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


\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 165, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/165\hfil Optimal control]
{Optimal control for the multi-dimensional viscous Cahn-Hilliard equation}

\author[N. Duan, X. Zhao \hfil EJDE-2015/165\hfilneg]
{Ning Duan, Xiufang Zhao}

\address{Ning Duan \newline
School of Science, Jiangnan University, Wuxi 214122, China}
\email{123332453@qq.com}

\address{Xiufang Zhao \newline
School of Science, Qiqihar  University,
Qiqihar 161006, China}
\email{17815358@qq.com}

\thanks{Submitted June 26, 2014. Published June 17, 2015.}
\subjclass[2010]{35K55, 49A22}
\keywords{Optimal control; viscous Cahn-Hilliard equation; 
\hfill\break\indent optimal solution;  optimality condition}

\begin{abstract}
 In this article, we study the multi-dimensional viscous Cahn-Hilliard equation.
 We prove the existence of optimal solutions and establish the optimality system.
\end{abstract}

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



\section{Introduction} \label{sect1}

In this article, we consider the  viscous Cahn-Hilliard equation
\begin{equation} \label{1-0}
 u_t-k\Delta u_{t}+  \gamma \Delta^2u=\Delta\varphi(u),\quad 
 (x,t)\in\Omega\times(0,T),
 \end{equation}
where $\Omega\subset\mathbb{R}^n(n\leq3)$ is a bounded domain with 
smooth boundary, the unknown function $u(x,t)$ is the concentration of one 
of the two phases, $\gamma>0$ is the interfacial energy parameter,
$k>0$ represents the viscous coefficient, $\varphi(u)$ is the intrinsic 
chemical potential.
The viscous Cahn-Hilliard equation, which was first propounded by 
Novick-Cohen \cite{Novic}, arises in the dynamics of viscous 
first order phase transitions in cooling binary solutions such as glasses, 
alloys and polymer mixtures (see\cite{Bai,Elliott}).
 Note that if we take $k = 0$, the equation becomes the well-known 
Cahn-Hilliard type equation (see \cite{Zheng,Zheng2}), which is originally 
proposed for modelling phase separation phenomena in a binary mixture, 
and it can be used to describe many other physical and biological phenomena, 
including the growth and dispersal in the population which is sensitive to
time-periodic factors.

During the past years, many papers were devoted to the viscous
Cahn-Hilliard equation. In \cite{Liu1}, Liu and Yin considered the global 
existence and blow-up of classical solutions for
viscous Cahn-Hilliard equation in $\mathbb{R}^n$ $(n\leq 3)$. 
In Grinfeld and Novick-Cohen's paper \cite{Grinfeld}, a
Morse decomposition of the stationary solutions of the 1D viscous
Cahn-Hilliard equation was established by explicit energy
calculations, and the global attractor for the viscous Cahn-Hilliard
equation was also considered. Li and Yin \cite{LiYin} investigate the
existence, uniqueness and asymptotic behavior of solutions to the 1D
viscous Cahn-Hilliard equation with time periodic potentials and
sources. We also noticed that some investigations of the viscous
Cahn-Hilliard equation were studied, such as in \cite{
Bonfoh,Dlotko2,Liu2, Rossi}.

In past decades, the optimal control of distributed parameter system had been
received much more attention in academic field.
Many papers have already been published to study the control
problems of nonlinear parabolic equations, for example 
\cite{Becker,Jeong,Ryu1, Tian2,Zheng,XP}.

In this article, we consider the  distributed optimal control
problem
\begin{equation} \label{1-4}
\min J(u,w)=\frac12\|Cu-z_d\|^2_S+\frac{\delta}{2}\|w\|^2_{L^2(Q_0)},
\end{equation}
subject to the initial boundary value problem for the viscous Cahn-Hilliard equation
\begin{equation}\label{1-5} 
\begin{gathered}
u_t-k\Delta u_{t}+ \gamma \Delta^2u-\Delta\varphi(u)=Bw,\quad 
(x,t)\in\Omega\times (0,T), \\
u(x,t)=\Delta u(x,t)=0,\quad (x,t)\in\partial\Omega\times(0,T) \\
u(0)=u_0,\quad x\in\Omega,
\end{gathered}
\end{equation}
where $\Omega\subset\mathbb{R}^n~(n\leq3)$ is a bounded domain with smooth 
boundary, $k>0$ and $\gamma>0$ are two constants, $\varphi(u)$ is an
intrinsic chemical potential with typical example as
$$
\varphi(u)=\gamma_2u^3+\gamma_1u^2-u,
$$
for some constants $\gamma_2>0$ and $\gamma_1$.

\begin{remark} \label{rmk1.1}\rm
The main difference between the viscous Cahn-Hilliard equation and the 
standard Cahn-Hilliard equation is the viscous term $k\Delta u_t$, which 
describe the viscosity of glasses, alloys and polymer. Note that the viscous 
term $k\Delta u_t$ is not only dependent on $x$ but also dependent on $t$. 
Because of  the existence of this term, we can obtain the results on the 
a prior estimates  more directed.
\end{remark}

\begin{remark} \label{rmk1.2} \rm
 In \cite{Zhao}, Zhao and Liu studied the optimal control problem for equation 
\eqref{1-0} in 1D case with $\varphi(s)=s^3-s$. Based on Lions' \cite{Lions} 
classical theory, they  proved the existence of optimal solution to the
equation.  Here, we consider the $n$-D case of equation \eqref{1-0}, where
$n\leq3$. 
We also established the optimality system, which was not established in \cite{Zhao}.
In fact, for the well-known Cahn-Hilliard equation,
using the same method as above, we can also obtain the  results  on the 
existence of optimal solutions and the optimality conditions.
\end{remark}

 The control target is to match the given desired state $z_d$ in $L^2$-sense 
by adjusting the body force $w$ in a control volume
$Q_0\subseteq Q=\Omega\times(0,T)$ in the $L^2$-sense.

In the following, we introduce some notations that will be used
throughout the paper. For fixed $T>0$, $V=H^2(\Omega)\bigcap H^1_0(\Omega)$ and
$H=L^2(\Omega)$, let $V^{*}$, $H^{*}$ be dual spaces of
$V$ and $H$. Then, we obtain
$$
V\hookrightarrow  H=H^{*}\hookrightarrow V^{*}.
$$
Clearly, each embedding being dense.

 The extension operator $B\in \mathcal {L}\left(L^2(Q_0), L^2(0,T;V^*)\right)$ 
which is called the controller is introduced as 
\begin{equation}
Bq= \begin{cases}
q,& q\in Q_0,\\
0,& q\in Q\setminus Q_0.
\end{cases}
\end{equation}
We supply $H$ with the inner product $(\cdot,\cdot)$ and the norm
$\|\cdot\|$, and define a space $W(0,T; V)$ as
$$
W(0,T;V)=\big\{v:v\in L^2(0, T;V),~\frac{\partial v}{\partial t}
\in L^2(0,T;V^{*})\big\},
$$
which is a Hillbert space endowed with common inner product.

This article is organized as follows. In the next section, we prove
the existence and uniqueness of the weak solution to problem
\eqref{1-5} in a special space and discuss the
relation among the norms of weak solution, initial value and control
item; In Section 3, we consider the optimal control problem and prove 
the existence of optimal solution; In the last section, the optimality 
conditions is showed and the optimality system is derived.

In the following, the letters $c$, $c_i$ ($i = 1, 2,\cdots$) will
always denote positive constants different in various occurrences.

\section{Existence and uniqueness of weak solution}\label{sec2}

In this section, we study the existence and uniqueness of weak
solution for the equation 
\begin{equation}\label{2-1}
u_t-k\Delta u_{t}+  \gamma \Delta^2u-\Delta\varphi(u)=Bw,\quad
\text{in } \Omega\times(0,T),
\end{equation}
with the boundary value conditions 
\begin{equation}\label{2-2} 
u(x,t)=\Delta u(x,t)=0,\quad \text{in }\partial\Omega\times(0,T),
\end{equation}
and initial condition
\begin{equation}\label{2-3} 
u(x,0)=u_0(x),\quad \text{in }\Omega,
\end{equation}
where $Bw\in L^2(0,T;V^*)$ and a control $w\in L^2(Q_0)$.

Now, we give the definition of the weak solution for 
problem \eqref{2-1}-\eqref{2-3} in the space $W(0,T;V)$.

\begin{definition} \label{def2.1} \rm
For all $\eta\in V,~t\in(0,T)$, the function $u(x,t)\in W(0,T;V)$ 
is called a weak solution to problem \eqref{2-1}-\eqref{2-3}, if 
\begin{equation}\label{2-4}
\frac{d}{dt}(u,\eta)+k\frac d{dt}(\nabla u,\nabla\eta)
+\gamma(\Delta u,\Delta\eta)+(\nabla\varphi(u),\nabla\eta)=(Bw,\eta)_{V^*,V}.
\end{equation}
\end{definition}

We shall give Theorem \ref{thm2.1} on the existence and uniqueness
of weak solution to problem \eqref{2-1}-\eqref{2-3}.

\begin{theorem} \label{thm2.1} 
Suppose $u_0\in V$, $Bw\in L^2(0,T;V^*)$, then the problem 
\eqref{2-1}-\eqref{2-3} admits a unique weak solution $u(x,t)\in W(0,T;V)$ 
in the interval $[0,T]$.
\end{theorem}

\begin{proof} Galerkin's method is applied for the proof.
Let $\{z_j(x)\}$ $(j=1,2,\cdots)$ be the orthonormal base in
$L^2(\Omega)$ being composed of the eigenfunctions of the eigenvalue
problem
$$
\Delta z+\lambda z=0,~~z(0)=z_0,
$$
corresponding to eigenvalues $\lambda_j~(j=1,2,\cdots)$.

Suppose that $u_n(x,t)=\sum_{j=1}^Nu_{nj}(t)z_j(x)$ is the Galerkin
approximate solution to the problem \eqref{2-1}-\eqref{2-3} require
$u_n(0,\cdot)\to u_0$ in $H$ holds true, where
$u_{nj}(t)$ $(j=1,2,\cdots,N)$ are undermined functions, $n$ is
a natural number.
By analyzing the limiting behavior of sequences of smooth function
$\{u_n\}$, we can prove the existence of weak solution to the
problem \eqref{2-1}-\eqref{2-3}.

Performing the Galerkin procedure for the problem \eqref{2-1}-\eqref{2-3}, 
we obtain 
\begin{equation}\label{2-5}
\begin{gathered}
\left(u_{nt}-k\Delta u_{nt}+\gamma\Delta^2u_n-\Delta\varphi(u_n),z_j\right)
=(Bw,z_j),\\
( u_n(\cdot,0),z_j)=(u_{n0}(\cdot),z_j),~j=1, 2,\cdots,N.
\end{gathered}
\end{equation}


Obviously, the equation in \eqref{2-4} is an ordinary differential equation and
according to ODE theory, there exists an unique solution to
the equation \eqref{2-4} in the interval $[0,t_n)$. what we should do is to
show that the solution is uniformly bounded when $t_n\to T$.
we need also to show that the times $t_n$ there are not decaying to
$0$ as $ n\to\infty$.

 There are four steps for us to prove it.
\smallskip

\noindent\textbf{Step 1.}
Multiplying both sides of the equation in \eqref{2-4} by $u_{nj}(t)$, summing up the
products over $j = 1, 2, \dots ,N$,  we derive that
\[
\frac12\frac d{dt}(\|u_n\|^2+k\|\nabla u_n\|^2)+\gamma\Delta u_n
+\int_{\Omega}\varphi'(u_n)|\nabla u_n|^2dx=(Bw,u_n)_{V^*,V}.
\]
By H\"{o}lder's inequality, we conclude that
\begin{align*}
(Bw,u_n)_{V^*,V}
&\leq\|Bw\|_{V^*}\|u_n\|_V\leq c_1\|Bw\|_{V^*}\|\Delta u_n\|\\
&\leq \frac{\gamma}2\|\Delta u_n\|^2+\frac{c_1^2}{2\gamma}\|Bw\|_{V^*}^2.
\nonumber
\end{align*}
Note that
$$
\varphi'(u_n)=3\gamma_2u_n^2+2\gamma_1u_n^2-1
\geq-\frac{\gamma_1^2}{3\gamma_2}-1=-c_2.
$$
Summing up, 
 %\label{2-6}
\begin{align*}
&\frac d{dt}(\|u_n\|^2+k\|\nabla u_n\|^2)+\gamma\|\Delta u_n\|^2\\
&\leq \frac{c_1^2}{\gamma}\|Bw\|_{V^*}^2+2c_2\|\nabla u_n\|^2
\\
&\leq \frac{c_1^2}{\gamma}\|Bw\|_{V^*}^2+\frac{\gamma}2\|\Delta u_n\|^2
+\frac{c_2^2}{\gamma}\|u_n\|^2
\\
&\leq \frac{c_1^2}{\gamma}\|Bw\|_{V^*}^2+\frac{\gamma}2\|\Delta u_n\|^2
 +\frac{c_2^2}{\gamma}(\|u_n\|^2+k\|\nabla u_n\|^2).
\end{align*}
Since $Bw\in L^2(0,T;V^*)$ is the control item, we can assume that
$\|Bw\|_{V^*}\leq M$, where $M$ is a positive constant. Then, we have
\[
\frac d{dt}(\|u_n\|^2+k\|\nabla u_n\|^2)
+\frac{\gamma}2\|\Delta u_n\|^2\leq\frac{c_1^2}{\gamma}M^2
+\frac{c_2^2}{\gamma}(\|u_n\|^2+k\|\nabla u_n\|^2).
\]
Using Gronwall's inequality, we obtain
\begin{equation}\label{2-7}
\begin{aligned}
\|u_n\|^2+k\|\nabla u_n\|^2
&\leq e^{\frac{c^2_2}{\gamma}t}(\|u_n(0)\|^2+k\|\nabla u_n(0)\|^2)
+\frac{c_1^2}{c_2^2}M^2 \\
&\leq e^{\frac{c^2_2}{\gamma}T}(\|u_n(0)\|^2+k\|\nabla u_n(0)\|^2)
+\frac{c_1^2}{c_2^2}M^2=c_3^2.
\end{aligned}
\end{equation}
By Sobolev's embedding theorem, we immediately obtain
\begin{equation}\label{2-7-}
\|u_n(\cdot,t)\|_p\leq c_4,\quad p\in\big(\frac n2,\frac{2n}{n-2}\big).
\end{equation}
\smallskip

\noindent\textbf{Step 2.}
 Multiplying both sides of the equation of \eqref{2-4} by $\lambda_ju_{nj}(t)$, 
summing up the products over $j = 1, 2, \cdots ,N$, we obtain
\[
\frac12\frac d{dt}(\|\nabla u_n\|^2+k\|\Delta u_n\|^2)
+\gamma\|\nabla\Delta u_n\|^2=-(\Delta\varphi(u_n),\Delta u_n)
-(Bw,\Delta u_n)_{V^*,V}.
\]
Note that
$$
\Delta\varphi(u_n)=(3\gamma_2u_n^2+2\gamma_1u_n-1)\Delta u_n
+(6\gamma_2u_n+2\gamma_1)|\nabla u_n|^2.
$$
Hence
\begin{align*}
&\frac12\frac d{dt}(\|\nabla u_n\|^2+k\|\Delta u_n\|^2)
+\gamma\|\nabla\Delta u_n\|^2+\gamma_2\|u_n\Delta u_n\|^2
\\
&=-2\gamma_2\int_{\Omega}u^2_n|\Delta u_n|^2dx
-2\gamma_1\int_{\Omega}u_n|\Delta u_n|^2dx+\|\Delta u_n\|^2\\
&\quad -6\gamma_2\int_{\Omega}u_n|\nabla u_n|^2\Delta u_n\,dx
-2\gamma_1\int_{\Omega}|\nabla u_n|^2\Delta u_n\,dx-(Bw,\Delta u_n)_{V^*,V}
\\
&\leq \gamma_2\int_{\Omega}u^2_n|\Delta u_n|^2dx
 +c_5(\|\Delta u_n\|^2+\|\nabla u_n\|_4^4+\|Bw\|_{V^*}^2+\|u_n\|^2)\\
&\quad +\frac{\gamma}4\|\nabla\Delta u_n\|^2.
\end{align*}
Using Nirenberg's inequality, we deduce that
$$
c_5\|\nabla u_n\|_4^4\leq c_4(c'\|\nabla\Delta u_n\|^{\frac n8}\|\nabla u_n\|^{1-\frac n8}+c{''}\|\nabla u_n\|)^4\leq\frac{\gamma}8\|\nabla\Delta u_n\|^2+c_6.
$$
On the other hand, we also have
\begin{align*}
c_5\|\Delta u_n\|^2
\leq\frac{\gamma}8\|\nabla\Delta u_n\|^2+\frac{2c_5^2}{\gamma}\|\nabla u_n\|^2
\leq\frac{\gamma}8\|\nabla\Delta u_n\|^2+\frac{2c_3^2c_5^2}{\gamma k}.
\nonumber
\end{align*}
Summing up, we derive that
\begin{equation}\label{leng1}
\frac d{dt}(\|\nabla u_n\|^2+k\|\Delta u_n\|^2)
+\gamma\|\nabla\Delta u_n\|^2\leq 2c_6+2c_3^2c_5
+\frac{4c_3^2c_5^2}{\gamma k}+2c_5\|Bw\|_{V^*}^2,
\end{equation}
which means
$$
\frac d{dt}(\|\nabla u_n\|^2+k\|\Delta u_n\|^2)
+\gamma\|\nabla\Delta u_n\|^2\leq 2c_6+2c_3^2c_5
+\frac{4c_3^2c_5^2}{\gamma k}+2c_5M^2.
$$
Therefore,
\begin{equation}\label{2-10}
\begin{aligned}
&\|\nabla u_n\|^2+k\|\Delta u_n\|^2\\
&\leq \|\nabla u_n(0)\|^2+k\|\Delta u_n(0)\|^2+(2c_6+2c_3^2c_5
 +\frac{4c_3^2c_5^2}{\gamma k}+2c_5M^2)T\\
&=(c'_6)^2.
\end{aligned}
\end{equation}
By \eqref{2-7-}, \eqref{2-10} and Sobolev's embedding theorem, we conclude that
\begin{equation}\label{2-10-}
\|u_n(\cdot,t)\|_{\infty}\leq c_7.
\end{equation}
Adding \eqref{2-7-} and \eqref{2-10} together gives
\begin{equation}\label{leng2}
\|u_n(x,t)\|_{L^2(0,T;V)}^2
\leq c\int_0^T(\|u_n\|^2+\|\nabla u_n\|^2+\|\Delta u_n\|^2)dt
\leq c_8^2.
\end{equation}
\smallskip

\noindent\textbf{Step 3.}
 We prove a uniform $L^2(0,T;V^*)$ bound on a sequence $\{u_{n,t}\}$. 
Set $y_n=u_n-k\Delta u_n$, by \eqref{2-4} and Sobolev's embedding theorem, we
obtain
\begin{equation}\label{leng3}
\begin{aligned}
\|y_{n,t}\|_{V^{*}}&=\sup_{\|\psi\|_V=1}(y_{n,t},\psi)_{V^{*},V}
\\
&\leq \sup_{\|\psi\|_V=1}\{(Bw,\psi)_{V^{*},V}+\gamma
|(\Delta u_{n},\Delta\psi)| +|(\varphi(u_n),\Delta\psi)|\}
\\
&\leq c(\|B^{*}\bar{\omega}\|_{V^{*}}+\|\Delta u_{n}\|+\| u_{n}\|)
\\
&\leq c(M+\|\Delta u_{n}\|+\| u_{n}\|).
\end{aligned}
\end{equation}
Integrating \eqref{leng3} with respect to $t$ on $[0,T]$, we obtain
\[
\|y_{n,t}\|_{L^2(0,T;V^{*})}^2
\leq c(M^2T+\|\Delta u_{n}\|_{L^2(0,T;H)}+\| u_{n}\|_{L^2(0,T;H)}).
\]
Hence
\begin{equation} \label{leng4}
\|u_{n,t}\|_{L^2(0,T;V^{*})}^2=\|(I-k\Delta)^{-1}y_{n,t}\|_{L^2(0,T;V^{*})}^2
\leq c_{9}^2.
\end{equation}
\smallskip

\noindent\textbf{Step 4.}
 Integrating \eqref{2-10} with respect to $[0, T]$, combining its result and 
\eqref{leng2} together, we deduce that
\begin{equation}\label{qingjiao-5}
\|u_n\|_{L^2(0,T;H^3)}\leq c_{10}.
\end{equation}

By the compactness of the embedding
$L^{\infty}(0,T;H^2)\hookrightarrow L^{\infty}(0,T;H^1)$ and  of
$L^2(0,T;H^3)\hookrightarrow L^2(0,T;H^1)$, we find that there exist 
$ u\in L^{\infty}(0,T;H^1)$ and $u\in L^2(0,T;H^1)$ such that, 
up to a subsequence,
\begin{equation} \label{qingjiao-5b}
\begin{gathered}
u_n\to u\quad \text{strongly in }L^{\infty}(0,T;H^1),\\
u_n\to u\quad \text{strongly in }L^2(0,T;H^1).
\end{gathered}
\end{equation}
It then follows from \eqref{qingjiao-5} that
$$
\|u_n-u\|_{L^{\infty}(0,T;H^1)}\to 0,\quad 
\|\Delta u_n-\Delta u\|_{L^2(0,T;H^2)}\to 0.
$$
 According to the previous subsequences $\{u_n\}$, we conclude that 
$\Delta \varphi(u_n)$ weakly converges to $\Delta\varphi(u)$  
in $L^2(0,T;V^*)$. In fact, for any $w\in L^2(0,T;V^*)$, we have
\begin{equation}
\begin{aligned}
&\big|\int_0^T(\Delta\varphi(u_n)-\Delta \varphi(u),w)_{V^*,V}dt\big|\\
&\leq C\big|\int_0^T(\varphi(u_n)-\varphi(u))wdt\big|\\
&\leq C\big|\int_0^T\varphi{'}(\theta u_n+(1-\theta)u)(u_n-u)wdt\big|\\
&\leq C\int_0^T\|\varphi{'}(\theta u_n+(1-\theta)u\|_{\infty}\|u_n-u\|\|w\|dt \\
&\leq C\|u_n- u\|_{L^2(0,T;H)}\|w\|_{L^2(0,T;H)},
\end{aligned}\label{jiang-1}
\end{equation}
where $\theta\in(0,1)$. By \eqref{jiang-1}, we know that there exists a 
subsequence $\{u_n(x,t)\}$ such that $\Delta\varphi(u_n)$  converges weakly
to $\Delta\varphi(u)$ in $L^2(0,T;V^*)$. On the other hand, the 
subsequence $\{u_{n,t}\}$ weakly converge to $\{u_t\}$  in $L^2(0,T;V^*)$.

Based on the above discussion,  we conclude that there exists a function 
$u(x,t) \in W(0,T;V)$ which satisfies \eqref{2-4}. Since the proof of
uniqueness is easy, we omit it.
 Then, Theorem \ref{thm2.1} has been proved.
\end{proof}

For the relation among the norm of weak solution
and initial value and control item, basing on the above discussion, 
we obtain the following theorem immediately.

\begin{corollary} \label{coro2.3}
Suppose that $u_0\in V$, $Bw\in L^2(0,T;V^*)$,
then there exists positive constants $C'$ and $C''$ such that
\begin{equation}
\|u\|^2_{W(0,T;V)}\leq
C'(\|u_0\|_V^2+\|w\|^2_{L^2(Q_0)})+C{''}, \label{2-17}
\end{equation}
\end{corollary}

\section{Optimal control problem}

In this section, we consider the optimal control
problem associated with the viscous Cahn-Hilliard equation and
prove  the existence of optimal solution.

In the following, we suppose $L^2(Q_0)$ is a Hilbert space of
control variables, we also suppose 
$B\in \mathcal {L}(L^2(Q_0), L^2(0,T;V^*))$ is the controller and a control
 $w\in L^2(Q_0)$, consider the following control system
\begin{equation}\label{3-1}
\begin{gathered}
u_t-k\Delta u_{t}+  \gamma \Delta^2u-\Delta\varphi(u)=Bw,
\quad (x,t)\in\Omega\times (0,T),
\\
u(x,t)=\Delta u(x,t)=0,\quad (x,t)\in\partial\Omega\times(0,T)
\\
u(0)=u_0,\quad x\in\Omega.
\end{gathered}
\end{equation}
Here it is assume that $u_0\in V$. By  Theorem \ref{thm2.1}, we can define 
the solution map $w\to u(w)$ of $L^2(Q_0)$ into
$W(0,T;V)$. The solution $u$ is called the state of the control
system \eqref{3-1}. The observation of the state is assumed to be
given by $Cu$. Here $C\in \mathcal {L}(W(0,T;V), S)$ is an operator,
which is called the observer, $S$ is a real Hilbert space of
observations. The cost function associated with the control system
\eqref{3-1} is given by 
\begin{equation}\label{3-4}
J(u,w)=\frac12\|Cu-z_d\|_S^2+\frac{\delta}2\|w\|^2_{L^2(Q_0)},
\end{equation}
where $z_d\in S$ is a desired state and $\delta>0$ is fixed.
An optimal control problem about the viscous Cahn-Hilliard equation is 
\begin{equation}
\min J(u,w), \label{3-5}
\end{equation}
%%
where $(u,w)$ satisfies \eqref{3-1}.

Let $X=W(0,T;V)\times L^2(Q_0)$ and $Y=L^2(0,T;V)\times H$. We
define an operator $e=e(e_1,e_2):X\to Y$, where
\begin{gather*}
e_1=(\Delta^2)^{-1}(u_t-k\Delta u_{t}+
 \gamma \Delta^2u-\Delta\varphi(u)-Bw), \\
e_2=u(x,0)-u_0.
\end{gather*}
Here $\Delta^2$ is an operator from $V$ to $V^{*}$. Then, we write
\eqref{3-5} in the form
$$
\min J(u,w)\quad \text{subject to }e(y,w)=0.
$$

\begin{theorem}\label{thm3.1} 
Suppose that $u_0\in V$, $Bw\in L^2(0,T;V^*)$, then there exists 
an optimal control solution $(u^{*},w^*)$ to  problem \eqref{3-1}.
\end{theorem}

\begin{proof} 
Suppose that  $(u,w)$ satisfy the equation
$e(u,w)=0$. In view of \eqref{3-4},  we deduce that
$$
J(u,w)\geq\frac{\delta}2\|w\|^2_{L^2(Q_0)}.
$$
By Corollary \ref{coro2.3}, we obtain
that $\|u\|_{W(0,T;V)}\to\infty$ yields $\|w\|_{L^2(Q_0)}\to\infty$.
Therefore,
\begin{equation} \label{3-6}
J(u,w)\to\infty,\quad \text{when }\|(u,w)\|_X\to\infty.
\end{equation}
As the norm is weakly lower semi-continuous, we achieve that $J$ is
weakly lower semi-continuous. 
Since for all $(u,w)\in X$, $J(u,w)\geq 0$,  there exists $\lambda\geq 0$
defined by
$$
\lambda=\inf\{J(u,w): (u,w)\in X,\; e(u,w)=0\},
$$
which means the existence of a minimizing sequence
$\{(u^n,w^n)\}_{n\in\mathbb{N}}$ in $X$ such that
$$
\lambda=\lim_{n\to\infty}J(u^n,w^n)\quad\text{and}\quad
e(u^n,w^n)=0,\quad\forall n\in\mathbb{N}.
$$
From \eqref{3-6}, there exists an element
$(u^{*},w^{*})\in X$ such that when $n\to\infty$,
\begin{gather}\label{3-7} 
u^n\to u^{*},\quad \text{weakly},\; u\in W(0,T;V), \\
\label{3-8}
w^n\to w^{*},\quad \text{weakly},\; w\in L^2(Q_0).
\end{gather}

Since $u_n\in L^\infty(0,T; V)$, $u_{n,t}\in L^2(0, T; V^*)$, we also have 
$L^\infty(0,T; V)$ is continuously embedded into $L^2(0, T; L^{\infty})$.
Hence by \cite[Lemma 4]{Si} we have $u^n \to u^*$ strongly in 
$L^2(0, T; L^{\infty})$, as $n\to\infty$, $u^n \to u^*$ strongly
in $C(0, T; H)$, as $n\to\infty$.

As the sequence $\{u^n\}_{n\in\mathbb{N}}$ converges weakly, then
$\|u^n\|_{W(0, T; V)}$ is bounded. Based on the embedding theorem,
$\|u^n\|_{L^2(0, T;L^{\infty})}$ is also bounded.

 Because $u^n\to u^{*}$ in $L^2(0,T;L^{\infty})$ as
$n\to\infty$, we know that
$\|u^{*}\|_{L^2(0,T;L^{\infty})}$ is also bounded.

It then follows from \eqref{3-7} that
\[
\lim_{n\to\infty}\int_0^T(u^n_t(x,t)-u^{*}_t,\psi(t))_{V^{*},V}dt=0,\quad\forall\psi\in
L^2(0,T;V).
\]
and
\begin{align*}
& \lim_{n\to\infty}\int_0^T(\Delta u^n_t(x,t)-\Delta u^{*}_t,\psi(t))_{V^{*},V}dt
\\
&=\lim_{n\to\infty}\int_0^T(u^n_t(x,t)-u^{*}_t,\Delta\psi(t))_{V^{*},V}dt =0,
\quad\forall\psi\in L^2(0,T;V).
\end{align*}
 Using \eqref{3-8} again, we derive that 
\begin{equation}
\left|\int_0^T\int_{\Omega}(Bw-Bw^{*})\eta \,dx\,dt\right|\to
0,\quad n\to\infty,~~\forall \eta\in L^2(0,T;H).\nonumber
\end{equation}
By \eqref{3-7} again, we deduce that
\begin{align*}
&\big|\int_0^T\int_{\Omega}\left(\Delta\varphi(u^n)-\Delta\varphi(u^*)\right)\eta
\,dx\,dt\big|\\
&=\big|\int_0^T\int_{\Omega}\left(\varphi(u^n)-\varphi(u^*)\right)\Delta\eta 
\,dx\,dt\big|
\\
&=\big|\int_0^T\int_{\Omega}[\gamma_2((u^n)^3-(u^*)^3)
 +\gamma_1((u^n)^2-(u^*)^2)-(u^n-u^*)]\Delta\eta \,dx\,dt\big|
\\
&=\big|\int_0^T\int_{\Omega}\big[\gamma_2(u^n-u^*)((u^n)^2+u^nu^*+(u^*)^2)
 +\gamma_1(u^n-u^*)(u^n+u^*)\\
&\quad -(u^n-u^*)\big]\Delta\eta \,dx\,dt\big|
\\
&\leq c\big|\int_0^T\big(\|(u^n)^2+u^nu^*+(u^*)^2\|_{\infty}
 +\|u^n+u^*\|_{\infty}+1\big)\|u^n-u^*\|_H\|\Delta\eta\|_Hdt\big|
\\
&\leq \left(\|(u^n)^2+u^nu^*+(u^*)^2\|_{L^2(0,T;L^{\infty})}
 +\|u^n+u^*\|_{L^2(0,T;L^{\infty})}+1\right)\\
&\quad\times \|u^n-u^*\|_{C(0,T;H)}|\|\eta\|_{L^2(0,T;V)}
\to 0,\quad n\to\infty,\; \forall\eta\in L^2(0,T;V).
\end{align*}
Hence we have $u=u(\bar{\omega})$ and therefore
$$
J(u,\bar{\omega})\leq\lim_{n\to\infty}J(u^n,\bar{\omega}^n)=\lambda.
$$
In view of the above discussions, we obtain 
\[
e_1(u^{*},w^{*})=0,\quad \forall n\in\mathbb{N}.
\]
Noticing that $u^{*}\in W(0,T;V)$, we derive that $u^{*}(0)\in H$.
Since $u^n\to u^{*}$ weakly in $W(0,T;V)$, we can infer that
$u^n(0)\to u^{*}(0)$ weakly when $n\to\infty$. Thus,
we obtain
$$
(u^n(0)-u^{*}(0), \eta)\to 0,\quad n\to\infty,\; \forall\eta \in H,
$$
which means $e_2(u^{*},w^{*})=0$. Therefore, we obtain
$$
e(u^{*},w^{*})=0,\quad\text{in } Y.
$$
So, there exists an optimal solution $(u^{*},w^{*})$ to problem
\eqref{3-1}.
Then, the proof of Theorem \ref{thm3.1} is complete.
\end{proof}

\section{Optimality conditions}

 It is well known that the optimality conditions for
$w$ are given by the variational inequality
\begin{equation}\label{4-1} 
J'(u,w)(v-w)\geq 0,\quad \text{for all } v\in L^2(Q_0),
\end{equation}
where $J'(u,w)$ denotes the Gateaux derivative of $J(u,v)$ at $v=w$.
The following Lemma \ref{lem4.1} is essential in deriving necessary optimality
conditions.

\begin{lemma}\label{lem4.1} 
The map $v\to u(v)$ of $L^2(Q_0)$ into $W(0,T; V)$ is weakly Gateaux 
differentiable at $v=w$ and such the Gateaux derivative of
$u(v)$ at $v=w$ in the direction
$v-w\in L^2(Q_0)$, say
$z=\mathcal{D}u(w)(v-w)$, is a unique
weak solution of the  problem
\begin{equation}\label{4-11}
\begin{gathered}
z_t-k\Delta z_t+\gamma\Delta^2z-\Delta(\varphi'(u(w))z)
 =B(v-w),\quad (x,t)\in Q, \\
z(x,t)=\Delta z(x,t)=0,\quad (x,t)\in\partial\Omega\times(0,T),\\
z(0)=0,\quad x\in\Omega.
\end{gathered}
\end{equation}
\end{lemma}

\begin{proof}
 Let $0\leq h\leq 1$, $u_h$ and $u$ be
the solutions of \eqref{3-1} corresponding to $w+h(v-w)$ and $w$,
respectively. Then we prove the lemma in the following two steps:
\smallskip

\noindent\textbf{Step 1.} We prove that $u_h\to u$ strongly in $C(0,T;H_0^1)$ as
$h\to 0$. Let $q=u_h-u$, then
\begin{equation}\label{4-a}
\begin{gathered}
\frac{dq}{dt}-k\frac{d\Delta q}{dt}+\gamma\Delta^2q
-\Delta(\varphi(u_h)-\varphi(u))=hB(v-w),\quad 0<t\leq T,
\\
q(x,t)=\Delta q(x,t)=0,\quad x\in\partial\Omega,\\
q(0)=0,\quad x\in\Omega.
\end{gathered}
\end{equation}
Using Corollary \ref{coro2.3} and Sobolev's embedding, 
$$
\|u\|_{\infty}\leq c_1',\quad \|u_h\|_{\infty}\leq c_2'.
$$
Taking the scalar product of \eqref{4-a} with $q$, we have 
\[
\frac12\frac d{dt}(\|q\|^2+k\|\nabla q\|^2)+\gamma\|\Delta q\|^2
=(hB(v-w),q) +(\Delta(\varphi(u_h)-\varphi(u)),q).
\]
Noticing that
\begin{align*}
(\Delta(\varphi(u_h)-\varphi(u)),q)
&=(\gamma_2(u_h^3-u^3)+\gamma_1(u_h^2-u^2)-(u_h-u),\Delta q)\\
&=([\gamma_2(u_h^2+u^2+u_hu)+\gamma_1(u_h+u)-1]q,\Delta q)\\
&\leq \|\gamma_2(u_h^2+u^2+u_hu)+\gamma_1(u_h+u)-1\|_{\infty}\|q\|\|\Delta q\|
\\
&\leq c'_3\|q\|\|\Delta q\|\leq\frac{\gamma}2\|\Delta q\|^2
+\frac{(c'_3)^2}{2\gamma}\|q\|^2.
\end{align*}
Hence
\begin{align*}
\frac d{dt}(\|q\|^2+k\|\nabla q\|^2)+\gamma\|\Delta q\|^2
&\leq \frac{(c_3')^2}{\gamma}\|q\|^2+2h\|B(v-w)\|\|q\|\\
&\leq \Big( \frac{(c_3')^2}{\gamma}+1\Big)\|q\|^2+h^2\|B(v-w)\|^2,
\end{align*}
Using Gronwall's inequality, it is easy
to see that $\|q\|^2\to 0$ as $h\to 0$. Then,
$u_h\to u$ strongly in $C(0,T;H_0^1)$ as $h\to 0$.
\smallskip

\noindent\textbf{Step 2.} We prove that $\frac {u_h-u}{h}\to z$ strongly in
$W(0,T;V)$. Rewrite \eqref{4-a} in the following form
\begin{gather*}
\begin{aligned}
&\frac{d}{dt}\Big(\frac{u_h-u}{h}\Big)-k\frac{d}{dt}
 \Delta\Big(\frac{u_h-u}{h}\Big)+\gamma\Delta^2\Big(\frac{u_h-u}{h}\Big)
 -\Delta\Big(\frac{\varphi(u_h)-\varphi(u)}h\Big)\\
& =B(v-w),\quad 0<t\leq T,
\end{aligned}\\
\frac{u_h-u}{h}(x,t)=\Delta\Big(\frac{u_h-u}{h}\Big)(x,t)=0,\quad
(x,t)\in\partial\Omega\times(0,T),\\
\frac{u_h-u}{h}(0)=0,\quad x\in\Omega.
\end{gather*}
We can easily verify that the above problem possesses a unique weak
solution in $ W(0,T;V)$. On the other hand, it is easy to check that
the linear problem \eqref{4-11} possesses a unique weak solution
$z\in W(0,T;V)$. Let $r=\frac{u_h-u}{h}-z$, thus $r$ satisfies
\begin{equation}\label{4-9}
\begin{gathered}
\frac{d}{dt}r+k\frac d{dt}\Delta r+\gamma\Delta^2r
-\Delta\Big(\frac{\varphi(u_h)-\varphi(u)}h-\varphi'(u)z\Big)=0,\quad
 0<t\leq T, \\
r(x,t)=\Delta r(x,t)=0,\quad (x,t)\in\partial\Omega\times(0,T),\\
r(0)=0,\quad x\in\Omega.
\end{gathered}
\end{equation}
Taking the scalar product of \eqref{4-9} with $r$, we obtain 
\[
\frac12\frac{d}{dt}(\|r\|^2+k\|\nabla r\|^2)+\gamma\|\Delta r\|^2
=\Big(\Delta(\frac{\varphi(u_h)-\varphi(u)}h-\varphi'(u)z),r\Big).
\]
Noticing that
\begin{align*}
&\Big(\Delta(\frac{\varphi(u_h)-\varphi(u)}h-\varphi'(u)z),r\Big)\\
&=\Big(\frac{\varphi(u_h)-\varphi(u)}h-\varphi'(u)z,\Delta r\Big)\\
&\leq \|\frac{\varphi(u_h)-\varphi(u)}h-\varphi'(u)z\|\|\Delta r\|\\
&= \|\varphi'(u+\theta(u_h-u))\frac{u_h-u}h-\varphi'(u)z\|\|\Delta r\|\\
&\leq \frac{\gamma}2\|\Delta  r\|^2+c'_4\|\varphi'(u+\theta(u_h-u))
 \frac{u_h-u}h-\varphi'(u)z\|^2,
\end{align*}
where $\theta\in(0,1)$. We have $u_h\to u$ strongly
in $C(0,T;H_0^1) $ as $h\to 0$, then
\begin{align*}
&\|\varphi'(u+\theta(u_h-u))\frac{u_h-u}h-\varphi'(u)z\|^2\\
&\to\|\varphi'(u)(\frac{u_{h}-u}h-z)\|^2\\
&\leq c_5'\|r\|^2\quad \text{as }h\to 0.
\end{align*}
Therefore,
$$
\Big(\Delta(\frac{\varphi(u_h)-\varphi(u)}h-\varphi'(u)z),r\Big)
\leq \frac{\gamma}2\|\Delta r\|^2+c_4'c_5'\|r\|^2.
$$
Summing up, we obtain
$$
\frac d{dt}(\|r\|^2+k\|\nabla r\|^2)+\gamma\|\Delta r\|^2
\leq 2c_4'c_5'(\|r\|^2+k\|\nabla r\|^2).
$$
Using Gronwall's inequality, it is easy to check that
$\frac{u_h-u}{h}$ is strongly convergent to $z$ in $W(0,T;V)$.
Then, Lemma \ref{lem4.1} is proved.
\end{proof}

As in \cite{Lions}, we denote the $\Lambda$ the canonical
isomorphism of $S$ onto $S^{*}$,  where $S^{*}$  is the dual spaces of $S$.
 By calculating the Gateaux derivative of \eqref{3-4} via
Lemma \ref{lem4.1}, we see that the cost $J(v)$ is weakly
Gateaux differentiable at $w$ in the direction
$v-w$. Therefore, \eqref{4-1} can be rewritten as
\begin{equation}\label{4-2} 
(C^*\Lambda\left(Cu(w)-z_d),
z\right)_{W(V)^*,W(V)}+\frac{\delta}2(w,v-w)_{L^2(Q_0)}\geq
0,\quad \forall v\in L^2(Q_0),
\end{equation}
where $z$ is the solution of \eqref{4-11}.

Now, we study the necessary conditions of optimality. To avoid the
complexity of observation states, we consider the two types of
distributive and terminal value observations.
\smallskip

\noindent\textbf{Case 1.}  $C\in \mathcal{L}(L^2(0,T;V);S)$.
In this case, $C^*\in\mathcal{L}(S^*;L^2(0,T;V^*))$, \eqref{4-2}
may be written as
\begin{equation}\label{5-1}
\int_0^T(C^*\Lambda(Cu(w)-z_d),z)_{V^*,V}dt+\frac{\delta}2(w,v-w)_{L^2(Q_0)}\geq0,
\quad \forall v\in L^2(Q_0).
\end{equation}
We introduce the adjoint state $p(v)$ by
\begin{equation} \label{4-3}
\begin{gathered} 
-\frac d{dt}[p(v)+k\Delta p(v)]+\gamma
\Delta^2p(v)-\varphi'(u(v))\Delta p(v)=C^*\Lambda(Cu(v)-z_d),~~(x,t)\in Q,
\\
p(v)=\Delta p(v)=0,\quad x\in\partial\Omega,\\
p(x,T;v)=0.
\end{gathered}
\end{equation}
According to Theorem \ref{thm2.1}, the above problem admits a unique
solution (after changing $t$ into $T-t$).

Multiplying both sides of \eqref{4-3} (with $v=w$)
by $z$, using Lemma \ref{lem4.1}, we obtain
\begin{gather*}
\int_0^T\Big(-\frac d{dt}p(w),z\Big)_{V^*,V}dt
=\int_0^T\Big(p(w),\frac d{dt}z\Big)dt, \\
\int_0^T\Big(-\frac d{dt}\Delta p(w),z\Big)_{V^*,V}dt
=\int_0^T\Big(p(w),\frac d{dt}\Delta z\Big)dt, \\
\int_0^T\Big(\Delta^2p(w),z\Big)_{V^*,V}dt
=\int_0^T(p(w), \Delta^2z)dt, \\
\int_0^T\left(\varphi'(u(w))\Delta p(w),z\right)_{V^*,V}dt
=\int_0^T\left(p(w),\Delta(\varphi'(u(w))z)\big)\right)dt
\end{gather*}
Thus, we obtain
\begin{align*}
&\int_0^T(C^*\Lambda(Cu(w)-z_d),z)_{V^*,V}dt
\\
&=\int_0^T\Big(p(w),\frac d{dt}(z+k\Delta z)
 +\gamma \Delta^2 z-\Delta(\varphi'(u)z)x\Big)dt
\\
&=\int_0^T(p(w),Bv-Bw)dt\\
&=(B^*p(w),v-w).
\end{align*}
Therefore, \eqref{5-1} may be written as
\begin{equation}\label{5-2}
\int_0^T\int_0^1B^*p(w)(v-w)\,dx\,dt+\frac{\delta}2(w,v-w)_{L^2(Q_0)}\geq0,
\quad \forall v\in L^2(Q_0).
\end{equation}
Then, we have proved the following theorem.

\begin{theorem} \label{thm4.1} 
Assume that $C\in\mathcal{L}(L^2(0,T;V);S)$ and all conditions of 
Theorem \ref{thm3.1} hold. Then, the
optimal control $w$ is characterized by the system of two
PDEs and an inequality: \eqref{3-1}, \eqref{4-3} and \eqref{5-2}.
\end{theorem}
\smallskip

\noindent\textbf{Case 2.} $C\in \mathcal{L}(H;S)$.
In this case, we observe $Cu(v)=Du(T;v),~D\in
\mathcal{L}(H;H)$. The associated cost function is 
\begin{equation}
J(u,v)=\|Du(T;v)-z\|_S^2+\frac{\delta}2\|v\|_{L^2(Q_0)}^2,\quad
\forall v\in L^2(Q_0). \label{5-3}
\end{equation}
Then, for all $v\in L^2(Q_0)$, the optimal control $w$ for \eqref{5-3} is
characterized by
\begin{equation}\label{5-4}
(Du(T;w)-z,Du(T;v)-Du(T;w))_{V^*,V}+\frac{\delta}2(w,v-w)_{L^2(Q_0)}\geq
0.
\end{equation}
We introduce the adjoint state $p(v)$ by
\begin{equation}\label{5-5}
\begin{gathered} 
-\frac d{dt}[p(v)+k\Delta p(v)]+\gamma
\Delta^2p(v)-\varphi'(u(v))\Delta p(v)x=0,\quad (x,t)\in Q,
\\
p(v)=\Delta p(v)=0,\quad x\in\partial\Omega,
\\
p(T;v)=D^*(Du(T;v)-z_d).
\end{gathered}
\end{equation}
According to Theorem \ref{thm2.1}, the above problem admits a unique
solution (after changing $t$ into $T-t$).

Set $v=w$ in the above equations and scalar
multiply both side of the first equation of \eqref{5-5} by
$u(v)-u(w)$ and integrate from $0$ to $T$. A simple
calculation shows that \eqref{5-4} is equivalent to
\begin{equation}\label{5-6}
\int_0^T\int_0^1B^*p(w))(v-w)\,dx\,dt+\frac{\delta}2w,v-w)_{L^2(Q_0)}\geq0,
\quad \forall v\in L^2(Q_0).
\end{equation}
We obtain the following result.

\begin{theorem} \label{thm4.2} 
Assume that $D\in\mathcal{L}(H;H)$ and all conditions of Theorem \ref{thm3.1}
hold. Then, the optimal
control $w$ is characterized by the system of two PDEs
and an inequality: \eqref{3-1}, \eqref{5-5} and \eqref{5-6}.
\end{theorem}

\subsection*{Acknowledgements}
The authors would like to thank the anonymous referees
 and Dr. Xiaopeng Zhao for their valuable
comments and suggestions about this paper.


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\end{document}