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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 131, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/131\hfil Optimal control for Swift-Hohenberg equations]
{Optimal distributed control problem for the modified Swift-Hohenberg equations}

\author[B. Sun \hfil EJDE-2018/131\hfilneg]
{Bing Sun}

\address{Bing Sun \newline
School of Mathematics and Statistics,
Beijing Institute of Technology,
Beijing 100081, China. \newline
Beijing Key Laboratory on MCAACI,
Beijing Institute of Technology,
Beijing 100081, China}
\email{sunamss@gmail.com}

\dedicatory{Communicated by Goong Chen}

\thanks{Submitted May 21, 2018. Published June 27, 2018.}
\subjclass[2010]{35Q35, 49B22, 49K20}
\keywords{Maximum principle; optimal distributed control;
\hfill\break\indent necessary optimality condition; Swift-Hohenberg equation}

\begin{abstract}
 This article  concerns the optimal distributed control for the modified
 Swift-Hohenberg equation. Using the Dubovitskii and Milyutin functional
 analytical  approach, we prove the Pontryagin maximum principle of
 the controlled modified  Swift-Hohenberg equation.
 A necessary optimality condition is established for the problem in  fixed
 final time horizon case. Also we indicate how to utilize the obtained results.
\end{abstract}

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

\section{Introduction}

In 1977, Jack B. Swift and Pierre Hohenberg \cite{SwiftH77} derived a partial 
differential equation from the equations for thermal convection, 
 named as the Swift-Hohenberg equation (SH equation) thereafter,
when they considered the effects of thermal fluctuations
on the convective instability. It takes the form
\begin{equation}\label{0SHe}
u_t = - (1 + \Delta)^2 u - a u + N(u),
\end{equation}
where $u = u(x, t)$ or $u = u(x, y, t) \in \mathbb{R}$, $t \geq 0$, 
$x, y \in \mathbb{R}$, $a$ is a real constant, $\Delta$ is the Laplace operator,
 and $N(u)$ is some smooth nonlinearity.

This equation is noted for its pattern-forming behavior
and has been widely used as a model for the study of various issues in pattern 
formation.  These include the effects of noise on bifurcations, pattern selection, 
spatiotemporal chaos and the dynamics of defects. It also has been used to 
model patterns in simple fluids (e.g. Rayleigh-B\'enard  (RB) convection) 
and in a variety of complex fluids and biological materials, such as neural tissues. 
The motivation for the study which led to the SH equation was the analogy 
between bifurcations in the hydrodynamic behavior of fluids and the associated 
partial differential equations on the one hand, and continuous phase 
transitions in thermodynamic systems on the other hand \cite{SwiftH08}. 
As one of the universal equations used in the description of pattern formation 
in spatially extended dissipative systems, the SH equation can also be found 
in the study of convective hydrodynamics, plasma confinement in toroidal devices,
 viscous film flow and bifurcating solutions of the Navier-Stokes equation 
\cite{DuanG12,LaMRT75,ShlangS82}. 

Because of its wide applications in science, the SH equation has drawn
lots of interest among physicists and mathematicians since it is presented 
\cite{Polat09, SongZM10}.
Moreover, to model specific experimental effects, on the SH equation, there 
are lots of modifications introduced.
Specifically, the spontaneous formation of patterns in spatially extended
systems has attracted much attention over the past few decades. 
Many beautiful patterns such as spatially periodic
rolls, hexagonal cell structures, and spiral waves have been observed 
in experiments \cite{DoelmanSSS03}.
While at this very point, the SH equation receives its unusual popularity 
and attains the second life \cite{SwiftH08}.

As a phenomenological model for pattern-forming systems near the onset of 
instability, the paper \cite{DoelmanSSS03} presents a modified SH equation
\begin{equation}\label{0mSH}
u_t = - (1 + \Delta)^2 u - a u - b |\nabla u|^2 - u^3,
\end{equation}
in which $b$ is a real constant, $\nabla u$ is the gradient of $u$, and 
$u = u(x, y, t) \in \mathbb{R}$.
Note that we recover the usual SH equation if we set
$b = 0$. The additional term $b |\nabla u|^2$, reminiscent of the 
Kuramoto-Sivashinsky equation (\cite{Sun10}),
breaks the symmetry $u \to - u$. It is needed to obtain stable
hexagonal patterns. In fact, people can obtain stripes and hexagonal patterns 
from \eqref{0mSH} by adopting proper parameters in equation.
By this modified SH equation, \cite{DoelmanSSS03} investigates some of 
the interfaces between competing
spatially-periodic patterns. These interesting interfaces can be modelled as
 modulated fronts, i.e. as waves that are time-periodic in an appropriate 
co-moving coordinate frame. Both speed and shape of these interfaces therefore
 vary periodically in time. 
Furthermore, they prove the existence of modulated fronts that connect
stable with unstable patterns. These modulated fronts respectively describe
 (i)  stable hexagons that invade the unstable rest state at $u = 0$, 
 (ii) stable hexagons that invade unstable roll solutions, 
 (iii) stable hexagons that invade unstable hexagons, and lastly, 
 (iv) stable roll solutions that invade unstable hexagons. 

In this article, we are interested in the optimal control investigations of 
the modified SH equation \eqref{0mSH} in one spatial dimension. At this stage,
the present state of the research is entirely different with those 
investigations in other directions for
this said equation. To the best of our knowledge,
few results are known even on the control problem investigations of the
SH equation, let alone its optimal control. Optimal control
problems for the SH equation are largely unexplored
and need more attention. Here we try to give all related references.
Duan and Gao \cite{DuanG12} prove the existence and uniqueness of weak solution to
modified SH equation and the existence of optimal solution to 
 an optimal distributed control problem
of modified SH equation with a specific linear quadratic cost functional.

Taking $N(u) = \gamma u - \beta u^3$ in equation \eqref{0SHe}, Stanton
and Golovin \cite{StantonG07}
investigate the following SH equation
$$
u_t = - (1 + \Delta)^2 u - a u + \gamma u^2 - \beta u^3,
$$
which is used to model the nonlinear dynamics of
RB and Marangoni convection, and many other systems, and has been
 extensively studied.
Feedback control of systems described by a supercritical
SH equation in one dimension is first considered in
\cite{HandelG05}. It was shown that applying localized feedback at a few
spatial locations can stabilize both uniform and pattern
states. In \cite{StantonG07}, they investigate the possibility of
applying a global feedback control to a pattern-forming system
whose dynamics is described by a SH
equation in 2D by means of stability analysis and numerical simulations.


The Pontryagin maximum principle unifies calculus of variations and control 
theory of ordinary differential equations (\cite{Fattorini99}) and establishes 
the theoretical basis of the modern optimal
control theory along with the Bellman dynamic programming principle. In this paper,
we commit ourselves to infinite dimensional generalizations of the
maximum principle and aim at the optimal control theory of partial differential
equations, a subject of much theoretical and practical interest \cite{LiY95}.
In contrast to the finite dimensional setting, the maximum principle for the 
infinite dimensional system does not generally hold as a necessary condition 
for optimal control.
Paying particular attention to the time optimal and norm optimal problems,
Fattorini \cite{Fattorini05} has found some optimal controls
which either do not satisfy the Pontryagin maximum principle or satisfy it in
 a certain weak form, which justified this argument.

This paper is concerned with the optimal distributed control problem of 
the modified SH equation.
By the Dubovitskii and Milyutin functional analytical approach, the cone of
directions of decrease, the cone of feasible directions and the
cone of tangent directions as well as their dual cones are, respectively, derived.
Then the Pontryagin maximum principle of the
optimal distributed control problem is proven. The necessary optimality condition
is established for the problem in fixed final time horizon case.
In the end, a remark on how to use obtained results is also made as an illustration.


It is true that the feedback control of dynamical systems has many merits 
comparing to the open-loop control \cite{SunG15}.
However, an undeniable fact is that the latter, the open-loop control, 
has its own advantages in the investigation of the infinite dimensional systems, 
such as the efficiency and  accuracy of the open-loop
control algorithms as well as the robustness aspect of investigational systems 
\cite{SlossBSA98}. Just as
Ho and Pepyne \cite{HoP02} said in 
``The No Free Lunch Theorem of Optimization (NFLT)", a general-purpose
universal optimization strategy is impossible. Therefore the open-loop 
control investigation to the modified SH equation is both necessary and interesting.

The rest of this article is organized as follows. 
In section 2, we will present the main
results of this paper. An optimal distributed control problem is formulated 
and the weak solution of the
controlled state system is recalled.  The Pontryagin maximum principle
of the optimal control problem is established in fixed final time horizon case.
The proof of the main results is given in section 3.
In Section 4, we address the numerical solution and make a remark on how 
to use obtained necessary optimality condition. 
Section 5 concludes the paper with remarks.

\section{Main results}

Let $T > 0$ and $\Omega$ be an open connected bounded domain in $\mathbb{R}$. 
We investigate the  partial differential equation
\begin{equation}\label{mSH}
\begin{aligned}
&u_t(x, t) + k u_{xxxx}(x, t) + 2 u_{xx}(x, t) + a u(x, t)
+ b |u_x(x, t)|^2  + u^3(x, t) \\
&= \tilde{f}(x, t), \quad x \in \Omega, t \in (0, T),
\end{aligned}
\end{equation}
 which is the 1-D version of modified SH equation \eqref{0mSH}.
 As mentioned above, instead of considering the full RB problem or general 
reaction-diffusion systems, \cite{DoelmanSSS03} studies the modified SH 
equation \eqref{0mSH} as a phenomenological model for pattern-forming systems 
near the onset of instability (see a discussion in  \cite{LinGDE00} 
about the validity of the SH model for the RB problem). 
In \eqref{mSH}, $k$ is an arbitrary constant.
We supplement the equation with the initial data
\begin{equation} \label{initD}
u(x, 0) = u_0(x), \quad x \in \Omega,
\end{equation}
and the boundary condition
\begin{equation}\label{boundaryC}
u(x,t) = u_{xx}(x, t) = 0, \quad x \in \partial \Omega.
\end{equation}

In this article, we take $\Omega = (0, 1)$, 
$V = H^2_0(0,1)$, 
$U = H^1_0(0, 1)$ and $H=L^2(0,1)$. Take the Hilbert space
$$
W(0,T; V) = \{\varphi\in L^2(0,T;V): \varphi_t\in L^2(0,T;V^*)\}
$$
equipped with the norm 
$$
\|\varphi\|_{W(0,T; V)} =
\Big(\|\varphi\|^2_{L^2(0,T;V)} + \|\varphi_t\|^2_{L^2(0,T;V^*)}\Big)^{1/2}
$$
where $V^* = H^{-2}(0, 1)$ is the dual space of $V$. It is supposed that $V$ 
is dense in $U$ and $U$ is dense in $H$ such that, by identifying $V^*$, $U^*$ 
and $H^*$, we have
$$
V \hookrightarrow U \hookrightarrow H 
= H^* \hookrightarrow U^* \hookrightarrow V^*,
$$
each embedding being dense, in which $H^* = L^2(0, 1)$ being the dual space of $H$,
$U^* = H^{-1}(0, 1)$ the dual space of $U$ \cite{DautrayL92}.


Furthermore, we introduce a  definition of weak solution to the modified SH 
equation \eqref{mSH} with \eqref{initD}, \eqref{boundaryC}. 
A function $u(x, t)\in W(0,T; V)$ is a weak solution to
\eqref{mSH}, \eqref{initD}, \eqref{boundaryC}, if
\begin{equation}\label{weaks0}
\begin{aligned}
& \langle \frac{\partial}{\partial t} u(\cdot, t), \varpi(\cdot)\rangle_{H}
+ k \langle u_{xx}(\cdot, t), \varpi_{xx}(\cdot)\rangle_{H}
- 2 \langle u_x(\cdot, t), \varpi_x(\cdot) \rangle_{H} \\
&+ a \langle u(\cdot, t), \varpi(\cdot) \rangle_{H} 
+ b \langle |u_x(\cdot, t)|^2, \varpi(\cdot) \rangle_{H}
+ \langle u^3(\cdot, t), \varpi(\cdot) \rangle_{H} \\
&= \langle \tilde{f}(\cdot, t), \varpi(\cdot)\rangle_{H}
\end{aligned}
\end{equation}
for all $\varpi(\cdot)\in V$, $t\in [0,T]$ a.e.\ and
$u(\cdot, 0) = u_0(\cdot) \in V$.
Here and thereafter,
$\left\langle\cdot, \cdot\right\rangle_{\mathcal{X}}$ is the inner product
of Hilbert space $\mathcal{X}$.

Under the definition of such weak solution, suppose that $k$ is sufficiently large, 
$u_0 \in V$ and $\tilde{f}\in L^2(0, T; H)$,
then the equations \eqref{mSH}, \eqref{initD}, \eqref{boundaryC} admit a unique 
weak solution $u(x, t) \in W(0, T; V)$,
which is the result has been proven in \cite{DuanG12} by Galerkin method.
In this paper, unless otherwise stated, in what follows when we speak of
a solution of \eqref{mSH}, \eqref{initD}, \eqref{boundaryC}, we shall always 
mean the weak solution in the sense of \eqref{weaks0}.


Now we consider the optimal distributed control of investigated
system in fixed final time horizon case. For $T> 0$, take
 $\tilde{f}(x,t) = f(x,t) + \alpha(t)$, in which
$f(x,t)\in L^2(0,T; H)$. And $\alpha(t)\in L^2(0,T)$ plays the role of control.
Let $U_{ad}$  be a non-empty closed convex set of $L^2(0, T)$ and then take it as
the admissible control set.
Consider an optimal control problem for the system
\eqref{mSH}, \eqref{initD}, \eqref{boundaryC} with the general cost functional
\begin{equation}\label{op}
\min_{\alpha(\cdot)\in U_{ad}} J(u, \alpha) = \min_{\alpha(\cdot)\in U_{ad}}
\int^T_0\int^{1}_0 L(u(x,t), \alpha(t), x, t)\, dx\,dt.
\end{equation}
Here, the cost function $J$ is quite general in the sense that it
contains most practically concerned ones like quadratic
cost functional of the following form
\begin{equation}\label{lqc}
J(u, \alpha) = \int^T_0\int^{1}_0  \rho_1 \left|u(x,t) -
u^\dag(x,t)\right|^2 \,dx\,dt  +  \int^T_0 \rho_2 \alpha^2(t) \,dt,
\end{equation}
where $\rho_i > 0$, $i = 1, 2$ are constants, and  given $u^\dag$
is the desired optimal state. The last quadratic term reflects the cost of
control.  The control target is to drive the
state variable $u$ to match the given desired state $u^\dag$ by
 adjusting the control
function $\alpha$ with minimal energy and work, which is exactly the main object of
 interest in \cite{DuanG12}.

The objective of this paper is to study the optimal distributed control 
problem \eqref{op} in fixed final time horizon case, for the modified
SH equation \eqref{mSH}, \eqref{initD}, \eqref{boundaryC}.
Take $u(\cdot, t) \in W(0,T; V)$. The control space is $L^2(0,T)$
and the control function satisfies a convex constraint $\alpha(\cdot) \in U_{ad}$.
Here, we assume that the set $U_{ad}$ of admissible controls has the non-empty 
interior with respect to $L^2(0,T)$ topology, i.e., 
$\operatorname{int}_{L^2(0,T)}U_{ad} \neq \emptyset$.

The following two assumptions for the cost functional in \eqref{op} are assumed:
\begin{itemize}
\item[(A1)]   $L$ is a functional defined on $V \times U_{ad} \times
[0, 1] \times [0, T]$ and
$$
\frac{\partial L(u(x,t), \alpha(t), x, t)}{\partial u},\quad
\frac{\partial L(u(x,t), \alpha(t), x, t)}{\partial \alpha}
$$
exist for every $(u, \alpha)\in V \times U_{ad}$ and $L$ is
continuous in its variables.

\item[(A2]
$$
\int^{1}_0\big|\frac{\partial L(u(x,t), \alpha(t), x,
t)}{\partial u}\big|\,dx, \quad 
\int^{1}_0 \big|\frac{\partial
L(u(x,t), \alpha(t), x, t)}{\partial \alpha}\big|\,dx
$$
are bounded for $t\in [0,T]$.
\end{itemize}

Define $X_T = W(0,T; V) \times L^2(0,T)$.  Let $(u^*, \alpha^*)$ be the
solution to optimal control problem \eqref{op} subject to the
equation \eqref{mSH}, \eqref{initD}, \eqref{boundaryC}. Set
\begin{gather*}
\Omega_1=\{(u, \alpha)\in X_T : \alpha(t) \in U_{ad}, \; t\in
[0,T] \text{ a.e.}\},\\
\begin{aligned}
\Omega_2 = \Big\{&(u, \alpha) \in X_T:
u_t(x, t) + k u_{xxxx}(x, t) + 2 u_{xx}(x, t) + a u(x, t)\\
&+ b |u_x(x, t)|^2  
+ u^3(x, t) = f(x,t) + \alpha(t), \\
& u(0, t) = u(1, t) = u_{xx}(0, t) = u_{xx}(1, t) = 0, \\
& u(x, 0) = u_0(x), \quad u(x,T) = u^*(x,T)\Big\}.
\end{aligned}
\end{gather*}
Then  problem \eqref{op} is equivalent to questing for
$(u^*, \alpha^*) \in \Omega=\Omega_1 \cap \Omega_2$ such that
\begin{equation}\label{3.4}
J(u^*, \alpha^*) = \min_{(u, \alpha)\in {\Omega}} J(u, \alpha).
\end{equation}

Thus far, we have seen that problem \eqref{3.4} is an extremum problem on
the constraint $\Omega_1$ and the equality constraint
$\Omega_2$. In this situation, the Dubovitskii and Milyutin
functional analytical approach has been  turned out  to be
very powerful to solve such kind of extremum problems 
(see e.g. \cite{ChanG89, Girsanov72, Sun10}).
The general Dubovitskii and Milyutin
theorem for the extremum problem \eqref{3.4} can be stated as  follows.


\begin{theorem}[Dubovitskii-Milyutin] \label{thm1}
 Suppose the functional $J(u, \alpha)$ assumes a minimum at the point 
$(u^*, \alpha^*)$ in $\Omega$. Assume that $J(u, \alpha)$ is regularly decreasing at
$(u^*, \alpha^*)$ with the cone of directions of decrease $K_0$ and the
constraint $\Omega_1$ is regular at $(u^*, \alpha^*)$ with the cone of
feasible directions $K_1;$ and that the equality constraint $\Omega_2$ is
also regular at $(u^*, \alpha^*)$ with the cone of tangent directions
$K_2$. Then there exist continuous linear functionals
$f_0,f_1,f_2$, not all identically zero, such that $f_i\in
K^*_i$, the dual cone of $K_i$, $i=0,1,2$, which satisfy the condition
\begin{equation}\label{dmth}
f_0+f_1+f_2=0.
\end{equation}
\end{theorem}


In this article, using Theorem \ref{thm1}, we establish the necessary optimality
 condition of optimal control problem \eqref{op}
for the modified SH equation \eqref{mSH}, \eqref{initD}, \eqref{boundaryC}.
The main results of this paper are formulated as Theorem \ref{thm2} below.

\begin{theorem} \label{thm2}
Suppose $(u^*, \alpha^*)$ is a solution
to the optimal control problem \eqref{op}. Then there exist
$\kappa_0\geq 0$ and $v(x,t)$, not identically zero, such that the
following maximum principle holds:
\begin{equation}\label{3.32-th3}
\begin{gathered}
 \Big\{\int^{1}_0 \Big[\kappa_0 \frac{\partial L(u^*, \alpha^*,
x, t)}{\partial \alpha} - v(x,t) \Big]dx \Big\}
[\alpha(t) - \alpha^*(t)] \geq 0, \\
\forall \alpha(t)\in U_{ad},\; t\in [0,T] \text{ a.e.},
\end{gathered}
\end{equation}
where the function $v(x,t)$ satisfies the adjoint equation
\begin{equation}\label{3.24}
\begin{gathered}
 \begin{aligned}
& v_t(x,t) - k v_{xxxx}(x,t) - 2 v_{xx}(x, t) - a v(x, t)
+ 2 b \;\! [u^*_{xx}(x,t) v(x,t) \\
&+ u^*_{x}(x,t) v_x(x,t)] - 3 (u^*(x,t))^2 v(x,t) \\
&= \kappa_0 \frac{\partial L(u^*, \alpha^*, x, t)}{\partial u},
\end{aligned} \\
v(0,t) = v(1, t) = v_{xx}(0,t) = v_{xx}(1, t) = 0, \quad
v(x,T) = \psi(x).
\end{gathered}
\end{equation}
\end{theorem}

\section{Proof of Theorem \ref{thm2}}

To prove Theorem \ref{thm2}, in the direction of Theorem \ref{thm1}, we need to determine
the cone of directions of decrease $K_0$, the cone of
feasible directions $K_1$, the cone of tangent directions
$K_2$ and their respective dual cones $K_i$, $i=0,1,2$. 
Moreover, in these dual cones,
to derive the continuous linear functionals
$f_i$, $i=0,1,2$. Then, step by step, by the equation \eqref{dmth},
to derive the final result,
which is exactly the Pontryagin maximum principle.

Firstly,  find the cone of directions of decrease $K_0$.
By assumption, $J(u, \alpha)$ is differentiable at any
point $(u^0, \alpha^0)$ in any direction $(u, \alpha)$ and its directional
derivative is
\begin{align*}
& J'(u^0, \alpha^0; u, \alpha) \\
&=\lim_{\varepsilon\to 0+}\frac{1}{\varepsilon}
[J(u^0 + \varepsilon u, \alpha^0 +
\varepsilon \alpha) - J(u^0, \alpha^0)]\\
&= \lim_{\varepsilon\to 0+}\frac{1}{\varepsilon}
\Big\{\int^T_0\int^{1}_{0}[L (u^0 +
\varepsilon u, \alpha^0 + \varepsilon \alpha, x, t) - L(u^0, \alpha^0, x,
t)]\, dx\,dt\Big\}\\
&=\int^T_0\int^{1}_{0}\big[\frac{\partial L(u^0,
\alpha^0, x, t)}{\partial u}u + \frac{\partial L(u^0, \alpha^0, x,
t)}{\partial \alpha}\alpha \big]\,dx\, dt.
\end{align*}
The cone of directions of decrease of the functional $J(u, \alpha)$
at point $(u^*, \alpha^*)$  is thereupon determined by
\begin{align*}
K_0 & = \big\{(u, \alpha)\in X_T : J'(u^*, \alpha^*; u,
\alpha)<0 \big\}\\ 
&=\Big\{(u, \alpha)\in X_T : \int^T_0\int^{1}_{0}\Big[
\frac{\partial L(u^*, \alpha^*, x, t)}{\partial u}u + \frac{\partial
L(u^*, \alpha^*, x, t)}{\partial \alpha}\alpha \Big]\,dx\,dt < 0 \Big\}.
\end{align*}
If $K_0\neq \emptyset$, then for any $f_0\in K^*_0$, there exists
a $\kappa_0\geq 0 $ such that
$$
 f_0(u, \alpha)= -\kappa_0\int^T_0\int^{1}_{0}\Big[\frac{\partial
L(u^*, \alpha^*, x, t)}{\partial u}u + \frac{\partial L(u^*, \alpha^*, x,
t)}{\partial \alpha}\alpha  \Big]\,dx\,dt.
$$

Secondly, for the cone of feasible directions $K_1$, since 
$\Omega_1 = W(0,T; V) \times U_{ad}$, in which
$\operatorname{int}_{L^2(0,T)} U_{ad} \not= \emptyset$  by assumption, 
so the interior of $\Omega_1$ is not empty, i.e.\ ,
 $\mathring{\Omega}_1 \neq\emptyset$.
And at point $(u^*, \alpha^*)$, the cone of feasible
directions $K_1$ of $\Omega_1$ is determined by
\begin{align*}
K_1 & =\big\{\kappa\big(\mathring{\Omega}_1 -\, (u^*, \alpha^*)\big)
:\kappa>0 \big\}\\
&=\big\{h:h=\kappa(u - u^*, \alpha - \alpha^*),\;(u, \alpha)
\in\;\mathring{\Omega}_1, \; \kappa > 0\big\}.
\end{align*}
As a result, for an arbitrary $f_1\in{K^*_1}$, if there is an
$\bar{a}(t)\in L^2(0,T)$, such that the linear functional defined
by
\begin{equation}\label{f1}
f_1(u, \alpha) = \int_0^T \bar{a}(t) \alpha(t)\, dt
\end{equation}
is a support to $\tilde{\Omega}_1$ at point $\alpha^*$, then 
\begin{equation}\label{3.9}
\bar{a}(t)[\alpha(t) - \alpha^*(t)]\geq 0,\quad \forall\; \alpha(t)
\in U_{ad}, \; t\in[0,T] \text{ a.e.}
\end{equation}

We proceed in the next step to derive the cone of tangent directions $K_2$.
Define the operator $ G: X_T \to L^2(0,T; H) \times (L^2
(0, T))^4 \times (V)^2$ by
\begin{align*}
 G(u, \alpha) 
&=  \Big(\vartheta(x,t),
 u(0,t), u(1,t), u_{xx}(0,t), u_{xx}(1,t), u(x,0) - \phi(x), \\
&\quad  u(x,T) - u^*(x,T) \Big),
\end{align*}
in which $\vartheta(x,t) = u_t(x, t) + k u_{xxxx}(x, t) + 2 u_{xx}(x, t) + a u(x, t)
+ b |u_x(x, t)|^2  + u^3(x, t) - f(x,t)$ $ - \alpha(t)$. Then
$$
\Omega_2=\{(u, \alpha) \in X_T : G(u(x,t), \alpha(t))=0\}.
$$
The Fr\'echet-derivative of the operator $G(u, \alpha)$ is
$$
 G'(u, \alpha)(\hat{u}, \hat{\alpha}) =
\Big(\hat{\vartheta}(x,t), \hat{u}(0,t), \hat{u}(1,t),
\hat{u}_{xx}(0,t), \hat{u}_{xx}(1,t), \hat{u}(x,0),
\hat{u}(x,T) \Big)
$$
in which we define
\begin{align*}
\hat{\vartheta}(x,t) 
&= \hat{u}_t(x,t) + k \hat{u}_{xxxx}(x,t) 
+ 2 \hat{u}_{xx}(x,t)
+ a \hat{u}(x,t) + 2 b u_x(x,t)\hat{u}_x(x,t)\\
&\quad + 3 u^2(x, t)\hat{u}(x,t) - \hat{\alpha}(t).
\end{align*}
Since $(u^*, \alpha^*)$ is the solution to the problem \eqref{op},
it follows that $G(u^*,\alpha^*)=0$. Choosing arbitrary
$$
(g_1,g_2,g_3,g_4,g_5,g_6,g_7) \in L^2(0,T; H) \times (L^2(0, T))^4 \times (V)^2
$$ and
solving the equation
$$
G'(u^*, \alpha^*)(\hat{u}, \hat{\alpha}) =(g_1(x,t), g_2(t),
g_3(t), g_4(t), g_5(t),  g_6(x), g_7(x)),
$$ 
we obtain
\begin{equation}\label{3.19}
\begin{gathered}
\begin{aligned}
&\hat{u}_t(x,t) + k \hat{u}_{xxxx}(x,t) + 2 \hat{u}_{xx}(x,t)
+ a \hat{u}(x,t) \\
&+ 2 b u^*_x(x,t)\hat{u}_x(x,t) 
+ 3 (u^*(x, t))^2\hat{u}(x,t) - \hat{\alpha}(t) = g_1(x,t),
\end{aligned} \\
\hat{u}(0,t) = g_2(t), \;\hat{u}(1,t) = g_3(t),\quad
\hat{u}_{xx}(0,t) = g_4(t),\\
 \hat{u}_{xx}(1,t) = g_5(t), \quad
 \hat{u}(x,0) = g_6(x), \;\hat{u}(x,T) = g_7(x).
\end{gathered}
\end{equation}
Next, we assume that the linearized system
\begin{equation}\label{3.20}
\begin{gathered}
\begin{aligned}
&u_t(x,t) + k u_{xxxx}(x,t) + 2 u_{xx}(x,t)
+ a u(x,t) \\
&+ 2 b u^*_x(x,t)u_x(x,t) 
+ 3 (u^*(x, t))^2 u(x,t) = \alpha(t),
\end{aligned} \\
u(0,t) = u(1,t) = u_{xx}(0,t) = u_{xx}(1,t) = 0, \quad
u(x,0) = 0,
\end{gathered}
\end{equation}
is controllable. Then choose $\alpha(t) = \hat{\alpha}(t)\in L^2(0,T)$ such
that $u(x,T) = g_7(x) - \eta(x,T)$ and let $u$ be the solution to
the linearized system \eqref{3.20}.  Choose $\hat{u}(x,t) = u(x,t) +
\eta(x,t)$, where $\eta(x,t)$ satisfies the following equations
\begin{gather*}
\begin{aligned}
&\eta_t(x,t) + k \eta_{xxxx}(x,t) + 2 \eta_{xx}(x, t) + a \eta(x, t) \\
&+ 2 b u^*_x(x,t) \eta_x(x,t)
+ 3 (u^*(x,t))^2 \eta(x,t) = g_1(x,t),
\end{aligned} \\
\eta(0,t) = g_2(t), \quad \eta(1,t) = g_3(t),\quad
\eta_{xx}(0,t) = g_4(t),\\
\eta_{xx}(1,t) = g_5(t), \quad  \eta(x,0) = g_6(x).
\end{gather*}
In this way, it suffices for $(\hat{u}, \hat{\alpha})$ satisfying 
\eqref{3.19}. Therefore $G' (u^*, \alpha^*)$ maps the space $X_T$ onto
$L^2(0,T; H) \times (L^2(0, T))^4 \times (V)^2$. Moreover,
the cone of the tangent directions $K_2$ to the constraint
$\Omega_2$ at point $(u^*, \alpha^*)$ consists of the kernel of
$G' (u^*, \alpha^*)$, i.e., $(u,\alpha)$ satisfies the following
equations in $X_T$, 
\begin{equation}\label{3.21}
\begin{gathered}
\begin{aligned}
&u_t(x,t) + k u_{xxxx}(x,t) + 2 u_{xx}(x,t) + a u(x,t) \\
&+ 2 b u^*_x(x,t)u_x(x,t) 
 + 3 (u^*(x, t))^2 u(x,t) = \alpha(t),
\end{aligned} \\
u(0,t) = u(1,t) = u_{xx}(0,t) = u_{xx}(1,t) = 0, \quad
u(x,0) = 0,
\end{gathered}
\end{equation}
and
\begin{equation}\label{3.22}
u(x,T)=0.
\end{equation}

Let
\begin{gather*}
K_{21}=\{(u,\alpha)\in X_T : (u(x,t), \alpha(t)) \text{ satisfies }
\eqref{3.21}\}, \\
K_{22}=\{(u,\alpha)\in X_T : (u(x,t), \alpha(t)) \text{ satisfies }
\eqref{3.22}\}.
\end{gather*}
Then the cone of tangent directions $K_2=K_{21}\cap K_{22}$.
Consequently,
$$
K^*_2=K^*_{21}+K^*_{22}.
$$
For any $f_2\in K^*_2$, decompose $f_2 = f_{21}+f_{22},\;
f_{2i}\in K^*_{2i}$, the dual cone of $K_{2i},\; i=1,2$.  Then
$f_{21}(u,\alpha) = 0$ and for all $u(x,t) \in W(0,T; V)$ satisfying
$u(x,T) = 0$, there exists a $\psi(x) \in V^*$ such that
$$
f_{22}(u, \alpha) = \int^{1}_0 u(x,T)\psi(x)\, dx.
$$
It then follows from Theorem \ref{thm1} that there exist continuous linear
functionals, not all identically zero, such that
$$
f_0+f_1+f_{21}+f_{22}=0.
$$
Therefore, when selecting $(u,\alpha)$ satisfies \eqref{3.21},
$f_{21}(u,\alpha) = 0$.  Moreover,
\begin{align*}
 f_1(u, \alpha) & = -f_0(u, \alpha) - f_{22}(u, \alpha)\\
& =  \kappa_0\int^T_0\int^{1}_{0}\Big[\frac{\partial L(u^*,
\alpha^*,x,t)}{\partial u}u(x,t) + \frac{\partial L(u^*,
\alpha^*,x,t)}{\partial \alpha}\alpha(t) \Big]\,dx\,dt \\
& \quad - \int^{1}_0 u(x,T)\psi(x)\,dx.
\end{align*}

Now it is only one step away from obtaining the necessary optimality condition
and establishing the Pontryagin maximum principle for problem \eqref{op}. 
For this purpose, we need to formulate the adjoint system of \eqref{3.20}.
 Here, define the adjoint system as \eqref{3.24}.
As with \eqref{mSH}, \eqref{initD}, \eqref{boundaryC}, 
the existence and uniqueness of solution to
the adjoint system can be obtained similarly.



\begin{theorem} \label{thm3} 
The solution of system \eqref{3.20} and
that of its adjoint system \eqref{3.24} have the
following relationship
\begin{align*}
&\int^{1}_0 u(x,T)\psi(x) \,dx  - \kappa_0\int^T_0 \int^{1}_0 \frac{\partial L(u^*,
\alpha^*, x, t)}{\partial u} u(x,t)\, dx\,dt\\
&= \int^T_0 \int^{1}_0 \alpha(t) v(x,t)\,dx\,dt.
\end{align*}
\end{theorem}

\begin{proof} 
Multiply  equation \eqref{3.24} by $v(x,t)$ and integrate the product by parts  
over $[0,T]\times [0,1]$ with respect to $t$ and $x$ respectively. 
The proof then follows. 
\end{proof}

Next we give the proof of the main results in this paper.

\begin{proof}[Proof of Theorem  \ref{thm2}]
 By  Theorem \ref{thm3}, we can rewrite $f_1(u, \alpha)$ as
$$
f_1(u, \alpha) =  \int^T_0\Big\{\int^{1}_0 \Big[\kappa_0
\frac{\partial L(u^*, \alpha^*, x, t)}{\partial \alpha} - v(x,t) \Big]dx \Big\}
\alpha(t)\, dt.
$$
In view of \eqref{f1},
$$
 \bar{a}(t) = \int^{1}_0 \Big[\kappa_0 \frac{\partial L(u^*,
\alpha^*, x, t)}{\partial \alpha} - v(x,t) \Big]\,dx\,.
$$
Then \eqref{3.9} reads
\begin{equation}\label{3.28}
\begin{gathered}
 \Big\{\int^{1}_0 \Big[\kappa_0 \frac{\partial L(u^*,
\alpha^*, x, t)}{\partial \alpha} - v(x,t) \Big]dx
\Big\}[\alpha(t) - \alpha^*(t)] \geq 0,  \\
 \alpha(t)\in U_{ad},\quad  t\in [0,T] \text{ a.e.},
\end{gathered}
\end{equation}
where $\kappa_0$ and $v(x, t)$ are not identical to zero
simultaneously. Since otherwise, there are definitely $f_0 = 0,
f_1 = 0, f_{22} = 0$ and $f_{21} = 0$, which contradict with the
fact in Theorem \ref{thm1} that these continuous linear functionals are not
all identically zero.

On the other hand, if $K_0$ is a null set, then 
\[
 \int^T_0\int^{1}_0 \Big[\frac{\partial L(u^*, \alpha^*, x,
t)} {\partial u} u(x,t) + \frac {\partial L(u^*, \alpha^*, x,
t)}{\partial \alpha} \alpha(t)  \Big]\,dx\,dt = 0,
\]
for all $ (u, \alpha)\in X_T$.
In particular, if we choose $\kappa_0=1$ and $\psi(x)=0$, then
 from Theorem \ref{thm3} it follows that
$$
 \int^T_0 \int^{1}_0 \frac{\partial L(u^*, \alpha^*, x,
t)}{\partial u} u(x,t)\, dx\,dt = - \int^T_0 \int^{1}_0 v(x,t)\alpha(t) \,dx\,dt
$$
so
\[
 \int^T_0 \Big\{ \int^{1}_0 \Big[\frac{\partial L(u^*,
\alpha^*, x, t)}{\partial \alpha} - v(x,t)
\Big]\,dx\Big\}\alpha(t)\, dt = 0,  \quad
\forall \alpha(t)\in L^2(0,T),
\]
from which we obtain
\[
\int^{1}_0 \Big[\frac{\partial L(u^*, \alpha^*, x,
t)}{\partial \alpha} - v(x,t) \Big]\,dx = 0, \quad \forall
 t\in [0,T] \text{ a.e.}
\]
Therefore \eqref{3.28} still holds.

In addition, if there is a nonzero solution $\hat{v}(x,t)$
to the adjoint system
\begin{equation}\label{3.29}     
\begin{gathered}
\begin{aligned}
& \hat{v}_t(x,t) - k \hat{v}_{xxxx}(x,t) - 2 \hat{v}_{xx}(x, t) - a \hat{v}(x, t)
 + 2 b  [u^*_{xx}(x,t) \hat{v}(x,t) \\
& + u^*_{x}(x,t) \hat{v}_x(x,t)]
- 3 (u^*(x,t))^2 \hat{v}(x,t) \\
&= \kappa_0 \frac{\partial L(u^*, \alpha^*, x, t)}{\partial u},
\end{aligned} \\
\hat{v}(0,t) = \hat{v}(1, t) = \hat{v}_{xx}(0,t) = \hat{v}_{xx}(1, t) = 0, \quad
\hat{v}(x,T) = \psi(x)
\end{gathered}
\end{equation}
such that the following equality holds 
$$
\int^{1}_0 \hat{v}(x,t)\,dx = 0, \quad \forall  t\in [0,T] \text{ a.e.},
$$
then when we  choose $\kappa_0=0$ and $\psi(x) = \hat{v}(x,T)$,
\eqref{3.28} is still valid. Since otherwise, if for any nonzero
solution $\hat{v}$ of \eqref{3.29}, it has
$$
\int^{1}_0 \hat{v}(x,t)\,dx \not\equiv 0,
$$
in this case we say the situation is non-degenerate. Then the
linearized system \eqref{3.20} is controllable. In fact, if
\eqref{3.20} is not controllable, then there exist a $\psi(x)\in V^*$
such that
$$
\int^{1}_0 u(x,T)\psi(x)\,dx = 0,\quad \psi(x)\not\equiv 0.
$$
Choose $\kappa_0=0$, $\hat{v}$ to be the solution of \eqref{3.29}. 
Then it follows from Theorem \ref{thm3} that
$$
\int^T_0 \Big[\int^{1}_0 \hat{v}(x,t)\,dx \Big]
\alpha(t)\,dt = 0, \quad \forall  \alpha(t)\in L^2(0,T).
$$
Thus
$$
\int^1_0 \hat{v}(x,t) \,dx = 0, \quad \forall  t\in [0,T]\text{ a.e.}
$$
This is a contradiction. Under the case of
\eqref{3.29}, the system \eqref{3.20} is consequently controllable.

Combining the results above,  we have obtained the Pontryagin maximum
principle \eqref{3.32-th3} for the problem \eqref{op} subject to the system
\eqref{mSH}, \eqref{initD}, \eqref{boundaryC}. This completes the proof of
main results. 
\end{proof}

\section{An iterative algorithm}

In this section, we show how to use the results obtained
above for the numerical solutions to the investigational optimal control problem. 
That is to say, we will give, by the Pontryagin
maximum principle along with an iterative algorithm, the profile for numerically
solving the optimal distributed control problem of the modified
SH equation in fixed final time horizon
case, i.e., the problem \eqref{op}.

Essentially speaking, by the necessary optimality condition of
optimal control, such as the Pontryagin maximum principle,
a two-point boundary-value problem solution is an effective numerical method 
for solving optimal control problems.  Through necessary conditions for 
numerically solving optimal control problems,
there are two approaches available for now. It is
commonly believed that the indirect method, which is mainly the
multiple shooting method, is the most powerful numerical method.
By the Pontryagin maximum principle, one can construct a
two-point boundary-value problem.
The optimal control of the lumped parameter systems can be obtained
by solving this two-point boundary-value problem.
Of course, except for the complexity
when the original problem involves inequality constraints of both
state variables and controls, the difficulty for shooting method
additionally includes the ``guess" for the initial data to start
the iterative numerical process. It demands that the user
understands the essential of the problem well in physics. 
Unfortunately, in all likelihood it is no easy job. The gradient method 
is developed to overcome this difficulty;
and then the ``min-H'' approach, which corrected from
the gradient method (\cite{GibsonL74, XingZX03}), 
comes with the higher convergence rate. In the following, we  show how to
utilize the min-H iterative method to solve the extremum problem.

To this end, rewrite the Pontryagin maximum principle
\eqref{3.32-th3} as follows:
\begin{equation}\label{example-1}
\alpha^*(t) H_\alpha(u^*,\alpha^*) 
= \max_{\alpha(\cdot) \in U_{ad}} \alpha(t) H_\alpha(u^*, \alpha^*),
\end{equation}
where
$$
H(u, \alpha) = \int^1_0 [\alpha(t) v(x,t) - \kappa_0 L(u, \alpha, x, t)]\,dt.
$$
Therefore, the so-called ``min-H'' iterative algorithm is formulated
below.

First, give $\alpha^0(t)$ and solve the state equation \eqref{mSH}, \eqref{initD}, 
\eqref{boundaryC} to get $u^0(x,t)$. 
\begin{itemize}
\item[(I)]  By $\alpha^0(t),\; u^0(x,t)$, solve the adjoint equation 
\eqref{3.24} to get $v^0(x,t)$.

\item[(II)] In view of $u^0(x,t),\; v^0(x,t)$ and the Pontryagin maximum
 principle \eqref{example-1}, to
determine $\alpha^1(t)$.

\item[(III)]  Give $\alpha^1(t)$ and solve the state equation \eqref{mSH}, 
\eqref{initD}, \eqref{boundaryC} to get $u^1(x,t)$.

\item[(IV)] Calculate $J(u^1, \alpha^1)$. If it does not reach the minimum,
 replace $(u^0, \alpha^0)$ with $(u^1, \alpha^1)$
and redo the steps above until we get the proper $J(u^1, \alpha^1)$.

\end{itemize}

Subsequently, we can proceed the numerical computation using the
algorithm above after setting some parameters such as $k$, $a$, $b$, $u_0(x)$, 
$T$, $f(x,t)$, $L(u, \alpha, x, t)$ and so on. Furthermore, for the
convenience, the quadratic cost functional \eqref{lqc} is a good
choice.  Noting that it is an optimal control problem of
distributed parameter system governed by nonlinear partial
differential equations, to get the numerical solutions for the
optimal control-trajectory pair is not an easy job. Here, although
we do not give the detailed numerical simulation, the algorithm does
give the concrete steps so that people can follow and finish this
nontrivial work.

\section{Conclusions}

For the infinite dimensional system, the maximum principle
does not generally hold as a necessary condition for optimal control.
Thus, in the optimal control theory of partial
differential equation, an important and interesting problem
is the infinite dimensional generalization of the maximum principle.
The SH equation is a partial differential equation for a scalar field which 
has been widely used as a model for the study of various issues in pattern formation. 
Optimal control problems for the SH equation are largely unexplored and need more 
attention.
This paper investigates an optimal distributed control problem of modified 
SH equation and in the fixed final time horizon case,
establishes the necessary optimality condition, the Pontryagin maximum principle.
Furthermore, to show the application of
obtained results, a remark on how to use the obtained results is made and 
numerically solving the
investigated problem is briefly discussed.
We remark that an important goal of this paper
is to provide a framework for using functional analysis and
control techniques to analyze and optimize the distributed parameter systems.
This result may be applied to other much complex nonlinear partial differential 
equations.

 As a direct continuation of the present paper, the future work can include 
the investigations of optimal distributed control problem of the
modified SH equation in free final time horizon case under weaker additional 
conditions and derives further new results of current interests.
Moreover, in the free final time horizon case, we can cancel the assumptions 
imposed on the preceding fixed final time horizon problem. Namely, 
the admissible control set neither needs be convex nor contains interior 
points as well as the cost functional needs not be differentiable with respect 
to the control variable. Therefore, the admissible control set can be any set. 
An interesting case is that it is allowed to contain only finite many points. 
We can also consider the optimal boundary control problem of the SH equation 
in these two cases. People can refer to \cite{Girsanov72,Kotarski97,Sun10} 
for the general information. 

\subsection*{Acknowledgments}
This work was supported  by the National Natural Science Foundation of China
under Grant 11471036.

The author would like to thank the editor and the anonymous referee
for the very careful reading and constructive suggestions
 that substantially improve the manuscript.


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\end{document}