\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada.
\newline
{\em Electronic Journal of Differential Equations},
Conference 12, 2005, pp. 159--169.
\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{159}
\begin{document}
\title[\hfilneg EJDE/Conf/12 \hfil Approximate controllability of
distributed systems]
{Approximate controllability of distributed systems by distributed controllers}
\author[B. Shklyar, V. Marchenko \hfil EJDE/Conf/12\hfilneg]
{Benzion Shklyar, Vladimir Marchenko}
\address{Benzion Shklyar\hfill\break
Department of Sciences,
Holon Academic Institute of Technology, Holon,
Israel}
\email{shk\_b@hait.ac.il}
\address{Vladimir Marchenko\hfill\break
Department of Mathematics,
University of Bialystok, Poland}
\email{vmar@bstu.unibel.by}
\date{}
\thanks{Published April 20, 2005.}
\subjclass[2000]{93B05}
\keywords{Approximate controllability; abstract boundary control problems;
\hfill\break\indent
evolution equations; linear differential control systems with delays;
\hfill\break\indent
partial differential control equations}
\begin{abstract}
Approximate controllability problem for a linear distributed control system
with possibly unbounded input operator, connected in a series to another
distributed system without control is investigated. An initial state of the
second distributed system is considered as a control parameter.
Applications to control partial equations governed by hyperbolic controller,
and to control delay systems governed by hereditary controller are considered.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\section{Statement of the Problem}
Research in control theory started for single control systems. However, many
technical applications use control systems interconnected in many ways. The
goal of the present paper is to establish approximate controllability
conditions for a control system interconnected in a series with a second
homogeneous system without control in such a way that a control function of
the first control system is an output of the second one, so a control is
considered as an initial state of a second system.
Let $X,U,Z$ be Hilbert spaces, and let $A,C$ be infinitesimal generators of
strongly continuous $C_{0}$-semigroups $S_{A}(t) $ in $X$ and $S_{C}(t) $ in
$Z$ correspondingly in the class $C_{0}$ \cite{Hille&Philips,Krein}.
Consider the abstract evolution control equation
\begin{equation} \label{2.1}
\begin{gathered} \dot{x}(t) = Ax(t) +Bu(t) ,\quad x(0) =x_{0}, \\ u(t) =
Kz(t) ,\quad 0\leq t<+\infty , \end{gathered}
\end{equation}
where $z(t)$ is a mild solution of another evolution equation of the form
\begin{equation}
\dot{z}(t) =Cz(t) ,\quad z(0) =z_{0},\quad 0\leq t<+\infty . \label{2.2}
\end{equation}
Here $x(t)$, $x_{0}\in X$, $u(t) \in U$, $z(t) ,z_{0}\in Z$, $B:U\to X$ is a
linear possibly unbounded operator, $K:Z\to U$ is a linear possibly unbounded onto
operator.
Equation (\ref{2.2}) is said to be a controller equation. A control $u(t) $
is defined by $u(t) =Kz(t) $ as an output of controller equation (\ref{2.2}).
Let $x(t,0,x_{0},u(\cdot ) ) $ be a mild solution of (\ref{2.1}) with the
initial condition $x(0) =x_{0}$, and let $u(t,0,z_{0}) =Kz(t,0,z_{0}) $,
where $z(t,0,z_{0}) $ is a mild solution of equation (\ref{2.2}) with the
initial condition $z(0) =z_{0}$.
The initial data $z_{0}\in Z$ of equation (\ref{2.2}) is considered as a control.
Let $\mu \notin \sigma _{A}$. We will consider the spaces $W$ and $V$
defined as follows: $W$ is the domain of the operator $A$ with the norm $\|
x\| _\mu=\| (\mu I-A) x\| $; $V$ is the closure of $X$ with respect to the
norm $\| x\|_{-\mu }=\| (R_{A}(\mu ) x)\| $, where $R_{A}(\mu ) =(\mu I-A)
^{-1}$.
Obviously $W\subset X\subset V$ with continuous dense imbeddings. The
following facts are well known, see for example \cite%
{Hille&Philips,Lasieska,Krein,Pazy,Salamon,Weiss},
\begin{itemize}
\item For each $t\geq 0$ the operator $S_A(t)$ has a unique continuous
extension $\mathcal{S}_\mathcal{A}(t)$ on the space $V$. The family of
operators $\mathcal{S}_\mathcal{A}(t):V\to V$ is the semigroup in the class
$C_{0}$ with respect to the norm of $V$. The corresponding infinitesimal
generator $\mathcal{A}$ of the semigroup $\mathcal{S}_\mathcal{A}(t)$ is
the closed dense extension of the operator $A$ on the space $V$ with domain
$D(\mathcal{A})=X$.
\item The sets of the generalized eigenvectors of operators $\mathcal{A},%
\mathcal{A}^{\ast }$ and $A,\,A^{\ast }$ are the same.
\item For each $\mu \notin \sigma _{A}$ the operator $R_{A}(\mu)$ has a
unique continuous extension to the operator $\mathcal{R}_{A}(\mu ) :V\to X$.
\item A mild solution $x(t,0,x_{0},u(\cdot ) ) $ of (\ref{2.1}) with initial
condition (\ref{2.2}) is defined by the representation formula
\begin{equation}
x(t,0,x_{0},u(\cdot ))=S(t)x_{0}+\int_{0}^{t}\mathcal{S}(t-\tau )Bu(\tau
)d\tau , \label{2.3}
\end{equation}
where the integral in (\ref{2.3}) is understood in the Bochner sense \cite%
{Hille&Philips} with respect to the topology of $V$.
\end{itemize}
%We say that the control $u(\cdot ) \in L_{2}^{\mathrm{loc}}([ 0,+\infty )
%,U) $ is vanishing after time moment $t_{2}$ if $u(t) =0$ a.e. on $[
%t_{2},+\infty)$.
Denote
\begin{equation}
u_{t_{2}}(t,0,z_{0}) =%
\begin{cases}
u(t,0,z_{0}) & \mbox{if }0\leq t\leq t_{2}, \\
0 & \mbox{if } t>t_{2}.%
\end{cases}
\label{2.4}
\end{equation}
%\label{Basic definitions}
\begin{definition} \label{D2.1} \rm
Equation (\ref{2.1}) is said to be approximately controllable
on $[ 0,t_{1}] $ in the class of controls vanishing after time
moment $t_{2},00$
there exists a control $u(\cdot) \in L_{2}([ 0,t_{2}] ,U) ,u(t) =0$
a.e. on $[ t_{2},+\infty ) $, such that
\[
\| x_{1}-x(t_{1},0,0,u(\cdot ) ) \| <\varepsilon .
\]
\end{definition}
\begin{definition} \label{D2.2} \rm
Equation (\ref{2.1}) is said to be approximately controllable
on $[ 0,t_{1}] $ by controller (\ref{2.2}) if for each $x_{1}\in
X $ and $\varepsilon >0$ there exists $z_{0}\in Z$, such that
\[
\| x_{1}-x(t_{1},0,0,u_{t_{2}}(\cdot ,0,z_{0}) ) \| <\varepsilon .
\]
\end{definition}
\section{Assumptions}
\label{theassumptions}
\begin{enumerate}
\item The operators $A$ and $C$ have purely point spectrum $\sigma _{A}$ and
$\sigma _{C}$ with no finite limit points. Eigenvalues of both $A$ and $C $
have finite multiplicities.
\item Let the spectrum $\sigma _{A}$ of the operator $%
A $ be infinite and consists of numbers $\lambda _{j}$, $j=1,2,\dots$, with
multiplicities $\alpha_{j}$, enumerated in such a way that their absolute
values are non-decreasing with respect to $j$
(i.e. $\vert \lambda _{j}\vert\geq \vert \lambda _{j+1}\vert $).
The sequence
\begin{equation}
t^{k}\exp \lambda _{j}t,\quad j=1,2,\dots ,\quad k=0,\dots ,\alpha _{j}-1
\label{2.6}
\end{equation}
is minimal on $[0,\delta ]$ for some $\delta >0$, i. e., there exists a
sequence biorthogonal to the above sequence with respect to the scalar
product in $L_{2}[0,\delta ]$.
\item There exists $T\geq 0$ such that all mild solutions of the equation $%
\dot{x}(t) =Ax(t) $ are expanded in a series of generalized eigenvectors of
the operator $A$ converging (in the topology of $X$) for any $t>T$ uniformly
in each segment $[T_{1},T_{2}] ,T0$ and
$(Kz)(\cdot )\in L_{2}([0,t_{1}],U),\forall t_{1}>0$.
The operator
\begin{equation*}
Q:Z\to L_{2}([0,t_{1}],U),\quad Qz=u(t),\quad t\in [0,t_{1}]
\end{equation*}
is bounded for all $t_{1}>0$.
\end{enumerate}
\section{Main results}
Denote
\begin{align*}
&\mathop{\rm Range}\{\lambda I-A,\mathcal{R}_{A}(\mu ) B\} \\
&=\{y\in X:\exists x\in X,\,\exists u\in U,\,\mathrm{\,}y=(\lambda I-A)x+
\mathcal{R}_{A}(\mu ) Bu\}.
\end{align*}
\begin{theorem} \label{T3.1}
For equation \eqref{2.1} to be approximately controllable on $[ 0,t_{1}]$,
$t_{1}>T+\delta$, in the class of controls vanishing after time
moment $t_{1}-T$, it is necessary and sufficient that
\begin{enumerate}
\item The linear span of the generalized eigenvectors of the operator $A$
is dense in $X$.
\item The condition
\begin{equation}
\overline{\mathrm{Range}\{\lambda I-A,\mathcal{R}_{A}(\mu ) B\}}
=X,\quad \forall \lambda \in \sigma _{A},\;\forall \mu \notin \sigma _{A},
\label{3.1}
\end{equation}
holds.
\end{enumerate}
\end{theorem}
\begin{theorem}\label{T3.2}
For equation \eqref{2.1} to be approximately controllable on
$[ 0,t_{1}]$, $t_{1}>T$ by distributed controller \eqref{2.2}, it is necessary
that all the conditions of Theorem \ref{T3.1} hold.
If these conditions hold and the subspace $KS_{C}(\cdot ) Z$ of
$L_{2}([ 0,t_{2}] ,U)$ is dense in $L_{2}([0,t_{2}] ,U) $ for some $t_{2}>0$,
then equation \eqref{2.1} is approximately controllable on $[ 0,t_{1}]$,
$t_{1}>T+\delta$, by controller \eqref{2.2}.
\end{theorem}
\section{Approximate controllability of abstract boundary control problem by
abstract boundary controller}
Let $X,U,Z,Y_{1},Y_{2}$ be Hilbert spaces. Consider the abstract boundary
control problem
\begin{equation} \label{4.1}
\begin{gathered} \dot{x}(t) =Lx(t) , \\ \Gamma x(t) =Bu(t) , \\ x(0) =x_{0},
\\ u(t) =Kz(t) , \end{gathered}
\end{equation}
where $z(t) $ is a mild solution of the boundary-value problem
\begin{gather}
\dot{z}(t) =Mz(t) , \label{4.2} \\
Hz(t) =0, \notag \\
z(0) =z_{0}. \label{4.3}
\end{gather}
Equation (\ref{4.2})-(\ref{4.3}) is called boundary controller.
Here $L:X\to X$ and $M:Z\to Z$ are linear unbounded operators with dense
domains $D(L) $ and $D(M);B:U\to Y_{1}$ is a linear bounded one-to-one
operator, $K:Z\to U~$is a linear (possibly unbounded) onto operator, $%
\Gamma:X\to Y_{1}$ and $H:Z\to Y_{2}$ are linear operators satisfying the
following conditions:
\begin{enumerate}
\item $\Gamma $ and $H$ are onto, $\ker \Gamma $ is dense in $X,\ker H$ is
dense in $Z$.
\item There exists a $\mu \in \mathbb{R}$ such that $\mu I-L$ is onto and $%
\ker (\mu I-L) \cap $ $\ker \Gamma =\{0\} $.
\item There exists a $\mu \in \mathbb{R}$ such that $\mu I-M$ is onto and $%
\ker (\mu I-M) \cap \ker H=\{ 0\} $.
\end{enumerate}
Problems (\ref{4.1}) and (\ref{4.2})-(\ref{4.3}) are assumed to be
well-posed. Problem (\ref{4.1}) is an abstract model for classical control
problems described by linear partial differential equations of both
parabolic and hyperbolic type when a control acts through the boundary. The
control process is released by initial condition (\ref{4.3}) which is
considered as a control.
Now consider the space $W_{1}=\ker \Gamma $. We have $W_1\subset D(L)\subset X$ with
continuous dense injection. Define the operator $A:W_{1}\to X$ by
\begin{equation}
Ax=Lx\quad \text{for }x\in W_{1}. \label{4.4}
\end{equation}
For $y\in Y_{1}$ define
\begin{equation}
\hat{B}y=Lx-Ax,\,x\in \Gamma ^{-1}(y) =\{z\in D(L) :\Gamma x=y\}.
\label{4.5}
\end{equation}
Given $u\in U$ denote $\tilde{B}u=\hat{B}Bu$. The operator $B:U\to V$ is
bounded, but the operator $\hat{B}:Y_{1}\to X$ defined by (\ref{4.5}) is
unbounded, so the operator $\tilde{B}:U\to X$ is unbounded. It follows from (%
\ref{4.5}) that
\begin{gather}
Lx=Ax+\tilde{B}u, \label{4.6} \\
\Gamma x=Bu. \label{4.7}
\end{gather}
The same way is applied to the space $W_{2}=\ker H$. Again, we have $W_{2}\subset
D(M) \subset Z$ with continuous dense injection. Define the operator $%
C:W_{2}\to Z$ by
\begin{equation}
Cz=Mz\quad \text{for }z\in W_{2}. \label{4.8}
\end{equation}
Hence
\begin{equation} \label{4.9}
\begin{gathered} \dot{z}(t) =Cz(t), \\ z(0) = z_{0}, \end{gathered}
\end{equation}
We assumed all the hypotheses in section \ref{theassumptions} for the above
operators $A,C,K$ hold true. Together with equation (\ref{4.1}) consider
the abstract boundary-value problem
\begin{gather}
Lx =\mu x, \label{4.10} \\
\Gamma x =y . \label{4.11}
\end{gather}
Since problem (\ref{4.1}) is uniformly well-posed then for any $y\in Y_{1}$
there exists the solution $x_{\mu }=D_{A}(\mu ) y$ of equation
(\ref{4.10})-(\ref{4.11}), where $D_{A}(\mu ) :Y_{1}\to X$ is a linear bounded
operator (The operator $D_{\mu }$ is defined by well-known Green formula for
given boundary problem).
The next theorems follow from Theorems \ref{T3.1}-\ref{T3.2}.
\begin{theorem} \label{T4.1}
For equation \eqref{4.1} to be approximately controllable on $[ 0,t_{1}]$, $t_{1}>T+\delta $,
in the class of controls vanishing after time moment $t_{1}-T$, it is necessary
and sufficient that
\begin{enumerate}
\item The linear span of the generalized eigenvectors of the operator $A$
(i.e. eigenfunctions of the boundary problem $Lx=\lambda x$, $Gx =0$)
is dense in $X$
\item \begin{equation} \label{4.12}
\overline{\mathrm{Range}\{\lambda I-A,R_{A}(\mu ) \hat{B}B\}}
=X, \quad \forall \mu \notin \sigma _{A},\; \forall \lambda \in \sigma _{A},
\end{equation}
\end{enumerate}
\end{theorem}
\begin{theorem} \label{T4.2}
For equation \eqref{4.1} to be approximately controllable on
$[ 0,t_{1}] $ by boundary controller \eqref{4.2}-\eqref{4.3}, it is necessary
that
\begin{enumerate}
\item The linear span of the generalized eigenvectors of the operator $A$
is dense in $X$.
\item The condition (\ref{4.12}) holds.
\end{enumerate}
If these conditions hold and the set of functions $u(\cdot ), u(t) =Kz(t) $
with $z(t)$ a solution of boundary-value problem \eqref{4.2}-\eqref{4.3},
is dense in $L_{2}([ 0,t_{2}] ,U) $ for some $t_{2}>0$, then equation
\eqref{4.1} is approximately controllable on $[0,t_{1}] ,t_{1}>T+\delta $,
by boundary controller \eqref{4.2}-\eqref{4.3}.
\end{theorem}
\begin{theorem} \label{T4.3}
For equation \eqref{4.1} to be approximately controllable
on $[ 0,t_{1}] $ by boundary controller \eqref{4.2}-\eqref{4.3}, it is
necessary that
\begin{enumerate}
\item All generalized eigenvectors of the operator $A$ defined by
\eqref{4.4} are dense in $X$.
\item
\begin{equation}
\overline{\mathrm{Range}\{\lambda I-A,D_{A}(\mu ) B\}}
=X,\quad \forall \mu \notin \sigma _{A},\;\forall \lambda \in \sigma _{A}.
\label{4.13}
\end{equation}
\end{enumerate}
If these conditions hold and the set of functions $u(\cdot )$,
$u(t) =Kz(t)$ with $z(t)$ a mild solution of boundary-value
problem \eqref{4.2}-\eqref{4.3}, is dense in $L_{2}([ 0,t_{2}] ,U)$
for some $t_{2}>0$, then equation \eqref{4.1} is approximately controllable
on $[0,t_{1}], t_{1}>T+\delta $, by boundary controller \eqref{4.2}-\eqref{4.3}.
\end{theorem}
\section{Approximate controllability of partial differential
equations by a hyperbolic controller}
The results of the previous section can be applied to the investigation of
approximate controllability of linear partial differential control equation
with boundary control governed by distributed controller described by
partial differential equations.
Consider the parabolic partial differential equation
\begin{equation}
\frac{\partial y}{\partial t}(t,x)=\frac{\partial }{\partial x}
\Big(%
p_{1}(x) \frac{\partial y}{\partial x}(t,x)\Big) +p_{2}(x) y(t,x),
\quad
t\geq 0,\, 0\leq x\leq l, \label{5.1}
\end{equation}
with non-homogeneous regular boundary conditions \cite{Kamke}, \cite{Naimark}
%
\begin{gather}
a_{0}y(t,0)+a_{1}\frac{\partial y}{\partial x}(t,0) =
a_{2}u(t) , t\geq 0,
\label{5.2} \\
b_{0}y(t,l)+b_{1}\frac{\partial y}{\partial x}%
(t,l) = b_{2}u(t) , t\geq 0,
\label{5.3}
\end{gather}
subject to the initial conditions
\begin{equation}
y(0,x) =\varphi _{0}(x) ,\quad 0\leq x\leq l,
\label{5.4}
\end{equation}
where $p_{1}(x) \ $and $p_{2}(x) $ are real
functions, continuous in the
segment $[ 0,l]$;
\begin{gather*}
p_{1}(x)
>0,\quad p_{2}(x) \leq 0,\quad x\in [ 0,l] ; \\
\varphi _{0}(\cdot ) ,\quad
\varphi _{1}(\cdot ) \in L_{2} [ 0,l] ; \\
a_{j},b_{j}\in \mathbb{R}, j=0,1; \\
\vert a_{0}\vert +\vert a_{1}\vert \neq 0,\quad \\
\vert b_{0}\vert +\vert b_{1}\vert \neq 0,\quad \\
a_{0}a_{1}\leq 0,\quad b_{0}b_{1}\geq 0.
\end{gather*}
Here
\begin{equation*}
u(t)
=z(t,\alpha ) ,\quad t\geq 0, \quad \alpha \in [ 0,m] ,
\end{equation*}%
where $z(t,x)$,\quad $t\geq 0$, \quad $0\leq x\leq m$, is a mild solution
of the hyperbolic partial differential equation
\begin{equation}
\frac{%
\partial ^{2}z}{\partial t^{2}}(t,x)=\frac{\partial }{\partial x}%
\Big(%
q_{1}(x) \frac{\partial z}{\partial x}(t,x)\Big) +q_{2}(x) z(t,x),
\quad
t\geq 0,\; 0\leq x\leq m, \label{5.5}
\end{equation}
with
homogeneous regular boundary conditions
\begin{gather}
\alpha
_{0}z(t,0)+\alpha _{1}\frac{\partial z}{\partial x}(t,0) =0,
\label{5.6}
\\
\beta _{0}z(t,m)+\beta _{1}\frac{\partial z}{\partial x}(t,m) =0 \label{5.7}
\end{gather}
subject to the initial conditions
\begin{equation}
z(0,x) =\psi _{0}(x) ,\quad \frac{\partial y}{\partial x}(0,x) =\psi
_{1}(x)
,\quad 0\leq x\leq m, \label{5.8}
\end{equation}
where $q_{1}(x)$
and $q_{2}(x) $ are real functions, continuous in the
segment $[ 0,m]$;
%
\begin{gather*}
q_{1}(x) >0,\quad q_{2}(x) \leq 0,\quad x\in [ 0,m] ;
\\
\psi _{0}(\cdot ) ,\psi _{1}(\cdot ) \in L_{2}[0,m] ; \\
\alpha _{j},\beta _{j}\in \mathbb{R},\quad j=0,1; \\
\vert \alpha _{0}\vert +\vert\alpha_{1}\vert \neq 0,\quad \\
\vert \beta _{0}\vert +\vert\beta_{1}\vert \neq 0,\quad \\
\alpha _{0}\alpha _{1}\leq 0,\beta _{0}\beta _{1}\geq 0.
\end{gather*}
Partial differential equation (\ref{5.5}) with boundary condition
(\ref{5.6})-(\ref{5.7}) will be called a hyperbolic controller.
The pair
\begin{equation*}
(\psi {0}(x) ,\psi _{1}(x) ) ,\quad 0\leq x\leq m,
\end{equation*}
where $\psi _0(x)$ and $\psi _1(x)$ are defined by (\ref{5.8}),
is considered as a control of equation (\ref{5.1})-(\ref{5.3}) governed
by hyperbolic controller (\ref{5.5})-(\ref{5.7}).
One can rewrite equation (\ref{5.1})-(\ref{5.3}) in the form of
(\ref{4.1}) with the state space $X=L_{2}[ 0,l] \times L_{2}[0,l] $;
the corresponding operator $A$ generates a $C_{0}$-semigroup.
By the same way one can rewrite equation (\ref{5.5})-(\ref{5.7}) in the form of
(\ref{4.2})-(\ref{4.3}) with the state space $Z=L_{2}[ 0,m] \times L_{2}[0,m] $;
the corresponding operator $C$ generates a $C_{0}$-semigroup.
Here, conditions 1-4 of
section \ref{theassumptions} are valid for $A$ and $C
$ with $T=0$. The linear span of the eigenvectors of the corresponding selfadjoint
operator $A$ is dense in $L_{2}[ 0,l]$. The eigenvalues of the operator $A$
are negative and the corresponding functions (\ref{2.6}) are minimal on
$[ 0,\delta ]$ for all $\delta >0$ \cite{Fattorini&Russel}. We have
\begin{equation} \label{5.16}
D_{\mu
}Bu=\int_{0}^{l}G(x,\xi ,\mu ) (\omega _{0}(\xi ) a_{2} +\omega
_{l}(\xi )
b_{2}) ud\theta ,
\end{equation}
where $G(x,\xi ,\mu )$ is the Green
function of the boundary value problem
\begin{equation} \label{5.17}
%
\begin{gathered} (p_{1}(x) y'(x)) '+p_{2}(x) y(x) =\mu y(x) ,0\leq x\leq
l,
\\ a_{0}y(0)+a_{1}y'(0) = a_{2}u,\\ b_{0}y(l)+b_{1}y'(l) = b_{2}u,
\end{%
gathered}
\end{equation}
and
\begin{gather}
\omega _{0}(\xi ) =%
%
\begin{cases}
-\frac{p_{1}(0) }{\alpha _{1}}\delta (\xi ) , & \mbox{if }%
a_{1}\neq 0, \\
\frac{p_{1}(0) }{\alpha _{0}}\delta ^{\prime}(\xi) , &
\mbox{if } a_{0}\neq
0,%
\end{cases}
\label{5.18} \\
\omega _{l}(\xi )
=%
\begin{cases}
-\frac{p_{1}(l) }{b_{1}}\delta (\xi -l) , & \mbox{if }
b_{1}\neq 0, \\
\frac{p_{1}(l) }{b_{0}}\delta ^{\prime}(\xi -l) , & \mbox{%
if } b_{0}\neq 0.%
\end{cases}
\label{5.19}
\end{gather}
We have here $U=%
\mathbb{R}^{2}$; the operator $K:Z\mapsto U$ is defined for
given $\alpha
\in [ 0,m] $ by
\begin{equation} \label{5.20}
Kz(\cdot ) =z_{1}(\alpha )
, \quad \forall z(\cdot ) =
\begin{pmatrix}
z_{1}(\cdot ) \\
z_{2}(\cdot )%
\end{pmatrix}
\in L^{2}[ 0,m] \times L^{2}[ 0,m]
\end{%
equation}
\begin{theorem} \label{T5.1}
Condition \eqref{4.12} holds if and only if for each $\lambda $ the
boundary-value problem
\begin{equation}
(p_{1}(x) \varphi ') '+p_{2}(
x) \varphi -\lambda \varphi =0,\quad x\in [
0,l] ,\lambda \in \sigma _{A}
\label{5.21}
\end{equation}
subject to the
boundary conditions
\begin{gather}
a_{0}\varphi (0)+a_{1}\varphi '(0) = 0,
\label{5.22}\\
b_{0}\varphi (l)+b_{1}\varphi '(l) = 0 \label{5.23}%
\\
\int_{0}^{l}\varphi (\xi ) (\omega _{0}(\xi ) a_{2}+\omega _{l}(\xi
)
b_{2}) d\xi =0, \label{5.24}
\end{gather}
has only trivial solution.
%
\end{theorem}
Using
Theorems \ref{T4.1}-\ref{T4.3} and \ref{T5.1} one can prove the
following
statement.
\begin{theorem} \label{T5.2}
Let $\frac{a}{l}$ be an
irrational number.
For \eqref{5.1} to be approximately controllable on $[
0,t_{1}]$,
for all $t_{1}>l$ by boundary controller \eqref{5.2}-\eqref{5.3}%
,
it is necessary and sufficient that for each $\lambda \in \sigma _{A}$
the
boundary-value problem \eqref{5.21} subject the boundary conditions
%
\eqref{5.22}-\eqref{5.23} and
the boundary conditions:
\begin{gather*}%
p(0) \varphi (0) \frac{a_{2}}{a_{1}}+p(l) \varphi (l) \frac{b_{2}}{b_{1}}
= 0,
\quad a_{1}\neq 0 \& b_{1}\neq 0, \\
p(0) \varphi '(0) \frac{a_{2}}{%
a_{0}}
-p(l) \varphi (l) \frac{b_{2}}{b_{1}} =0,\quad
a_{0}\neq0 \&
b_{1}\neq 0, \\
p(0) \varphi (0) \frac{a_{2}}{a_{1}}-p(l) \varphi '(l)
\frac{b_{2}}{b_{0}}
=0,\quad a_{1}\neq 0\ \&b_{0}\neq 0, \\
p(0) \varphi
'(0) \frac{a_{2}}{a_{0}}
+p(l) \varphi '(l) \frac{b_{2}}{b_{0}}=0,
\quad
a_{0}\neq 0\& b_{0}\neq 0,
\end{gather*}
has only trivial
solution.
\end{theorem}
\subsection*{Remarks}
\textbf{1.} The problem
of approximate controllability of equation (\ref{4.1}%
) by parabolic
controller (\ref{4.2})-(\ref{4.3}) is still open. It means
that if there
exists a possibility to choose a distributed controller for
construction
then it is worthwhile to construct a hyperbolic controller.
\noindent
\textbf{2.} The results of this section can be extended to the
case of
partial differential hyperbolic equation
\begin{equation*}
\frac{\partial
^{2}y}{\partial t^{2}}(t,x)=\frac{\partial }{\partial x} \Big(%
p_{1}(x)
\frac{\partial y}{\partial x}(t,x)\Big)
+p_{2}(x) y(t,x),t\geq 0,0\leq
x\leq l,
\end{equation*}
subject to boundary conditions (\ref{5.2})-(\ref{5.3}) governed by
hyperbolic controller (\ref{5.5})-(\ref{5.8}).
\section{Approximate controllability of linear differential control systems
with delays by hereditary controller}
In this section we will investigate linear differential control systems with
delays governed by hereditary controller. These objects can be considered as
a particular case of equation (\ref{2.1}) with a bounded input operator
\cite{Hale,Krein,Manitius&Triggiani,Salamon,Shimanov} subject to the
distributed controller of the form (\ref{2.2}), so the results of the
previous section can be applied.
Consider a linear differential-difference system \cite{Bellmann&Cooke}
\begin{gather}
\dot{x}(t)=\sum_{k=0}^{m}A_{k}x(t-h_{1k})+B_{0}u(t) , \label{6.1} \\
0 = h_{10}0$ and for any final state $(x_{1},\psi (\cdot) ) \in X$
there exists $(u_{0},\xi (\cdot)) \in Z$ such that the corresponding solution
$x(t) $ of system \eqref{6.1} satisfies the inequality
\[
\| (x_{1},\psi (\cdot ) ) -(x(t_{1})
,x(t_{1}+\cdot ) ) \| <\varepsilon
,-h_{1m}\leq \tau \leq 0
\]
(The norm is considered in the space $X$).
\end{definition}
It is well-known \cite{Krein, Hale,Manitius&Triggiani,Salamon,Shimanov}
that systems (\ref{6.1}) and (\ref{6.3}) can be written in the form
(\ref{2.1})-(\ref{2.2}) with the state spaces $X$ and $U$ defined above, and
the linear space of the eigenvectors of the corresponding operator $A$ is
dense in $X$ if and only if $\mathrm{rank}A_{p}=r$.
Here Assumptions 1-4 of section \ref{theassumptions} for the corresponding
operators $A, C$ and $K$ are valid with $T=nh$
\cite{Bellmann&Cooke,Banks&Yakobs&Langenhop,Shklyar,Shklyar_2};
the corresponding functions (\ref{2.6}) are minimal on $[ 0,\delta ]$, for
all $\delta >0$ \cite{Shklyar_3}. It was proved \cite{Shklyar_2} that
condition (\ref{3.1}) for equation (\ref{2.1}) is equivalent to the condition
\begin{equation*}
\mathop{\rm rank}\big\{ \lambda
I-\sum_{k=0}^{m}A_{k}e^{-h_{1k}},B_{0}\big\} %
=n\,,\quad \forall \lambda
\in \sigma _{A}.
\end{equation*}
and the density of the linear span of the generalized eigenvectors of operator
$C$ implies the density of the corresponding subspace $KS_{C}(\cdot ) Z$ in
$L_{2}([0,t_{1}-T], U)$.
\begin{theorem} \label{T6.1}
For equation \eqref{6.1} to be approximately controllable on $[ 0,t_{1}] $
by boundary controller \eqref{6.3}, it is necessary that
\begin{gather}
\mathop{\rm rank}A_{m}=n. \label{6.5} \\
\mathop{\rm rank}%
\big\{ \lambda I-\sum_{k=0}^{m}A_{k}e^{-h_{1k}},B_{0}\big\}
=n\,,\quad
\forall \lambda \in \mathbb{C}. \label{6.6}
\end{gather}
When these conditions hold and $\mathop{\rm rank}C_{p}=r$, system
\eqref{6.1} is approximately controllable on $[ 0,t_{1}]$, $t_{1}>nh_{m}$,
by hereditary controller \eqref{6.3}.
\end{theorem}
\subsection*{Approximate controllability of linear differential control
systems with delays by scalar hereditary controller}
Consider system (\ref{6.1}) with one delay and one input, subject to scalar
hereditary regulator ($r=1$) with one delay, namely
\begin{gather}
\dot{x}(t) = A_{0}x(t) +A_{1}x(t-h_{1})+B_{0}u(t) ,
\label{7.1} \\
h_{1} > 0, \notag \\
x(0) = x^{0},x(\tau )=\varphi (\tau ) \quad\text{a.e. on }[ -h_{1},0] ,
\label{7.2}
\end{gather}%
where
\begin{gather}
\dot{u}(t) =C_{0}u(t) +C_{1}u(t-h_{2}) , \label{7.3} \\
h_{2} > 0, \notag \\
u(0) = u^{0},u(\tau ) =\psi (\tau ) \quad \text{a.e. on }[ -h_{2},0] .
\label{7.4}
\end{gather}
Here
\begin{%
gather*}
x(t) ,x^{0} \in \mathbb{R}^{n},\quad \varphi (\cdot ) \in L_{2}([
-h_{1},0] ,%
\mathbb{R}^{n}) ,\, \\
u(t) ,u^{0} \in \mathbb{R},\quad \psi (\cdot ) \in L_{2}[ -h_{2},0] ;
\end{gather*}
$A_{j}$, $j=0,1$, are constant $n\times n$ matrices, $B_{0}$ is a constant
column-vector, $C_{j},j=0,1,$ are scalars.
We consider the Hilbert spaces
\begin{equation*}
X =\mathbb{R}^{n}\times L_{2}([ -h_{1},0] ,\mathbb{R}^{n}) ,
\quad Z =\mathbb{R}\times L_{2}[ -h_{2},0]
\end{equation*}
as the state spaces of systems (\ref{7.1}) and (\ref{7.3}) respectively;
$U=\mathbb{R}, K(u_0,\psi (\cdot ))=u_0, \forall (u_0,\psi (\cdot ))\in Z$.
\begin{corollary} \label{C4.1}
System \eqref{7.1} is approximately controllable on $[0,t_{1}] ,t_{1}>nh$,
by hereditary controller \eqref{7.3} if and only if
\begin{enumerate}
\item $\mathop{\rm rank}\big\{ \lambda I-A_{0}-A_{1}e^{-h_{1}},B_{0}\big\} =n$,
for all $\lambda \in \mathbb{C}$.
% \label{7.5}
\item $\mathop{\rm rank}A_{1}=n$ and $C_{1}\neq 0$.
\end{enumerate}
\end{corollary}
\subsection*{Remark}
Many ideas of the proofs of the theorems presented above, are imported from
\cite{Marchenko&Shklyar,Shklyar&Marchenko}, where closed problems of
approximate null-controllability for distributed equations governed by
distributed controller were considered. Complete proofs of the theorems will
be presented in the full version of the paper.
\begin{thebibliography}{99}
%
\bibitem{Banks&Yakobs&Langenhop} H. Banks, M. Jakobs, and C.
Langenhop;
Characterization of the controlled states in $W_{2}^{(1) }$ of
linear
hereditary systems, \textsl{SIAM Journal of Control,} \textbf{13}%
(1975),
611--649.
\bibitem{Bellmann&Cooke} R. Bellmann, K. Cooke;
\textsl{%
Differential-Difference Equations}, New York Academic Press
London, 1963.
\bibitem{Fattorini&Russel} H. Fattorini, and D. Russel;
Uniform bounds on
biorthogonal functions for real exponentials with an
application to the
control theory of parabolic equations, \textsl{Quart.
Appl. Math.}, \textbf{%
32}(1974), 45--69.
\bibitem{Hale} J. Hale,
\textit{Theory of Functional Differential Equations}%
, Springer-Verlag,
New York Heidelberg Berlin, 1977.
\bibitem{Hille&Philips} E. Hille and R.
Philips; \textsl{Functional Analysis
and Semi-Groups}, AMS, 1957.
%
\bibitem{Hutson&Pum} V. Hutson and J. Pum; \textsl{Application of
Functional
Analysis and Operator Theory}, Academic Press, 1980.
\bibitem{%
Kamke} E. Kamke; \textsl{Differential Gleichunden}, L\"{o}%
sungsmethoden
und L\"{o}sungen, Leipzig 1959.
\bibitem{Krein} M. Krein; \textsl{Linear
Differential Equations in Banach
Spaces}, Moscow, Nauka Publisher, 1967.
%
\bibitem{Lasieska} A. Lasieska; Unified theory for abstract
parabolic
boundary problems -- a semigroup approach, \textsl{Appl. Math.
Optimiz}%
\textrm{.}, \textbf{10(}1983), 225--286.
\bibitem{%
Manitius&Triggiani} A. Manitius, R. Triggiani; Function
space
controllability of linear retarded systems: a derivation from
abstract
operator conditions, \textsl{SIAM J. Control and Optimiz.},
\textbf{6}%
(1978),599--643.
\bibitem{Marchenko&Shklyar} V. Marchenko
and B. Shklyar; Approximate
controllability of distributed systems with
bounded input operator by
distributed controller, \textsl{Functional
Differential Equations}, to
appear.
\bibitem{Nagel} R. Nagel; \textsl{%
One-parameter semigroups of positive
operators,} Lecture Notes in Notes in
Mathematics, 1184, Springer-Berlag,
Berlin, New York, 1984.
\bibitem{%
Naimark} M. Naimark; \textsl{Linear Differential Operators, Part I}%
, F.
Ungar, New York, 1967.
\bibitem{Pazy} A. Pazy; \textsl{Semigroups of
Linear Operators and
Applications to Partial Differential equations},
Springer-Verlag, New York
Heidelberg Berlin Tokyo, 1983.
\bibitem{Salamon} D. Salamon;
Infinite dimensional linear systems with
unbounded control
and observation: a functional analytic approach, \textsl{%
Trans. Amer.
Math. Soc.}, \textbf{300}(1987), 383--431.
\bibitem{Shimanov} S. Shimanov;
To the theory of linear differential
equations with aftereffect,
\textit{Differential Equations,} \textbf{1}
(1965), 102--116 (in
Russian).
\bibitem{Shklyar} B. Shklyar, On Attainable Set and
Controllability for
Asbtract Control Problem with Unbounded Input Operator,
\textsl{Proceeding
of the Institute of Mathematics, NAS of Belarus},
\textbf{7}(2001), 142--152.
\bibitem{Shklyar_2} B. Shklyar; On the
approximate controllability of
distribute-parameter systems, Differential
Equations, \textbf{27}(1991),
326--334.
\bibitem{Shklyar_3} B. Shklyar,
Observability of objects of neutral type by
means of linear operations.
Differential Equations, \textbf{24}(1988),
196--202.
\bibitem{Shklyar&Marchenko} B. Shklyar and V. Marchenko; Approximate
controllability
of distributed systems with unbounded input operator by
distributed
controller, \textsl{Functional Differential Equations}, to
appear.
\bibitem{Weiss} G. Weiss; Admissibility of unbounded control operators,
\textsl{SIAM J. Contr. and Optimiz}., \textbf{27}(1989), 527--545.
\end{thebibliography}
\end{document}