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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 248, pp. 1--22.\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-2015/248\hfil
 Stabilization of ODE-Schr\"odinger cascaded systems]
{Stabilization of ODE-Schr\"odinger cascaded systems
subject to boundary control matched disturbance}

\author[Y.-P. Guo, J.-J. Liu \hfil EJDE-2015/248\hfilneg]
{Ya-Ping Guo, Jun-Jun Liu}

\address{Ya-ping Guo \newline
School of Mathematics and Statistics, Beijing Institute of Technology,
 Beijing 100081, China}
\email{guoyaping0904@163.com}

\address{Jun-Jun Liu \newline
Department of Mathematics, Taiyuan university of technology,
Taiyuan 030024, China}
\email{liujunjun@tyut.edu.cn}

\thanks{Submitted August 13, 2015. Published September 23, 2015.}
\subjclass[2010]{93C20}
\keywords{Cascade systems; disturbance; backstepping ; boundary control;
\hfill\break\indent  active disturbance rejection control}

\begin{abstract}
 In this article, we consider the state feedback stabilization of
 ODE-Schr\"odinger cascaded systems with the external disturbance.
 We use the backstepping transformation to handle the unstable part
 of the ODE, then design a feedback control which is used to cope with
 the disturbance and stabilize the Schr\"odinger part.
 By active disturbance rejection control (ADRC) approach,
 the disturbance is estimated by a constant high gain estimator, then
 the feedback control law can be designed.
 Next, we show that the resulting closed-loop system is practical stable,
 where the peaking value appears in the initial stage and the stabilized result
 requires that the derivative of disturbance be uniformly bounded.
 To avoid the peak phenomenon and to relax the restriction on the disturbance,
 a time varying high gain estimator is presented and asymptotical stabilization
 of the corresponding closed-loop system is proved. Finally, the effectiveness
 of the proposed control is verified by numerical simulations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
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\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
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\section{Introduction}

Environmental disturbances (e.g., noise, wave, and wind)
and modeling uncertainties (e.g., unknown plant parameters) are often
encountered problems in practical engineering systems which
reduce the system quality, lead to limited productivity and result in premature
fatigue failure.
As noted in \cite{Wang(2015)}, there is an input disturbance in
heat PDE-ODE cascade which causes variations in system dynamic characteristics,
and makes systems unstable.
To suppress vibrations of the systems, many control approaches have been
developed to cope with system uncertainty or disturbance.
For instance, the result of control design to the systems with unknown
plant parameters by adaptive control method is given in \cite{Krstic(2008)a},
with external disturbance
by sliding mode control is presented in \cite{Wang(2012)a},
and with input disturbance by active disturbance rejection control (ADRC)
is investigated in \cite{Guo(2013)c}.
However, it is noticed that stabilized result of \cite{Guo(2013)c} requires that
the derivative of disturbance is uniformly bounded.
Furthermore, the time varying
feedback control design has been recently proposed for the unstable wave
system in \cite{Feng(2014)a}, which relaxes the restriction
on the disturbance.


Actuator appears in many control applications such as
electromagnetic coupling \cite{Krstic(2008)}, chemical engineering
\cite{Wang(2015)a}, and industrial oil-drilling plants \cite{Bekiaris-Liberis(2014)}.
Some practical systems with actuator
are modelled by ordinary/partial differential equation (ODE/PDE)-PDE cascade,
in which the original system is considered as the
ODE/PDE part, while the actuator is regarded as a PDE part.
Much attention has been dedicated in the past years to
the control of unstable systems with infinite-dimensional actuator dynamics.
For example, the diffusion PDE-ODE cascaded system
is considered in \cite{Krstic(2008)}, where
 the compensating actuator dynamics dominated by first-order hyperbolic PDE systems.
The results in \cite{Krstic(2008)} are extended to the case of
actuator dynamics modelled by heat \cite{Krstic(2009)},
wave \cite{Bekiaris-Liberis(2014)} or Schr\"odinger \cite{Ren(2013)} systems.
In all these works, without the disturbances and uncertainties,
the predictor feedback control law is designed
for the actuator and the systems achieve stabilization.
More recently, the feedback control law is designed for the cascaded ODE-heat
system with the input disturbance in \cite{Wang(2015)} using sliding
mode control and backstepping method.

To the best of our knowledge, the predictor feedback control law designing for
ODE-Schr\"odinger cascades with the external disturbances is still open.
The Schr\"odinger equation is challenging due to its complex state and the
fact that all of its eigenvalues are on the imaginary axis
\cite{ Ren(2013), Wang(2012)}.
When we take into account the input disturbances, the stabilization of the cascaded
Schr\"odinger-ODE systems is difficult .

In this article, we consider the stabilization of the cascaded
ODE-Schr\"odinger systems with input disturbances:
\begin{equation}\label{01}
\begin{gathered}
\dot{X}(t)=AX(t)+Bu(0,t),\quad t>0,\\
u_{t}(x,t)=-ju_{xx}(x,t),\quad x\in (0,1), \; t>0,\\
u_x(0,t)=0,\\
u_x(1,t)=U(t)+d(t),
\end{gathered}
\end{equation}
where
$X\in\mathbb{C}^{n\times1}$ and $u$ are the states
of ODE and PDE respectively, $A\in\mathbb{C}^{n\times n}$,
$B\in\mathbb{C}^{n\times1}$;
$U(t)$ is the control input; $d(t)$ is the input
disturbance at the control end; $X_0$ and $u_0(x)$ are the initial
value of ODE and PDE respectively.
The whole system is depicted in Figure \ref{fig1}.

\begin{figure}[htb]
\begin{center}
\unitlength 0.9mm
\linethickness{0.4pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi
\begin{picture}(118,45)(5,100) % (118.75,78)(5,100)
\put(38.5,128){\framebox(46.75,18.25)[]{}}
\put(39.5,100.25){\framebox(46.75,18.25)[]{}}
\put(45,141){Schr\"odinger PDE} % system}
\put(55,108.25){ODE} % system}
\put(48,132.75){(actuator)}
\put(85.5,135.25){\line(1,0){5}}
\put(90.5,135.25){\line(0,-1){23}}
\put(90.5,112.25){\vector(-1,0){4.5}}
\put(90.5,124.25){$B u(0, t)$}
\put(18,110){$X(0)$}
\put(8,108.75){\vector(1,0){31.6}}
\put(86.25,108.25){\vector(1,0){31.0}}
\put(95.25,110.25){$X(t)$}
\put(12,136.25){\vector(1,0){26.5}}
\put(85.5,140.25){\vector(1,0){29.5}}
\put(12,140.5){$U(t)+d(t)$} % initial temperature}
\put(90.5,142.25){$u(x,t)$}
\end{picture}
\caption{Block diagram for the coupled ODE-PDE system}
\label{fig1}
\end{center}
\end{figure}

Our aim is to design a feedback control law
such that the resulting closed-loop cascaded system is asymptotic
stable.
First, our design method is based on two-step invertible backstepping
transformations that deal with the unstable part of the system, so that
the feedback control law is only used to deal with the disturbance and stabilize
the PDE part.
Second, using ADRC approach, the disturbance is estimated by a
constant high gain estimator and time varying high gain estimator respectively;
the feedback controllers are designed for the system.
Furthermore, we show that the solution of the resulting closed-loop cascaded system
is practical stable and asymptotic stable respectively.
Finally, the numerical simulation results are provided to illustrate the
effectiveness of the proposed design method.

We proceed as follows.
In Section \ref{sect2}, the two-step backstepping design is developed using
the invertible Volterra integral transformation.
In Section \ref{sect3},
we consider the well-posedness of the system obtained
from original system \eqref{01} by designing a feedback control law.
In Section \ref{sec4}, we design a constant high gain estimator by the ADRC approach
and show the practical stability of the resulting closed-loop system.
In Section \ref{sec5}, we design a time varying disturbance estimator
and obtain the asymptotic stability of the corresponding closed-loop system.
 In Section \ref{sec6}, the numerical simulation
results are provided to show the effectiveness of the proposed
method.


\section{Backstepping design}\label{sect2}

We first introduce the following transformation
to stabilize the ODE part \cite{Ren(2013)},
\begin{equation}\label{02}
\begin{gathered}
X(t) = X(t), \\
 w(x,t)= u(x,t)-\int_0^xq(x,y)u(y,t)dy-\gamma(x)X(t),
\end{gathered}
\end{equation}
where
\begin{gather*}
q(x,y)=\int_0^{x-y}j\gamma(\sigma)Bd\sigma, \\
\gamma(x)=\begin{bmatrix}K &0 \end{bmatrix}
\exp\Big( \begin{bmatrix}
 0 & jA \\
 I & 0 \\
 \end{bmatrix}
x\Big)
\begin{bmatrix}
 I \\
 0 \\
 \end{bmatrix}.
\end{gather*}
The transformation changes system \eqref{01} into
\begin{equation}\label{04}
\begin{gathered}
\dot{X}(t)=(A+BK)X(t)+Bw(0,t),\\
w_{t}(x,t)=-jw_{xx}(x,t),\\
w_x(0,t)=0,\\
w_x(1,t)=U(t)+d(t)-\int_0^1q_x(1,y)u(y,t)dy-\gamma'(1)X(t),
\end{gathered}
\end{equation}
where $K$ is chosen such that $A+BK$ is Hurwitz.
By \eqref{04}, if the PDE part is stable, then the ODE part is also stable.

The transformation \eqref{02} is invertible,
and the inverse transformation $w\mapsto u$
is postulated as follows \cite{Ren(2013)}:
\begin{equation}\label{021}
u(x,t)=w(x,t)+\int_0^xl(x,y)w(y,t)dy+\psi(x)X(t),
\end{equation}
where
\begin{gather*}
l(x,y)=\int_0^{x-y}j\psi(\xi)Bd\xi, \\
\psi(x)=\begin{bmatrix} K &0\end{bmatrix}
\exp \Big(\begin{bmatrix}
 0 & j(A+BK) \\
 I & 0 \\
 \end{bmatrix}
 x\Big)
\begin{bmatrix}
 I \\
 0 \\
 \end{bmatrix}.
\end{gather*}
For increasing the decay rate,
we define the  transformation \cite{Krstic(2008)}
 \begin{equation}\label{020}
\begin{gathered}
X(t)=X(t),\\
 z(x,t)=w(x,t)-\int_0^xk(x,y)w(y,t)dy,
\end{gathered}
\end{equation}
where
\begin{equation}
k(x,y)=-cjx \frac{I_1(\sqrt{cj(x^2-y^2)})}{\sqrt{cj(x^2-y^2)}},\quad
0\leq y\leq x\leq1,
\end{equation}
and $I_1$ is the modified Bessel function.
Transformation \eqref{020} changes system \eqref{04} to the system
\begin{equation}\label{04-4}
\begin{gathered}
\dot{X}(t)=(A+BK)X(t)+Bz(0,t),\\
z_{t}(x,t)=-jz_{xx}(x,t)-cz(x,t),\\
z_x(0,t)=0,\\
\begin{aligned}
z_x(1,t)&=U(t)+d(t)-\int_0^1q_x(1,y)u(y,t)dy-\gamma'(1)X(t)\\
 &\quad-k(1,1)w(1,t)-\int_0^1k_x(1,y)w(y,t)dy .
\end{aligned}
\end{gathered}
\end{equation}

The inverse of transformation \eqref{020} can be found as follows:
\begin{equation}\label{inverse2}
w(x,t)=z(x,t)+\int_0^x p(x,y)z(y,t)dy,
\end{equation}
where
\begin{equation}
p(x,y)=-cjx \frac{J_1(\sqrt{cj(x^2-y^2)})}{\sqrt{cj(x^2-y^2)}},\quad
0\leq y\leq x\leq1,
\end{equation}
where $J_1$ is a Bessel function.

Therefore, under the two
transformations \eqref{02} and \eqref{020}, systems \eqref{01}
and \eqref{04-4} are equivalent. So we only need consider system
\eqref{04-4}.
Denote
\begin{equation} \label{2.9}
\begin{aligned}
U_0(t)&=U(t)-\int_0^1q_x(1,y)u(y,t)dy-\gamma'(1)X(t)
 -k(1,1)w(1,t)\\
&\quad -\int_0^1k_x(1,y)w(y,t)dy.
\end{aligned}
\end{equation}
By \eqref{2.9}, the system \eqref{04-4} can be rewritten as follows
\begin{equation}\label{2.10}
\begin{gathered}
\dot{X}(t)=(A+BK)X(t)+Bz(0,t),\\
z_{t}(x,t)=-jz_{xx}(x,t)-cz(x,t),\\
z_x(0,t)=0,\\
z_x(1,t)=U_0(t)+d(t).
\end{gathered}
\end{equation}

\section{Well-posedness of \eqref{2.10}}\label{sect3}

Let us consider systems \eqref{01} and \eqref{04-4} in the state space
$\mathcal{H}=\mathbb{C}^{n}\times L^2(0,1)$,
equipped with the usual inner product:
\begin{equation} \label{05}
\langle(X,f)^{\top},(Y,g)^{\top}\rangle_\mathcal{H}
=X^{\top}\overline{Y}+\int_0^1f(x)\overline{g(x)}dx,
\quad
 \forall (X,f)^{\top}, (Y,g)^{\top}\in \mathcal{H}.
\end{equation}
Define the system operator $\mathcal{A}_0: D(\mathcal{A}_0)(\subset \mathcal{H})\to \mathcal{H}$ as
\begin{equation}\label{27}
 D(\mathcal{A}_0)=\big\{(X,f)\in \mathbb{C}^n \times H^2(0,1)| f'(0)=0, f'(1)=0
 \big\},
\end{equation}
and for any $Z=(X,f)^{\top}\in D(\mathcal{A}_0),$
\begin{equation}\label{28}
 \mathcal{A}_0Z=  ((A+BK)X+Bf(0), -jf''-cf).
\end{equation}
We compute $\mathcal{A}_0^*$, the adjoint of $\mathcal{A}_0$, to obtain
\begin{equation}\label{29}
\begin{gathered}
\mathcal{A}_0^*(Y,g)=(\overline{(A+BK)^{\top}}Y, jg''-cg), \quad \forall (Y,g)\in D(\mathcal{A}_0^*),
\\
D(\mathcal{A}_0^*)=\{(Y,g)\in \mathbb{C}^n \times H^2(0,1): g'(0)
 =j\overline{B^{\top}}Y,\; g'(1)=0\}.
\end{gathered}
\end{equation}
Define the unbounded operator $\mathcal{B}_0$ by
\begin{equation}\label{30}
 \mathcal{B}_0=(0,-\delta(x-1))^{\top}.
 \end{equation}
Then system \eqref{2.10} can be written as an abstract evolution equation 
in $\mathcal{H}$,
\begin{equation}\label{abstract}
 \frac{d}{dt}Z(t)=\mathcal{A}_0Z(t)+j\mathcal{B}_0(U_0(t)+d(t)),
 \end{equation}
where $Z(t)=(X(t),z(\cdot,t))$.

\begin{lemma}\label{L2.1}
Let $\mathcal{A}_0$ be given by \eqref{27} and \eqref{28}.
Then $\mathcal{A}_0^{-1}$ exists and is compact
on $\mathcal{H}$ and hence $\sigma(\mathcal{A}_0)$, the spectrum of $\mathcal{A}_0$, 
consists of isolated eigenvalues of finitely algebraic multiplicity only.
\end{lemma}

\begin{proof}
For any given $(X_1, z_1)\in \mathcal{H}$, solve
\begin{equation}\label{2.16}
 \mathcal{A}_0(X,z)=((A+BK)X+Bz(0), -jz''(x)-cz(x))=(X_1, z_1).
\end{equation}
We obtain
\begin{equation}\label{2.17}
\begin{gathered}
(A+BK)X+Bz(0)=X_1. \\
-jz''(x)-cz(x)=z_1(x), \\
z'(0)=0, z'(1)=0,
\end{gathered}
\end{equation}
with solution
\begin{equation}\label{2.18}
\begin{gathered}
X=(A+BK)^{-1}(X_1-Bz(0)),
 \\
 z(x)=  c_0\left(e^{c_{\lambda}x}+e^{-c_{\lambda}x}\right)-
 \frac{1}{2c_{\lambda}}\int_0^x
\left(e^{c_{\lambda}(s-x)}-e^{-c_{\lambda}(s-x)}\right)jz_1(s)ds,
 \\
c_0=-\frac{1}{2c_{\lambda} \left(e^{c_{\lambda}}-e^{-c_{\lambda}}\right)}
 \int_0^1 \left(e^{c_{\lambda}(s-1)}+e^{-c_{\lambda}(s-1)}\right)jz_1(s)ds, \\
 c_{\lambda}=\sqrt{cj}.
\end{gathered}
\end{equation}

Hence, we have the unique $(X,z)\in D(\mathcal{A}_0)$.
Then, $\mathcal{A}^{-1}_0$ exists and
is compact on $\mathcal{H}$ by the Sobolev embedding theorem.
Therefore, $\sigma(\mathcal{A}_0)$ consists of isolated eigenvalues of finite algebraic
multiplicity.
\end{proof}

Now we consider the eigenvalue problem of $\mathcal{A}_0$.
Let $\mathcal{A}_0 Y=\lambda Y$, where $Y=(X,z)$, then we have
\begin{equation}\label{2.19}
\begin{gathered}
(A+BK)X+Bz(0)=\lambda X, \\
-jz''-cz=\lambda z(x), \\
z'(0)=z'(1)=0.
\end{gathered}
\end{equation}

\begin{lemma}\label{L2.2}
Let $\mathcal{A}_0$ be given by \eqref{27} and \eqref{28}, let
 $\lambda_k^0, k=1,2, \dots,n$  be the simple eigenvalue of $A+BK$
with the corresponding eigenvector $X_k$, and assume that
\begin{equation}\label{2.20}
\lambda^0_k \notin \{\lambda^p_m, m \in \mathbb{N} \},
\end{equation}
where
\begin{equation}\label{2.21}
\lambda^p_m=-c+m^2\pi^2 j.
\end{equation}
Then the eigenvalues of $\mathcal{A}_0$ are
\begin{equation}\label{2.22}
\{\lambda^0_k,\; k=1,2,\dots,n\}\cup\{\lambda^p_m,\; m=0,1,2,\dots\}
\end{equation}
and the eigenfunctions corresponding to $\lambda_k^0$
and $\lambda_m^p$ are respectively
\begin{gather}\label{2.23}
 W_k=(X_k,0), \quad k=1,2, \dots ,n; \\
\label{2.24}
 W_m(x)=([\lambda_m^pI-(A+BK)]^{-1}B, z_m(x)), \quad m\in \mathbb{N};
\end{gather}
where
\begin{equation}\label{2.25}
 z_m(x)=\cos m \pi x,\quad  m\in \mathbb{N}.
\end{equation}
\end{lemma}

\begin{proof}
Since $A + BK$ is Hurwitz, we have
\begin{equation}\label{2.26}
 \operatorname{Re}\lambda^0_k<0,\quad  k=1,2, \dots ,n.
\end{equation}
A simple computation shows that the eigenvalue problem
\begin{equation}\label{2.27}
\begin{gathered}
-jz''(x)-cz=\lambda z(x), \\
z'(0)=0, z'(1)=0,
\end{gathered}
\end{equation}
has the nontrivial solutions
\begin{equation}\label{2.28}
 (\lambda_m^p, z_m(x)),\quad  m\in \mathbb{N},
\end{equation}
where $\lambda_m^p$ and $z_m(x)$ are given by \eqref{2.21} and \eqref{2.25}
 respectively.

Next, we look for the eigenvalues for \eqref{2.19}.
Let $\lambda=\lambda^0_k, ~k=1,2,\dots,n$, since $B\neq0$ and
$(A+BK)X_k+Bz(0)=\lambda_k^0 X_k$, we have $z(0)\equiv 0$. Moreover,
\begin{equation}\label{2.29}
\begin{gathered}
-jz''-cz=\lambda_k^0 z, \\
z(0)=z'(0)=z'(1)=0,
\end{gathered}
\end{equation}
only has trivial solutions. So we obtain that $\lambda^0_k, k=1,2, \dots ,n$
are the eigenvalues of \eqref{2.19} and have the corresponding eigenfunctions
$(X_k, 0)$, as \eqref{2.23}.

On the other hand, when $\lambda=\lambda_m^p$, $(\lambda_m^p,z_m(x))$
satisfies the second and third equations of \eqref{2.19} and $z_m(0)=(-1)^m \neq0$.
 By the first equation of \eqref{2.19}, we have
$$
X_m^p=[\lambda_m^pI-(A+BK)]^{-1}B.
$$
So $\lambda_m^p, m\in \mathbb{N}$ is the eigenvalue of \eqref{2.19} and
has the corresponding eigenfunction
$$
([\lambda_m^pI-(A+BK)]^{-1}B, \cos(m\pi x)).
$$
 The proof is complete.
\end{proof}

\begin{theorem}\label{T2.1}
Let $\mathcal{A}_0$ be given by \eqref{27} and \eqref{28}, let $\lambda_k^0$ be the
simple eigenvalue of $A + BK$ with the corresponding eigenvector $X_k$.
Then, there is a sequence of eigenfunctions of $\mathcal{A}_0$
which forms a Riesz basis for $\mathcal{H}$. Moreover, the following conclusions
hold:
\begin{enumerate}
 \item $\mathcal{A}_0$ generates a $C_0$-semigroup $e^{\mathcal{A}_0 t}$ on $\mathcal{H}$.

 \item The spectrum-determined growth condition $\omega(\mathcal{A}_0)=s(\mathcal{A}_0)$
holds true for $e^{\mathcal{A}_0}t$ , where
$\omega(\mathcal{A}_0) = \lim_{t\to\infty}
\|e^{\mathcal{A}_0t}\|/t$ is the
growth bound of $e^{\mathcal{A}_0}t$ , and
$s(\mathcal{A}_0)=\sup\{\operatorname{Re} \lambda|\lambda \in\sigma(\mathcal{A}_0)\}$ is the
spectral bound of $\mathcal{A}_0$.

 \item The $C_0$-semigroup $e^{\mathcal{A}_0}t$ is exponentially stable in the sense
 $\|e^{\mathcal{A}_0t}\|\leq M_1e^{-c_1t}$,
 where $M_1>0$ and $c_1$ is an arbitrary pre-designed decay rate.
\end{enumerate}
\end{theorem}

\begin{proof}
It is noted that $\{X_k, k=1,2, \dots, n\}$ is an orthogonal basis
in $\mathbb{C}^n$ and $\{z_m(x), m\in \mathbb{N}\} $ given by \eqref{2.25} forms
an orthogonal basis in $L^2(0, 1)$. We have
$$
\{F_k,F_m(x): k=1,2, \dots, n, m\in\mathbb{N}\},
$$
which forms an orthogonal basis in $\mathcal{H}$ with $F_k=(X_k,0)$ and
$F_m(x) =(0,z_m(x))$. It follows from \eqref{2.23}, \eqref{2.24} that
\begin{equation}\label{2.30}
 \sum^n_{k=1}\|W_k-F_k\|^2+\sum^\infty_{m=0}\|W_m(x)-F_m(x)\|^2
 =\sum^\infty_{m=0}\|\left(\lambda_m^pI-(A+BK)\right)^{-1}B\|_{\mathbb{C}^n}^2,
\end{equation}
where $\|\cdot\|_{\mathbb{C}^{n}}$ denotes the norm in $\mathbb{C}^n$.
A simple computation gives
\begin{align*}
&\|[\lambda_m^pI-(A+BK)]^{-1}B\|_{\mathbb{C}^n}^2\\
 &=\overline{([\lambda_m^pI-(A+BK)]^{-1}B)^{\top}}([\lambda_m^pI-(A+BK)]^{-1}B )\\
 &=\overline{B^{\top}}\overline{([\lambda_m^pI-(A+BK)]^{-1})^{\top}}([\lambda_m^pI-(A+BK)]^{-1})B\\
 &=\frac{1}{|\lambda_m^p|^2}\overline{B^{\top}}
 \overline{\Big(\big[I-\frac{1}{\lambda_m^p}(A+BK)\big]^{-1}\Big)^{\top}}
 \big[I-\frac{1}{\lambda_m^p}(A+BK)\big]^{-1}B.
\end{align*}
It follows from \eqref{2.22} that when $m\to\infty$, $\lambda^p_m\to-\infty$,
there is a positive number $N$ such that for $m>N$,
$$
 \big[I-\frac{1}{\lambda_m^p}(A+BK)\big]^{-1}
=I+\mathcal {O}\Big(\frac{1}{|\lambda_m^p|}\Big), \quad m>N.
$$
Thus
$$
\|[\lambda_m^pI-(A+BK)]^{-1}B\|_{\mathbb{C}^n}^2
=\frac{\|B\|_{\mathbb{C}^n}^2}{|\lambda_m^p|^2}
\Big(1+\mathcal {O}\Big(\frac{1}{|\lambda_m^p|}\Big)\Big).
$$
Hence, it follows from \eqref{2.30} that
\begin{align*} %\label{jun19}
& \sum^n_{k=1}\|W_k-F_k\|^2+\sum^\infty_{m=0}\|W_m(x)-F_m(x)\|^2 \\
&=\sum^\infty_{m=0}\|[\lambda_m^pI-(A+BK)]^{-1}B\|_{\mathbb{C}^n}^2\\
&=\sum^N_{m=0}\|[\lambda_m^pI-(A+BK)]^{-1}B\|_{\mathbb{C}^n}^2+
 \sum^\infty_{m=N+1}\frac{\|B\|_{\mathbb{C}^n}^2}{|\lambda_m^p|^2}
\Big(1+\mathcal {O}\Big(\frac{1}{|\lambda_m^p|}\Big)\Big)<\infty.
\end{align*}
Therefore, by Bari's theorem,
$$
\{W_k,W_m(x): k=1,2,\dots,n,m=0,1,\dots\}
$$
forms a Riesz basis for $\mathcal{H}$. Moreover, by Lemma \ref{L2.1}, $\mathcal{A}_0$ generates
a $C_0$-semigroup $e^{\mathcal{A}_0t}$ on $\mathcal{H}$ and the spectrum-determined growth
condition $\omega(\mathcal{A}_0)=s(\mathcal{A}_0)$ holds true for $e^{\mathcal{A}_0t}$.
Finally, by the eigenvalues
of $\mathcal{A}_0$ given by \eqref{2.22}, there is a positive constant
$M_1>0$ and $c_1$ such that
\begin{equation}\label{2.32}
 \|e^{\mathcal{A}_0t}\|\leq M_1 e^{-c_1t}, \quad\quad\quad \forall t\geq0.
\end{equation}
The proof is complete.
\end{proof}

\begin{lemma}\label{L2.3}
Let $\mathcal{A}_{0}$, $\mathcal{B}_{0}$ be defined by \eqref{27}-\eqref{28} and \eqref{30} respectively.
Then $\mathcal{B}_{0}$ is admissible to the semigroup generated by $\mathcal{A}_{0}$.
\end{lemma}

\begin{proof}
Now we show that $\mathcal{B}_{0}$ is admissible for $e^{\mathcal{A}_{0} t}$, or equivalently,
$\mathcal{B}_0^*$ is admissible for $e^{\mathcal{A}_{0}^* t}$.
 To this end, we consider  the dual system of \eqref{abstract},
\begin{equation}\label{dual-A}
\begin{gathered}
\dot{X^*}(t)=\overline{(A+BK)^{\top}}X^*(t),\\
z^*_{t}(x,t)=jz^*_{xx}(x,t)-cz^*(x,t),\\
z^*_x(0,t)=\overline{B^{\top}}jX^*,\\
z^*_x(1,t)=0,\\
y(t)=\mathcal{B}_{0}^*\begin{pmatrix}
 X^* \\
 z^*(x,t) \\
 \end{pmatrix}=-jz^*(1,t).
\end{gathered}
\end{equation}

From Lemma \ref{L2.2}, we claim that  $\bar{\lambda}^0_j$,
$j=1,2,\dots,n $ is the simple eigenvalue of $\overline{(A+BK)^{\top}}$ with
the corresponding eigenvector $X^*_j$.
In a similar way as Lemma \ref{L2.2}, we can find the spectrum
$\sigma (\mathcal{A}_{0}^*)$ of the adjoint operator $\mathcal{A}_{0}^*$,
\begin{equation}\label{2.33}
\sigma (\mathcal{A}_{0}^*)=\{\bar{\lambda}^0_j: j=1,2,\dots,n\}
\cup\{\bar{\lambda}^p_m: m=0,1,2,\dots\},
\end{equation}
and the eigenvectors corresponding to $\bar{\lambda}^0_j$ and
$\bar{\lambda}^p_m$ are respectively
\begin{equation}\label{2.34}
 Z^*_j=(X^*_j,z^*_j),\; j=1,2, \dots ,n,\quad
 Z^*_m(x)=(0, \cos (m \pi x)),\; m\in \mathbb{N},
\end{equation}
where
\begin{equation}\label{2.35}
z^*_j=\frac{\overline{B^{\top}}iX_j^*}{m(e^{-m}-1)}
\big\{e^{m(1+x)}+e^{-m(1-x)}-e^{mx}+e^{m(1-x)}
\big\},
\end{equation}
$m=i\sqrt{i(c+\lambda_j^0)}$.
Moreover, $\{Z_j,Z_m(x), j=1,2, \dots,n, m\in \mathbb{N}\}$
forms a Riesz basis for $C^n\times L^2(0,1)$
and $\mathcal{A}^*$  generates a $C_0$-semigroup on $\mathcal{H}$.
Hence, for any $Z^*(\cdot,0)\in \mathcal{H}$, we suppose that
$$
Z^*(x,0)=\sum_{k=1}^na_k Z_k+\sum_{m=0}^\infty b_mZ_m(x).
$$
Then the solution of \eqref{dual-A} is
$$
[X^*, z^*]
=Z^*(x,t)=e^{\mathcal{A}^*_0 t}Z^*(x,0)=
\sum_{k=1}^na_ke^{\lambda^0_k t} Z_k+ \sum_{m=0}^\infty b_me^{\lambda^p_m t}Z_m(x),
$$
where
$$X^*=\sum _{k=1}^n a_ke^{\lambda^0_kt}X_k^*, \quad
z^*=\sum _{k=1}^n a_ke^{\lambda^0_kt}z^*_k
 +\sum_{m=0}^\infty b_me^{\lambda^p_m t}Z^*_m(x),
$$
hence
$$
y(t)=\sum_{k=1}^na_ke^{\lambda^0_k t}z^*_k+ \sum_{m=0}^\infty b_me^{\lambda^p_m t}.
$$
By Ingham's inequality \cite[Theorems 4.3]{Komornik(2005)}, there exists a
$T>0$, such that
\begin{equation}
\int_0^T|y(t)|^2dt
\leqslant  C_{T_1}\sum_{k=1}^n|a_kz^*_k|^2+C_{T_2}
\sum_{m=0}^\infty |b_m|^2 \leqslant D_T\|Z^*(\cdot ,0)\|^2,
\end{equation}
for some constants $C_{T_1},C_{T_2}, D_T$ that depend on $T$.

On the other hand, for any given $(X,f)^{\top}\in \mathcal{H}$, we solve that
$$
\mathcal{A}_{\hat{\omega}}^{\ast}(Y,g)^{\top}=(X,f)^{\top}.
$$
Combine the definition of $\mathcal{A}_{\hat{\omega}}^{\ast}$ with its boundary condition
 to obtain
\begin{gather*}
 \overline{(A+BK)^{\top}}Y=X,\quad jg''-cg=f,\\
 g'(0)=jB^{T}Y, \quad g'(1)=0.
\end{gather*}
A direct computation gives the solution of the above equations
\begin{gather*}
 Y=[(A+BK)^{T}]^{-1}X,\\
 g(x)= c_1e^{c_{\lambda}x}-c_2e^{-c_{\lambda}x}+\frac{1}{2c_{\lambda}}
 \int_0^x\left(e^{-c_{\lambda}(s-x)}-e^{c_{\lambda}(s-x)}\right)jf(s)ds,
 \\
c_1=\frac{1}{c_{\lambda}(e^{-c_{\lambda}}-e^{c_{\lambda}})}
\left\{\frac{1}{2}
\int_0^1 \left(e^{-c_{\lambda}(s-1)}-e^{c_{\lambda}(s-1)}\right)
jf(s)ds +jB^{T}Ye^{c_{\lambda}}\right\},
\\
c_2=\frac{1}{c_{\lambda}(e^{-c_{\lambda}}-e^{c_{\lambda}})}\left\{\frac{1}{2}
\int_0^1 \left(e^{-c_{\lambda}(s-1)}-e^{c_{\lambda}(s-1)}\right)jf(s)ds
+jB^{T}Ye^{-c_{\lambda}}\right\}, \\
 c_{\lambda}=j\sqrt{c j}.
\end{gather*}
We  obtain
$$
\mathcal{B}_0^{\ast}(\mathcal{A}_{\hat{\omega}}^{\ast})^{-1}(X,f)^{\top}=
 \mathcal{B}_0^{\ast}(Y,g)^{\top}=-g(1),
$$
which is bounded from $\mathcal{H}$ to $\mathbb{C}$.
This shows that $\mathcal{B}_0^*$ is admissible for $e^{\mathcal{A}_0^* t}$ and so is $\mathcal{B}_0$ for
$e^{\mathcal{A}_0 t}$ .
The proof is complete.
\end{proof}

\begin{proposition}\label{Pro.Guo}
The operator $\mathcal{A}_0$ defined by \eqref{27} and \eqref{28} generates an
exponential stable $C_0$-semigroup on $\mathcal{H}$, and the control operator
$\mathcal{B}_0$ is admissible to the semigroup $e^{\mathcal{A}_0 t}$. Hence,
for any $Z(x,0)\in \mathcal{H}$, there exists a unique (weak) solution to
\eqref{abstract}, which can be written as
\begin{equation}\label{revisedguo} Z(\cdot,t)=e^{\mathcal{A}_0
t}Z(\cdot,0)+j\int_0^t e^{\mathcal{A}_0 (t-s)}\mathcal{B}_0[U_0(s) +d(s)]ds,
\end{equation}
for all $U_0(s) +d(s)\in L^2_{loc}(0,\infty)$;
that is,
\begin{equation}\label{GuoA03}
\frac{d}{dt}\langle Z(\cdot,t),\rho\rangle=\langle
Z(\cdot,t),\mathcal{A}_0^*\rho\rangle +j[U_0(t)+d(t)]\mathcal{B}_0^* \rho, \;
\forall\; \rho\in D(\mathcal{A}_0^*).
\end{equation}
\end{proposition}

\section{Constant high gain estimator based feedback}\label{sec4}

In this section, we propose a state disturbance estimator with constant high gain
based on the ADRC approach.
It is supposed that $d$ and its derivative are
uniformly bounded, i.e., $|d(t)|\leq M_1$ and $|\dot{d}(t)|\leq M_2
$ for some $M_1, M_2>0$ and all $t\geq0$.
Taking specially $\rho(x)=(0, 2x^{3}-3x^{2})^{\top}$ in \eqref{GuoA03},
 we obtain
\begin{equation}\label{Dtest}
\dot{y}_0(t)  =  U_0(t)+d(t)+y_1(t),
\end{equation}
where
\begin{gather}\label{test}
y_0(t)=-j\int_{0}^{1} (2x^{3}-3x^{2}) z(x,t) dx, \\
\label{2.12}
y_1(t)=\int_{0}^{1}(-12x+6+2cjx^3-3cjx^2)z(x,t) dx.
\end{gather}
Then we are able to design
an extended state observer to estimate both $y_0(t)$ and $d(t)$
\cite{Guo(2011)} as follows:
 \begin{equation}\label{2.13}
\begin{gathered}
\dot{\hat{y}}_{\epsilon}(t)= U_0(t)+\hat{d}_{\epsilon}(t)+y_1(t)-
\frac{1}{\epsilon}(\hat{y}_{\epsilon}-y_0),
\\
\dot{\hat{d}}_{\epsilon}(t)=-\frac{1}{\varepsilon^{2}}\{ \hat{y} _{\epsilon}-y_0\},
\end{gathered}
 \end{equation}
where $\epsilon$ is the tuning small parameter and $\hat{d}_{\epsilon}(t)$
is regarded as approximation of $d(t)$.
We have the following result.

\begin{lemma}\label{L-2.1}
Let $(\hat{y}_{\epsilon}, \hat{d}_{\epsilon})$
be the solution of \eqref{2.13} and $y_0$ be defined as \eqref{test}.
The followings hold.
\begin{enumerate}
 \item For any $\alpha>0$,
 \begin{equation}\label{2.14}
 |\hat{y}_{\epsilon}(t)-y_0(t)|+|\hat{d}_{\epsilon}(t)-d(t)|\to 0, \quad
\text{as $\varepsilon\to 0$,
  uniformly } t\in [\alpha, \infty).
 \end{equation}

 \item For any $\alpha>0$,
 \begin{equation}\label{2.15}
 \int_{0}^\alpha  |\hat{y}_{\epsilon}(t)-y_0(t)|+|\hat{d}_{\epsilon}(t)-d(t)|dt
 \text{ is  uniformly bounded as } \epsilon\to 0.
 \end{equation}
 \end{enumerate}
\end{lemma}

\begin{proof}
Suppose the errors
\begin{equation}\label{17}
\tilde{y}_{\epsilon}(t) =\hat{y}_{\epsilon}(t)-y_0(t),\;\;
\tilde{d}_{\epsilon}(t) =-\hat{d}_{\epsilon}(t)+d(t),
\end{equation}
satisfy
\begin{equation}\label{18}
\begin{gathered}
\dot{\tilde{y}}_{\epsilon}(t)= -\tilde{d}_{\epsilon}(t)
 -\frac{1}{\varepsilon} \tilde{y}_{\epsilon}(t),\\
\dot{\tilde{d}}_{\epsilon}(t)=\frac{1}{\varepsilon^{2}} \tilde{y}_{\epsilon}(t)
 +\dot{d}(t).
\end{gathered}
\end{equation}
Then \eqref{18} can be written as an evolution equation:
\begin{equation}\label{19}
 \frac{d }{ds}
 \begin{pmatrix} \tilde{y}_{\epsilon} \\ \tilde{d}_{\epsilon} \end{pmatrix}
 =A_{1}\begin{pmatrix} \tilde{y}_{\epsilon} \\ \tilde{d}_{\epsilon}
\end{pmatrix}
+D_{1}(s),
\end{equation}
where
\begin{equation}
A_{1}=
\begin{pmatrix} -\frac{1}{\epsilon}& -1 \\
\frac{1}{\epsilon^{2}}& 0\\
 \end{pmatrix},\quad
D_{1}(s)=\begin{pmatrix} 0 \\ 1
 \end{pmatrix}.
\end{equation}
A simple exercise shows that the eigenvalues of the matrix $A_{1}$ are
$$
\lambda_{1}=-\frac{1}{2\epsilon}+\frac{\sqrt{3}}{2}j,\quad
\lambda_{2}=-\frac{1}{2\epsilon}-\frac{\sqrt{3}}{2}j,
$$
which satisfy
\begin{equation}\label{guonewjia}
\begin{gathered}
 e^{A_1t} =\begin{pmatrix}
 \frac{\lambda_1}{\lambda_2-\lambda_1}e^{\lambda_1
 t}-\frac{\lambda_2}{\lambda_2-\lambda_1}e^{\lambda_2 t} &
 \frac{\lambda_1\lambda_2}{C_\varepsilon
 (\lambda_2-\lambda_1)}\left(e^{\lambda_2 t}-e^{\lambda_1 t}\right)
 \\
 \frac{C_\varepsilon}{\lambda_2-\lambda_1}\left(e^{\lambda_1t}-e^{\lambda_2t}\right) &
 -\frac{\lambda_2}{\lambda_2-\lambda_1}e^{\lambda_1
 t}+\frac{\lambda_1}{\lambda_2-\lambda_1}e^{\lambda_2t}
\end{pmatrix},
 \\
 e^{A_1t}D_1=-\begin{pmatrix}
 \frac{\lambda_1\lambda_2}{C_\varepsilon(\lambda_2-\lambda_1)}\left(e^{\lambda_2
 t}-e^{\lambda_1 t}\right)
 \\
 -\frac{\lambda_2}{\lambda_2-\lambda_1}e^{\lambda_1 t}+
 \frac{\lambda_1}{\lambda_2-\lambda_1}e^{\lambda_2t}
 \end{pmatrix}, \quad
 C_\varepsilon=\frac{1}{\varepsilon^2}.
\end{gathered}
\end{equation}
The above three equations imply that there exist
 constants $\hat{L}$ and $\hat{M}$ such that
 \begin{equation}\label{gu}
\|e^{A_{1}t}\|\leq \frac{\hat{L}}{\epsilon}e^{-\frac{1}{2\epsilon}t},\quad
\|e^{A_{1}t}D_{1}\|\leq \hat{M}e^{-\frac{1}{2\epsilon}t}.
 \end{equation}
Since
\begin{equation}\label{21}
 \begin{pmatrix} \tilde{y}_{\epsilon}(t) \\
\tilde{d}_{\epsilon}(t) \end{pmatrix}
 =e^{A_{1}t}
 \begin{pmatrix} \tilde{y}_{\epsilon}(0) \\
\tilde{d}_{\epsilon}(0) \end{pmatrix}
 +\int_{0}^{t}e^{A_{1}(t-s)}
D_{1}\dot{d}(s)ds,
\end{equation}
the first term above can be arbitrarily small as $t\to \infty$
by the exponential stability of $e^{A_{1}t}$, and the second term can
also be arbitrarily small as $\epsilon\to 0$
due to boundedness of $\dot{d}$ and the expression of $e^{A_{1}t}D_{1}$. As
a result, the solution $(\tilde{y}_{\epsilon},\tilde{d}_{\epsilon})$
of \eqref{18} satisfies
\begin{equation}\label{error}
(\tilde{y}_{\epsilon},\tilde{d}_{\epsilon})\to 0, \quad\text{as }
 t\to\infty,\epsilon\to 0.
 \end{equation}
By estimating \eqref{gu}, we can obtain \eqref{2.15}.
The proof is completed.
\end{proof}

By Lemma \ref{L-2.1}, we deduce that the design of ESO
\eqref{2.13} is based on the arbitrary decay rate of $\|e^{A_1t}\|$
and the special structure of $D_1$.
In this way, the ADRC is not well adapted to PDEs
because it is hard that a PDE
system has the arbitrary decay rate.
That also explains
why $\dot{d}$ must be uniformly bounded.

In \eqref{2.15}, it is worth pointing out that
$\int_{0}^\alpha |\hat{d}_{\epsilon}(t)-d(t)|dt$
is uniformly bounded in $\epsilon$ for any fixed
$\alpha>0,$ while $\int_{0}^\alpha |\hat{d}_{\epsilon}(t)-d(t)|^2dt$
is unbounded in $\epsilon$.
Then we could find that
the $L^2$ unboundedness of $\int_{0}^\alpha |\hat{d}_{\epsilon}(t)-d(t)|^2dt$
brings trouble to PDEs (see \eqref{add2}).
To avoid this phenomenon, the feedback controller for system \eqref{2.10}
is proposed as follows:
\begin{equation}\label{22}
 U_0(t)=-\operatorname{sat}(\hat{d}_{\epsilon}(t)),
 \end{equation}
where
\begin{equation}\label{23}
\operatorname{sat}(x)= \begin{cases}
M_1, & x\geq M_1+1, \\
-M_1, & x\leq- M_1-1, \\
x,& x\in (- M_1-1,M_1+1).
\end{cases}
 \end{equation}
Combining $|d(t)|\leq M_1$ with \eqref{17}, for any given $\alpha>0$,
when $\epsilon$ is sufficiently small,
we obtain $U_0(t)=-\hat{d}_{\epsilon}(t),$ for all $t\in [\alpha, \infty)$.

Under the feedback \eqref{22},  system \eqref{2.10} becomes
\begin{equation}\label{24}
\begin{gathered}
 \dot{X}(t)=(A+BK)X(t)+Bz(0,t),  \\
 z_{t}(x,t)=-jz_{xx}(x,t)-cz(x,t),  \\
 z_x(0,t)=0,  \\
 z_x(1,t)=-\operatorname{sat}(\hat{d}_{\epsilon}(t))+d(t),  \\
 \dot{\hat{y}}_{\epsilon}(t)= -\operatorname{sat}(\hat{d}_{\epsilon}(t))
 +\hat{d}_{\epsilon}(t)+y_1(t)
 -\frac{1}{\epsilon}(\hat{y}_{\epsilon}-y_0),
 \\
 \dot{\hat{d}}_{\epsilon}(t)=-\frac{1}{\varepsilon^{2}}\{ \hat{y} _{\epsilon}-y_0\}.
\end{gathered}
 \end{equation}
Using the error dynamics defined in \eqref{17},
we see that \eqref{24} is equivalent to:
\begin{equation}\label{25}
\begin{gathered}
 \dot{X}(t)=(A+BK)X(t)+Bz(0,t), \\
 z_{t}(x,t)=-jz_{xx}(x,t)-cz(x,t), \\
 z_x(0,t)=0, \\
 z_x(1,t)=\operatorname{sat}(\tilde{d}_{\epsilon}(t)-d(t))+d(t), \\
 \dot{\tilde{y}}_{\epsilon}(t)= -\tilde{d}_{\epsilon}(t)
-\frac{1}{\varepsilon} \tilde{y}_{\epsilon}(t), \\
 \dot{\tilde{d}}_{\epsilon}(t)=\frac{1}{\varepsilon^{2}}
\tilde{y}_{\epsilon}(t)+\dot{d}(t).
\end{gathered}
 \end{equation}
By \eqref{17}, it is seen that $(\tilde{y}_{\epsilon}(t),\tilde{d}_{\epsilon}(t))$
is independent of the ``$(X, z)$-part", which can be
arbitrarily small as $t\to \infty, \varepsilon \to 0$
by Lemma \ref{L-2.1}. Hence, we only need to consider the ``$(X, z)$-part"
which is rewritten as
\begin{equation}\label{26}
\begin{gathered}
 \dot{X}(t)=(A+BK)X(t)+Bz(0,t), \\
 z_{t}(x,t)=-jz_{xx}(x,t)-cz(x,t), \\
 z_x(0,t)=0, \\
 z_x(1,t)=\operatorname{sat}(\tilde{d}_{\epsilon}(t)+d(t))+d(t)
 \triangleq \tilde{d}(t).
\end{gathered}
 \end{equation}
System \eqref{26} can be written as the following abstract evolution equation
in $\mathcal{H}$:
\begin{equation}\label{31}
 \frac{d}{dt}Z(t)=\mathcal{A}_0Z(t)+j\mathcal{B}_0\tilde{d}(t),
 \end{equation}
where $Z(t)=(X(t),z(\cdot,t))$,
$\mathcal{A}_0$ and $\mathcal{B}_0$ are given respectively by
\eqref{28} and \eqref{30}.

\begin{lemma}\label{L3.1}
Assume that $|d(t)|\leqslant M_1$ and $\dot{d}(t)$ is measurable,
$|\dot{d}(t)|\le M_2$ for all $t\ge 0$. Then for any initial value
$(X(0),z(\cdot,0))\in \mathcal{H}$, the
 closed-loop system \eqref{26} admits a unique solution
$(X,z)^\top\in C(0,\infty;\mathcal{H})$, and
$$
 \lim_{t\to \infty,\, \varepsilon\to 0}
\|(X(t),z(\cdot,t),\hat{y}_\varepsilon
 (t),\hat{d}_\varepsilon(t)-d(t))\|_{\mathcal{H} \times \mathbb{C}^2}=0.
 $$
\end{lemma}

\begin{proof}
By Theorem \ref{T2.1} and Lemma \ref{L2.3}, for any initial value
$(X(0),z(\cdot,0))\in \mathcal{H}$, there exists a unique (weak) solution
$(X,z)\in C(0,\infty;\mathcal{H})$ which can be written as
\begin{equation}\label{abssolution}
Z(\cdot, t)
=e^{\mathcal{A}_0 t}Z(\cdot, 0)+j\int_0^te^{\mathcal{A}_0(t-s)} \mathcal{B}_0\tilde{d}(t)ds.
\end{equation}
By \eqref{error}, for any given $\varepsilon_0>0$, there exist $t_0>0$
and  $\varepsilon_1>0$ such that
 $$
|\tilde{d}(t)|=
 |-\operatorname{sat}(\tilde{d}_\varepsilon(t)+d(t))+d(t)|<\varepsilon_0,
$$
for all $t>t_0$ and $0<\varepsilon<\varepsilon_1$.
 We rewrite the solution of \eqref{abssolution},
\begin{equation}\label{abssolution0}
\begin{aligned}
 Z(\cdot, t)
& = e^{\mathcal{A}_0 t} Z(\cdot, 0) +
 je^{\mathcal{A}_0(t-t_0)}\int_0^{t_0}e^{\mathcal{A}_0 (t_0-s)}
 \mathcal{B}_0 \tilde{d}(s)ds \\
&\quad + j\int_{t_0}^{t}e^{\mathcal{A}_0 (t-s)}\mathcal{B}_0 \tilde{d}(s)ds.
\end{aligned}
\end{equation}
According to the admissibility of $\mathcal{B}_0$, we obtain
\begin{equation}\label{add2}
\begin{aligned}
\|\int_0^{t_0}e^{\mathcal{A}_0 (t-s)}\mathcal{B}_0 \tilde{d}(s)ds\|_\mathcal{H}^2
&\leqslant C_{t_0}\|\operatorname{sat}(\tilde{d}_\varepsilon+d)+d\|^2_{L^2(0,t_0)}\\
&\leqslant t_0 C_{t_0}(2M_1+1)^2,
\end{aligned}
\end{equation}
where the constant $C_{t_0}$ is independent of $\tilde{d}_\varepsilon$ and $d$.
 Since $e^{\mathcal{A}_0 t}$ is
exponentially stable and $\mathcal{B}_0$ is admissible to $e^{\mathcal{A}_0 t}$ with
$L^2_{loc}$ control, $\mathcal{B}_0$ is admissible to $e^{\mathcal{A}_0 t}$ with
$L^\infty_{loc}$ control.
It follows from \cite[Proposition 2.5]{Weiss(1989)} that
\begin{equation}\label{add3}
\begin{aligned}
\|\int_{t_0}^t
 e^{\mathcal{A}_0 (t-s)}\mathcal{B}_0 \tilde{d}(s)ds\|
&=\|\int_0^t e^{\mathcal{A}_0 (t-s)}\mathcal{B}_0 (
 0\mathop{\Diamond}\limits_{t_0} \tilde{d}(s)ds\|
 \\
&\leqslant L \|\tilde{d}(s)\|_{L^\infty(0,\infty)} \\
&\leqslant L \|[\operatorname{sat}(\tilde{d}_\varepsilon+d)+d]\|_{L^\infty(0,\infty)}
 \leqslant L\varepsilon_0,
\end{aligned}
\end{equation}
where $L$ is a constant that is independent of
$\tilde{d}_\varepsilon$ and $d$, and \cite{Weiss(1989)}
\begin{equation}\label{3.36}
(d_1 \mathop{\Diamond}_\tau d_2)(t)
=\begin{cases}
d_1(t), 0\leqslant t\leqslant \tau,\\
d_2(t-\tau), t>\tau.
\end{cases}
\end{equation}
 Assume that $\|e^{\mathcal{A}_0 t}\|\leqslant L_0e^{-\omega t}$ for some $L_0,
\omega>0$. By \eqref{abssolution0}, \eqref{add2}, and \eqref{add3},
\begin{equation}\label{add5}
 \|
 Z(\cdot,t)\|\leqslant L_0e^{-\omega t} \|
 Z(\cdot,0) \|+L_0t_0(2M_1+1)^2C_{t_0}e^{-\omega
(t-t_0)}+L\varepsilon_0.
\end{equation}
 This implies that $\|Z(\cdot,t)\|_{L^2(0,1)}\to 0$ as $t\to \infty$.
Consequently, by \eqref{test},
$y_0(t)=-j\int_{0}^{1} (2x^{3}-3x^{2}) z(x,t) dx\to 0$ as
$t\to \infty$. The result then follows with \eqref{17} and
\eqref{error}.
 The proof is complete.
\end{proof}

Returning to system \eqref{01} by the inverse transformations
\eqref{021} and \eqref{inverse2}, we have the following theorem.

\begin{theorem}\label{Th3.1}
 Assume that $|d(t)|\leqslant M_1$ and $\dot{d}(t)$ is measurable,
$|\dot{d}(t)|\le M_2$ for all $t\ge 0$. Then for any initial value
$(X(0),u(\cdot,0),\hat{y}_\varepsilon(0),\hat{d}_\varepsilon(0))\in
\mathcal{H}\times \mathbb{C}^2$, the
 closed-loop of system \eqref{01} as follows:
\begin{equation}\label{closedadrc1}
\begin{gathered}
 \dot{X}(t)=AX(t)+Bu(0,t),\quad t>0,  \\
 u_{t}(x,t)=-ju_{xx}(x,t),\quad x\in (0,1), \; t>0,  \\
 u_x(0,t)=0,  \\
 u_x(1,t)=U(t)+d(t), \\
\dot{\hat{y}}_{\epsilon}(t)= -\operatorname{sat}(\hat{d}_{\epsilon}(t))
 +\hat{d}_{\epsilon}(t)+y_1(t)
 - \frac{1}{\epsilon}(\hat{y}_{\epsilon}-y_0), \\
\dot{\hat{d}}_{\epsilon}(t)=-\frac{1}{\varepsilon^{2}}\{ \hat{y} _{\epsilon}-y_0\},
\end{gathered}
\end{equation}
admits a unique solution $(X,u,\hat{y}_\varepsilon,
\hat{d}_\varepsilon)^\top\in C(0,\infty;\mathcal{H}\times \mathbb{C}^2)$, and
$$
\lim_{t\to \infty,\, \varepsilon\to 0} \|(X(t),u(\cdot,t),\hat{y}_\varepsilon
 (t),\hat{d}_\varepsilon(t)-d(t))\|_{\mathcal{H} \times \mathbb{C}^2}=0,
$$
where the feedback control is
\begin{equation}
\label{closedadrccontrol}
\begin{aligned}
 U(t)&=\int_0^1q_x(1,x)u(x,t)dx  +\gamma'(1)+k(1,1)w(1,t)\\
&\quad +\int_0^1k_x(1,x)w(x,t)dx-\operatorname{sat}(\hat{d}_\varepsilon(t)),
\quad t\geqslant 0,
\end{aligned}
\end{equation}
and
\begin{gather*}
 y_0(t)=-j\int_{0}^{1} (2x^{3}-3x^{2}) z(x,t) dx, \\
 y_1(t)=\int_{0}^{1}(-12x+6+2cjx^3-3cjx^2)z(x,t) dx,\\
 z(x,t) = w(x,t)-\int_{0}^{x}k(x,y)w(y,t)dy, \\
 k(x,y)=-cxj\frac{I_{1}(\sqrt{cj(x^{2}-y^{2})})}{\sqrt{cj(x^{2}-y^{2})}}, \\
 w(x,t) = u(x,t)-\int_{0}^{x}q(x,y)u(y,t)dy-\gamma(x)X(t), \\
 q(x,y)=\int_{0}^{x-y}j\gamma(\sigma)Bd\sigma.
 \end{gather*}
\end{theorem}

\section{Time varying high gain estimator based feedback} \label{sec5}

In this section, we stabilize system \eqref{2.10} by
ADRC with a time varying high gain state feedback disturbance estimator
which is different from that in Section \ref{sec4}.
The advantage of using the disturbance estimator by time varying high gain
lies in four aspects:
\begin{itemize}
\item[(a)] the stability of system \eqref{closedadrc} we obtain in Theorem \ref{Th4.1}
 is irrelevant to gain $\epsilon$,
it is different from Theorem \ref{Th3.1};

\item[(b)] the boundedness of derivative of
disturbance is relaxed in some extent by choosing properly the
time varying gain;

\item[(c)] the peaking value is reduced significantly;

\item[(d)] the possible non-smooth control \eqref{22} becomes smooth.
\end{itemize}

Now, we design the following extended state observer with time varying
high gain for  $y_0(t)$ and $d(t)$:
 \begin{equation}\label{TVS}
\begin{gathered}
\dot{\hat{y}}(t)=U_0(t)+\hat{d}(t)+y_1(t)-g(t)[\hat{y}(t)-y_0(t)],\\
\dot{\hat{d}}(t)=-g^2(t)[\hat{y}(t)-y_0(t)],
\end{gathered}
\end{equation}
where $g \in C^1[0, \infty)$ is a time varying gain real value function
satisfying
\begin{equation}\label{gain}
\begin{gathered}
g(t)>0, \quad  \dot{g}(t)>0, \quad \forall t\geq 0,\\
g(t) \to \infty \quad\text{as } t\to \infty, \quad
\sup_{t\in[0,\infty)}\big|\frac{\dot{g}(t)}{g(t)}\big|<\infty.
\end{gathered}
\end{equation}
In addition, we assume that the disturbance $d(t)\in
H^1_{loc}(0,\infty)$ satisfies
\begin{equation}\label{gain2}
 \lim_{t\to \infty} \frac{|\dot{d}(t)|}{g(t)}=0.
\end{equation}
By \eqref{gain2}, $\dot{d}(t)$ is allowed to grow exponentially at any rate by
choosing properly the gain function $g(t)$. This relaxes the
condition in Section \ref{sec4} where $\dot{d}(t)$ is
assumed to be uniformly bounded.
We use $\hat{d}(t)$ to estimate $d(t)$, then the
convergence is stated in the following lemma.

\begin{lemma}\label{Le4.1}
Let $(\hat{y},\hat{d})$ be the solution of \eqref{TVS}. Then
\begin{equation}
 \lim_{t\to \infty}|\hat{y}(t)-y(t)|=0, \quad
\lim_{t\to \infty}|\hat{d}(t)-d(t)|=0.
\end{equation}
\end{lemma}

\begin{proof}
Set
\begin{equation}\label{liu2}
 \tilde{y}(t)=g(t)[\hat{y}(t)-y_0(t)], \quad
 \tilde{d}(t)=\hat{d}(t)-d(t).
\end{equation}
Then the error $(\tilde{y},\tilde{d})$ is governed by
\begin{equation}\label{error-system}
\begin{gathered}
\dot{\tilde{y}}(t)=-g(t)[\tilde{y}(t)-\tilde{d}(t)]
+\frac{\dot{g}(t)}{g(t)}\tilde{y}(t),\\
\dot{\tilde{d}}(t)=-g(t)\tilde{y}(t)-\dot{d}(t).
\end{gathered}
\end{equation}
The existence of the local classical solution to \eqref{error-system}
is guaranteed by the local Lipschitz condition of the right side
of \eqref{error-system}.
We consider the stability of this ODE. To this end,
we introduce the following Lyapunov function (in addition,
 the global solution is ensured by the following Lyapunov function argument).
 Define
 \begin{gather}\label{lya}
 E(t)=|\tilde{y}(t)|^2+\frac{3}{2} |\tilde{d}(t)|^2, \quad
 \rho(t)=\tilde{y}(t)\overline{\tilde{d}(t)}, \\
V(t)= E(t)-\operatorname{Re}\rho(t). \nonumber
 \end{gather}
Then
\begin{equation}\label{lya1}
 \frac{1}{2}E(t)\leq V(t) \leq 2E(t).
\end{equation}
Differentiating $E(t)$ and $\rho(t)$ with respect to $t$, we obtain
\begin{align*}
\dot{E}(t)
&=2\operatorname{Re}[\overline{\tilde{y}(t)}\dot{\tilde{y}}(t)]
 +3\operatorname{Re}[\overline{\tilde{d}(t)}\dot{\tilde{d}}(t)]\\
&=-2g(t)|\tilde{y}(t)|^2+2\frac{\dot{g}(t)}{g(t)}|\tilde{y}(t)|^2
 -g(t)\operatorname{Re}[ \overline{\tilde{y}(t)} \tilde{d}(t)]
 -3\overline{\tilde{d}(t)}\dot{d}(t),
\end{align*}
 and
\begin{align*}
\dot{\rho}(t)
&=\dot{\tilde{y}}(t)\overline{\tilde{d}(t)}
 +\tilde{y}(t)\overline{\dot{\tilde{d}}(t)}\\
&=-g(t)|\tilde{y}(t)|^2+g(t)|\tilde{d}(t)|^2
-g(t) \tilde{y}(t)\overline{\tilde{d}(t) }+
\frac{\dot{g}(t)}{g(t)}\tilde{y}(t) \overline{\tilde{d}(t) }
-\tilde{y}(t)\overline{\dot{d}(t)}.
 \end{align*}
Then
 \begin{equation}\label{lya2}
\begin{aligned}
\dot{V}(t)
&=[-g(t)+\frac{2\dot{g}(t)}{g(t)}]|\tilde{y}(t)|^2
 -g(t)|\tilde{d}(t)|^2-\frac{\dot{g}(t)}{g(t)}\operatorname{Re}\rho(t)\\
&\quad +\operatorname{Re}
 [-3\dot{d}(t)\overline{\tilde{d}(t)}+
 \tilde{y}(t)\overline{\dot{d}(t)}]
\\
&\leq -\frac{1}{2}\kappa(t)V(t)+m_0|\dot{d}(t)|\|(\tilde{y}(t),\tilde{d}(t))\|\\
&\leq -\frac{1}{2}\kappa(t)V(t)+m_0|\dot{d}(t)|\sqrt{V(t)},
\end{aligned}
\end{equation}
where $m_0$ is a constant,
\begin{equation}\label{k(t)}
 \kappa(t)=g(t)-\sup_{t\in[0,\infty)}
|\frac{3\dot{g}(t)}{g(t)}|\to \infty \quad \text{as } t\to \infty,
 \end{equation}
and there exists $t_0>0$ such that
$$
 \kappa(t)>0, \quad \forall  t\geq  t_0.
$$
This, together with \eqref{lya2}, yields
 \begin{equation}
 \frac{d\sqrt{V(t)}}{dt}\leq -\frac{1}{4}\kappa(t)\sqrt{V(t)}
+\frac{m_0}{2}|\dot{d}(t)|, \quad
 \forall  t\geq t_0.
 \end{equation}
We deduce that
\begin{equation}\label{liu1}
 \sqrt{V(t)} \leq \sqrt{V(t_0)}e^{-\frac{1}{4}\int_{t_0}^t\kappa(s)ds}
 + e^{-\frac{1}{4}\int_{t_0}^t\kappa(s)ds}\frac{m_0}{2}
 \int_{t_0}^t|\dot{d}(s)|e^{\frac{1}{4}\int_{t_0}^s\kappa(\tau)d\tau}ds.
 \end{equation}
The first term on the right-hand side of \eqref{liu1} is obviously
convergent to zero as $t\to \infty$ owing to \eqref{k(t)}.
Applying the L'Hospital rule to the second term on the right-hand side
of \eqref{liu1}, we obtain
 \begin{equation}\label{L'Hospital rule}
\begin{aligned}
 \lim_{t\to\infty}
 \frac{ m_0\int_{t_0}^t|\dot{d}(s)|e^{\frac{1}{4}\int_{t_0}^s\kappa(\tau)d\tau}ds}
 { 2 e^{\frac{1}{4} \int_{t_0}^t\kappa(s)ds}}
 &=\lim_{t\to\infty}
 \frac{m_0}{2 }\frac{|\dot{d}(t)|e^{\frac{1}{4}\int_{t_0}^t\kappa(\tau)d\tau}}
 { \frac{1}{4}e^{\frac{1}{4} \int_{t_0}^t\kappa(s)ds}\kappa(t)} \\
 &=\lim_{t\to\infty}
 2m_0\frac{|\dot{d}(s)|}
 { g(t)}\cdot\frac{g(t)}{\kappa(t)}=0,
\end{aligned}
 \end{equation}
which implies
$\lim_{t\to \infty}\sqrt{V(t)}=0$; that is,
\begin{equation}
 \lim_{t\to \infty }[|\tilde{y}(t)|^2+|\tilde{d}(t)|^2]=0.
\end{equation}
The proof is complete.
\end{proof}

By Lemma \ref{Le4.1}, we design the feedback control
\begin{equation}\label{newcontrol}
U_0(t)=-\hat{d}(t),
\end{equation}
then we can rewrite  the closed-loop of system \eqref{2.10} as
\begin{equation}\label{closed-loop}
\begin{gathered}
 \dot{X}(t) =  (A+BK)X(t)+Bz(0,t),\\
 z_{t}(x,t) =  -jz_{xx}(x,t)-cz(x,t),\\
 z_x(0,t) =  0,\\
 z_x(1,t) =  -\hat{d}(t)+d(t),\\
 \dot{\hat{y}}(t) =  y_1(t)-g(t)[\hat{y}(t)-y_0(t)],\\
 \dot{\hat{d}}(t) =  -g^2(t)[\hat{y}(t)-y_0(t)].
\end{gathered}
\end{equation}

\begin{proposition}\label{Pro4.1}
Assume that the time varying gain $g(t)\in C^1[0, \infty)$ satisfies
\eqref{gain} and the disturbance $d(t)\in H^1_{loc}(0,\infty)$
 satisfies \eqref{gain2}.
Then for any initial value
$(X(0),z(0),\hat{y}(0), \hat {d}(0)) \in \mathcal{H} \times \mathbb{C}^2$,
 there exists a unique solution
$(X,z,\hat{y},\hat{d})\in C(0,\infty; \mathcal{H} \times \mathbb{C}^2 )$ to
 system \eqref{closed-loop} and system \eqref{closed-loop} is asymptotically
stable in the sense that
 $$
 \lim_{t\to \infty}\|(X(t),z(\cdot,t),\hat{y}(t),
\hat{d}(t)-d(t))\|_{\mathcal{H} \times \mathbb{C}^2}=0
 $$
\end{proposition}

\begin{proof}
By the error variables $(\tilde{y},\tilde{d})$ defined in
\eqref{liu2}, we have the following equivalent system for
system \eqref{closed-loop},
\begin{equation}\label{closed-loop2}
\begin{gathered}
\dot{X}(t) =  (A+BK)X(t)+Bz(0,t),\\
 z_{t}(x,t) =  -jz_{xx}(x,t)-cz(x,t),\\
 z_x(0,t) =  0,\\
 z_x(1,t) =  -\hat{d}(t)+d(t),\\
\dot{\tilde{y}}(t) =-g(t)[\tilde{y}(t)-\tilde{d}(t)]
 +\frac{\dot{g}(t)}{g(t)}\tilde{y}(t),\\
\dot{\tilde{d}}(t) =-g(t)\tilde{y}(t)-\dot{d}(t).
\end{gathered}
\end{equation}
 The ``ODE part'' of \eqref{closed-loop2} is just the system
\eqref{error-system}, which is shown to be convergent by Lemma
\ref{Le4.1}. The ``$(X,z)$ part'' of \eqref{closed-loop2} is similar
to \eqref{2.10}, hence the proof becomes similar to the proof of
Lemma \ref{L3.1}.
 The details are omitted.
\end{proof}

Returning  to system \eqref{01} by the inverse transformations
\eqref{021} and \eqref{inverse2},  we prove
the following theorem.

\begin{theorem}\label{Th4.1}
 Assume that the time varying gain $g(t)\in C^1[0, \infty)$ satisfies
\eqref{gain} and the disturbance $d(t)\in H^1_{loc}(0,\infty)$
 satisfies \eqref{gain2}. Then for any initial value
$(X(0),u(\cdot,0),\hat{y}(0),\hat{d}(0))\in
\mathcal{H}\times \mathbb{C}^2$, the
 closed-loop of system
\begin{equation}\label{closedadrc}
\begin{gathered}
\dot{X}(t) =AX(t)+Bu(0,t),\quad t>0\\
u_{t}(x,t) =-ju_{xx}(x,t),\quad x\in (0,1), \; t>0,\\
u_x(0,t) =0,\\
\begin{aligned}
u_x(1,t) &=-\hat{d}(t)+\int_0^1q_x(1,y)u(y,t)dy+\gamma'(1)+k(1,1)w(1,t)\\
&\quad + \int_0^1k_x(1,y)w(y,t)dy +d(t),
\end{aligned}\\
\dot{\hat{y}}(t)  = y_1(t)-g(t)[\hat{y}(t)-y_0(t)],\\
 \dot{\hat{d}}(t)  = -g^2(t)[\hat{y}(t)-y_0(t)],
\end{gathered}
\end{equation}
admits a unique solution
$(X,u,\hat{y},\hat{d})^\top\in C(0,\infty;\mathcal{H}\times \mathbb{C}^2)$, and
system \eqref{closedadrc} is asymptotically stable
$$
\lim_{t\to \infty}
\|(X(t),u(\cdot,t),\hat{y}(t),\hat{d}(t)-d(t))\|_{\mathcal{H}\times \mathbb{C}^2}=0,
$$
where
\begin{gather*}
y_0(t)=-j\int_{0}^{1} (2x^{3}-3x^{2}) z(x,t) dx, \\
y_1(t)=\int_{0}^{1}(-12x+6+2cjx^3-3cjx^2)z(x,t) dx, \\
z(x,t) = w(x,t)-\int_{0}^{x}k(x,y)w(y,t)dy, \\
k(x,y)=-cxj\frac{I_{1}(\sqrt{cj(x^{2}-y^{2})})}{\sqrt{cj(x^{2}-y^{2})}},\\
w(x,t) = u(x,t)-\int_{0}^{x}q(x,y)u(y,t)dy-\gamma(x)X(t), \\
q(x,y)=\int_{0}^{x-y}j\gamma(\sigma)Bd\sigma.
\end{gather*}
\end{theorem}


\section{Numerical simulation}\label{sec6}


\begin{figure}[htb]
\begin{center}
\subfigure[Real part]{\label{Fig.1a}
 \includegraphics[width=6cm]{fig2a}} %  condisre
\subfigure[Imaginary part]{\label{Fig.1b}
 \includegraphics[width=6cm]{fig2b}} % condisim
\end{center}
\caption{The $\hat{d}(t)$ and disturbance $d(t)=2\sin(2t)+2j\cos(2t)$
 by constant high gain} \label{fig2}
\end{figure}

\begin{figure}[htb]
\begin{center}
\subfigure[Real part ]{\includegraphics[width=6cm]{fig3a}} % gtdisre
\subfigure[Imaginary part]{\includegraphics[width=6cm]{fig3b}} % gtdisim
\end{center}
\caption{The $\hat{d}(t)$ and disturbance $d(t)=2\sin(2t)+2j\cos(2t)$
by time varying gain} \label{fig3}
\end{figure}

\begin{figure}[htb]
\begin{center}
\subfigure[Real part]{\includegraphics[width=6cm]{fig4a}} % conode
\subfigure[Imaginary part]{\includegraphics[width=6cm]{fig4b}} % gtode
\end{center}
\caption{The ODE state $X(t)$ for systems \eqref{closedadrc1}
and \eqref{closedadrc}} \label{fig4}
\end{figure}

\begin{figure}[htb]
\begin{center}
\subfigure[Real part ]{\includegraphics[width=6cm]{fig5a}} % conpdere
\subfigure[Imaginary part]{\includegraphics[width=6cm]{fig5b}} % conpdeim
\end{center}
\caption{The Schr\"odinger sate $u(x,t)$ with constant high gain controllers}
\label{fig5}
\end{figure}

\begin{figure}[htb]
\begin{center}
\subfigure[Real part ]{\includegraphics[width=6cm]{fig6a}} % gtpdere
\subfigure[Imaginary part]{\includegraphics[width=6cm]{fig6b}} % gtpdeim
\par\end{center}
\caption{The Schr\"odinger sate $u(x,t)$ with time varying high gain
 controllers} \label{fig6}
\end{figure}

\begin{figure}[htb]
\begin{center}
\subfigure[Real part ]{\includegraphics[width=6cm]{fig7a}} % undisre
\subfigure[Imaginary part]{\includegraphics[width=6cm]{fig7b}} % undisim
\end{center}
\caption{The $\hat{d}(t)$ and the disturbance 
$d(t)=2\sin(t^{3/2})+2j\cos(t^{3/2})$
by time varying gain} \label{fig7}
\end{figure}

\begin{figure}[htb]
\begin{center}
\subfigure[Real part ]{\includegraphics[width=6cm]{fig8a}} % unpdere
\subfigure[Imaginary part]{\includegraphics[width=6cm]{fig8b}} % unpdeim
\end{center}
\caption{The Schr\"odinger sate $u(x,t)$ for system \eqref{closedadrc}} 
\label{fig8}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=6cm]{fig9} % unode
\end{center}
\caption{The ODE state $X(t)$ for system \eqref{closedadrc}}\label{fig9}
\end{figure}

In this section, we present some numerical simulations to show
visually the effectiveness of the proposed controllers for
systems \eqref{closedadrc1} and \eqref{closedadrc} respectively.
We choose the initial values and the parameters as following:
$u(x,0)=-x+3x j$, $A=2$, $ B=-6$,
$K=4$, $\epsilon=0.01$, $g(t)=10+12t^2$.
To estimate the unknown disturbance
$d(t)$, we assume  that the time varying gain
function $g(t)$ satisfies assumption \eqref{gain}. However,
from the practice standpoint,
the increasing $g(t)$ can not be applied in extended
time interval.
 A recommended scheme
is to use the time varying gain first to reduce the peaking
value in the initial stage to a reasonable level and
then use the constant high gain. To this end, we take the gain function
\begin{equation}\label{value}
 \hat{g}(t)= \begin{cases}
 g(t), & t\leq t_0,  \\
 g(t_0), & t> t_0,
 \end{cases}
\end{equation}
where $t_0>0$ and $\epsilon=1/g(t_0)$.
Using combined varying gain
\eqref{value} with $t_0\simeq 2.8$.

On the one hand, the disturbance is taken as $d(t)=2\sin(2t)+2j\cos(2t)$.
It is seen that with the constant high gain, the peaking value is observed
in the initial stage in Figure \ref{fig2},
whereas with the time varying gain, the peaking value
is dramatically reduced as shown in Figure \ref{fig3}.
This is an advantage of the time varying gain approach.
The price is that the convergence by the time varying gain is
slightly slow which is observed from Figure \ref{fig3}.
The ODE state $X(t)$ of the
systems \eqref{closedadrc1} and \eqref{closedadrc} are shown in Figure \ref{fig4}(a)
and Figure \ref{fig4}(b), respectively.
The PDE part of solutions of systems \eqref{closedadrc1} and \eqref{closedadrc}
are plotted in Figure \ref{fig5} and Figure \ref{fig6} respectively.

On the other hand, the disturbance is taken as $d(t)=2\sin(t^{3/2})+2j\cos(t^{3/2})$.
The solutions of system \eqref{closedadrc} are plotted in
Figure \ref{fig7}, \ref{fig8}, \ref{fig9}, respectively.
In spite of the derivative of disturbance is unbounded, we see that
convergence of the state is satisfactory with the time varying gain approach.
This is another advantage of the
time varying gain approach.


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


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