\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 267, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/267\hfil Finite volume methods of elliptic optimal control]
{A priori error estimates of finite volume methods for general
elliptic optimal \\ control problems}

\author[Y. Feng, Z. Lu, L. Cao, L. Li, S. Zhang \hfil EJDE-2017/267\hfilneg]
{Yuming Feng, Zuliang Lu, Longzhou Cao,  Lin Li,  Shuhua Zhang}

\address{Yuming Feng \newline
Key Laboratory of Intelligent Information Processing and Control,
Chongqing Three Gorges University, Wanzhou, Chongqing, 404100, China. \newline
Chongqing Engineering Research Center of Internet of Things
and Intelligent Control Technology,
Chongqing Three Gorges University,
Wanzhou, Chongqing, 404100, China}
\email{yumingfeng25928@163.com}

\address{Zuliang Lu (corresponding author)\newline
Key Laboratory for Nonlinear Science and System Structure,
Chongqing Three Gorges University,
Chongqing 404100, China. \newline
Research Center for Mathematics and Economics,
Tianjin University of Finance and Economics,
Tianjin 300222, China}
\email{zulianglux@126.com}

\address{Longzhou Cao \newline
Key Laboratory for Nonlinear Science and System Structure,
Chongqing Three Gorges University,
Chongqing 404100, China}
\email{caolongzhou@126.com}

\address{Lin Li \newline
Key Laboratory for Nonlinear Science and System Structure,
Chongqing Three Gorges University,
Chongqing 404100, China}
\email{linligx@126.com}

\address{Shuhua Zhang \newline
Research Center for Mathematics and Economics,
Tianjin University of Finance and Economics, 
Tianjin 300222, China}
\email{szhang@tjufe.edu.cn}


\dedicatory{Communicated by Goong Chen}

\thanks{Submitted August 9, 2017. Published October 27, 2017.}
\subjclass[2010]{49J20, 65N30}
\keywords{A priori error estimates; general elliptic optimal control problems; 
\hfill\break\indent finite volume methods; optimal-order}

\begin{abstract}
 In this article, we establish a priori error estimates for the finite
 volume approximation of general elliptic optimal control problems.
 We use finite volume methods to discretize the state and adjoint
 equation of the optimal control problems. For the variational
 inequality, we use the variational discretization methods to
 discretize the control. We show the existence and the uniqueness of
 the solution for discrete optimality conditions. Under some
 reasonable assumptions, we obtain some optimal order error estimates
 for the state, costate and control variables. On one hand, the
 convergence rate for the state, costate and control variables is
 $O(h^2)$ or $O(h^2 \sqrt{|\log(\frac{1}{h})|})$ in the sense of $L^2$
 norm or $L^{\infty}$ norm. On the other hand, the convergence rate
 for the state and costate variables is $O(h)$ or $O(h{|\log (\frac{1}{h})|})$
 in the sense of $H^1$ norm or $W^{1,\infty}$ norm.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

In recent years, optimal control problems have attracted substantial
interest due to their applications in aero-hydrodynamics,
atmospheric, hydraulic pollution problems, combustion, exploration
and extraction of oil and gas resources, and engineering. They must
be solved successfully with efficient numerical methods. Finite
element methods are an important numerical method for the problems
of partial differential equations and widely used in the numerical
solution of optimal control problems. There have been extensive
studies in convergence of finite element approximation for optimal
control problems. Let us mention two early papers devoted to linear
optimal control problems by Falk \cite{F} and Geveci \cite{TG}. A
systematic introduction of finite element method for optimal control
problems can be found in \cite{ChenHuang, YL1, YL2, YLH, RWHT, JL,
WY, LC, LC1, LCZ}, but there are very less published results on this
topic for finite volume methods for optimal control problems.
 Recently, the adaptive finite element method has been
investigated extensively and become one of the most popular methods
in the scientific computation and numerical modeling. In
\cite{hiis}, the authors studied a posteriori error estimates for
adaptive finite element discretizations of boundary control
problems. A posteriori error estimates and adaptive finite element
approximation for parameter estimation problems have been obtained
in \cite{KV, KL}. Some related works can also be found in
 \cite{Tingwen1,Tingwen2}.

Finite volume methods have a long history as a class of important
numerical tools for solving differential equations. Because of their local
conservative property and other attractive properties such as the
robustness with the unstructured meshes, the finite volume methods
are widely used in computational fluid dynamics. In general, two
different functional spaces are used in the finite volume methods,
one for the trial space and one for the test space. Owing to the two
different spaces, the numerical analysis of the finite volume
methods is more difficult than that of the finite element methods
and finite difference methods. So, the analysis of finite volume
methods lags far behind that of finite element and finite difference
methods. Early work for the finite volume methods can be found in
\cite{Bank, Cai, Chat, ChenLiZhou, ChouLi, Ewing}. In \cite{Bank},
Bank and Rose obtain the result that the finite volume approximation
is comparable with the finite element approximation in $H^1$ norm.
The optimal $L^2$ error estimate is obtained in \cite{ChenLiZhou}
under the assumption that $f\in H^1$. In \cite{Ewing}, Ewing obtain
the $H^1$ norm and maximum-norm error estimates. In \cite{Chat}, the
author proposes a nonconforming finite volume element method and
obtains the $L^2$ norm and $H^1$ norm error estimates. Chou and Ye
propose a discontinuous finite volume element method. Unified error
analysis for conforming, nonconforming and discontinuous finite
volume method is presented in \cite{ChouYe}. High order finite
volume methods can be found in \cite{ChenLong, ChenWuXu}. For other
recently development, we refer reader to see \cite{Cartensen, DuJu,
Kumar, ShuYuHuang}.


For optimal control problems, the state and costate variables are
discretized by continuous linear elements and the control variable
by piecewise constant or piecewise linear polynomials in most
references. The convergence rate of the control variable is $O(h)$
or $O(h^{3/2})$ in the sense of $L^2$ norm or $L^{\infty}$ norm in
\cite{Meyer2}. In \cite{Hinze}, Hinze proposes a variational
discretization methods for optimal control problems with control
constraints. With the variational discretization concept, the
control variable is not discretized directly, but discretized by a
projection of the discrete costate variable. The convergence rate of
the control variable is $O(h^2)$. There are two approaches to find
the approximate solution of the optimal control problems governed by
partial differential equation. One is of the
optimize-then-discretize type. One first applies the Lagrange
multiplier methods to obtain an optimal system, at the continuous
level, consisting of the state equation, an adjoint equation and an
optimal condition. Then one use some numerical method to discretize
the resulting system. The other is of the discretize-then-optimize
type. One first discretizes the optimal control problems by some
means and then applies the Lagrange multiplier rule to the resulting
discrete optimization problem. The two discrete systems, determined
by the two approaches, are the same when finite element method is
used. In general, these discrete systems are not the same. In
\cite{YanZhou}, the authors also use the optimize-then-discretize
approach to solve the optimal control problem governed by convection
dominated diffusion equation.

Recently, in \cite{LCH}, the authors discussed distributed optimal
control problems governed by elliptic equations by using the finite
volume element methods. The objective functional was
$\frac{1}{2}||y-
y_{d}||^2_{L^2(\Omega)}+\frac{1}{2}||u||^2_{L^2(\Omega)}$.
They used finite volume methods to discretize the state and adjoint
equation of the optimal control problems. Under some reasonable
assumptions, they obtained some error estimates. In this paper, we
will use the optimize-then-discretize methods to discretize general
elliptic optimal control problems. We consider the elliptic optimal
control with objective functional $g(y)+j(u)$. We show the existence
and the uniqueness of the solution for discrete optimality
conditions. Finally, we obtain some optimal order error estimates
for the state, costate and control variables.

For $1\leq p<\infty$ and $m$ a
 nonnegative integer let
$W^{m,p}(\Omega)=\{v\in L^{p}(\Omega);\ D^{\alpha}v\in L^{p}(\Omega)
\ \textrm{if}\ |\alpha|\leq m\} $ denote the Sobolev spaces endowed
with the norm $\| v \|_{m,p}^{p}=\sum_{\mid\alpha\mid\leq
m}\| D^\alpha v\|_{L^{p}(\Omega)}^{p},$ and the semi-norm $\mid
v\mid_{m,p}^{p}=\sum_{\mid\alpha\mid= m}\| D^\alpha
v\|_{L^{p}(\Omega)}^{p}$. We set $W_0^{m,p}(\Omega)=\{v\in
W^{m,p}(\Omega):
 v\mid_{\partial \Omega}=0\}$. For $p$=2, we denote
$H^m(\Omega)=W^{m,2}(\Omega)$, $H_0^m(\Omega)=W_0^{m,2}(\Omega)$,
and $\|\cdot\|_{m}=\|\cdot\|_{m,2}$, $ \|\cdot\|=\|\cdot\|_{0,2}.$

We consider the general elliptic optimal control problems
\begin{gather}
 \min_{u\in U}\{g(y)+j(u)\},\label{1e1}\\
-\operatorname{div}(A\nabla y)=f+u,\quad \text{in }\Omega,\label{1e2}\\
y=0,\quad \text{on }\partial\Omega,\label{1e3}
\end{gather}
 where $\Omega\subset \mathbb{R}^2$ is a convex bounded polygon with boundary
$\partial\Omega$, $g$ and $j$ are convex functionals,
$f\in H^1(\Omega)$, $U$ is denoted by $U=\{u\in L^2(\Omega): a\leq
u(x)\leq b,\ a.e.\ {\rm in}\ \Omega,\ a,b\in \mathbb{R}\}$. Furthermore, we
assume that the coefficient matrix $A(x)=(a_{i,j}(x))_{2\times 2}\in
(W^{2,\infty}({\Omega}))^{2\times 2}$ is a symmetric positive
definite matrix and there is a constant $c>0$ satisfying for any
vector $\mathbf{X}\in \mathbb{R}^2$, $\mathbf{X}^{t}A\mathbf{X}\geq c\|
\mathbf{X}\|_{\mathbb{R}^2}^2$.


This article is organized as follows.
In next section, we describe the
finite volume methods briefly and apply the piecewise linear finite
volume elements to the optimal control problems
\eqref{1e1}-\eqref{1e3}.
 In Section 3, we prove the existence and
the uniqueness of the solutions for discrete optimality conditions.
And then the optimal order error estimates in $L^2$ norm are derived
for the state, costate and control variables in Second 4. We
estimate the error of the numerical solutions of control, state and
costate in $L^{\infty}$ norm. Finally we estimate $W^{1,\infty}$ and
$H^1$ errors for the state and costate variables in Second 5.

\section{Finite volume element methods}

For the convex polygon $\Omega$, we consider a quasi-uniform
triangulation $\mathcal {T}_h$ consisting of closed triangle
elements $K$ such that $\bar{\Omega}=\cup_{K\in\mathcal
{T}_h}K$. We use $N_h$ to denote the set of all nodes or vertices of
$\mathcal {T}_h$. To define the dual partition $\mathcal {T}_h^*$ of
$\mathcal {T}_h$, we divide each $K\in\mathcal {T}_h$ into three
quadrilaterals by connecting the barycenter $C_K$ of $K$ with line
segments to the midpoints of edges of $K$ as is shown in Figure \ref{fig1}.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig1} % TriDu-2.eps
\end{center}
\caption{Dual partition of a triangular $K$.}
\label{fig1}
\end{figure}

The control volume $V_i$ consists of the quadrilaterals sharing the
same vertex $z_i$ as is shown in Figure \ref{fig2}.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig2} % Vi-1.eps
\end{center}
\caption{Control volume $V_i$ sharing the same vertex $z_i$.}
\label{fig2}
\end{figure}

The dual partition $\mathcal {T}_h^*$ consists of the union of the
control volume $V_i$. Let $h=\max\{h_K \}$, where $h_K$ is the
diameter of the triangle $K$. As is shown in \cite{Ewing}, the dual
partition $\mathcal {T}_h^*$ is also quasi-uniform, i.e., there
exists a positive constant $C$ such that
\[
C^{-1}h^2\leq \operatorname{meas}(V_i)\leq Ch^2,\quad \forall V_i \in\mathcal
{T}_h^*.
\]

We define the finite dimensional space $V_h$ associated with
$\mathcal {T}_h$ for the trial functions by
\begin{equation*}
V_h=\{v\in C(\Omega): v|_K\in P_1(K),\; \forall K\in \mathcal
{T}_h,\; v|_{\partial\Omega}=0 \},
\end{equation*}
and define the finite dimensional space $Q_h$ associated with the
dual partition $\mathcal {T}_h^*$ for the test functions by
\begin{equation*}
Q_h=\{q\in L^2(\Omega): q|_V\in P_0(V),\; \forall V\in \mathcal
{T}_h^*;\; q|_{V_z}=0,\; z\in\partial\Omega\},
\end{equation*}
where $P_l(K)$ or $P_l(V)$ consists of all the polynomials with
degree less than or equal to $l$ defined on $K$ or $V$.

To connect the trial space and test space, we define a transfer
operator $I_h: V_h\to Q_h$ as follows:
\begin{equation*}
I_hv_h=\sum_{z_i\in N_h} v_h(z_i)\chi_i ,\quad
I_hv_h|_{V_i}=v_h(z_i),\quad \forall V_i \in\mathcal {T}_h^*,
\end{equation*}
where $\chi_i$ is the characteristic function of $V_i$. For the
operator $I_h$, it is well known that there exists a positive
constant $C$ such that for all $v\in V_h$,
\begin{equation}
\|v-I_hv\|_{0,\Omega}\leq C h \|v\|_{1,\Omega}.\label{2e1}
\end{equation}

To address the finite volume methods clearly, we consider the
 problem
\begin{gather}
-\operatorname{div}(A\nabla \varphi)=f,\quad \text{in }\Omega,\label{2e2}\\
\varphi=0,\quad \text{on }\partial\Omega, \label{2e3}
\end{gather}
where $A$, $\Omega$, $\partial\Omega$ are the same as in
\eqref{1e2}-\eqref{1e3}, $f\in L^2(\Omega)$ or $H^1(\Omega)$.

The finite volume approximation $\varphi_h$ of
\eqref{2e2}-\eqref{2e3} is defined as the solution of the problem:
find $\varphi_h\in V_h$ such that
\begin{equation}
a(\varphi_h,I_hv_h)=(f,I_hv_h),\quad \forall v_h\in V_h,\label{2e4}
\end{equation}
where the bilinear form $a(\varphi_h,I_hv_h)$ is defined by
\begin{equation*}
a(\varphi,I_hv)=-\sum_{z_i\in N_h}v(z_i)\int_{\partial V_i}A
\nabla\varphi\cdot \mathbf{n}ds,\quad \varphi,v\in H^1_0(\Omega),
\end{equation*}
where $\mathbf{n}$ is the unit outward normal vector to $\partial
V_i$. The bilinear form $a(\cdot,\cdot)$ is not symmetric though the
problem is self-adjoint. Then for all $w_h,v_h\in V_h$, there exist
positive constants $C$ and $h_0\geq 0$ \cite{ChouLi} such that for
all $0<h<h_0$,
\begin{equation}
|a(w_h,I_hv_h)-a(v_h,I_hw_h)|\leq C h \|w_h\|_{1,\Omega}\
\|v_h\|_{1,\Omega}.\label{2e5}
\end{equation}


It is well known \cite{JL, YL0} that the optimal control problems
\eqref{1e1}-\eqref{1e3} have a solution $(y,u)$, and that if a pair
$(y,u)$ is the solution of \eqref{1e1}-\eqref{1e3}, then there is a
co-state $p\in H^1_0(\Omega)$ such that the triplet $(y,p,u)\in
H^1_0(\Omega)\times H^1_0(\Omega)\times U$ satisfies the
optimality conditions:
\begin{gather}\label{3e1}
(A\nabla y, \nabla w)=(f+u,w),\quad \forall w\in
H^1_0(\Omega),\\
\label{3e1-1}
(A\nabla p,\nabla q)=(g'(y),q),\quad \forall q\in H^1_0(\Omega),\\
\label{3e1-2}
(j'(u)+p,v-u)\geq 0,\quad \forall v \in U.
\end{gather}
If $y\in H^1_0(\Omega)\cap C^2(\Omega)$ and
$p\in H^1_0(\Omega)\cap C^2(\Omega)$, then optimality conditions
\eqref{3e1}-\eqref{3e1-2} can be written as
\begin{gather} \label{3e2}
-\operatorname{div}(A\nabla y)=f+u, \quad \forall x\in \Omega,\\
y(x)=0,\quad \forall x\in \partial\Omega,\\
\label{3e2-1}
-\operatorname{div}(A\nabla p)=g'(y), \quad \forall x\in \Omega,\\
p(x)=0,\quad \forall x\in \partial\Omega,\\
\label{3e2-2}
(j'(u)+p,v-u)\geq 0,\quad \forall v\in U.
\end{gather}
We use finite volume methods to discretize the state and costate
equation directly. Then the optimality condition
\eqref{3e2}-\eqref{3e2-2} can be approximated by: find
$(y_h,p_h,u_h)\in V_h\times V_h\times U$ such that
\begin{gather}
a(y_h,I_hw_h)=(f+u_h,I_hw_h), \quad\forall w_h\in V_h,\label{3e31}\\
a(p_h,I_hq_h)=(g'(y_h),I_hq_h), \quad\forall q_h\in V_h,\label{3e32}\\
(j'(u_h)+p_h,v-u_h)\geq 0,\quad \forall v \in U.\label{3e33}
\end{gather}

For simplicity of notation, let $j(u)=\frac{1}{2}\|u\|_{L^2(\Omega)}^2$,
then we derive $(j'(u),v-u)=(u,v-u)$ and
$(j'(u_h),v-u_h)=(u_h,v-u_h)$. Then the variational inequality
\eqref{3e2-2} can be restated as
\begin{equation}
(u+p,v-u)\geq 0,\quad \forall v\in U. \label{3e2-2-xin}
\end{equation}
Similarly, the variational inequality \eqref{3e33} can be rewritten
by
\begin{equation}
(u_h+p_h,v-u_h)\geq 0,\quad \forall v \in U.\label{3e33-xin}
\end{equation}
Now, we introduce a projection \cite{Hinze}:
\begin{equation}
 P_{[a,b]}(f(x))=\max(a,\min(b,f(x))),
\end{equation}
 we can denote the variational inequality
\eqref{3e2-2-xin} by
\begin{equation}
 u(x)=P_{[a,b]}(-{p}).\label{3e4}
\end{equation}
 And the variational inequality \eqref{3e33-xin} is equivalent to
\begin{equation}
 u_{h}(x)=P_{[a,b]}(-{p_{h}}). \label{3e5}
\end{equation}
Then the discrete optimality conditions can be rewritten by: find
$(y_h,p_h,u_h)\in V_h\times V_h\times U$ such that
\begin{gather} \label{3e68}
a(y_h,I_hw_h)=(f+u_h,I_hw_h), \quad\forall w_h\in V_h,\\
\label{3e68-1}
a(p_h,I_hq_h)=(g'(y_h),I_hq_h), \quad\forall q_h\in V_h,\\
\label{3e68-2}
u_{h}(x)=P_{[a,b]}(-{p_{h}}).
\end{gather}
For $\varphi\in W_h$, we shall write
\begin{equation}\label{phi}
g(\varphi)-g(\rho)=-\tilde{g}'(\varphi)(\rho-\varphi)
=-g'(\rho)(\rho-\varphi)
+\tilde{g}''(\varphi)(\rho-\varphi)^2,
\end{equation}
where
\begin{gather*}
\tilde{g}'(\varphi)=\int_0^1g'(\varphi+s(\rho-\varphi))ds,\\
\tilde{g}''(\varphi)=\int_0^1(1-s)g''(\rho+s(\varphi-\rho))ds
\end{gather*}
are bounded functions in $\bar{\Omega}$ \cite{FA}.

\section{Existence and uniqueness}

In this section, we show the existence and uniqueness of the
solutions for discrete optimality conditions. We can easily see that
the optimality conditions \eqref{3e68}-\eqref{3e68-2} are the finite
volume approximation of \eqref{3e1}-\eqref{3e1-2}. Now we show the
existence and the uniqueness of the solution for
\eqref{3e68}-\eqref{3e68-2}. Let $y_h(u)$ be the solution of
\begin{equation}
a(y_h(u),I_hw_h)=(f+u,I_hw_h), \quad\forall w_h\in V_h, \label{3e6}
\end{equation}
and $p_h(y)$ be the solution of
\begin{equation}
a(p_h(y),I_hq_h)=(g'(y),I_hq_h), \quad\forall q_h\in V_h.\label{3e7}
\end{equation}
For $y_h(u)$ and $p_h(y)$, note that $y_h=y_h(u_h)$ and
$p_h=p_h(y_h)$, we have the following results.

\begin{lemma}\label{lem9}
Assume that $y_h(u), p_h(u)$ are the solutions of \eqref{3e6} and
\eqref{3e7}, respectively. Then
\begin{equation}
\|p_h(y)-p_h\|_{1,\Omega}\leq C \|y-y_h\|_{0,\Omega},\quad
\|y_h(u)-y_h\|_{1,\Omega}\leq C \|u-u_h\|_{0,\Omega}.\label{3e81}
\end{equation}
\end{lemma}

\begin{proof}
Subtracting \eqref{3e32} from \eqref{3e7}, and by using \eqref{phi},
we have
\begin{equation}
a(p_h(y)-p_h,I_hq_h)=(g'(y)-g'(y_h),I_hq_h)=(\tilde{g}''(y)(y-y_h),I_hq_h),\quad
\forall q_h\in V_h.\label{3e81-0}
\end{equation}
Let $q_h=p_h(y)-p_h$, by using \cite[Lemma 2.2]{Ewing} and the
Cauchy-Schwarz's inequality, we can easily obtain that
\begin{equation}
\|p_h(y)-p_h\|_{1,\Omega}\leq C \|y-y_h\|_{0,\Omega}.\label{3e81-1}
\end{equation}
Similarly, subtracting \eqref{3e31} from \eqref{3e6}, we have
\begin{equation}
a(y_h(u)-y_h,I_hw_h)=(u-u_h,I_hw_h), \quad\forall w_h\in V_h, \label{3e81-2}
\end{equation}
let $w_h=y_h(y)-y_h$, we derive
\begin{equation}
\|y_h(u)-y_h\|_{1,\Omega}\leq C \|u-u_h\|_{0,\Omega}.\label{3e81-3}
\end{equation}
This completes the proof.
\end{proof}


\begin{lemma}\label{lem8}
The optimality conditions \eqref{3e31}-\eqref{3e33} admit an unique
solution for sufficiently small $h$.
\end{lemma}

\begin{proof}
We first introduce a projection $P_k: L^2(\Omega)\to U$ defined by
\begin{equation}
\|z-P_k(z)\|_{0,\Omega}=\min_{z_h\in U}\|z-z_h\|_{0,\Omega}.\label{czwy1}
\end{equation}
The projection $P_k$ has the property that
\begin{equation}
\|P_k(z')-P_k(z'')\|_{0,\Omega}\leq \|z'-z''\|_{0,\Omega}, \quad
\forall z',z'' \in L^2(\Omega).\label{czwy2}
\end{equation}
For a given $v_h\in L^2(\Omega)$, let $(y_h(v_h),p_h(v_h))$ be the
solution of the following auxiliary problem: find
$(y_h(v_h),p_h(v_h))\in V_h\times\ V_h$ such that
\begin{gather}
a(y_h(v_h),I_hw_h)=(v_h+f,I_hw_h), \quad \forall w_h\in V_h,\label{czwy3}\\
a(p_h(v_h),I_hq_h)=(g'(y_h(v_h)),I_hq_h), \quad \forall
q_h\in V_h.\label{czwy4}
\end{gather}
Define a mapping $\Phi: L^2(\Omega)\to L^2(\Omega)$ by
\begin{equation}
\Phi(z_h)=z_h-\rho( z_h+p_h(z_h)), \quad\forall z_h\in
L^2(\Omega),\; \rho>0.\label{czwy5}
\end{equation}
Let $T(z_h)=P_k\Phi(z_h)$, then the existence and uniqueness of
\eqref{3e31}-\eqref{3e33} is to show that $T(z_h)$ is a contractive
mapping. It follows from \eqref{czwy2} that for all
$z'_h,z''_h\in L^2(\Omega)$,
\begin{align*}
\|T(z'_h)-T(z''_h)\|_{0,\Omega}^2
&=\|P_k(\Phi(z'_h))-P_k(\Phi(z''_h))\|_{0,\Omega}^2\\
&\leq \|\Phi(z'_h)-\Phi(z''_h)\|_{0,\Omega}^2
 =(\Phi(z'_h)-\Phi(z''_h),\Phi(z'_h)-\Phi(z''_h)).
\end{align*}
Note that
\begin{align*}
&(\Phi(z'_h)-\Phi(z''_h),\Phi(z'_h)-\Phi(z''_h))\\
&=(1-2\rho)(z'_h-z''_h,z'_h-z''_h)
 -2\rho(z'_h-z''_h,p_h(z'_h)-p_h(z''_h))\\
&\quad +\rho^2\|z'_h-z''_h+p_h(z'_h)-p_h(z''_h)\|_{0,\Omega}^2.
\end{align*}
Then we have
\begin{equation}
\begin{aligned}
&\|T(z'_h)-T(z''_h)\|_{0,\Omega}^2\\
&=\leq(1-2\rho)(z'_h-z''_h,z'_h-z''_h)
-2\rho(z'_h-z''_h,p_h(z'_h)-p_h(z''_h)) \\
&\quad+\rho^2\|z'_h-z''_h+p_h(z'_h)-p_h(z''_h)\|_{0,\Omega}^2.
\end{aligned}\label{czwy6}
\end{equation}
For $z'_h,z''_h\in L^2(\Omega)$, it follows from
\eqref{czwy3}-\eqref{czwy4} and \eqref{phi} that
\begin{gather*}
a(y_h(z'_h)-y_h(z''_h),I_hw_h)=(z'_h-z''_h,I_hw_h), \quad\forall w_h\in V_h,\\
a(p_h(z'_h)-p_h(z''_h),I_hq_h)=(\tilde{g}''(y_h(z'_h))(y_h(z'_h)-y_h(z''_h)),
I_hq_h),\quad \forall q_h\in V_h.
\end{gather*}
Let $w_h=p_h(z'_h)-p_h(z''_h)$ and $q_h=y_h(z'_h)-y_h(z''_h)$, we
have
\begin{align*}
&(z'_h-z''_h,p_h(z'_h)-p_h(z''_h))\\
&=(\tilde{g}''(y_h(z'_h))(y_h(z'_h)-y_h(z''_h)), I_h(y_h(z'_h)-y_h(z''_h)))\\
&\quad +a(y_h(z'_h)-y_h(z''_h),I_h(p_h(z'_h)-p_h(z''_h)))\\
&\quad -a(p_h(z'_h)-p_h(z''_h),I_h(y_h(z'_h)-y_h(z''_h)))\\
&\quad +(z'_h-z''_h,(p_h(z'_h)-p_h(z''_h))-I_h(p_h(z'_h)-p_h(z''_h)))\\
&\geq a(y_h(z'_h)-y_h(z''_h),I_h(p_h(z'_h)-p_h(z''_h)))\\
&\quad -a(p_h(z'_h)-p_h(z''_h),I_h(y_h(z'_h)-y_h(z''_h)))\\
&\quad +(z'_h-z''_h,(p_h(z'_h)-p_h(z''_h))-I_h(p_h(z'_h)-p_h(z''_h))),
\end{align*}
where we have used the fact that $(v_h,I_hv_h)\geq 0$.
Using \cite[Lemma 2.4]{ChouLi} and Lemma \ref{lem9}, we have
\begin{equation}
\begin{aligned}
&a\big(y_h(z'_h)-y_h(z''_h),I_h(p_h(z'_h)-p_h(z''_h))\big) \\
&-a\big(p_h(z'_h)-p_h(z''_h), I_h(y_h(z'_h)-y_h(z''_h))\big) \\
&\geq -c_0h\|p_h(z'_h)-p_h(z''_h)\|_{1,\Omega}\cdot
 \|y_h(z'_h)-y_h(z''_h)\|_{1,\Omega} \\
&\geq -c_0c_1h\|z'_h-z''_h\|_{0,\Omega}^2.
\end{aligned} \label{czwy8}
\end{equation}
Note that by \eqref{2e1} and Lemma \ref{lem9}, we have
\begin{equation}
\begin{aligned}
&(z'_h-z''_h,(p_h(z'_h)-p_h(z''_h))-I_h(p_h(z'_h)-p_h(z''_h)))
\\
&\geq -c_2h\|p_h(z'_h)-p_h(z''_h)\|_{1,\Omega} \cdot\|z'_h-z''_h\|_{0,\Omega} \\
&\geq -c_2c_3h\|z'_h-z''_h\|_{0,\Omega}^2.
\end{aligned} \label{czwy9}
\end{equation}
Combining \eqref{czwy8} and \eqref{czwy9}, we deduce that
\begin{equation}
(z'_h-z''_h,p_h(z'_h)-p_h(z''_h))
\geq -(c_0c_1+c_2c_3)h \|z'_h-z''_h\|_{0,\Omega}^2. \label{czwy10}
\end{equation}
Now, it is easy to see that
\begin{equation}
\|z'_h-z''_h+p_h(z'_h)-p_h(z''_h)\|_{0,\Omega}^2\leq c_4
\|z'_h-z''_h\|_{0,\Omega}^2.\label{czwy11}
\end{equation}
Then it follows from \eqref{czwy6}, \eqref{czwy10}, and
\eqref{czwy11} that
\begin{equation}
\|T(z'_h)-T(z''_h)\|_{0,\Omega}^2\leq C\|z'_h-z''_h\|_{0,\Omega}^2.
\end{equation}
For sufficiently small $h$ we can ensure $0<C<1$. Therefore $T(z_h)$
is a contractive mapping and hence the optimality conditions
\eqref{3e31}-\eqref{3e33} admit an unique solution.
\end{proof}

\section{Optimal-order $L^2$ error estimates}

In this section, we derive an optimal-order $L^2$ error estimates
for the finite volume methods with the minimal regularity assumption
for the exact solution $u$. Owing to the property of the variational
inequality, we first estimate the error of the approximate control
in $L^2$ norm. Using the properties of the control, we then estimate
the errors of the numerical solutions for the state and the costate.

\begin{theorem}\label{thm1}
Let $(y,p,u)\in (H^2(\Omega) \cap H^1_0(\Omega))\times (H^2(\Omega)
\cap H^1_0(\Omega))\times U$ and
$(y_h,p_h,u_h)\in V_h\times\ V_h\times U$ be the solutions of
\eqref{3e1}-\eqref{3e1-2} and \eqref{3e31}-\eqref{3e33},
respectively. Assume that $u\in H^1(\Omega)$. Then there exists an
$h_0>0$ such that for all $0<h\leq h_0$,
\begin{equation}\label{4e1}
\|u-u_h\|\leq Ch^2(\|y\|_{2,\Omega}+\|p\|_{2,\Omega}).
\end{equation}
\end{theorem}

\begin{proof}
Let $v=u$ in \eqref{3e33} and $v=u_h$ in the variational inequality
of \eqref{3e2-2}, by using \eqref{3e81-2}, \eqref{3e2-2-xin}, and
\eqref{3e33-xin}, then we have
\begin{equation}
\begin{aligned}
 (u-u_h,u-u_h)\leq& (p-p_h, u_h-u)\\
 =&(p-p_h(y), u_h-u)+(p_h(y)-p_h, u_h-u)\\
 =&(p-p_h(y), u_h-u)+(I_h(p_h(y)-p_h),
u_h-u)\\ &+((p_h(y)-p_h)-I_h(p_h(y)-p_h), u_h-u)\\
 =&(p-p_h(y),u_h-u)+a(y_h-y_h(u),I_h(p_h(y)-p_h))\\
&+((p_h(y)-p_h)-I_h(p_h(y)-p_h), u_h-u).
\end{aligned}\label{4e1-0}
\end{equation}
By using \eqref{phi} and \eqref{3e81-0}, the second term on the
right hand side of \eqref{4e1-0} can be written by
\begin{equation}
\begin{aligned}
 &a(y_h-y_h(u),I_h(p_h(y)-p_h))\\
&=a(y_h-y_h(u),I_h(p_h(y)-p_h))-a(p_h(y)-p_h,I_h(y_h-y_h(u)))\\
&\quad +a(p_h(y)-p_h,I_h(y_h-y_h(u)))\\
&= a(y_h-y_h(u),I_h(p_h(y)-p_h))-a(p_h(y)-p_h,I_h(y_h-y_h(u)))\\
 &\quad +(\tilde{g}''(y)(y-y_h),I_h(y_h-y_h(u)))\\
&= a(y_h-y_h(u),I_h(p_h(y)-p_h))-a(p_h(y)-p_h,I_h(y_h-y_h(u)))\\
&\quad +(\tilde{g}''(y)(y-y_h(u)),I_h(y_h-y_h(u))) \\
&\quad  -(\tilde{g}''(y)(y_h-y_h(u)),I_h(y_h-y_h(u)))\\
&\leq (\tilde{g}''(y)(y-y_h(u)),I_h(y_h-y_h(u)))+a(y_h-y_h(u),I_h(p_h(y)-p_h))\\
&\quad -a(p_h(y)-p_h,I_h(y_h-y_h(u))),
\end{aligned} \label{4e1-1}
\end{equation}
where we have used that
$(\tilde{g}''(y)(y_h-y_h(u)),I_h(y_h-y_h(u)))\geq0$.
Connecting \eqref{4e1-0} and \eqref{4e1-1}, we obtain
\begin{equation}
\begin{aligned}
&\alpha(u-u_h,u-u_h) \\
&\leq (\tilde{g}''(y)(y-y_h(u)),I_h(y_h-y_h(u)))+(p-p_h(y),u_h-u) \\
&\quad +((p_h(y)-p_h)-I_h(p_h(y)-p_h), u_h-u) \\
&\quad +a(y_h-y_h(u),I_h(p_h(y)-p_h))-a(p_h(y)-p_h,I_h(y_h-y_h(u))) \\
&\equiv E_1+E_2+E_3+E_4.
\end{aligned} \label{4e3}
\end{equation}
Note that
\begin{equation}
(\tilde{g}''(y)(y-y_h(u)),I_h(y_h-y_h(u)))
=(\tilde{g}''(y)(y-y_h(u)),y_h-y_h(u)).\label{4e9-0}
\end{equation}
By using Lemma \ref{lem9} and \eqref{4e9-0}, we have
\begin{equation}
\begin{aligned}
E_1&=(\tilde{g}''(y)(y-y_h(u)),I_h(y_h-y_h(u))) \\
&\leq \|y-y_h(u)\|_{0,\Omega}\cdot \|y_h-y_h(u)\|_{0,\Omega} \\
&\leq\|y-y_h(u)\|_{0,\Omega}\cdot \|u_h-u\|_{0,\Omega}\leq Ch^2
\|y\|_{2,\Omega} \cdot\|u_h-u\|_{0,\Omega}.
\end{aligned} \label{4e9}
\end{equation}
Now, we can easily obtain
\begin{equation}
\begin{aligned}
E_2
&=(p-p_h(y),u_h-u) \\
&\leq \|p-p_h(y)\|_{0,\Omega}\cdot \|u_h-u\|_{0,\Omega} \\
&\leq \|p-p_h(y)\|_{0,\Omega}\cdot\|u_h-u\|_{0,\Omega} \\
&\leq Ch^2\|p\|_{2,\Omega}\cdot \|u_h-u\|_{0,\Omega},
\end{aligned}\label{4e8}
\end{equation}
where we have used the estimate in \cite[Theorem 3.5]{Ewing}.
Furthermore, by using Lemma \ref{lem9}, \eqref{2e1}, and the
triangle inequality, we derive
\begin{equation}
\begin{aligned}
E_3
&=((p_h(y)-p_h)-I_h(p_h(y)-p_h), u_h-u) \\
&\leq Ch \|p_h(y)-p_h\|_{1,\Omega}\cdot \|u_h-u\|_{0,\Omega} \\
&\leq Ch \|y-y_h\|_{0,\Omega}\cdot \|u_h-u\|_{0,\Omega} \\
&\leq Ch \|y-y_h\|_{1,\Omega}\cdot \|u_h-u\|_{0,\Omega} \\
&\leq Ch (\|y-y_h(u)\|_{1,\Omega}+\|y_h(u)-y_h\|_{1,\Omega})
 \cdot \|u_h-u\|_{0,\Omega} \\
&\leq Ch (Ch\|y\|_{2,\Omega}+\|u_h-u\|_{0,\Omega}) \|u_h-u\|_{0,\Omega} \\
&\leq Ch \|u_h-u\|_{0,\Omega}^2.
\end{aligned}\label{4e6}
\end{equation}
Using \eqref{2e5} and Lemma \ref{lem9}, we have
\begin{equation}
\begin{aligned}
E_4&=(a(y_h-y_h(u),I_h(p_h(y)-p_h))-a(p_h(y)-p_h,I_h(y_h-y_h(u))) \\
&\leq Ch \|y_h-y_h(u)\|_{1,\Omega}\cdot \|p_h(y)-p_h\|_{1,\Omega} \\
&\leq Ch \|u_h-u\|_{0,\Omega}\cdot \|y-y_h\|_{0,\Omega} \\
&\leq Ch \|y-y_h\|_{1,\Omega}\cdot \|u_h-u\|_{0,\Omega} \\
&\leq Ch (\|y-y_h(u)\|_{1,\Omega}+\|y_h(u)-y_h\|_{1,\Omega})\cdot \|u_h-u\|_{0,\Omega} \\
&\leq Ch (Ch\|y\|_{2,\Omega}+\|u_h-u\|_{0,\Omega})\ \|u_h-u\|_{0,\Omega} \\
&\leq Ch \|u_h-u\|_{0,\Omega}^2.
\end{aligned}\label{4e7}
\end{equation}
Hence, the estimate \eqref{4e1} follows from \eqref{4e3} and
\eqref{4e9}-\eqref{4e7}.
\end{proof}

\begin{theorem}\label{thm2}
Let $(y,p,u)\in (H^2(\Omega) \cap H^1_0(\Omega))\times (H^2(\Omega)
\cap H^1_0(\Omega))\times U$ and
$(y_h,p_h,u_h)\in V_h\times V_h\times U$ be the solutions of
\eqref{3e1}-\eqref{3e1-2} and
\eqref{3e31}-\eqref{3e33}, respectively. Assume that
$u\in L^2(\Omega)$. Then there exists an $h_0>0$ such that for all
$0<h\leq h_0$,
\begin{equation}\label{4e10}
\|y-y_h\|_{0,\Omega}+\|p-p_h\|_{0,\Omega}
\leq Ch^2(\|y\|_{2,\Omega}+\|p\|_{2,\Omega}).
\end{equation}
\end{theorem}

\begin{proof}
Using the triangle inequality, we have
\begin{gather*}
\|y-y_h\|_{0,\Omega}\leq \|y-y_h(u)\|_{0,\Omega}+\|y_h(u)-y_h\|_{0,\Omega},\\
\|p-p_h\|_{0,\Omega}\leq \|p-p_h(y)\|_{0,\Omega}+\|p_h(y)-p_h\|_{0,\Omega}.
\end{gather*}
Lemma \ref{lem9} implies that
\begin{equation}
\begin{aligned}
 \|y-y_h\|_{0,\Omega}&\leq \|y-y_h(u)\|_{0,\Omega}+C\|y_h(u)-y_h\|_{1,\Omega}\\
&\leq \|y-y_h(u)\|_{0,\Omega}+C\|u-u_h\|_{0,\Omega},
\end{aligned} \label{4e12}
\end{equation}
and
\begin{equation}
\begin{aligned}
 \|p-p_h\|_{0,\Omega}&\leq
\|p-p_h(y)\|_{0,\Omega}+C\|p_h(y)-p_h\|_{1,\Omega}\\
&\leq \|p-p_h(y)\|_{0,\Omega}+C\|y-y_h\|_{0,\Omega}.
\end{aligned} \label{4e13}
\end{equation}
By using \cite[Theorem 3.5]{Ewing}, we can easily obtain
\begin{equation}
\|y-y_h(u)\|_{0,\Omega}\leq Ch^2\|y\|_{2,\Omega}.\label{4e14}
\end{equation}
From \eqref{4e12}, \eqref{4e14}, and Theorem \ref{thm1}, we derive
\begin{equation}
\|y-y_h\|_{0,\Omega}\leq Ch^2\|y\|_{2,\Omega}.\label{4e15}
\end{equation}
Connecting \eqref{4e13}, \eqref{4e15}, and
$\|p-p_h(y)\|_{0,\Omega}\leq Ch^2\|p\|_{2,\Omega}$, we have
\begin{equation}
\|p-p_h\|_{0,\Omega}\leq Ch^2\|p\|_{2,\Omega}.\label{4e16}
\end{equation}
From \eqref{4e15}-\eqref{4e16} we can immediately obtain
\eqref{4e10}.
\end{proof}

\section{Optimal-order maximum-norm and $H^1$ error estimates}

In this section, we first estimate the errors of the numerical
solutions of control, state and costate in $L^{\infty}$ norm. Then
we estimate $W^{1,\infty}$ errors for the state and costate
variables.

\begin{theorem}\label{thm4}
Let $(y,p,u)\in (H^2(\Omega) \cap H^1_0(\Omega))\times (H^2(\Omega)
\cap H^1_0(\Omega))\times U$ and $(y_h,p_h,u_h)\in V_h\times
V_h\times U$ be the solutions of \eqref{3e1}-\eqref{3e1-2} and
\eqref{3e31}-\eqref{3e33}, respectively. Assume that
$u\in H^1(\Omega)$. Then there exists an $h_0>0$ such that for all
$0<h\leq h_0$,
\begin{equation}
\|u-u_h\|_{0,\infty}+\|y-y_h\|_{0,\infty}+
\|p-p_h\|_{0,\infty}\leq Ch^2 \sqrt{|\log
(\frac{1}{h})|}.\label{4e22}
\end{equation}
\end{theorem}

\begin{proof}
Using the definition of $P_{[a,b]}(\cdot)$ and \eqref{3e4}-\eqref{3e5}, we have
\begin{equation}
\begin{aligned}
 \|u-u_h\|_{0,\infty}
&\leq C\|p-p_h\|_{0,\infty}\\
 &\leq C(\|p-p_h(y)\|_{0,\infty}+\|p_h(y)-p_h\|_{0,\infty})\\
 &\leq C\|p-p_h(y)\|_{0,\infty}+C\sqrt{|\log (\frac{1}{h})|}\|p_h(y)-p_h\|_{1,\Omega}\\
 &\leq C\|p-p_h(y)\|_{0,\infty}+C\sqrt{|\log (\frac{1}{h})|}\|y-y_h\|_{0,\Omega}\\
&\leq C h^2\sqrt{|\log (\frac{1}{h})|},
\end{aligned}
\end{equation}
where we have used the inverse inequality, Lemma \ref{lem9},
\cite[Theorem 3.11]{Ewing}, and Theorem \ref{thm1}. Similarly, we obtain
\begin{equation}
\begin{aligned}
 \|y-y_h\|_{0,\infty}
&\leq \|y-y_h(u)\|_{0,\infty}+\|y_h(u)-y_h\|_{0,\infty}\\
 &\leq \|y-y_h(u)\|_{0,\infty}+C\sqrt{|\log (\frac{1}{h})|}\|y_h(u)-y_h\|_{1,\Omega}\\
 &\leq \|y-y_h(u)\|_{0,\infty}+C\sqrt{|\log (\frac{1}{h})|}\|u-u_h\|_{0,\Omega}\\
&\leq C h^2\sqrt{|\log (\frac{1}{h})|}.
\end{aligned}
\end{equation}
Then we complete the proof of \eqref{4e22}.
\end{proof}


\begin{theorem}\label{thm5}
Let $(y,p,u)\in (H^2(\Omega) \cap H^1_0(\Omega))\times (H^2(\Omega)
\cap H^1_0(\Omega))\times U$ and
$(y_h,p_h,u_h)\in V_h\times V_h\times U$ be the solutions of
\eqref{3e1}-\eqref{3e1-2} and \eqref{3e31}-\eqref{3e33}, respectively.
Assume that $u\in H^1(\Omega)$. Then there exists an $h_0>0$ such that for all
$0<h\leq h_0$,
\begin{equation}
\|p-p_h\|_{1,\infty}+\|y-y_h\|_{1,\infty}\leq Ch {|\log
(\frac{1}{h})|}.\label{4e25}
\end{equation}
\end{theorem}

\begin{proof}
Using the inverse inequality, Lemma \ref{lem9}, and
\cite[Theorem 3.10]{Ewing}, we have
\begin{equation}
\begin{aligned}
 \|\nabla(p-p_h)\|_{0,\infty}
&\leq
\|\nabla(p-p_h(y))\|_{0,\infty}+\|\nabla(p_h(y)-p_h)\|_{0,\infty}\\
 &\leq\|\nabla(p-p_h(y))\|_{0,\infty}+Ch^{-1}\|\nabla(p_h(y)-p_h)\|_{0,\Omega}\\
 &\leq\|\nabla(p-p_h(y))\|_{0,\infty}+Ch^{-1}\|y-y_h\|_{0,\Omega}\\
&\leq Ch\ {|\log (\frac{1}{h})|}+Ch\leq Ch\
{|\log (\frac{1}{h})|}.
\end{aligned}
\end{equation}
Similarly, we obtain
\begin{equation}
\begin{aligned}
 \|\nabla(y-y_h)\|_{0,\infty}&\leq \|\nabla(y-y_h(u))\|_{0,\infty}
+\|\nabla(y_h(u)-y_h)\|_{0,\infty}\\
 &\leq \|\nabla(y-y_h(u))\|_{0,\infty}+Ch^{-1}\|y_h(u)-y_h\|_{0,\Omega}\\
 &\leq \|\nabla(y-y_h(u))\|_{0,\infty}+Ch^{-1}\|u-u_h\|_{0,\Omega}\\
&\leq Ch\ {|\log (\frac{1}{h})|}+Ch\leq Ch\
{|\log (\frac{1}{h})|}.
\end{aligned}
\end{equation}
Then we complete the proof of \eqref{4e25}.
\end{proof}

Now, we consider the errors of the state and costate in $H^1$ norm.

\begin{theorem}\label{thm3}
 Let $(y,p,u)\in (H^2(\Omega) \cap H^1_0(\Omega))\times
(H^2(\Omega) \cap H^1_0(\Omega))\times U$ and $(y_h,p_h,u_h)\in
V_h\times\ V_h\times U$ are the solutions of
\eqref{3e1}-\eqref{3e1-2} and \eqref{3e31}-\eqref{3e33},
respectively. Then there exists an $h_0>0$ such that for all
$0<h\leq h_0$,
\begin{equation}\label{4e18}
\|y-y_h\|_{1,\Omega}+\|p-p_h\|_{1,\Omega}\leq
Ch(\|y\|_{2,\Omega}+\|p\|_{2,\Omega}).
\end{equation}
\end{theorem}

\begin{proof}
Using the triangle inequality, we have
\begin{gather*}
\|y-y_h\|_{1,\Omega}\leq \|y-y_h(u)\|_{1,\Omega}+\|y_h(u)-y_h\|_{1,\Omega},\\
\|p-p_h\|_{1,\Omega}\leq
\|p-p_h(y)\|_{1,\Omega}+\|p_h(y)-p_h\|_{1,\Omega}.
\end{gather*}
Lemma \ref{lem9} implies
\begin{gather}
\|y-y_h\|_{1,\Omega}\leq \|y-y_h(u)\|_{1,\Omega}+C\|u-u_h\|_{0,\Omega},\label{4e19}\\
\|p-p_h\|_{1,\Omega}\leq
\|p-p_h(y)\|_{1,\Omega}+C\|y-y_h\|_{0,\Omega}.\label{4e20}
\end{gather}
By using \cite[Theorem 3.3]{Ewing}, we obtain
\begin{equation}
\|y-y_h(u)\|_{1,\Omega}\leq Ch \|y\|_{2,\Omega},\quad
\|p-p_h(y)\|_{1,\Omega}\leq Ch\|p\|_{2,\Omega}.\label{4e21}
\end{equation}
From Theorem \ref{thm2} and \eqref{4e19}-\eqref{4e21} we can easily
obtain \eqref{4e18}.
\end{proof}

\section{Conclusion and future works}

In this article, we presented the finite volume approximation of general
elliptic optimal control problems. We prove the existence and the
uniqueness of the solution for discrete optimality conditions. Under
some reasonable assumptions, we obtain some optimal order error
estimates for the state, costate and control variables. The
convergence rate for the state, costate and control variables is
$O(h^2)$ or $O(h^2 \sqrt{|\log
(\frac{1}{h})|})$ in the sense of $L^2$
norm or $L^{\infty}$ norm. The convergence rate for the state and
costate variables is $O(h)$ or $O(h {|\log
(\frac{1}{h})|})$ in the sense of $H^1$ norm
or $W^{1,\infty}$ norm.

We presented a priori error estimates for the finite volume approximation of
general elliptic optimal control problems. To our best knowledge in
the context of optimal control problems, these priori error
estimates for the general elliptic optimal control problems are new.

In the future, we shall consider the finite volume approximation of
parabolic optimal control problems. Furthermore, we shall consider a
posteriori error estimates and super-convergence of the finite volume
solutions for parabolic optimal control problems.

\subsection*{Acknowledgments}

This work is supported by the National Basic Research Program (2012CB955804),
by the Major Research Plan of National Natural
Science Foundation of China (91430108),
 by the National Science Foundation
of China (11201510, 11171251),
by the Innovation Team Building at Institutions of Higher Education in
 Chongqing (CXTDX201601035),
by the China Postdoctoral Science Foundation (2015M580197), Chongqing
Research Program of Basic Research and Frontier Technology
(cstc2015jcyjA20001),
by the Science and Technology Project of Wanzhou
District of Chongqing (2013030050),
by the Ministry of education Chunhui
projects (Z2015139), Major Program of Tianjin University of Finance
and Economics (ZD1302),
by the Research Foundation of Chongqing Municipal
Education Commission (KJ1710253, KJ1501004),
by the Chongqing Municipal Key Laboratory of Institutions of Higher
 Education ([2017]3), and by the
Chongqing Development and Reform Commission (2017[1007]).


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