Omega))}&\leq
C\|F\|_{L^{2}((\hat{t},T);L^{2}(\Omega))}\\\ &\leq
C(\|f\|_{L^{2}((\hat{t},T);L^{2}(\Omega))}+\|u\|_{L^{2}((\hat{t},T);L^{2}(\Omega))})\\\
&\leq
C(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}+\|u_{0}\|_{L^{2}(\Omega)}),\end{split}$

where we have used the estimate (9). From the above inequality we obtain

$\displaystyle\|{u}\|_{L^{2}((\tilde{t},T);H^{2}(\Omega)\cap
H_{0}^{1}(\Omega))}+\|{u}\|_{H^{1}((\tilde{t},T);L^{2}(\Omega))}$
$\displaystyle=$
$\displaystyle\|\tilde{u}\|_{L^{2}((\tilde{t},T);H^{2}(\Omega)\cap
H_{0}^{1}(\Omega))}+\|\tilde{u}\|_{H^{1}((\tilde{t},T);L^{2}(\Omega))}$
$\displaystyle\leq$
$\displaystyle\|\tilde{u}\|_{L^{2}((\hat{t},T);H^{2}(\Omega)\cap
H_{0}^{1}(\Omega))}+\|\tilde{u}\|_{H^{1}((\hat{t},T);L^{2}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}+\|u_{0}\|_{L^{2}(\Omega)}).$

Therefore, we complete the proof of the estimate (10). ∎

With the help of Theorem 3, the identity (8) in Definition 2 is equivalent to
the following one

(14)
$\displaystyle(u,g)_{*}+(u(T),z_{T})_{L^{2}(\Omega)}=\int_{I}(f,z)_{L^{2}(\Omega)}dt+\langle
q,z\rangle_{\bar{I}_{c}\times\omega}+(u_{0},z(0))_{L^{2}(\Omega)}$

for any $(g,z_{T})\in S\times L^{2}(\Omega)$, where
$(u,g)_{*}:=(u,g)_{L^{2}(I;L^{2}(\Omega))}$ for $S:=L^{2}(I;L^{2}(\Omega))$
and $(u,g)_{*}:=\langle
g,u\rangle_{L^{2}(I;H^{-1}(\Omega)),L^{2}(I;H_{0}^{1}(\Omega))}$ defined by

(15) $\langle
g,u\rangle_{L^{2}(I;H^{-1}(\Omega)),L^{2}(I;H_{0}^{1}(\Omega))}:=\int_{I}\langle
u,-\partial_{t}z\rangle_{H^{1}(\Omega),H^{-1}(\Omega)}+(\nabla u,\nabla
z)_{L^{2}(\Omega),L^{2}(\Omega)}dt$

for $S:=L^{2}(I;H^{-1}(\Omega))$, where $z\in C(\bar{I}_{c};L^{2}(\omega))$
satisfies (5) with the right-hand side $g$ and $z_{T}\in L^{2}(\Omega)$.

In fact, taking $\hat{z}=z-\tilde{z}$, where $\tilde{z}$ is the solution of
equation (5) with $g=0$ and initial data $\tilde{z}(T)=z(T)=z_{T}$, then
$\hat{z}$ satisfies (5) with the right-hand side $g$ and initial data
$z_{T}=0$. In other words, $\hat{z}$ can be chosen as a test function in
Definition 2, i.e,

$\displaystyle(u,g)_{L^{2}(I;L^{2}(\Omega))}=$
$\displaystyle\int_{I}(f,\hat{z})_{L^{2}(\Omega)}dt+\langle
q,\hat{z}\rangle_{\bar{I}_{c}\times\omega}+(u_{0},\hat{z}(0))_{L^{2}(\Omega)}$
$\displaystyle=$ $\displaystyle\int_{I}(f,{z})_{L^{2}(\Omega)}dt+\langle
q,{z}\rangle_{\bar{I}_{c}\times\omega}+(u_{0},{z}(0))_{L^{2}(\Omega)}$
$\displaystyle-\left(\int_{I}(f,\tilde{z})_{L^{2}(\Omega)}dt+\langle
q,\tilde{z}\rangle_{\bar{I}_{c}\times\omega}+(u_{0},\tilde{z}(0))_{L^{2}(\Omega)}\right)$
$\displaystyle=$ $\displaystyle\int_{I}(f,{z})_{L^{2}(\Omega)}dt+\langle
q,{z}\rangle_{\bar{I}_{c}\times\omega}+(u_{0},{z}(0))_{L^{2}(\Omega)}-\mathscr{L}(z_{T}),$

where $\mathscr{L}(z_{T}):=\int_{I}(f,\tilde{z})_{L^{2}(\Omega)}dt+\langle
q,\tilde{z}\rangle_{\bar{I}_{c}\times\omega}+(u_{0},\tilde{z}(0))_{L^{2}(\Omega)}$.
It is easy to check that $\mathscr{L}$ is a bounded linear functional of
$z_{T}\in L^{2}(\Omega)$. Therefore, there exists a unique $\theta\in
L^{2}(\Omega)$ such that $\mathscr{L}(z_{T})=(\theta,z_{T})_{L^{2}(\Omega)}$
by the Riesz representation theorem. Obviously, $\theta=u(T)$. Then, the
identity (14) holds.

In order to show that the optimal control problem (1) has a unique solution,
we have to provide a continuity property of the control-to-observation mapping
under the weak-* topology.

######  Proposition 4.

Let
$\\{q_{n}\\}_{n\in\mathbb{N}_{+}}\subset\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$
be a sequence of control variables such that
$q_{n}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}q$ in
$\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$. Assume that $u_{n}:=u(q_{n})$ and
$u:=u(q)$ are the corresponding solutions to the state equation (2) associated
with $q_{n}$ and $q$, respectively. Then we have

$\displaystyle\|u_{n}-u\|_{L^{2}(I;L^{2}(\Omega))}\rightarrow 0\ \ \mbox{\rm
and}\ \ \|u_{n}(\cdot,T)-u(\cdot,T)\|_{H^{1}(\Omega)}\rightarrow
0\quad\mbox{\rm for}\ n\to\infty.$

###### Proof.

The main idea of the proof is to apply the definition of very weak solutions
to $u-u_{n}$. To this end, we note that $u-u_{n}$ satisfies the equation (2)
with $f=0$, $u_{0}=0$, and $q$ replaced by $q-q_{n}$. Therefore, taking any
$g\in L^{2}(I;H^{-1}(\Omega))$ and $z_{T}=0$, let $z\in
C(\bar{I};L^{2}(\Omega))$ be the solution of equation (5). Using (14) there
holds

(16) $\displaystyle\langle
g,u-u_{n}\rangle_{L^{2}(I;H^{-1}(\Omega)),L^{2}(I;H_{0}^{1}(\Omega))}=\langle
q-q_{n},z\rangle_{\bar{I}_{c}\times\omega}\to 0\quad\mbox{\rm as}\ \
n\to\infty,$

which implies that $u_{n}\rightharpoonup u$ in $L^{2}(I;H^{1}_{0}(\Omega))$.
Furthermore, we obtain $\|u-u_{n}\|_{L^{2}(I;L^{2}(\Omega))}\rightarrow 0$ by
the compact embedding $L^{2}(I;H^{1}_{0}(\Omega))\hookrightarrow
L^{2}(I;L^{2}(\Omega))$. This proves the first statement.

Below we will verify the second statement. Since $u-u_{n}$ satisfies the
equation (2) with $f,\,u_{0}$ and $q$ replaced by $0$, $0$, $q-q_{n}$,
respectively, then applying the estimate (10) in Theorem 3 to $u-u_{n}$ yields
the following estimate:

$\displaystyle\|(u-u_{n})(T)\|_{H^{1}(\Omega)}\leq
C\|u-u_{n}\|_{L^{2}((\hat{t},T);L^{2}(\Omega))}\leq
C\|u-u_{n}\|_{L^{2}(I;L^{2}(\Omega))}\to 0\quad\mbox{\rm as}\ \ n\to\infty,$

where we have used the first statement. This finishes the proof. ∎

##  3 Optimal control problems

With the above preparations, we are in the position to study the existence and
uniqueness of solutions to the optimal control problem (1)-(2), and derive the
first order optimality system and regularity results of the solution.

###  3.1 Well-posedness of the optimal control problem

Recall that $X:=L^{2}(I;L^{2}(\Omega))\times L^{2}(\Omega)$ is the observation
space, then we introduce the control-to-observation operator
$S:\mathcal{M}(\bar{I}_{c};L^{2}(\omega))\to X$ as

$Sq:=(S_{1}q,S_{2}q),$

where $S_{1}q:=u_{q}$ and $S_{2}q:=u_{q}(T)$, and $u_{q}$ solves equation (2)
with the control variable $q$ on the right-hand side. Theorem 3 and
Proposition 4 imply that the operator $S$ is well-defined, affine linear and
bounded, and weak continuous under the weak-∗ topology in
$\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$. Obviously, the operator $S$ is
injective since $I_{c}\subseteq I$. With the help of the control-to-
observation operator $S$ the reduced cost functional of (1) can be defined as

(17) $j(q):=J_{1}(q)+J_{2}(q)\quad\forall
q\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega)),$

where

$J_{1}(q):=\frac{1}{2}\|S_{1}q-u_{d}\|^{2}_{L^{2}(I;L^{2}(\Omega))}+\frac{\beta}{2}\|S_{2}q-u_{T}\|^{2}_{L^{2}(\Omega)},\quad\
J_{2}(q):=\alpha\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}.$

$J_{1}(q)$ is a quadratic functional of tracking type, which is continuous
under the weak-∗ topology in $\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ and
strictly convex by the weak-∗ continuity and injection of the operator $S$. On
the other hand, $J_{2}(q)$ is weakly-∗ lower semicontinuous in space
$\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$. Therefore, we conclude that the
reduced functional $j$ is also weakly-∗ lower semicontinuous and strictly
convex. With this observation we can provide the following result.

######  Theorem 5.

The optimal control problem (1)-(2) admits a unique solution
$(\bar{u},\bar{q})\in X\times\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$, where
$\bar{q}$ is an optimal control that minimizes the reduced cost functional
$\eqref{reduced_j}$ and $\bar{u}$ is the optimal state that solves the state
equation (2) associated with $\bar{q}$.

###### Proof.

According to Theorem 3, the objective functional $j$ is well defined on
$\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$. For the existence of solutions, we
follow the standard arguments. Since $j\geq 0$ is bounded from below on
$\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$, we can find a minimizing sequence
$\\{q_{n}\\}$ with

$\lim_{n\to\infty}j(q_{n})=\mathop{\rm
inf}_{q\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}j(q)=j^{*}\quad{\rm
and}\quad\|q_{n}\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\leq\frac{1}{\alpha}j(q_{n})\leq
C.$

On the other hand, the predual space ${C}(\bar{I}_{c};L^{2}(\omega))$ is
separable, then the bounded set in $\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ is
weakly-∗ compact by the Banach-Alaoglu theorem. Hence, we can extract a
weakly-∗ convergent subsequence, still denoted by $\\{q_{n}\\}$, such that
$q_{n}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\bar{q}$ in
$\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ for some
$\bar{q}\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$. Let $u_{n}$ and $\bar{u}$
be the state corresponding to $q_{n}$ and $\bar{q}$, respectively. Then
$u_{n}\to\bar{u}$ in $X$ since the operator $S$ is weakly-∗ continuous. It is
easy to check that $(\bar{u},\bar{q})$ is an optimal pair. In fact, $j$ is
weakly-∗ lower semicontinuous, then

$j(\bar{q})\leq\mathop{\lim\inf}_{n\to\infty}\ j(q_{n})=j^{*},$

which means that $\bar{q}$ is optimal, i.e., $(\bar{u},\bar{q})$ is an optimal
pair.

Furthermore, the control-to-observation mapping $S$ is injective, thus the
objective functional $j$ is strictly convex. Therefore, the optimal pair
$(\bar{u},\bar{q})$ is unique. ∎

######  Remark 3.1.

As pointed out in [35], if the state observation is of the form
$\chi_{I_{o}\times\Omega_{o}}(u-u_{d})\in L^{2}(I_{o};L^{2}(\Omega_{o}))$ with
${\rm dist}(I_{o},I_{c})>0$ and $\beta=0$ where $I_{o}\subset I$ is an
observation time window. The objective functional is no longer strictly convex
since the control-to-observation operator is not injective, and thus the
optimal pair of the optimization problem (1)-(2) is not unique. However, in
the current paper we do not consider this case and focus only on the situation
of $I_{o}=I$, i.e., $I_{c}\subset I_{o}$.

###  3.2 First order optimality system

Below, we are in the position to derive the first order optimality condition.

######  Theorem 6.

A control $\bar{q}\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ and an associated
state $\bar{u}\in L^{2}(I;H^{1}_{0}(\Omega))\cap L^{\infty}(I;L^{2}(\Omega))$
are an optimal pair of the optimal control problem (1)-(2), if and only if
there exists an adjoint state $\bar{\varphi}\in L^{2}(I;H_{0}^{1}(\Omega))\cap
H^{1}(I;L^{2}(\Omega))\hookrightarrow C(\bar{I};L^{2}(\Omega))$ satisfying

(18) $\displaystyle\left\\{\begin{aligned}
-\partial_{t}\bar{\varphi}-\Delta\bar{\varphi}&=\bar{u}-u_{d}&&\mbox{in}\,\,\,\Omega\times(0,T),\\\
\bar{\varphi}&=0&&\mbox{on}\,\,\Gamma\times(0,T),\\\
\bar{\varphi}(T)&=\beta(\bar{u}(T)-u_{T})&&\mbox{in}\,\,\Omega,\end{aligned}\right.$

where $u_{T}\in L^{2}(\Omega)$, $u_{d}\in L^{2}(I;L^{2}(\Omega))$, such that
the following subgradient condition holds:

(19)
$0\in\bar{\varphi}|_{\bar{I}_{c}\times\omega}+\alpha\mathscr{\partial}\|\cdot\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}(\bar{q})\quad\mbox{\rm
in}\ (\mathcal{M}(\bar{I}_{c};L^{2}(\omega)))^{*}$

i.e.,

(20) $-\langle
p-\bar{q},\bar{\varphi}\rangle_{\bar{I}_{c}\times\omega}+\alpha\|\bar{q}\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\leq\alpha\|p\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\quad\forall
p\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega)),$

where
$\mathscr{\partial}\|\cdot\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}(\bar{q})$
denotes the set of subgradients of
$\|\cdot\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}$ at $\bar{q}$, which is
nonempty since $\|\cdot\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}$ is a
convex functional on $\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$.

Furthermore, from the condition (20) we can easily conclude the following
relation between the optimal control $\bar{q}$ and the adjoint state
$\bar{\varphi}$:

(21)
$\displaystyle\alpha\|\bar{q}\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}+\langle\bar{q},\bar{\varphi}\rangle_{\bar{I}_{c}\times\omega}$
$\displaystyle=0,$ (22)
$\displaystyle\|\bar{\varphi}\|_{{C}(\bar{I}_{c};L^{2}(\omega))}\left\\{\begin{aligned}
&=\alpha&&\mbox{if}\,\,\bar{q}\neq 0,\\\
&\leq\alpha&&\mbox{if}\,\,\bar{q}=0.\end{aligned}\right.$

###### Proof.

We split the reduced cost functional $j$ into the sum of two parts in (17),
where $J_{1}$ is differentiable and $J_{2}$ is subdifferentiable. We use
$J_{1}^{\prime}(\bar{q})$ and $\mathscr{\partial}J_{2}(\bar{q})$ to denote the
Fréchet derivative of $J_{1}$ at $\bar{q}$ and subgradients of $J_{2}$ at
$\bar{q}$, respectively. By the calculus rules of subdifferentials for convex
functions, there holds (see, e.g., [23, Section 5.3]):

(23) $j(\bar{q})=\\!\\!\\!\mathop{\rm
min}_{p\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}j(p)\quad\mbox{\rm if and
only if}\quad 0\in J_{1}^{\prime}(\bar{q})+\mathscr{\partial}J_{2}(\bar{q}),$

where for any $p\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$,
$J_{1}^{\prime}(\bar{q})$ and $\mathscr{\partial}J_{2}(\bar{q})$ satisfy

(24)
$\begin{split}J_{1}^{\prime}(\bar{q})(p-\bar{q})=&(\bar{u}-u_{d},u_{p}-\bar{u})_{L^{2}(I;L^{2}(\Omega))}+\beta(\bar{u}(T)-u_{T},u_{p}(T)-\bar{u}(T)),\\\
\forall\xi\in\mathscr{\partial}J_{2}(\bar{q}),\quad&\langle\xi,p-\bar{q}\rangle_{(\mathcal{M}(\bar{I}_{c};L^{2}(\omega)))^{*},\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\leq
J_{2}({p})-J_{2}(\bar{q}),\end{split}$

respectively, where $(\mathcal{M}(\bar{I}_{c};L^{2}(\omega)))^{*}$ denotes the
topological dual of the space $\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ and
$\langle\cdot,\cdot\rangle_{(\mathcal{M}(\bar{I}_{c};L^{2}(\omega)))^{*},\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}$
denotes the duality pairing between
$(\mathcal{M}(\bar{I}_{c};L^{2}(\omega)))^{*}$ and
$M(\bar{I}_{c};L^{2}(\omega))$, $u_{p}$ is the solution of problem (2) with
$q$ replaced by $p$. In order to give an explicit representation of
$J_{1}^{\prime}(\bar{q})(p-\bar{q})$ with respect to $p-\bar{q}$, let
$\bar{\varphi}$ be the solution of equation (5) with $g=\bar{u}-u_{d},\
z_{T}=\beta(\bar{u}(T)-u_{T})$, and then apply the identity (14) to the
difference $u_{p}-\bar{u}$ to deduce

(25)
$\displaystyle(\bar{u}-u_{d},u_{p}-\bar{u})_{L^{2}(I;L^{2}(\Omega))}+\beta(\bar{u}(T)-u_{T},u_{p}(T)-\bar{u}(T))=\langle
p-\bar{q},\bar{\varphi}\rangle_{\bar{I}_{c}\times\omega}$

for any $p\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$. Furthermore, we obtain

$J_{1}^{\prime}(\bar{q})(p)=\langle
p,\bar{\varphi}\rangle_{\bar{I}_{c}\times\omega}\quad\forall
p\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega)),$

which means that
$J_{1}^{\prime}(\bar{q})=\bar{\varphi}|_{\bar{I}_{c}\times\omega}$. Therefore,
combining with (23) we deduce the optimality condition (19) which claims that
$-\bar{\varphi}|_{\bar{I}_{c}\times\omega}\in\alpha\mathscr{\partial}\|\cdot\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}$,
i.e., (20).

Testing (20) with $p=2\bar{q}$ and $p=0$ we arrive at (21). Furthermore, it
follows from setting $p=\bar{q}-r$ in (20) for arbitrary
$r\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ that

$\displaystyle\langle r,\varphi\rangle_{\bar{I}_{c}\times\omega}\leq
J_{2}(\bar{q}-r)-J_{2}(\bar{q})\leq
J_{2}(r)=\alpha\|r\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\quad\forall
r\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega)).$

Hence, we obtain

(26)
$\displaystyle{\|\varphi\|_{{C}(\bar{I}_{c};L^{2}(\omega))}=\sup\limits_{\|r\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\leq
1}\langle r,\varphi\rangle_{\bar{I}_{c}\times\omega}\leq\alpha},$

this verifies (LABEL:adjoint_prop) in view of (21). ∎

In the following we will derive the sparsity structure in time of $\bar{q}$.

######  Theorem 7.

Let $\bar{q}$ be the optimal control of the optimization problem (1)-(2) and
$\bar{\varphi}$ be the optimal adjoint state defined by equation (18), then
there holds

(27) $\displaystyle{\rm
supp}|\bar{q}|\subset\\{t\in\bar{I}_{c}:\|\bar{\varphi}(t)\|_{L^{2}(\omega)}=\alpha\\},$
(28)
$\displaystyle\bar{q}^{\prime}(t,x)=-{1\over\alpha}\bar{\varphi}(t,x)\quad\mbox{in}\
L^{1}(I_{c},|\bar{q}|;L^{2}(\omega)),$

where $d\bar{q}=\bar{q}^{\prime}d|\bar{q}|$ denotes the polar decomposition of
$\bar{q}$.

###### Proof.

The idea of proof follows from [35, Theorem 2.12], see also [5, Theorem 3.3].
Here we sketch it for completeness. Applying the polar decomposition of $q$ in
(21) we have

(29)
$\displaystyle\int_{\bar{I}_{c}}(\alpha+(\bar{q}^{\prime}(t),\bar{\varphi})_{L^{2}(\omega)})d|\bar{q}|(t)=0.$

On the other hand, it follows from (3) and (LABEL:adjoint_prop) that

(30)
$\displaystyle(\bar{q}^{\prime}(t),\bar{\varphi})_{L^{2}(\omega)}\geq-\|\bar{q}^{\prime}(t)\|_{L^{2}(\omega)}\|\bar{\varphi}\|_{L^{2}(\omega)}\geq-\alpha\quad\
|\bar{q}|-\mbox{a.e.}\ t\in\bar{I}_{c},$

i.e., the integrand in (29) is nonnegative. Thus, it must be zero $\
|\bar{q}|$\- almost everywhere, that is,

(31)
$\displaystyle-(\bar{q}^{\prime}(t),\bar{\varphi})_{L^{2}(\omega)}=\alpha\quad\mbox{for}\
|\bar{q}|-\mbox{almost\ all}\ t\in\bar{I}_{c}.$

Therefore, in view of (30) we have the identity:

(32)
$\displaystyle(\bar{q}^{\prime}(t),\bar{\varphi})_{L^{2}(\omega)}=-\|\bar{q}^{\prime}(t)\|_{L^{2}(\omega)}\|\bar{\varphi}\|_{L^{2}(\omega)}=-\alpha,\quad\mbox{for}\
|\bar{q}|-\mbox{almost\ all}\ t\in\bar{I}_{c},$

which is equivalent to

(33)
$\displaystyle\|\bar{\varphi}(t)\|_{L^{2}(\omega)}=\alpha\quad\mbox{and}\quad\bar{\varphi}(t,x)=-\alpha\bar{q}^{\prime}(t,x)$

for $|\bar{q}|$-almost all $t\in\bar{I}_{c}$ and a.e. $x\in\omega$. Thus, we
finish the proof of (28).

In view of (3), we have

$\|\bar{\varphi}(t)\|_{L^{2}(\omega)}=\alpha\|\bar{q}^{\prime}(t)\|_{L^{2}(\omega)}=\alpha\quad\mbox{for}\
|\bar{q}|\ \mbox{-almost all}\ t\in\bar{I}_{c}.$

Namely,

$|\bar{q}|(\bar{I}_{c})=|\bar{q}|(\mbox{supp}\
|\bar{q}|)=|\bar{q}|(\\{t\in\bar{I}_{c}:\|\bar{\varphi}(t)\|_{L^{2}(\omega)}=\alpha\\}),$

which means that $\mbox{supp}\
|\bar{q}|\subseteq\\{t\in\bar{I}_{c}:\|\bar{\varphi}(t)\|_{L^{2}(\omega)}=\alpha\\}$.
This finishes the proof. ∎

In view of (27) in Theorem 7, we find that the optimal control
$\bar{q}\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ has sparsity pattern in
time that is independent of the spatial domain. Moreover, if
$\|\bar{\varphi}(t)\|_{L^{2}(\omega)}=\alpha$ holds for a finite set of time
instances, namely,
$\\{t\in\bar{I}_{c}:\|\varphi(t)\|_{L^{2}(\omega)}=\alpha\\}=\\{\tau_{i}\\}_{i=1}^{N}$,
then $\bar{q}$ has the representation
$\bar{q}(t,x)=\sum\limits_{i=1}^{N}\chi_{\omega}\bar{q}_{i}(x)\delta_{\tau_{i}}$
such that $\bar{q}_{i}\in L^{2}(\omega)$ (cf. [35]). This is exactly the
impulse control problem, studied extensively in the literature, e.g., [21, 22,
46, 47, 52], and the references therein.

######  Proposition 8.

There exists $\alpha_{0}>0$ such that the optimal control $\bar{q}=0$ when
$\alpha>\alpha_{0}$.

###### Proof.

The main idea follows from [5, Corollary 3.5], see also [4, Proposition 2.2],
and we sketch it here. Note that

$\displaystyle\frac{1}{2}\|\bar{u}-u_{d}\|_{L^{2}(I;L^{2}(\Omega))}^{2}+\frac{\beta}{2}\|\bar{u}(T)-u_{T}\|_{L^{2}(\Omega)}^{2}\leq
J(\bar{q})\leq J(0),$

where $J(0)$ is independent of $\alpha$. Then we have obtained a uniform upper
bound of $\|\bar{u}-u_{d}\|_{L^{2}(I;L^{2}(\Omega))}$ and
$\|\bar{u}(T)-u_{T}\|_{L^{2}(\Omega)}$ with respect to $\alpha$. For any
$u_{d}\in L^{2}(I;L^{2}(\Omega)),\ u_{T}\in L^{2}(\Omega)$, let
$\bar{\varphi}\in H^{1}(I;H^{-1}(\Omega))\cap
L^{2}(I;H^{1}_{0}(\Omega))\hookrightarrow C(\bar{I};L^{2}(\Omega))$ be the
optimal adjoint state defined by equation (18) with the following estimate:

$\displaystyle\|\bar{\varphi}(t)\|_{L^{2}(\Omega)}\leq
C(\|\bar{u}-u_{d}\|_{L^{2}(I;L^{2}(\Omega))}+\beta\|\bar{u}(T)-u_{T}\|_{L^{2}(\Omega)})\leq
2CJ(0),$

where we have used Lemma 1. Setting $\alpha_{0}=2CJ(0)$, it follows from (27)
in Theorem 7 that $\bar{q}=0$ for all $\alpha>\alpha_{0}$. This finishes the
proof. ∎

###  3.3 The regularity of solutions

In this subsection we prepare to state the regularity of solutions to the
optimality system, which will be used in the finite element approximation to
the optimal state and adjoint state.

######  Theorem 9.

For any $u_{d},\,f\in L^{2}(I;L^{2}(\Omega)),\,u_{0},\,u_{T}\in
L^{2}(\Omega)$, let $(\bar{q},\bar{u},\bar{\varphi})$ be the optimal solution
of the optimal control problem (1)-(2), where $\bar{q},\bar{u}$ and
$\bar{\varphi}$ are the optimal control, optimal state and adjoint state,
respectively. Then there hold

$\displaystyle\bar{u}\in L^{2}(I;H^{1}_{0}(\Omega))\cap
L^{\infty}(I;L^{2}(\Omega)),\quad\bar{\varphi}\in H^{1}(I;H^{-1}(\Omega))\cap
L^{2}(I;H^{1}_{0}(\Omega)),\quad\bar{q}\in\mathcal{M}(\bar{I}_{c};H^{1}(\omega)),$
$\displaystyle\bar{u}|_{(\tilde{t},T)}\in
L^{2}((\tilde{t},T);H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\cap
H^{1}((\tilde{t},T);L^{2}(\Omega))\hookrightarrow
C([\tilde{t},T];H^{1}_{0}({\Omega})),$ $\displaystyle\bar{\varphi}\in
L^{2}(I_{c};H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\cap
H^{1}(I_{c};L^{2}(\Omega))\hookrightarrow C(\bar{I}_{c};H^{1}_{0}({\Omega})),$

where $(\tilde{t},T)\cap I_{c}=\emptyset$, and there hold the following
stability estimates

$\displaystyle\|\bar{u}\|_{L^{2}(I;H^{1}_{0}(\Omega))}+\|\bar{u}\|_{L^{\infty}(I;L^{2}(\Omega))}$
$\displaystyle+$
$\displaystyle\|\bar{\varphi}\|_{H^{1}(I;H^{-1}(\Omega))}+\|\bar{\varphi}\|_{L^{2}(I;H^{1}_{0}(\Omega))}+\|\bar{\varphi}\|_{C(\bar{I};L^{2}(\Omega))}+\|\bar{q}\|^{\frac{1}{2}}_{\mathcal{M}(\bar{I}_{c};H^{1}(\omega))}$
$\displaystyle+$
$\displaystyle\|\bar{u}\|_{H^{1}((\tilde{t},T);L^{2}(\Omega))}+\|\bar{u}\|_{L^{2}((\tilde{t},T);H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))}+\|\bar{u}\|_{C([\tilde{t},T];H^{1}_{0}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C\left(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{d}\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{T}\|_{L^{2}(\Omega)}+\|u_{0}\|_{L^{2}(\Omega)}\right).$

Moreover, if $u_{T}\in H^{1}_{0}(\Omega)$, then the adjoint state has the
following improved regularity:

$\bar{\varphi}\in H^{1}(I;L^{2}(\Omega))\cap L^{2}(I;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))$

with the estimate

$\displaystyle\|\bar{\varphi}\|_{H^{1}(I;L^{2}(\Omega))}+\|\bar{\varphi}\|_{L^{2}(I;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))}+\|\bar{\varphi}\|_{C(\bar{I};H^{1}_{0}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C\left(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{d}\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{T}\|_{H^{1}(\Omega)}+\|u_{0}\|_{L^{2}(\Omega)}\right).$

###### Proof.

To begin with, we first consider the regularity of $\bar{u},\,\bar{\varphi}$
and derive the associated estimates. By using Theorem 3 and Lemma 1, $\bar{u}$
and $\bar{\varphi}$ have the above mentioned regularity, but the local
regularity of $\bar{\varphi}$ and the associated stability estimates depend on
$\|\bar{q}\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}$. Therefore, we first
provide the estimate for
$\|\bar{q}\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}$.

Let $\tilde{u}_{0}$ be the solution of equation (2) with $q=0$. Using the
identity (21) there holds

$\displaystyle\alpha\|\bar{q}\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\leq$
$\displaystyle J(\bar{u},\bar{q})\leq J(\tilde{u}_{0},0)$ $\displaystyle\leq$
$\displaystyle
C(\|u_{d}\|^{2}_{L^{2}(I;L^{2}(\Omega))}+\|f\|^{2}_{L^{2}(I;L^{2}(\Omega))}+\|u_{0}\|^{2}_{L^{2}(\Omega)}+{\beta}\|u_{T}\|^{2}_{L^{2}(\Omega)}).$

Below, we show that $\bar{\varphi}\in H^{1}(I_{c};L^{2}(\Omega))\cap
L^{2}(I_{c};H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\hookrightarrow
C(\bar{I}_{c};H^{1}_{0}(\Omega))$ and
$\bar{q}\in\mathcal{M}(\bar{I}_{c};H^{1}(\omega))$. Since
$I_{c}=(t_{1},t_{2})\subseteq I$ is relatively compact, there exist
$\tilde{t}_{1}$, $\tilde{t}_{2}$ satisfying
$\tilde{t}_{1}<t_{1}<t_{2}<\tilde{t}_{2}$, such that
$I_{c}\subseteq(\tilde{t}_{1},\tilde{t}_{2})$. Let $\tilde{\omega}$ be a
smooth cut-off function satisfying

$\displaystyle\tilde{\omega}(t)\in[0,1]\quad\forall
t\in[0,T];\quad\tilde{\omega}(t)=1\quad\forall t\in
I_{c};\quad\tilde{\omega}(t)=0\quad\forall
t\in\bar{I}\backslash(\tilde{t}_{1},\tilde{t}_{2}),$

and $\tilde{\varphi}:=\bar{\varphi}\tilde{\omega}$, then $\tilde{\varphi}$
satisfies the following equation:

(34) $\displaystyle\left\\{\begin{aligned}
-\partial_{t}\tilde{\varphi}-\Delta\tilde{\varphi}&=-\partial_{t}\tilde{\omega}\bar{\varphi}+\tilde{\omega}(\bar{u}-u_{d})&&\mbox{in}\,\,\,\Omega\times(\tilde{t}_{1},\tilde{t}_{2}),\\\
\tilde{\varphi}&=0&&\mbox{on}\,\,\Gamma\times(\tilde{t}_{1},\tilde{t}_{2}),\\\
\tilde{\varphi}(\tilde{t}_{2})&=0&&\mbox{in}\,\,\Omega.\end{aligned}\right.$

Therefore, by using Lemma 1 we obtain $\tilde{\varphi}\in
H^{1}((\tilde{t}_{1},\tilde{t}_{2});L^{2}(\Omega))\cap
L^{2}((\tilde{t}_{1},\tilde{t}_{2});H^{2}(\Omega)\cap H^{1}_{0}(\Omega))$,
which implies that $\bar{\varphi}\in H^{1}(I_{c};L^{2}(\Omega))\cap
L^{2}(I_{c};H^{2}(\Omega)\cap H^{1}_{0}(\Omega))\hookrightarrow
C(\bar{I}_{c};H^{1}_{0}(\Omega))$ and the following estimate holds:

$\displaystyle\|\bar{\varphi}\|_{H^{1}(I_{c};L^{2}(\Omega))}+\|\bar{\varphi}\|_{L^{2}(I_{c};H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))}+\|\bar{\varphi}\|_{C(\bar{I}_{c};H^{1}_{0}(\Omega))}$
$\displaystyle\leq$
$\displaystyle\|\tilde{\varphi}\|_{H^{1}((\tilde{t}_{1},\tilde{t}_{2});L^{2}(\Omega))}+\|\tilde{\varphi}\|_{L^{2}((\tilde{t}_{1},\tilde{t}_{2});H^{2}(\Omega))}+\|\tilde{\varphi}\|_{C([\tilde{t}_{1},\tilde{t}_{2}];H^{1}_{0}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C(\|\bar{\varphi}\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{d}\|_{L^{2}(I;L^{2}(\Omega))}+\|\bar{u}\|_{L^{2}(I;L^{2}(\Omega))})$
$\displaystyle\leq$ $\displaystyle
C(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{d}\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{T}\|_{L^{2}(\Omega)}+\|u_{0}\|_{L^{2}(\Omega)}).$

Finally, we prove that $\bar{q}\in\mathcal{M}(\bar{I}_{c};H^{1}(\omega))$. In
view of Theorem 7, there holds the relation
$d\bar{q}=\bar{q}^{\prime}d|\bar{q}|$, where
$\bar{q}^{\prime}(t,x)=-\frac{1}{\alpha}\bar{\varphi}(t,x)$ with
$\bar{\varphi}\in C(\bar{I}_{c};H_{0}^{1}(\Omega))$. Therefore, for any
$\psi\in C(\bar{I}_{c};(H^{1}(\omega))^{*})$, there holds

$\displaystyle\langle\bar{q},\psi\rangle_{\bar{I}_{c}\times\omega}=$
$\displaystyle\int_{I_{c}}\langle\psi,d\bar{q}\rangle=-\frac{1}{\alpha}\int_{I_{c}}\langle\psi,\bar{\varphi}\rangle
d|\bar{q}|$ $\displaystyle\leq$
$\displaystyle\frac{1}{\alpha}\|\psi\|_{C(\bar{I}_{c};(H^{1}(\omega)^{*}))}\|\bar{\varphi}\|_{C(\bar{I}_{c};H^{1}(\omega))}\|\bar{q}\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}.$

Thus, $\|\bar{q}\|_{\mathcal{M}(\bar{I}_{c};H^{1}(\omega))}\leq
C\frac{1}{\alpha}\|\bar{\varphi}\|_{C(\bar{I}_{c};H^{1}(\Omega))}\|\bar{q}\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}$
(cf. [45]). This combining with the estimate (3.3) yields the conclusion. We
thus finish the proof. ∎

##  4 Finite element approximations

In this section we consider the space-time finite element approximation for
optimal control problems.

###  4.1 Notations for finite element methods

Let $\\{\mathcal{T}_{h}\\}_{h>0}$ be a family of quasi-uniform and shape
regular triangulations of $\Omega$ in the sense of Ciarlet [16], such that
$\bar{\Omega}=\cup_{K\in\mathcal{T}_{h}}\bar{K}$, where $h$ is the mesh
parameter. Define the piecewise linear and continuous finite element space

$V_{h}:=\left\\{v_{h}\in H_{0}^{1}(\Omega):\ v_{h}|_{K}\in P_{1}(K),\ \forall
K\in\mathcal{T}_{h}\right\\},$

where $P_{1}(K)$ denotes the space of linear functions in $K$.

For simplicity, we assume that $\omega\subseteq\Omega$ is polygonal and the
restriction of $\mathcal{T}_{h}$ on $\omega$ gives a partition of $\omega$.
Thus, we define $U_{h}:=V_{h}|_{\omega}$ consisting of piecewise linear and
continuous functions in $\omega$.

Next, we divide $[0,T]$ into a family of subintervals
$I_{m}:=(t_{m-1},t_{m}],\,m=1,2,\cdots,M-1,\,I_{M}:=(t_{M-1},t_{M})$ with step
size $\tau_{m}=t_{m}-t_{m-1}$ such that
$[0,T]=\\{0\\}\cup\mathop{\cup}\limits_{i=1}^{M}I_{m}\cup\\{T\\}$, where
$0=t_{0}<t_{1}<\cdots<t_{M}=T$. We assume that there exist $1<k_{1}<k_{2}<M$
such that
$\bar{I}_{c}=[t_{1},t_{2}]=\\{t_{k_{1}-1}\\}\cup\mathop{\cup}\limits_{m=k_{1}}^{k_{2}}I_{m}$,
this can be achieved by setting $M$ sufficiently large. The maximal time step
is defined by $\tau:=\mathop{\max}\limits_{1\leq m\leq M}{\tau_{m}}$.

Now, we are ready to define two time semi-discrete finite element spaces
consisting of either piecewise constant or piecewise linear and continuous
Ansatz. Define (cf. [20])

$\displaystyle P_{\tau}:=\\{v\in C(\bar{I};H_{0}^{1}(\Omega)):\
v|_{I_{m}}\in\mathcal{P}_{1}(I_{m};H_{0}^{1}(\Omega)),\ m=1,2,\dots,M\\},$
$\displaystyle Y_{\tau}:=\\{v\in L^{2}(I;H_{0}^{1}(\Omega)):\
v|_{I_{m}}\in\mathcal{P}_{0}(I_{m};H_{0}^{1}(\Omega)),\ m=1,2,\dots,M,\
v(T)\in H_{0}^{1}(\Omega)\\},$

where $\mathcal{P}_{i}(I_{m};H^{1}_{0}(\Omega))$ ($i=0,1$) denotes the set of
polynomial functions of degree at most $i$ on time interval $I_{m}$ and valued
in $H^{1}_{0}(\Omega)$, and let

$P_{\tau}^{0}:=\\{v_{\tau}\in P_{\tau}:\ v_{\tau}(T)=0\\}.$

The notation $\sigma=(\tau,h)$ denotes the vector of two discretization
parameters $\tau$ and $h$. In order to introduce the Petrov-Galerkin scheme
for parabolic equations, we also need to define the following two time-space
finite element spaces:

$\displaystyle P_{\sigma}:=\left\\{v\in
P_{\tau}:\,v|_{I_{m}}\in\mathcal{P}_{1}(I_{m};V_{h}),\
m=1,2,\dots,M\right\\},$ $\displaystyle Y_{\sigma}:=\left\\{v\in Y_{\tau}:\
v|_{I_{m}}\in\mathcal{P}_{0}(I_{m};V_{h}),\ m=1,2,\dots,M,\ v(T)\in
V_{h}\right\\},$

where the definition of $\mathcal{P}_{i}(I_{m};V_{h})$ ($i=0,1$) is similar to
$\mathcal{P}_{i}(I_{m};H^{1}_{0}(\Omega))$ ($i=0,1$). We set

$P_{\sigma}^{0}:=\\{v_{\sigma}\in P_{\sigma}:\ v_{\sigma}(T)=0\\}.$

It is clear that $P_{\sigma}\subset L^{2}(I;H^{1}_{0}(\Omega))\cap
H^{1}(I;L^{2}(\Omega))$ and $Y_{\sigma}\subseteq L^{2}(I;H^{1}_{0}(\Omega))$.

Let $\\{x_{j}\\}_{j=1}^{N_{h}}$ be the interior nodes of the mesh
$\mathcal{T}_{h}$, and $\\{\psi_{x_{j}}\\}_{j=1}^{N_{h}}$,
$\\{e_{t_{m}}\\}_{m=0}^{M}$ be the nodal basis functions of $\mathcal{P}_{1}$
elements of $H^{1}(\Omega)$ and $H^{1}(I)$, respectively. Obviously,
$P_{\sigma}$ and $Y_{\sigma}$ can be rewritten as

$\displaystyle P_{\sigma}={\rm span}\\{\psi_{x_{j}}\otimes e_{t_{m}}:1\leq
j\leq N_{h},\ 0\leq m\leq M\\},$ $\displaystyle Y_{\sigma}={\rm
span}\\{\psi_{x_{j}}\otimes\chi_{m}:1\leq j\leq N_{h},\ 1\leq m\leq M\\},$

where $\chi_{m}$ denotes the characteristic function on $I_{m}$. We define the
indices

$\displaystyle\mathcal{I}_{\sigma}=\Big{\\{}(j,m):(x_{j},t_{m})\in\bar{\omega}\times\bar{I}_{c},1\leq
j\leq N_{h},\ 0\leq m\leq M\Big{\\}},\
\mathcal{I}_{\tau}=\big{\\{}m:t_{m}\in\bar{I}_{c},0\leq m\leq M\big{\\}},$

and the discrete spaces

$\displaystyle U_{\tau}={\rm
span}\\{\delta_{t_{m}}:m\in\mathcal{I}_{\tau}\\}\subset\mathcal{M}(\bar{I}_{c}),\quad
V_{\tau}={\rm
span}\\{e_{t_{m}}|_{\bar{I}_{c}}:m\in\mathcal{I}_{\tau}\\}\subset{C}(\bar{I}_{c}),$
$\displaystyle U_{\sigma}={\rm
span}\\{\psi_{x_{j}}|_{\bar{\omega}}\otimes\delta_{t_{m}}:(j,m)\in\mathcal{I}_{\sigma}\\},\quad
V_{\sigma}={\rm span}\\{\psi_{x_{j}}\otimes
e_{t_{m}}|_{\bar{\omega}\times\bar{I}_{c}}:(j,m)\in\mathcal{I}_{\sigma}\\}.$

For $q_{\tau}\in U_{\tau}$ and $v_{\tau}\in V_{\tau}$ we identify them with
$\vec{q}_{\tau}=(q_{1},\dots,q_{M_{c}})$ and
$\vec{v}_{\tau}=(v_{1},\dots,v_{M_{c}})$ where $M_{c}$ is the cardinality of
$\mathcal{I}_{\tau}$. The linear functions in $V_{\tau}$ attain their maximum
and minimum at the nodes. Therefore, for $v_{\tau}\in V_{\tau}$ we define

$\displaystyle\|v_{\tau}\|_{L^{\infty}(I_{c})}=\max\limits_{1\leq m\leq
M_{c}}|v_{m}|.$

Similarly, we have for $q_{\tau}\in U_{\tau}$ that

$\displaystyle\|q_{\tau}\|_{\mathcal{M}(\bar{I}_{c})}=\sup\limits_{v\in{C}(\bar{I}_{c}),\|v\|_{L^{\infty}(I_{c})}\leq
1}\sum\limits_{m=1}^{M_{c}}q_{m}\langle\delta_{t_{m}},v\rangle_{\bar{I}_{c}}=\sum\limits_{m=1}^{M_{c}}|q_{m}|.$

Given two functions $u\in L^{2}(I;H^{1}_{0}(\Omega))$ with $u(T)\in
L^{2}(\Omega)$, $v\in L^{2}(I;H^{1}_{0}(\Omega))\cap H^{1}(I;H^{-1}(\Omega))$,
we define a bilinear form $A(u,v)\to\mathbb{R}$ as follows:

$\displaystyle A(u,v):=-\langle
u,\partial_{t}v\rangle_{L^{2}(I;H^{1}_{0},H^{-1})}+\int_{I}(\nabla u(t),\nabla
v(t))dt+(u(T),v(T)),$

where $\langle\cdot,\cdot\rangle_{L^{2}(I;H^{1}_{0},H^{-1})}$ denotes the
duality pairing between $L^{2}(I;H^{1}_{0}(\Omega))$ and
$L^{2}(I;H^{-1}(\Omega))$.

If, in addition, $u_{\tau}\in Y_{\tau}$ and $v_{\tau}\in P_{\tau}$, applying
integration by parts to the bilinear form $A(u_{\tau},v_{\tau})$ we can obtain
the following dual representation:

(36)
$A(u_{\tau},v_{\tau})=\sum_{m=1}^{M}([u_{\tau}]_{m},v_{\tau}(t_{m}))_{L^{2}(\Omega)}+(u_{\tau,0}^{+},v_{\tau}(0))_{L^{2}(\Omega)}+\int_{I}(\nabla
u_{\tau}(t),\nabla v_{\tau}(t))dt,$

where

$\displaystyle u_{\tau,m+1}=u^{+}_{\tau,m}:=\mathop{\rm
lim}\limits_{\epsilon\to 0^{+}}u_{\tau}(t_{m}+\epsilon),\quad
u_{\tau,m}=u^{-}_{\tau,m}:=\mathop{\rm lim}\limits_{\epsilon\to
0^{+}}u_{\tau}(t_{m}-\epsilon),$
$\displaystyle[u]_{m}:=u^{+}_{\tau,m}-u^{-}_{\tau,m},\ m=1,2,\cdots,M-1,\
[u_{\tau}]_{M}=u_{\tau}(T)-u^{-}_{\tau,M},$

and $u_{\tau,m}:=u_{\tau}|_{I_{m}},\ m=1,2,\cdots,M$.

In view of the identity (14), the very weak solution to the state equation (2)
reads: Find $u\in L^{2}(I;H^{1}_{0}(\Omega))$ such that

$\displaystyle A(u,v)=\\!\\!\int_{I}(f,v)dt+\langle
q,v\rangle_{\bar{I}_{c}\times\omega}+(u_{0},v(x,0))\quad\forall v\in
L^{2}(I;H^{1}_{0}(\Omega))\cap H^{1}(I;H^{-1}(\Omega)).$

Similarly, the weak formulation of the backward equation (5) can be rewritten
as: Find $z\in L^{2}(I;H^{1}_{0}(\Omega))\cap H^{1}(I;H^{-1}(\Omega))$ such
that

(37) $\displaystyle A(v,z)=\int_{I}(g,v)dt+(v(T),z_{T})\quad\forall v\in
L^{2}(I;H^{1}_{0}(\Omega))\cap H^{1}(I;H^{-1}(\Omega)).$

###  4.2 Interpolation and projection operators

In the following we introduce some interpolation operators defined in
$\bar{I}_{c}$ and give their properties whose proofs are very similar to [4,
Theorem 3.1], see also [8, Proposition 4.1].

######  Lemma 10.

Let the linear operators $\Lambda_{\tau}$ and $\Pi_{\tau}$ be defined as
follows:

$\displaystyle\Lambda_{\tau}:\mathcal{M}(\bar{I}_{c})\rightarrow
U_{\tau}\subset\mathcal{M}(\bar{I}_{c}),\quad\Lambda_{\tau}q:=\sum\limits_{m\in\mathcal{I}_{\tau}}\delta_{t_{m}}\int_{\bar{I}_{c}}e_{t_{m}}dq,$
$\displaystyle\Pi_{\tau}:{C}(\bar{I}_{c})\rightarrow
V_{\tau}\subset{C}(\bar{I}_{c}),\quad\Pi_{\tau}v:=\sum\limits_{m\in\mathcal{I}_{\tau}}v(t_{m})e_{t_{m}}.$

Then for every $q\in\mathcal{M}(\bar{I}_{c})$, $v\in{C}(\bar{I}_{c})$ and
$v_{\tau}\in V_{\tau}$, there hold

(38) $\displaystyle\langle
q,\Pi_{\tau}v\rangle_{\bar{I}_{c}}=\langle\Lambda_{\tau}q,v\rangle_{\bar{I}_{c}},$
(39)
$\displaystyle\|\Lambda_{\tau}q\|_{\mathcal{M}(\bar{I}_{c})}\leq\|q\|_{\mathcal{M}(\bar{I}_{c})},$
(40)
$\displaystyle\Lambda_{\tau}q\stackrel{{\scriptstyle*}}{{\rightharpoonup}}q\
\mbox{\rm in}\ \mathcal{M}(\bar{I}_{c})\quad\mbox{and}\
\|\Lambda_{\tau}q\|_{\mathcal{M}(\bar{I}_{c})}\stackrel{{\scriptstyle\tau\rightarrow
0^{+}}}{{\longrightarrow}}\|q\|_{\mathcal{M}(\bar{I}_{c})},$ (41)
$\displaystyle\|q-\Lambda_{\tau}q\|_{(H^{1}(I_{c}))^{\prime}}\leq
C\tau^{1\over
2}\|q\|_{\mathcal{M}(\bar{I}_{c})},\quad\|q-\Lambda_{\tau}q\|_{(W^{1,\infty}(I_{c}))^{\prime}}\leq
C\tau\|q\|_{\mathcal{M}(\bar{I}_{c})}.$

###### Proof.

The proofs of (38)-(40) are trivial, thus we only provide the proof for (41).
For an arbitrary $v\in H^{1}(I_{c})$, using (38) and the standard Lagrange
interpolation error estimate we have for $s>1$ that

$\displaystyle\langle q-\Lambda_{\tau}q,v\rangle_{\bar{I}_{c}}=\langle
q,v-\Pi_{\tau}v\rangle_{\bar{I}_{c}}\leq\|q\|_{\mathcal{M}(\bar{I}_{c})}\|v-\Pi_{\tau}v\|_{{C}(\bar{I}_{c})}\leq
C\tau^{1-{1\over s}}\|q\|_{\mathcal{M}(\bar{I}_{c})}\|v\|_{W^{1,s}(I_{c})}.$

From the duality we can obtain two desired results by setting $s=2$ and
$s=\infty$. ∎

Let $\pi_{h}:\,L^{2}(\omega)\to V_{h}|_{\omega}$ be the usual
$L^{2}$-projection defined by

$(v-\pi_{h}v,\varphi_{h})_{L^{2}(\omega)}=0\quad\forall\varphi_{h}\in
V_{h}|_{\omega},$

then there holds

$\|v-\pi_{h}v\|_{L^{2}(\omega)}+h\|v-\pi_{h}v\|_{H^{1}(\omega)}\leq
Ch^{m}\|v\|_{H^{m}(\omega)}\quad\forall v\in H^{m}(\omega),\ m=1,2,$

where $C>0$ is a constant independent of $h$ and $v$. Note that the
application of the operator $\pi_{h}$ to time-dependent arguments has to be
understood pointwisely in time. Below, we will extend $\pi_{h}$ to a negative
exponent Sobolev space that includes $L^{2}(\omega)$.

Let $V:=H^{1}(\omega)$ (resp. $H_{0}^{1}(\Omega)$), $H:=L^{2}(\omega)$ (resp.
$L^{2}(\Omega)$), then the inclusion $V\subseteq H$ is dense and continuous.
Note that $V\hookrightarrow H=H^{*}\hookrightarrow V^{*}$ is a Gelfand triple,
where $H\hookrightarrow V^{*}$ is given by $y\in
H\to(y,\cdot)_{L^{2}(\Omega)}\in H^{*}\subseteq V^{*}$. Therefore, we extend
in the following definition the usual projection $\pi_{h}$ from $H$ to
$V^{*}$.

######  Definition 11.

Define the action of the $L^{2}$-projection $\pi_{h}$ on $V^{*}$ as

$\displaystyle\pi_{h}:V^{*}\to V_{h}|_{\omega}\quad v\to\pi_{h}v,$

where $\pi_{h}v\in V_{h}|_{\omega}$ satisfies

(42) $(\pi_{h}v,\varphi_{h})_{H}=\langle
v,\varphi_{h}\rangle_{V^{*},V}\quad\forall\varphi_{h}\in
V_{h}|_{\omega}\subset V.$

Furthermore, we also define the following two interpolation operators:

$\displaystyle\Lambda_{\sigma}:\mathcal{M}(\bar{I}_{c};H)\rightarrow
U_{\sigma}\subset\mathcal{M}(\bar{I}_{c};H),\quad\Lambda_{\sigma}q:=\pi_{h}(\Lambda_{\tau}q)=\sum\limits_{(j,m)\in\mathcal{I}_{\sigma}}q_{j,m}\delta_{t_{m}}\otimes\psi_{x_{j}}|_{\bar{\omega}},$
$\displaystyle\Pi_{\sigma}:{C}(\bar{I}_{c};V^{*})\rightarrow
V_{\sigma}\subset{C}(\bar{I}_{c};V^{*}),\quad\Pi_{\sigma}v:=\pi_{h}(\Pi_{\tau}v)=\sum\limits_{(j,m)\in\mathcal{I}_{\sigma}}v_{j,m}e_{t_{m}}\otimes\psi_{x_{j}}|_{\bar{I}_{c}\times\bar{\omega}},$

where $q_{j,m}:=\pi_{h}(\int_{\bar{I}_{c}}e_{t_{m}}dq)(x_{j})$ and
$v_{j,m}:=\pi_{h}(v(t_{m}))(x_{j})$.

It is easy to check that there exists a unique $\pi_{h}v\in V_{h}|_{\omega}$
such that the identity (42) holds for any $v\in V^{*}$. Moreover,
$\pi_{h}|_{H}:H\to V_{h}|_{\omega}$ is consistent with the usual
$L^{2}$-projection and $\pi_{h}$ is stable, i.e.,

$\displaystyle\|\pi_{h}v\|_{V^{*}}\leq\|v\|_{V^{*}},\quad\|\pi_{h}{v}\|_{H}\leq\|{v}\|_{H}$

for $v\in V^{*}$ and ${v}\in H$, respectively. The definition of above two
interpolation operators $\Lambda_{\sigma}$ and $\Pi_{\sigma}$ are very similar
to [5, Theorem 4.2], see also [8, Proposition 4.2].

######  Lemma 12.

For every $q_{\sigma}\in U_{\sigma}$ and $v_{\sigma}\in V_{\sigma}$ one have

(43)
$\displaystyle\Lambda_{\sigma}q_{\sigma}=q_{\sigma}\quad\mbox{and}\quad\Pi_{\sigma}v_{\sigma}=v_{\sigma}.$

Moreover, there hold

(44) $\displaystyle\langle
q,\Pi_{\sigma}v\rangle_{\bar{I}_{c}\times\omega}=\langle\Lambda_{\sigma}q,v\rangle_{\bar{I}_{c}\times\omega}\quad\forall\,(q,v)\in\mathcal{M}(\bar{I}_{c};S)\times{C}(\bar{I}_{c};S^{*}),$
(45)
$\displaystyle\|\Lambda_{\sigma}q\|_{\mathcal{M}(\bar{I}_{c};S)}\leq\|q\|_{\mathcal{M}(\bar{I}_{c};S)},\quad\|\Pi_{\sigma}v\|_{{C}(\bar{I}_{c};S^{*})}\leq\|v\|_{{C}(\bar{I}_{c};S^{*})},$
(46)
$\displaystyle\Lambda_{\sigma}q\stackrel{{\scriptstyle*}}{{\rightharpoonup}}q\in\mathcal{M}(\bar{I}_{c};H),\quad\
\|\Lambda_{\sigma}q\|_{\mathcal{M}(\bar{I}_{c};H)}\stackrel{{\scriptstyle|\sigma|\to
0}}{{\longrightarrow}}\|q\|_{\mathcal{M}(\bar{I}_{c};H)},$

where $S=H$ or $V$.

###### Proof.

A simple calculation gives

(47) $\displaystyle\langle q,\Pi_{\sigma}v\rangle_{\bar{I}_{c}\times\omega}$
$\displaystyle=$ $\displaystyle\langle
q,\pi_{h}(\Pi_{\tau}v)\rangle_{\bar{I}_{c}\times\omega}=\big{\langle}q,\pi_{h}\big{(}\sum_{m}v(t_{m})e_{t_{m}}\big{)}\big{\rangle}_{\bar{I}_{c}\times\omega}$
$\displaystyle=$
$\displaystyle\Big{\langle}\sum_{m}\pi_{h}(v(t_{m})),\int_{I_{m}}e_{t_{m}}dq\Big{\rangle}_{S^{*},S}$
$\displaystyle=$
$\displaystyle\Big{\langle}\sum_{m}v(t_{m}),\pi_{h}(\int_{I_{m}}e_{t_{m}}dq)\Big{\rangle}_{S^{*},S}$
$\displaystyle=$
$\displaystyle\Big{\langle}v,\sum_{m}\pi_{h}(\int_{I_{m}}e_{t_{m}}dq)\otimes\delta_{t_{m}}\Big{\rangle}$
$\displaystyle=$
$\displaystyle\langle\Lambda_{\sigma}q,v\rangle_{\bar{I}_{c}\times\omega}$

for any $(q,v)\in\mathcal{M}(\bar{I}_{c};S)\times{C}(\bar{I}_{c};S^{*})$.
Moreover, by using (43) and (44) we have $\langle
q,v_{\sigma}\rangle_{\bar{I}_{c}\times\omega}=\langle
q,\Pi_{\sigma}v_{\sigma}\rangle_{\bar{I}_{c}\times\omega}=\langle\Lambda_{\sigma}q,v_{\sigma}\rangle_{\bar{I}_{c}\times\omega}$.

For the second inequality in (45), we use the stability of the projection
$\pi_{h}$ to conclude

$\displaystyle\|\Pi_{\sigma}v\|_{{C}(\bar{I}_{c};S^{*})}$ $\displaystyle=$
$\displaystyle\sup\limits_{t\in\bar{I}_{c}}\|\pi_{h}(\Pi_{\tau}v)(t)\|_{S^{*}}\leq\max\limits_{1\leq
m\leq M_{c}}\|\pi_{h}v(t_{m})\|_{S^{*}}$ $\displaystyle\leq$
$\displaystyle\max\limits_{1\leq m\leq
M_{c}}\|v(t_{m})\|_{S^{*}}\leq\|v\|_{{C}(\bar{I}_{c};{S^{*}})}.$

Next, we prove the first inequality in (45)

$\displaystyle\|\Lambda_{\sigma}q\|_{\mathcal{M}(\bar{I}_{c};S)}$
$\displaystyle=$ $\displaystyle\sup\limits_{\|v\|_{{C}(\bar{I}_{c};S^{*})}\leq
1}\langle\Lambda_{\sigma}q,v\rangle_{\bar{I}_{c}\times\omega}=\sup\limits_{\|v\|_{{C}(\bar{I}_{c};S^{*})}\leq
1}\langle q,\Pi_{\sigma}v\rangle_{\bar{I}_{c}\times\omega}$
$\displaystyle\leq$
$\displaystyle\sup\limits_{\|v\|_{{C}(\bar{I}_{c};S^{*})}\leq
1}\|q\|_{\mathcal{M}(\bar{I}_{c};S)}\|\Pi_{\sigma}v\|_{{C}(\bar{I}_{c};S^{*})}$
$\displaystyle\leq$ $\displaystyle\|q\|_{\mathcal{M}(\bar{I}_{c};S)},$

where we have used the second inequality in (45). This proves the first
inequality in (45), while (46) is obvious. This finishes the proof. ∎

Next, we will introduce the following interpolation and projection operators
in time (cf. [20]). Define the $L^{2}$-projection
$\mathcal{P}_{Y_{\tau}}:L^{2}(I;H_{0}^{1}(\Omega))\rightarrow Y_{\tau}$ such
that $\mathcal{P}_{Y_{\tau}}z\in Y_{\tau}$ satisfies

$\displaystyle\mathcal{P}_{Y_{\tau}}z|_{I_{m}}:=\frac{1}{\tau_{m}}\int_{I_{m}}zdt\quad\forall
z\in L^{2}(I;H_{0}^{1}(\Omega)),\ m=1,\cdots,M,\ \mbox{and}\
\mathcal{P}_{Y_{\tau}}v(T):=0.$

In addition, we also need the following interpolation operators. Define the
Lagrange interpolation operator
${\Pi}_{P_{\tau}}:C(\bar{I};H_{0}^{1}(\Omega))\rightarrow P_{\tau}$ and the
piecewise constant in time interpolation operator
${\Pi}_{Y_{\tau}}:C(\bar{I};H_{0}^{1}(\Omega))\rightarrow Y_{\tau}$ such that
${\Pi}_{P_{\tau}}v\in P_{\tau}$, ${\Pi}_{Y_{\tau}}v\in Y_{\tau}$ satisfy

$\displaystyle{\Pi}_{P_{\tau}}v:=\sum\limits_{m=0}^{M}v(t_{m})e_{t_{m}},\quad{\Pi}_{Y_{\tau}}v:=\sum\limits_{m=1}^{M}v(t_{m})\chi_{I_{m}},\quad{\Pi}_{Y_{\tau}}v(T):=0,$

where $\\{e_{t_{m}}\\}$ and $\\{\chi_{I_{m}}\\}$ are the families of node
basis functions of $\mathcal{P}_{1}$ and $\mathcal{P}_{0}$ elements on the
time interval $I$, respectively.

######  Lemma 13.

For arbitrary $v\in L^{2}(I;H^{1}_{0}(\Omega))\cap H^{1}(I;L^{2}(\Omega))$,
there hold the following standard interpolation error estimates (cf. [20,
44]):

(48)
$\displaystyle\left\|v-\mathcal{P}_{Y_{\tau}}v\right\|_{L^{2}(I;L^{2}(\Omega))}\leq
C\tau\left\|\partial_{t}v\right\|_{L^{2}(I;L^{2}(\Omega))},$ (49)
$\displaystyle\|v-{\Pi}_{P_{\tau}}v\|_{L^{2}(I;L^{2}(\Omega))}\leq
C\tau\|\partial_{t}v\|_{L^{2}(I;L^{2}(\Omega))},$ (50)
$\displaystyle\|v-{\Pi}_{P_{\tau}}v\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C\tau^{1\over 2}\|\partial_{t}v\|_{L^{2}(I;L^{2}(\Omega))}.$

We also have the following half an order interpolation error estimates for any
$v\in H^{1}(I;L^{2}(\Omega))\cap L^{2}(I;H^{2}(\Omega))$ (cf. [35, Lemma
3.13])

(51) $\displaystyle\|v-{\Pi}_{Y_{\tau}}v\|_{L^{2}(I;H_{0}^{1}(\Omega))}\leq
C\tau^{1\over 2}(\|\partial_{t}v\|_{L^{2}(I;L^{2}(\Omega))}+\|\Delta
v\|_{L^{2}(I;L^{2}(\Omega))}),$ (52)
$\displaystyle\|v-{\Pi}_{P_{\tau}}v\|_{L^{2}(I;H_{0}^{1}(\Omega))}\leq
C\tau^{1\over 2}(\|\partial_{t}v\|_{L^{2}(I;L^{2}(\Omega))}+\|\Delta
v\|_{L^{2}(I;L^{2}(\Omega))}),$

where $C>0$ is a constant independent of $\tau$ and $v$.

###  4.3 Discretization of the optimal control problem

With the above preparation the discrete optimal control problem reads:

(53)
$\displaystyle\min_{q\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))\atop{u_{\sigma}\in
Y_{\sigma}}}J_{\sigma}(u_{\sigma},q):=\frac{1}{2}\|u_{\sigma}-u_{d}\|_{L^{2}(I;L^{2}(\Omega))}^{2}+\frac{\beta}{2}\|u_{\sigma}(T)-u_{T}\|_{L^{2}(\Omega)}^{2}+\alpha\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))},$

where $u_{\sigma}\in Y_{\sigma}$ is the discrete state variable satisfying the
following discrete state equation:

(54) $\displaystyle A(u_{\sigma},v_{\sigma})=\int_{I}(f,v_{\sigma})dt+\langle
q,v_{\sigma}\rangle_{\bar{I}_{c}\times\omega}+(u_{0},v_{\sigma}(0))\quad\forall
v_{\sigma}\in P_{\sigma}.$

Note that in the above discrete optimal control problem (53) the control
variable is not explicitly discretized, but in the following we will see that
the discretization of the adjoint state indeed automatically yields the
discretization of the control variable, which is the so-called variational
discretization for optimal control problems proposed in [33].

Similar to subsection 3.1, we can easily check that the discrete optimal
control problem (53) exists at least one optimal pair. However, the optimal
control is not unique in general, since the discrete control-to-state operator
is not injective. In fact, we have
$\langle\Lambda_{\sigma}q,v_{\sigma}\rangle_{\bar{I}_{c}\times\omega}=\langle
q,v_{\sigma}\rangle_{\bar{I}_{c}\times\omega}$ for any $v_{\sigma}\in
P_{\sigma}$, which implies that the cost functional $J_{\sigma}$ is not
strictly convex on $\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$. Fortunately, we
find that the discrete optimal control problem (53) does indeed have a unique
optimal control in the subspace
$U_{\sigma}\subseteq\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$.

To derive the discrete first order optimality condition, we denote the
solution of (54) by $u_{\sigma}(q)\in Y_{\sigma}$, then we can obtain the
following reduced optimization problem:

(55)
$\displaystyle\min_{q\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}j_{\sigma}(q):=j_{\sigma,1}(q)+j_{\sigma,2}(q),$

where

$\displaystyle
j_{\sigma,1}(q):=\frac{1}{2}\|u_{\sigma}(q)-u_{d}\|_{L^{2}(I;L^{2}(\Omega))}^{2}+\frac{\beta}{2}\|u_{\sigma}(q)(T)-u_{T}\|_{L^{2}(\Omega)}^{2},\quad
j_{\sigma,2}(q):=\alpha\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}.$

Obviously, $j_{\sigma,1}:\mathcal{M}(\bar{I}_{c};L^{2}(\omega))\to\mathbb{R}$
is a quadratic functional of tracking type that is differentiable, and
$j_{\sigma,2}:\mathcal{M}(\bar{I}_{c};L^{2}(\omega))\to\mathbb{R}$ is convex
and subdifferentiable.

By straightforward calculations, we obtain for any
$q,\,p\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ that

$\displaystyle
j^{\prime}_{\sigma,1}(q)p=\left(u_{\sigma}(q)-u_{d},\tilde{u}_{\sigma}(p)\right)_{L^{2}(I;L^{2}(\Omega))}+\beta\left(u_{\sigma}(q)(T)-u_{T},\tilde{u}_{\sigma}(p)(T)\right),$

where $\tilde{u}_{\sigma}(p)\in Y_{\sigma}$ is the finite element
approximation of the state equation (2) with $f=0,\,u_{0}=0,\,q=p$, i.e.,
satisfying equation (54) with the assumed data. On the other hand, we define a
discrete adjoint variable as follows: Find $\varphi_{\sigma}\in P_{\sigma}$
such that

(56) $\displaystyle
A(w_{\sigma},\varphi_{\sigma})=\int_{I}(u_{\sigma}(q)-u_{d},w_{\sigma})dt+\beta(u_{\sigma}(q)(\cdot,T)-u_{T},w_{\sigma}(T))\quad\forall
w_{\sigma}\in Y_{\sigma}.$

Then, we obtain for any $q,\,p\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ that

(57) $\displaystyle j^{\prime}_{\sigma,1}(q)p$
$\displaystyle=\left(u_{\sigma}(q)-u_{d},\tilde{u}_{\sigma}(p)\right)_{L^{2}(I;L^{2}(\Omega))}+\beta\left(u_{\sigma}(q)(T)-u_{T},\tilde{u}_{\sigma}(p)(T)\right)$
$\displaystyle=A(\tilde{u}_{\sigma}(p),\varphi_{\sigma})=\langle
p,\varphi_{\sigma}\rangle_{\bar{I}_{c}\times\omega},$

which implies that
$j^{\prime}_{\sigma,1}(q)=\varphi_{\sigma}|_{\bar{I}_{c}\times\omega}$ with
$\varphi_{\sigma}$ given by (56).

Now we are in the position to derive the first order optimality condition for
the discrete optimization problem (55).

######  Theorem 14.

Let $\hat{q}_{\sigma}\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ be an optimal
control and $\bar{u}_{\sigma}\in Y_{\sigma}$ be the corresponding optimal
state of the optimal control problem (53). Then there exists an adjoint state
$\bar{\varphi}_{\sigma}\in P_{\sigma}$ solving (56) with $u_{\sigma}(q)$
replaced by $\bar{u}_{\sigma}$ on the right-hand side, such that the following
subgradient condition holds

(58) $\displaystyle
0\in\bar{\varphi}_{\sigma}|_{\bar{I}_{c}\times\omega}+\alpha\mathscr{\partial}\|\cdot\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}(\hat{q}_{\sigma})\
\ \mbox{\rm in}\ \,(\mathcal{M}(\bar{I}_{c};L^{2}(\omega)))^{*},$

i.e.,

(59) $\displaystyle-\langle
p-\hat{q}_{\sigma},\bar{\varphi}_{\sigma}\rangle_{\bar{I}_{c}\times\omega}+\alpha\|\hat{q}\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\leq\alpha\|p\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\quad\forall
p\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega)),$

where
$\mathscr{\partial}\|\cdot\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}(\hat{q}_{\sigma})$
denotes the set of subgradients for
$\|\cdot\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}$ at $\hat{q}_{\sigma}$
which is nonempty since $\|\cdot\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}$
is convex on $\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$, and
$(\mathcal{M}(\bar{I}_{c};L^{2}(\omega)))^{*}$ denotes the topological dual of
$\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$.

Furthermore, from the above condition (59) we can easily conclude the
following relation between $\hat{q}_{\sigma}$ and $\bar{\varphi}_{\sigma}$:

(60)
$\displaystyle\alpha\|\hat{q}_{\sigma}\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}+\langle\hat{q}_{\sigma},\bar{\varphi}_{\sigma}\rangle_{\bar{I}_{c}\times\omega}=0,$
(61)
$\displaystyle\|\bar{\varphi}_{\sigma}\|_{{C}(\bar{I}_{c};L^{2}(\omega))}\left\\{\begin{aligned}
&=\alpha&&\mbox{if}\,\,\hat{q}_{\sigma}\neq 0,\\\
&\leq\alpha&&\mbox{if}\,\,\hat{q}_{\sigma}=0.\end{aligned}\right.$

In addition, since the discrete control-to-state mapping has an infinite-
dimensional kernel, the discrete optimal control problem (53) admits more than
one optimal control
$\hat{q}_{\sigma}\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ corresponding to
the identical optimal state $\bar{u}_{\sigma}\in Y_{\sigma}$. Among these
optimal controls there exists a unique one $\bar{q}_{\sigma}\in U_{\sigma}$,
such that for any other optimal control
$\hat{q}_{\sigma}\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ there holds
$\bar{q}_{\sigma}=\Lambda_{\sigma}\hat{q}_{\sigma}$, i.e.,

$j_{\sigma}(\bar{q}_{\sigma})=j_{\sigma}(\hat{q}_{\sigma})=\min\limits_{q\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}j_{\sigma}(q).$

In other words, $(\bar{q}_{\sigma},\bar{u}_{\sigma})\in U_{\sigma}\times
Y_{\sigma}$ is the unique optimal pair of the following fully discrete optimal
control problem:

(62) $\displaystyle\min_{q_{\sigma}\in U_{\sigma}\atop{u_{\sigma}\in
Y_{\sigma}}}J_{\sigma}(u_{\sigma},q_{\sigma}):=\frac{1}{2}\|u_{\sigma}-u_{d}\|_{L^{2}(I;L^{2}(\Omega))}^{2}+\frac{\beta}{2}\|u_{\sigma}(T)-u_{T}\|_{L^{2}(\Omega)}^{2}+\alpha\|q_{\sigma}\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))},$

where $u_{\sigma}\in Y_{\sigma}$ is the discrete state variable satisfying
(54) with the discrete control $q_{\sigma}\in U_{\sigma}$, which can be
computed in practice.

###### Proof.

Similar to Theorem 6, we can easily obtain (58)-(LABEL:adjoint_prop_h_0) by
recalling that
$j^{\prime}_{\sigma,1}(\bar{q}_{\sigma})=\bar{\varphi}_{\sigma}|_{\bar{I}_{c}\times\omega}$
in (57).

Next, we state the non-uniqueness of optimal controls and the unique
solvability of problem (62). Let
$\hat{q}_{\sigma}\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ be any optimal
control of problem (53). Setting
$\bar{q}_{\sigma}:=\Lambda_{\sigma}\hat{q}_{\sigma}$, it follows from (44) and
(45) that
$\langle\bar{q}_{\sigma},v_{\sigma}\rangle_{\bar{I}_{c}\times\omega}=\langle\hat{q}_{\sigma},v_{\sigma}\rangle_{\bar{I}_{c}\times\omega}$
for any $v_{\sigma}\in P_{\sigma}$ and

$\displaystyle J_{\sigma}(\bar{q}_{\sigma})\leq J_{\sigma}(\hat{q}_{\sigma}).$

This means $\bar{q}_{\sigma}\in U_{\sigma}$ is also optimal, and
$\bar{q}_{\sigma}\neq\hat{q}_{\sigma}$ unless $\hat{q}_{\sigma}\in
U_{\sigma}$. On the other hand, the functional $j_{\sigma}$ is not strictly
convex on $\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$. Therefore, the discrete
optimization problem (53) admits more than one solution in
$\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$.

Obviously, the control-to-state mapping is injective on $U_{\sigma}$, and thus
the discrete cost functional $J_{\sigma}$ is strictly convex on $U_{\sigma}$.
Therefore, the optimization problem (62) admits a unique optimal pair
$(\bar{q}_{\sigma},\bar{u}_{\sigma})\in U_{\sigma}\times Y_{\sigma}$. The
uniqueness of optimal controls for the cost functional $J_{\sigma}$ in
$U_{\sigma}$ ensures that any other optimal control
$\hat{q}_{\sigma}\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ satisfies
$\bar{q}_{\sigma}=\Lambda_{\sigma}\hat{q}_{\sigma}$. This finishes the proof.
∎

##  5 Error estimates for the state and adjoint equations

In this section we intend to derive a priori error estimates for finite
element solutions of the discrete state equation (54) and the discrete adjoint
equation (56). Here we assume that the data of the optimal control problem (1)
satisfy the following assumptions.

######  Assumption 5.1.

Assume that $f\in L^{2}(I;H^{1}(\Omega)),\ u_{d}\in L^{2}(I;L^{2}(\Omega)),\
u_{0}\in H^{1}_{0}(\Omega)$ and $u_{T}\in H^{1}_{0}(\Omega)$.

In the following we also need the stability of the semi-discrete in time
approximation to the state equation (2). Therefore, we introduce here the
semi-discrete state equation for given $f$, $u_{0}$ and $q$: Find $u_{\tau}\in
Y_{\tau}$ such that

(63) $\displaystyle A(u_{\tau},v_{\tau})=\int_{I}(f,v_{\tau})dt+\langle
q,v_{\tau}\rangle_{\bar{I}_{c}\times\omega}+(u_{0},v_{\tau}(0))\quad\ \forall
v_{\tau}\in P_{\tau}.$

Similarly, for any given $g$ and $z_{T}$, the semi-discrete approximation to
problem (5) reads: Find $z_{\tau}\in P_{\tau}$ such that

(64) $\displaystyle
A(v_{\tau},z_{\tau})=\int_{I}(g,v_{\tau})dt+(v_{\tau}(T),z_{T})\quad\ \forall
v_{\tau}\in Y_{\tau}.$

###  5.1 Stability estimates for the discrete state and adjoint state

We first give a stability result for the semi-discrete approximation of the
backward parabolic equation (5).

######  Lemma 15.

Let $z_{\tau}\in P_{\tau}$ solve (64) for given $g\in L^{2}(I;H^{-1}(\Omega))$
and $z_{T}\in L^{2}(\Omega)$, then there exists a constant $C>0$, independent
of $\tau$, such that

$\displaystyle\|z_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}\leq
C\sqrt{T}\|z_{\tau}\|_{C(\bar{I};L^{2}(\Omega))},$ (65)
$\displaystyle\|z_{\tau}\|_{C(\bar{I};L^{2}(\Omega))}+\|\partial_{t}z_{\tau}\|_{L^{2}(I;H^{-1}(\Omega))}\leq
C\|g\|_{L^{2}(I;H^{-1}(\Omega))}+\|z_{T}\|_{L^{2}(\Omega)}.$

In addition, if $g\in L^{2}(I;L^{2}(\Omega))$ and $z_{T}\in
H^{1}_{0}(\Omega)$, then there hold

$\displaystyle\|\nabla z_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}\leq
C\sqrt{T}\|\nabla z_{\tau}\|_{C(\bar{I};L^{2}(\Omega))},$ (66)
$\displaystyle\|\nabla
z_{\tau}\|_{C(\bar{I};L^{2}(\Omega))}+\|\partial_{t}z_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}\leq
C\|g\|_{L^{2}(I;L^{2}(\Omega))}+\|\nabla z_{T}\|_{L^{2}(\Omega)}.$

###### Proof.

For any fixed $m_{0}\in\\{1,2,\dots,M\\}$, setting $v_{\tau}|_{I_{m}}=0$ for
$m=1,\cdots,m_{0}-1$,
$v_{\tau}|_{I_{m}}=-\partial_{t}(-\Delta)^{-1}z_{\tau}|_{I_{m}}$ for
$m=m_{0},\cdots,M$ and $v_{\tau}(\cdot,T)=z_{\tau}(T)=z_{T}$ in (64), it is
clear that such $v_{\tau}\in Y_{\tau}$ and we have

$\displaystyle A(v_{\tau},z_{\tau})$
$\displaystyle=\|\nabla\partial_{t}(-\Delta)^{-1}z_{\tau}\|_{L^{2}(I^{\prime};L^{2}(\Omega))}^{2}+{1\over
2}(\|z_{\tau}(t_{m_{0}-1})\|_{L^{2}(\Omega)}^{2}+\|z_{\tau}(T)\|_{L^{2}(\Omega)}^{2})$
$\displaystyle=-\int^{T}_{t_{m_{0}-1}}\langle
g,\partial_{t}(-\Delta)^{-1}z_{\tau}\rangle dt+\|z_{T}\|^{2}_{L^{2}(\Omega)}$
$\displaystyle\leq C\|g\|^{2}_{L^{2}(I^{\prime};H^{-1}(\Omega))}+{1\over
2}\|\nabla\partial_{t}(-\Delta)^{-1}z_{\tau}\|_{L^{2}(I^{\prime};L^{2}(\Omega))}^{2}+\|z_{T}\|^{2}_{L^{2}(\Omega)},$

where $I^{\prime}:=\cup_{m=m_{0}}^{M}I_{m}$, this implies that

$\displaystyle\|\nabla\partial_{t}(-\Delta)^{-1}z_{\tau}\|_{L^{2}(I^{\prime};L^{2}(\Omega))}^{2}+\|z_{\tau}(t_{m_{0}-1})\|_{L^{2}(\Omega)}^{2}\leq
C\|g\|^{2}_{L^{2}(I;H^{-1}(\Omega))}+\|z_{T}\|_{L^{2}(\Omega)}^{2}.$

That is, for any $t_{m}$ $(m=0,\cdots,M-1)$ we have the stability

$\displaystyle\|z_{\tau}(t_{m})\|_{L^{2}(\Omega)}\leq
C\|g\|_{L^{2}(I;H^{-1}(\Omega))}+\|z_{T}\|_{L^{2}(\Omega)},$

i.e., $\|z_{\tau}\|_{L^{\infty}(I;L^{2}(\Omega))}\leq
C\|g\|_{L^{2}(I;H^{-1}(\Omega))}+\|z_{T}\|_{L^{2}(\Omega)}$. This finishes the
proof of (65).

Next, we prove the estimate (66). Similar to the above procedure, for any
fixed $m_{0}\in\\{1,2,\cdots,M\\}$, setting $v_{\tau}|_{I_{m}}=0$ for
$m=1,\cdots,m_{0}-1$, $v_{\tau}|_{I_{m}}=-\partial_{t}z_{\tau}|_{I_{m}}$ for
$m=m_{0},\cdots,M$ and $v_{\tau}(\cdot,T)=z_{\tau}(T)=z_{T}$ in (64), it is
clear that $v_{\tau}\in Y_{\tau}$ and we have

$\displaystyle A(v_{\tau},z_{\tau})$
$\displaystyle=\|\partial_{t}z_{\tau}\|_{L^{2}(I^{\prime};L^{2}(\Omega))}^{2}+{1\over
2}(\|\nabla z_{\tau}(t_{m_{0}-1})\|_{L^{2}(\Omega)}^{2}-\|\nabla
z_{\tau}(T)\|_{L^{2}(\Omega)}^{2})+\|z_{\tau}(T)\|_{L^{2}(\Omega)}^{2}$
$\displaystyle=-\int^{T}_{t_{m_{0}-1}}(g,\partial_{t}z_{\tau})dt+\|z_{\tau}(T)\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq C\|g\|^{2}_{L^{2}(I^{\prime};L^{2}(\Omega))}+{1\over
2}\|\partial_{t}z_{\tau}\|_{L^{2}(I^{\prime};L^{2}(\Omega))}^{2}+\|z_{\tau}(T)\|_{L^{2}(\Omega)}^{2}.$

This implies that

$\displaystyle\|\partial_{t}z_{\tau}\|_{L^{2}(I^{\prime};L^{2}(\Omega))}^{2}+\|\nabla
z_{\tau}(t_{m_{0}-1})\|_{L^{2}(\Omega)}^{2}\leq
C\|g\|^{2}_{L^{2}(I^{\prime};L^{2}(\Omega))}+\|\nabla
z_{T}\|_{L^{2}(\Omega)}^{2},$

then the estimate (66) follows. This finishes the proof. ∎

Then we will prove the following stability result for the semi-discrete state
approximation.

######  Lemma 16.

Let $u_{\tau}\in Y_{\tau}$ solve (63) for given $f\in L^{2}(I;L^{2}(\Omega))$,
$u_{0}\in L^{2}(\Omega)$ and $q\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$.
Then there exists a constant $C>0$, independent of $\tau$, such that

(67) $\displaystyle\|\nabla
u_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{\tau}(T)\|_{L^{2}(\Omega)}\leq
C(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}+\|u_{0}\|_{L^{2}(\Omega)}).$

Moreover, if $f\in L^{2}(I;H^{1}(\Omega))$, $u_{0}\in H^{1}_{0}(\Omega)$ and
$q\in\mathcal{M}(\bar{I}_{c};H^{1}(\omega))$, then there holds

(68) $\displaystyle\|\Delta u_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}+\|\nabla
u_{\tau}(T)\|_{L^{2}(\Omega)}\leq
C(\|f\|_{L^{2}(I;H^{1}(\Omega))}+\|q\|_{\mathcal{M}(\bar{I}_{c};H^{1}(\omega))}+\|u_{0}\|_{H^{1}(\Omega)}),$

where $C>0$ is a constant independent of $\tau$.

###### Proof.

We first prove the estimate (67). Recall that
$\|g\|^{2}_{L^{2}(I;H^{-1}(\Omega))}=\|\nabla
u_{\tau}\|^{2}_{L^{2}(I;L^{2}(\Omega))}=\langle-\Delta
u_{\tau},u_{\tau}\rangle_{L^{2}(I;H^{-1},H^{1})}$ with $g=-\Delta u_{\tau}$,
and $\|h\|_{L^{2}(\Omega)}=\|u_{\tau}(T)\|_{L^{2}(\Omega)}$ with
$h=u_{\tau}(T)$. We denote by $z_{\tau}\in P_{\tau}$ the semi-discrete
approximation defined in (64) with $g=-\Delta u_{\tau}$ and
$z_{T}=u_{\tau}(T)$, and test (64) with $u_{\tau}$, then

$\displaystyle\int_{I}\langle g,u_{\tau}\rangle
dt+(z_{T},u_{\tau}(T))=A(u_{\tau},z_{\tau})$ $\displaystyle=$
$\displaystyle\int_{I}(f,z_{\tau})dt+\langle
q,z_{\tau}\rangle_{\bar{I}_{c}\times\omega}+(u_{0},z_{\tau}(0))$
$\displaystyle\leq$
$\displaystyle\|f\|_{L^{2}(I;L^{2}(\Omega))}\|z_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{0}\|_{L^{2}(\Omega)}\|z_{\tau}(0)\|_{L^{2}(\Omega)}+\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\|z_{\tau}\|_{{C}(\bar{I}_{c};L^{2}(\omega))}$
$\displaystyle\leq$ $\displaystyle
C(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{0}\|_{L^{2}(\Omega)}+\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))})\|z_{\tau}\|_{{C}(\bar{I};L^{2}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{0}\|_{L^{2}(\Omega)}+\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))})(\|g\|_{L^{2}(I;H^{-1}(\Omega))}+\|z_{T}\|_{L^{2}(\Omega)}),$

where we used the scheme (63) and the estimate (65) in Lemma 15. Therefore, we
obtain the estimate (67).

Now, we prepare to show (68). First, we denote by
$u_{\tau,m}:=u_{\tau}|_{I_{m}}$, $m=1,2,\cdots,M$, and
$u_{\tau,M+1}:=u_{\tau}(T)$. Thus there holds (cf. [20])

(69) $\begin{split}&(u_{\tau,1}-u_{0},v)+{1\over 2}\left(\tau_{1}\nabla
u_{1},\nabla v\right)=(\tilde{f}_{0},v)_{L^{2}(I_{1};L^{2}(\Omega))},\\\
&(u_{\tau,m+1}-u_{\tau,m},v)+{1\over 2}\left(\tau_{m+1}\nabla
u_{\tau,m+1}+\tau_{m}\nabla u_{\tau,m},\nabla v\right)=(\tilde{f}_{m},v),\\\
&(u_{\tau,M+1}-u_{\tau,M},v)+{1\over 2}\left(\tau_{M}\nabla u_{M},\nabla
v\right)=(\tilde{f}_{M},v)\end{split}$

for arbitrary $v\in H_{0}^{1}(\Omega)$, where

$\displaystyle\tilde{f}_{0}=(f,e_{t_{1}})_{L^{2}(I_{1})},\quad\tilde{f}_{M}=(f,e_{t_{M}})_{L^{2}(I_{M})},$
$\displaystyle\tilde{f}_{m}=(f,e_{t_{m}})_{L^{2}(I_{m}\cup
I_{m+1})}+\int_{(I_{m}\cup I_{m+1})\cap\bar{I}_{c}}e_{t_{m}}dq(t)\in
L^{2}(\Omega)\quad m=1,2,\cdots,M-1.$

Since $f\in L^{2}(I;H^{1}(\Omega))$ and
$q\in\mathcal{M}(\bar{I}_{c};H^{1}(\omega))$, we have $\tilde{f}_{m}\in
H^{1}(\Omega)$ for $m=0,1,\cdots,M$. Then from the first two expressions in
(69) it follows that $u_{\tau,m}\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$
($m=1,2,\cdots,M$) by the regularity of elliptic equations. By the last
expression, there has $u_{\tau,M+1}=\tilde{f}_{M}+u_{\tau,M}+{1\over
2}\tau_{M}\Delta u_{\tau,M}$, which implies that $u_{\tau,M+1}\in
L^{2}(\Omega)$. Therefore, we have $-\Delta u_{\tau,m}\in L^{2}(\Omega)$,
$1\leq m\leq M$ and $-\Delta u_{\tau,M+1}\in H^{-2}(\Omega)$ in the sense of
distributions. Using the similar idea we can show that for any $g\in
L^{2}(I;L^{2}(\Omega))$ and $z_{T}\in H^{1}_{0}(\Omega)$, the semi-discrete
solution of (64) satisfies $z_{\tau}\in L^{2}(I;H^{2}(\Omega))\cap
C(\bar{I};H^{1}(\Omega))$ and $z_{\tau}(0)\in H^{1}(\Omega)$.

With the above spatial regularity of semi-discrete solutions to (64) and (63)
we can rewrite the semi-discrete schemes (64) and (63) as: Find $u_{\tau}\in
Y_{\tau}$ such that for any $v_{\tau}\in P_{\tau}$

(70) $\int_{I}-(u_{\tau},\partial_{t}v_{\tau})-(\Delta
u_{\tau},v_{\tau})dt+(u_{\tau}(T),v_{\tau}(T))=\int_{I}(f,v_{\tau})dt+\langle
q,v_{\tau}\rangle_{\bar{I}_{c}\times\omega}+(u_{0},v_{\tau}(0)),$

and find $z_{\tau}\in P_{\tau}$ such that for any $w_{\tau}\in Y_{\tau}$

(71) $\int_{I}-(w_{\tau},\partial_{t}z_{\tau})+(w_{\tau},-\Delta
z_{\tau})dt+(w_{\tau}(T),z_{\tau}(T))=\int_{I}(g,w_{\tau})dt+(z_{T},w_{\tau}(T)).$

Since there are no spatial derivatives for the test functions in schemes (70)
and (71), the formulations (70) and (71) hold not only for all $v_{\tau}\in
P_{\tau}$ and $w_{\tau}\in Y_{\tau}$, but also hold, by the dense of
$H^{1}_{0}(\Omega)$ in $L^{2}(\Omega)$, for all $v_{\tau}\in\tilde{P}_{\tau}$
and $w_{\tau}\in\tilde{Y}_{\tau}$, respectively, where

$\displaystyle\tilde{P}_{\tau}:=\\{v_{\tau}\in
C(\bar{I};L^{2}(\Omega)):v_{\tau}|_{I_{m}}\in\mathcal{P}_{1}(I_{m};L^{2}(\Omega)),\
m=1,2,\cdots,M\\},$ $\displaystyle\tilde{Y}_{\tau}:=\\{v_{\tau}\in
L^{2}(I;L^{2}(\Omega)):v_{\tau}|_{I_{m}}\in\mathcal{P}_{0}(I_{m};L^{2}(\Omega)),m=1,2,\cdots,M,\
v_{\tau}(T)\in H^{-1}(\Omega)\\}.$

We denote by $z_{\tau}$ the semi-discrete approximation to the backward
equation (5) defined by (64), or equivalently, (71), for arbitrary $g\in
L^{2}(I;L^{2}(\Omega))$ and $z_{T}\in C_{0}^{\infty}(\Omega)$. Similarly, we
test (71) with $-\Delta u_{\tau}$, then

$\displaystyle\int_{I}(g,-\Delta u_{\tau})dt+\langle z_{T},-\Delta
u_{\tau}(T)\rangle=A(-\Delta u_{\tau},z_{\tau})$ $\displaystyle=$
$\displaystyle A(u_{\tau},-\Delta z_{\tau})$ $\displaystyle=$
$\displaystyle\int_{I}\langle f,-\Delta z_{\tau}\rangle dt+\langle q,-\Delta
z_{\tau}\rangle_{\bar{I}_{c}\times\omega}+\langle u_{0},-\Delta
z_{\tau}(0)\rangle$ $\displaystyle\leq$
$\displaystyle\|f\|_{L^{2}(I;H^{1}(\Omega))}\|\nabla
z_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}+\|\nabla u_{0}\|_{L^{2}(\Omega)}\|\nabla
z_{\tau}(0)\|_{L^{2}(\Omega)}$
$\displaystyle+C\|q\|_{\mathcal{M}(\bar{I}_{c};H^{1}(\omega))}\|\nabla
z_{\tau}\|_{{C}(\bar{I}_{c};L^{2}(\Omega))}$ $\displaystyle\leq$
$\displaystyle
C(\|f\|_{L^{2}(I;H^{1}(\Omega))}+\|u_{0}\|_{H^{1}(\Omega)}+\|q\|_{\mathcal{M}(\bar{I}_{c};H^{1}(\omega))})\|\nabla
z_{\tau}\|_{{C}(\bar{I};L^{2}(\Omega))}$ $\displaystyle\leq$ $\displaystyle
C(\|f\|_{L^{2}(I;H^{1}(\Omega))}+\|u_{0}\|_{H^{1}(\Omega)}+\|q\|_{\mathcal{M}(\bar{I}_{c};H^{1}(\omega))})(\|g\|_{L^{2}(I;L^{2}(\Omega))}+\|\nabla
z_{T}\|_{L^{2}(\Omega)}),$

where we have used the scheme (70) and the estimate (66) in Lemma 15. By the
density of $C_{0}^{\infty}(\Omega)$ in $H_{0}^{1}(\Omega)$ we obtain

$\displaystyle\|\Delta u_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}+\|\Delta
u_{\tau}(T)\|_{H^{-1}(\Omega)}\leq
C(\|f\|_{L^{2}(I;H^{1}(\Omega))}+\|u_{0}\|_{H^{1}(\Omega)}+\|q\|_{\mathcal{M}(\bar{I}_{c};H^{1}(\omega))}),$

which confirms the estimate (68) by using the fact $\|\Delta
u_{\tau}(T)\|_{H^{-1}(\Omega)}\approx\|\nabla u_{\tau}(T)\|_{L^{2}(\Omega)}$.
This finishes the proof. ∎

Furthermore, the fully discrete finite element approximation of the backward
parabolic equation (5) can be defined as: Find $z_{\sigma}\in P_{\sigma}$ such
that

(72) $\displaystyle
A(v_{\sigma},z_{\sigma})=\int_{I}(g,v_{\sigma})dt+(v_{\sigma}(T),z_{T})\quad\
\forall v_{\sigma}\in Y_{\sigma}.$

Similar to Lemma 15 we have the following stability result on the fully
discrete approximation of backward parabolic equations.

######  Lemma 17.

Let $z_{\sigma}\in P_{\sigma}$ solve (72) for given $g\in
L^{2}(I;H^{-1}(\Omega))$ and $z_{T}\in{L^{2}(\Omega)}$. Then there exists a
constant $C>0$, independent of $\sigma$, such that

(73)
$\displaystyle\left\|\nabla\partial_{t}(-\Delta_{h})^{-1}z_{\sigma}\right\|_{L^{2}(I;L^{2}(\Omega))}+\|z_{\sigma}\|_{C(\bar{I};L^{2}(\Omega))}\leq
C\left(\|g\|_{L^{2}(I;H^{-1}(\Omega))}+\|z_{T}\|_{L^{2}(\Omega)}\right),$

where $-\Delta_{h}:V_{h}\rightarrow V_{h}$ is defined as
$(-\Delta_{h}v_{h},w_{h}):=(\nabla v_{h},\nabla w_{h})$ for any $w_{h}\in
V_{h}$.

###### Proof.

Setting $v_{\sigma}\in Y_{\sigma}$ such that
$v_{\sigma}|_{I_{m}}=-\partial_{t}(-\Delta_{h})^{-1}z_{\sigma}|_{I_{m}}\in
Y_{\sigma}$ ($m=m_{0},\cdots,M$), $v_{\sigma}|_{I_{m}}=0\
(m=1,2,\cdots,m_{0}-1)$ and $v_{\sigma}(T)=z_{\sigma}(T)$ in (72) for
arbitrary $1\leq m_{0}\leq M$, there holds

$\displaystyle A(v_{\sigma},z_{\sigma})$
$\displaystyle=\|\nabla\partial_{t}(-\Delta_{h})^{-1}z_{\sigma}\|_{L^{2}(I^{\prime};L^{2}(\Omega))}^{2}+{1\over
2}(\|z_{\sigma}(t_{m_{0}-1})\|_{L^{2}(\Omega)}^{2}+\|z_{\sigma}(T)\|_{L^{2}(\Omega)}^{2})$
$\displaystyle=\int_{t_{m_{0}-1}}^{T}\langle
g,-\partial_{t}(-\Delta_{h})^{-1}z_{\sigma}\rangle dt+(z_{T},z_{\sigma}(T))$
$\displaystyle\leq C\|g\|^{2}_{L^{2}(I^{\prime};H^{-1}(\Omega))}+{1\over
2}\|\nabla\partial_{t}(-\Delta_{h})^{-1}z_{\sigma}\|_{L^{2}(I^{\prime};L^{2}(\Omega))}^{2}+C\|z_{T}\|^{2}_{L^{2}(\Omega)}+{1\over
2}\|z_{\sigma}(T)\|^{2}_{L^{2}(\Omega)},$

which implies that

$\displaystyle\|\nabla\partial_{t}(-\Delta_{h})^{-1}z_{\sigma}\|_{L^{2}(I^{\prime};L^{2}(\Omega))}+\|z_{\sigma}(t_{m_{0}-1})\|_{L^{2}(\Omega)}\leq
C(\|g\|_{L^{2}(I^{\prime};H^{-1}(\Omega))}+\|z_{T}\|_{L^{2}(\Omega)})$

for any $1\leq m_{0}\leq M$. Therefore, we can derive the conclusion. This
finishes the proof. ∎

Now we prove the following stability for the fully discrete approximation to
the state equation.

######  Lemma 18.

For given $f\in L^{2}(I;L^{2}(\Omega))$, $u_{0}\in L^{2}(\Omega)$ and
$q\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$, let $u_{\sigma}(q)\in
Y_{\sigma}$ solve (54). Then there holds the following estimate:

$\displaystyle\|\nabla
u_{\sigma}(q)\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{\sigma}(q)(T)\|_{L^{2}(\Omega)}\leq
C(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}+\|u_{0}\|_{L^{2}(\Omega)}),$

where $C>0$ is a constant independent of $\sigma$.

###### Proof.

Similar to Lemma 16, we denote by $z_{\sigma}$ the discrete approximation
defined in (72) with $g=u_{\sigma}$ and $z_{T}=u_{\sigma}(T)$, then

$\displaystyle\int_{I}(g,u_{\sigma})dt+(z_{T},u_{\sigma}(T))=A(u_{\sigma},z_{\sigma})$
$\displaystyle=$ $\displaystyle\int_{I}(f,z_{\sigma})dt+\langle
q,z_{\sigma}\rangle_{\bar{I}_{c}\times\omega}+(u_{0},z_{\sigma}(0))$
$\displaystyle\leq$
$\displaystyle\|f\|_{L^{2}(I;L^{2}(\Omega))}\|z_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{0}\|_{L^{2}(\Omega)}\|z_{\sigma}(0)\|_{L^{2}(\Omega)}+\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\|z_{\sigma}\|_{{C}(\bar{I}_{c};L^{2}(\omega))}$
$\displaystyle\leq$ $\displaystyle
C(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{0}\|_{L^{2}(\Omega)}+\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))})\|z_{\sigma}\|_{{C}(\bar{I};L^{2}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{0}\|_{L^{2}(\Omega)}+\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))})(\|g\|_{L^{2}(I;H^{-1}(\Omega))}+\|z_{T}\|_{L^{2}(\Omega)}),$

where we have used Lemma 17. Then we can obtain the result by canceling the
common term. This finishes the proof. ∎

###  5.2 A priori error estimates for the state and adjoint equations

In this subsection we are now able to give a priori error estimates for the
finite element solutions to the state and adjoint equations.

######  Theorem 19.

For arbitrary $f\in L^{2}(I;H^{1}(\Omega))$,
$q\in\mathcal{M}(\bar{I}_{c};H^{1}(\omega))$ and $u_{0}\in H^{1}_{0}(\Omega)$,
let $u\in L^{2}(I;L^{2}(\Omega))$ be the solution to problem (2) and
$u_{\sigma}\in Y_{\sigma}$ be its discretization defined in (54). Then there
exists a positive constant $C$, independent of $\sigma$, such that

(75)
$\displaystyle\|u-u_{\sigma}(q)\|_{L^{2}(I;L^{2}(\Omega))}+\|(u-u_{\sigma}(q))(T)\|_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle C(h+\tau^{1\over
2})(\|f\|_{L^{2}(I;H^{1}(\Omega))}+\|q\|_{\mathcal{M}(\bar{I}_{c};H^{1}(\omega))}+\|u_{0}\|_{H^{1}(\Omega)}).$

###### Proof.

We split the fully discrete error estimate into the temporal and spatial
parts. To begin with, let $u_{\tau}\in Y_{\tau}$ be the semi-discrete solution
to problem (63). Then we estimate respectively
$\|u-u_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}+\|u(T)-u_{\tau}(T)\|_{L^{2}(\Omega)}$
and
$\|u_{\tau}-u_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{\tau}(T)-u_{\sigma}(T)\|_{L^{2}(\Omega)}$.
To prove the two estimates we use the duality argument (cf. [5, 26]).

We first prove the estimate for
$\|u-u_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}+\|u(T)-u_{\tau}(T)\|_{L^{2}(\Omega)}$.
Let $z\in H^{1}(I;L^{2}(\Omega))\cap L^{2}(I;H^{2}(\Omega)\cap
H^{1}_{0}(\Omega))$ be the solution to the backward parabolic equation (5)
with $g:=u-u_{\tau},\ z_{T}:=u(T)-u_{\tau}(T)$. Setting
$\tilde{z}_{\tau}:={\Pi}_{P_{\tau}}z\in P_{\tau}$, then there holds
$(z-\tilde{z}_{\tau})(T)=0$ and

$(v_{\tau},\partial_{t}(z-\tilde{z}_{\tau}))_{L^{2}(I;L^{2}(\Omega))}=0\quad\forall
v_{\tau}\in Y_{\tau}.$

From (37) and (63) we have

$\displaystyle\int_{I}(g,u-u_{\tau})dt+(z_{T},u(T)-u_{\tau}(T))=A(u,z)-A(u_{\tau},z_{\tau})$
$\displaystyle=$ $\displaystyle
A(u,z)-A(u_{\tau},z)-\int_{I}(f,\tilde{z}_{\tau})dt-\langle
q,\tilde{z}_{\tau}\rangle_{\bar{I}_{c}\times\omega}-(u_{0},\tilde{z}_{\tau}(0))+A(u_{\tau},\tilde{z}_{\tau})$
$\displaystyle=$ $\displaystyle\int_{I}(f,z-\tilde{z}_{\tau})dt+\langle
q,z-\tilde{z}_{\tau}\rangle_{\bar{I}_{c}\times\omega}+(u_{0},z(0)-\tilde{z}_{\tau}(0))-A(u_{\tau},z-\tilde{z}_{\tau})$
$\displaystyle=$ $\displaystyle\int_{I}(f,z-\tilde{z}_{\tau})dt+\langle
q,z-\tilde{z}_{\tau}\rangle_{\bar{I}_{c}\times\omega}-\int_{I}(\nabla
u_{\tau},\nabla(z-\tilde{z}_{\tau}))dt$ $\displaystyle\leq$
$\displaystyle\left(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|\Delta
u_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}\right)\|z-\tilde{z}_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}+\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\|z-\tilde{z}_{\tau}\|_{C(\bar{I}_{c};L^{2}(\omega))}$
$\displaystyle\leq$ $\displaystyle
C\tau\left(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|\Delta
u_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}\right)\|z\|_{H^{1}(I;L^{2}(\Omega))}+C\tau^{\frac{1}{2}}\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\|z\|_{H^{1}({I}_{c};L^{2}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C\tau\left(\|f\|_{L^{2}(I;L^{2}(\Omega))}+\|\Delta
u_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}\right)\left(\|g\|_{L^{2}(I;L^{2}(\Omega))}+\|z_{T}\|_{H^{1}(\Omega)}\right)$
$\displaystyle+C\tau^{\frac{1}{2}}\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\left(\|g\|_{L^{2}(I;L^{2}(\Omega))}+\|z_{T}\|_{L^{2}(\Omega)}\right)$
$\displaystyle\leq$ $\displaystyle C\tau
H\left(\|g\|_{L^{2}(I;L^{2}(\Omega))}+H\right)+C\tau^{\frac{1}{2}}\|q\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\left(\|g\|_{L^{2}(I;L^{2}(\Omega))}+\|z_{T}\|_{L^{2}(\Omega)}\right)$
$\displaystyle\leq$ $\displaystyle
C\tau(1+\tau)H^{2}+C\tau\|q\|^{2}_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}+{1\over
2}(\|g\|^{2}_{L^{2}(I;L^{2}(\Omega))}+\|z_{T}\|^{2}_{L^{2}(\Omega)}),$

where we have used Lemma 16, Theorem 3 and
$H:=\|f\|_{L^{2}(I;H^{1}(\Omega))}+\|q\|_{\mathcal{M}(\bar{I}_{c};H^{1}(\omega))}+\|u_{0}\|_{H^{1}(\Omega)}$.
Therefore, we have obtained

$\displaystyle\|u-u_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}+\|u(T)-u_{\tau}(T)\|_{L^{2}(I;L^{2}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C\tau^{\frac{1}{2}}(\|f\|_{L^{2}(I;H^{1}(\Omega))}+\|q\|_{\mathcal{M}(\bar{I}_{c};H^{1}(\omega))}+\|u_{0}\|_{H^{1}(\Omega)}).$

Next, we estimate
$\|u_{\tau}-u_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{\tau}(T)-u_{\sigma}(T)\|_{L^{2}(\Omega)}$.
Note that there exists the following splitting:

$\displaystyle
u_{\tau}-u_{\sigma}=u_{\tau}-\mathcal{R}_{h}u_{\tau}+\mathcal{R}_{h}u_{\tau}-u_{\sigma}:=\eta_{\sigma}+\xi_{\sigma},$

where $\mathcal{R}_{h}:H_{0}^{1}(\Omega)\rightarrow V_{h}$ is the standard
spatial Ritz projection (cf. [16]).

Let $z_{\sigma}\in P_{\sigma}$ be the fully discrete solution to problem (72)
with $g:=\xi_{\sigma}$ and $z_{T}:=\xi_{\sigma}(T)$. Taking
$v_{\sigma}=\xi_{\sigma}\in Y_{\sigma}$ in the scheme (72) and applying the
Galerkin orthogonality, one obtains

$\displaystyle\int_{I}(g,\xi_{\sigma})dt+(z_{T},\xi_{\sigma}(T))=A(\xi_{\sigma},z_{\sigma})=-A(\eta_{\sigma},z_{\sigma})$
$\displaystyle=$
$\displaystyle\int_{I}(\eta_{\sigma},\partial_{t}z_{\sigma})-(\nabla\eta_{\sigma},\nabla
z_{\sigma})dt-(\eta_{\sigma}(T),z_{\sigma}(T))$ $\displaystyle=$
$\displaystyle\int_{I}(\eta_{\sigma},\partial_{t}z_{\sigma})dt-(\eta_{\sigma}(T),z_{\sigma}(T))$
$\displaystyle\leq$
$\displaystyle\|\nabla\eta_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}\|\nabla\partial_{t}(-\Delta_{h})^{-1}z_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}+\|\eta_{\sigma}(T)\|_{L^{2}(\Omega)}\|z_{\sigma}(T)\|_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
C(\|\nabla\eta_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}+\|\eta_{\sigma}(T)\|_{L^{2}(\Omega)})(\|g\|_{L^{2}(I;L^{2}(\Omega))}+\|z_{T}\|_{L^{2}(\Omega)}),$

where we have used Lemma 17. Therefore, we have

$\displaystyle\|u_{\tau}-u_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{\tau}(T)-u_{\sigma}(T)\|_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
C(\|\nabla\eta_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}+\|\eta_{\sigma}(T)\|_{L^{2}(\Omega)})$
$\displaystyle\leq$ $\displaystyle Ch(\|\Delta
u_{\tau}\|_{L^{2}(I;L^{2}(\Omega))}+\|\nabla u_{\tau}(T)\|_{L^{2}(\Omega)})$
$\displaystyle\leq$ $\displaystyle
Ch(\|f\|_{L^{2}(I;H^{1}(\Omega))}+\|q\|_{\mathcal{M}(\bar{I}_{c};H^{1}(\omega))}+\|u_{0}\|_{H^{1}(\Omega)}),$

where we have used Lemma 16. Combining the above two estimates we finish the
proof. ∎

######  Theorem 20.

For any $z_{T}\in H^{1}_{0}(\Omega)$ and $g\in L^{2}(I;L^{2}(\Omega))$, let
$z_{\sigma}\in P_{\sigma}$ be the solution of the discrete scheme (72), and
$z\in H^{1}(I;L^{2}(\Omega))\cap L^{2}(I;H^{2}(\Omega)\cap H^{1}_{0}(\Omega))$
be the solution of equation (5). Then there exists a positive constant $C>0$,
independent of $\sigma$, such that

(78) $\|z-z_{\sigma}\|_{C(\bar{I};L^{2}(\Omega))}\leq
C(h+\tau^{\frac{1}{2}})(\|g\|_{L^{2}(I;L^{2}(\Omega))}+\|z_{T}\|_{H^{1}(\Omega)}).$

###### Proof.

Let
$e_{\sigma}:=z-z_{\sigma}=(z-\pi_{h}\Pi_{P_{\tau}}z)+(\pi_{h}\Pi_{P_{\tau}}z-z_{\sigma})=:\eta_{\sigma}+\zeta_{\sigma}$,
then by the Galerkin orthogonality there holds for any $v_{\sigma}\in
Y_{\sigma}$ that

$\displaystyle A(v_{\sigma},\zeta_{\sigma})=-A(v_{\sigma},\eta_{\sigma})$
$\displaystyle=$
$\displaystyle\int_{I}(v_{\sigma},\partial_{t}\eta_{\sigma})-(\nabla
v_{\sigma},\nabla\eta_{\sigma})dt-(v_{\sigma}(T),\eta_{\sigma}(T))$
$\displaystyle=$
$\displaystyle\sum_{m=1}^{M}\Big{(}v_{\sigma}|_{I_{m}},(z-\pi_{h}\Pi_{P_{\tau}}z)(t_{m})-(z-\pi_{h}\Pi_{P_{\tau}}z)(t_{m-1})\Big{)}$
$\displaystyle-(v_{\sigma}(T),\eta_{\sigma}(T))-\int_{I}(\nabla
v_{\sigma},\nabla\eta_{\sigma})dt$ $\displaystyle=$
$\displaystyle-\int_{I}(\nabla v_{\sigma},\nabla\eta_{\sigma})dt,$

i.e., $\zeta_{\sigma}$ satisfies the following variational problem: Find
$\zeta_{\sigma}\in P^{0}_{\sigma}$ such that

(80) $\displaystyle\int_{I}-(v_{\sigma},\partial_{t}\zeta_{\sigma})+(\nabla
v_{\sigma},\nabla\zeta_{\sigma})dt=-\int_{I}(\nabla
v_{\sigma},\nabla\eta_{\sigma})dt\quad\ \forall v_{\sigma}\in Y_{\sigma}.$

For arbitrary $1\leq m_{0}\leq M$, taking the test function $v_{\sigma}$
satisfying
$v_{\sigma}|_{I_{m}}=-\partial_{t}(-\Delta_{h})^{-1}\zeta_{\sigma}|_{I_{m}}$,
$m=m_{0},m_{0}+1,\cdots,M$ and $v_{\sigma}|_{I_{m}}=0$,
$m=1,2,\cdots,m_{0}-1$, $v_{\sigma}(T)=0$ in the above identity (80), then we
have

$\displaystyle\left\|\nabla\partial_{t}(-\Delta_{h})^{-1}\zeta_{\sigma}\right\|^{2}_{L^{2}(I^{\prime};L^{2}(\Omega))}+\frac{1}{2}\|\zeta_{\sigma}(t_{m_{0}-1})\|^{2}_{L^{2}(\Omega)}$
$\displaystyle=$
$\displaystyle\int_{t_{m_{0}-1}}^{T}(\nabla\partial_{t}(-\Delta_{h})^{-1}\zeta_{\sigma},\nabla\eta_{\sigma})dt$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\|\nabla\partial_{t}(-\Delta_{h})^{-1}\zeta_{\sigma}\|^{2}_{L^{2}({I^{\prime}};L^{2}(\Omega))}+\frac{1}{2}\|\nabla\eta_{\sigma}\|^{2}_{L^{2}({I^{\prime}};L^{2}(\Omega))}.$

Therefore, we obtain

$\displaystyle\|\zeta_{\sigma}(t_{m_{0}-1})\|^{2}_{L^{2}(\Omega)}\leq\|\nabla\eta_{\sigma}\|^{2}_{L^{2}({I^{\prime}};L^{2}(\Omega))}$

for arbitrary $1\leq m_{0}\leq M$, i.e.,

$\displaystyle\|\zeta_{\sigma}\|_{C(\bar{I};L^{2}(\Omega))}=\max_{1\leq m\leq
M-1}\|\zeta_{\sigma}(t_{m})\|_{L^{2}(\Omega)}\leq\|\nabla\eta_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}.$

Combining with the expression $e_{\sigma}=\eta_{\sigma}+\zeta_{\sigma}$ one
deduces

(81)
$\displaystyle\|e_{\sigma}\|_{C(\bar{I};L^{2}(\Omega))}\leq\|\eta_{\sigma}\|_{C(\bar{I};L^{2}(\Omega))}+\|\nabla\eta_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}.$

Then it suffices to bound the two terms on the right-hand side.

The first term on the right-hand side of (81) can be bounded by

(82) $\displaystyle\|\eta_{\sigma}\|_{C(\bar{I};L^{2}(\Omega))}$
$\displaystyle=$
$\displaystyle\|z-\pi_{h}\Pi_{P_{\tau}}z\|_{C(\bar{I};L^{2}(\Omega))}$
$\displaystyle\leq$
$\displaystyle\|z-\pi_{h}z\|_{C(\bar{I};L^{2}(\Omega))}+\|\pi_{h}(z-\Pi_{P_{\tau}}z)\|_{C(\bar{I};L^{2}(\Omega))}$
$\displaystyle\leq$
$\displaystyle\|z-\pi_{h}z\|_{C(\bar{I};L^{2}(\Omega))}+C\|z-\Pi_{P_{\tau}}z\|_{C(\bar{I};L^{2}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
Ch\|z\|_{C(\bar{I};H^{1}(\Omega))}+C\tau^{\frac{1}{2}}\|z\|_{H^{1}(I;L^{2}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C(h+\tau^{\frac{1}{2}})(\|z\|_{L^{2}(I;H^{2}(\Omega))}+\|z\|_{H^{1}(I;L^{2}(\Omega))})$
$\displaystyle\leq$ $\displaystyle
C(h+\tau^{\frac{1}{2}})(\|g\|_{L^{2}(I;L^{2}(\Omega))}+\|z_{T}\|_{H^{1}(\Omega)}),$

where we have used the stability of the $L^{2}$-projection $\pi_{h}$. On the
other hand, $\|\nabla\eta_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}$ can be
estimated by

(83) $\displaystyle\|\nabla\eta_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}$
$\displaystyle=$
$\displaystyle\|\nabla(z-\pi_{h}\Pi_{P_{\tau}}z)\|_{L^{2}(I;L^{2}(\Omega))}$
$\displaystyle\leq$
$\displaystyle\|\nabla(z-\pi_{h}z)\|_{L^{2}(I;L^{2}(\Omega))}+\|\nabla\pi_{h}(z-\Pi_{P_{\tau}}z)\|_{L^{2}(I;L^{2}(\Omega))}$
$\displaystyle\leq$
$\displaystyle\|\nabla(z-\pi_{h}z)\|_{L^{2}(I;L^{2}(\Omega))}+\|\nabla(z-\Pi_{P_{\tau}}z)\|_{L^{2}(I;L^{2}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
Ch\|z\|_{L^{2}(I;H^{2}(\Omega))}+C\tau^{\frac{1}{2}}(\|z\|_{L^{2}(I;H^{2}(\Omega))}+\|z\|_{H^{1}(I;L^{2}(\Omega))})$
$\displaystyle\leq$ $\displaystyle
C(h+\tau^{\frac{1}{2}})(\|g\|_{L^{2}(I;L^{2}(\Omega))}+\|z_{T}\|_{H^{1}(\Omega)}),$

where we have used the $H^{1}$-stability of the $L^{2}$-projection $\pi_{h}$
and the estimate (52) in Lemma 13. Combining the two estimates (82) and (83)
we finish the proof. ∎

###  5.3 Error estimates for the optimal control problem

At first we prove a plain convergence for the solution of problem (53) to that
of problem (1) as $|\sigma|:=\tau+h\rightarrow 0^{+}$.

######  Theorem 21.

Let $\\{\hat{q}_{\sigma}\\}\subseteq\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ be
the set of optimal controls for the discrete optimal control problem (53), and
$\bar{u}_{\sigma}\in Y_{\sigma}$ be the unique discrete optimal state
associated to $\\{\hat{q}_{\sigma}\\}$. Let
$(\bar{q},\bar{u})\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))\times X$ be the
unique optimal pair of the continuous problem (1), where $\bar{q}$ is the
optimal control and $\bar{u}$ is the optimal state. Then we obtain

(84)
$\displaystyle{q}_{\sigma}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\bar{q}\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))\quad\
\forall q_{\sigma}\in\\{\hat{q}_{\sigma}\\},$ (85)
$\displaystyle\|q_{\sigma}\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\rightarrow\|\bar{q}\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\quad\
\forall q_{\sigma}\in\\{\hat{q}_{\sigma}\\},$ (86)
$\displaystyle\|\bar{u}_{\sigma}-\bar{u}\|_{L^{2}(I;L^{2}(\Omega))}+\|(\bar{u}_{\sigma}-\bar{u})(T)\|_{L^{2}(\Omega)}\rightarrow
0,\quad$ (87) $\displaystyle J_{\sigma}(q_{\sigma})\rightarrow
J(\bar{q})\quad\ \forall q_{\sigma}\in\\{\hat{q}_{\sigma}\\},$

when $|\sigma|\rightarrow 0^{+}$.

###### Proof.

The main ideas follow from [5, Theorem 4.9] and [31, Theorem 1.2], see also
[4, Theorem 3.5]. Similar to Theorem 9, since $q_{\sigma}$ is optimal, we can
easily show that the sequence $\\{q_{\sigma}\\}$ is uniformly bounded in
$\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ with respect to $\sigma$. Then there
exists a subsequence, still denoted by $\\{q_{\sigma}\\}$, such that
$q_{\sigma}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\tilde{q}$ in
$\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ for some $\tilde{q}$ when
$\sigma\rightarrow 0^{+}$. Below, we show that $\tilde{q}=\bar{q}$.

Let $u_{\tilde{q}}$ be the solution of equation (2) with $q$ replaced by
$\tilde{q}$. We first show that

(88)
$\displaystyle\|u_{\tilde{q}}-\bar{u}_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}+\|(u_{\tilde{q}}-\bar{u}_{\sigma})(T)\|_{L^{2}(\Omega)}\rightarrow
0\quad\mbox{as}\ |\sigma|\rightarrow 0.$

In fact, from the triangle inequality we have

(89)
$\displaystyle\|u_{\tilde{q}}-\bar{u}_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}+\|(u_{\tilde{q}}-\bar{u}_{\sigma})(T)\|_{L^{2}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\|u_{\tilde{q}}-u_{q_{\sigma}}\|_{L^{2}(I;L^{2}(\Omega))}+\|(u_{\tilde{q}}-u_{q_{\sigma}})(T)\|_{L^{2}(\Omega)}$
$\displaystyle+$
$\displaystyle\|u_{q_{\sigma}}-\bar{u}_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}+\|(u_{q_{\sigma}}-\bar{u}_{\sigma})(T)\|_{L^{2}(\Omega)}.$

Applying Proposition 4 to the first two terms and Theorem 19 to the latter two
terms on the right-hand side of the above estimate yields the result.

Next, there holds

(90) $\displaystyle j(\tilde{q})\leq\liminf\limits_{|\sigma|\rightarrow
0}j_{\sigma}(q_{\sigma})\leq\limsup\limits_{|\sigma|\rightarrow
0}j_{\sigma}(q_{\sigma})\leq\limsup\limits_{|\sigma|\rightarrow
0}j_{\sigma}(\Lambda_{\sigma}\bar{q})=j(\bar{q}),$

where in the first inequality we have used the weakly-∗ lower semicontinuity
of the cost functional $j$, in the third inequality we used the optimality of
$q_{\sigma}$, while in the last equality we have used (46) and (88).
Therefore, $\tilde{q}$ is also optimal, so $\tilde{q}=\bar{q}$ since $\bar{q}$
is unique, i.e.,
$q_{\sigma}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\bar{q}\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$,
then $\bar{u}=u_{\tilde{q}}$ by the unique solvability of the state equation,
which proves (84) and (86). Obviously, (87) can be concluded from (90).
Lastly, (85) follows from (86) and (87). ∎

A second convergence result concerns the convergence order of the objective
functional.

######  Theorem 22.

Let
$q_{\sigma}\in\\{\hat{q}_{\sigma}\\}\subseteq\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$
be any optimal control to the discrete problem (53) and
$\bar{q}\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$ be the unique optimal
control for the continuous problem (1). Then there exists a constant $C>0$,
independent of $\sigma$, such that

(91) $\displaystyle|j(\bar{q})-j_{\sigma}(q_{\sigma})|\leq C(h+\tau^{1\over
2}).$

###### Proof.

The proof uses the approach of [5, Theorem 5.1], see also [4, Theorem 4.1]. It
follows from the optimality of $\bar{q}$ and $q_{\sigma}$ that

$\displaystyle j(\bar{q})-j_{\sigma}(\bar{q})\leq
j(\bar{q})-j_{\sigma}(q_{\sigma})\leq j(q_{\sigma})-j_{\sigma}(q_{\sigma}),$

which means that

(92)
$\displaystyle|j(\bar{q})-j_{\sigma}(q_{\sigma})|\leq\max\\{|j(\bar{q})-j_{\sigma}(\bar{q})|,|j(q_{\sigma})-j_{\sigma}(q_{\sigma})|\\}.$

Now it remains to estimate the two terms on the right-hand side.

For arbitrary $\tilde{q}\in\mathcal{M}(\bar{I}_{c};L^{2}(\omega))$, we denote
by $u_{\tilde{q}}$ and $u_{\sigma}(\tilde{q})$ the unique solutions to
problems (2) and (54), respectively. Then we obtain from Theorem 19 that

$\displaystyle\|u_{\tilde{q}}-u_{\sigma}(\tilde{q})\|_{L^{2}(I;L^{2}(\Omega))}+\|(u_{\tilde{q}}-u_{\sigma}(\tilde{q}))(T)\|_{L^{2}(\Omega)}\leq
C(h+\tau^{1\over 2}).$

It is straightforward to show that

$\displaystyle|j(\tilde{q})-j_{\sigma}(\tilde{q})|$ $\displaystyle\leq$
$\displaystyle{1\over
2}\Big{|}\|u_{\tilde{q}}-u_{d}\|_{L^{2}(I;L^{2}(\Omega))}^{2}-\|u_{\sigma}(\tilde{q})-u_{d}\|_{L^{2}(I;L^{2}(\Omega))}^{2}\Big{|}$
$\displaystyle+{\beta\over
2}\Big{|}\|u_{\tilde{q}}(T)-u_{T}\|_{L^{2}(\Omega)}^{2}-\|u_{\sigma}(\tilde{q})(T)-u_{T}\|_{L^{2}(\Omega)}^{2}\Big{|}$
$\displaystyle\leq$ $\displaystyle
C(\|u_{\tilde{q}}\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{\sigma}(\tilde{q})\|_{L^{2}(I;L^{2}(\Omega))}+\|u_{d}\|_{L^{2}(I;L^{2}(\Omega))})\|u_{\tilde{q}}-u_{\sigma}(\tilde{q})\|_{L^{2}(I;L^{2}(\Omega))}$
$\displaystyle+C(\|u_{\tilde{q}}(T)\|_{L^{2}(\Omega)}+\|u_{\sigma}(\tilde{q})(T)\|_{L^{2}(\Omega)}+\|u_{T}\|_{L^{2}(\Omega)})\|(u_{\tilde{q}}-u_{\sigma}(\tilde{q}))(T)\|_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle C(h+\tau^{1\over 2}),$

where we have used Lemma 18 and Theorem 3. By setting $\tilde{q}=q$ and
$\tilde{q}=q_{\sigma}$ in the above error estimate we finish the proof by
considering (92). ∎

The last convergence result is about the approximation of the state equation.

######  Theorem 23.

Let $\bar{u}\in L^{2}(I;L^{2}(\Omega))$ be the optimal state of the continuous
optimal control problem (1) and $\bar{u}_{\sigma}\in Y_{\sigma}$ be the
discrete optimal state of the discrete optimization problem (53). Then there
exists a constant $C$, independent of $\sigma$, such that

(93)
$\displaystyle\|\bar{u}-\bar{u}_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}^{2}+\beta\|(\bar{u}-\bar{u}_{\sigma})(T)\|_{L^{2}(\Omega)}^{2}\leq
C(h+\tau^{1\over 2}).$

###### Proof.

In order to obtain the above estimate (93), we first introduce two auxiliary
variables. The first one is the finite element approximation to the state
equation (2) with the optimal control $\bar{q}$: Find $\hat{u}_{\sigma}\in
Y_{\sigma}$ such that

$A(\hat{u}_{\sigma},v_{\sigma})=\int_{I}(f,v_{\sigma})dt+\langle\bar{q},v_{\sigma}\rangle_{\bar{I}_{c}\times\omega}+(u_{0},v_{\sigma}(0))\quad\
\forall v_{\sigma}\in P_{\sigma},$

while the second is the finite element approximation to the adjoint equation
(18) with the optimal state $\bar{u}$: Find $\hat{\varphi}\in P_{\sigma}$ such
that

$A(\omega_{\sigma},\hat{\varphi}_{\sigma})=\int_{I}(\bar{u}-u_{d},\omega_{\sigma})dt+\beta(\bar{u}(T)-u_{T},\omega_{\sigma}(T))\quad\
\forall\omega_{\sigma}\in Y_{\sigma}.$

Taking $p=\hat{q}_{\sigma}$ in the continuous optimality condition (20) and
$p=\bar{q}$ in the discrete optimality condition (59), where
$\hat{q}_{\sigma}$ is any optimal control for the discrete optimization
problem (55), then adding them up we obtain

$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\langle\hat{q}_{\sigma}-\bar{q},\bar{\varphi}-\bar{\varphi}_{\sigma}\rangle_{\bar{I}_{c}\times\omega}$
$\displaystyle=$
$\displaystyle\langle\hat{q}_{\sigma}-\bar{q},\bar{\varphi}-\hat{\varphi}_{\sigma}\rangle_{\bar{I}_{c}\times\omega}+\langle\hat{q}_{\sigma}-\bar{q},\hat{\varphi}_{\sigma}-\bar{\varphi}_{\sigma}\rangle_{\bar{I}_{c}\times\omega}$
$\displaystyle=$
$\displaystyle\langle\hat{q}_{\sigma}-\bar{q},\bar{\varphi}-\hat{\varphi}_{\sigma}\rangle_{\bar{I}_{c}\times\omega}+A(\bar{u}_{\sigma}-\hat{u}_{\sigma},\hat{\varphi}_{\sigma}-\bar{\varphi}_{\sigma})$
$\displaystyle=$
$\displaystyle\langle\hat{q}_{\sigma}-\bar{q},\bar{\varphi}-\hat{\varphi}_{\sigma}\rangle_{\bar{I}_{c}\times\omega}+(\bar{u}-\bar{u}_{\sigma},\bar{u}_{\sigma}-\hat{u}_{\sigma})+\beta((\bar{u}-\bar{u}_{\sigma})(T),(\bar{u}_{\sigma}-\hat{u}_{\sigma})(T))$
$\displaystyle=$
$\displaystyle\langle\hat{q}_{\sigma}-\bar{q},\bar{\varphi}-\hat{\varphi}_{\sigma}\rangle_{\bar{I}_{c}\times\omega}+(\bar{u}-\bar{u}_{\sigma},\bar{u}_{\sigma}-\bar{u})+\beta((\bar{u}-\bar{u}_{\sigma})(T),(\bar{u}_{\sigma}-\bar{u})(T))$
$\displaystyle+(\bar{u}-\bar{u}_{\sigma},\bar{u}-\hat{u}_{\sigma})+\beta((\bar{u}-\bar{u}_{\sigma})(T),(\bar{u}-\hat{u}_{\sigma})(T)).$

Therefore, there holds

$\displaystyle\|\bar{u}-\bar{u}_{\sigma}\|_{L^{2}(I;L^{2}(\Omega))}^{2}+\beta\|(\bar{u}-\bar{u}_{\sigma})(T)\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\leq$ $\displaystyle
2\langle\hat{q}_{\sigma}-\bar{q},\bar{\varphi}-\hat{\varphi}_{\sigma}\rangle_{\bar{I}_{c}\times\omega}+\|\bar{u}-\hat{u}_{\sigma}\|^{2}_{L^{2}(I;L^{2}(\Omega))}+\beta\|(\bar{u}-\hat{u}_{\sigma})(T)\|^{2}_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
2\|\hat{q}_{\sigma}-\bar{q}\|_{\mathcal{M}(\bar{I}_{c};L^{2}(\omega))}\|\bar{\varphi}-\hat{\varphi}_{\sigma}\|_{C(\bar{I}_{c};L^{2}(\Omega))}+\|\bar{u}-\hat{u}_{\sigma}\|^{2}_{L^{2}(I;L^{2}(\Omega))}+\beta\|(\bar{u}-\hat{u}_{\sigma})(T)\|^{2}_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle C(h+\tau^{1\over 2}),$

where we have used Theorems 19 and 20. This finishes the proof. ∎

Acknowledgement The first author was supported by the Strategic Priority
Research Program of Chinese Academy of Sciences under grant No. XDB41000000,
the National Key Basic Research Program (2022YFA1004402) and the National
Natural Science Foundation of China under grant 12071468.

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