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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 155, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/155\hfil Optimal control]
{Optimal control of a modified Swift-Hohenberg equation}

\author[N. Duan, W. Gao \hfil EJDE-2012/155\hfilneg]
{Ning Duan, Wenjie Gao}

\address{Ning Duan \newline
College of Mathematics, Jilin University, Changchun 130012, China}
\email{duanningjlu@qq.com}

\address{Wenjie Gao \newline
College of Mathematics, Jilin University, Changchun 130012, China}
\email{gaowj@jlu.edu.cn}

\thanks{Submitted April 2, 2012. Published September 7, 2012.}
\thanks{Supported by grant 11271154 from NSFC.}
\subjclass[2000]{35K55, 49A22}
\keywords{Optimal control; modified Swift-Hohenberg equation;
\hfill\break\indent
 optimal solution}

\begin{abstract}
 In this article, we present the optimal control for the  modified
 Swift-Hohenberg equation, under certain boundary conditions, and
 show the existence of an optimal solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}\label{sec1}

 This article concerns the 1-D modified Swift-Hohenberg equation
 that was proposed by Doelman et al \cite{Arjen}:
\begin{equation}\label{1-1}
u_t+ku_{xxxx}+2u_{xx}+au+b|u_x|^2+u^3=0,\quad x\in\Omega,\; t\in(0,T).
\end{equation}
On the basis of physical considerations,  \eqref{1-1} is
supplemented with the  boundary value condition
\begin{equation} \label{1-2}
u(x,t)=u_{xx}(x,t)=0\quad \text{for }x\in\partial\Omega,
\end{equation}
and the initial condition
\begin{equation}
u(x,0)=u_0 (x),\quad x\in\Omega, \label{1-3}
\end{equation}
where $\Omega$ is an open connected bounded domain in $\mathbb{R}$,
$k$, $a$ and $b$ are arbitrary constants. $u_0(x)$ is a given
function from a suitable phase space.

The Swift-Hohenberg equation is one of the universal equations used
in the description of pattern formation in spatially extended
dissipative systems, (see \cite{Song}), which arise in the study of
convective hydrodynamics \cite{Swift}, plasma confinement in
toroidal devices \cite{Quey}, viscous film flow and bifurcating
solutions of the Navier-Stokes \cite{Shang}. Note that, the usual
Swift-Hohenberg equation \cite{Swift} is recovered for $b=0$. The
additional term $b|\nabla u|^2$, reminiscent of the
Kuramoto-Sivashinsky equation, which arises in the study of various
pattern formation phenomena involving some kind of phase turbulence
or phase transition, (see \cite{K,  Polat,S}), breaks the symmetry
$u\to -u$.


During the past years, many authors have paid much attention to the
Swift-Hohenberg equation (see, e.g. \cite{ Lega, Peletier,Swift}).
However, only a few people dovoted to the modified Swift-Hohenberg
equation. It were A. Doelman et al.\cite{Arjen} who first studied
the modified Swift-Hohenberg equation for a pattern formation system
with two unbounded spatial directions that is near the onset to
instability. M. Polat\cite{Polat} also considered the modified
Swift-Hohenberg equation. In his paper, the existence of a global
attractor is proved for the modified Swift-Hohenberg equation as
\eqref{1-1}-\eqref{1-3}. Recently, L. Song et al.\cite{Song}
studied the long time behavior for modified Swift-Hohenberg equation
in $H^k~(k\geq 0)$ space. By using an iteration procedure,
regularity estimates for the linear semigroups and a classical
existence theorem of global attractor, they proved that problem
\eqref{1-1}-\eqref{1-3} possesses a global attractor in Sobolev
space $H^k$ for all $k\geq 0$, which attracts any bounded subset of
$H^k(\Omega)$ in the $H^k$-norm.

The optimal control plays an important role in modern control
theories, and has a wider application in modern engineering. Two
methods are used for studying control problems in PDE: one is using
a low model method, and then changing to an ODE model \cite{Ito};
the other is using a quasi-optimal control method \cite{Atwell}. No
matter which one is chosen, it is necessary to prove the existence
of optimal solution and establish the optimality system. Many papers
have already been published to study the control problems of
nonlinear parabolic equations. For example, Yong and
Zheng\cite{Zheng}, Tian et al.\cite{Tian2,Tian3}, Ryu and Yagi
\cite{Ryu1, Ryu2}, Zhao and Liu\cite{Zhao} and so on.


This article concerns the distributed optimal control
problem
\begin{equation} \label{1-4}
\text{minimize} \quad J(u,
w)=\frac12\|Cu-z_d\|^2_S+\frac{\delta}{2}\|w\|^2_{L^2(Q_0)},
\end{equation}
subject to
\begin{equation} \label{1-5}
\begin{gathered}
\frac{\partial u}{\partial t}+ku_{xxxx}+2u_{xx}+au+b| u_x|^2+u^3=Bw,\quad
 (x,t)\in\Omega\times (0,T),\\
u(x,t)=u_{xx}(x,t)=0,\quad x\in\partial\Omega,\\
u(x,0)=u_0(x),\quad x\in\Omega.
\end{gathered}
\end{equation}
 The control target is to match the given desired state $z_d$ in the
$L^2$-sense  by adjusting the body force $w$ in a control volume
$Q_0\subseteq Q=(0,1)\times(0,T)$ in the $L^2$-sense.

Assume that $V=\{u\in H^2(0,1)\big| u(0,t)=u(1,t)=0\}$,
$U=H_0^1(0,1)$ and $H=L^2(0,1)$. Assume further that $V'$, $U'$ and
$H'$ are dual spaces of $V$, $U$ and $H$. Then, we obtain
$$
V\hookrightarrow U\hookrightarrow H=H'\hookrightarrow
U'\hookrightarrow V'.
$$
Each embedding being dense. The extension operator $B\in \mathcal
{L}(L^2(Q_0), L^2(0,T;H))$ which is called the controller is
introduced as %%
\[
Bw= \begin{cases}
w,& q\in Q_0,\\
0,& w\in Q\setminus Q_0.
\end{cases}
\]
We supply $H$ with the inner product $(\cdot,\cdot)$ and the norm
$\|\cdot\|$, and define a space $W(0,T; V)$ as
$$
W(0,T;V)=\{y:y\in L^2(0, T;V),~y_t\in L^2(0,T;V')\},
$$
which is a Hilbert space endowed with common inner product.

This paper is organized as follows. In the next section, we prove
the existence and uniqueness of weak solution to the equation in a
special space. We also 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; Finally in Section 4, conclusions are obtained.

\section{Existence and uniqueness of weak solutions}

In this section, we prove the existence and uniqueness of weak
solution for problem \eqref{1-5},
where $x\in (0,1)$, $t\in [0,T]$, $Bw\in L^2(0,T;H)$ and a control
$w\in L^2(Q_0)$.
Now, we give the definition of the weak solution in the space
$W(0,T;V)$.

\begin{definition}\label{def2.1} \rm
For all $\eta\in V$, a function $u(x,t)\in W(0,T;V)$
is called a weak solution to problem \eqref{1-5}, if
\begin{equation} \label{2-2}
 (\frac{\partial u}{\partial t},\eta)+k(u_{xx},\eta_{xx})
-2(u_{x},\eta_x)+a(u,\eta)+b(|u_x|^2,\eta)+(u^3,\eta)=(Bw,\eta).
\end{equation}
\end{definition}

We shall give a theorem on the existence and uniqueness
of weak solution to problem \eqref{1-5}.

\begin{theorem} \label{thm2.1}
Suppose that $k$ is sufficiently large, $u_0\in V$,
$Bw\in L^2(0,T;H)$, then  \eqref{1-5} admits a unique
weak solution $u(x,t)\in W(0,T;V)$.
\end{theorem}

\begin{proof}
Galerkin's method is applied for this proof.
Denote $\mathbb{A}=(-\partial _{x}^2)^2$ as a differential operator,
let $\{\psi_i\}_{i=1}^{\infty}$ denote the eigenfunctions of the
operator $\mathbb{A}=(-\partial _{x}^2)^2$. For $n\in N$, define the
discrete ansatz space by
$$
V_n=\operatorname{span}\{\psi_1,\psi_2,\dots,\psi_n\}\subset V.
$$
Let $u_n=\sum_{i=1}^{n}u^n_i(t)\psi_i(x)$ require
$u_n(0,\cdot)\to u_0$ in $H$ to hold true.


By analyzing the limiting behavior of sequences of smooth function
$\{u_n\}$, we can prove the existence of a weak solution to the
modified Swift-Hohenberg equation.

Performing the Galerkin metod for \eqref{1-5}, we obtain
\begin{equation} \label{2-3}
\begin{gathered}
\begin{aligned}
&(x\frac{\partial u_n}{\partial t},\eta)+k(u_{n,xx},\eta_{xx})
 -2(u_{n,x},\eta_x)+a(u_n,\eta)+b(|u_{n,x}|^2,\eta)+(u_n^3,\eta)\\
&=(Bw,\eta),\quad \forall \eta\in V,~(x,t)\in Q,
\end{aligned} \\
(u_n(x,0),\eta)=(u_0(x),\eta),\quad \forall \eta\in V, \; x\in(0,1)
\end{gathered}
\end{equation}

Then the equation of problem \eqref{2-3} is an ordinary differential
equation and according to ODE theory, there exists a unique solution
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$.

Then, we shall prove the existence of solution in the following
steps.

Step 1, multiplying the equation in \eqref{2-3} by $u_n$,
integrating with respect to $x$ on $(0,1)$, we deduce that
\begin{equation} \label{2-4}
\begin{split}
&\frac12\frac{d}{dt}\|u_n\|^2+k\|u_{n,xx}\|^2+\|u_n\|_4^4 \\
&\leq |a|\|u_n\|^2+2\|u_{nx}\|^2+|b|(|u_{nx}|^2,u_n)+(Bw,u_n).
\end{split}
\end{equation}
By Nirenberg's inequality,
$$
\|u_{nx}\|_{8/3}\leq c_0\|u_{nxx}\|^{1/2}\|u_n\|_4^{1/2}.
$$
Then
$$
|b|(|u_{nx}|^2,u_n)\leq |b|\|u_{nx}\|_{8/3}^2\|u\|_4\leq
c_0^2|b|\|u_{nxx}\|\|u_n\|^2_4\leq
\|u_n\|_4^4+\frac{c_0^4b^2}4\|u_{nxx}\|^2.
$$
On the other hand, we have
\begin{gather*}
2\|u_{nx}\|^2=-2(u_n,u_{nxx})\leq\|u_n\|^2+\|u_{nxx}\|^2,\\
(Bw,u_n)\leq\|Bw\|\|u_n\|\leq\frac12\|Bw\|^2+\frac12\|u_n\|^2.
\end{gather*}
 Summing up, we have
$$
\frac d{dt}\|u_n\|^2+(2k-\frac{c_0^4b^2}2-2)\|u_{nxx}\|^2
\leq(2|a|+3)\|u_n\|^2+\|Bw\|^2,
$$
where $k$ satisfies $2k-\frac{c_0^4b^2}2-2>0$. Since
$Bw\in L^2(0,T;H)$ is the control item, we can assume $ \|Bw\|\leq M$,
where $M$ is a positive constant. Then
\begin{eqnarray}
\label{2-5} \frac
d{dt}\|u_n\|^2+(2k-\frac{c_0^4b^2}2-2)\|u_{nxx}\|^2\leq(2|a|+3)\|u_n\|^2+M^2.
\end{eqnarray}
Using Gronwall's inequality, we obtain
\begin{equation}\label{2-6}
\begin{split}
\|u_n\|^2&\leq e^{(2|a|+3)t}\|u_{n,0}\|^2+\frac{M^2}{2|a|+3}\\
&\leq e^{(2|a|+3)T}\|u_{n,0}\|^2+\frac{M^2}{2|a|+3}= c_1^2,\quad t\in[0,T].
\end{split}
\end{equation}
Integrating \eqref{2-5} with on $[0,T]$,
\begin{equation} \label{2-7}
\begin{split}
&\int_0^T\|u_{n,xx}\|^2dt\\
&\leq \frac2{4k-c_0^4b^2-4}
\Big((2|a|+3)\int_0^T\|u_n\|^2dt+M^2T+\|u_{n,0}\|^2\Big)\\
&\leq \frac2{4k-c_0^4b^2-4}
\big( (2|a|+3)c_1^2T+M^2T+\|u_{n,0}\|^2\big)=c_2^2.
\end{split}
\end{equation}
Multiplying the equation in \eqref{2-3} by $u_{nxx}$, integrating
with respect to $x$ on $(0,1)$, we deduce that
\begin{equation}
\begin{split}
&\frac12\frac d{dt}\|u_{n,x}\|^2+k\|u_{n,xxx}\|^2\\
&=2\|u_{nxx}\|^2-a\|u_{nx}\|^2+((u_n)^3,u_{nxx})
+b(|u_{nx}|^2,u_{nxx})-(Bw, u_{n,xx}).
\end{split} \label{zxpp}
\end{equation}
Noticing that
$$
2\|u_{nxx}\|^2=-2(u_{nx},u_{nxxx})\leq\frac
k{12}\|u_{nxxx}\|^2+\frac{12}k\|u_{nx}\|^2,
$$
and
\begin{align*}
-(Bw,u_{nxx})
&\leq \|Bw\|\|u_{nxx}\|\leq\frac{M^2}2+\frac12\|u_{nxx}\|^2\\
&\leq \frac {M^2}2+\frac12(\frac
k6\|u_{nxxx}\|^2+\frac{3}{2k}\|u_{nx}\|^2).
\end{align*}
By Nirenberg's inequality,
$$
\|u_n\|_6\leq c_0\|u_{nxxx}\|^{1/9}\|u_n\|^{8/9}, \quad
\|u_{nx}\|_4\leq c_0\|u_{nxxx}\|^{5/12}\|u_n\|^{7/12}.
$$
Hence
\begin{align*}
((u_n)^3,u_{nxx})
&\leq 2\|u_{nxx}\|^2+\frac18\|u_n\|_6^6 \\
&\leq \frac k{12}\|u_{nxxx}\|^2+\frac{12}k\|u_{nx}\|^2+\frac
k{12}\|u_{nxxx}\|^2+c(c_1)  \\
&= \frac{12}k\|u_{nx}\|^2+\frac k{6}\|u_{nxxx}\|^2+c(c_1),
\end{align*}
 and
\begin{align*}
|b|((u_{nx})^2,u_{nxx})
&=|b|\int_0^1(u_{nx})^2u_{nxx}dx
\leq\frac{|b|^2}8\|u_{nx}\|_4^4+2\|u_{nxx}\|^2\\
&\leq  \frac{k}{12}\|u_{nxxx}\|^2+c(c_1)+\frac
k{12}\|u_{nxxx}\|^2+\frac{12}k\|u_{nx}\|^2  \\
&= \frac k6\|u_{nxxx}\|^2+c(c_1)+\frac{12}k\|u_{nx}\|^2.
\end{align*}
Summing up,
 $$
\frac d{dt}\|u_{nx}\|^2+k\|u_{nxxx}\|^2\leq
(\frac{72}k+2|a|+\frac3{2k})\|u_{nx}\|^2+2c(c_1)+M^2.
$$
Using Gronwall's inequality, we deduce that
\begin{equation} \label{2-9}
\begin{split}
\|u_{n,x}\|^2
&\leq e^{(\frac{72}k+2|a|+\frac3{2k})t}\|u_{n,x}(0)\|^2
 +\frac{2k(2c(c_1)+M^2)}{144+4k|a|+3} \\
&\leq e^{(\frac{72}k+2|a|+\frac3{2k})T}\|u_{n,x}(0)\|^2
 +\frac{2k(2c(c_1)+M^2)}{144+4k|a|+3}=c_3^2,\quad t\in[0,T].
\end{split}
\end{equation}
Then, by \eqref{2-6}, \eqref{2-7} and \eqref{2-9}, we obtain
$$ \int_0^T\|u_n(x,t)\|_{H^2}^2dt\leq c.
$$
Using Sobolev's embedding theorem, we also have
\begin{equation} \label{xp1}
\|u_{n}\|_{\infty}\leq c_4.
\end{equation}


Step 2, we prove a uniform $L^2(0,T;V')$ bound on a sequence
$\{u_{n,t}\}$. In order to obtain the result, we first establish the
$H^2$-norm estimate for problem \eqref{2-3}.

Multiplying the equation in \eqref{2-3} by $u_{nxxxx}$, integrating
with respect to $x$ on $(0,1)$, we deduce that
\begin{equation}
\begin{split}
&\frac12\frac d{dt}\|u_{n,xx}\|^2+k\|u_{n,xxxx}\|^2\\
&= 2\|u_{nxxx}\|^2-a\|u_{nxx}\|^2-((u_n)^3,u_{nxxxx})
-b(|u_{nx}|^2,u_{nxxxx})+(Bw, u_{nxxxx}).
\end{split} \label{zxppp}
\end{equation}
By Nierenberg's inequality,
$$
\|u_{nx}\|_4\leq
c_0\|u_{nxxxx}\|^{1/12}\|u_{nx}\|^{11/12}.
$$
Therefore,
$$
b((u_{nx})^2,u_{nxxxx})\leq\frac
k{10}\|u_{nxxxx}\|^2+\frac{5|b|^2}{2k}\|u_{nx}\|_4^4\leq\frac
k5\|u_{nxxxx}\|^2+ c(c_2).
$$
On the other hand, we have
\begin{gather*}
((u_n)^3,u_{nxxxx})
\leq \sup_{x\in[0,1]}|u_n|^3\cdot\|u_{nxxxx}\|_1\leq\frac
k{10}\|u_{nxxxx}\|^2+c(c_4),
\\
2\|u_{nxxx}\|^2=-2(u_{nxx},u_{nxxxx})\leq\frac
k{10}\|u_{nxxxx}\|^2+\frac{10}k\|u_{nxx}\|^2,
\\
(Bw,u_{nxxxx})\leq\|Bw\|\|u_{nxxxx}\|\leq\frac
k{10}\|u_{nxxxx}\|^2+\frac{5M^2}{2k}.
\end{gather*}
Summing up,
$$
\frac d{dt}\|u_{nxx}\|^2+k\|u_{nxxxx}\|^2\leq
(\frac{20}k+2|a|)\|u_{nxx}\|^2+\frac{5M^2}k+2c(c_2)+2c(c_4).
$$
Using Gronwall's inequality, we derive that
\begin{equation}
\begin{aligned}
\|u_{nxx}\|^2
&\leq e^{(\frac{20}k+2|a|)t}\|u_{nxx}(0)\|^2
+\frac{5M^2+2k(c(c_2)+c(c_4))}{20+2k|a|}
\\
&\leq e^{(\frac{20}k+2|a|)T}\|u_{nxx}(0)\|^2
+\frac{5M^2+2k(c(c_2)+c(c_4))}{20+2k|a|}\\
&= c_5^2,\quad \forall t\in[0,T].
\end{aligned}\label{xp2}
\end{equation}
It then follows from \eqref{2-6}, \eqref{2-7} and \eqref{xp2} that
\begin{equation} \label{xp3}
\|u_{nx}\|_{\infty}\leq c_6.
\end{equation}
Notice that
\begin{gather*}
(u_{nxxxx},\eta)=(u_{nxx},\eta_{xx})\leq\|u_{nxx}\|\|\eta_{xx}\|
\leq\|u_{nxx}\|\|\eta\|_V,
\\
(|u_{nx}|^2,\eta)\leq\sup_{x\in[0,1]}|u_{nx}|\cdot(u_{nx},\eta)
\leq c_6\|u_{nx}\|\|\eta\|\leq c_6\|u_{nx}\|\|\eta\|_V,
\\
((u_n)^3,\eta)\leq\sup_{x\in[0,1]}|u_n|^2\cdot(u_n,\eta)
\leq c_4^2\|u_n\|\|\eta\|\leq c_4^2\|u_n\|\|\eta\|_V,
\\
(u_{nxx},\eta)=(u_n,\eta_{xx})\leq\|u_n\|\|\eta_{xx}\|\leq\|u_n\|\|\eta\|_V,
\quad (u_n,\eta)\leq\|u_n\|\|\eta\|\leq\|u_n\|\|\eta\|_V,
\end{gather*}
 Therefore,
\begin{align*}
 \|u_{nt}\|_{V'}
& \leq k\|u_{nxx}\|+2\|u_n\|+|a|\|u_n\|+|b|c_6\|u_{nx}\|+c_4^2\|u_n\|+\|Bw\|
\\
&\leq (kc_5+2c_1+|a|c_1+|b|c_6c_3+c_4^2c_1+M).
\end{align*}
Hence,
\begin{equation}
\|u_{n,t}\|_{L^2(0,T;V)}\leq (kc_5+2c_1+|a|c_1+|b|c_6c_3+c_4^2c_1+M)T=c_7. \label{2-10}
\end{equation}

Collecting the previous results, we obtain:

(1) For every $t\in [0,T]$, the sequence $\{u_n\}_{n\in N}$ is
bounded in $L^2(0,T;V)$, which is independent of the dimension of
ansatz space $n$.

(2) For every $t\in[0,T]$, the sequence $\{u_{n,t}\}_{n\in N}$ is
bounded in $L^2(0,T;V')$, which is independent of the dimension of
ansatz space $n$.


By the above discussion, we obtain $u(x,t)\in W(0,T;V)$. It is easy
to check that $W(0,T;V)$ is compactly embedded into $C(0,T;H)$ which
denote the space of continuous functions. We concludes convergence
of a subsequences, again denoted by $\{u_n\}$  weak into $W(0,T;V)$,
weak-star in $L^{\infty}(0,T;H)$ and strong in $L^2(0,T;H)$ to
functions $u(x,t) \in W(0,T;V)$.

Since the proof of uniqueness is easy, we omit it.
 Then, Theorem \ref{thm2.1} is proved.
\end{proof}

Now, we shall discuss the relation among  the norm of the weak solution,
the initial value, and the control item.

\begin{theorem}\label{thm2.2}
Suppose that $k$ is sufficiently large, $u_0\in V$,
$Bw\in L^2(0,T;H)$, then there exists positive constants $C_1$ and
$C_2$ such that
\begin{equation}
\|u\|^2_{W(0,T;V)}\leq C_1(\|u_0\|^2_{V}+\|w\|^2_{L^2(Q_0)})+C_2, \label{2-15}
\end{equation}
\end{theorem}

\begin{proof}
 Clearly, \eqref{2-15} implies
\begin{equation} \label{2-16}
\|u\|^2_{L^2(0,T;V)}+\|u_t\|^2_{L^2(0,T;V')} \leq
C_1(\|u_0\|^2_{V}+\|Bw\|^2_{L^2(H)})+C_2.
\end{equation}
Passing to the limit in \eqref{2-4}, we obtain
\begin{equation}\label{2-17}
\frac12\frac{d}{dt}\|u\|^2+k\|u_{xx}\|^2+\|u\|_4^4\leq
|a|\|u\|^2+2\|u_{x}\|^2+|b|(|u_{x}|^2,u)+(Bw,u).
\end{equation}
Using the same method as in the proof of the above theorem, we derive that
\begin{equation} \label{zxp1}
\frac d{dt}\|u\|^2+(2k-\frac{c_0^4b^2}2-2)\|u_{xx}\|^2\leq(2|a|+3)\|u\|^2+\|Bw\|^2.
\end{equation}
Then, by Gronwall's inequality,
\begin{equation}
\begin{split}
 \|u\|^2&\leq  e^{(2|a|+3)t}\|u_0\|^2+\frac1{2|a|+3}\|Bw\|^2\\
&\leq  c_8\|u_0\|^2+c_9\|Bw\|^2,~~\forall t\in[0,T].
\end{split} \label{dn1}
\end{equation}
Therefore,
\begin{equation}  \label{bj}
\|u\|^2_{L^2(0,T;H)}\leq c_8T\|u_0\|^2+c_9\|Bw\|_{L^2(0,T;H)}^2.
\end{equation}
Integrating \eqref{zxp1} with respect to $t$ on $[0,T]$, we obtain
\begin{align*}
&\|u(T)\|^2-\|u_0\|^2+(2k-\frac{c_0^4b^2}2-2)\|u_{xx}\|^2_{L^2(H)}\\
&\leq \|Bw\|^2_{L^2(H)}+(2|a|+3)\|u\|_{L^2(H)}^2.
\end{align*}
By \eqref{bj} and the above inequality,
\begin{equation} \label{z4}
\begin{split}
&\|u_{xx}\|_{L^2(H)}^2\\
&\leq \frac2{4k-c_0^4b^2-4}\Big(\|Bw\|_{L^2(H)}^2+(2|a|+3)
 (c_8T\|u_0\|^2+c_9\|Bw\|_{L^2(H)}^2)+\|u_0\|^2\Big)\\
&\leq c_{10}\|Bw\|_{L^2(H)}^2+c_{11}\|u_0\|^2.
\end{split}
\end{equation}
Passing to the limit in \eqref{zxpp}, we obtain
\begin{align*}
&\frac12\frac d{dt}\|u_{x}\|^2+k\|u_{xxx}\|^2\\
&=2\|u_{xx}\|^2-a\|u_{x}\|^2+((u)^3,u_{xx})+b(|u_{x}|^2,u_{xx})-(Bw,u_{xx}).
\end{align*}
Using the same method as in the proof of the above theorem, we derive that
$$
\frac d{dt}\|u_{x}\|^2+k\|u_{xxx}\|^2\leq
2c(c_1)+\|Bw\|^2+(\frac{72}k+2|a|+\frac3{2k})\|u_{x}\|^2.
$$
By Gronwall's inequality,
\begin{equation}
\begin{split}
\|u_x\|^2
&\leq e^{(\frac{72}k+2|a|+\frac3{2k})t}\|u_{x0}\|^2
 +\frac{4kc(c_1)}{144+4k|a|+3}+\frac{2k}{144+4k|a|+3}\|Bw\|^2 \\
& \leq c_{12}\|u_{x0}\|^2+c_{13}\|Bw\|^2+c_{14}.
\end{split}\label{dn2}
\end{equation}
Therefore,
\begin{equation} \label{z5}
\|u\|_{\infty}\leq c,\quad
\|u_x\|_{L^2(H)}^2\leq c_{12}T\|u_{x0}\|^2+c_{13}\|Bw\|^2_{L^2(H)}+c_{14}T.
\end{equation}
 Adding \eqref{bj}, \eqref{z4} and \eqref{z5}
 gives
\begin{equation} \label{2-19}
\|u\|^2_{L^2(0,T;V)}\leq c_{15}(\|Bw\|^2_{L^2(0,T;H)}+\|u_0\|_U^2)+c_{16}.
\end{equation}
On the other hand, passing to the limit in \eqref{zxppp}, a simple
calculation shows that
$$
\frac d{dt}\|u_{xx}\|^2+k\|u_{xxxx}\|^2\leq
(\frac{20}k+2|a|)\|u_{xx}\|^2+\frac{5}k\|Bw\|^2+2c(c_2)+2c(c_4).
$$
Using Gronwall's inequality,
\begin{equation}
\begin{split}
\|u_{xx}\|^2
&\leq e^{(\frac{20}k+2|a|)t}\|u_{xx0}\|^2
 +\frac{5\|Bw\|^2}{20+2k|a|}+\frac{kc(c_2)+kc(c_4)}{10+|k|a|}\\
&\leq  c_{17}(\|Bw\|^2+\|u_{xx0}\|^2)+c_{18}.
\end{split} \label{dn3}
\end{equation}
It then follows from \eqref{dn1}, \eqref{dn2} and \eqref{dn3} that
$$
\|u_x(x,t)\|\leq c.
$$
On the other hand, we have
\begin{gather*}
(u_{xxxx},\eta)=(u_{xx},\eta_{xx})\leq\|u_{xx}\|\|\eta_{xx}\|
\leq\|u_{xx}\|\|\eta\|_V,
\\
(|u_{x}|^2,\eta)\leq\sup_{x\in[0,1]}|u_{x}|\cdot(u_{x},\eta)\leq
c\|u_{x}\|\|\eta\|\leq c\|u_{x}\|\|\eta\|_V,
\\
((u)^3,\eta)\leq\sup_{x\in[0,1]}|u|^2\cdot(u,\eta)\leq
c^2\|u\|\|\eta\|\leq c^2\|u\|\|\eta\|_V.
\\
(u_{xx},\eta)=(u,\eta_{xx})\leq\|u\|\|\eta_{xx}\|\leq\|u\|\|\eta\|_V,
\quad (u,\eta)\leq\|u\|\|\eta\|\leq\|u\|\|\eta\|_V,
\end{gather*}
 Therefore,
\begin{align*}
&\|u_{t}\|_{V'}\\
& \leq k\|u_{xx}\|+2\|u\|+|a|\|u\|+|b|c\|u_{x}\|+c^2\|u\|+\|Bw\|\\
&\leq k(c_{17}(\|Bw\|^2+\|u_{xx0}\|^2)+c_{18})^{1/2}
 +(2+|a|+c^2)(c_8\|u_0\|^2+c_9\|Bw\|^2)^{1/2}\\
&\quad +|b|c(c_{12}\|u_{x0}\|^2+c_{13}\|Bw\|^2+c_{14})^{1/2} +\|Bw\|
\end{align*}
Hence,
\begin{equation}
\|u_{n,t}\|_{L^2(0,T;V)}^2\leq c_{19}(\|u_0\|_V^2+\|Bw\|^2)+c_{20}.
\label{dn4}
\end{equation}
By \eqref{2-19}, \eqref{dn4} and the definition of extension
operator $B$, we obtain \eqref{2-16}. Then, Theorem \ref{thm2.2} is
proved.
\end{proof}

\section{Optimal control problem}

 In this section, we consider the optimal control
problem associated with the fourth-order parabolic equation and
prove  the existence of optimal solution basing on J. L. Lions' theory
(see \cite{Lions}).

In the following, we suppose $L^2(Q_0)$ is a Hilbert space of
control variables, we also suppose $B\in \mathcal{L}(L^2(Q_0),
L^2(0,T;H))$ is the controller and a control $w\in L^2(Q_0)$,
consider the following control system
\begin{equation} \label{3-1}
\begin{gathered}
\frac{\partial u}{\partial t}+ku_{xxxx}+2u_{xx}+au+b| u_x|^2+u^3=Bw,
\quad (x,t)\in(0,1)\times(0,T),\\
u(x,t)=u_{xx}(x,t)=0,\quad x=0,1,\\
u(x,0)=u_0(x),\quad x\in(0,1).
\end{gathered}
\end{equation}
Here, it is assumed 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 problem \eqref{3-1} is
\begin{equation}
\text{minimize} \quad J(u,w).\label{3-5}
\end{equation}

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=G=(\Delta^2)^{-1}(\frac{\partial u}{\partial t}
 +ku_{xxxx}+2u_{xx}+au+b| u_x|^2+u^3-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
$$
\text{minimize}\quad  J(u,w) \quad \text{subject to }e(u,w)=0.
$$

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

\begin{proof}
 Suppose $(u,w)$ satisfy $e(u,w)=0$. In view of \eqref{3-4},  we deduce that
$$
J(u,w)\geq\frac{\delta}2\|w\|^2_{L^2(Q_0)}.
$$
By Theorem \ref{thm2.2}, we obtain
$\|u\|_{W(0,T;V)}\to\infty$ yields $\|w\|_{L^2(Q_0)}\to\infty$.
Therefore,
\begin{equation} \label{3-6}
J(u,w)\to\infty,\quad\text{when}\quad \|(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 imlies the existence of a minimizing sequence
$\{(u^n,w^n)\}_{n\in N}$ 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}, \quad u\in W(0,T;V),\\
\label{3-8} w^n\to w^{*},\quad \text{weakly},\quad w\in L^2(Q_0).
\end{gather}
Using \eqref{3-7}, we obtain
\begin{gather*} % 3-10
\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), \\
\lim_{n\to\infty}\int_0^T(u^n(x,t)-u^{*},\psi(t))_{V',V}dt=0,\quad
\forall\psi\in L^2(0,T;V), \\
\lim_{n\to\infty}\int_0^T(u^n_{xx}(x,t)-u^{*}_{xx},\psi(t))_{V',V}dt=0,\quad
\forall\psi\in L^2(0,T;V),
\end{gather*}

Since $W(0, T; V)$ is compactly embedded into
$L^2(0, T; L^{\infty})$, we have $u^n\to u^*$ strongly in $L^2(0, T;  L^{\infty})$.
On the other hand, we know that
$u_n\in L^\infty(0,T; V)$ and $u_{n,t}\in L^2(0, T; V^*)$.
Hence by \cite[Lemma 4]{Si}
we have $u^n \to u^*$ strongly in $C(0, T; L^\infty)$, $u_x^n \to u_x^*$ 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. And
$\|u^n\|_{L^2(0, T; L^{\infty})}$ is also bounded based on the embedding theorem.

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

By \eqref{3-7}, we deduce that
\begin{align*}
\Big|\int_0^T\int_0^1\big((u_x^n)^2-(u_x^*)^2\big)\eta \,dx\,dt\Big|
&= \Big|\int_0^T\int_0^1(u_x^n+u_x^*)(u_x^n-u^*_x)\eta \,dx\,dt\Big|
\\
&\leq \Big|\int_0^T\|u_x^n+u_x^*\|_{L^{\infty}}\|u_x^n-u_x^*\|_H\|\eta\|_Hdt\Big|
\\
&\leq \|u_x^n+u_x^*\|_{L^2(L^{\infty})}\|u_x^n-u_x^*\|_{C(H)}\|\eta\|_{L^2(H)}
\\
&\to 0,\quad n\to\infty,\; \forall\eta\in L^2(0,T;H).
\end{align*}
and
\begin{align*}
&\Big|\int_0^T\int_0^1\left((u^n)^3-(u^*)^3\right)\eta\,dx\,dt\Big|
\\
&\leq \Big|\int_0^T\int_0^1((u^n)^2+u_nu^*+(u^*)^2)(u^n-u^*)\eta \,dx\,dt\Big|
\\
&\leq \Big|\int_0^T\|(u^n)^2+u_nu^*+(u^*)^2\|_{L^{\infty}}\|u^n-u^*\|_H
\|\eta\|_Hdt\Big|
\\
&\leq \|(u^n)^2+u_nu^*+(u^*)^2\|_{L^2(L^{\infty})}\|u^n-u^*\|_{C(H)}\|\eta\|_{L^2(H)}
\\
&\to 0,\quad n\to\infty,\; \forall\eta\in L^2(0,T;H).
\end{align*}
 Using \eqref{3-8} again,
\[
\Big|\int_0^T\int_0^1(Bw-Bw^{*})\eta \,dx\,dt\Big|\to
0,\quad \text{as }n\to\infty,\; \forall \eta\in L^2(0,T;H).
\]
In view of the above discussions,
\[
e_1(u^{*},w^{*})=0,\quad \forall n\in N.
\]
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 as $n\to\infty$. Thus,
$$
(u^n(0)-u^{*}(0), \eta)\to 0,\quad\text{as }n\to\infty,\;\forall\eta \in H,
$$
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, Theorem \ref{thm3.1} is proved.
\end{proof}

\section{Conclusions}
 The modified
Swift-Hohenberg equation is an important mathematical physical
model. Because of the complexity of nonlinear parts of the equation,
there has been no research on the optimal control and boundary
control of this equation. In this paper, we study the distributed
optimal control problem for problem \eqref{1-1}-\eqref{1-3} using a
series of mathematical estimates. Our research is motivated by the
study of the optimal control problem for the viscous
Degasperis-Procesi equation, viscous Camassa-Holm equation
\cite{Tian2,Tian3}, and the existence theory of optimal control of
distributed parameter systems. We also prove the existence of an
optimal solution to problem \eqref{1-1}-\eqref{1-3}. In order to
realize optimal solutions of optimal control problems in practice
one must be able to recompute the optimal solutions in the presence
of disturbances in real time unless one gives up optimality. We will
use mathematical theory and related numerical methods to solve that
problem numerically, which is our intention in the future.

\subsection*{Acknowledgements}
The author would like to thank the anonymous referee for the valuable
comments and suggestions on the original manuscript.

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\end{document}